o utions
anua
for •
• ~
as-Colell,
inston, and Green
Prepared by: Chiaki Hara Cambridge University
Ilya Segal University of California, Berkeley
Steve Tadelis Han-·ard University
New York
Oxford
OXFORD UNIVERSITY PRESS
1997
We could never overestimate the
of work which
to be done to
complete this solution t:..:>Ok, but the satisfaction in seeing the finished product more than makes up for the many hours of work. both
co~x:ise
We have tried to be
and exhaustive, and we hope that these two objectives do not
conflict too often ln this solution book. On rare occasions. we refer the reader to a book or uticle containing a well-presented solution.
Chiaki Hara has done a lion•s share of the work, preparing solutions for parts I and IV. Ilya Segal has provided the solutions for chapters 10 and 22. Steve Tadelis has
the solutions for chapters 11, 13, 14, 21 and
23, and has completed the work on part II. Finally, llya and Steve. have
together prepared solutions for chapter 12. Some of the early work on the solutions had been done by Marc foundations for solutions for
and we thank him for laying down II and chapter 12.
We would like to thanJc: Andreu Mas-Colell and Mike Whinston for many hours of
We are also
to the many teaching fellows who taught
Ec:2010a/b, the Harvard graduate sequence in microeconomic theory, for their
input ove:- the years when earlier versions of the textt:..XJk had been used for instNction. The list of these teaching fellows is too long to inc:lude here. While some errors surely remain, we hope that both students and teachers will benefit from our solutions, which make an excellent textbook even more useful.
Finally, some personal thanks: Chiaki Hara thanks his parents and colleagues at various places where he has spent the last four years.
Ilya
Segal thanks Olga. for her unwavering support during all times. Steve Tadelis thanks lrit, for always being there to encourage and support. Chiaki Hara Ilya Segal Steve Tade!is
June 1996
CHAPTER 1
Since y >- z implies y >- z, the transitivity implies that x >- z.
l.B.l
-
Suppose that z >- x.
-
-
Since y ;t- z, the transitivity then implies that y >- x.
-
But this contradicts x >- y.
Thus we cannot have z >- x.
-
By the completeness, x >- x for every x e X.
l.B.2
Hence there is no x e X
-
such that x >- x.
Hence x >- z.
Suppose that x >- y and y >-
2,
then x >- y >- z.
By (iii) of
-
Proposition l.B.l, which was proved in Exercise I.B.l, we have x >- z. is transitive.
Property
(i)
Hence >-
is now proved.
As for (ii), since x >- x for every x e X, x - x for every x e X as welL
-
Thus - is refiexive. and z ;t- y. z.
Suppose that
x -
y and y -
z. Tnen x
By the transitivity, this implies that x
Hence - is transitive.
>- x and x >- y.
-
-
Suppose x that - y.
Hence y - x.
~ 2
and
>- y, y >- z.
-
-
x.
2 ~
-
Property
-
(ii)
->- X .
Thus x -
Then x >- y and y >- x.
Thus - is symmetric
V
~
Thus y
is now
proved.
Le~
l.B.3
u(x}
:!::
x e X and y e X.
u(y).
Since
f( ·)
Since u( ·) represents
is strictly increasing, u(x)
~·
i!:
x
?: y ;:·
,:;.:~d
only if
u(y) if and only if
•
Hence x >- y
\I{X) :!:: Y{y).
-
if and only if
v(x) :!:: v(y).
Therefor-e
v( ·
l
represents >-.
-
Suppose first that x >- y.
l.B.4
-
u(x) = u(y). u(x)
>
u(y).
If, furthermore, y >- x, then x - y and hence
If, on the contrary, we do not have y
>- x, -
then x >- y.
Hence
Thus, if x >- y, then u(x) <: u(y).
-
Suppose conversely that u(x)
:!::
u(y).
1-1
If, furthermore, u{xl = u(yl, then
x - y and hence x ::; y. hence x >- y.
-
l.B.S
If, on the contrary, u(x) > u(y), then x >- y, and
Tnus, if u(x)
l!
u(y), then x >- y.
-
So u{ ·) represents >-.
-
First, we shall prove by induction on the number N of the elements of X
that, if there is no indifference between any two different elements of X, If N = 1, there is nothing to prove:
then there exists a utility function.
So let N > 1 and suppose that
Just assign any number to the unique element. the above assertion is true for N - 1.
N.
We will show that it is still true for
By the induction hypothesis, >- can be
-
represented by a utility function u( ·) on the subset {xl' ... ,xN-l} .
Without
•
loss of generality we can assume that u(x ) > uCx ) > ... > 1 2
u(~-l ).
Consider the following three cases: Case 1: For every i < N, xN >- xi" Case 2: For every i < N, x
1
>-
~·
Case 3: There exist i < N and j < N such that x. >- XN >- x .. 1 • . J Since there is no indifierence between two different elements, these _three cases are are exhaustive and mutually exclusive.
We shall now show how the
in each of the three cases, for u(·) to
value of u(xN) should be
-
reoresent >on the whole X. . If Case 1 applies, then take u(xN) to be larger than u(x ). 1 applies, take applies.
u(~)
to be smaller than u(xN_ ). 1
If Case 2
Suppose now that Case 3
Let I = {i e {1, ... , N - 1}: xi >- xN+l} and J = {j e {1, ... , N - 1}:
XN t >x .}. ) .... • J •
Completeness and the assumption that there is no indifference
•
implies that I v J = {1, ... , N - 1}.
The transitivity implies that both I and
J are "inte;vals," in the sense that if i e I and i' < i. then i' e I; and if j e J and j' > j, then j' e J.
Let i• = max I, then i• + 1 = min J.
1-2
Take
u{xN) to lie in the open interval (u(x .•
J,u(x .• )).
1 +1
•
Then it is
1
-··•
tc see
~-c:.::.y
-
that u( · ) reoreser:ts >on the whole X. . Suppose next that there may be indifference between some two elements of For each n = 1, ... ,N, define X
= {x
n
m
E
X: x
= X. transitivity of - (Proposition l.B.Hiill, if X
·
n
let M be a subset of {!, ... ,N} such that X = u ~
and any n e M with m
~
>-• X if and only if x >- x . n m- n
r.
rr:
X , then X " X rn n m
me
MX
m
Then,
Also, bv• the
and X
m
Define an relation >-• on {X : m m
n.
- x }.
= e.
= Xn E
So
for- any m e M
M} by letting X
m
In fact, by the definition of M, there is no
indifference between two different elements of {X : m e M}. m
Thus,
by the
•
preceding result, there exists a utility function u•( ·) that represents Then define u: X
-7
lR by u(x ) = u•(x ) if m e M and x
n
m
n
e X .
m
It is easv to •
show that, by the transitivity, u( ·) represents >-.
-
l.C.l
If y e C({x,y,z}), then the \VA would imply that y e C({x,y}).
contradicts the equality C({x,y}) = {x}.
Hence y
~
C((x,y,z}).
B:.1t
Tl'rJs
C({x,y,z}) e {{x},{z},{x,z}}.
I.C.2 If B X
The property in the question are equivalent to the followir.g property:
e :E, B' e !13,
X E
B. y
e C(B') and -v e C(BL
e B.
X
e B', y e 8', x e C(B), and y e C{B'), then
We shall thus prove the equivalence between this
propeny and the Weak Axiom. Suppose first that the Weak Axiom is satisfied.
Assume tha: B e 23, B' e
13, x e 8, y e 8, x e 8', y e B', x e C(B), and y e C(B' ).
Weak Axiom twice, we obtain x e C(B') and y e C(Bl. is also satisfied.
1-3
If we apply the
Hence the above property
Suppose conversely that the above property is satisfied.
Furthermore, let 8' e :B, x e B', y e 8', and
8, y e B. x e B', and x e C(B). y e C(B').
Let B e 13. x e
Tnen the above condition implies that x e C(B') (and y e C(Bl).
Thus the Weak Axiom is satisfied.
l.C.3
(a} Suppose that x >-• y, then there is some B e :B such that x e 8, y e Suppose that y
Thus x >-• y.
B, x e C(B), and y fl! C(B).
-
exists B e 13 such that x e B, y e B and x e C(Bl.
-
>-••
Hence x
-
x. then there
But the Weak Axiom implies
Hence if x >-• y, then we cannot have
that y e C(B), which is a contradiction. y >-• x.
>-•
y.
Converseiy, suppose that x >-•• y, then x >-• y but not y >-• x.
-
-
Hence
there is some B e :8 such that x e B, y e B, x e C(B) and if x e B' and y e B' for any B' e :B, then y
E
C(B'}.
In particular, x e C(B) and y E C(B).
Thus
The equa!ity of the two relation is not guaranteed without the WA. car. be seen from the above proof, the WA is not necessary to guarantee 6
x >-•• y, the:-1 x >-
following example.
y.
tha~
if
But the converse need not be true, as sho·wn by the
Define X
and C({x,y,z}) = {y}.
As
= {X".y,z},
:B
= {{x,y},{x,y,z}},
Then x >-• y and y >-• x.
C({x,y}) = {x}.
But neither x >-• y nor y >-• x.
(b) The relation >-• need not be transitive, as shown by the following example. Define X = {x,y,z}, :B
= {{x,y},(y,z}},
C({x,y}) = {x} and C({y,z}} = (y}.
But we do not have x >-• z (because neither of the two
-
sets in 13 in::ludes {x,z}} and hence we do not have x >-• z either.
(c) According to the proof of Proposition 1.0.2, if :B includes all threeeler:1ent subset of X. then >-• is transitive.
-
1-4
By Proposition l.B.Hil, >-•• is
Since >-• is equal to >-••, >-• is also transitive.
transitive.
An alternative proof is as follows: Let x e X, y e X, z e X, x >-• y, and
y >-• z. Then (x,y,z} neither- y >-•
-
x nor z
4: ~
and, by (a), x >-•• y, and y >-•• z.
>-• y.
-
Hence we have
Since >-• rationalizes (!B,C( • )), this implies that
-
y fl! C({x,y,z}) and z - C((x.y,z}).
Since C({x,y,z})
:;e "· C{{x,y,z})
= {x).
Thus x >-• z.
The simplest example is X = {x,y}. !B = {{x},{y}}, C({x}) = {x}, C{{y})
1.0.1
relation of X rationalizes C( · ).
= {y}. Then UJ rational
1.0.2
By Exercise l.B.S, let u( ·) be a utility representation of
is finite, for any B c X with B
~ 0,
t·
Since X
there exists x e B such that u(x)
for a!l y e B. Then x e c-(B,_t} and hence c•(B.tl
:;e "·
l!:
u(y)
(A direct proof with
no use of utility representation is possible, but it is essentially the same
as the proof
m Exercise 1.8.5.)
Su:~cose ••
1.0.3
If x e C(X), then x e C( {x,z} },
that the Weak Axiom holds.·
which cor:tradicts the equality C({x,z}) = {z}. C({x.y}), whid contradicts C({x,y}) = {x}.
which cont:radkts C( {x,z}) = {y}.
Thus
v CCB ), the~ x ~ y becango 8 u B 1 2 2
;::l
If z e C(X), ther.. z e C{ {v, z}), •
(~,C( ·))
Let t ntionali~ C{ ·) relative to !B.
1.0.4
If y e C(X), then y E
must violate the Weak Axiom.
Let x e C(B v B l and y e C\B ) 2 1 1
C(B l v CCB ). 1 2
Tnus x e C(C(B l v 1
CCB )). 2 Le: x e ctCCB l v CCB )) and y e B v B • then there a:-e four cases: 2 1 2 1 Case 1.
x e C(B ), y e Bl" 1 •
1-5
Case 2.
x·e C(B ), y
Case 3.
X
Case 4.
x e C(B l. y e 82. 2
1
E
B . 2
e CCB J. y e Br 2
If either Case 1 or 4 is true, then x >- y follows directly fro:-::.
-
rationalizability.
If Case 2 is true, then pick any z e
cm2 >.
Then z >- y.
-
•
>-
-
z.
Hence, .by the transitivity,
X
>- y.
If
-
•
Case 3 is true, then pick any z e C(B l and do the same argurne:tt as for Case 1 2.
l.D.S
(a) Assign probability 1/6 to each of the six possible preferences,
which are x >- y >- z, x >- z >- y, y >- x >- z, y >- z >- x, z >- x >- y, and z >- y >-
x. (b) If the given stochastic choice function were rationalizable, then the probability that at least one of x >- y, y >- z, and z >- x holds would be at most 3 x {1/4)
= 3/4.
But, in fact, at least one of the three relations
always holds, because, if the first two do not hold, then y >- x and z >- y. Hence the transitivity implies the third.
Thus. the given stochastic choice
function is not rationalizable . •
{c) T!le same argument as in (b) can be used to show that a:
2:
1/3.
Since
C({x,y}) = C({y,z}} = C({z.x}) = (a:, 1 - a:) is equivalent to C({y,x}l = C( {z,y})
= C( {x,z}) = (1 - et, et), if we apply the same argurner.t as in (b) to y
>- x, z >- y, a:1.d x >- z. then we can establish 1 - «
~
1/3, that is. a: =: 2/3.
Thus. in order for the given stochastic choice function is rationalizable, it is necessary that a: e [1/3,2/3].
Moreover, this condition is actually
suffic!e:1.t: Fo:- any a: e [1/3,2/3), assign probability a: - 1/3 to each of x >-
1-6
y >- z, y >- z >- x, and z >- x >- y; assign probability 2/3 - a to each of x >- z >-
y, y >-
>- z, and z >- y >- x.
Then we obtain the given stochastic choice
function.
1-7
CHAPTER 2 2.0.1
Let p
be the price of the consumption good in period 2. measured in
2
units of the consumption good in period 1. levels in periods 1 and 2. respectively. 2
•
set 1s equal to {x e IR+: x
1
Let x , x 1
be the
2
consum:~tion •
Then his lifetime Walrasian budget
+ PzXz :s w} .
2
2.D.2
{(x,h) e IR : h :s 24, px + h
2.0.3
(a) No.
+
:::!:
24}.
In fact, the budget set consists of the two points, each of
which is the intersection of the budget line and an axis.
(b) Let x e B
p,w
, x' e B
p,w
, and ~ e {0,1).
X is convex, x" e X.
Moreover, p·x" =
=
.
W.
2.0.4
"T"\.
!HUS X
..
E B
p,w
Write x" = AX + (1 -
~(p·x) + (1 -
~)(p·x'J :::!:
~)x'.
Since
i\.w + (l - i\.)w
It follows from a direct calculation that consumption level M can be
attained by (8 + (M - Ss)/s') hours of labor.
It follows from the definition
that (24,0} and {16 - (M - Ss)/s', M) are in the budget set. •
•
But their convex
•
combination of these two consumption vectors Wltn rat1o M - 8s
8
s' -----;-M~-~8::::-s-
8
+
' 8 +
s'
M- 8s
s'
is net in the budget set: the amount of leisure of this combination equals to 16 (so the labor is eight hours), but the amount of the consumption good is 8
M--~-~
8 +
M - 8s s
8 > M --.......,M...,...---~8::-s- = M M/s 8 + s 8
2-1
= 8s.
2.£.1 The homogeneity can be checked as follows: tl.W
tl.W
x (ctp ctw) = 3
•
ctpl
a.w
ctp 1 + ctp2 + a.p3
w
-
p
= x,(jJ,wl, l
1
w
-
w
-
To see if the demand function satisfies Walras' law, note that
Hence p • x(p, w) = w if and only if (3 = 1.
Therefore the demand function
satisfies Walras' law if and only if J3 = 1.
2.E.2 Multiply by plc/w both sides of (2.E.4), then we obtain l=l(Pr£'P·w)/wH8xl(p,w)/81\)(pk/xt'p,w)} Hence
+
pkxk(p.w)/w =
o.
l=lbl(p,wklk(p,w} + bk(p,w} = 0.
By (2.E.6),
l=l(plxl(p,w)/wHaxt(p,w)law)(w/xip,w)} = l.
Hence
l=l0!(p,w)clw(p, w) = 1.
2.E.3. Tnere are two ways to verify that p · D x(p, wlp = - w. p
One way is to post-multiply (2.E.5) by p, then p·Dpx(p,w} p
+
w = 0 by
Walt as' law.
Tne othe:- way is to pre-multiply (2.E.l) by pT, then p·DwJdp.w)w = 0.
p·D x{p,w)p p
+
By Proposition 2.E.3, this is equal to p·Dpx(p,wjp + w = 0.
An inte:-pretation is that, when all prices are doubled, in order for the
to stay a: the same consumption, it is necessary to increase his wealth bv• w .
2-2
By differentiating the equation x(p,cxw} = cxx(p,w) with respect to a and
2.E.4
evaluating at a = l, we obtain wD wx(p, w} = x(p, w).
Hence D\\' x(o,w) = .
(1/w)x(p,w). Hence ctw = (8xip,w)/8w)(w/xip,w)) = 1.
This means that an
•
one-percent increase in wealth will increase the consumption level for- all goods by one per-cent. Since (1/w)x(p, w) = x(p,l} by the homogeneity assumption, D x(p,w) is a w
function of p only. path, E
p
2.E.S
The assumption also implies that the wealth expansion
= {x(p, w): w > 0}, is a ray going through x(p,l).
Since x(p, w) is homogeneous of degree one with respect to w, x(p,o:w) =
o:x(p, w) for every o: > 0. 8cpip)/8pk = 0 whenever k we can write xt(p,w) =
Thus xt'p,w) :;e
= xip,l)w.
t, xt(p,l) is actually a function of pl alone.
xt(pl
2.E.6
So
Since x(p,w} is homogeneous of degree zero,
xl(pl) must be homogeneous of degree - 1 (in pl). such that xt(pl} =
Since 8xefp.l)/cpk =
Hence there exists o:l > 0
«lPt· By Walras' law, Lt.Pt(alpt)w = w[lc::i =
When a = l, Walras' law and homogeneity hold.
w.
We
must
Hence the conclusions of •
Propositions 2.E.l - 2.E.3 hold.
2.E.7
Bv Walras' law, •
Tnis demand function is thus homogeneous of degree zero.
2-3
2.E.8
For the fil"st part, note that
Thus, by the chain rule, ~
oxt
a
d(ln xl(p,w))
pk
-
d(ln p,)
axl
a
(p, w) • exp On pk)
pk xl(p,w}
-
•
xt(p,w)
iC
{p,w)·pk
-
~ lk(p, w) .
Simiiarlv, -
cHIn x l( p, wl l d(ln\r,:}
8xl
-
aw
axl
aw (p, w)w
(p, w) · exp(ln w)
=
xl(p,w) •
Since o:
= dOn
1
= d(ln
xip,w))ld(ln p ), o: 1 2
=
xip,w))/d(ln p l, and a: 2 3
d(ln x
iP· w) )/d(ln
Z.F.l
We proved in Exercise l.C.Z that Definition l.C.l and the prope:-ty -in
w), the assertion is established.
the exercise is equivalent.
It is easy to see that the latter is equivalent
to the foUowing property: For every B
E
:B and B' e
~.
if C(B) " 8'
" C\8') .: e, t!ien C(Bi "B' c C(B') and B" CCB') c CCB).
:;c 0
and B
If C{ · l is
· , , · t;.en ~ · · property 1s · equ1va · 1ent to t h e f o 11 owmg . smg.:.e-va.uec, trus one: F o:o-
-
ever-•: 8
E
73 anci B' e B, if C(8) c B' and B c C(B' ), then C(B) = C(B' ). .
.
'
In the
contex: of Walrasiar: demand functions, this can be resta-;ed as follows: For any (p,v:) and (p',w'), if p·x(p',w') :s wand p'·x(p,w) :s w', the:t x(p,w) = x(p', \II'). 2 . -t
He:1ce Definitions l.C.l and Z.F.l are equivalent.
1 ....
2.F.2 •
But this is the contraposition of the property stated in Definition
It is straightfor-ward to check that the Weak Axiom holds. •
I
. .
I
8 anc
0 • • X"' :!: •
P
-
... -e.,
'-e·~~:~~ & .....,, I
J
-
·~ ~'W
x
l
•
•
•
•
•
=
j, then pJ·x
. .. Iy r S lr:lllar"
.
1
Stnce
= 9. p
I
·X
In fact, if
2 1 . 2 S mce p ·x = 8, x is revealed
3
= 8,
2-4
X
I .
15
revealed
prefe~:-ed
3
to x.
3
But, since p · x
2.F.3
2
= 8, x
3
2
i's revealed preferred to x .
[First prin~ing errata: Add the sentence "Assume that the weak axiom is
satisfied." in (b) and (c).]
Denote the demand for good 2 in year 2 by y.
(a) His behavior violates the weak axiom if 100 · 120 + lOOy
::5
100 ·100 + 100 · 100
and 100·100 + 80·100
::5
100·120 + SOy.
That is, the Weak Axiom is violated if y e [75,80).
(b) The bundle in year 1 is revealed preferred if 100 ·120 + 100y
::5
100 ·100 + 100 ·100
and 100 ·100 + 80 ·100 > 100 ·120 + SOy,
that is, y < 75.
(c) The bundle in year 2 is revealed preferred if 100·100 + 80 ·100
::5
100 ·120 + SOy
and 100 ·120 + lOOy > 100 ·100 + 100 ·100,
that is, •v > 80 . (d) For any value of y, we have sufficient information to justify exactly one of (a), (b). and (c).
(e) We shall prove that if y < 75, then good 1 is an inferior good. suppose that y < 75.
Then
2-5
Sc
100·"120 + 100y :s 100·100
+
100·100
and 100 ·100 + 80 •100 > 100 ·120 + 80y. Hence the real wealth decreases from year 1 to 2. good 1 increases. 100.
Also the rela:ive price of
But the demand for good 2, y, decreases because y < 75 <
This means that the wealth effect on good 1 must be negative.
Hence it
is an inferior good.
(f) We shall prove that if 80
suppose that 80 < y < 100.
< y < 100, then good 2 is an inferior good.
So
Then
100·100 + 80·100 :s 100 ·120 + 80v • and 100 ·120 + 100y > 100 · 100 + 100 · 100. Hence the real wealth increases from year 1 to 2. good 2 dec:-eases.
Also the relative price of
But the demand for good 2, y, dec::-eases because y < 100.
This mea..ls tha-t the wealth effect on good 2 must be negative.
.m: er1or . ~
Hence it is an
. gooc.
the co:1sume::- has a revealed preference for x
over x . 1 0
(b) If PQ >!,then (p ·x )/(p ·x ) > 1 and hence p ·x > p ·x . 1 0 1 1 1 0 1 1
consume:- has a revealed preference for x
Thus the
over x . 0 1 = i\.
Hence, by taking i\
larger or smalle; than one, we can make EQ larger or smaller than one. this obviously does not have any revealed preference relationship.
2-6
But
2.F.5
We shall first prove the discrete version.
By the homogeneity of
degree one with respect to wealth, it is enough to show that (p' - p) · (x(p' ,1) - x(p,1})
~
0 for every p and p'.
Since x(p' ,1) - x(p.ll =
p . X p, +
. 1
(x(p, p' ·x(p,1) ) - x(p,l)),
it is sufficient to show that (p' - p) • (x(p' ,p' · x(p,l)) - x(p,l))
~
0,
and 1
(p'- p}·(x(p,-p-;-·-.x......,-(p-,-:-~ )- x(p,l)) :s 0 .
11
•
For the first inequality, note that (p'-
p)·(.~c(p',p'·x(p,1))-
x(p,1)) =- p·x(p',p'·x(p,ll) + 1.
If x(p',p' ·x{p,1)} = x(p,l), then the value is equal to zero. x(p',p'·x(p,l)l 1.
;=
If
x(p,l), then the weak axiom implies that p·x(p',p'·x(p,lll >
Hence the above value is negative. As for the second inequality, ---.-___;,.----,..,.... ) - x( p ,1 })
p ·x p,
1
-
p' · x(p,ll
= 2 - (p' ·x(p,l) :s 2 - 2
+
+
1•
1
p' ·x(p,l) )
(p' ·x(p,l))(
1
p' ·x(p,l} )
= 2 - 2 = 0.
The infinitesimal version goes as follows.
By differentiating x(p,aw) =
ax(p, w) with respect to a and evaluating at a = 1, we obtain Dwx(p, w)w =
x(p, w).
Hence
2-7
S(p, w) =· D x{p. w) + D x(p, w)x(p, w)
T
w
p
= D x(p, w) + (1/w)x{p,w)x(p,w)T. p
Thus D x(p,w) = S(p,w) - (1/w)x(p,w}x(p,w)T. p
By Proposition 2.F.2, S(p,w) is negative semidefinite.
Moreover, since
2
v·(x(p,w)x(p,w}T)v =- (v·x(p,w)) , the matrix- (1/w)x(p,w)x(p,w)T is also negative semidefinite.
2.F.6
Thus D x(p,w) is negative semidefinite. • p
Clearly the weak axiom implies that there exists w > 0 such that for
every p, p', and w', if p·x(p',w')
~wand
x(p',w')
:;e
x(p,w), then p'·x(p,w) >
w. •
Conversely, suppose that such a w > 0 exists and that p · x{p', w'} x(p',w')
:;e
Let a: = w' /w.
x(p,w).
~
w and
-1
Then x(p' ,w') = x(p' ,a:w) = X(IX p', w} by
the homogeneity assumption, and p·x(a:
-1
p',w}
~
w and x{a:
-1
p',w)
:;e
x{p,wl.
But
this implies that (IX-lp')·x(p,w} > w, or, equivalently, p'·x!p,w) > a.w = w'. Tnus the weak axiom holds.
2.F.7
By Propositions 2.E.2 and 2.E.3,
-
p·.S(p,w)
= p·Dpx(p,w)
+
•
.
T p·Dwx(p,w)x(p,w) = p·Dpx(p,w)
+
T x(p,\'11")
=0
By Proposition 2.E.l and Walras' law,
T
D x(p,w}o D x(p,w)p + D x(o, w)w = 0 . S(c w)c = ... Dwx(p,w)x(p,w) p = • • • 0 • p
•
pk
2.F.8
-
OXi. --.--..... -::::--- ( p, w) xl ( p, w) 8pk •D.lC
= c //' (0. w) + '"K"
w
xi.(p,w)
= clk(p,w) + ci.w(p,w)bk(p,w).
2-8
w
.
. TAT ( TA )T TA . A . . . r" . .~ ( a ) S mce x x = x x = x x, a matr1x 1s negative ae. m1te 1.
2 . F •9
and only if x TAx T
+
n
x TATx < 0 for every x e IR \{0).
. TA S 1nce x .. x
+
T
TAT x x = T
x (A + A }x, this is equivalent to the negative definiteness of A + A . A is negative definite if and only if so is A + AT_
Thus
The case of negative
definiteness can be proved similarly. The following examples shows that the determinant condition is not - 1
Let A =
sufficient for the nonsynunetric case.
_
3
0 1
, then A
11
= - 1
•
and A
22
= 1.
- 1 3
But U, 1)
0
1 1
- 1
Hence A is not negative
• 1.
semidefinite. By Proposition 2.F.3, S(p, w)p = 0
(b) Let S(p, w) be a substitution matrix.
and hence s
= (-
Cp,w) = (- P/P >s (p,w). 12 2 11
p 1p ls Cp,w). 1
2
11
Thus s
22
(p,w)
=
Also p·S(p,w) = 0 and hence s
21
2 2 (p /p >s (p,w). Thus, for every v 1
2
11
Cp.w)
=
vl v
2
•
v·S(p,w)v
Now, su:ppose that S(p, w) is negative semidefinite and of rank one. to (•J, the
nega~ive
semidefiniteness implies that s (p,w} 11
rank one implies that s
11
(p,w}
:;!:
0.
Hence s (p,w) < 0. 11
Conversely, let s Cp.w) < 0, then, by (•), v·S(p,w)v 11
2.F.10 •
~
~
0.
Tnus s
Ac::ording
Being o:· 22
Cp,w) < 0.
0 for every v.
(a) If p = (1,1,1) and w = 1, then, by a straightforward caicu!ation, •
we ootam 1
0
0
- 1
1
1
0
- 1
S(p,w) = (1/3)
1
2-9
•
•
Hence S(p, w) is not symmetric. -
- 1 0
Hence
1
1
0
I - 1
Note that =-v
- 1
2
2
2
+ v v - v = - (v 2 1 1 1 2
is negative definite.
3v /4. 2
Thus, by Propositi"on 2.F.3 and
Theorem M.D.4(iii), S(p,w) is negative semidefinite .
(b) Let p = (l,l,d and w
= 1.
... Let S(p,w} be the 2 x 2 submatrix of S(p,wl
obtained by deleting the last row and column.
By a straightforward
calculation, we obtain
- 2 - c
S(p, w)
0
1 + 2c •
- 3c
Thus, (1, 4, O)S(p,w)
if
£
1 4 0
A
= (1, 4}S(p, w)
> 0 is sufficiently small.
1
4
2
= (2 + cl- £2 - 4lc) > 0,
Then S(p, w} is not negative semidefinite and
hence the demand function in Exercise 2.E.l does not satisfy the Weak Axiom.
2.F.ll
By Pro;>osition 2.F.3, S(p, w}p = 0 and hence s
(- P/P )s (p,w). 2
11
Also p:S(p,w} = 0 and hence s
12
(p, w) =
(p,w) = (- p /p >s £p,w). 21 1 2 11
(We sa\'1 this in the answer for Exercise 2.F. 9 as well.)
Thus s
12
(p, w) =
s21 (p, w).
2.F.l2
By a;:~plying Proposition 1.0.1 to the Walrasian choice s:ructure, we
know that x(p, w} satisfies the weak axiom in the sense of Defir.itior. 1. C.l. By Exercise 2.F.l. this implies that x(p,w} satisfies the weak axiom in the sense of Definition 2. F .1.
2-10
2.F.l3
[First printing errata: In the last part of condition (•) of (b), the
inequality p · x > w should be p' · x > w'.
Also, in the las7. part of (c), the
relation x' e x(p,wl should be x' E x(p,w).}
(a) We say that a Walrasian demand correspondence satisfies the weak axiom if the following condition is satisfied: For any (p, w} and (p', w' ), if x e .x(p,w), x' e x(p',w'J, p'·x::!: w', and p·x'::!: w, then x' e x(p,w). equivalently, for any (p,w} and (p',w'), if x
E
Or
x(p,w), x' e x(p',w'), p·x' ::!:
w, and x' E x(p, w), then p' · x > w'. (b) If x e x(p,w), x' e x(p',w'), and p·x' < w, then x' E x(p,w) by Walras'
law.
Thus p'·x > w'.
(c) If x e x(p,w), x' e x(p',w'), and p'·x p·x'.
= w',
then (p'- p)·(x'- xl
= w-
If, furthermore, x' e x(p,w), then Walras' law implies that p·x' = w.
Hence (p' - p) · (x' - x) = 0.
If, on the contrary, x' E x(p, w), then the
generalized weak axiom implies that p·x' > w.
Hence (p' - p) · (x' - x) < 0.
(d) It can be shown in the same way as in the small-type discussion of the p::-oof of Proposition 2.F.l-that, in.order to verify the
asse::-•.:~·: ..
it is
sufficient tc show that the generalized weak axiom holds fer aH compensated price changes. ::!: w.
So suppose that x e x(p,w), x' e x(p',w'l, p' ·x = w', a:1d p·x'
Then (p' - p) · (x' - x) = w - p · x'
2::
0.
Hence, by the gene.-alized
co::npensated law of Demand, we must have (p' - p) · (x' - x} = 0 and x' e x(p, w}.
2.F.l4
Let p »
0, w
2!:
o, and a> 0.
Since p·x(p,w)::!: w an:i (o:pl·x(o:p,o:w) ~
o:w, we have o:p·x{p,wl ::!: o:w and p·x(a.p,o:w) ::!: w.
2-11
.
.
The wea:< ax1o:r: r:ow
.1mpnes ..
that x(p, w) = x(a:p,o:w).
Since oxip, w)/8w = 0 for both l = 1,2, we have slk(p, w) =
2.F.15
8x£'p,w)/8pk for both l
= 1,2
and k = 1,2.
-S(p,w)
Hence, let
be the 2 x 2
submatrix of S(p,w) obtained by deleting the last row and column, then S(p,w)
-
-
1
0
1 - 1
•
Tnis matrix is negative definite because
-
l
0
1
vl
- 1
v2
2 v 2Hence, by Theorem
(We saw this in the answer to Exercise 2.F.lO(a).)
M.D.4(iii), v·S(p,w)v < 0 for all v not proportional to p.
Since
-S(p,w)
is
net S)-m.me:ric, S(p,w) is not symmetric either. •
2.F.l6
(a) The homogeneity can be checked as follows: xl (a:p,a:w) = o:p2/a:p3 = p2/p3 = xl (p, w),
x 2 £o:p,a.w) = x
3
o:p/a:p
3
(c:;::,o:w) = IXW/o:p
3
= - P/P
= w/p
3
= x Cp,w), 3 2
= x (p,w). 3
As for Wal:-as' law, ~ v • ./,~ ::'-. y.-_
.;
(b)
p = (1,2,1), w = l, p'
= (1,1,1),
(2, - l, 1} and x(p', w') = (1, - 1, 2).
p · x(p', w') = 1
(c) Denote by obtained by
= w.
•••. 'IY
..)
and w' = 2, then x(p,w) = • • Thus p' · x(p, w) = 2 = \V anc
Hence the Weak Axiom is violated.
Dx{p, wl
de~eting
--
the 2 x 2 submatrix of the Jacobian matrix Dx(p, w)
the last row and column, then
-
-
1
0
1 - 1
•
Let S(p, w} be the 2 x 2 submatrix of S(p, w) obtained by deleting the last row
2-12
and column, then
-S(p, w)
-
-
= Dx(p, w) = 0/p )
1
1 - 1
because ax (p,w)/8w = 0 3 1 2 Note that v·S(p,w)v = 0 for every v e ~ . Now let v E !R 3 • "
Note that v = (v - Cv /p )p) 3 3
+
.,
1
A
-
Cv /p )p and the third coordinate of v 3 3
-
2
So denote its first two coordinates by v e IR •
-
...
...
Then, by Proposition 2.F.3, v·S(p,w)v = v·S(p,w)v = 0.
(c) Suppose that p' ·x(p,w)
~
w' and p·x(p' ,w')
~
w.
The first inequality
implies that C[tP£lw/C[lpt) ~ w', that is, w/([tpt) :s w' /([tPe>·
Tne second
inequality implies similarly that C[tpt)w' /([tpt) :s w, that is, w' /([tpt) :s w/([lpl).
Therefore w/([lpl) = w'/([tpt).
Hence x{p,w) = x(p',w').
Thus the
weak axiom holds.
(d)
Bv• calculation, we obtain 1
D x(p,w)
p
-
(-
2
w/([tpt) )
• • •
1 • •
• •
•
•
1
• • •
•
1
1 • • •
= x(p, w).
1
Hence S(p, w) = 0.
It is symmetric, negative semidefinite, but not negative
definite.
2-13
CHAPTER 3
•
3.B.l
y.
(a) Assume that t is strongly monotone and x » y.
-
Then Uy - x II
»
y.
£,
-
~
= Min {x - y , ... , 1 1
£
z•u
<
By the local nonsatiation, •
£
- yL} > 0, then,
and z• >- y.
By x » z• and the
By Proposition l.B.Uiiil (which is implied by the
-
transitivity). x >- y.
rnus >- is locally nonsatiated. .
then x » z.
e X such that lly -
weak monotonicity, x >- z•.
3.8.3
0.
c and y >- x.
$
Define
for every z e X, if hy - zH <
z•
>
L Let e = (1, ... ,1) e IR and
•
Suppose that x
there exists
-
-
(b) Assume that >- is monotone, x e X, and £
3.B.2
v and x ;:
~
Thus >- is monotone.
Hence x >- y.
y = x + (c/v'L)e.
Then x
Thus >- is monotone.
-
Foliowing is an example of a convex, locally nonsatiated preference 2
relation that is not monotone in IR . +
-
For example, x » y but y >- x. •
•
X
•
y 0
i
0
XI
Figure 3.B.3
•
•
3-1
•
Let ~ be a lexicographic ordering.
3.C.l
To prove the completeness, suppose
Thus y >- x.
-
To prove the transitivity, suppose that x ?: y and y ?: z.
Then x
1
~
y
1
•
z1• then x ?: z.
and
Hence x
!:::
2
z . 2
z , then X = 1 1 Thus x >- z.
-
To show that the strong monotonicity, suppose that x
i!:
y and y
:;c
x.
This
•
either case x >- y. To show the strict convexity, suppose that y >- x, z >- x, y
-
(0,1). Without loss of generality, assume that x
:;e
y.
-
:;c
z, and o: e
By the definition of
the lexicographic: ordering, we have either "y > x " or "y = x a.'ld y >. x ". 1 1 1 1 2 2
z:
On the other hand, since z V
J
2'
II
Hence, we have either "a:y
and a:y
3.C.2
2
+
(1 - cdz
> x . 2 2
1
+ (1 - o:)z
Thus a.y
II
+ (1 -
Take a sequence of pairs
x. and yn .. y. impiies that u(xl
3.C.3
x, we have either "z > x " or "z = x and x 1 1 1 1 2
Then u(xnl !:::
u(y).
"!:::
1
>
X/'
or "o:y
+ (1 - cdz
1
1
= x
1
o:}z >- x.
such that x
n
n >- y for all n,
-
X
n
u'(yn) for all n, and the continuity of u( ·)
Hence x >- y.
-
Thus >- is continuous.
-
•
•
One way to prove the assertion is to assume that >- is rnor.otone and
-
notice that the p:-oof actually make use only of the closedness of uppe:- and lower contour sets.
Tnen the proposition is applicable to >-, implying that it
-
has a continuous utility function.
Thus, by Exercise 3.C.2, >- is continuous.
-
A mere direct proof [without assuming monotonicity or using a utility function) goes as follows.
Suppose that there exist two sequences {xn} and
3-2
i!:
{yn} in X such that xn ?:' yn for every n, xn
-?
x e X, yn -+ y e X. a:1d y >- x.
Since {z: y >- z} is open, there exists a positive integer N for every n > Nl"
1
such that y >-
X
n
Since {z: z >- x} is open there exists a positive integer 1':
n such that y >- x for every n > N .
2
Conceivably, there are two cases on the
2
•
n .sequence {y }: Case 1: There exists a positive integer N .
such that yn >- y for every n > N .
3
-
3
. k(n) k(n) Case 2: There ex1sts a subsequence (y ) such that y >- y for every n. If Case 1 applies, then, by Proposition l.B.Hiii), we have y
n
>- x
n
for every
•
n > Max {N ,N }. 1
3
This is a contradiction.
open, the:-e exists a positive integer N By x .
Smce
{
n
?:' y
n
--
such that y
and Proposition l.B.l(iii), x
k(m)} .
z: z >- v
4
If Case 2 applles, then there
1s closed, x >- y
-
k(m)
n
>- y
n
k(m) >- y ) fo:- every n
k(m)
>
. for every n > N . · 4
But, since k(m) > N , this is a
.
2
. con ....ra d'1ct1on.
3.C.4
We p:-cvid.e two examples.
satisfies
mono~:)nicity,
Exar:1-:>le -1.
Let X = IR
The first one is simpler, but the second o:1.e
which the first does not .
-
+
•
.
and define u( · ): IR
+
-+ IR by letting u(xl = 0 fo:- x < ;,,
u(x) = 1 for x > 1, and u(l) be any number in (0,1}. preference relation represented by u(- ). continuous.
Denote bv >- the
--
We shall now prove that >- is not
-
Ir:. fact, if u(l) > 0, then consider a sequence {xn} with xn = 1 -
1/n fer everv r...
-
Although x
n
- 0 for every n and x
n
-+ l, we have 1 >- 0.
If
u( l) < l, the:l consider a a sequence {xn} with xn = 1 + 1/n for every n. Although xn - 2 fer every n and xn ~ 1, we have 2 >- 1. then all lower contour sets are closed. sets are
Note that if u(x) = 0,
If u(l) = 1, then all uppe:- contou:-
. Close ..... ~
3-3
•
2.
-
following rule: Case 1.
If x + x :s 2 and x 2 1
Case 2.
If min{x ,x > 3!::: 1 and x 1 2
Case 3.
If x
1
+
x
:;e
0,1), then u(x) = x + x . . 1 2 :;e (1,1),
then u(x) = min{x ,x } 1 2
2.
+
> 2., min{xl'x } < 1, and x > x , then 2
2
2
1
•
•
Case 5.
u(l,l) e [2,31.
The indifference curves of the prefezoence relation >- represented by u( · ) are
-
•
described in the following picture:
•
2
I
... ·----····· . •
•0 •
•• •
•
0
•
0
0
)
2 Figure 3.C.4
XI
It follows from this construction that u( ·) is continuous at every x
Tne preference >- is convex and monotone.
-
1/n, 1 - l/n)
-7
(l,ll and (1 + 1/n, 1 + 1/n)
u(l -
ln fact,
-7 (1,1)
1/n, 1 - lin) = 2 - 2/n -+ 2;
3-4
(1,1).
But, whateve: the choice of the
value of u(l,l) is, it cannot be continuous at (1,1). (1 -
:;e
as n
-7 cc,
and
+ 2 -+ 3.
u(l + 1/n, 1 + 1/n) = 1 + 1/n
Hence, if 2 < u(l,l), then (2,0) >- (1 - 1/n, 1 - 1/n) but (l,ll >- (2,0}; if
-
u(l,l) < 3, then (l + 1/n, 1 + 1/n) >- (2,1) but (2,1) >- (1,1).
-
•
= 3,
If u(l,l)
...
then all upper contour sets of >- are closed; if u(l,l) = 2, then all lower contour sets of >- are closed.
-
•
•
3.C.S
(a) Suppose first that u( ·) is homogeneous of degree one and let o:
L L xe!R,y e IR +, and
X -
+
y.
homogeneity, u(o:x) = u(o:y).
Then u(x) = u(y) and hence cw(x) = o:u(y).
0,
l!::
By the
Thus o:x - a.y.
Suppose conversely that >- is homothetic.
-
We shall prove that the utility
function constructed in the proof of Proposition 3.C.I is homogeneous of •
degree one.
Let x e IRL and o: > 0, then u(x)e - x and u(ax)e - ax.
Since >- is
-
+
homothetic, o:u{x)e - o:x.
By the transitivity of - (Proposition l.B.l(ii)), •
u(ax)e - au(x)e.
T.'lus u(ax) = o:u(x).
(b) Suppose fir-st that t is represented by a utility function of the form u(x) Let o: e IR,
X E
~
and hence u(x} + a = u{y} + a.
L L IR , y e IR , and +
+
X -
y.
Then u(x) = u(v) •
the functional form,
u(xj + o:
= (a
+
~(x , .... xL)
= u(x
+ a.e 1,
u(yl
= (a
+ y ) + ~(y , ... ,yL)
= u(y
+
+ a
x } + 1
2 2
1
where e. = (l,O, ... ,0} e
1
ae ). 1
ae
L
1
- y +
Suppose conver-sely that ::;- is quasilinear with respect to the first c~mmodity.
T."!e idea of the proof of this direction is the same as in (a) or
Propositi or.. 3. C.l, in that we reduce comparison of commodity bundles on a line by fining ot:t i.-:.different bundles and then assigning utility levels along the
line.
But this proof turns out to exhibit more intricacies, partly because it
3-5
1 depends crucially on the connectedness of IRL- , which appears in X = {+
•
L-1 x lR + .
a:,
m)
(Connectedness was mentioned in the first small-type discussion in
the proof of Proposition 3.C.l.)
The proof will be done in a series of steps.
First, we show that comparison of bundles can be reduced to a line parallel to Then we show that the quasilinearity of >- implies the given functional
-
form.
•
-u{ ·)
Let >- be a quasilinear preference and a utility function
-
-u( ·)
The existence of such a
>-.
-
is guaranteed by Proposition 3.C.l, but, of A
For each x e
course, it need not be of the quasilinear form. •
-
,.
represent
L-1 G< + , define
...
I(x) = {u(x .x> e IR: x e IR}, then J(x} is a nonempty open interval, by the 1 1 •
continuity and the strong monotonicity of •
L-1
along e . 1
,.
L-1 e IR • i.f J(x) :;e l(y), then Hxl ,... l(y) +
and y
x e IR +
Step 1: For every
=
...
~
0.
Proof: Suppose that
l(x)
l(y).
:;e
Without loss of generality, we can assume
... that there exists u e J(x) such that u IC l(y). u
::5
infl{yl.
simila.:-ly.)
Suppose that u Then let
(y (y
1 1
-
- x
,y).
1
-uCx
..
u =
-u(xi,x),
then, for every y
~.
...
e IR, 1
.x)
1
> u(y l'y).
Hence
l(x)
n
l(y)
L-1 : l(x) , define E(x) = {y e IR +
= 0.
=
l(y)}.
L-1 "' · . L-1 , E(x} is open tn IR For every x e IR •
Step 2:
+
-
-
-
+
-
L-1 Let x e IR , x e IR, and u = u(xl'x) e Hx). 1 +
Proof:
(u - c, u + d c !(x).
.u- -v e - L . -1
-
(The other case can be treated
By the quasilinearity, this implies that (x ,x) >1
L-1
For each x e
~ sa~isfy.
s•;p I(y} or•
supl(y).
2:
2:
..
+ xi, y).
Thus
xi e
...
Then either u
~"~
+
•
-
Since
...
an c. llx - •vii <
~.
-u( ·)
then
Let c > 0 satisfy
is continuous, there exists
I u(xl'xJ
•
•
3-6
-
-u(x ,yl- I 1
<
£.
o> 0 Hence
such that
•
I(x)
l(y)
"
Thus, by Step 1, l(x) = l(y), c E(xl. Thus E(x) is open. Step 3:
:~
(u - c, u + c) r.
... or y
-
L-1 Hence {y e IR+ : fix - vii < o}
E(x).
E
Hyl = 0. -
For every -
L-1
L-1 anc! y e IR , we
It is sufficient to show that for every x e IR +
Proof:
.
have E(x) = £(y). -
.
...
Suppose not, t h en th ere ex1st x e
+
IRL-1
an
+
d ...
y e
~
L-1 +
such
-
L-1 "' that E(x) E(y), then the complement IR 'E(x) is nonempty. By Step 1, + L-1 "' " ... L 1 lR+ ~(x) is equal to the union of those E(y) for which y e IR +- ~(x). By
=
•
IRL-l~(;)
Step 2. this implies that ...
L-1
partition {E(x},IR + open.
is open.
+
...
~(x)}
L-1
of IR +
,
Hence we have obtained a
both of whose
This contradicts the connectedness of IRL-I. •
+
-
e~ements
are nonempty and
E(~) =
Hence
E(;) for
... L-1 L-1 every x e IR . and •v e IR +
+
I(x)
By Ste? 3,
-
-
L-1 I(O) for every x e IR . +
Thus for every x e IR
...
exists a unique a e IR such that o:e - (O,x}. 1
-
-L-1 e ;:'1 ... •
L-1
Define f/J: IR+
Define u: X -+ IR by u(x) = x
L-1 +
, there ...
-+ IR by ¢(x)e
1
-
fc:-
1
every x e X. •
Step 4: The function
u.(
•
·l represents >-. •
Proof: Suppose that x e X, y equivalent to (x
1
E
-
X, and x >- y.
-
By the quasilinearity, this is
- y , x , ... , xL) ~ (O,y , ... ,yL}.
1
2
2
¢( ·}. this is equivalen-:: to Cx
By the definition of
- y , x , ... , xL} ~ ¢(y , ... ,yL)el" 2 1 1 2
the quasilinearity, this is equivalent to
Again by the definition of ¢( · ), this is equivalent to ¢(x2, ... , XL)el ?= (¢(y2, ... ,yL) + yl - xl)e1.
Hence ¢(x , ... , xL) 2
2:
q)(y
2
, ... ,yL) + y
- x , that is, u(x) 1 1
•
•
3-7
2:
u(yl.
Again by
These properties of u( · ) are cardinal, because they are not preserved under some monotone transformation, such as f(u(x)) = u(x?.
3.C.6
(a} For p = 1, we have u(x) = o: x + o: x . 1 1 2 2
Thus the indifference
•
curves are linear. •
(b) Since every monotonic ·transformations of a utility function represents the same preference, we shall consider
By L'Hopital's rule, lim
-u(x)
p .. O
-
1 im (o: x~lnx p ... O
Since expUet
1
1
1
we have obtained a Cobb-Douglas utility
+
function. T..1ere is an alternative proof to this proposition: Since both the CES and •
the Cobb-Dougias utility functions are continuously differentiabie and •
• •
•
homothetic, i: is sufficient to check the convergence of the marginal rate of substitu:ion at eve:y point.
The marginal rate of substitution at (xl'x l 2
p-1 p-1 with respect tc the CES utility function is equal to o: x lc:: x . 1 1 2 2
•
The
mar gina! rate of substitution at (x ,x ) with respect to the Cobb-Dougias 1 2
as p
~ 1.
(In fact, a: x
p-1
p-1
let x 2 2 1 1
o: x 11X x when p = 1.) 1 1 2 2
is well defined for eve:-y p a!1d is equal to
The proof is thus completed.
St:-ictly speaking. there is a missing point in both proofs: We proved the conver-ge:.1ce
o:~
2
preferences on the strictly positive orthant {x e IR : x » 0},
3-8
•
but we did not prove the convergence on the horizontal and vertical axes.
In
fact, the convergence on the axes are obtained in such a way that all vectors To be more specific, compare, for example, x =
there tend to be indifferent.
= (yl'O}
Cx ,o> and y 1
with x
1
> y > 0. 1
According to the CES utility •
function, x is preferred to y, regardless of the values of p.
But, according
to the Cobb-Douglas utility function, x and y are indifferent..
Futhermore,
•
the following is true: If x is in the strictly positive orthant and y is on an axis, then x is preferred to y for every p sufficiently close to 0.
To see
.
»
this, simply note that if x = (x ,o) with x > 0 and y 1 1 and a ~ + a ~ -+ a
1
2
1
a .
+
0, then
The implication of this fact is that, as p -+ 0,
2
every vector in the strictly positive orthant becomes preferred to all vectors on the axes.
That is, unconditional preference towards strictly positive .
vectors tends to hold, as it is true for the Cobb-Douglas utility function.
(c) Suppose that x
1
:s x .
We want to show that
2
x
Let p < 0.
Since x
?! (o:
x
p
1 1
+
1
~
.
p l/p
= 11m £o: x + o: x l 1 2 2 1 1 p-t -co
0 and x p 1/p .
a: x ) 2 2
p
.
2
~
-
0,
Thus
'
•
On the other hand, smce x :s x . 1 2 p
(o:l + o:2)xl.
Therefore,
Hence al Letting p
.1lffi . p-+- a:
3.0.1
'
\C::l
-+ -
l/p
>
xi -
("'
xp
Yo! 1
p)l/p
+ a2x2 p
CXl,
i!:
(
p)l/p
we obtain lim Co: x + a x 2 1 1 2 p-+ -co X •. L
To check condition (i),
3-9
al + a:2
)1/p
xl.
= xl' because 1 im P.... -co
x (i\p,i\w} 2
= (1
...: o:.HAw)/(Ap )
2
= (1
- a:)w/p
= x 2 (p,w).
2
To check condition (ii), •
plxl(p,w) + p2x2(p,w) = pla:w/pl + p2(1 - cx)w/p2
= w.
Condition (iii) is obvious.
•
3.D.2.
To check condition (i),
•
•
•
v(i\p,ll.w)
= a:lna: + Cl - a:)ln(l - a:) + IM.w - cxlllAp 1
= a:lna:
+
(1 - o:.)ln(l - et} + InA + lnw
- a:lni\ - cdnp - (1 - «)lnA - (1 - cxHnp 1 2
= edna: + (1 - a:HnCl - a:) + lnw - cdnp - (1 - a)lnp 1
c
2
•
v(p,w).
To check condition (ii), 8v(p, w)/8w = 1/w > 0, cv(p, w)/8pl = - alp! < 0,
ov(p, w)/8p2 = - (1 - a:)/p2 < 0.
Condition (iv) follows the functional fox·m of v( • ). In order- to ver-ify i iii)', by 'Property (i}, it is sufficient tha:., for anv v e .
-
-;,~
!)!"'ove
~ and w > 0, the set {p e IRL++ : v(p,w) ~ v} is convex.
Since
the logari:hrni:: function is concave, the set
is convex for eve':"y v e IR.
Since the other terms, a:lna: + (1 - a.)ln(l - IX) + lnw,
de not depend on p, this implies that the set {p e IR
3.0.3
(a) We shall prove that for every p e •
3-10
L
w
++
i!:
: v(p, w)
~
v} is convex.
~L 1"f 0, a i!: 0, and x e "' +'
x
= x(p,w),
= x(p;aw).
then ax
affordable at (p,aw). u(IX
-1
yl :s u(x).
Note first that p· (ax) :s aw, that is, ax is
Let y e IR
L +
and p · y :s a:w.
Then p· (a
Tnus, by the homogeneity, u(y) :s u(crx).
•
-!
yl :s w.
Hence
Hence a:x = x(p,a:w) . •
By this result, v(p,o:w) = u(x(p,a.w)} = u(cxx(p,w))
= «X"(x(p,w)) = av(p,w).
Thus the indirect utility function is homogeneous of degree one in w.
wx(p)
•
Given the above results, we can write x(p,w) = wx(p,l) =
wv(p).
= wv(p,l) =
C JJ
C.T/v•
lS
and v(p,w)
•
Exercise 2.E.4 showed that the wealth expansion path
-x(p).
{x(p,w): w > 0} is a ray going through •
•
The wealth elasticity of demand
eoual to l. •
(b) \Ve firs: prove that for every p e IRL , w ++
ax(p,w}.
:.?:.
0, and a. :: 0, we· have X(p,o:w) =
In fact, since v(·,·) is homogeneous of degree one in w, Vpv(p,cxw) =
aVpv(p,w) and Vwv(p,IXW) = Vwv(p,w).
Thus, by Roy's identity, x(p,aw) =
ax{p, w}. NOYl
,.. 1
X E
-~
L
IR • x' +
e IRL, u(x) = u(x'), and o: :: 0. +
Since u( · l is strictly
quasiconcave, by the supporting hyperplane theorem (Theorem M.G.3), the::-e exist p e x(p',w'}.
1
r;.io ••
, w :: '0, and w' ' e !RL P ++ ++ ,
i!::
0 such that x = x(c,w) and x' = .
Then u(x) = v(p,w) and u(x') = v(p',w').
Hence vlp,wi = v(p',w'} .
•
Tnu·s, by the homogeneity, v(p,a:w) = v(p' ,aw' ). lXX
and x(p' ,aw') = ax'.
u(IXX) = u.(ax').
3.D.4
But as we saw above. x(p,o:w) =
Hence v(p,o:w) = u(a:x) and v(p',aw') = u(cxx' ).
Ther-efore u(x) is homogeneous of degree one.
L
(a) Le: e = (1,0, ... ,0) e IR . 1
We shall prove that for every p e
1 w e R, a e iR, and x e (- co, co) x IR:- • if x = .x(p,w}, then x x(p, w + c::l.
Thus
Note fir-st that, by p
is affordable a! (p, w + a.e L 1
1
= 1, p· (o:x L
+
ae
1
=
e l :: o:w, that is, x 1
Let y e IR + and p · y :s w + a.
3-11
+
+
ae
1
Then p · (y - IXe l 1
< -
w•
Hence·x
x + o:e
1
~
y- o:.er
= x(p, w +
1
>- y .
-
Hence
o:).
Therefore, for every
XiP· w' ).
Thus, by the quasilinearity, x + ae
t e
w e IR, and w' e rR, xl(p,wl =
{2, ... ,L},
That is, the Walrasian demand functions for goods 2, ... ,L are As for good 1, we have 8x(p, w)/Bw = 1 for every (p, w).
independent of wealth.
That is, any additional amount of money is spent on good 1. •
•
(b) Define t/J(p) = u(x(p,O)).
Since x(p,w) • x(p,O) + we and the preference 1
relation can be represented by a utility function of the quasilinear form u(x)
= x
1
+
-u(x
2
, .... ;_l (Exercise 3.C.S}, we have
v(p,w) = u(x(p,w))
•
-u(x = w + x (p,O) + 1
2
(p, w), ... ,xL (p, w)).
-u(x
2
(p,O), ... ,xL(p,O))
= w + u(x(p,O}) = w
+
t/J(p).
(c) The non-negativity constraint is binding if and only if p x Cp,Ol > w. 2 2 Note that x (p,O) = (1)') 2
-1
(p l, because p = 1. 2 1
binding if and only if p (1J') 2 by x(p,w) = (O,wlp l. 2
-1
Cp ) > w. 2
Hence the const:-aint • is
If so, the Walrasian demand is given •
Tlrus,, 'as w changes, the consumption level of the first
good is unchanged and the consumption of the second good changes at rc.te llp with w until the non-negativity constraint no longer binds.
3.0.5
(a) Si!lce any monotone transformation of a utility function represents •
the same prefer-ence relation, we may as well choose
By the first-or-der condition of the UMP with x(p,w) = •
•
3-12
-u( · ),
2
where
= p/(p - 1) e (- co, 1).
Plug this into u( · ), then we obtain
•
v(p,w)
(b) To check the homogeneity of the
x(IXp,o:w) = (aw/[(o:p ) 1
= (a.·a:
~-1
~
function, ~
~-1
+ (o:p } )((o:p } 2 1
, (ap l 2
tS-1
)
~
tS ~ tS-1 ~-1 /(X }(w/(pl + PzlHpl ·P2 > •
= x(p,w}. To check Walras' law, p·x(p,w)
= w.
The uniqueness is obvious. To check the homogeneity of the indirect utility function, v(o:p,aw) = a:w/( (o:pl)
~
+
+
~)1/~
Pz
= v(p,w)
To check the monotonicity, ~
~
8v(p,w)/8w = 1/(pl + Pz)
1/o
> 0, •
~
-
~-1
ov(p,w)/opt = - wpl
~
0 1/~+1
/(pl + p2)
. immediately from
< 0.
•
The continuity follows •
the derived functional form .
•
In orde:- to prove the quasiconvexity, by property the homogeneity, it is 2
sufficient to prove that, for any v e IR and w > 0, the set {p e IR : v(p, w) :s
. v, 1s convex. \
If
o=
0, then the utility function is a Cobb-Douglas one, and
the quasiconcavity was already established in Exercise 3.0.2.
f( p ) = (f( pJ.~>1/o
. . vt !S
~
f convex or every v.
2
Since v(p,wl = w/f(p}, this
implies that {p e IR : v(p,w) :s v} is convex for every v and w.
•
3-13
So we consider
If 1/f(p) =
o<
o, then
f(p)
o 1/C-a> (f(p} }
=
:s v}
Ill:
this implies that {p e
(c)
~
a p
.1s
+
1
~
•
Pz is a convex function.
convex for every v.
Hence {p
E
2
IR :
Sir.ce v(p,w) = w/f(p),
v(p,w) :s v} is convex for every v and w.
For the linear indifference curves. we have
x(p, w}
(w/p , 0}
if pl < p2.
(0. w/p2)
•r
1
-
h
{(w/p1)(A.. 1 -A.): A e 10,1)}
.1.
.,.
pl > P2' p ; p1 2
v(p,w) = max{w/p ,wlp }.
1
•
2
For the Leontief preference.
•
As for the limit argument with respect to p. with p < 1 and p -+ 1.
Then
a
First consider the case
= p/(p - 1) -+ - co as p -+ 1.
a )
..n ... ,.. S' __
1. we have (p ;p -+ 0. Thus 2 1 w/pl Cl-1 ~ 0 1 • ... l IH p w/Cp + p l = 1 im 1 2 1 ~...;. - = o-+- co 1 + •
1, we have (p{p l 2 ~-1
1 im P2 o-+ -co
~
~
w/(pl +
-+ co .
Thus
•
a Pz>
w/p2 1 im 0 o-+-co (p1/p2) + 1
-
0.
Thus the CES Walrasian demands converge to the Walrasian demand of the linear indiffe::"e!"lce curves.
As for the indirect utility functions, we showed in the
. ~ 0 1/~ answer to Exercise 3.C.6(c) that (p + p ) -+ p for p :s p . 2 1 1 1 2
Hence the
CES indirect u"tilities converge to the indirect utility of the linea:indifference curves. Case 2.
p
1
> p . 2
De the sa:ne argument as in the Case 1.
•
3-14
•
Case 3. ln this case, [w/2.p Hl,l}.
This consumption bundle belongs to the set of the Walrasian
1
demal'lds cf the linear indifference curves when p = p . 1 2
As for the indirect
utility functions, we showed in the answer to Exercise 3.C.6(c) that ~
(pl
~ 1/~
+ Pz>
-+ Pl for PI :s
Pz·
•
Let's next consider the case p -+ - co. So just plug
1.
~
Note that
~
= p/( p - 1) -+ 1 as p -+
= 1 into the CES Walrasian demand functions and the .
indirect utility functions.
We then get the Walrasian demand function and the
indirect utility function of the Leontief preference.
(d) From the calculation of the Walrasian demand functions in (a) we get
x (p, w)/x (p, w) = (p /p ) 1
2
1
~-1
2
•
,
£x (p,w)/x (p,w)}/(p /p > = (P/Pzl 1 2 1 2
~-2
,
d(x (p, w)/x (p, w)]/d[p /p J = (~ - 1Hp /p 1 2 1 2 1 2
Thus ~
12
~lZ(p,w}
(p,w) :::
Hence,
- p).
0 for the Leontief, and
~
12
-
•
(a) Define
&' !
7' = 1 a:1.d
12
£p,w)
= cz:
for the linear,
= I for the Cobb-Douglas u!ility
Cp,w)
•
•
-u(x)
bl)
with a' = a/(a + {3 + a-l. {3' = {3/(a. + f3 + +
~
•
... f unc-.1cns.
3.0.6
= 1/(1
= - (~ - 1)
o-2 ) .
-u( · l
unction u -+ u
r>.
a'
(x2
'l' = 7/(a + f3 + a-).
Tnen c::'
represents the same preferences as u( · ), because the f
1/( a+f3+1') .
.
1s a monotone trans ormation.
wi:hout loss of generality that a.
+
f3
+
Tnus we can assume
a- = 1.
(b) Use anothe:- monotone transformation of the given utility function,
•
+
3-15
W
The first-order condition of the UMP yields the demand function x(p,w) = (b ,b ,b ) + (w - p·b)(ct!p ,,a/p ,7/p J, 1 2 3 1 2 3
-=
where p · b
p b + p b + p b • 1 1 2 2 3 3
Plug this demand function to
u( · ),
then we
obtain the indirect utility function v(p,w) = (w •
(c) To check the homogeneity of the demand function, • x(A.p,hw) = Cb ,b ,b ) + CA.w - i\p· b)(a/i\pl',S/A.p .r/i\.p ) 1 2 3 2 3
= (b ,b ,b ) + [w- p·b}(a/p ,,S/p ,,./p ) = x(p,w). 1 2
3
1
2
3
To check Walras law,
= p·b + (w - p·b)(o: + f3 + 7} = w.
•
The uniqueness is obvious. To check the homogeneity of the indirect utility function, v(i\p,hW)
= (w -
= v(p, w).
To check the m:>notonicity,
> 0, Civ(p,w)/op
1
= v(p,w}·(- a/p ) < 0, 1
Civ(p,w)/c3p
= v(p,w)·(- {3/p ) < 0, 2 2
8v(p,w)/8p
= v(p,w}• (- 0 /p ) < 0. 3 3
The continui:y follows directly from the given functional form.
In order to
prove the quasiccnvexity, it is sufficient to prove that, for any v e !R and w 3
> 0, the set (p e IR : v(p. w) ::= v} is convex.
Consider
lnv(p, wl = IX!niX + {3in(3 + 7ln'1 + ln[w - p· b) - cdnp
Since the logarithmic function is concave, the set
•
3-16
1
- ,Slnp
- a-lnp . 2 3
3
{p e IR : lnlw - p·b) - cdnp - (3lnp 1
2
- 'llnp
3
~ v}
is convex for every v e IR.
Since the other terms, edna: + (3lr:/3 + 'lin)', do not 3 . depend on p, this implies that the set {p e ~ : lnv(p,w) :s v} is convex. 3 Hence so is {p e IR : v(p,w) :s v)
.
1
(a) Smce p · x
3.0.7
Thus x
1
0
< w
1
and x
1
:;e x
0
,
o. the weak axiom implies p · x > w . 0
1
has to be on the bold line in the following figure.
8 6
4 ··-········--·· • •' •' •• ••
0
• '•' ••
• • '•• • ••
2
4
8 Figure 3.D.7(a)
Xl
•
•
•
•
... _... be a In the following four question; we assume the given preferer.::..: ···"'..,
differentiable utility function
u( • ).
•
(b) If the pre!"erence is quasilinear with respect to the first good, then we
can take a utility function u( ·) so that 8u(x)/8x 3.C.5(b)).
each
t
1
= 1 for every x CExe:-cise
Hence the first-order condition implies 8u(xt)/8x~ = p~/p~ for
= 0,!.
0 0 S ...:t .. .. p /p < 2 1
.
I
concave, x
... . .
to be on the bold line in the following figure .
•
3-17
0
2
1
> x .
2
.
T nus x
1
has
8
4
•• •
0
•• •
•• •• •• • • '• '
•• •
2
4
'' ••' •
•
•
•
8
10 3.D.7(b)
XI
(c} If the preference is qnasilinear with respect to the second good, then then we can take a utility function u( ·) so that 8u(x)/8x = 1 for every x 2 Hence the first-order condition implies c3u(xt)/8x~ =
(Exercise 3. C.S(b)).
0,1.
Thus x
1
0 0 Since P 1P 1 2
>
1 1 . p 1p and u( · l 1s concave, we must have 1 2
has to be on the bold line in the following figu:-e. •
•
••
•
8 5.5 ..... ::~.:;:-:: • •• • • •• ••
.................... • ••• •• •• •
••• • •• •••
2
4
•
__.__.___ ___________ 0
1
(d) Since p · x
0
....;..._
8 Figure 3.D.7(c)
1
X!
<w and the relative price of good 1 decreased,
•
3-18
x~
has to
increase if good 1 is normal.
If good 2 is normal, then the wealth effect
(positive) and th~ substitution effect (negative) go in opposite direction which gives us no additional infonnation about x .
2
Thus x
1
has to be on the
bold line in the following figure.
• •
8
................ • •• ••• •• ••'
•
•
••
0
•
2
••• • •• •• •• ••
4
•
8 Figure 3.D.7(d)
Xl
(e) If the preference is homothetic, the the marginal rates of substitution at • •
all vectors on a ray are the same, and they becomes less steep as the ray 1
becomes flatte:-.
> P/P 2 . x 0
to be on the right side of the ray that goes through x . the bold line in the following figure .
•
•
1
•
•
3-19
Thus x
1
1
has
has to be on
8 •'
••
••
•
••
••
•• •
...
•••
•••
••
• • •••
• •
5.2
•
• •• • •• •••
····--··
•
••
•• ••
••
0
2
4 5.2
8
X!
3.D.7(e)
3.0.8
By Proposition 3.D.3(i}, v(ttp,tXW} = v(p,w} for all
IX
> 0.
differentiating t.'lis equality with respect to « and evaluating at obtain 'V v(p,wl·p + wc3v(p,w}/8w = 0. p
Bv
-
et •
1, we
Thus w8v(p,w)/8w = - 'V v(p,wl·p. p
3.E.l The EMP is equivalent to the following maximization problem: Max
- p·x
s.t.
u(x)
~
u and x
~
0.
Tne Kul:m-Tucker condition (Theorem M.K.2) implies that the fi!'st-order conditions ar-e that ther-e exists and IJ. • x• = 0.
~
> 0 and IJ. e IR
L +
such that p = A'ii'u(x•) + 1-1
That is, for some
This is the same as that of the UMP.
3.E.2
To check the homogeneity of the expenditure function, e(Ao,u) •
= e(p,ul. To check the monotonicity,
3-20
•
-«
cx-1 « 1-a. Be(p, u)/au = a (1 - «} p p > O, 1 2 ~ 1-IX et-1 a-1 1-a oe(p,u)/8p = et (1 - a.) p p > O, 1 1 2 ~
oe(p,u)/8p
2
= «
(X
(1 - o:J
« «
p p
-(X
> o.
1 2
2
To check the concavity, it is easy to actually calculate Dpe(p,u) and then apply the condition in Exercise 2.F.9 to show that . semidefinite.
2
op e(p,u)
is negative
An alternative way is to only calculate •
_2 2 1-« « o:-2 1-a. o e(p,u)l8p = - a. (1 - a.) p p < 1 1 2 2 Then note that the homogeneity implies that D e(p,u)p = 0. p
o. Hence we can apply
2 Theorem M.D.4(iii) to conclude that D e(p,u) is negative semidefinite. p
The
continuity follows from the functional form. •
To check the homogeneity of the Hicksian demand function, 1-a. 1-o: h 1 (ll.p, u) =
(1 -
•
u
« )ll.p1
=
•
(1 -
tt)p1" •
(1 -
h 2 (ll.o u} = . t
a.)ll.p1
(1 -
u
a.ll.p2
=
a.)pl
a.p2
To check the no excess utility, 1-o:
u(h(p,u)) =
(1 - cx)p
-
u
.cxp2
1 •
( 1 - o:)p
(1 - a)p1
u
-
1-a.
{1 ~(X )a.-(X(l-(X)
u
et+(l-a.)
= u.
1
•
The unioueness is obvious . •
3.E.3
Then A
-
Let x e IR :;e
-x
0 by
that of {x e
~
L
:
L +
u(xl
and
e A. u!x)
i!::
u.
Define A = {x e IRL: p·x s +
p·x
and u(x)
i!:: u}.
Furthermore, A is compact: The closedness follows from i!::
u}
L
and of IR +'• the boundedness follows from the
inclusion L
-
A c {x e IR : 0 s xl s p·xlpt for every l = l, ... ,L}.
3-21
Now consider the truncated EMP: p·x
Min
s.t.
x e A.
Since p · x is a continuous function and A is a compact set, this problem has a solution, denoted by x• e A. the original EMP.
We shall show that this is also a solution to L
If x e A, then p · x :=: p· x• because
Let x e IR + and u(x) o= u.
x• is a solu:ion to the truncated EMP.
If x t! A, then
p· x > p · x
and hence p ·-x
•
> p · x•.
Thus x• is a solution of the original EMP.
Suppose first that >- is convex and that x e h(p,u) and x' e h(p,u}.
3 •.E.4
-
Then p·x = p·x' and u(x) (1 -
2:
u, u(x'}
0!::
Let a. e [0,1] and define x" = ax +
u.
Tnen p · x" = ap · x + (1 - a:)p · x' = p · x = p · x' and, by the convexity
a}x'.
of >-, u(x"l
-
~
u.
Thus x" e h(p,u).
Suppose next that >- is strictly convex and that x e h(p,ul, x' e h(p,u},
-
= x',
and u(x) o= u(x' l
0!::
u.
By the argument above, x" = ax + (1 - o:)x' with
a e (0,1) satis:ies p·x" = p·x = p·x' and, by the strict convexity of >-, we
-
have x" >- x'. to L
Since >- is continuous, (3x" >- x' for any (3 e (0 ,ll close enough
-
But this implies that p · ((3x") < p · x and u((3x" l > u(x')
cont:;:-adicts the fact that x. is a sol\ltion of the EMP.
3.E.S
uh{p),
[Fi:-st pfir..tinsz errata: The equality h(p,u) = because u is a scalar and
tha:, for every p » 0, u h(p,o:ul.
2:
0, a
-h(p) l!:.
is a vector.]
0, and x
2:
Hence
-h(p)u
2: u,
which
1 ~ ···.---
l mu-· ;l!L. be a
.... : --
shoulc be h(o,u} = •
We shall first prove
0, if x = h(p,c), ther. ax =
In fact, note that u(ax) = a.u(x) o= au, that is, ax satisfies the
constrain~
Hence p· (a
of -1
t~e
y) l!:.
EMP for em. p·x.
Let y e IR
Thus p·y
L +
~ p·(ax).
3-22
and u(y)
l!:.
o:u.
Hence ax = h(p,au].
-.•
u(o: ·y) :=: u.
The:-e:!'ore
h(p,u) is homogeneous of degree one Jn u. By this result, e(p,o:u) = p•h(p,o:u) = p·(ah(p,u)} = o:(p·h(p,u)) = o:e(p,ul. Thus the expenditure function is homogeneous of degree one in u. Now define e(p,u) =
-h(p)
= h(p,l) and
-e(p)
ue(p) .
= e(p,l), then h(p.u} =
uh{p)
and
•
•
3.£.6
Define
~
= p/(p - 1}, then the expenditure function and the Hicksian
demand function are derived from the first-order conditions of the EMP and they are as follows: h(p,u) = uCp 1 e,p,u} =
~
1
o u(p 1
+ +
P
p
~
2 ~
2
)
c1-a)la
ua ) .
(p
~-1
1
.p
CJ-1
2
),
To check the homogeneity of the expenditure function,
a ·I/o c ~ a. u pl
CJ J11a _ o:e(p, u). + Pz -
To check the monotonicity,
o (p +
~
oe(p,ul/au =
1
~
p l
I/a
2
> o,
> 0.
oe(p,u}/opl = ~
•
To check the concavity. it is a
. bit lengthy
::..:1.lculate but easy to actuai lv 2 D e(o,u) p •
is negative semidefinite.
An alternative way is to only calculate
a>I/~-1
+ p2
bv cS < 1. •
+
a-1£ a up! PI
..,_ 1
a,I/~-2(1/~ - l)Clpa • + Pz 1
2 The:-t note that the homogeneity implies that D e(p,u)p = 0. p
semidefinite.
Tne continuity follows from the functional form.
3-23
Hence
To cheek the homogeneity of the Hicksian demand function, h(ap.u) = u((ap )
~
1
= ex
+
~ (1-~}/~
(o:p ) ) 2
cao-~>t~l+(o-u
((ap l 1
~-1
,(exp l
c3-l
2
)
c a up
1
= h(p,u).
To check no excess utility, u (h( p, u )) =
a u (p 1
a + Pz
>(1-ol/oc ca-op . P
ca-Ilp>l/p + P 2
1
Since (c3 - l)/c3 = - 1/p, we obtain u(h(p,u)) = u.
The uniqueness is obvious .
•
3.E.7
In Exercise 3.C.S(b), we showed that every quasilinear preference with
respect to good 1 can be represented by a utility function of the
=
u(x)
•
x
1
+
-u(x
2
We shall prove that for every
, ... ,XL). •
p » 0 with p x + ae
1
1
= l, u e IX, ex e IR, and x e (- .,, .,) x IR
= h(p, u +
Note first that u(x + cte )
a).
1
:2::
satisfies the constraint of the EMP for (p, u + «). c::.
L-1 +
, if x
~
h(p,u), then
u + o:, that is, x + ae Let y e !R
L +
and u(y)
~
Then u(y - ae.) :: u . •
Hence x + ae
1
= h(p, u
+ a).
Therefore, for e.,.·e:-y l e {2, ... ,L}, u e IR, and u' e IR, hl(p, u) =
.
That is, the Hick!>ian' demand functions for goods 2., ... ,L a -· independent of utiiity levels. =
-h(p)
+
Thus, if we define
-h(p)
~
= h(p,O), then h(p,u)
uel'
Since h(p, u + a) = h(p,u} + ae , we have e(p, u + a) = e(p,u) + a. 1
Thus, if we define
3.E.8
-e{p)
= e(p,O), then e(p,u) =
We use the utility function u(x} =
-e(p)
+
u.
To prove (3.E.1l,
o:-1 a 1-o: « 1-a -a a-1 e(p, v(p, w)) = a (1 - o:) p p (o: (1 - ctl p p w) c w, 1 2 1 2 a-1 a 1-a a 1"« -a o:-1 -a {p,e(p,u)) = o: (1 - «} Pl Pz (a (1 - «) P1P2 u) = u .
-a
•
3-24
1 u +
To prove (3.t.3), x(p,e(p,u)) = (a:-c::(l 1-IX
-
u,
= h(p,u),
u
1-a:
a>
h(p,v{p,w))
1-IX -a: a:-1
PI Pz w
(1 - o:)p
•
0:.
1
•
3.E.9
First, we shall prove that Proposition 3.0.3 implies Proposition 3.E.2
via (3.E.Il.
Let p » 0, p'
»
0, u e IR, u' e IR, and a
~
0.
•
> 0.
(i) Homogeneity of degree one in p: Let a:
v(p,w) by the second relation of (3.E.l).
Define w = e(p,u), then u =
Hence
e(o:p,u} = e(a:p,v(p,w)) = e(a:p,v(a:p,a:w))
= a.w
whe:-e the second equality follows from the homogeneity of
= o:e(p,u}, v( • , • }
and the third
from the first relation of (3. E.l). (ii) ~fonotonicity:
Let u' > u.
v(p, w) a.'1d u' = v(p,w').
Define w = e(p,u) and w' = e{p,u'}, t!\en
By the monotonicity of •
•
v( ·, ·)
L:
=
in w, we must have w'
•
> w, that is, e{p' ,u) > e(p,u). Next let p' =: p.
Define w = e(p,u) and w' = e(p' ,u), then, by the second •
relation of (3.E.l), u = v(p,w) = v(p' ,w').
By the monotonicity of
v( ·, · ), we
must have w' =:. w, that is, e(p' ,u) :=: e(p,u). (iii} Concavity: Let o: v(p,w) = v{p',w).
e [0,1].
Define w = e(p,u) and w' = e(p',u). then u = •
Define p" = a.p + (1 - o:.)p" and w" = a:w + (1 - IX)w'.
by the quasiconvexity of v( ·, • ), v(p", w") v( • , ·
~
u.
Hence, by the monotonicity of
l in w and the second relation of (3.E.l), w"
~
e(p•, ul. that is,
e(ap + (1 - o:)p', u) :=: o:e(p,u) + (1 - a.)e(p' ,ul.
3-25
Then,
•
(iv) Continuity: It is sufficient to prove the following statement: For any
n n)}co . ( n n sequence {(p ,u n=l wtth p .u )
~
. n n (p,u) and any w, 1f e(p ,u ) :s w for
every n, then e(p,u) ~ w; if e(pn.un} =::: w for every n, then e(p,u) == w. n n Suppose that e(p , u ) :s w for every n.
Then, by the monotonicity of v( ·,.) in
w, and the second relation of (:l.E.ll, we have. un ~ v(pn,w) for every n. the continuity of
v( ·, • ),
u :> v(p,w).
By the second relation
~f
By
(3.E.ll and
•
the monotonicity of v( ·, ·) in w, we must have e(p,u) :s w.
The same argument
can be applied for the case with e(pn,un) =::: w for every n.
Let's next prove that Proposition 3.E.2 implies Proposition 3.0.3 via (3.E.ll.
Let p
»
0, p'
Homogeneity: Let
(i)
»
0, w e R, w' e IR, and a.
Define u = v(p, w).
:> 0.
of (3.E.l}, e(p,u} = w.
2:
0.
Then, by the first relation
Hence
v(o:p,o:w) = v(«p,cte(p,w)} = v(o:p,e(o:p,u)} = u = v(p,w}, where the
se~ond
equality follows from the homogeneity of e{ ·, · l and the thi::-d
from the second relation of (3.E.l). (ii)
Monotonicity: Let w' > w.
e(p,u)
=w
and e(p,u'}
must have u'
= w'.
Define u = v(p,w) and u' = v(p,w'), then By the monotonicity of e( ·, ·) and w' > w, we
> u, that is, v(p, w') > v(p, w).
Nex:, assume that p' e(p,u} = e(p',u'} = w.
2:
p.
Define u = v(p, w) and u' = v(p', wl, then
By the monotonicity of e(·,·l and p'
2::
p, we must have
u' :s u, that is, v(p,wl == v(p',w). (iii}
Quasiconvexity: Let o: e [0,1}.
Then e(p,u) = w and e(p,u') = w'. 2:
u.
Define p''
= o:p
+ (1 -
Define u = v(p,w) and u' = v(p',w'). Without loss of generality, assume that u'
a:)p' and w = o:w
+ (1 -
e{p" ,u') == ae(p,u'} + (1 - o:)e(p' ,u')
3-26
ct)w'.
Then
i!::
ae(p, u) + (1 - ct)e(p', u')
= o:w
+ (1 -
ct)w'
= w",
where the first inequality follows from the concavity of e( · ,u), the second from the monotonicity of e( ·, ·) in u and u'
i!::
u.
We must thus have v(p", w")
~
•
u'.
(iv} Continuity: It is sufficient to prove the following statement.
For anv•
•
nnca. nn nn sequence {(p ,w )}n=l Wlth (p ,w ) -+ (p,w) and any u, if v(p ,w )
::5
u for
every n, then v(p,w} ~ u; if v(pn,wn)
i!::
u.
n
n
Suppose that v(p , w )
i!::
u for every n, then v(p,w)
· u for every n.
~
Then, by the monotonicity of e( ·, ·) in
u and the first relation of (3.E.l), we have wn ~ e(pn,u) for eve:-y n. continuity of e( ·, · ), w
~
e(p, u).
We must thus have v(p, w}
~
u.
By the
The same
argument can be applied for the case with v(pn,wn) :=: u for every n.
An alternative, simpler way to show the equivalence on the concavity/
quasiconvexity and the continuity uses what is sometimes called the epig:-aph. For the concavity/quasiconvexity, the concavity of e( · ,u) is equivalent to the convex!:y of the v( ·:.
u.
s~.t
{(p,w):. e(p,u)
i!::
w}
and the quasi-convexity of
is equivalent to the convexity of the set {(p,w): v(p,w) :s u} for eve!"y B•Jt (3.E.l} and the monotonicity imply that v(p,w)
e(p,\.i)
i!::
w.
::5
u if and only if ·
Hence the two sets coincide and the quasiconvexity of
v( ·)
is
equi\•alent to the concavity of e( · ,u). As for the continuity, the function e( ·) is continuous if and only if both {(p,w,u): e(p,ul ~ w} and {(p,w,u): e(p,u) :=: w} are 'closed sets. function v( ·} is continuous if and only if both {(p, w, u): v(p, w) Hp. v:,u}: v(p, w)
::5.
u} are closed sets.
i!::
u} and
But, again by (3.£.1) and the
monot~nici ty,
3-27
The
{(p,w,u):· e(p,u}
~
w) = {(p,w,u): v(p,w) 2: u};
{(p,w,u): e(p,u)
2:
w} = {(p,w,u): v(p,w) s ti}.
Hence the continuity of e( ·) is equivalent to that of v( •).
[Fi:-st printing errata: Proposition 3.E.4 should be Proposition
3.E.10 3.E.3.] •
Let's first prove that Proposition 3.0.2 implies Proposition 3.E.3 via the relations of (3.E.l) and (3.E.4). (i) Homogeneity: Let a.
relation of (3.E.l).
> 0.
L
Let p e R
++
Define w
and u e R.
= e(p.u),
then u
= v(p, w)
by the second
Hence .
h(a.p,u)
= x(a.p,e(a.p,u)) = x(a.p,a.e(p,u)) = x(p,e(p,u)) = h(p,u), •
where the first equality follows from by the first relation of (3.E.4), the second from the homogeneity of e( · ,u), the third from the homogeneity of x( ·, · }, and the last from by the first relation of (3.E.4l. (ii) No excess utility: Let (p,u) be given and x e h(p,u}.
x(p,e(p,ull by the fir-st relation of (3.E.4).
Then x e
Thus u(xl = v(p,e(p,u)) = u bv •
the second relation of (3.£.1). (iii}
Convexity/Uniqueness:. Obvious.
Let's first prove that Proposition 3.E.3 implies Proposition 3.0.2. via the relations of (3.E.l) and (3.E.4).
Let p e IR
L ++
and w e IR.
(il Homogeneity: Let a. > 0 and define w = e(p,u), then v(p,w) = u.
Hence
x(a.p,a.w) = h(a.p,v(a.p,a:w)) = h(a:p,v(p,w)) = h(p,v(p,w)) = x(p,w}, where the fi!"st equality follows from the second relation of (3.£.4), the second from the homogeneity of v( · ), the third from the homogeneity of h( · l in p, and the last from the first relation of (3.E.4). ( ii) Walras' law: Let (p, w) be given and x e x(p, w}.
3-28
Then x e h(p, v(p, w) l by
the second relation of C3.E4}.
Thus p·x = e(p,v(p,w)} == w by the definition
of the Hicksian demand and the first relation of (3.E.ll. (iii)
Convexity/Uniqueness: Obvious.
3.F.l
Denote by A the intersection of the half spaces that includes K. then
clearly A
::l
K.
To show the inverse inclusion, let
-x
e K, then, since K is a
•
closed convex set, the separating hyperplane theorem (Theorem M.G.2) implies that there exists a p L
Then {z e IR : p·z
i!!:
:;e
0 and c, such that
p •x
< c < p · x for every x e K.
c} is a half space that includes K but does not contain x.
Hence
-x
3.F.2
If K is not a convex set, then there exists x e K and y e K such that
0/2)x
+ {l/2)y E
E
A.
Thus K
::l
A.
K, as depicted in the figure below.
The intersection of all
the half -spaces containing K (which also means containing x and y) will contain the point (l/2)x
+ (l/2)y,
since half-spaces are convex and the
intersection of convex sets is convex.
Therefore, the point (112lx
cannot be separ-ated from K.
X
(II2)x + (112}y
y Figme 3.F.2
•
3-29
K
+
{1/2)y
As for the second statement, if K is not convex, then there exist x e K, y e K, and o: e [0,11 such that o:x + (1 - o:)y e K.
Since every half space that
includes K also contains o:x + (1 - et)y, it cannot be separated from K.
3.G.l
Since the identity v(p,e(p,u)) = u holds for all p, differentiation
with respect to p yields V v(p,e(p,u)) + (Bv(p,e(p,u))/8w)V e(p,u) = 0.
p
p
By Roy's identity,
(8v(p,e(p, u))/Bw)(- x(p,e(p, u)) + V e(p,u)) • 0. p By ov(p,e(p,u))/Bw > 0 and h(p,u} = x(p,e(p,u}}, we obtain h(p,u) = V e(p,u}. p •
•
3.G.2
from Examples 3.D.1 and 3.E.l, for the utility function u(x)
we obtain a/pl [I - o:)/p2
•
2 - o:w/pl D x(p,w) = p
0
0 - (1 - a:)w/p2
Ve(p,ul = D e(p,u) p
•
2
•
•
= Dp h(p,u) - cx(l
o:(l - (X)/pr P2
o:(l - o:)/plp2
2
•
- o:(l - IX)/p2
1-o: . The indirect utility function for u(x) = x x 1s 1 2 0:
« v(p,w) = Cp /et) (p /0 1
2
a.))
1-o:
w.
(Note he:-e that the indirect utility function obtained in Example 3.0.2 is for the utility function u(x) = o:lnx
1
+ (1 -
o:)lnx .) 2
Thus
vpv(p,w) = v(p,w)(- a/pl. - (1 - o:)/p2)'
3-30
Vwv(p,w) = v(p,w)/w. Hence: h[p,u) = V e[p,u}, p
2 o e(p,u) = D h(p,u), which is negative semidefinite and symmetric,
p
p
D h[p, u)p = 0,
p
D h(p,u) = D x(p,w) + D x(p,w)x(p,w)T, p p w •
x l(p,w) = - (8v(p,u)/Bpt)/(8v(p,u)/8w).
3.G.3
(a) Suppose that o: + f3 + '¥ = 1.
Note that - b ) + "¥ln{x - b l . 2 3 3 2
lnu(x) = cdn(x - b ) + 13lnCx 1
1
•
By the first-order condition of the EMP,
Plug this into p·h(p,u), then we obtain the expenditure function
•
To check the homogeneity of the expenditure function, (-,. I e,,.p,u,
To check the monotonicity, assume b
1
~
0, b
2
i!!:
0, and b
3
i!!:
>
0.
0,
> 0, > 0.
2 to To c!leck the concavit.y, we can show that D e(p,u) is eaual • p
3-31
Then
-
af3/plp2
- 13 c1
and then apply the condition in
-
2
a ltp2
•
ise 2.F. 9 to show •
negative semidefinite.
An alternative way is to only calculate the 2 x 2
2 submatrix obtained from o e(p,u) by deleting the last row and the last column p
and apply the condition in Exercise 2.F.9 to show that it is negative definite.
Then note that the homogeneity implies that
semidefinite.
2
op e(p,u)p
= 0.
Hence
The continuity follows from the functional form.
To che:::k the homogeneity of the Hicksian demand function. h( i\.p, u}
= (bl,b2,b3) + ui\.
tt+a+;-1
tt
13
;
(pl/tt) (p2//3) (p3/ir) (tt/pl,/3/p2' 71p3)
= h(p,u).
To che:::k no excess utility,
= u. The uniqueness is obvious.
[b) We cal:::ulated the derivatives 8e(p,u)/8pl in (a).
If we compare them with
hip,u). then we can immediately see 8e(p,u)/8pl = hip,u).
(c) Bv (b), D h(v,ul • p .
2
= D e(p,u). p
2 In (a), we calculated D e(p,ul. p
3.0.6, we showed
3-32
In Exercise
2 o:/pl D x(p, w) p
-
-(w-p·b)
0
0 2
0
-
0
f3/p2
0
a:/pl
2
0
1'1P3
f3/p2
(bl,b2,b3).
7/p3
Using these results, we can verify the Slutsky equation . •
(d) Use D h(p,u) p
and the explicit expression of
(a).
•
2 (e) This follows from S(p,u) = D h(p,u) = D e(p,u) and (a). in which we showed p
p
2
•
that D eCp, u) is negative semidefinite and has rank 2. p
3.G.4
Let a > 0 and b e IR.
(a)
-u{
each l,
IR+ ~ IR by
-ut(xt}
~:
Define
= aut(xt)
u(x) = artut(xt} + b
+
IRL +
~
b/L.
= I:t'aul(xt)
1R bv u(x) = au(x} + b and, for •
Then +biLl
= r_i~l(xe).
Thus any linear (to be exact, affine} transformation of a separable utility function is again separable. l\ext, we prove that if a monotone transformation of a sepa:-able utility function is again separable, then the monotone tra."l.sformatior. must be linea• (affine). monotone. ul(~+}
To do this, let's assume that each u£' ·) is continuous
strongly
Tnen, for each l, the range ut(IR) is a half -open inter-val.
or
some be is equal to + co, then b and c a!"e +
Then uCKL) = [a, b). +
and the utility function
..u( ·)
So let
Define cl = bl - at > 0, a = Ltal' b =
= [al' bel, where bl may be + co.
L:ebt' and c = Etc . 2 well.)
Lid
(I)
as
Suppose that /: [a, b) ... IR is strongly monotone defined by
..u(x)
= f(u(x}) is sepa:-abie.
simplify the proof, let's assume that f( ·) is differentiable. -+ IR bv•
g(v) = f(v + a) - f(a),
3-33
To
Define g: [O,c)
then gCO) = 0, g( ·) is diffel"'entiable, and g(u(x} - u(O)) = f(u(x)) - f(u(Q)) L
for every x e IR ,..•
Thus, in OI"'der to piOVe that
sufficient to prove that g( ·) is lineal"'.
f( ·) is linear (affine), it is
For' this, it is sufficient tc show
that the first-order derivatives g'(v) do not depend on the choice of v e [O,c).
To this end, we shall fil"'st pi"'OVe that if vl e [O,ct) for' each t, then g([tvt) = Ltg(vl).
For' this, it is sufficient to prove that g(u(x) - u(O)) ==
L
for' every x e IR • +
In fact, by the separability assumption, for each i., there
exists a monotone utility function
-ut,< ·}
such that
-u(x)
= Ltut'xt)
for every x •
. I. L I. l e IR . Fix an x e IRL and, for each t, defme y e IR+ by Yt = xi and yk = 0 for + + I. l any k :;c t. Since u(y ) = f(u(y )), L
Subtracting
' Lk=luk(O)
=
-u(O)
= f(u(O)) from both sides and noticing that
ul(xl) + Lk:;cfuk(Q} = uixtl - u,_!O) +
k=luk(O), we obtain
Summing over- l, we obtain
Since
we have g(u(x) - u(O)) ==
Et~Cut(xt)
- u(O)).
We have thus proved that g([lvl) == [,_gCv,_L To prove that the g'(v) do not depend on the choice of v e [O,c), note first that if vl e [O,cl) for each l and v == Etvl e [O,c), then g'(vl = g'(vll
3-34
for each
t.
This can be established by differentiating both sides of g([lvll
= Ltg(vl} with respect to vt" So let v e (O,c) and v' e [O,c}, then, for each l, there exist vf. e
Then g'(v) = g'(v ) and g'(v') = g'(vil·
vi
+
-v e [O,cl. 2
Now, for some
-v
e [O,c ), consider v + 2 1 2
-v
1
2
e [O,c) and
•
Then
= g'(v + 1
= g'(v' + 1
=
g'(v2}'
=
g'(v2).
Thus g'(v ) = g'(v') and hence g'(v) = g'(v'). 1 1 Note that the above proof by means of derivatives is u!:'lderlain by the •
cardinal proper.:y of additively separable utility function, which is that, when moving from one commodity vector to another, if the loss in utility from some commodity is exactly compensated by the gain in utility from another, then this must be the case for any of its monotone transformations resulting in
anothe~
additively separable utility function.
(This fact is often much
mo:-e shortly put into as: utility differences matter.) X
e IR
L
• •
anc
X'
For example, consider-
e
•
implies that u(x) = u(x'). g(u (x l - u. (0)} 1 1 1
+
By the equality g([lvl} = [lg(vtl•
g(u Cx >- u CO)) = g(u (xi> - u Wll + g~u <xzl - u (0)). 2 2 2 2 1 1 2
Hence
We have st:.own tha:, under the differentiability assumption, if this holds for
(b) Define S = {l, ... ,L} and let T be a subset of S.
3-35
The commodity vectors
for those in·s are represented by z
1
=
the like, and the
commodity vectors for those outside S are represented by z and the like. and z2 e IR +
~#T
We shall prove that for
L-#T
,
2
= {zf)lET
E ~'I.
.,.
I
1
2
IR
+
L-#T z 2 e IR+ ,
(z ,z l ?: (zi,z 1 if and only if Cz .z;21 ?: (zi,zil. 1 2
E
L-#T
In fact,
since u( ·) represents ?:• (zl"z l ?: (zj.z l if and only if 2 2 LteTut(zl) + Lt!CTut(zt)
·::!!::
LteTut(zt) + LtETul(zt).
Likewise, Cz ,zz.) ?: (zi,z2_) if and only if 1
LteTut(zl} + Lt!C~lCz£>
l!:
LteTu£'zt) + LteTulCz£l-
But both of these two inequalities are equivalent to
Hence they are equivalent to each other.
{c) Suppose that the wealth level w increases and all prices remain unchanged. Tne::-1 the demand for at least one good (say, good l) has to increase by the
Wali"as' law.
From (3.0.4) we know that uk_(xk(p,w)) = (pk/pelu£(xl(p,w)) for
eve:-y k = l, ... ,L.
Since xl(p,w) increased and ul( ·) is strictly concave, the
right hand side will decrease. xk(p,w) will have to
increase.
Hence, again since uk( ·) is strictly concave, •
Thus all goods are normal.
(d) The first-o::-der condition of the UMP can be written as ll.(p,w)pt = u'(X£'P· w}), wher-e the Lagrange multiplier ll.(p, w) is a differentiable function of (p, wl: This can be easily seen in the proof of ·the differentiability of Wah· asian demand functions. which is contained in the Appendix. By diffe:--entia:ing the above first-order condition with respect to Pf we obtain
3-36
... (Ci~(p,w)/Bpt)pl + ~(p,w)
= u"(xip,w))(ch::ip,w)/cpl
By differentiating the above first-order condition with respect to pk (k
:;o:
ll,
we obtain
...
(8~(p, w)/8pk)pt
= u"(xip,w))(Bxip,w)/8pk).
Thus d[p · x(p, w} }/dp,
iC
= xk(p,w) + pk(Bxk(p,w)/8pk) + Lt~kpiBxip,w)/pk}.
2
{8~(p,w}/Bpk)pk
+ ~(p,w)pk
...
+ Lt:;ek
u" (xk(p, w))
u"(xl(p,w))
2
>.(p,w)pk
Pt
+ (BA.(p, w)/Bpk)[l -=-~---.
-
u"(xl(p,wll
..
= u'(xk(p, w)) and hence this equals 2 u'(xk(p,w)) xk(p,w)u'(xk(p,w)} pl '"""::_:------- ( ... + 1) + (o>.(p, w)/8pk}Lt -::...: - - - - - . u"(xk(p,w)) u"(xk(p,w)) u''(xl(p,w))
By the first-order condition,
~[p, w)pk
A
By Walras' law, this equals zero.
By the strong monotonicity and the strict
...
2
u'[x. (p,w)) •
-
concavity,
-u ( . ) •
K
..
< 0 .and
-
Pt
Lt -,..~--- . u"(xl(p,w))
< 0.
xk(p, w)u' (xk ( p, w)) +
-u'(xk(p,wll
1 > 0.
Hence
... xk(p,w)u'(xk(p,w))
-.=-----. ( ---,. ,..------. u"(xk(p,w)} We must thus have 8:\(p,w)/Bpk < 0.
3.G.5
Bv on • the assumction •
+ 1)
< 0.
u"(xk(p,w)) Hence, by (•), Bxip,w)/cpk > 0.
(a) We shall show the following two statements: First, i: (x•,z•J is a
solution to
•
3-37
Max(
x,z
)
-u(x,z)
s.t. p·x
a:z ::= w,
+
then there exists y• such that q · y• ::= z• and (x• ,y• J is a solution to Max(x,yl u(x,y}
s. t. p· x
+
(aq l · y :: w;
+
(a:q l·y:: w,
0
second, if (x•,y•) is a solution to •
Max(
x,y 1
u(x,y)
s.t. p·x
0
then (x• ,q · y•) is a solution to 0
Max(x,z)
-u(x,zl
s.t. p·x
a:z :: w.
+
Suppose first that (x•,z•) is a solution to a:z :: w.
Then, by the definition of
z• and u(x•,y•) =
-u(x•,z•).
-u( • ),
X,Z )
u(x,y) ::
0
+
(aq )·y :: w. 0
inequality follows from p·x + a(q ·y) = p·x 0
Then
+
-u( · )
.
anc the second
(a:q )·y:: w and the definition 0
The first statement is thus established. se~ond
As for the u(x,yl
p·x +
·y) :: u(x•,z•) = u(x•,y•),
where the first inequality follows from the definition of
of (x•, z•).
s.t.
there exists y• such that q0 · y• -<
Let (x,y) satisfy p·x
-u(x,q
-u(x,z)
s.t.
p·x
+
one, suppose that (x•,y•) is a solution to Max( ) x,y
(aq )·y:: w. 0
For every y, if q ·y:: q ·y•, then 0 0 .
p·x•
+
(aq )·y = p·x• + a(q ·y):: p·x• 0
•
Thus p·x + o:z:: w. Thus p·x
. u(x•,y•)
0
=
-u(x•,q
0
+
a:(q ·y•):: w . 0
Now let (x,zl satisfy
·y•).
Tnen there exists y such that q ·y:: z and u(x,yl = 0
+ ( IXQ 0 ) • .V
= p • x + o:( q · y l :s p · x + az :: w. 0
-u(x,z}
= u(x,y)
::5
u(x•,y•) =
-u(x•,q
-u(x,z}.
Hence 0
· y•).
(b) (c) Tnese are immediate consequences of the fact that the Walr·asian demand functions anc the Hicksian demand functions are derived in the sta:!'ldard way, by taking
3.G.6
-u( · )
to be the (primitive) direct utility function.
(a) By applying Walras' law, we obtain x
3-38
3
- x p llp . 3 2 2 1 1
= (w - x p
(b) Yes.
In fact, for every
100 - 5Apl/~p a +
1
3
~Ap 1Ap
3
~
2
> 0,
+ ~Ap /Ap
3
+ ~AWIApJ
= 100 - Sp 1p
1 3
+ ~p 1p
2 3
+ oAP 1Ap + ~~w/~p = a + Sp 1p + oP 1p 2 3 3 1 3 2 3
+ ~w/p .
3
+ ~w1p
3
.
•
(c) By Proposition 3.G.2 and 3.G.3, the Slutsky substitution matrix is symmetric.
Thus •
f3/p3 + (~/p3)(a. + flp1/p3 +
= (3/p3
+ (~/pJ)(lOO
7Pz1 P3
+ ~w/p3)
- Spl/pJ + ~p2/p3 + ~w/p3) . •
Hence by putting p
3
= 1 and rearranging terms, we obtain +
Since this equality must hold for all pl' p
f3 +
o:o
= f3 +
2
100~. 13~
Hence a == 100, f3 = - 5, and 71 = - 5.
2 ~ w.
and w, we have &
-
sa. ,.-a
= {3a.
Therefore,
•
Recall also that all diagonal entries of the Slutsky matrix must be nonoositive. We shall now derive from this fact that • the first diagonal element is equal to • .. • - 5 + ~(100 - Sp
I r~ o~
ilt
o, . . ·n_....._. o~
2
>
o
a!;,ove value is positive.
(d) Si::1.ce x
(x
1
= x
2
~
= 0.
Let p_ = l, then
•
1
an d h ence we can a 1ways We must thus have
o
r·m d
= 0.
(p ,p . w l su:::n · ·h • a~• t'ne 1 2 In conclusion,
for a!l prices, the consumer's indifference curves in the
,x )-plane must be L-shaped ones, with kinks on the diagonal.
1
.J
2
restricted preferen.ce is the Leontief one.) following fig•.:.re.
3-39
(So the
They are depicted in the
••
...
••
••
••
••
•• • ••
••
••
• • •
••
•
••
,•
. · -------• •• •
:..·
••
•• •• •
••
••
• .~---------------
••
••
•• ••
•• • •
••
0
XI
Figure 3.G.6(d)
(e) By (d}, for a fixed x , the preference for goods 1· and 2 can be 3 Moreover, there is no wealth effect on the demands •
for- goods 1 and 2.
We must thus have
or a monotone transformation of this.
3.G.7
By the first-order condition of the UMP, there exists A > 0 such that
hg(x) = 'i7u(x).
P::-emultiply both sides by x, then .Ax·g(x) = x·'i7u(x).
Walras' law, x·g{x) = 1 and he'nce A = x·Vu(x). -1
By
Thus 1
g(x) = A 'Qu(x) = ·x·Vulx) Vu(x) By Exercise 3.0.8, we have 8v(p,l)/8w = - p·V v(p,l). By Proposition p 1 1 3.G.4, x(p) = - av(p,l)/dw Vpv(p,l) = p·ti v{p,"l) Vpv(p,l}. p
•
3.G.8
Differentiate the equality v(p,a:w) = v(p, w) + lna with respect to a and
evaluate at a: = l, then we obtain (Bv(p,w)/8w)w = 1. By Proposition 3.G.4, x(p,ll = - 'i7 v(p,l). p
3-40
Hence 8v(p,ll/8w = 1.
3.G.9
Let p
•
» 0 and w > 0.
All the functions and derivatives below are
evaluated at (p, w).
By differ-entiating Roy's identity with respect to pk, we obtain
2
2
[8 v/8ptapk)(av/8w) -
(8vl8pl)(8 vlcplaw)
(a v/8w)
•
2
Or, in the matrix notation,
1
2
- - - - :::::- (V vD v - V vD V Dx=p 2
w
v)
(V
w
p
p w
p
v)
e IR
LxL
.
L
(Recall that V v is a colmnn vector of IR , and D v and D V v are row vectors p
p
of IRL (Section M.A).}
p w
By differentiating Roy's identity with respect to w, we
obtain
(8 ------------------2~----------.
(Bv/8w)
Or, in matrix notation, 1
- - -...:.~::::- (V Dw X y)2
('17
L
e R .
vV V v w p w
w
Hence
s
=-
(v
w
- •
2
1
1
•
(v
w
2 \/}
cv w vD pv
2 (v vD v w p 2 v}
- V vD V
p
-
•
-
p w •
y
-
2
1
vw v
(V vV V v)( D v) + p 'ii v w p w w
vw2 ·. vw v
(V vD 'i7 v + V V vD v) + p p w p w p
~
p
V v( p
1
v w \!
••iJ v). •D
It is noteworthy that we can know directly from Roy's identity and this eq•Jality that the Slutsky matrix S has all the properties stated in Proposition 3. G. 2.
To see this, note first that both V v(p, w) and V v(p, w}
c•
are homogeneous of degree - 1 (in (p,w)) by Theorem M.B.l.
w
Hence x(p,w) is
homogeneous of degree 0 (where we are regarding Roy's identity as defining x(o,w) from 'i7 v(o,w} and V v(p,w)). . p . w Proposition 2.E.l.
Thus (2.E.2) follows, as proved in
On the other hand, by Exercise 3.0.8, p·x(p,w) = w.
3-41
D v p
Hence, as proved in Propositions 2.£.2 and 2.£.3, this implies (2.E.5) and (2.E.7).
Now, as proved in Exercise 2.F.7. (2.E.2), (2.E.Sl, and (2.E.7l
together imply that S(p,w)p
=0
and p·S(p,w)
= 0.
2 Tne mati"ix S(p, w) is symmetric because D v. 'V vD 'V v + 1/ 'V vD v, and p
'V vD v are symmetric. p p
following way.
p
p w
p w
p
The negative semidefiniteness can be shown in the
Since v( ·) is quasiconvex, for every price-wealth pair (q,b), •
2
if D vq + 'V vb = 0, then (q,b)·D v (q,b) p w
= 0 if and onlv• if b = - Dp vq/'V wv.
2
2
q • D vq + 2b(D 'V vq) + V v b p pw w
2:
0 (Theorem M.C.4).
But D vq + 'V · vb p
w
2 Plug this into (q,b) · D v(q,b). then
2 2
2 = q· D vq p
-
1
'V
w
\1
20 'V vq (D vq) --:P::--_w__ D vq + V2 v -..:.P__·-::;-1/wv p w ('V v)2 w
2 (('V v)(q · D vq) - 2(D V vq)(D vq) + w p p w p
2 (D vq) p 2 ). ( 'V v) p
Hence the quasiconvexity of v( • ) implies that the above expression is nonnegative for every q.
On the other hand,
CD vql
1 2 q·S(p,w}q = - ----::::-({1/ v)(q·D vq)- 2£0 V vq)(D vq) ('V v) 2 w p p w p w
Thus q · S{p, w)q ::= 0.
3.G.l0
2 2 ).
.p
( v'IN \1)
There1=ore· S(p, w) is negative semidefinite.
We shall prove that a(p) is a constant function and b(p) is
homogeneous of degree - l, quasiconvex, and satisfies b(p)
:!::
0 a.11.d Vb(p)
~
0
for every p >> 0. We shall fi:-st pro\"e that a(p) must be homogeneous of degree zero. v(p, w) is homogeneity of degree zero, we must have a(ll.p) b(p)w fo:- all p » ~
0, w
2:
0, and ll. > 0.
+
Since
b(ll.p)hw = a(p) +
If there are p and i\ for which a(hp}
a(p), then this constitutes an immediate contradiction with w = 0.
3-42
Thus
a(p) is homogeneous of degree zero. Next, we show by contradiction that 'i7a(p)
0 for every p » 0.
~
Sir.ce
v(p, w) is nonincreasing in p, we must have 'i7a(p) + 'i7b(p)w ~ 0 for every p
and w
~
0.
If there are p
» 0 and t for which Ba(p)/8pl > 0, then this
constitutes an immediate contradiction with w = 0.
»
» 0
Thus a(pl == 0 for every p
0.
We shall now prove that the only function a(p) that is homogeneous of degree zero and satisfies Va(p) :s 0 for every p
»
0 is a constant function.
In fact, by differentiating both sides of a(i\p) = a(p) with respect to i\ and evaluating the:n at h = 1, we obtain Va(p)p = 0.
Since p » 0 and va(p) :s 0,
•
we must have Va(p) = 0.
Hence a(p)
be constant.
Given this result, the homogeneity of v(p, w) implies that b(p) is homogeneous of degree - 1. function of p.
Since v(p,w) is quasiconveX, so is a(p) as a
Finally, since 'i7pv(p,w)
= 'i7b(p)w
must have b(p} :: 0 and 'i7b(p) ::= 0 for every p
:s 0 and 'i7wv(p,w) = b(p), we
» 0.
This result implies that, up to a constant, v(p, w) = b(p)w and hence, if • the unde:-lying ctility function is quasiconcave, then it must be homogeneous
-
of degree one.
.
On the other hand, according to Exercise 3.D.4(b), if the
underlying utility function is quasilinear with respect to good l. then, for all w and p »
0 with p = 1, v(p,w) can be written in the form 41Cp •... ,pL) + 1 2
w.
You will thus wonder why we have ended up excluding this quasilinear case.
Th~
reason is that, when we derived v(p,w) = (/l(pl + w, we assumed that the
. . ( consumouon set 1s .
then Bv(p, w)/Bp
1
Ill,
) IRL-1 Ill x + .
Thus x (p,w) can be negative and, if so, 1
is positive, which we excluded at the beginning of our
analysis, following Proposition 3.0.3.
(Note that, when we established
Bv(p, w)/Bpe ::= 0 in Proposition 3.0.3, we assumed that the ccnsu:::ption set is
3-43
L
As x 1(p, w) is positive for sufficiently large w > 0, the c;uasil!nea:-
1R +.)
case could be accommodated in our analysis if we assume that the ineaualitv . ~
8v(p, w)/Bpl ~ 0 applies only for sufficiently large w > 0.
(lr:
this case, we
can show that a(p) is homogeneous of degree zero, and b(p) is homogeneous of degree - 1, quasiconvex, and satisfies b(p) ~ 0. and V'b(p) :s 0 for every p » 0.)
3.G.ll
~y
Suppose that v(p,w) = a(p) + b(p)w.
Roy's identity,
x(p, w) = - (1/b(p))V' a(p} - (w/b(p))V' b(p). p
p
Thus the wealth expansion path is linear in the direction of 'iJ b!p) and p intercept (- 1/b(p))V a(p). p
•
3.G.l2
Note first that, according to Exercise 3.G.ll, the wealth expansion
path is linear in the direction of V' b(p) and intercept (- 1/b(p))V' a(p). p
p
If the uncedying preference is homothetic, then (- 1/b(p))V'pa(p) = 0. Hence a(p) must be a constant function.
If the underlying utility function is
homogeneous of degree one in w, then v(p,w) must be homogeneous of degree one ~
in w by Exe:-cise 3.0.3(a). -Hence atpl = 0 for every p »
0.
If the pre!'e:-ence is quasilinear in good l, then please first go back to . the proviso given at the end of the answer to Exercise 3.G.IO.
After doing
so, note that, since the demand for goods 2, ... ,L do not depend on w,
c-
Ilb(pllV'Pb(p) =
c-
l/p ,o, ... ,o>. 1
or (8b(p)/op, l/b(pl = 1/p and 8b(p)/Bpl = 0 for eve:-y l > 1. 1 {3p~ +
r
for some 13
:;e
0, p :~: 0, and 7 e IR.
But, by Exercise 3.G.l0, b(p) must
be homogeneous of degree - 1, positive, and nonincreasing. 0, and (3 > 0.
Tr.a: is, b(p) = f3/p
1
with {3 > 0.
3-44
Hence b(p} =
Hence p
=-
1, 1· =
If the underlying utility
function is in the qnasilinear fonn x
+
1
-ucx
.
2
, ... ,xL), then, by Exercise
J.D. 4(b), v(p, w) must be written in the form (j)Cp , ... ,pL) + w for all p 2
with p
.3.G.l3
1
= 1.
»
0
= 1•
Thus (3
For each i e {0,1, ... ,n}, let a. be a differentiable function defined 1
on the strictly positi've ort~ant {p be an indirect utility flDlction.
» 0}. Let v(p, w)
IR : p
E
= ~ =Oai (p)w
. 1
Denoting the corresponding Walrasian demand
By Roy's identity,
function by x(p.w}. 1
V
) V v(p,w) wv p,w p
x(p,w) = -
(
J
=
L
ril
.
.
i
•
Li=liai{pJw
1-
1
I::t=ow~a.(p) 1
=
r:;=O ___ w_·--:. l . J-
...,....1
va.(p). 1
Hence, for any fixed p, the wealth expansion path is contained the linear
As for the interpretation, recall from Exercise 3.G.ll that, an indirect uti!ity function in the Gorman form exhibits linear wealth expansion curves. But the Gorman form is a polynomial of degree one on w and a linear wealth expansion curve is contained' in a· linear subspace of dimension two.
Hence
the above result implies that the indirect utility functions that are polyr:omials on w is a natural extension of the Gorman form.
3.G.14
Define
a.
b,
c. d. e, and
f so that
a
b
c
- 4
d
3
e
f
- 10
•
Since the substitution matrix is symmetric, we know b = 3, a = c, and e == d. Thus c
By Propositions 2.F.3, p·S(p, w) = 0.
= - 4 and a = c = - 4.
= 0.
For the
3-45
Hence e
s:::
= 0. 3
p f
2 and d = e = 2: Thus f = - 7/6.
Finally, for the third column, we have p 3 + p 2 + 1 2 Hence we have - 10
- 4
3
-
4
- 4
2
3
2
•
- 7/6
The matrix has all the properties of a substitution matrix, which are •
synuuetry, negative semidefiniteness, S[p, w)p
= 0,
•
and p • S(p, w)
= 0.
(For
•
negative semidefiniteness, apply the determinant test of Exercise 2.F.l0 and Theorem M.D.4Uii).}
•
2
2
•
•
To verify Roy's identit:'. 8v(p ,p ,w)/op = (w/p 1 2 1 1 A
•
+
-112 2 4w/p ) (- w/p ), 2 1
ov(pl,p2,w)/8w = (w/pl + 4w/p2)
3.G.l6
,.,.....e ... ,_
-1/2
(1/pl + 4/p2).
(a) I! is easv to check that •
Since e(p,u) is nondecreasing in p, this must be nonnegative for all (p,u). But, if ak < 0 and llpll is sufficiently small, then this becomes negative. Also, if
~k
< 0 and llpll is sufficiently big, then this becomes negative. 3-46
Tnerefore (I)
ak
~
0 and ak
0 for all k.
:!::
It is a little bit manipulation to show that
•
•
Since e(p,u) is homogeneous of degree one with respect to p, they must be equal for every (p,u) and A > 0.
Take, for example, p = (1, ..• ,1) and u = 1.
Then
1]3,_ log e(p,u) = CL(llllogi\ + A. log A.e(p, u) = logi\
1.
+
They must be equal for every i\ > 0.
(2)
Lf'Xt
,
Therefore
= 1,
Lf3t
=
o.
T:"'US
Hen~e
(3)
the expenditure function now takes the simplified form:
.
at
e(p,u) =·(expu)(Uipt
);
Lt_«t
= I,
o..t
~
0.
This is ir:.c:-easing with respect to u and concave in p.
(b) By equatior. (3.£.1), w =
Hence
(4)
(c) By differentiating e(p,u) with respect to p, we obtain
Since x(p,w) = h(p,v(p,w)} and e(p,v(p,w)) = w, (5)
x(p,w) = w(«/Pl·····~/pL).
Use equations (3), (4}, and (5) and follows the same method as in the answer
3-47
to Exercise 3.G.2 to verify Roy's identity and the Slutskf equation . •
hand side of the indirect utility function should be deleted.
That is, it
shoulo be v(p,w) = Also, in (b), the minus sign in front of the first term d the right-hand side of the expenditure function should be deleted.
That is, it should be
e(p,u) = p uexp(bp/p ) - (1/bHap + ap{b + p cl. 2 2 1 2 Finally. in (c), the minus sign in front of the first tel m af the right-hand •
side of the Hicksian function should be deleted.
That is. it should be
(a) Use
anc apply Roy's formula. •
•
•
(b) According to (3.E.l}, we· can obtain the expenditure fmctio:1 by solving •
(c)
A~ly
Pr-oposition 3.G.l to obtain the given Hicksian dm:land function for
the fit st good. •
3.G.l8 We p::--cve the assertion by contradiction.
Suppo2 that the!'e exist l e
{!, ... ,L} and k e {1, ... ,L} such that there is no chain of Slbstitutes
connecting t and k.
Define J = (f} u {j e {1, ... ,U: there is a chain of
substitutes connecting l and j}.
Since t e J and k IC J, mth J and its
•
•
3-48
Moreover, for any j e J ar:c any j'
complemst {l, ... ,U\J are nonempty. 8h j( p, u )lip
r
<
o.
because, otherwise, j' e J.
Let u: IRL -+ IR be the underlying utility function. +
in Exercise 3.G.S, define
u:
IR
2 +
h: ol+."
Follo·.ving the hint, as
-+ IR by
-
Let
J
E
L
u{yl'y J = Max{u(x): x E IR+, L· Jp.x.:; Yl' L·dJo.x. ~ Yz}. 2 JE J J J"" • J J 2 2 X ~CIR ) ~ IR be the Hicksian demand function derived from +
+
e IR
2 +
•
is the solution to
-u(yl'y
s. t.
lie.).
2
l
i!:
u. if j e _J and
Define We shall prove that
J.
•
-u(t.
Then
p .x~.r '.-#JP .x~) ~ u(x•} = u.
JE1 J J
Jo::
J J
constraint of the cost minimization problem is satisfied.
Hence the
Suooose that -. •
t.:1ere
J J
JE
+
~::ost
Thus u(x)!:. u and hence .. by-the pCa ,a l· x". 1 2
Jfl!
J J
minimization of x•, p(etl'a l·x 2
2:.
Tnls is equivalent to saying that alyl + a:2y2
= C[. Jp .x~. JE J J
2:.
a:lQ:jeJP /jl + a2([je:JP jxjl-
L·Jfl!1p J.x~). J
By tiis equa!ity and the chain rule (M.A.l),
8h 1(')-a 2 , ul/oa 2
= '[ jeJP }Ekfl!iBh}pCa1,a2 ),u)/8pk)pk)
= LjeJLk!CJP lk(Bh }PCIX1,a2 l,u)/8pk l. We naw derive a contradiction from this equality evaluated a' Co: ,a l = 1 2 Bh }p,u)/8pk < 0 for every j e J a:~d every k
. . On the or.e hand, since
( .;.; ' 1 )
J. we mus1 have
oh Ci,l,ul/8a 1
2
< 0.
On the other hand, note that
3-49
-h( · l
fl!
is the
Hicksian demand function of
8h
-u( ·)
for two (composite) goods.
Since
Ca ,et ,u)/8et ~ 0 by the negative semidefiniteness, and 1 1 2 1
we must have
ah U,l,u)/8et 1
0.
2::
2
We have thus obtained a contradiction.
By Proposition 3.H.l, e(p,u) = Min{p·x: x e Vu}.
3.H.l
Thus, to complete the
•
proof, it is sufficient to show that V = {x: u(x) u and only if Sup {t: x e Vt} Clearly, if
X
2::
u}.
Assume that Sup {t: x e Vt}
> u,
2::
X
e Vt}
ii!::
u.
Define u• = Sup {t: x e Vt}.
u.
such that x e Vt.
u
e(p, u ) for all p. n
e(p,u) for all p.
3.H.2
Let n ..
If u• = u, then,. for
e (u - 1/n, u) such that x e V
u
then u
a~,
n
..
11
If u•
Since e( ·) is
increasing in utility levels, Vu .::> Vt and hence x e V . n
vu if
u.
2::
e Vt then Sup {t:
every n e IN, there exists u
That is, x e
, that is, p·x =:::
n
and, by continuity of e(p,u}, p • x
2::
Thus x e V . u
We show the contrapositive of the assertion.
If a preference is not
convex, then there exists at least one nooconvex upper contour set. be its corres;:>ollliing utility level.
Let u e
We can choose a price vector p so that
h(p,u) consists of more than one elements, as the following figure shows. According to Proposition 3.F.l, e( ·) is not differential at (p,u}.
•
3-50
[R
0
XI
Figure 3.H.2
3.H.3
By (3.E.l), for each p, take the inverse of e(p,u) with respect to .u.
3.H.4
The following method is analogous to that of "Recovering the
Expenditure Function from Demand" for L = 2. 0 0 b' . d . . . P lCK an a: ltrar-y consumpuon vector x an ass1gn a utilitv• value u to X
0
.
0
x .
. h . We will now recover the ,indifference -curve {x: u(x) = u 0} gomg t• :rougn
-
•
Assuming s-.:rong mono tonicity, this is equivalent to finding a function for
eve~y
x1 > 0.
•
Differ-entiate
8u(x
1
.~Cx
1
))/ax
1
+
(8u(x
1
,~£x
1
)}/Bx
2
l~'(x
1
l
= 0.
Hence
= -
eu(xl.~( xl) )/ox 1 Since
-
Bu(x .~£x ) }/8x 2 1 1 gl(xl.~(xl)) _
_
, we have
gz(xl.~(xl}}
3-51
•
gl(xl.~(xl})
-
~'(x l
1
= -
gzcxl.~(xl>l
•
or, by replacing
-x
by x , ·we obtain 1 1 ~·ex
gl(xl.~(xl))
) = 1
-
g2(xl.~(xl))
.
By solving this differential equation, we obtain the indifference curve going 0
through x .
By (3.E.l), we can recover the expenditure function by simply inverting
3.H.S
the indirect utility function. To recover the direct utility function, define u: IRL +
~
IR by u(x)
=
Min{v(p, w): p · x :s w}.
We shall prove that u(x) is the direct utility function
that generates v(p, w).
So let x•(p, w) be the demand function and v•(p, w) be It is sufficient to show
the indirect utility function generated by u(x). that v•(p, w) = v(p, w) for all p Let p » v(p, w).
0 and
w
:2::
»
:2::
It thus remains to show that v•(p, w)
Then x e
IR~ ~
0.
0, then p·x•(p,w) = w and hence v•(p,w) = u(x•(p,w)) ::
x = -
V
1 (
wv p,w
:2::
v(p, w).
Define
} V v(p, w).
p
the
by the monotonici ty.
homogeneity, p· x = w. let p'
0 and w
» 0 and
w~
It • is thus sufficient to show that u(x1 ::::: v(p,w). •
:: 0 satisfy p' ·x = w'.
Then (p' - p) · x = w' - w, or, by
the· definition of x. V v(p,w)·{p' - p} + V v(p,w)(w' - w) = 0. p
w
quasiconvexity of v(p,w}, v(p',w') :: v(p,w).
3.H.6
So
Hence, by the
Thus u(x) :: v(p,w).
Let's first prove that the symmetry condition on S(p,w) is satisfied.
By equation (3.H.2}, c3e(p,u)/8pl =
o:.~(p,u)/pt
for every l.
By
differ-entiating both sides with respect to pk' we obtain
On the ether- hand, by differentiating b~th sides of Be(p, u}/c3pk = '\e(p, u)/pk 3-52
•
with respect to pl' we obtain
Hence the symmetry condition is satisfied. We can thus apply the iterative method explained in the small-type discussion at the end of Section 3.H to derive the expenditure function. First, we shall prove by induction that, for every 1., there exists a function •
f t(pl+l' ... ,pL' u) such that lne(p,u) = Suppose first that l = 1.
Since (8e(p,u)/8p )/e(p,u) = «{p . 1 1
Hence, by
integrating both sides with respect to p , we obtain 1
•
Suppose next that l > 1 and the
Thus the equality is verified for I. = 1. equality holds for t - 1.
with
respec~
By differentiating both sides of
tc pl. we obtain ae(p,u)/Bpl = Bfl.-l(Pt····•PL'u)/8pt.
Since ce(p,u)/Bpt =
alpt1 thjs is _equivalent to
Hence, by integrating both sides with respect to Pt.• we know that there exists fl(pi+l' ... ,pL,ul such that o:llnpl = ft-l(pl'" .. ,pL,u) - ft.'Pt+!' ... ,pL,ul By plugging this into
we obtain lne(p,ul = Lk~t'1clnpk + ft(pl+I'"'"'pL,u). If t = L, ther1 this equality becomes lne(p,u) = Lk:sLa:klnpk + fL(ul. •
3-53
Or,
equivalently,· e(p,u) = In what follows, for every increasing function /L(u), we shall find the utility function that generates the expenditure function e(p, u) = To sta.-t, consider the utility function u•(xl = which appeared in Example 3.E.1 for the case of L = 2.
Denote its expenditure
function by e•(p,u), then
(We considered a similar expenditure function in Exercise 3. G.16.
Note that,
these similarities incidentally show that, for every increasing function
proper-ties of expenditure functions in Proposition 3.E.2, because it
a.t
corresponds one of the monotone transfonuations of u•(x) = ntxt
.)
Let g(u•)
be an monotone transformation and denote by e (p,u) the expenditure function
g
of the utility function (gou•)(x).
Then
-a.,_
-1
e g{p,u) :: e•(p,g (u)) = mflt
)(IIlpl
cxl
-1 )g
(ul.
a. =
By comparing u(x) if
This equality is equivalent to g
-1
. u • -- g-l(ul L ettmg
(u)
and solving this with respect to u•, we obtain
Thus -1
-1
u(x) = g(u•(x)l = /L (lnu•(x) - Lfltlna.l) = fL ([lcxl(lnxl- lna:l)).
a. to
Of = u •
3-54
3.H.7
(a) Let
-p = (1, ... ,1).
Since
x(p,L)
=
(1, ... ,1) and u(l, ... ,l)
according to Propositions 3.E.l and 3.G.l, D
p
e(p,l)
=
h(p,ll
= 1,
= (1, ... ,1).
Hence e(p,l) = L.
2
On the other hand, S(q, w) = D e(q,w) = 0 for every q p
2.F.l7(d).
»
0 by Exercise
•
Hence e(p,u) - e(q,u) = D e(q,u)(p - ql. p
for every q
»
0 and p
»
0.
Now take q =
-p,
then e(p,l) - L =
-p· (p
-
-p).
Thus e(p,ll = •
(b) The uppe:- contour set is equal to
•
•
L {x e IR : p·x :=: +
X ~ (1, ... ,1)}.
By the same method as deriving equation (3.1.3), we obtain
3.1.1
.01 01 11 £1i(p ,p ,w) = e[p ,u) - e(p ,u) 01
= e(p ,u l -
100
01
p1 0 0 0" 1 . . = J 0 h(pl'p2,p3, ... ,pL,u ) dpl +
p.!
•
100
I
01
11
e(p ,u l
p2 1 0 0 1 0 h(pl,p2,p3, ... ,pL,u l dp2; p2
.01 00 10 Cv(p ,p ,w) = e!p ,u ) - e(p •. u )
= e(p =
J
00
1 pl 0
Pr
100 00 100 00 10 ,u ) - e(p ,p ,p , ... ,pL,u ) + e(pl'p ,p , ... ,pL,u } - e{p ,u l 1 2 3 2 3 l 0 0 0 0 p2 1 0 0 0 h(pl'p2,p3, ... ,pL,u ) dpl + I 0 h(pl'p2,p3, ... ,pL,u ) dp2. P2
If there is no wealth effect for either good, then, by the first reiatior. of
(3.E.4l,
3-55
3.1.2
Denote the deadweight loss given in equation (3.1.5) by DW ( t) and that 1
in equation (3.1.6) by DW (t}. Then 0 1 0 1 0 DWi(t) = h[p + t, p_ , u ) - (h(p + t, p_l' u } + 1 1 1 0 I = - t·ahCp + t, p_ • u >!apr 1 1
+ t, •
•
> 0 for every p > 0,
Thus
1
>
0 for ever-y t
0.
It can be similarly shown that DW (0) = 0 and, if
0
0 Sh(pl'p _ ,u )/ap > 0 for every pi > 0, then DW £t) > 0 for ever-y t > 0. 1
0
1
A possible interpretation of this result is that the first-order •
derivatives of the deadweight loss at t = 0 may be a bit misleading approximation.
In fact, their being zero means that, approximately, there is
no deadweight loss.
Henc~
3.1.3 p
0
:;c
On the other hand, since those derivatives are positive
the deadweight losses are in fact positive.
Wr-ite u 1 p .
0
0
= v(p , w) and u
1
1
= v(p ,w), then u
0
< u
1
because p
0
1
c: p and
Thus
for every Pt. > 0.
Since good
t
is inferior.
By the first relation of (3.£.4). this is equivalent to 0
0
0
1
hiPt•P -l'u ) > ht'Pt•P -l'u ). 0 1 0 1 0 1 Hence, by (3.1.3), (3.1.4), and Pt < pl. we have CV(p ,p ,w) > EV(p ,p ,w).
3.1.4
We
shai~
give two examples, both of which have two commodities. •
3-56
The
first one is simpler, while ·the second one is more illustrative. In the first example, we consider a preference with "L-shaped"
indifference curves such that the vectors of indifference curves.
= 1.
Let u(l,l)
(1,1), (4,2), and (5,3) are kinks
Note that if one of the two prices
is equal to zero, then the demand is not a singleton.
We thus need to
•
consider a dema."ld correspondence x( p, w}. ·But this does not essentially change· •
•
our argument because we are working on expenditure functions, which is single-valued by its definition. Let p
0
1
= (1,1), p = U/2,0), p
1 (1,1), x(p ,w)
i!)
2 (4,2), x(p ,w)
2
112 and e(p ,ll = 2/3.
2
= (0,2/3), and w = 2.
2 (5,3), and v(p ,w)
>
0
Then x[p ,w)
1 v(p ,w).
i!)
1
But e(p ,1) =
•
Thus
•
0
1
0
2
CV(p ,p ,w) = 2 - 1/2 = 3/2, CV(p ,p ,w) = 2 - 2/3 = 4/3.
0
1
0
2
Hence CV(p ,p , w) > CV[p ,p , w). It is wo::-thwhi!e to remark that, although the given preference is neither
smooth, strongly monotone, nor strictly convex, it can be approximated by such one. In the se:ond example, we consider a utility function c{x' ·.vnich is •
quasilinear with respect to the first conunodity. corresponding indirect utility function.
Let v(p, w) b!! the
Starting from p
0
= (l,l) and w > 0,
. 1 1 2 2 we consider two other pnce vectors p = (pl"l) and p = (1,p ) such that 0 < 2 1 2 . 0 0 1 1 2 pl < 1. 0 < P2 < 1. and v(p ,w) = v(p ,w). Wr1te u = v(p ,w) and u = 1
2
v{p , w) = v(p ,w).
0
1
0
2
Then EV(p ,p ,w) = EV(p ,p ,w). •
0
1
0
2
We shall now show .. hat CV(p ,p , w) < CV(p ,p , w).
By p
1
1 < 1, CV(p ,p ,w)
1 0 2 0 2 Also, by the quasilinearity, CV(p ,p ,w} = EV(p ,p ,w)
(Exercise 3.!.5).
0
1
0
2
Hence CV(p ,p ,w) < CV(p ,p ,w).
3-57
0
0
1 0 2 It is worthwhile to remark that, although £V(p ,p ,w} = £rtp ,p ,w}, we 0
1
0 2 can obtain the strict reverse inequality EV(p ,p ,w) > £V(p ,p ,w). while
. 01 02 . 2 preservmg CV(p ,p ,w) < CV(p ,p ,wl, by decreasmg p only slightly.
2
3.1.5
According to Exercise 3.E. 7, we can write the expenditure function
-eCp
e(p,u) =
2
0
, ... ,pL) + u for p
1
1
= L
Hence
0
1 0 0 EV(p ,p ,w) = e(p ,u ) - e(p ,u )
- 0 0 1 - 0 0 0 = (e(p ..... pL) + u l - (eCp , ...• pL) + u ) 2 2 = u 0
1
0
- u .
1
1 1 1 0 CV(p ,p ,w) = e(p ,u } - e(p ,u )
- 1 1 1 = (e(p2, ... ,pL) + u ) -
= u
3.1.6
Let
0
1.!.
1
=
I
0
- u .
0
v.(p ,w.). 1
1
If
1 0 So define w: = e.(p ,u. ), then 1
3. I. 7
1
1
.CV. p ,p ,w.) - 0, then
11
[.w~ 11
1
.w. -
ll
1
.e.(p ,u. ). 1
0
1
0
s 'l.w. and v.(p ,w:) = u. = v.(p ,w.). ~1
1
1
1
1
1
(a) By appiying Walras' law and the homogeneity of degr-ee ze:"'o, we can
obtain the demand functions for all three good defined over the whole domain {(p, w)
e !R
.-.,
x IR: p
»
0}.
Thus we can obtain the whole 3 x 3 Slutskv matrix
as well from the demand function.
-
The 2 x 2 submatrix of the Slutsky matr-ix
that is obtained by deleting the last row and the last column is equal to b
e
c g
•
•
By the homogeneity and Walras' law, the 3 x 3 Slutsky r.1atrix
is symmetric if and only if this 2 x 2 matrix is symmetric.
Moreover, just as
in the oroof of Theorem M.D.4(iii), we can show that the 3 x 3 Siutsky matr-ix • 3
is negative semidefinite (on T , and hence on the whole IR ) if and cnlv- if the p
3-58
2 x 2 matrix is negative semidefinite.
2 that c = e, b =" 0, g s 0, and bg - c
Hence, utility maximization implies 0.
j!:
(b) First, we verify that the corresponding Hicksian demand functions for the first two commodities are independent of utility levels and, as functions of the pr-ices of the first two commodities alone, they are equal to the given Let p be any price vector and u, u' be any two
Walrasian demand functions. utility levels.
By (3.E.4), hip,u) = xip,e(p,u)) and hl(p,u') =
xip,e(p,u'}) for l
= 1,2.
Since the
xe
xi·}
do not depend on wealth,
Hence ht'p,u} = hip,u' ).
Thus the ht(p,u} do
not depend on utility level and they are the same as the xl(p, w). ~
If the prices change following the path (1,1)
(2,1) .. (2,2), then the
equivalent variation is 11
22
1
211 1 = J x(p,l,w)dp + 1
22
s1
2
2
2
x (2,p ,w)dp
2
= (a + (3/2)b + c) + (d + 2e + (3/2)g).
If the prices change following the path (1,1) .. (1,2)
~
(2,2), then the
equi,.·alent variation is 2 2 2" 2 J 1 h (l,p ,u)dp + 22 2 2
= (d
+ e +
2 1 1 1 J h (p ,2,u)dp 1 11 1
(3/2)g) + (a + (3/2)b + Zc).
These two equivalent variations are the same if and only if c = e.
(c) As we saw above,
+ c + (3/2)g,
EV
= (a
+ (3/2)b + c) + (d + Ze + (3/2)g)
3-59
= a + (3/2)b + 3c + d + (3/2)g.
Hence EV - (EV
+ EV ) = c. 2 1
The sum EV + EV does not contain the effect on equivalent variation due 2 1 •
to the shift of the graph of the demand function for the second commodity when p
1
•
goes up to 2 (or equivalently, the shift of the graph of the demand
function for the first commodity when p
2
goes up to 2).
Graphically,
•
letting c = e > 0, EV contains the shaded area below but EV
1
+ EV
2
does not:
2 '.. --... ---· ·--•
1
····~--~--··--···------
0
XI
d+2e
d+e
3.1.7(c)
(d) Since x,(2,l.w) = a
•
to this.
Thus DW, •
= (a
2b + c, the tax revenue from the first good is equal
+
+ (3/2}b + c) -
(a + 2b + c)
=-
•
b/2 .
Since x (1,2, w} = d + e + 2g, the tax revenue from the second good is 2 equal to this.
Thus
nw 2
= (d
+
e + (312}g) - (d + e + 2g) = - g/2.
Since x C2,2,w) = a + 2b + 2c and x (2,2,w) = d + 2e + 2g, the tax 2 1 revenue frcr.l both commodities is (a
+
2b + 2c) + (d + 2e + 2g) = a + 2b + 4c + d + 2g.
Thus DW = (a + (3/2}b + 3c + d + (3/2}g) - (a + 2b + 4c + d + 2gl.
3-60
•
= - b/2 - c - g/2.
(e) Our problem is
•
s.t . •
Here, DW(t ,t } = £V(tl't ) 1 2 2
TR(~ ,t )
1 2
= e(I + t , 1 + t , u} - e(l,l,u) 1 2
l.=lht'l + t , 1 + 1
t
2
, u}tt"
Set up the Lagrangean by L(\,t ,i\) = DW(t ,t > + i\(R - TR£t ,t )). 2 1 2 1 2
Then the
first-orde:- condition with respect to tl is 8DW(t ,t l/8tl- i\STR£t .t l18tl 1 2 1 2 •
= 0.
But,
- ., /• b) ce1 ~ ~
+
+
~
1
•
ana
•
1/"'"" "'TR(· • 0 0 "I' ~2' '"l -
Her:.ce the fi:-s: -order condition is written as
•
+ i\hil + t , 1 + t , ul = 0
1
for both f.
-
i:O.
= 1,2.
From this and R
=
2
l.=lhil + t , 1 + t • u)tl, we obtain 1
-
=
3-61
2
3.1.8 (b) As in Exercise 3.1.7(a), the symmetry implies c
1
+ d
p = b + d p for all p > 0 and p > 2 2 1 1 1 2 2
Thus c = b , d = d = 0. 1 2 2 1
o.
Then the negative semidefiniteness implies that =:;
0.
•
(c) Since the Walrasian demand functions and Hicksian demand functions are the •
same as we saw m Exercise 3. I. 7(b), we can define p' p'
cv P1
Pz
or, equivalently,
cv
•
{d) By the same calculation as in Exercise 3.1. 7(c), we obtain
EV In this case. EV (b)
~
EV
1
= 1/2, EV
+ EV . 1 2
= 1/2, EV3 2
= 3/2.
In the general case in which the conditions in
hold,
EV EV
= 2 3
az
= a
1
+ b2. + (3/2)c2, + (3/2)b
1
+ c
1
+ a
2
+ 2b
Her.ce EV + EV = EV if and only if b = c = 0. 3 1 2 2 1
+(3/2lc . 2 2
This condition is
equivalent tc saying that any change in the price of one good does not have any (cross) effect on the demand for the other.
the other components are zero. after-re~ate
For each t, define p(t) = p + tet"
Then the
income w(t) with tax t satisfies w(t) = w + xt(p(t),w(t)lt.
3-62
Hence p·x(p(t),w(t))
= (p(t')
- tet)·x(p(t),w(t))
= w(t)
Therefore x(p(t),w(t)) is at most as good as x(p,w).
- xl(p(t),w(t))t = w. In order to prove that
x(p(t),w(tll is strictly less preferred to x(p,w), it is sufficient to prove that these two are different, because the demand function is assumed to be single-valued. •
Now SU;:Jpose that there exists at> 0 such that x(p(t),w(t)) = x(p,w) . •
Let u = v(p,w). hip,u).
Tnen we have h(p(t),u)
= h(p,u).
Since the Hicksian demand function s
~
In particular, ht(p(t),u) ht(p(s),u) is nonincreasing,
this equality implies that hip(s),u} = hl(p,u) for every s e [0, t]. d[hl(p(s),u)J/ds = 8hl(p(s),u)/8pl. = sU(p, wl "" 0.
h(p(t),w(tll
3.1.10
~
=
But
Evaluating at s = 0, we have Sht(p,u)/8pt
This violates the assumption that sU(p, wl < 0.
Hence
h(p,w) for every t > 0.
We consider an example of a consumer who face the choices over two
goods and whose preference t and demand function x(pl'p ,wl satisfy the 2 following condition: Fo:-- every p
e [1,2), x(p ,1,2) 1 1 •
for e·1rery p
e fl,2}, x(l;p ,2} 2 2
= {(1 - c)/p1• 1 + c); = (1 - c, (1 + cVr:,::i;
• •
((1 - cl/2, 1 + c) >- (1 - c, (1 + c}/2} .
By using the figure below, you can convince yourself, perhaps with some
application of the weak axiom, that there actually exists such a preference.
3-63
2
J+E
I--
1 •
•• •• •
1-
0
Define p
0
1
E
1
= (2,1) and p = 0,2).
2
XI
Then
0
x£p .21 = ((1 - c)/2, 1 + c), 1
x£p ,21 = (1 - c, (1 + c}/2).
0
1 Thus x(p ,2) >- x(p ,2).
price-change path p 0
0
However, the area variation measure following the
-+ (1,1) -+ p
1
is
1
AV(p ,p ,2) =
•
•
. = - f(l - clln
p
2 1
1 p1=1
+ [(1 + din p
2
2 J • pl=~
= 2£ln2 > 0.
0 . . nk 1 Hence the area var1at1on measure ra s p over p .
3.1.11
If (p
1
0
- p )·x
1
0
1
> 0, then w > p · x .
The local non-satiation implies
•
that x
0
1 is oref erred to x . -
1 Hence the consumer must be worse off at (-c , w}.
As for the interpretation in term of the first-order approximation, since e(p,u) is concave in p, , 0
elp ,u
1)
1 1 1 1 0 ::s e(p ,u) + Ve(p ,u )·(p •
. 3-64
1 1 0 0 1 1 1 1 Since Ve(p ,u ) · Cp - p ) < 0, e(p ,u > < e(p ,u ) = w.
0 0 Thus u = v(p ,wl >
1
u.
This test is depicted in the picture below:
p
X
0
XI
Figure 3.1.11
3.1.12
Let u
0
0
0
= v(p ,w ) and u
0011 EV(p ,w ;p ,w )
0
0
1 "1
1
1
1
= v(p ,w ). 01
Then we define 00
0
01
= e(p ,u ) - e(p ,u ) = e(p ,u ) - w • .
1 1 1 0 1 1 0 CV(p ,w ;p ,w ) = .e(p ,u ) - e(p ,u ) = w - e(p ,u }.
1
The "partial information" test can be extended as follows: If p · x I
1
the consumer is better off at (p • w ).
0
1
<w •
This can be proved in three ways.
The first one is the same revealed-preference argument as in the proof of Proposition 3.I.l. •
The second way is to use the indirect utility function.
Since v(p, w} is
. . quas1convex, li (p
1
0
- p ). v
p
00 1 0 00 v(p • w } + (w - w )8v(p •w )/ aw > 0,
1 1 0 0 then we can conclude that v(p ,w ) > v(p ,w ).
3-65
But, by Roy's identity, this
sufficient condition is equal to - (p
1
00 1 0 00 - p )·(av(p ,w )/8w)x(p ,w) + (w - w Hov(p ,w )/8w) 0-00
0
0
0
0
= (Bv(p .w )/8w)(- p = (8v(p ,w )/Bw)(w
1
1
·x
0 1
+ w
0
- p ·x )
0
+ w
1
0
- w )
> 0.
. 1 o 1 1 1 o o· Hence, 1f p ·x < w , then v(p ,w ) > v(p ,w ).
The third way is to use the expenditure function.
1 1 0 0 v(p , w ) > v(p , w ) if
•
. . 111 I 00 ana only 1f e(p ,v(p ,w )) > e(p ,v(p ,w )). 100 <10 e(p , v(p , w )) - p • x . 1
1
v(p ,w )
3.J.l
0
10
Hence, if p • x
0
111
1
But e(p ,v(p ,w )) = w and
1 < w , then we can conclude that
.
> v(p ,w ).
[First printing errata: The difficulty level should probably be B.]
It
follows immediately from the definition that if x(p, w) satisfies the strong axiom, then it satisfied the weak axiom.
Conversely, if x( p, w ) satisfies
the weak axiom (in addition to the homogeneity of degree zer-o and Wa1ras' law), ther. the Slutsky matrix is negative semidefinite and, by Exercise 2. F .11, symmetric.
Hence x(p, w) is integrable, implying that there exists a
preference relation that generates x( p, w ).
Thus x{p, w) satisfies the strong
axiom as wel:.
3.AA.l
If (p,w)
= (1,1,1),
then x(p,w)
= (0,1).
The locally cheaper
condition is not satisfied since B p,w
1}
y such that p · y < w, as depicted in the following figure.
3-66
and there is no
'
'
0
XI
Figure 3.AA.l
To check that the demand function is not continuous at (1,1,1). consider the n
n sequence (p ,w ') = (1 - 1/n, 1, 1 - 1/n).
n
n
n
Then (p ,w ) -. (1,1,1) and x(p· ,w )
= (1,0), but x(l,l,l) = (0,1).
This discontinuous change in demands arises
because the budget set B
consists of (1,0) for every n, but B(l,l),l = n
n p ,w
+ x a~
suddenly
3.AA.2
2
= 1}, so that the commodity bundle (0,1) becomes available
p = {l,l}.
!Firs: printing errata: The upper hemicontinuity of h(p,u) cannot be
guaranteed a: p
l!:
0, because the local boundedness condition in t.he definition
of uppe:- hemicontinuity need not be satisfied.
Hence the clause in the
bracket "eve::-: i: we replace minimum by infimum and allow p unde:-stood as concerning only with e(p,u).] .lS
' .:.UDD-. • •
h em.-~., . . . .Jl•. n-vUS. ,.-nr.~,
"i!:
e B
S'...i.C!":
that
-u
i!:
u for eve:-y (p, u l e B.
0} x IRl.
u for every (p,u) e B. For each l, define
3-67
i!:
0" should be
We shall first pr-ove that h(p, u)
Let 8 be a compact subset of the
(which is, in tu:-n, a subset of {p e IRL: p »
(p, u)
n
Let
XE
do:r.~ain
of h(p, u)
Ther. the:-e exists a h(p,u), then u(x)
i!:
-u
-Yt and
-Y = (yl, - ... ,yL) -
h(p, u), we ha·;e
-y
e IRL.
x.
=
e IR+: (p,u) e B}
We now show that, for every (p, u) e B and x e
+
l!::
Max{p·x/pl
u(x)
If fact, since
p·x
u,
l!::
Since p » 0 and
p · x.
i!:
•
•
Divide both side by pi, then we ootam
p · x/pl
-y l
xt and hence
i!:
i!:
xi..
We have therefore established the lo:a.l n
n
Next, let {(p ,u )}
boundedness condition of upper hemicontinuity.
be a
n
sequence of pairs of price vectors and utility levels, converging to (p,u). Let {xn}
n n n • IRL n be a sequence 1n , x e h(p ,u ) for every n, and x ... x.
n
+
Since u(xn)
sufficient to prove that x e h(p, u}. continuity, we obtain u(x) at (p,u}.
i!:
u.
u(y} > u, then u(y) > u
0
E
+
n
Hence p ·y l!::
p · x.
u.
i!:
p
i!:
If
n
n
·x for
S•Jppose then that
By the local nonsatiation, there exists a sequence {yn} n in IRL such +
n
-+
x.
Hence there exists a subseouence •
n n)} nsuch tatuy h ( n {( ? k(n) ,u k(n)),.tnof {'\p,u 1 k(n~
•
IRL and u(y l
for any sufficiently large n.
that u(y l > u(x) for every n and y
·x
un and u(x l -+ u(x) by the
Hence x satisfies the constraint of the EMP
n
o•
n
By taking the limit as n ~ co, we obtain p · y
u(y) = u.
k!n)
i!:
To show that it is cost-minimizing, let y
such n.
It is
B".,' taking the limit as n •
•
•
•
~ co,
l!::
u
k(n) •
Hence p
we obtain p · y
l!::
k(nl
p· x .
·y
n
l!::
Hen:e x is
•
cost-mmmuzmg. We now turn to the continuity of e(p,u). every p »
In fact. its continuity at
0 can be derived immediately the continuity of h(p,u), as the
latter is weH defined at every p »
0.
Thus the essential part of the
following proof is the case of nonnegative, but not strictly vectors.
posi~ive,
price
We sl':all establish the continuity with respect to p and that with
respect to u separately. Let u e IR be a utility level, p e IRL be a price vector, and {pr:} be a +
seque~ce
L . t of p;ice vectors in IR convergmg o p. +
3-68
We need to prove that
n
e(p ,u) -+ e(p,u).
As a preliminary result, let's first prove that if the
sequence {e(pn,u)} in IR+ converges, then it must do so to e(p,u). n
Let x e IR
the limit of {e(p ,u)}. every n.
L +
and u(x)
~
n
u.
Then p · x
Taking the limit as n -+ co, we obtain p·x
every x e IRL with u(x) +
~
•
subsequence {p
:5
k(n)
.
L
w, we use the concavity of e(p,u) in p e IR+.
k(n)
n
} of {p } such that (Pt
k(m)
)(
~
- Pt Pt
at most 2L sign patterns. - pl
Pt
~
Take a 0 for all
t e
That is, for each t e {l, ... ,L}, we •
Such a subsequence does actually exist because each p
k(n)
e(p ,ul for
~ u, we have e(p,u) ~ w. To prove the reverse
{1, ... ,L} and positive integers n and m.
zero).
n
Since this holds for
w.
•
inequality e(p,u)
l!:'
Let w be
n
- p has one of
Now, for each t e {l, ... ,U, let vl = 1 if •
k(n)
0 for every n; and vt = - 1 1f Pt
- Pt => 0 for every n.
Then,
•
k(n}
Pt So let
;!:
k(n)
= Pt
+
Jpt
- Ptlvt' l = vl and v lc = 0
0
for every k ;:: t, then k(n)
P
Now, define
. k(n)
S mce p
+
Ltzlv.
n
1 -
l=lzt, then p
.
= p
n l
k(n)
.
=
n ... p, zl -+ 0 for every
t e
{l, ...• U.
n Thus z 0
1.
Hence, for
. · 1arge n, z n > 0 an d p k(n) 1's a convex com b"matton · 1 every su ff 1c1ent.y oi- z n , 0 0
z7, ... ,
z~.
The:-efore, by the concavity, k(n) ) e (p ,u ~ z~e(p.u)
...... w. ..- ce e(pn,U'J ........ ...., .... , e(pk(nl,u) ...., 5.;••
Th e r1g · ht - h an d s1'de converges t o e(p . u) .
Therefore w s e(p,u). We have thus proved our preliminary fact that if the sequence in !R
+
converges. then it must do so to e(p,u).
3-69
Let's now prove by
n {e(p ·,u)}
0
contradiction that this implies that e(p ,u) -+ e(p,u).
So suppose not, then
there are a ~ > 0 and a subsequence {pk(n)} of {pn} such that Je(pk(n),u) - e(p,u)J ~ Cl for all n.
Since the subsequence {e(p
convergent subsequence. because I e(p
k[n)
k(n)
. , u)} 1s bounded, it has a further,
On the one hand, the limit can never be e(p,u),
,u) - e(p,u) I
~
o for
all• n.
On the other hand, our
preliminary result implies that the limit must be e(p,u). contradiction.
This is a
n
We must thus have e(p ,u) -+ e(p,u).
Let's now turn to the continuity of e(p,u) with respect to u.
Let p
iR
E
L +
be a price vector, u e IR be a utility level, and {un} be a sequence of utility •
levels in IR converging to u.
.
n
We need to prove that e[p,u ) -+ e(p,u).
before, it is sufficient to prove that if the sequence {e(p, un)} in IR converges, then it must. do so to e(p.u). L
Let c > 0, x e IR , u(x) +
~
By the local nonsatiation,
we can make u(x) > u while preserving p·x < e(p,u)
pos!~ive intege:- N such that u(x) > u i!::
n
e(p,'J ).
Take the limit as n-+
-
•
Ill,
.
+
Let w be the limit of {e(p,un)}.
u, and p·x < e(p,u) + c.
0
Just as
c.
+
for every n > N.
Then there exists a Thus, for such n, p · x
Thus e(p,u) + c > w .
then p·x :: w.
Since this holds for every c > 0, we must have e{p,u) reverse inequality e(p,u) :s w, we can assume that u
n
~
w.
To show the
:s u for every n.
reason is as follows: If there is a subsequence such that u
0
(The
:s u for every n
in the subsequence, then we can apply this case to subsequen=e.
If there is
no such subsequence, then there is a subsequence such that un
u for e·;e::-v n •
in the subsequence.
. n Hence e(p,u} :: e(p,u } for such n. -
L IR and u(x) i!!: u. +
i!!:
Taking the limit, we
obtain e(p,ul :: w.)
Now let x e
then B is compa·:::t.
This and un :s u implies that the truncated EMP Min p·x
s.t u(x) :: u
3-70
Define B = {x e
n
-X
->
' X),
has a solution, denoted by xn e B.
Then xn e h(p,un), that is, xn a solution
to the original, untruncated EMP, because p is a conver-gent subsequence {xk(n)} of {x
0
e[p,u
) and p·x
ldn)
.. p·x.
IR + • Since B is compact, there
E
Denote its limit bv x. -
}.
u( ·) is continuous, u[x) a: u and hence p·x k[n)
L
Moreover, p·xk(n) =
e[p,u).
i2::
Since
Thus w = p·x and hence w
2::
e[p,u).
Suppose that u(x) is strictly quasiconcave, twice continuously differentiable and that Vu(x)
*
0 for all x.
Then we know that h(p,u) is a
function and the Lagrange multiplier i\ of the EMP must be positive.
The
first-order condition for the EMP can be considered as a system of L + 1 eouations and L •
+ 1
unknowns: p - i\Vu(x) = 0 u(x) - u
0
.o::: •
By the implicit function theorem [Theorem M.E.ll, the solution h(p,u) as a function of the parameters (p,u) of the system is differentiable if the Jacobian of this system has a nonzero determinant 2 - D u(x) Vu(x)
- p T
~
0
0
•
But, then, p = i\.'i7u(x} and hence
at (p,x) satisfying the above· two eqllations. this condition is equivalent to
2
- D u(x)
'i7u(x)
- Vu(x) :;1!:
T
0.
0
that is,
o
2
uCx)
Vu(x)T
Vu(x)
:;1!:
0.
0 2
By Theorem M.D.3Ci), this inequality holds if o u(x} is negative definite on I
{y e ~-= V'u(x) · y = 0}.
Tnis sufficient condition is a stronger differential
version of quasic-oncavity, as the latter is equivalent to the condition that
3-71
2 D uCx) is negative semidefinite on {y e IRL: Vu(x) · y = 0}.
3-72
CHAPTER 4
By Roy's identity (Proposition 3.G.4) and vi(p,wi) = ai(p) + b(p)wi"
4.B.l
x (p,w) = i
1
I
V
w.
( ) 'il v.(p,w.) v.p,w. p1 1 1
l
Thus IJ
= -
1
I
x.(p w.) = - -:--:-(-p~) 'il p b (p) for all i. 'IN, 1 1 0
Since the right-hand side is
1
1
identical fo:r every i, the set of consumers exhibit parallel, straight expansion paths. As for the second part, by (3.E.I), e.(p,u.) = (u. - a.(p))/b(p). 1
1
1
1
Hence, by letting c(p) = I/b(p) and d.(p) = - a.(p)/b(p), we obtain e.(n, u. l = 1 • 1 1
1
c(p)u. + d.(p). 1
4.B.2
1
(a) Let p e IR
L
be a price vector and w
Consider two consumers, i and j.
!::
Consider two wealth distributions
Cw , ... ,w } a."':d (wi, ... ,wjl such that wi = wj = w 1 1 and wk = 0 for any k for eve:-y k.
:;c
j.
0 be an aggregate wealth.
2:
..... 0, wk = 0 f or any k ~ 1,
Since the preferences are homothetic, x(p,O,s. ) = 0 K
Thus the aggregate demand with (wl' ... ,w ) is x(p,w,si) and the 1
agg:-egate demand with (wi, ... ,wjl is x(p,w,sj).
Since aggregate demand
depends only on prices and aggregate wealth, we have x(p,w,s.) = x(p,w,s .). I J Since p and w were arbitrarily chosen. this means that i and j have the same demand fun::tion.
Hence they have the same preference.
Since i and j were
arbit::-a!"'ilv c:.:csen, we conclude that all consumers have the same preference.
-
(b) By analogy to the Gorman form, consider the following form of indirect utilitv funct!ons:
-
4-1
v.(p-,w.,s.} = a.(p) + b(p)w. + c(p)s .. 1
1
1
1
1
Note that b(p) and c(p) do not depend on i.
1
By this and Roy's ider.tity
(Proposition 3.G.4), x(p,w.,s.) 1
1
=-
1
V
w.
1
-( ) V v.(p,w.) v.p,w. pl 1 1
1
w.
1 b(p) Vai(p} -
--
s. I
1
'"";"b-;(-p~) 'i7 c (p )
b( pl Vb(p} -
Thus
[.w. [.x(p,w.,s.) 1
I
1
t::
1
b( p) Vb(p} -
-
1
L·1s.1
b(p) Vc(p).
Thus the aggregate demand depends only on r.w. and r.s. (and pl. 1 1
1 1
By the definition of a directional partial derivative,
4.C.I
Dpx(p,w)dp =
limE;.~(I/c)(x[p +
cdp, w) - x(p,w)).
Hence cp·Dpx(p,w}dp = dp·(limc.-+O(l/E;)(x(p + cdp, w)- x(p,w)))
=
lim£~(1/c)dp·(x(p +
cdp, w) - x{p,w))
But the ULD property implies that dp· (x(p + cdp, w} - x(p, v:)) 0.
Hence, by taking the limit c .-+ 0, we obtain dp · D x(p, w)dp p
i!::
:S
0 for all c >
0.
Tnus
D x(p,wl is negative semidefinite. p We shall p:-ove the converse by contradiction.
Suppose that the Jacobian L
D x(p, w) is negative definite for all (p, w) and that there exist p e IR , p' e p
iRL, and w. e !R suc:h that x.(p,w.) -:t: x.(p',w.) and 1
1
1
1
1
(p' - p) · (x.(p' ,w.) - x.(p,w.)} 1
1
1
i!::
1
0.
Let ~ > 1 be sufficiently close to 1 that for every A e [O.~J. demand is well defined at (1 - A)p + Ap'. price vectors, [O.~].)
-A is
(If demand is well defined at strictly positive
determined so that (1 - i\.)p + ll.p'
Define _o(i\.) == (1 - A)p + i\.p' and
4-2
»
0 for every A e
w .(h) = (p' - p)·(x.(p(~),w.)- x.(p,w.)). 1
1
1
1
1
~
Then the function w( ·} is differentiable, w.(O) = 0, w.(l) 1
= [p'
w~(~) 1
- p)·D
1
x.(p(~).w.)(p' p 1 1
0, and
- p).
We consider two cases: Case 1: w.(ll.) ::s: 0 for every 1
~
e
[O,ll.].
Then w.(l) = 0 and it is a maximum. 1
Thus w:(l) = 0, that is, 1
(p' - p) · D x(p', w)(p' - p)
•
p
= 0.
This is a contradiction to the negative definiteness. Case 2: w.(h) > 0 for some ll. e 1
(O,ll.].
Then, by the mean-value theorem, there exists 11.• e w.(O) = w:(ll.•)(h - O). 1
!
(0,~)
such that w.(h) 1
By w.(O) = 0 and w.(ll.) > 0, w:(ll.•) > 0. 1
1
1
(p' - p) · D x(p(h •), w)(p' - p) p
This is a contradiction to the negative definiteness.
That is,
> 0. Our proof is thus
completed.
4.C.2
It D x.(p,a..w) is negative definite on the whole IR p 1 1
L
for every i, then
the sum [.D x.(p,cr..w) is negative definite on the whole IRL. 1 p 1 1 ~.D x.(p,o:.w), L.1 p l 1
Since D x(p, w) = p I
D p x(o,w) is negative definite on the whole R-. i:-:.:). lyin.e: that . . ~
x(p,wl satisfies the ULD property.
To establish the WA, one way is simply to
notice that the ULD property implies the WA, as the latter conside:--s only compensated pr-ice changes. Another- \..,·ay is to pr-ove the given differential sufficien! condition. Let's ass;,1me that w > 0.
Define H = {v e IRL: v·x(p,w) = 0}, that is, H is the
hyperplar.e with normal x(p, w} that goes through the origin. because p · x(p, w) = w > 0.
Then P E H
Thus, if v e IRL and v is not proportional to p, IR
L
such that v
4-3
1
e H, v
1
~
0, v
2
is
?roportional to p, and v = v
1
+
v . 2
Since S(p,w)v
= 0 and v ·S(p,w) = 0 by 2 2
Proposition 2..F.3, we have v·S(p,w}v = Cv + v J·S(p,w)(v + v l 2 1 2 1
= v ·S(p,w)v + v ·S(p,wlv + v •S(p,w)v 1
1
2
1
2
1
+
v ·S(p,w)v 2 2
= v ·S(p,wlv . 1
1
But here, by v e H, 1
T
S(p,wlv = (Dpx(p,w) + Dwx(p,w)x(p,w) lv = Dpx(p,w)vl" 1
1
Hence v ·S(p,w)v = v ·Dpx(p,w)v < 0 because v 1
definite.
4.C.3
1
1
1
1
~
•
0 and Dpx(p,w) is negat1ve
Thus the WA holds.
A Giffe;1 good will be a most familiar example.
In the figure below,
good 1 is a Giffen good.
•• ••• •• •• ••
•
••
'•
0
XI
This example shows that the ULD property is actually not derived from the utility maximization.
4.C.4
It is a restriction on preferences.
The L-shaped indifference curves imply that for every strictiy positive
4-4
price vector, the consumer's demand is always be at the corner of a upper contour se!.
Hence no compensated price change will change the demand.
= D
S.(p,w.) = 0 for every (p,w.) and D x.(p,w.) 1 1 1 p 1 I
that there exists (p,w.) such that D
w.
1
p·D
w.
x.(p,w.) 1
1
1
:r::
w.
1
1
1
l
1
1
1
1
p·£1/w.)x(p,w.) = 1, this implies that D 1
w.
Suppose
1
1
Since
x.(p,w.) and
1
Hence there exists a v e IR
1
v·D
1
x.(p,w.) ;e (1/w.)x.(p,w.l.
1
x.(p,w.) are not proportional. 1
W,
T x.(p,w.)x.(p,w.) .
Thus
2
1
1
such that
x.(p,w.) < 0 and v·x.(p,w.) > 0, as illustrated in the figure below:
1
1
1
1
1
DwOO(p, Wi)
p
0
XI
Figure 4.C.4
v Thus v·D x.(p,w.)v = - v·D p
1
1
T x.(p,w.)x.(p,w.) v = - (v·D
w. 1 1
1
1
1
w.
1
which implies that the ULD property is not satisfied. satisfied, the:l we must have D
w.
x.(p,w.))(v·x.(o,w.)) > 0, 1
1
1.
!
Thus, if it is in fact
x.(p,w.) = (1/w.)x.(p,w.) for every (c,w.).
1
1
1
1
1
1
.
1
Thus the unique \':ealth expansion path, which is the set of the corners of the upper contour set.s, is a ray going through the origin. is homothetic.
4-5
Hence the prefe:-ence
Following the hint, we fix w = 1 and write x.(p) = x.(p.1).
4.C.S
I
1
1
the indirect demand function g.(x) =
V ( ) Vu.(x).
x· u. x
1
1
1
Consider
Since x.(p) = x if and 1
only if g.(x) = p, the tJLD property of x.(p) is equivalent to the following 1
1
property: if x property,
i~
:;e
.
y, then (g.(x) - g.(y)) · (x - y) < 0. 1
1
For this latter
2 is sufficient to show that D g.(x) is negative definite. 1
We
shall now establish this. By the chain rule (Appendix M.A), •
2 2 2 Dg.(x) = (x·Vu.(x))- (Cx·Vu.(x))D u.(x) - Vu.(x)V'u.(x)T - V'u.(x}xTo u.(x)) 1
1
1
1
1
1
l
1
2
Let q = V'u.(x) and C = D u.(x}, then this can be rewritten as 1
1
2 Dg.(x) = (x·ql- ((x·q)C - qqT- qxTC). 1
We need to show that v · Dg.(x}v < 0 for every v 1
1 v·Dg.(x)v = (x·q( v·Cv < 0. 1
-:;e
0.
If v · q = 0, then ·
(This property is equivalent to the negative •
definiteness of the bordered Hessian of u.( ·) and used to guarantee the 1
differentiability of the demand function, as explained in the Appendix to Chapter 3}.
Sc suppose that v · q
*
0.
By multiplying a scalar to v if
necessary, we can assume _that v · Q. = x · q.
•
Then
v·Dg.(x)v = (x·q)-l(v·Cv- v·q- x·Cvl. 1
> 0, we need to show that v·Cv- v·q - x·Cv < 0.
By x·q
Sb::e C is svmmetric, •
v·Cv - x·Cv = (v - (l/2)x)·C(v - (l/2)x) - (l/4lx·Cx. Since u. ( · l is co;,.cave, C is negative semidefinite and the first term in the l
above expression is non-positive.
Thus,
v·Cv - x·Cv
~
- (l/4)x·Cx.
v·Cv - v·q - x·Cv
~
x . - (1/4)x·Cx - c. •
Hence
Since -
x·Cx q·x
< 4, the right-hand side is negative.
4-6
Hence so is the left-
hand side.
4.C.6
By differentiating both sides of u(>.x) = i\.u(x) with respect to i\. and
taking i\. = 1, we obtain 'i7u(x) · x = u(x).
Then by differentiating both sides of
2
o
this equality with respect to x, we obtain
•
u(x)x + 'i7u(x) = 'i7u(x). Tnus
2
D u£x)x = 0 and hence cr(x) = 0 . •
4.C.7
Suppose that the distribution of wealth has a differentiable, •
nonincreasing density function /( ·) over the interval ;~t
[0, w ).
Let v e IR
L
and v
0, then, just as in the proof of Proposition 4.C.4, we have
v·Dx(p)v = I
-w
-w
o(v·S(p,w)v)f(w)dw - I o(v·Dwx(p,w))(v·x(p,w))f(w)dw.
Here, the first term is negative, unless v is proportional to p.
(This
property is equivalent to the negative definiteness of the bordered Hessian of w.( ·) l
and used to guarantee the differentiability of the demand function, as
explained ir. the Appendix to Chapter 3).
As for the second term. just as in
the proof of P;-oposition 4.C.4,
-
-
-
2 f(w}dw .
.
By integration by parts and
-
-x(p,O) .= •
-w
2 (l/2)(v·x(p,w)) f(w) - (l/2)J o(v·x(p,w)} f'(w)cw The
firs~
2
part of this is always nonnegative, and it is positive when v is
proportional to p. f'(w) :::
-
0, this is equai to
The integral of the second part is nonpositive because
0.
0.
Hence v·Dx(p)v < 0.
To see that there are unimodal density functions for which the conclusions o: this propositions do not hold. recall that
-
v · Dx(o)v = iw(v · S(p w)v)f(w)dw 0 •
w
1
-w
- .Ir 0 (v·D w x(o,w))(v·x(o,wj)j(w)dw. . .
IRL t then To be scecifi::, let v = (l, 0, ... , 0) e • •
4-7
-
-
-+ x
Suppose a!so that the graph of the function w
(p, w) is as shown below . 1
•• •• •• •
•• •• •• •• •• •• •• • •
w
w•
0
Figure 4.C.7
Dwx Cp,w)
Then
1
(w•,w].
c:: 0 for every w e [Q,w•] and
Dwx (p,w) 1
-w
:s 0 for every w e
w• Tnus I Dwx (p,w}x (p,w)f(w)dw:: 0 and Jw.Dwx (p,w)x Cp,w)f(w)dw s 0. 0
1
1
1
1
If the density function f( • ) is nonincreasing, the weight placed on the interval [O,w•J is sufficient to dominate the negative effect of the interval
-
[w•.~·), implying that J~Dw;l(p,w); Cp,w)f(w)dw c:: 0. •
•
1
If the dist::-ibution
function is not noninc:reasing, then the weight on the interva: i ·;:",~·I may
-
dominate that on the interval It could even dominate the substitution effect, in which case we have v·Dx(p}v =
4.C.8
J
-\'4/ 0
(v·S(p,w)v)f(w)dw- J
-w 0
(v·Dwx(p,w)){v·x(p,w))f(w}dw > 0.
By substitu-.ing (4.C.6) into (4.C.S), we obtain
S(p,w} = }:.S.(p,o:.wl - t.D l
1
1
x.(p,o:.w)x.(p,o:.w) + Ct.o:.D
1 W. 1
1
1
1
1
T
x.(p,o:.wl)x(p,wl .
1 1 W. 1 1
1
On the other- hand, the right-hand side of (4.C. 7) equals to r.s.(p,c:.w) 1 1
1
[,D . X;(p,o:.w)x.(p,o:.w) •
w. 1
1
1
. 1
4-8
+
([.o:.D 1 1
w.
x.(p,a..w))x(p,w)
1
1
1
T
· T Dwx(p,w)rf'.x.(p,a.w})
+
'L11
1
- r.o x.Cp,cx..w)x.(p,a:.w) l w. 1 1 1 1
= f.S.(p,a.w) "'i 1 1
T xfp,w))x(p,w)
- ([.a:.)D
11W
(~.o:.D x.fp,a..w)lx(p,w)T L.l 1 w. 1 1 .
+
1
1
We h• thas proved (4.C.8).
4.C.9 The homotheticity implies that D hence\\ r{p.w) = I:.o:.D ..
1 1
w.
x.(p,o:.w) = (1/a.w).x.(p,a.w) and
W. 1 1
1
. I
x.(p,a:.wl = (1/w}x(p,w). 1
1
1
1
Thus
I
ap.w) = [.o:.((l/a.w)x.(p.o:.w) - (1/w)x(p,w))((l/o:.)x.(p,o:.w) - x(p, 11
1
1
I
11
1
w)?
T = [.(a./w)((l/a.)x.(p,o:.w) - x(p,w})((l/o:.)x.(p,«.w) - x(p,w)) . 11
111
111
L
Tnus, Car every v e IR ,
•
v·C(p,w)v = l:.Co:./w)((Vo:.)x.(p,a.w) - x(p,wll·v) 1
Thereftre.
4.C.l0
ctp. w)
~t's
1
1
1
1
2
2:
0
•
is positive semidefinite.
start by formulating our continuum-of-consttue!"S situation.
take t2 iutu·val [0,21 as the set of (the names of) the Clll51tll1ers. popul
density is equal to 1/2 uniformly on [0,2).
pro
We
Tne
We assume that the
. af each consumer's wealth to the average weallh (whic!-1 is, in this
situatilll, a counterpart '"of .the aggregate wealth) is const.-:. r--::.;::c.rdiess of the
af the aggregate wealth.
wealths
to
-w
d7) =
Since
We further assume 1hat. when the "average" •
> 0, the wealth of consumer 11 e [0.21 is equa" to
-w,
the term "average" is justified.
demand is then defined by x(p,w) =
Slutskymatri."<: con
J:.~st
S(p, w)
TjW.
is defined as in (4.C.4).
. 11 is denoted by S(p,l'!Wl.
2J 0 .xtp,Tiw)(I/2ldTI and the
The Slutsky mat:-ix of
We then define C(p,
w) by
as in the small-type discussion following the proof of Proposition 4.C.4, •
4-9
we have
C(p,w) 2
-
--
-
J 0 U/2)Dwx(p,7Jw)x(p,lJw)dlJ
=
= I
-
-T
- D x(p,w)x(o,wj
w
.
2 -2 T 00/2)Dwx(p,7JW}x(p,l)W)dlJ - (f o(l)/2JDwx(p,l)W)dT1 )x(p, ~·) .
Thus, for each v e IRL,
v · C(p, w)v
For the first term, we know that
2 2 J Cl/2Hv·Dwx(p,1Jw))(v·x(p,lJw})dlJ = (l/4w)(v·x(p,2w)) . 0
As for the se::::ond tenn, by integration by parts, 2
-
-
I o(T1/2)v. Dwx(p,l)W)dlJ
=
(l/w}(v·x[p,2w))
-
(1/w)(v·x(p,w}).
Hence
(v • x(p,2w))(v · x(p, w) l 2 = (l/w)[(l/Z}(v·x(p,2w)) - v·x(p,w))
C(p, w)
Thus
4.C.ll
l!:
+
2 (v·x(p,w)) }
0.
is positive semidefinite.
(a) \\"ben deriving individual
from the first-orde:- conditions
of utility maximization, we will neglect the nonnegativity constraints (which is investigated in Exercise 3.D.4(c)). prices and wealths
und~r
In fact, we will later see that, fo:-
consideration, the demands are always in the interior
of the nonnegative orthant.
It follows di::-ectly from the first-order conditions that
2
2
x (p,w/2) = Cx (p,w/2),x (p,w/2)) = (w/2p - 4P/Pz• 4p 1p l. 1 2 11 21 1 1 2 2 x (p,w/2) = (x cp,w/2),x (p,w/2)) = (4p 1p , w/2p - 4p 1p l, 2 1 2 2 1 12 2 22
4-10
Hence x(p, w)
(b) Denote the (l,k) entry of the Slutsky matrix S.(p, w) of consumer i by 1
5 lki(p, w).
2
Since
ax21 Cp,w/2)/Bw 1
= 0, s
221
Cp,w/2) =
ax21 (p,w/2)/Bp2 =
3
2
- spl/Pz· Hence by Proposition 2.F.3, s 211 Cp,w/2) = s 121 Cp,w/2) = 8P/Pz· and hence s
Cp,w/2) = - 8/p . 111 2
Thus
s1(p,w/2)
=
•
Similarly, we can show that
•
•
We can also apply Proposition 2.F.3 to derive the Slutsky matrix S(p, w) of the aggregate demand function: - w/4p S(p, w) =
2 1
2
-
w/4pl p2 + 6pl/p2 2 - w/4p 2 2
By Exercise 2.F. 9(b} (and S(p,w)p = 0), if dp e IR , dp
is not proportional to p, then dp·S(P,w)dp < 0.
;~t
0, and dp
Thus, according to the
small-type discussion after Proposition 2.F.3, the aggregate demand function ' ' sat1s.1es ' ~· •h x,p,w, .. e WA .
(c)
By substituting p = Cl,ll, we obtain w/4 - 4
C(p,w) =
~Si(p,w/2)
4 - W/4
- S(p,w) =
•
4 - w/4
w/4 - 4
Thus, if w > 16, then it is positive semidefinite, and if 8 < w < 16, then it is negative semidefinite.
(This can be shown by applying Theorem M.D.2(iil,
or by noticing that C(p,w)p = 0 and p·C(p,w) = 0 and then applying the
4-11
For example, if w = 12 and v =
argument in the proof of Exercise 2. F. 9(b}.} (1,0), then v·C(p,w)v =- 1.
Thus C(p,w) is not positive semidefinite.
saw in (b) that the aggregate demand function satisfies the WA.
We
We can thus
conclude that in order for an aggregate demand function to satisfy the WA, it •
. is not necessary that the matrix C(p, w) is positive semidefinite . •
(d) Here is a figure depicting the wealth expansions paths of the two consumers for p = (1,1).
= x (1,1,8l = (4,4). 2
They intersect each other at (4,4) because x (1,1,8) 1
Note that if 8 < w < 16, the Engel curves resemble those
of figure 4.C.2(b); if w > 16, the Engel curves resemble those of figure 4.C.2(a).
consumer 2 •
4
0
consumer 1
4
x:
Figure 4.C.ll(d)
4.C.l2
As suggested in the hint, our example is a two-commodity, two-consumer-
economy in whlc.h the two consumers have the same preference and the wealth dist;:-ibutior:. rule is such that when the agg1·egate wealth is equal to 4, the wealth of consumer 1 is equal to 1 and the wealth of consumer 2 is equal to 3. The example here is essentially the same as Example 4.C.l (which is
4-12
illustrated in Figure 4.C.l).
Example 4.C.l is not directly applicable to the
present context, because the two consumers have the same wealth but the different demands.
However, if wealth levels are different between the two
consumers, then it is possible that they have the same preference and yet c;iemands similar to those in Figure 4.C.l, yielding a violation of the WA in the aggregate.
This is illustrated in the figure below: X2
6
4
3
2 1 0
1
4
3
2
6
8
X:
Figure 4.C.12
4.C.l3
We consider a two-commodity, two-consumer- economy with the given
wealth distribution rule w (p,w} = wp/(p + p l and w (p,w) = wp /(p + P l. 2 1 2 2 1 1 2 The preferences of the consumers are represented by the following utility
functions: u £x ) = Min <x ,2x
1
11
1
u £x ) = Min {2x
2
2
12
21
,x
}.
22
}.
So the prefer-ences are homothetic and have L-shaped indifference curves. unique wealth expansion paths are depicted in the figu:-e below:
4-13
The
consumer 2 4 .................... . •
••• •• •
consumer 1
•
••
•• .,••......
...... 2 ·----· ••
•• •
•
••
•
••• ••
•••
2
4
••
•
0
XI
Figure 4.C.13 Just as we saw in the answer to Exercise 3.D.S(c), the individual demand functions are xl(p,wl) = (2.wl/(2.pl + p2)' wl/(2.pl + p2)), x2(p,w2) = (w2/(pl + 2.p2)' 2.w2/(pl + 2.p2)). Tne aggregate demand function is given by
We claim that the aggregate demand function does not satist"y the WA. . . d ~t~·1ne see tn1s. 6 and w = 1.
A
.~..:
= { p e IR2 : p ++
Tnen w.(p,wl I
1
= p.1
+
Pz
To
= 1} and our restrict attention to p e
fJJr both i and hence
limp-+(l,O)x(p,l) = limp-+[l,O)x Cp,w Cp,l)) = (U/:.::;, 1 1 limp-+(O,l)x(p,l}
Thus, if p e 1!, q e 1!, and p and p · x(q,ll < l.
2
= limp-+(O,l)')C:2 (p,w 2 (p,lll = (1/2,1}. and q
1
are sufficiently small, then q · x(p,l) < 1
Hence the aggregate demand function does not satisfy the
WA. Proposition 4.C.l does not apply to this example because the wealth distribution rule of this example depends on prices, while the proposition cioes r.ot aliow this.
4-14
It is easy to check from the budget constraints that th~ distribution
4.0.1
(x (p, w (p, w}), ... ,x (p, w (p, w))) of commodity bundles satisfies the
1
1
1
1
constraints of the maximization problem in this exercise, and, from the definition of v[p, w).
th~
indirect utility functions, that it attains the value
It thus remains to show that if (x , ... ,xi) satisfies the constraints 1 •
of the maximization problem in this exercise, then WCu (x ), ... ,u (xi}) :s 1
v(p,w).
Fo:-- each i, define
W. 1
= p•X.,1
then r.w.
11
= [.p·X. 1 1
1
1
= p•([.X.) 11
:S W,
that
is, Cw , ... ,w l satisfies the constraint of the maximization problem of 1 1
Hence, by the definition of v(p,w).
(4.0.1).
u.(x.) On the other ha.-ld, by the definition of the indirect utility functions, . 1 1
:s v.(p,w.) for ever-y 1
1
•
1.
This and the increasingness of W( • ) imply that
W(u (x ), ... ,u £x )) :s W(v £p,w ), .... v (p,w )J. 1 1 1 1 1 1 1 1
cx
Hence W(u £x ), ... ,u ll :s v(p,w). 1 1 1 1
4.0.2
This completes our proof.
To check that v(p, w) is increasing in w, let p be a price vector, w and
w' be two aggregate wealth levels with w :s w', and (w , ... ,\\'t} be a solution 1
to the social welfare maximization problem of (p,w).
\ < d • \' < AI so, L..w.- wan nence ~....w. - w. I
!
1
1
1
Then
Thus, by the definition of v( p, w'),
Hence v(p, wl :s v(p, w'). To check that v(p,w) is nonincreasing in p, let w be an aggregate wealth level, p and p' be two price vectors with and p'
;!:
p, and let (w , ... , wi) be a 1
solution to the social welfa•e maximization problem of (p', w).
•
4-15
Then
Also, v.(p' ,w".> ~ v.(p,w.) for every i because p' 1
1
1
i!::
1
p.
Since W( ·) is
increasing, this implies that W(v Cp',w ), ... , v (p',w }) 1 1 1 1
W{v Cp,w ), ... ,v (p,w )J. 1 1 1 1
~
By the definition of v(p,w),
Hence v(p',w)
~
v(p,w).
To verify the homogeneity of degree zero and the quasiconvexity, we apply the equivalence of the two maximization problems established in Exercise 4.0.1.
For any (p,w} and A > 0, the two price-wealth pairs (p,w) and (i\p,Aw)
give the same constraints to the maximization problem of Exercise 4. D.l. •
Hence v(p, w) = v(i\p,i\w), implying the homogeneity. let
11
e lR.
As for the quasiconvexity,
Let (p,w) and (p',w') be two price-wealth pairs such that v(p,w)
v and v(p' ,w') :> v.
Let i\ e [0,1} and define p"
= Ap
+ (1 - A)p' and w"
~
= AW
and hence AD· C[.x. l + (1 - i\)p' · (E.x.l ~ etw + ( 1 - o:}w'. •
1 1
We must thus have either w •
either WCu Cx ), ... ,u Cx ll 1 1 1 1
~
1 1
i!::
.
p· Ct.x.) or w' 1 1 •
p' · £l:.x.). 1 1
Hence we
v(p,w) or W(u £x ), ... ,u Cx )) :: v(p',w'). 1 1 1 1
eithe:-- case, we have W(u £x ), ... _,u cx ll 1 1 1 1
4.0.3
i!!:
:>
v.
mus~
have
In
Hence v(p,w) is quasiconvex.
The welfar-e maximization problem is now rewritten with nannegativity
constraints: •
W£v (p, w ), ... ,v Cp, w )} 1 1 1 1
s. t.
[.w. I
w
I
1
:> i!::
w, 0 for all i.
We assume tha: xi(p,O) = 0 for every i and every p
4-16
» 0. This assumption is
satisfied
• the consumption
0 for
»
p
sees are
all IRL. +
Then 9 v .(p.O) =
p
1
The Kuhn-1-.:ter conditions for the
0.
{Theorem M.K.2) ~that there exist A > 0 and IJ.
social
1
0 such t
~
.) - A + IJ. = 0 for~ i = 1, ... ,I,
. 1
1
= 0
~t.w. 1 1
where all
a.e
for every i =l, .... I.
evaluated at the
(w ..... w )
= {w (p,w)...••wlp.w)). 1 1
1
The En
em M. L 1) implies
•
/Bu. )(Bv ./ap,_L 1
Now
w)
(8W /8u.H
> 0}.
Since p.. • 0 far every i e J, 8v/8w = 1
Since BvlfPL = 0 for any i I! J, ovlapl =
i e J.
1
caw / IE 1
1
[.
. Thus. •
(BW/Bu. )(Bv./8pDJ 1
-
1
v.
l
Hence the
w.
.1
1
~
I·
=- LieJ (c3W/Ciu."iUlv./8w.)
Bv/Bw ilv/BP l a /8
(8W/8u.)(8v./8pl.) 1
1
1
= l . .x.(p.w.{p,w)) = [.x.(p,w.(p,w)). L..j.eJ
1
1
1 1
1
function derived from v(p,wl equals [ixi(p,wi(p,w))
and the
4.D.4
(a)
-+ W(u £x ), 1 1
Maximum If the
and W( ·) are monotone. lben so is u( ·).
-1lso continuous.
Thus.
a,
the Theorem of the
• , u( ·) is also con tin ....... • •
and
we . ) are
mc:w#one and concave. then
u{.)
is
•
4-17
concat This can be proved as follows. Definer"
= ll.x
Let x e IRL, x' e IRL, and A e [0,1].
+ (1
= each i Then
[.x~ ::5 1 1
Alx:
1
for
1
Ax + (1 - ll.)x' and hence u(~x + (1 - ~)x•) ~
By the
~x. + (1 -
WCu Cxi), ... ,u Cxl)) .. 1 1
vity of the u.( ·} and the monotonicity of W( • ), 1
(xi), ... ,u Cxjll 1
!:ltA.u Cx 1 + (1 - i\)u Cxil• ...• ;w (x > + 1 1 1 1 1
{1 -
A}u fxjll. 1
By theccavity of W( · ),
I{Au cx > + (1 - ~)u Cxi>·
1
1 1
... ,
~ cx > + (1 - ~lu Cxjl)
1 1
1
!:lu.(x) + (1 - A)u(x').
Hence
~ + (1 -
ll.)x')
ll.u(x) + (1 - i\)u[x').
Hence u( ·) is concave.
Its worthwhile to point out that the quasiconcavity of W( ·) and the ui ( · ) dll5 not imply that of u( · ).
2
2
As an example, let L
x 1 :· uJ&zl = x 22 • and WCu1,u2 ) = u1
+
= 2,
I
= 2,
u Cx ) = 1 1
u2 . Then
W(2,0) = W(u C2,0),u CO,O)) = W£4,0) = 4, 2 1 W(0,2)
= WCu1CO,O),u2 (0,2)) = W(0,4)
= 4,
W(l,l) = W(u U,O),u CO,l)) = W(l,l) = 2. 2 1
_LI ::-;
\..:-:-·1es tu..sat.~.::~.
~ L..x.
1 1
::5
· a so1ut1on · . . . x an d 1s to t h e maxmuzauon pro bl em of
Exer-cisr4.D.l, then x is a solution to the maximization problem of this part. •
1s a soiutioall the ma..ximization problem of Exercise 4.0.1.
4-18
Let x' e i:1
L
and p · x'
solu1ir 1D the maximization problem of this part.
Thus, by the above result, [.x.(p,w.(p,w)J is a
Pi"Oblem.
1 1
1
•
til
the maximization problem of this part.
Hence the Wal!·asian demand
prated from it is equal to the aggregate demand· function.
4.0.5 1Jere is no positive
consumer if the WA is violated.
4.Cl thus serves as an e-vaul'le for this exercise.
E
welfare maximization problem is now v.rritten as
4.0.6.
[.w.
s.t.
1
~
1
w.
conditions are that there exists A > 0 such that
The
.} = A for every i, where all derivatives are evaluated at a
1
solut•tw •... , w 1
>. 1
By the definitiCil of WC • ), 8W lou. = a./v.(p, w.). 1
1
1
Bv ~
l
•
l.D.3(b), v.(p,w.) is
Ex
1
c3·dp,~. 1 t 1
of degree one in w.
1
1
= v.(p,w.)/w.. 1
1
~•d
hence
Hence the left-hand sides of the abo...-e
1
mrsditions equal (o:./v.(p,w.))(v.(p,w.)/w.) = a.lw. for eve:-v i.
firs~
1111
Th '.J s .. = a..ll. 1
I
Since [.o:.
1 1
= 1 and li_Wi = W,
11
W
= 1/i\..
II
•
Hence w = a • w . l
Thus
I
w. (p, ..Pc.w. 1
J.
welfare maximizatim problem is now written as
4.0.7
r.a.(p) + b(p)~.w.) •... , w I ) 1 1 'Li. 1
s..t.
[.w. 1
1
~
w.
with [.w. = w is a solution to this problem and v(p,w) = 1
1
4--19
+ b(p)w.
U
Suppose that (p I w"lJIB the potential 1
I
e fi such that '"· w: L.l 1
Then there exists
fiR
definition of v(p
1 I
over
W
1
:S
w and v.(p' w:) > 1
1
•
•
l
1
WCv Cp ,wi), ... ,v Cp',-I)}· 1 1
) :!:
•
1
Hence
•
f,r} > v(p w). 1
Define A. = {x.:u.(x..llllill. then A = r:.A.. 1
1
1
1· 1
I
Ths, for every p
1 l
»
\x E -/- is a solution to ·
p·y anly if there exists
A x . . . x A such tiat 1 1
E
r,.xi
= x
C fer every i, x. is a soblidll. 1
. eA •
1 ~
Win{p·y: y e A} =
1
1
y. e A.}. 1
y.
1
i.
p·x .. Bv the d ~
1
E A.} = e.(p 1 u.(x.)) 1
1
1
1
Hence
[e.(p,u.(x.ll.
e A} =
1 1
l
1
ather hand, by the J e B} =
k A c B. Min{p • y: y e A)
·y: y e B}. •
g(p) p.
-
1 1
.-rmite.
ii'lG.JI s
- r:.s:( 1 1
~
-
1
1
::5
0
Hence, by the
amy condition for a ~(Section
2 Since D g(p)
Hence
r:.e.(p,u.(x.ll
= 0.
Bv the •
e(p,u(x)).
2 M.K), D g(pl is aqative
r.s.(p,u.(x.ll 1 1
~negative
4-20
1
1
by
semidefinite.
3.G.3
4.0.10
As a saw in the answer to 2
is symmetric. and if dp e IR , dp dp · S( P, w }dp c 0. representati~
:;e
4.C.ll(b), for every lp.w), S(p, w) 0, and d.p is not proportional tap, then
Thus, according to
3.H, there exists a psitiw
But accordill( tD 'Exercise 4.C.ll(c), if I< w < 16 •
consumer.
•
then the matrix C(p, w} is not positive
e.
Hence, accordiag to the
small-type djsr:nssion in Section 4.0, tll!r-e- is no normative represmtatrn consumer.
4.0.11
We sbil give an example in wbkh L = 3 and
c12 Cp,w)
:;e
the definitioa.
Here,
by parts,
2
-
I C7JI2lax (p,,.,wl/Bw)dl'l
0
1
2 - = I 0("'1)/iW)(dxl(p,TIW)/dl))dT)
•
=
cuwJif.zw> -
Clll;>x (p.w>. 1
H..-n,..·· •
--
-
-(1/w)x (p,2wlx
-
(llw)x
1
2
(p, w) + (l/2w}x (p, w)x 1
2
(j), w).
Similarly,
Again by in
-(p,w)x
1
2
(p,2w) + OJZ.wlx Cp,wlx
by parts,
(p. nw )( ax
2
(p,1Jw)/8w)dll
1
2
Cp,w}.
c (p,w). 21
By
2 = J (1/2)X l (p, l)W)(l/w)(dx (p, l)W)/dl))dl) 0 2 - - - 'J=2 2 = ((1/2W)Xl (p, 'J)W)x £p,1JW}l7)=Q - JQ(l/2w)(dx (p,l)W)/dl))X (p, "')W)dl) 2 1 2
2 = (l/2w)x £p,2wlx [p,2wl - J' {1/2)(dx (p,l)w)/8w)x (p,l)W)dlJ 2
1
0
2
1
an example in which
It is thus sufficient to
(p, wlx
-
(l/w)x
+
-U/w)x (p,2wlx
1
2
(p,2w),
or, equivalently,
;t.
Sc
--(112w)x (p,2w)x 1
2
(p,2w)
1
•
2
(p,w)
-
-(l/w)x (p,w)x 1
ider a preference, a price vector p, and the average wealth
-w
2
(p,2w} .
such
that:
-xip,2w} x£'p,wl
= 2 for both l = 1,2, 2 = S0 £1/2lxt(p;qwld'J)
= 1 for
(Another resti"'iction will be given shortly.
both t = 1,2. The demand for good 3 is
detetmined by Walras' law.) Then the right-hand side of the above inequality is equal to
2/w.
It is then sufficient to show that
But if the graphs of the functions 1'1 -+
-x l(p,l)w} -
are as in the figure below,
then the first term can be wade as close to zero as needed. inequality can thus be established.
4-22
The above
JU{p,T)W)
2 --··------·-······· ••
'•'
•• •• •• •• •• • •• ••• ••• •• •• ••• •• •
w
2w
••
xr{p,TJw) • ••
•• •• •
••'
0
Figure 4.0.11
4-23
w
CHAPTER 5
S.B.l
The first example violates irreversibility and the secane! one satisfies
this property.
0
0
Figm e S.B.l
5.8.2
L-1 Let z e ~+
Suooose first that Y exhibits constant returns to scale. ••
and a: > 0.
The~
(- o:z, af(z}} e Y. place of z and f(o:z) :s c::f{z).
(- z, f(z)) e Y.
By the constant retUI"ns to scale,
Hence etf(z) :s f(o:z).
By applying this inequality to o:z in
1 i::J place of o:, we obtain et- /(o:z} Hence f(o:z) = o:f(z).
:'5
1 f(a- (o:z)) = f(z), or
The homogeneity of degree one is thus
obtained. Suppose con•;ersely that /( ·) is homogeneous of degree one. e Y and a
i!:
0, then q :s f(zl and hence o:q ::= a:f(z) = f(o:z}.
(- a:z, f(o:z)) e Y, we obtain (- o:z, o:q) e Y.
Let (- z. q)
Since
The constant returns to scale is
thus obta:nec.
S.B.3
Sunoose firs: that Y is convex. ••
Let
5-1
Z
•
Z
•
e lRL-1 and a e lO,l]. then +
(- z, f(zl) e Y and (- z', f(z'))
E
Y.
By the convexity,
(- (a:z + U - «lz}. a.flzl + U - a.)f(z)) e Y.
Thus, cxf(z) + U - o:)f(z) == f(o:z + (1 - «)z).
Hence f(z) is concave. Let (q, - z) e Y, (q'. - z') e
Y, and
11 E
[0,11, then q
:£
f(z) and q'
a.q + (1 - cr)q' :s
~
/(z'J.
Hence
..rrz> + U
- o:)f{z' }.
By the concavity, a.f(z) + (1 - «)f(z'l s ttuz + (1 - a:)z').
Thus
aq + (1 - a.)q' :s f(a.z + (1 - a}z').
Hence
= «(- z, q) +
(- (a:z + ( l - «)z'), o:q + [1- «)q')
(1 -a)(- z', q') e Y.
Therefore Y is convex.
S.B.4
Note first tha-: if Y itself is additive, then Y+ = Y by the definition
of the additive closure. 5. B. 7, and S.B.8.
This applies to
5.8.4, S.B.6(a), S.B.6(b),
So we shall not depict
Let's now take up the cases in
the production set is not additive.
Note first that Y+ is equal to the set of
of IKL that can be •
represented a.s the sum of finitely
of Y.
If the production set
is convex, as in Figures S.B.l. S.B.2fa), 5.B.2(b), 5.8.3(a), S.B.3(b), and
5. B.S(a) (which is the same as S.B-2(b}), then we have the following stronger production plans.
of all , far
positive integer n, define nY c IRL by nY = {n:y e
We then claim that Y+ = vr:o nY. n=1
Y.
~L:
y e Y}.
In fact, if 1 e nY for some n, then (1/n)y e
By the definition, (1/n)y e Y+ and h
vr:sJ nY. n= 1
To be more
y = n((l/n)y) e Y+.
Thus Y+ ::;:,
To m-nve that Y+ c vm_ nY, it is sufficient to show that um nY is ~n-1 · n= 1
5-2
additive, by the def'inition of Y+. ·
So let y e vm
n=l
nY and
v' J
the:--e exist positive integers n and n' such that y e nY and y' E n'Y. 0/n)y e Y and (Vn')y' e Y.
Thus
Since Y is convex and n/(n + n') + n'/(n + n') =
1, (nl(n + n')){(lJn)y) +
/{n + n'))((l/n'}y') e
Thas y + y' e (n + n')Y.
That is, (1/(n + n'))(y + y') e Y.
as in Figure S.B.S(b),
If the pa oduction set is not m
+
longer have Y = un=InY. +
have Y :> v
As we can see in the above proof, +
~
m
= nY, we need not have Y c un=lnY. 0 1
some production plans in Y+ can be
('i·'i} be
. . d. max1mtzc
we no
we still
That is, it may be true that
only by allocating
inputs to di!fe-eot p&oduction units. follows. Let
Y.
point can be foz·mulated as •
•
plan at which the average return is
the
is, Y211Yi1 > y 1ly 1 for any other y = (y ,y l e Y. 2
that the function that associates each
quasiconcave.
1 2
1
Assume
r 1 < 0 to the average retmn at y1 is It is e;uivalent to
(This appears to be true from the figure.
Let y < 0 be an 1
saying that the average return functiCil is single-peaked.}
(aggr-egate) input level and n be the positive integer such that n.Ji < y
1
~
allocation •f a;gregate inpu:
Yr
(The number J of productim onits to be used is also being
•
optimized here.)
We shall prove that aoe of the following three li'ses must
then apply: (i)
J = n - 1 and ylj
(ii l J (iii)
=n
and ylj
c:
y /(n - I) far eruy j. 1
= Y{n for every j.
J = n and there exists k
y j = Yj for any j 1
E
{l, ... ,J} sucb tha.t ylk = y
~ k.
To prove this, note first that we y : ::= Yi for everJ j. 1...
1
- (n • IlJi md
have either y lj ?!
Yi
for- ere.ry j, or
In fact, if neither of these applies, then a small
5-3
•
input reassignment from a production unit with y . < y• to one with v . . 1 1J . ~ 1J increases the (aggregate) output level, because the average return increases. If ylj
~
Yi for every j, then the average return is decreasing at every ylj
and hence the y,. must be all the same. ~J
maximum integer that satisfies ( i)
applies.
at every ylj'
If y lj
~ y~
Yi
By the quasiconcavity, J must be the
= Lllj (and ylj :s.
Yil.
Hence J = n -
1 and
for every j, then the average return is increasing
If the y j are all the same, then, by the quasiconvexity, J 1
must be the minimum integer that satisfies
= n and (ii) applies.
Yi = Lllj
(and ylj
!:
Yi).
So suppose that some of the ylj are d"ifferent.
there exist two production units for which y lj >
Yi.
Hence J If
then a small input
reassignment from one to the other would increase the output level, because the average return is increasing. which ylk > Yi· applies.
Hence there mnst exist at most orie k for
By the quasiconcavity, y j = 1
Yi
for any j
~~~'
k.
Thus
(iii)
This la5t case happens when the average return decreases very fast
after the input level goes beyond
Yj·
For Figure S.B.l
For Figure 5.B.2(a)
5-4
For Figme 5.B.2(b)
••
••
••
••
••
••
........
•• •
•'
• •• •
• • •• • • •• •• ••• •
•• •••
•••
•• •• •
••
••
For Figure S.B.3(a)
5-S
·.....• •
•• •
••
•
........... •
...... . •
••
••
•• • • • • - · ...... • •• • - - - - · • • • · •..••••• •r •• •• •• •• •• •• •
•• • ••
·-·····-~·-•
••
••'
••
•'
For Figure 5.B.3(b)
•• •
•• ••
• • • • • ••
•• •
'
•
For Figure S.B.S(b)
S-6
•
Let y E Y and v e - IRL.
S.B.S
e Y bv Y c - iR + •
Since Y is convex,
~
(1 -
1/n)y + (1/n)(nv) = (1 - 1/n)y + v e Y.
Since Y is closed, y + v = lim
S.B.6
+
+
L
•
Then, for every n e IN, r.v e - c:tL and hence nv
n..-
((1 - 1/n)y + v) e Y.
(a) From the given functions tf>.( ·) (i = 1,2), the production set is 1
defined as Y = {(yl'y ,q): there exist q 1 2 and - y.
1
2:.
t/). (q.) for both i}. 1
•
0 and q
2:
2
0 such that q
2:
1
+ q
2
2:
q
A three-dimensional production set is depicted
l
in the following picture, assuming that. the
~.( 1
·) are convex.
output input 2 input 1
Figure 5.B.6(a)
•
(b)
We ciaim that the condition that ¢.(q. + q: l 11
1
!S
t/>. (q.l 11
is su:'ficient fer additivity.
+ t/).(q:) 11
for all q.
1
0, q:
1
2:
0, and i = 1,2,
In fact, let (y ,y ,ql e Y and (Yi.Y2_.q'l e Y. 1 2
Then there exist (ql'qZ) such that q + ~ 1
2:
sue h ~· ..na t q • 1
Then
+
2:.
q . -> q • ana. - Y•i
2
"" ( qi') · 2: .,..i
q and - yi
2:
¢i(qi). and (qi· q2_l
and -
(V'. + y:) 2: "".(q.) + .,._(q:) 2:. 1/>.(q. + q:) . • 1
1
'f'l
1
'f'1
5-7
l
1
1
1
Thus (y 1
Yj·
+
Yz
+
q + q') e Y, establishing the additivity.
(c) Let the output price be p.
The first-order necessary conditions for
profit maximization are that, for both i,
0.
w.t/J~(q.} ~ 1 1 1
p, with equalitv if ~
qi >
The interpretation is that marginal cost (in monetary term) due to a unit
increase of output level must be smaller than or equal to the output price, and the former must be equal to the latter if the output level is positive . •
If both ¢ ( ·) and ¢ ( ·) are convex (so that the corresponding production 1 2 M.K.J, these first-order necessary
set is convex}, then, according to conditions are also sufficient.
(d) Let q
>
By renumbering i = 1.2 if necessary, we can assume that
0.
w ¢ (q) ::s: w ¢ £q}. 2 2 1 1
> wl¢1(q). ql/q
+
In order to prove the statement, it is sufficient to show •
In fact, since the ¢.( ·} are strictly concave (and ¢.(0) = 0) and 1
1
q2/q = l, +
w2¢2(q2) > wl(ql/q)¢l(q) + w2(q2/q)¢2(q} = (ql/q)wlt/Jl(q) + ~q2/q)w2¢2(q)
The statement is th'.ls proved.
~ w,¢,(q). •
•
The strict concavity of the ¢. ( · } is 1
inte.r:;:::retec as the increasing returns to scale, which makes the statement qui:e plausib!e: tJnde:- the increasing returns to scale, it is bette:- to concentrate on one technique.
The
isoq~ants
of the input use is drawn in the following figu:-e.
Note
that the additive se;:>arability imposes the same restriction on the isoquants as that alluded to in Exercise 3.G.4(b).
5-8
input2
•
input 1
0 Figme 5.B.6(d)
If the:-e is a production plan y e Y with p· y > 0, then, by us"ing a.y e Y
S.C.l
with a lar-ge o: > l, it is possible to attain any sufficiently large profit Hence rr(p) = co.
level.
:s 0.
If, on the contrary, p·y :s 0 for all y e Y, then rr{p)
Thus we ha•te either n(p} = +
S.C.2
Let p >> 0, p' »
rr{p) and p' • y :s n{p').
e~~
or ll(p) :s 0.
0, o: e [0,1}, and y e y(o:p + (1 - o:)p' ), then p • y :s
Thus,
(a.p + Cl - o:)p') · y = o:p · y + (1 - o:)p' · y :s o:rr(p) + (1 - o:m(p' l.
Since (o:p
(1 - o:}p')·y
+
= n(a.p
+ (1 - a.)p'),
n(o:p + (1 - cx)p') :s «ll(p) + (I - o:)n{p'). Hence n( • } is convex.
S.C.3
The hcmoge:1eity of c( ·} in q is implied by that of z( ·).
prove this latte:- homogeneity only. Let w z(w,q).
»
0, q
~
0, and o: > 0.
Since f( ·} is homogeneous of degree one, f(o:z) = o:f(z) ,L-1
every z' e =-
+
,
·~
1J
f(z'}
~
-1
-1
c:q, then f(o: z') = o: f(z')
S-9
\v'e shall thus
~
q.
?!
Let z e o:q.
Fer
Thus, by z e
-1
z(w,q), w·(o: z') a:: w·z.
Hence w•z' a:: w·(a:z).
Thus a:z e z{w,a:q).
So a:z(w,q)
•
c z(w,o:q).
By applying this inclusion to
-1
a.
-1
in place of a: and o:q in place of
-1
q, we obtain o: z(w,a:q) c z(w,a: (a:q)), or z(w,a.q) c o:z(w,q) and thus conclude that zlw,a:q) = az{w,q). We next prove property {viii). [0,11.
Let w e IR
Let z e z(w,q) and z' e z(w,q').
w·z.• and c(w,q'l = w·z'.
L-1 ++
, q a:: 0, q' a:: 0, and o: e
Then f(z)
q, /(z' l a:: q', c(w,q) =
k
Hence
cxc(w,q) + (I - a:)c(w,q') = a:(w·z) + (1 - o:Hw·z') = w·(a:z
+ (1 -
a:)z').
Since f( ·) is concave, f(a.z + (1 - o:)z') a:: o:f(z) + (1 - a:)f(z') a:: o:q + (1 - o:)q'.
Thus w · (o:z +
(1 -
o:lz') a:: c(w, o:q + (1 - «)q' ).
ac(w, q)
S.C.4
•
+ (1 -
o:}c(w, q')
ii!::
Tnat is,
c(w, o:.q
+ (1 -
a.)q').
[First printing errata: When there are multiple outputs, the function
f(z) need not be well defined because it is conceivably possible to produce
different combinations of outputs from a single combination of inputs. Assuming that Li.e first L - M commodities are inputs and the last M corr.modities are outputs, we should thus understand the set {z as
' {Z
z, q) Yfa) • •
E
Y}. 1 For each q
= {z e IR
L-M +
: (- z, q)
i!:
0: f(z) a:: q}
0, define
L-M e Y} = {z e IR : f(z} +
Then c( · ,q) is the support function of Y(q) for every q. foHows from the discussion of Section 3.F.
~
i!: G}.
Hence prope:--ty (ii l
Moreover, according to Exercise
3.F.l, if Y(q) is closed and convex. then
L-M Y(q) = {z e IR : w·z a:: c(w,ql for all w » 0}. + Since Y = {(- z, q}: q a:: 0 and z e Y(q)}, this implies property (iii}. To prove property (iv), let w » 0, q Y(q).
Then w·z ::s w·z'.
i!:
0, a > 0, z e z(w,q), and z'
Hence (a:w)·z ::s (o:w)·z'.
5-10
Thus z e z(a:w,q).
E
Therefore
z(w,q) c z(a:w,q).
By applying this inclusion to o:w in place of w and 1
place of o:, we obtain z(o:w,q) c: z(o:- Co:w),q) = z(w,q).
0:
-1 .
1n
Property (iv) thus
follows. Property (iv) implies the homogeneity of degree one of c{ ·) in w, which As for its second part, let w » 0, q
is the first part of property (i). q'
i!!:
0, and q'
2:
Then Y(q') c Y(q).
q.
~
0,
Since c( ·,q) and c( · ,q'} a:-e the
support functions, this inclusion implies that c(w,q')
2:
c(w,q).
Hence c( ·)
is nondecreasing in q. As for property (v}, note that z(w,q) = Y(q)
every w >> 0 and q
~
0.
n {z e IR : w ·z = c(w,q)} for
Since both of the two sets on the right-hand side is
convex, so is the intersection, and hence so is z(w,q). single-valuedness, let q
2:
L
0., w
As for the
» 0, z e z(w,q) , z' e z(w,q), and z
Also suppose that Y(q) is strictly convex.
:;t
z'.
By the convexity of z(w,q},
0/2)z + (1/Z)z' e z(w,q).
By the strict convexity of Y(q), there exists a z" e Y(q) such that
(l/2)z + (1/2)z' » z". Hence w·((l/2)z + (l/2)z') > w·z", which contradicts (l/2lz + (112}z' e z(w,q}.
Tnus z{w,q} must be single-valued.
Property (vi) foliows from the fact that c( · ,q) is the support function of Y(q) and the c•Jality theorem (Proposition 3.F.l). Property (vi) implies that if z( · ,q) is differentiable at -
-
2
-
D z(w,q) = D ('i7 c(w,q)) = D c(w,q). w w w w
By property (ivl,
Dwz(w,q}w
=
Property (vii) is thus established.
S.C.S
e lR
then
As a Hessian matrix, this is svmmetri::. -
By property (ii}, i<; is negative semidefinite.
0.
-w,
L-1 +
If the p!"oduction fun:::tion f( ·} is quasiconcave, then the set Y(q) = {z : f(z)
q) is convex for any q, and .thus property (iii) holds.
5-11
(Note
that we used the convexity of Y(q), not of Y.) If there is a single input and the production function is given by f(z) = 2
z , then it is quasiconcave but the corresponding production set exhibits increasing returns to scale.
Quasiconcavity is thus compatible with
•
•
S.C.6
Throughout the following answers, the input prices are denoted by w »
mcreasmg returns.
0 and the output price is denoted by p > 0. z( p, w)
the input demands at prices (p, w ).
For convenience, we denote by As a preliminary result, by using
the implicit function theorem (Theorem M.E.l), we shall prove that z( ·) is a continuously differentiable function and give its derivatives in terms of f{ · ).
(To be rigorous, we need to assume that f( ·) is twice continuously
differentiable and the input demands are always strictly positive, so that the nonnegativity constraints never bind.) 2 Since D f(z) is negative definite for all
z, the first-order necessary
and sufficient conditior. (Theorems M.K.2 and M.K.3) for profit maximization is •
then that z is an input demand vector at prices (p, w) if and only if
p'i7f(z} - w = 0. If we regard the left-hand side as defining the function defined over (p, w,z).
the::1 the function is continuously differentiable and its derivative with 2
resoect to z is eaual to pD f(z). • • inve:-se matrix (at every z).
It is negative definite, and hence has the
Thus, by the implicit function theorem (Theorem
M.E.ll, for each (p, w), there is a unique z_ for which p'i7f(z) - w = 0 and the mapping from (p. wl to z is continuously differentiable. saying tha: z( ·) is a continuously differentiable function. function
theor~:n
also tells us that
az(p w)/op • •
2
= (- 1/p}D f(z(p,w))
-1
5-12
'i7f(z{p,w)),
This is equivalent to The imol icit •
2
Vwz(p,w) = (1/p)D f[z(p,w))
-1
.
2
Note here that, since D t(z(p, w)) is negative definite, so is its in..;erse
o
2
t(z(p, w))-l bJ Theorem M.D.l(iii).
(a) By the chain rule (Section M.A), d dp {f(z(p,w)}] = Vf(z(p,w}~
az
(p,w) 2
1
= (- l/pJVf(z(p,w))·D t£z(p,w)f V't{z{p,w))
2 -1 Since D f(z{p,w)) is negative definite, d[/(z(p,w)}]/dp > 0.
(b) Since
8f{zVaz,_ >=
0 for all
t
and d[/(z(p, w))]/dp > 0, as the output price
increases, the demand for some input must increase. 2 -1 (c) Since V' w2ip,w) = (1/p)D f(z(p, w)) , 8z(p, w)/8pl is equal to the lth
2
1 diagonal entry of (1/plD t£z(p,w))- .
Since
definite, the diagonal entry is negative.
5.C.7
[Fi:-st
2
o
t(z(p,w))-l is negative
Hence so is Bz(p, w)lopl.
er-rata: That is, all inputs are complementary to one anothe:-.1
As
we saw in the answer to Exercise S.C.6,
a·z
2
Bw (p, w) = (1/p)(D f(z(p, w})]
-1
and liz
-=-
2
(1/p)[D f(z(p,w))J
-1
Vf(z(p,w)).
Hence, in order to prove that Bzip,w)/Bwk < 0 for all k
azl
ap (p,w) > 2 _, 0 for all l, it is sufficient to show that all entries of [D /(z(p,w))] • are negative.
:;e
land
1nis is an immediate consequence of the celebrated Hawkins-Sim-::m
condition, which
be found, for example, in "Convex Structu:-es and Economic
Theory" by Hukukane Nikaido.
Here, instead of simplemindedly quoting the
Hawkins-Simon condition, we shall provide a direct proof tha:
reli~s
on the
2 symmetry of o t£zfp,w)) (which is not assumed in the Hawkins-Simon condition).
5-13
2 Write H = D tCz(p,w)).
To show that all entries of H-l are negative, it
is sufficient ta prove that, for every v e RL-l. if Hv
«
i!:
In fact. then, for each l. we can choose v e IRL-
0.
vector whose l:th
txM dinate
0 and Hv 1
;t
O, then v
so that Hv is the
is equal to one and the other coordinates are
-1
-1
equal to zero. then H (Hv) is equal to the lth. column of H .
Of course, it
•
is also equal to v, which is claimed to be strictly negative. I column of H- is strictly negative.
Tnus every •
-1 • Hence all entries of H are negat1ve.
For this prupez ty, in turn, it is sufficient to prove that for every v e IRL-l. if Hv ~ 0. then v ~ 0. if there cYists a v e IR some l.
Then '
;t.
inequalities suffice.}
such that Hv
::!::
0, Hv * 0, v
0, and "l = 0 for
:5
~
0.
We shall now prove by contradiction that for every v e IR then v
l.
~
0.
In fact,
0 and hence
Hv
which
L-1
(That is.
Then there exists v e IR
L-1
such that Hv
i!:
L-1
, if Hv
i!:
0,
0, anc vl > 0 for some
By re-ordering the inputs if necessary, we can assume that the fir-st M
entries of v au: positive and the last L - 1 - M entries are nonpositive Write write v =
write H =
HI
whose entries
0, x
2
e IR
L-1-M
,
0. Also,
, where H is an M x M matrix, H is M x (L - 1 - M} matrix 2 1 all positive, H is an (L - 1 - M) x M matrix whose entries 3
are all positi"R. and H is an (L - 1 - Ml x (L - 1 - M) mat:-ix. 4 0, y
2
e IR
L-1-M
=
,
H1x1 + H2x2
H x + H x 4 2 3 1
5-14
Then
-
y
1 •
Let Hv =
positive. liDs. by x » 0, • 1
x s 0, which is a contradiction
1 1
•
definiteness of H. aud heDce that of H, .
to the
~C.B
~ ·H
•
mat
12'G:St
in .uat1l 95' is 2·55 + 2·40 = 190, but it
AI
could
output
with a lower cost by using the input
-
•.
comb
"' '
is that.
due to
aodlor some restrictions that AI faced
the
plans are not observed to
oatside 12
~
8llf ;md it is
ba.ve
Thus the problem we wili encounter
•
recover i1s
those observations in order to 5.C.2(iii) or S.C.l(iii) .
based oa
S.C.9. To lid 1d ·} and y( ·) for (a}. the fil:st-order condition (5.C.2) is •
DOt
rmd
very •u1.
constraint binds.
a.use one d the
l
Tt(.) . . y(.)
•
for (b), it is not even applicable because f(. )
1s
Also, to not
In both cases, bowe91r, btcanse of the nature of the
diff
productia.la::tions, it is quia easJ to salve their CMP (which is similar to those in
S.C.lO. ), and the c:ast fmv:tions c( · ) turn out t::> be
c!ff er-entiliR witb respect to
levels q.
a-der c01llli'ilrll5.C.6} (which
' aoly the differentiabilit::
.~:· ~he
cost
· to output levels} to f"lod profit maximizing p;:-oduction
function levels,
We can thus apply the first-
= the
and sapply correspondences.
profit
the answer, til! output priee is fixed to be equal to one. (;.} rr( w)
V4'11 if w s 1 1 = V4-w if w > 2 1
2
wi wz
l-l/4wl, 0, 112w
J(w}
-
~--~·
» 1
- z2' l/2W l): ,
i!.
0, ~
5-IS
2:
2 0, zl + z = l/4w } 2 1
·r
1·
w
<w· 2' 1
•r l!
w
1
w2;
(b) n(w) = 1/4(wl +
= (- l/4(wl
y(w)
wz>2
+ wz) •
(c) Note first that this production
• x:bibits constant r
Moreover, if p < 1, then the
scale.
CUilSti
If p = l, then this production
to
aint does not bind.
rise to the satne ·
that of (a}, and hence one of the DODDegativity cousu·aints binds.
as It is thus
easy to apply (5.C.6).
If p < 1, then
Tr(W)
ID
·r 1
0
if
o/(p-1)
w-1
+
=
o/(p-1) < 1·
W"z
•
) :!:: 1.
p-1) +
1
'f
0
=
y(w)
1
V(p-1) ( w { a, 1
p/(p-1) +
w1
·r p/(p-1) 1 w 1
+
o/(p-1)
.w-2
< 1
o/(p-I} _
w2
- 1;
{0)
If p = l, then 1l(w) =
0 if Min{wl'w } 2 (I) if Min{w ,w }
2
1
=
1
<1 . , '''" {w 1 . '"' .: 1•
0
if Min{w ,w } < 1; 1 2
I. 0, 1): a
ii!::
0}
if 1 = w l
•
S.C.lO qwl
if wl ~ wz;
qw
if w
= 2
1
< wz;
if w > w = 1. 2 1
{o:(O, -1, 1): IX~ 0}
(a} c(w,q)
w2 } -> 1·,
{0)
{a:(-
y(w}
:!::.
>w . 2
5-16
•
Cq, 0) z(w,q)
2
=
e IR :
if w
+
= w : 1 2
if WI z{w,q) = (q,q).
(b) c(w,q) = (wi + w )q. 2
.PI(p-1) _ ( (c ) c (..,q ) - q y; 1 +
z ( W,Q )
S.C.Il
a
(
D/(p-1)
Q W"I
+
> w2.
p-1) (1-I/p)
1
2
.
o/(p-1))(-Vp)( 1/(p-1)
91'2
Wl
•
ll{p-1))
,W2
•
.ue that c( ·) is t1rice continuously diff'a Ultiable.
By Proposition
5. C.2(vi). %{ ·} is continuously differentiable and
= (Btaq)(8c(w,q)/Bwl) = (a/awl)(8c(w,q)/Bq).
Bzl(w,ql/Bq
Hence Bzl(w,q)/Bq > 0 if and only if (8/Bwll(BC(w,q)/Bq) > 0, that is, marginal rost is increasing in w t:
S.C.l2 Suppose first that y
= 0.
E
E
"J
at p, then {p,n(p))· (y, - 1)
y' e Y, (p,n(p))·(fdy', -Ill = a(p·y' - tr{p))
Also, for every a. c: 0
Converseiy, if y
Y n1aximizes profit
and {y, .- 1) maximizes profit in Y' at (p,pL+l ), •
then (p.Il.+l)·(y,- 1) = p·y -1\.+l = 0 by the coostant returns to scale. Also, for every y' e Y, (p, pL·t"" [y:, - 1) = p · y' - 1\.+l
!S
0.
Hence y
•
m
profit in Y at
p and Jr(p)
= p· y = 'l.+r
S.C.l3 Denote the production fum. lion of the firm by f( · ), then its
optimization problem is
Tnis is analogous to the utility maximization problem i."l Section 3.0 and the function R( ·) corresponds to de indirect utility fmw:tion.
5-17
Hence,
analogously to lo}'s identity (Proposition l.G.4), the input demands are obtained as 1
S.D.l
We shall \Ee the differentiabilitJ of Cl..·l aniy at
differentiabilitJ is not necessary.
AC" (q)
The everywhert
By the M"Ulition,
1:
at q = q,
Thus, if the awnge cost is
C' (q) = C(q)/q a
q.
then
AC'(q) =
0 and hence
ICtq).
5.0.2
.•
·"
....
•• ••
••••
·-··
___ .. ....
•••••
•
••
• ••'
• • • -----------~ •• •• •• • •• •• • • • • • • • • • • •• •• •• • •• • • •• • •• ••
q
C'(q)
A
A
Figure 5.D.2
•
•
5.0.3
Let q(w,pl bt the profit-maximizing
are w and the initial output
price is p.
pri£z. z be •
wbe tie initial
input prices,
the initial long-run ilp"' demand at
output level at
be the initial p for every p (
Let
level when the input pric(
(w,p).
that q("w,p) > 0).
(w,p).
-p
be the
and
-q
By (5.C.6), Bc(w,q(w,p})/8q =
Thm. 'by differentiating both si• •
5-18
wilh
p,
:: to P 3lll tlaaaluating at p =
we obtain
2
(l•.q(w,iin;aq Haqcw.p)lap> -1
) On the other
t.t.
= 1.
.
the short-run cost function function
qs(w,q!z
aut the .
s ._did -a.bo"'e, we
1
in the hint.
) as 1
- - -
2 - - -
aqs{w,p I z1 )/8p ={aciw,q I z 1)/8
•
c s { w, q Iz ) 2
Just
q)
-1
•
cfw,ql
Now, by the
--c {w,qlz ).
Hence te flldan g(q) =
1
5
-q 3llld,
by the
5
c(w,q)
1
-
cs(w ~)
'a .
=2
cost.
GltJ. J
J
t
and c:idilm
g"(q)
:s 0,
costs (and C(O} . IC(q .).
J=
J
multiple firms aDI
=Yiliin{q ,q }, ·ther: C(·) exhibits dec:-easing ray 1
~
But le!-, =Ul,
2
= (8,1}, and q = q + q = (9,9). 2 1
Hwce C(q} > C(q l + C('lz)· 1
Hence
C{ ·
Then
l is
1 additive.
(cl We shall first prae • • q
r,ctqJ. In fact, J
M.Kl,
over j, we obtain C(q) :s
C(<Jt = C(Ciz) = 1 andf:tqJ=l. not
is maximized at q =
By the decreasing
q .•
is no w;r ta hit up the production of q
(b) Let M
=
1
S:D.4 (a) Suppose tilt (q ./q)C(q) :s J
c(w,q)
) for all q and
condition (see
2 that is. acl<w.q)la ,
-
:s c
(w,q I z
l
=
.q . and c . » 0 far everv j, then C( q l ~ L.J J 'J • 't"
f«.ry j, there exists 1. > 0 such that 7.q. » J J J
th~
> ctq).
By lie •
ctal. •J
•
eXIstS Cl ...
J
theu E.lltk. = I and :J J ~
C(q).
By lie:
Tnus, by the
and C(q) >
SEh that C(ajqj) = C(q}. 'Define f3 =
5-19
~
L}laj,
Thus, by the quasi::onvexity.
.} = {1/fl)q. J.
ss, f3
q.
1.
Hence, by the decreasing ray
average cost. I:.C(q.) ~ I:.U/o:JC(o:.q.) = l:.U/a.)C{q} = SC(q) ~ C(c). J J :.J J JJ J J .
For the general
in which some q. J
-applJ the above result to the q. J
and then
+ £e,
where
~
0
?!
not be strictly positive,
> 0 and e = 0,!, ... ,1) e IRM,
£
The continuity of C( ·} then implies that
the limit as c -+ 0.
•
:j=lC(qj)
S.D.S
i!!
C(q).
(a) The prcdTCtim function f( ·) exhibits
only if f(Az}
i!!
Af(z) for all z and all 1 i!! 1.
returns if and
Helice, lf z'
?!
z > 0, then
(1/z')f(z') = (1/z')/((z'/z)z) 2: U/z'Kz'/z)f(z) = (1/z)/(z).
Thus the average product is nondecreasing.
The ma!'ginal product may however
be decreasing on some region of output levels, as the following example shows:
q
.r··
••
• • ••
.•
•••••
• • ••
•
••
••
•••
••
••••
..... ··· •
•
0
(b)
SD.S(a)
Mathctaatically, the t:nmmmer's
ion p-ablem is
ma.xz!:O u(f(z1J -
The first-order
rewritten as u'(f(z)) 1
z
z.
condition is u'(f(z))f'(zl = 1, which ca::1 be
1 = rczl- . Since the
cost
1
is given by z =
f- (q), the marginal cost is equal to f'(z}- and the equality of marginal
5-20
cost and marginal utility is thus a necessary condition for a maximurr... Economically, the consumer will choose the output
l~vc!
at which the
marginal utility of an extra unit of the output is exactly equal to the disutility incurred by giving up the necessary amounts of input to produce it. But •
IS
~
lcnter is nothing but the marginal cost.
Hence the marginal utility
to the marginal cost.
(c) "Ibis assertion is wrong.
As the following figure shows. even if marginal
cost ami ma.-ginal utility are equal at an input level.
may be another
input level at which the consumer attains higher utility:
l..z)
A. q
•
Curves
0
The
z
SD.S(c)
for suboptimality is that the first-order necessary conditions
are DOt S".Uficie:-:.t when the production function exhibits increasing ret
(which gives rise to nonconvexity of the feasible set).
S.E.l By applying Hotelling's lemma (Proposition S.C.l(vi)) twice, we get y•(p} = tTn•(p}
= V([ .11" .(p)) J J
=
E.tTfl. .(p} = LY .(p). J
J
I J
S.E.2 "This is just a matter of going through the proof of Proposition S.E.l
5-21
and checking that convexity was never used.
Its interpretation was given
before the stat~ment of the proposition (p. 148).
It is a consequence of the
very definition of the aggregate production set, that is, it is the sum of the
J firms' production sets.
It is thus independent of convexity o:- any other
properties of the firms' production sets.
[First printing errata: We should assume that there is a p• »
S.E ..3
l!:
r.v .. J J
Otherwise, denoting the
L
LJ.Y.J
profit function of
p•. y for every y e
by n•( · ), the set {y e IR : p·y
~
n•(p) fo:r all p »
may be empty, and all we can obtain is the equality between p · y :s n•(p) for all p
l!:
0).
This assumption is also
validity of Proposition S.C.l(iii).
L/
necess~ry
0}
L
j and {y e IR :
for the
In fact, its proof should· go as follows:
It is sufficient to prove that, for every z e that p · z > n(p).
0 and y•
IR~Y.
there exist a p
»
0 such
Since Y is closed and convex, the separating hyperplane
theorem implies the existence of such a nonzero vector p. property implies that p must actually be nonnegative.
The free disposal
If it is not strictly
positive, then t";.ce the convex combination (1 - c}p + cp• with a sufficiently small
£
> 0.
1 ne:-:
it is strictly positive, and satisfies ( (1 - c l:=
+ ~p· J • z
>
nUl - c )p + £p*) by the upper semi continuity of n( • }, which is implied by its
conveX:itv• .l
Since each Y. is convex and satisfies the free disposal prope!'ty, J
[.Y. is also convex and satisfies the free disposal property. J J
Since it is
also assumed to be closed, Proposition S.C.l(iii) implies that L [jYj = {y e IR: p·y :s n•(p) for all p »
But here, by
Pr~p~sition
r J.Y.J
0}.
S.E.l, n•(p) = Ljn j(p) and hence
= {v e IRL: p· y :s -
LJ:nJ.(p)
for all p » 0}.
of the technoiogy with
S.E.4
5-22
istics z = (~;z_Ja
{(- 'r-~ y (w) 2
l)}
((-., -itr
=
ct}: ct
e (O,I])
if
•fi + wiz.2
if Wft +
{0}
rcr which
area of the
w-r2
= 1,
>
1.
the output .., be one is depicted
in the following pict'IJI'e.
- - - - - - - (10,
IIW1
• 0
)/WI
5.E.4
"More gener ally• 5ladd. be deleted. J
(h) [First printing
with
the profit of the by -.: (1(118
z
n (w)= z
nws.
•n --rz.
1 -
if w z + 1 1
0
the aggregate (or,
the integral of 1 - w \ 1
s
1}, which is depicted rr(w)
=
l/w 0
if w lzl + w2z2 ~ 1,
wr-2 >L
average) profit is
-n_•* area
Cll ~
rlgUre.
1
{z
E
(O.lOh
Thus the
by taking
wlzl + w2z2
profit is
=l/600w1w 2 .
I
5-23
(c) The agg1 egate input demand can also be obtained by integr-ating the input demands of the individual firms, but the following point is noteworthy: If a firm has characteristic z
= £z 1,z2 l
with w z + w z 11 2 2
input demands at input prices w = (w .w ). 1 2
= 1,
then it has multiple
But those firms constitute only a
negligible pmtion in the whole populaU.. Hence it is harmless to assume that such a firm has input demand z =
(~·~)
(in absolute values).
Hence the
aggregate demands are 11w 1
0
1
0
/w
1
2"
600w w 1 2
0
It is easy to check that these aggregate input demand functions can also be obtained by applying Hotelling's lemma to the aggregate profit function, which was obtained in (b).
(d) We need to find an aggregate production function whose input demand
function is the same as the aggregate input demand function in (c).
Denote
such a production function by f( · ), then the first order-conditions for profit maximization a1 e of(zrz )1azr = w. and Bf(zl'z l1Bz = w . 2 2 2 1 2
These
expressions, when evaluated at the inputs demanded, must hold for all w. 1
--
L "".
2
z = l/600w w and 1 2 1
Zz •
then
Thus
•
5-24
2 = l/600w w , 1 2
I
Thus
Therefore, f(z .z l 1 2
a:
3Cz z /600) 1 2
1/3
.
The aggregate production function is a
Cobb-Douglas one exhibiting decreasing returns to scale.
(a) Plant j"s marginal cost is MC .{q.) = « + ~ .q .. J J J J
S.E.S
Since f3. > 0 for J
every j, the first-order necessary and sufficient. conditions for cost minimization are that
L .q. J J
q and MC .{q.) = MC ., (q ., ) for all j and j". J J J J
a::
From
these, we obtain qj = (q/{3 j}/(l:hl/,Sh).
(b) (c) In both cases, it is cost-mininlizing to concentrate on plants with the •
smallest
.S.
J
< 0, because the average cost is decreasing at the highest rate at
such plants.
S.F.l
The production plan y in Figure S.F.Hbl is not efficient but it
maximizes profit for p = (0,1).
Throughout the answer, we fix the price of the input at one and denote
S.G.l
Suppose that there are I consumer
the price of the output by p.
Denote their shares by 9. > 0. .. , • 1
indexed by i = 1, ... ,I.
Of course,
r.e. = 1. 1 1
Since they have quasilinear utility functions, by Exercise 3.D.4(b), their indirect utilitv functions can be written as v.(p,w.) = w. + ¢.{pl. -
1
1
1
1
Note that
the demand fun:tior:. x.( ·) for the output of consumer i does not depend on the 1
weal~h
and satisfies xi(p) = - ¢i(p) by Roy's identity.
(a) Wher. the input is z, the utility level of consumer i is e.(p(z)f(z) - zJ + f/>.(p(z)). 1
1
(Her-e we are ass-
But this does not affect our results below, because
demancs fo: the output does not depend on their weal~h levels. Thus, if
5-25
a."l input level z maximizes his utility level,
it satisfies the following
first-order condition: ei(p"(z)f(z) + p(z)r(z) - 1)
+
+iCp(zJ)p' (z) =
If an input level z is unanimously agreed, then must be satisfied at some z for all i.
By
x. (p(z)) = -
condition over i and (p'lz)f(z)
+
·pcz>rCzl - 11 -
But, since f(z) = lri(p(z)), this implies
first-order condition the summation of the
cf>~(p(z)). 1
1
o.
we obtain
~I(p(z))p'(zl
p(z)r~
- 1 = 0.
= o. Plugging this
into the first-order condition, we obtain e.p'(z)f(z) - x.(p(z)Jp"lz) = 0. l
1
Thus B. = x.(p(z))/f(z). 1
1
(b) We know from (a) that, if ownership share are identical, then, in order
for consumer-owners to unanimously agree on a production plan, it is necc ssary tha! they all cor.smne the same amount of the l:mput.
But if their tastes are
different for the output, then their consumption levels will be different. Hence they will instruct managers to carry out different output levels. (c) If p:-efer-ences and ownership
r~>hares
are
, then the first-order
conditions are also identical and hence the cons.aer-owners unanimously agree
unanimous agreement is that p(z) = Vf'(z).
The right-hand side is the
inverse of the marginal return, and hence equal to the marginaL nothing p( zl
bu~
profit maximization with respect to iaput., when the output
is taken as giveo.
S.AA.l
This is
Fr-::~m
the unit isoquant,
5-26
pri~
w . < w2, 1 ·r - w2, I• w
(2,1) •
-
z(w,ll
'#"
1&
{A(2,1) + U - A)(l,2)
IR: i\
E
E
[O,ll}
1
( 1,2)
if w
1
> ....... 2.
Thus 2w + w 2 1 w + 2w 1 2
c(w,l) =
if w
:.o;
w ,
1 2 if w > w . 1 2
This is differentiable at w = lwrw ) if and only if w ':f:. w . 2 1 2
Moreover-,
•
(a) We shall first prove that if t3 e IRL-l, (I - Alf3
S.AA.2 ':f:.
0, then b•f3 > 0, and that if {J
RL-l and (I -
E
A)~ =
0, and (I - A}{3
i!::
0, then b·t3 = 0.
In
fact, in the proof of Proposition S.AA.l, we showed that since A is
1
the inverse matrix U - Af exists and ail its entries are
productive, •
Thus, if {3 e
nonnegative.
(I - A)-l({l - A)(3} i!:: 0
e IR
L-1
1
~- • U -
and f3 • 0.
A)f3
l!:
Since b
0, and
»
(I -
A)f3
i!::
(I - A)o:' and (I - A)o:
AHa - o:') :;: 0.
A}o:' • tnen (I - A)(o: - a'} = 0.
anv• case, it is i:::1:>ossible that • •
I - A -
I - A -
•
0
':f:.
L-1 +
a •.
and a:'
E
If (I - Ala:=
Hence b • (o: - cr.') = 0, or r· - e:· = b·a:'. •
I - A - b
0: i!::
ln
I - A , d _ b o: an
Efficiency is thus established.
(b) By (a) and Proposition S.F.2, any production plan with o:
»
0 is profit-
L-1 be To establish its uniqueness, let p E IR
maximizing at some price vector.
+
•
a suppor-ting p&ice vector.
L-1 IR . +
(l - A)o:', then (I- A}(o: - o:') i!:: 0 and
Hence b·(c- o:') > 0, or b·o: > b·«'.
•
•0
If
and (I - Alt3 = 0, then f3 = 0 and hence b·f3 = 0.
If (I - A)o:
(I -
t3 =
0, then
0, this implies b·(3 > 0.
To derive efficiency from the above result, let a E IR
(I -
;t
By a:
»
0 and the zero-profit condition for
activities beir.g actcally used, we must have p· (I - Al = b, that is, p = ((I - Al-llTb (where b is now a column vector).
This implies the uniqueness
•
and the strict positivity of p,
all entries of (I - A)
5-27
-1
are
nonnegative, all its diagonal entries of are positive, and b » 0 .
•
•
(c) For each l, denote by et the vector in IR
the other components are zero.
L-1
whose lth camoonent ts one and • •
As we saw in the rema!'k following Proposition
5.AA.1, the total amounts of the producible goods necessary to realize a net output vector el e IR
L-1 +
is equal to (
Hence the total •
amount of labor embodied in these necessary amounts of the producible goods n
•
equals b· b ·(I
- A)
A )el = b ·(I - A) -1
-1
et = Pt·
=
Thus the price (row) vector p
can be interpreted as the amounts of primary factor directly or
indirectly embodied in the production of one unit of each producible good.
I - A' - b'
(d) Let
productive. (! - A')
-1
Lx[L-1) be 1 . h ' f .. ' e IR any a ternat1ve c otce o actiVities that is
, -. By the productivity, the inverse matrices (I - A) • and So denote them by C = [c
exis-:. and are nonnegative.
1
... cL_
1 and 1
C' = [c~ ... cL-l ), where the cl and ci (l = 1, ... , L - 1) are (L - ll-
dimensional column vectors.
Then
(I -
A)cl
= el
and (I - A' lc£
for each c > 0, define dl(c) = cl + c([k~lck) and dl(c) Then die) -+ c< and dt(c) -+
cl
as
£
= cl
+
= et
Now,
dl:k;!:lck).
Moreover, dl(c) >> 0, dt(£) »
-+ 0.
0,
and
Bv (a''• ~
1- A
-
By this strict positivity and the
- b
assumptior. tha: the activities
I - A - b
have been singled out by the
nonsubstitution theo&em, we must have b · dt'c) :s b' · dt(c).
Taking the limit as
assertion now fellows from (c).
5.AA.3
(a) Denote the activity levels by o: and o: . 1 2
for labor is a
1
+
2a:
= 10, or a 110 1 2
+
a. 15 = L 2
5-28
The resource constraint
Since the production level
10 •
is given by
-s
- 5
• the production possibility frontier is as
5
follows:
Y1 5 •
10 •• i y1 •• ••• ••
Figure S.AA.3(a)
(b) By Exercise S.AA.2(c), the equilibrium price vector is b ·(I - A)
-1
-
(4,6).
(c) The amount of labor embodied in each commodity equals its price, as shown in Exercise S.AA.2(c).. (d} The locus of amounts of good 1 aM labor necessary to produce one uni't of good 2 is equal to 2
{A(1,2) + (1 - i\)(l/2,fJ): i\ e {0,1]} - IR , +
assuming free disposal.
It is represented in the following figure:
5-29
-----·· ~
•••
------·!·-···· • ••• ••
•• •• •• •• • ••
---····----·--· •• •• •• •• •• •• •• •• •• ••
• ••
•• •• •
0
••• •• •• •• •• •• •• •• •• ••
••• ••• •• •
••• ••
1/2
1
••
0
13>2
s
(e) In the context of (d), the nonsubstitution theorem says that it is possible to choose one of the two techniques to produce good 2 (or a combination of the two with a fixed proportion) in such a way that any efficient production plan with positive net outputs of the two producible goods can be attained by using the technique chosen for good 2. We could determine which of the two techniques (or their mixtures) is efficient by actually plotting the frontier of the feasible output combinations from one unit of labor. 'In the following, however, we shall identify an effi::ie:1t technique based on Exercise S.AA.2(d). 0
For each i\ e {0,1], define A{i\) = (1 - i\)A + i\A,
112 b(i\) =
So let A' =
(1 -
1 A.)b + i\b', and p(i\) = b(i\) · (l - A(i\))- (where p(i\) is a row •
vector).
According tc Exercise S.AA.2(d), if i\• e [0,1) a.-'ld the convex
combination of the fir-st and the second technique with weight 1 - i\ • and i\ • is efficient, then p(i\)
~
p(i\ •) for every i\ e [0,1).
Hence the switch of
efficient techniques occurs precisely when the value of i\ • switches as varies.
We shall now find a value of
~
at which the value of i\• switches.
Just as ir.. (b), we can calculate
5-30
2
p(i\} =
2
+
((3 - 2)i\ + 4 (2(j - S)i\ + 6
213 - 8
and Dp(i\) =
1•
2
(i\ + 2)2
•
Hence: if (3 < 4, then A• = 0; if 13 > 4, then .A• = I: and if f3 = 4, then .A• can Thus the switching occurs at (3 = 4.
be any value in [0,1}.
More precisely:
if (3 < 4, then it is efficient to continue using the first technique: if (3 > 4, then it is efficient to switch to the second technique; and if 13 = 4, every •
mixtu•e of the two technique is efficient .
•
But y and y are not. 5 1
these three vectors are in the production set. this, suppose that y By a
2
1
::!'
L.o: .a.
Suppose next that Ys :s By a
(b) If p = (1,3,3,2), then p · a .
J
profit at p.
3
= 0.
LJ.o:J.a.J
But there is no cx
with a. := 0. J
= 0, according to ·good 4, a
4
According to good 2, o: = a. = 0. 1 2
with ex. := 0. J J J J
= 0, according to good 3, a
:5
To see
= 0.
3
4
!:
0 for which y
1
:5
According to good 1, a · • 1 But there is no
0 for all j and p · y = 0.
ct
2
:= 0 for
Hence y maximizes
By Proposition S.F.l, y is efficient.
(c) Since y = a , y is feasible.
But, since a
feasible, y ca..·mot be efficient.
(Note that a
1
•
2
+ a
2
3
+ a
3
+ a +
4
a
4
= (2, - 1, 0, 0} is represents a
round-about production of good 1 out of good 2.)
S.AA.S
[First p:-!nting
=·· The last elementary activity as ::
(- 2, - 4, 5, 2) should be a 3 = (- 2 , - 4, 5, - 2).]
(a) The set Y is defined as
y =
La. a e ... m m
o: a
m m
4 e IR : «
m
Y, and y' = }"1n o:'ma m e Y. Ay + (l - i\)y' =
!:
0 for each m}.
Then
Lm(i\Clrn +
5-31
(1 - i\)o:' )a
.
m m
Let A e [O,ll,
Since ~cr. m + (1 - ~)o:m' ~ 0 for every m, ~Y + (1 - A.)y' e Y.
Thus y is convex.
(b) Since all the activities use commodities 1 and 2 as an input, in order- to
produce any commodity in positive quantity, it is necessary to use commodities 1 and 2 as an inputs.
The no-free-lunch property thus follows .
•
(c) Note that it is impossible to dispose of one of commodities 3 and 4 without increasing the output· of the other, and that it is impossible to dispose of any one of commodities 1 and 2 without disposing the other.
Hence
Y does not satisfy the free-disposal property and it is necessary to add the four disposal activities to the given elementary activities in orde!"' for the free-disposal property to be satisfied.
(d) Note that 3a 2a , and a 6
7
:;e
1
~
2a . 6
a , 3a 5 1
:;e
Hence a , a , a 5
(e) We can che:k that (4/3)a + 3
a
3
(f)
+ (1/2)a
1
~
a , and a 2
3
~
a , a 5 2 4
8
a , a 4 2
and a
(5/24)~ ~
+ (1/2)~
* a . 2
6
:;e
a , 2a 3 4
~
a , 2a 3 8
*
a , 8
~ ~
are not efficient.
a , (4/3la + (5/24la_, :;: a , 1 3 1 Hence a
1
and a
2
are not efficient.
We shall pr-o':le that the set of the efficient production vectors is equal
production vector- belongs to this set.
Conversely, we can check that the
production vectcr-s in this set cannot be dominated by each othe:-.
Hence they
are all efficient and the set of the efficient production vectors is equal to cr._ ~ 0, 0:7 == 0} . .j
{g}
Since cr. a 3 3
+ cx._a..., I I
= (- cr.
3
problem of max:wizing the net production of the third commodity is as follows:
5-32
s.t.
::
~so.
5~ ~
300,
10~ ::
o.
a~
0,
a..,
~
0.
(h) The feasible set is shaded in the
CX7
(o:
3
.a_,l-space below.
2w + Sa., = 300
60
CXl
12 0
120 ISO
+ 8a.7 = 480 480
Figure S.AA.S(h)
The solut.ior. to this problem satisfies 2« + 3
5-33
SU.,
c=
300, o:
3
=
10~.
Thus
CHAPTER 6
6.8.1
Suppose fir-st that L >- L'.
A first application of the independence
axiom (in the "only-if" dil'ection in Definition 6.8.4) yields cd. + (1 - a)L" ..., >- aL' + (1 - o:)L" .
rr these two compound lotteries were indifferent, then a second application of the independence
(in the "if" direction) would
yield L' >- L, which contradicts L >- L'.
We must thus have
-
cd. + (1 - a)L" >- cd." + (1 - a)L".
>- aL' +
Suppose conversely that aL +
(1 - «)L"
independence axiom, L >- L'.
If these two simple lotteries were indifferent,
-
(1 -
a)L", then, by the
then the independence axiom would imply o:L' a contradiction.
+ (1 -
a)L" >- aL
-
+ (1 -
alL",
We must thus have L >- L'.
Suppose next that L - L', then L >- L' and L' >- L.
-
-
Hence by applying the
independence axiom twice (in the "only if" direction), we obtain aL +
(1 -
cdL" - aL' + (1 - o:)L" . •
Conversely, we can show that·if o:l.. + •
(1 - a)L" - cxL' + (1 - Gt!L"",
then L - L' .
For the last part of the exercise, suppose that L >- L' and L" >- L'", then, by the independence axiom and the first assertion of this exercise, aL
+ (1 -
o:}L" >- «L' + (1 - a)L"
and «L' + (1 - a)L" >- cxL' + (1 - o:)L"'.
Thus, by the transitivity of >- (Proposition l.B.l(i)), aL + (1 - o:)L" >- «L' +
6-1
(1 -
o:)L "'.
Assume that the preference relation
6.B.2
•
~
-
is represented by an v.N-M
expected utility function U(L) = Lounpn for every L = (p , ... ,pN) E !f.. 1
e !f., and Tnen L ~ L' if and only if \' u p ~ \' u p'. "nnn '1lnn cl() u p ) + (l - ex)() u p")
'Tlnn
'Tlnn
~
e
(0,1}.
This inequality is equivalent to
o:() u p') + "nnn
Hence L ~ L' if and only if cxL + (1 - a:)L'' ~ o:L' +
-
a.) (\' u p"}.
(1 -
(1 -
~nn
~
This latter inequality holds if and only if cd. + (1 - a)L"
-
et
Let L
-
o:L' + C1 - a.)L".
o:)L".
Thus the
independence axiom holds.
6.B.J
Since the set C of outcomes is finite, there are best and worst
outcomes in C.
Let
-L
be the lottery that yields a particular best outcome
with probability one and L be the lottery that yields a particular worst
-
We shall now prove that
outcome with probability one.
-L
~
-
L
~
L for every L e
- -
!f. by applying the following lemma:
Lem..:na: Let L ,L , ... , 0
. b'l' . prooa 1 1t1es
1
~
be (1 + K) lotteries and (o:l' ... ,~)
. h ~l( 1 Wit~ L..k=l~ = .
If
~
~ 0
be
t L0 for every k, then •
•
Pr-oof of Lemma: We shall 'prove th1s lemma by induction on K.
If K = 1, there
•
is nothing to prove. l.
So let K > 1 and -suppose that the lemma is true for K -
Assume tha: Lk ?: L
0
for every k.
By the definition of a compound lottery.
-1 ~ et. L = (1 - o:.,) k=l KK .IIi. k=l 1 - IX
By the induction hypothesis,
-1
a:k
K
+ a L k K K'
Hence, as our first •
application of the independence axiom, we obtain o:k a.K) k=l-=----1
(! -
L
Applying the axiom once again, we obtain
6-2
k=la.kLk .t L . 0
Hence, by the transitivity, verified.
The case of L
The first statement is thus
.t Lk can similarly be verified.
0
Now, for each n, let L n be the lottery that yields outcome n with Then L
probability one. outcomes.
Let
{First
-
L
because both of
can be identified with sure n L=}oL.
L
the above lemma.
6.B.4
~
n
"'n·n
Thus,
The same argument can be used to prove that L
:prin~~pg
~
-L
~
-
L
L.
--
errata: On the the 11th and the 12th line of the
exercise, the phrase "the lottery of B with probability q and D with •
•
probability 1 - q" should be "the lottery of A with "probability q and D with probability 1 - q".
Also, in the description of Criterion 2, the phrase "an
unnecessary evacuation in 57." should be "an unnecessary evacuation in 15'7.".1
(a} We can choose an assign utility levels (uA''1ruc·~) so that uA = 1 and u 0 = 0 as a normalization (Proposition 6.8.2).
uC = q·l +
(1-
Then u
8
= p ·1
+ (1 - p} · 0
= p and
q)·O = q.
{b) The probability distribution under Criterion 1 is
The orobabilitv distribution under Criterion 2 is • •
The expected utility under Criterion 1 is thus 0.891 + 0.099p + 0.009q. expected utility under Criterion 2 is thus 0. 8415 + 0.1485p + 0. 0095q. the age:1cy
wo~ld
The Hence
prefer Criterion 1 if and only if 99 > 99p + q, and it would
prefer Criterion 2 if and only if 99 < 99p + q.
6-3
6.B.S
(a) This follows from Exercise 6.8.1.
(b) The equivalence of the betweenness axiom and straight indifference curves can be established in the same way as in the part of Section 6. B on pp. 175-176 that explains how the independence axiom implies straight indifference •
curves .
(Note that the argument there does not use the fully fledged
independence axiom; as it is concerned with two indifferent lotter-ies, the betweenness axiom suffices.)
'I hose straight lines need not be parallel,
because the betweenness axiom imposes restrictions only on straight indifferent curves and nothing on the relative positions of different indifference lines.
In fact, the argument for Figure 6.B.S(c) is not
applicable to the betweenness axiom. (c) Any preference represented by straight. but not parallel indifference curves satisfies the betweenness axiom but does not satisfy the independence axiom.
Hence the former is weaker than the latter.
(d) Here is an example of a preference relation and its indiffer-ence map that satisfies the betweenness axiom and yields the choice of the Allais paradox.
6-4
(2 500 000 dollars) •
•• ••
•• ••
•
•••• •• •• •• •
••
·-....,
•••
••
•• ••
••
••
•••
•••
••
L'2 ..•
•
••
•• •
•
•• •
L1 (0 dollars)
(500 000 dollars) Figure 6.B.5(d)
6.B.6 U(p) = Max{p·c e IR: c e C} = - Min{p·c e IR: c e - C}. Hence U( ·) is equal to - J.l_c( • ), the support function (Section 3.F) of - C multiplied by - 1, where the domain of the support function is restricted to N
the simplex {p e IR : } p +
is convex.
(A
"'n n
= 1}.
Since any support function is concave, U( ·)
more direct proof is possible, which is essentially the same as
the proof of concavity of support functions in Section 3.F.) As ar. example of a nonlinear Bernoulli utility function, consider A = 8 = {1,2} and d~fine
u U) = u (2) = 1 and u 1
2
then U(L) = Max {p ,p }. 1 2
6.8.7
2
(1) =
u (2) = 0. 1
Let L = (p ,p l,
(This is essentially the same as Example 6.8.4. l
Sin::e the individual prefers L to L' and is indifferent between L and
xL and bet·.vee!:: L' and XL' by Proposition l.B.Hiii), he orefers • the monotcnicity, this this is equivalent to XL > xL,.
6.C.l
1 2
If « :::: D > 0 (complete insurance), then
6-5
Bv •
- q(l - n)u'(w - o:q) + n(l - q)u'(w - D + o:(l - q)} •
= - q(l - n)u'[w - Dq)
+ n(l -
q)u'(w - Dql
= u'(w - DqHnU - q) - q(l - n}) < 0 = u'(w - Dq)(n - q) < 0
by q > n.
Thus the first-order condition is not satisfied at o: = D.
Hence
the individual will not insure completely . •
6.C.2
(a) Let F( ·) be a distribution function, then
Ju(xldF(x} = j((3x
2
+ 7XlclF(x)
= (3(mean of F>
2
+
2
= tJJx dF(x) + 7IxdF(x) tJ(variance of F)
+
7(mean of F).
(b) We prove by contradiction that U( ·) is not compatible with any Be:-noulli
utility function.
So suppose that there Is a Be:-noulli utility function u( · )
such that U(F) = Ju(x)dF(x) for every distribution function F( ·).
Let ·x and y
be two amounts of money, G( ·) be the distribution that puts probability one at x. and H( • l be the distribution that puts probability one at y. u(x)
= U(G)
u(yl = U(H)
Thus, x
:!:::
= (mean of G) - (variance of G)
=x
- 0
= x.
= (mean of H) - (variance of Hl = y - 0 = y.
y if and only if u(xl
2:
u(y).
Hence u( ·) is strict:'·• ;;·onatone.
let F ( ·) be the distribution that puts probability one on 0 and 0 distribution t."'!at puts probability 1/2 on 0 and on 4/r > 0. and the va:-ia..."'l.ce of F ( ·) are zero, U(F 1 = 0. 0 0 u( ·) thus implies that U(F) > 0. 2
vai"iance is 4/r . contradiction. ..... l
Then
F( ·)
1\iow
be the
Since the mean
The strict monotonicity of
However, the mean of F( ·) is 2/r and the
Hence U(F) = 2/r - r(4//) = - 2/r < 0, which is a
Hence U( ·} is not compatible with any Bernoulli utility
.. . tmc .. lon. An example of two lotteries with the property requested in the exercise
was given in the above proof of incompatibility.
6-6
(Note that if all we need to
show were the incompatibility of U( ·) and any Bernoulli utility function, the equality u(x) = x obtained above would be sufficient to complete the proof, because this implies the risk neutrality, which contradicts the fact that the variance of F( • l is subtracted in the definition of U( • ). )
6.C.3
Suppose first that condition (i) ·holds.
Let x
and
E IR
> 0.
E:
Let F( ·)
be the dist:;ibution that puts probability 1/2 on x - c and on x + c, and F ( ·) be the distribution that puts probability 1/2 - n(x,c,u) on x - c and
c
1/2 + n(x,c,u) on x + c.
That is,
F(z) =
0
if
l/2
if
c
(z)
=
1/2 -
X -
£ ::S Z
<
X
+ C,
C,
z.
~
<
X -
C :S Z
<
X + C,
if
n(X, E:,U)
if
1
if X + C
X -
C,
Z
0
x and Ju(z)dF(z)
Then JzdF(z) =
<
if X + C
1
F
X -
Z
~
Z.
::s u(x) = Ju(z)dF (z) bv (i l.
-
c
But
fu{z)dF(z} = (l/2}u(x - c) + (l/2)u(x + c), fu(z)dF c(z) = (1/2 - n(x,c,u))u(x - c) + (1/2 + n(x,c,u))u(x + c).
= (l/2)u(x Since u(x + ~
0.
Thus
E:) -
(i)
c) +
(l/2)u(x + c) + n(x,c,u))(u(x + c) - u(x - c)).
u(x - c} > 0, the above inequality is
De~ine
:.o Tr(x,c,ul
implies (iv).
Suppose conversely that condition (iv) holds. z.
equiva~~:n
Let y e IR, z
E
!R, and y >
x = (y + z)/2 and c = (y - z)/2, then y = x + c, z = x -
£,
u{x) = (1/2 + rr(x,c,u))u(y) + (1/2 - n(x,c,u))u(z)
= (1/2)u(y) Since rr(x,c,u}
~
+
(l/2}u(z) + n(x,c,u)(u(y) - u(z)).
0 and u(yl
~
u(z), this implies
(l/2)u(y) + (1/2)u(z) ::: u(x) = u((l/2)y + (l/2)z). Although we orr.it the proof, this is sufficient for the concavity of u( · ).
6-7
and
Hence (iv) implies (ii).
Since the equivalence of (i), (ii), and (iii) have
already been established, this completes the proof of the equivalence of all four conditions.
• - ( • ex -
•)
cxr····«N
6.C.4 \ ex z "-T.nn
IDN d > • h e "'+' an o: - o:, t en
ex' z for almost every realization
i!! \
returns are non.-'l.egative with probability one. implies that u() ex z ) '"nnn
!:
u(\ ex' z ) with probability one. "nnn
Ju(Lnexnzn)dF(z , ... ,~) 1 that is, U(o:)
i!!
Since u( ·) is increasing, this
i!!
Hence
juCLx,ex~zn)dF(z , ... ,zN)'
1
U(o:' }. N e IR , o:' = +
and i\ e [0,1), then, by
the concavity of u( · ), u(\ (i\.o: "'n
n
+ (1 - i\.)ex' )z ) = u(i\.\ o: z
n n
"nnn
~
+ (1 - i\.)\ o:' z )
i\.u(\ ex z) "n n n
for almost every realization (z •... ,zn). 1
'"nnn
+ (1-
i\.)u(\ a:'z) 'n n n
Hence
U(i\o: + (1 - i\)o:')
= Ju(Ln (i\.o: • n
+ (1 -
i\.)o:' )z )dF(z , ... ,zN) • n -n . 1 •
~
j(i\.u(\ o: z ) + ""r.nn
i\.)u(\ ex'z ))dFCz , ... ,zN) 1 'nnn I'
(1 -
= i\.JuCL.,exnzr.ldF(zl' ... ,zN)
+ (1 -
i\.)Ju(Lnex~zn))dFCz , ... ,z1\l
1
= i\.U(o:) + (1 - i\.)U(o:').
•
be a sequence in IRN convergmg to o: e IRN, then the:-e exists a +
+
positive number B such that o:m :s (B, ... ,B) for every m. .1s f.m1.e. .T
Of course, l'(B, ... ,Bl
But this is equivalent to saying that the (measurable) function z
-+ u(LnBzn) is
in~eg:-able.
Since u( ·) is monotone and all the returns are
nonnegative with probability one, u(\ a.mz ) :s u(\ Bz ) for every rr. and for "nnn 'n n every realization (z , ... ,zN). 1
Moreover, since u( ·) is continuous, u(\ a:mz ) "11 n n
6-8
converges to u(Lnanzn) for almost every realization Cz •... ,zN).
Hence, by
1
Lebesgue's dominated convergence theorem, m
Ju(Lncr.nxn)dFCx , ... ,xN) .. Ju(Lncr.nxn}dF(xl' ... ,xN). 1 That is, U(cr.ml ~ U(o:l.
L
6.C.S
L
(a) Let x e IR , y e IR and i\ e (0,1}. + +
In analogy with expression
(6.C.I) the value i\u(x) + (1 - i\)u(y) can be considered as the expected
utility from the lottery that yields x with probability i\ and y with probability 1 - i\.
On the other hand, the value u(i\x + (1 - i\)y) is the
utility from consuming the mean i\x + (1 - i\)y of the lottery with probability one.
The concavity of u( · } would then imply that consuming the mean bundle of
the L commodities with probability one is at least as gooc as entering into •
the lottery.
But this is the defining property of risk aversion in Definition
6.C.l.
(b) [First
p~ir.ting
errata: The Bernoulli utility function u( ·) for wealth
should be denoted by another symbol, say
-u( · ),
original utility function u( ·) defined on IR:.}
to avoid confusion with the
Let p
» 0 be a fixed price
vector, w and w' be two wealth levels, and i\ e (0,11.
Denote the demand
func:ior- by x:·l and let x = x(p,w} and x' = x(p,w'), then p·(i\x + !5
i\w + (1 - i\)w'.
Thus u(i\x + (1 - i\)x') :s
-u(i\w
+ (1 -
i\)w').
(1-
i\)x'l
If u( ·l is
concave, the:-:. u{i\x + (I - i\lx' l Hence
-u(i\w
+ (1-
i\)w')?!
;!:
i\u(x) + (1 - i\)u(x')
i\u(w)
+ (1-
i\)u(w').
= i\u(w) + Thus u(·)
(I -
i\)u(w' }.
also exhibits risk
•
avers1on. T.'"le following interpretation can be given to this result.
Although, in
the text, we are mainly concerned with the cases where outcomes are monetary •
6-9
amounts, in many cases in economic theory, utilities do not direc:l.Y come from money, but from physical commodities.
It is therefore desirable to derive
risk aversion of Bernoulli utility functions for money from the properties of the underlying utility function for the commodities.
The above result says
that, if an individual has a risk-averse utility function for commodities, the •
his Bernoulli utility functions exhibits risk aversion. (c) We shall give an example with the properties stated in the exe:-cise.
consider the price vector p = (1,2), then, for each w Hence ~(w) =
./W,
:!!::
Let
0, x(p, w} = (w,O).
which is concave and exhibits risk aversion.
this example is that, in order to obtain the risk aversion of
The lesson from
-u( ·)
fa:- a fixed
price vector, all that matters is the risk attitude along the wealth expansion
...
pa~n.
6.C.6
(a) Suppose that condition (ii) is true and let F( ·) be any
distribution function, then
Since 1/J( • }
•
lS
concave, Ju lxldF(x) = Jl/l(u (x))dF(x) 2
~
Thus rp{u £c{F,u ))) 1
2
1
ljJ(Ju (x)dF(x}). 1
~ ~(Ju
1
(x)dF(xl ).
Since rp( ·} is increasing, this implies
that u (c(F,u )l 1 2
~
Ju £x)dF(x). 1
Since Ju (x)dF(x)= u (c(F,u )l, this implies 1 1 1
that u (c(F,u )) 1 2
~
u (c(F,u )). 1 1
Since u ( ·) is increasing, we obtain c(F.u l 2 1
~
dF,u }. 1
Condition (ii!) is thus established.
Converse:y, suppose that (iii) is true. We shall prove that
6-10
Let x e ~. y e iR, and A.
E
[0,1].
Let F( ·) be the distribution function that puts probability A. on x and probabiiity 1- A. on y.
Then, A.u Cx) + (1- A.lu (y) = u (c(F,u ll and hence 1 1 1 1
ljl(i\u (x) + (1 - A.lu (y)) = u (c(F,u )). 1 2 1 1
On the other hand, by the definition,
A.ljl(u (x)) + (1 - i\)ljl(u (y)) = i\u (x) + (1 - i\lu £y) = u £c(F,u )J. 1 1 2 2 2 2 By (iii) and the increasingness of u ( · ), we ot)tain 2 lji(A.u (x) + (1 - i\.)u (y)} 1 1
i!:
A.I/J(u (x)) + (1 - A.)ljl(u 1
(y)) .
1
•
(b) Suppose first that condition (iii) holds.
u Cc£F,u )) 2 2
i!:
u
2
£x).
Hence u £c(F,u J) 1
i!:
1
Thus c[F,u ) 2 u
Cx),
1
(v)
i!:
u
(x}.
2
holds, then Ju CxldF(x) 1
Ju CxldF(x) = u Cc(F,u ll, we have u (c(F,u )) 1 1 1 1 1
i!:
then
-x.
i!:
Thus condition (v) holds.
1
•
i!:
u
::1:
By condition Ciii), c(F,u > 1
or Ju (x}dF(x)
1
Suppose next that
i!:
-x.
If Ju Cx)dF(x) 2
Cx),
i!:
u (c(F,u 1L 1 2
Since
u (c(F,u J) and hence c(F,u J 2
1
1
c(F,u ). 2
6.C.7
Sucoose first that condition (iii) holds. ••
Let x e IR and
~
> 0.
Denote
by F{ ·) the ciis!ribution function that puts probability 1/2 - n(x,c,u l on 2 x -
~
and 112 +
n(x.~.u >
2
~-
on x +
That is,
0
F(z) =
if
1/2 ... n(x,c,u l 2
By (iii), c(F,u ) 1
if i!:
x.
<
X -
£,
if x - E :s z < x + c,
1
Then c(F,u J = x. 2
Z
X + C <: Z.
Thus u (c(F,u }) 1 1
i!:
u (x).
1
Bu:
here, we have u (c(F,u )J
1
1
:;; (1/2 - n{x,c,u J>u Cx - c) + Cl/2 + 1l"(X,E,u lh.: Cx + c) 2 1 2 1 = (l/2}L..::(x - c) + (l/2)u Cx + ~) + n(x,c,u Hu £x + c) - u Cx - Ell 1 1 2 1
and Ldx} •'
6-11
= (l/2lu Cx - £) + (l/2)u Cx + c} + Tr(x,c,u Hu (x + c) - u (x - c)). 1 1 1 1 1
Thus the last inequality is equivalent to n(x,c,u ) 2
~
Tr(x,c,u/
Hence
condition (iv) holds. Suppose now that condition (iv) holds. (iv) implies that 8Tr(x,o,u )/Bc 2
i!:
Since n(x,o,u ) 1
= rr(x,o,u 2 ) = o,
Since r A (x,u ) =
cm(x,o,u }/Bc. 2
1
4drr(x,o,u )/Cic and r A (x,u ) = 4Bn(x,o,u )/8c, (i) follows. 1 2 2
Let w and w be two wealth levels such that w > w and define u (z) = 1 2 2 1 1
6.C.8
•
u(w + z) and u Cz) = u(w 1 2 2
+ z),
u
It was shown in Example 6.C.2 continued that the
C·)
1
by Proposition 6.C.3.
then u ( ·) is a concave transformation of 2
demand for the risky asset of u ( ·) is greater than that of u ( · ). 1
2
This means
that the demand for the risky asset of u( ·} is greater ·at wealth level w than 1
at w . 2
6.C.9
[First printing errata: The function u( ·) on the left-hand side of the
equality on the fifth line should be denoted by a different symbol, because. on the right-hand side, u( ·) is used for the utility function on th! first period. I (a) The first-orde:- condition for the first problem is u'(w - x l = v'(x L 0
0
For the second problem, let's first define a function ¢( ·) by t/J(x) = u(w - x) Then 1/J'(x) = - u'(w - x) + E[v'(x
+ y})
Note also tha: ¢'(x•) = 0 and 1/J"(x) ¢'(x) > o,
the~
¢'Cx l 0
x• > x.
=-
::5
+
E[v(x
+ y)J.
and t/J"(x) = u"(w - x) + E:v"(x
+ y)).
0 for every x, which impiies that if
Now, since E[v'Cx
0
u.'(w - x ) + E[v'Cx + y)) 0 0
6-12
+ y)]
=-
> v'Cx ), 0
v'Cx ) 0
+ E!v'Cx
0
+ y))
> 0.
•
(b) Define two functions 'rl ( ·) and 'rl £ ·) by 'rl {x) = - vi(xl and TJ (x) = 1
Tnen 71 £ ·) and 11
- vz.(x).
1
2
2
1
2
are increasing and the coefficients of absolute
C·)
prudence of v ( ·) and of v £ ·) are equal to the coefficients of absolute risk 1 2 aversion of 11 ( · ) and of 11 £ • ).
Thus, if the coefficient of absolute prudence
2
1
of v ( · ) is not larger than that of v { • ) , then the coefficient of absolute 1 2 risk aversion of 1'1 ( ·) is not larger than that of 1'1 £·). . Moreover, since 1 2 E(viCx
0
+ y)]
> vi(x }, we have Ei71 Cx0 0
1
Proposition 6.C.2 to 11 £·) and 1
Eivi(zo
+
1)
2
+
y)} < lJ (x J. 1
0
£ • ), we obtain EflJ tx
2
0
Thus, by applying + y)]
< 11 Cx l. 2
0
Hence
yll > v2(zol.
The implication of this fact to part (a) is that, if the coefficient of absolute prudence of the first is not larger than that of the second, and if .
the risk y induces the first individual to save more, then it also induces the second to do so.
Hence coefficients of absolute prudence measure how much
individuals are willing to save when faced with a risk in the future.
> 0, then
(c) If .,.."'(x) •
aver-s1on.
Tl"(x)
= - v'(x) <
0
and hence 71( ·) exhibits risk
Thus E['f1(X + y)} < lJ(x), that is, E[vi(x + y)] >
.
\ V. ( X;,
1
(d) Since r.~(X,Y)
- -
v"'(x )v' (x) v' ( xl
2
. 2 v" ( x)
v"(x) (V' (X}
-
v"'(x) v"( xl
+
V '' (X)
y' (X)
} < 0,
the assertion follows.
6.C.l0
Throughout this answer, we let x
such that x
1
> x
1
and x
2
be two fixed wealth levels
a.'ld define u (z) = u(x + z) and u (z) = u(x + z). 2 2 2 1 1
It is
sufficient t·::! •orove that each of the five conditions of Prooosition 6.C.3 is • eq-:.1ivalent to its counterpart of Proposition 6.C.2.
6-13
of Proposition 6.C.3 is equivalent to (i} of Proposition 6.C.2. Property
(ii)
of Proposition 6.C.3 is nothing but a restatement of (ii}
of Proposition 6.C.2. As for property (iii), note that ) = u((c
Ju (z)dF(z) = Ju(x + z)dF(z) = u(c 1
1
and likewise for u C· ). 2
XI
-
X ) + X }
1
Xl
1
=
U (c
1 x
-
1
Thus the certail}tY equivalent for u ( ·) is smaller 1
< c
x2
- x . Thus property 2
(iii)
of Proposition £.C.3 is equivalent to (iii) of Proposition 6.C.2. As for property (iv), since
we have u (0) = (1/2 - tr(xl'c,u))u (- c) 1
1
Hence rrCx ,c,u) = tr(O,c,u ). 1
+ U/2 + 7r(x ,c,u))u (c).
Similarly, tr£x ,E:,u)
1
2
1
1
z:
tr(O,c,u L 2
Hence (iv} of
Proposition 6.C.3 is equivalent to (iv) of Proposition 6.C.2. Note that J'u£x + z)dF(z) 1
and likewise for u ( · ). 2
2:
Thus
u£x ) if and only if Ju £z)dF(z) 1
1
~ u
CO),
1
(v) of Proposition 6.C.3 is equivalent
to ( v) of Pro'Cosition 6. C. 2 . •
6.C.ll
Fer any wealth level x, denote by ;(x) the optimal
invested in the risky asset.
propor~ion
of x
We shall give a direct proof that if the
coefiicie::lt of relative risk aversion is increasing, then a-'(x) < 0; along the same line of proof. we can show that if it is decreasing, then a-'(x} > 0. shown in Exe:-cise 6.C.2, 7(x) is positive and satisfies the following first-o:-de:- condition for every x: Ju'{(l - 7(x) + ;(x)z)x}(z - l)xdF(z) = 0. Hence
6-14
As
'() _- Ju"((l- 7(x} + ;(x)z)x)(l- ;(x) 1
X
a-(x)z)(z- l)xdF(z) 2 2 . Ju"((l- ;(x) + ;(x)z)x)(z- 1) x dF(z)
-
+
Since the denominator is negative, it is sufficient to show that the numerator •
•
•
1s pos1uve. By the definition of the coefficient of relative risk aversion,
- u"((l - 7(x) + ;(x)z}x)(l - ;(x) + ;(x)z)x = rR((l - 7(x)
for every realization z. x by a-(x) > 0.
+
a-(x)z)x)u'((l - ;(x) + 7(x)z)x)
Note also that if z > 1, then (1 - ;(x) + ;(x)z)x >
Since the coefficient of relative risk aversion is increasing,
this implies that rR((l - a-(x) + ;(x)z)x) > r R(x).
Hence
- u"((l - 7Cxl + a-(x)z)x)(I - a-(x} + ;(x)z)x
> rR(x)u'((l - a-(x) By
a-Cx)z}x).
+
z - 1 > 0, - u"((l - a-(x) + a-(x)z)x)(l - ;(x) + a-(x)z)x(z - 1} > r R (x)u' Ul - a-(x) + J(X)z)x)(z - 1). •
We can sim.Ea!"ly show that this last inequality also holds for every z < 1.
- -· . .
Th ,._,.'o'"e .
- Ju .. (,t
- 7\XI +
> Jr R{x)u'((l - 7{x)
7(X)z)x)(l - a-(x) + 7(x)z)x(z - l}dF{z) +
a-(x)z)x)(z - l)dF(z)
= rR(x)Ju'((I - a-(x) + 7(X)z)x)(z- lldF(z) = 0 by the first-order condition.
(a) {First printing
6.C.l2
< 1 ar..d negative if p > 1. ... na ?. . ,
• •
:..,. .a
u ( x) = 1Qx .~
1-p
+
relative risk aversion p.
· The coefficient {3 should be positive if p This makes u( ·) increasing.) with p
~ 1
It is easy to check
and ; e IR, then u( ·) exhibits constant
Suppose conversely that u( ·) exhibits constant risk
aversicr! p, then u"(x)/u'(x) = - pix.
Thus ln (u'(x)) = - pln x + c 1 for some •
6-15
Hence u(x) = (exp c )x
1-p
1
for
2
Letting f3 = ( exp c )/(1 - p} and 7 = c , we complete the proof. 1 2
i?..
E
/(1 - p) + c
if u(x) =
(b) It is easy to check that,
~ln
x + '1 with f3 > 0 and 7 e ~. then
u( · J exhibits constant relative risk aversion one.
The other dir-ection can be
•
shown in the same way as in (a).
(c) By L'hopital's rule, [' 1-p . 1ll:lp-+l \X - 1)/(1 - p}]
Let
6.C.13
rt( • )
=
= ln x.
be the profit function and F( ·) be the distribution function
of the random price.
Since
Jensen's inequality.
n( ·)
is convex, Jn(p)dF(p}
?!
n(JpdF(p)) by
But the left-hand side is the expected payoff from the
uncertain p:-ices and the right-hand side is the utility of the vector.
price
Tnus the firm prefers the uncertain prices.
Define a functior! g( • ) by g(o:) = ko: + v(u
6.C.l4
ex;>e::~ed
+ v(x) = u•(x).
-1
(et)},
then g(u(x}) = ku(x)
It is thus sufficient to show that g( ·) is concave.
For
tnis, in tu:-n, it is sufficient to prove that (vou -l)( ·} is concave. Let c::, ,'3 e !R and A e [ 0.11. is convex.
Thus u
Since
v( •
v{ ·
-1
(Aet + ( 1 - i\.){3) :s i\.u -l(o:) + (1 - i\.)u
-1
(8).
l is nonincr-easing, this implies v(u
Since
Since u( ·) is increasing and concave, u
-1
~(i\.et + (l
- i\.)(l)) ~ v(i\.u
-1
(o:.) + (I - i\.)u
-1
((l)}.
l is concave,
•.
-1
( cd + (1 - i\.}u -l(f3)) ?!. i\.v(u -l(o:)) + (1 - i\.)v(u -l(f3)}.
Thus,
6-16
-1
(. )
v(u
-1
(i\a + (1 - i\){3)) ~ i\v(u
-1
(a)) + (1 -
_,
i\)v(u ·({3}).
or. eaui vale:J.tlv. . -
(b} [Fir-st printing errata: The entire interval (0, + co) should be [0, + ao).} Suppose that we have u•(x) = ku(x) + v(x) for a non-constant v{ · ).
Since
v( • )
is decreasing and concave, v(x + 1) - v(x} is negative and decr-easing with x. On the othe:- hand, since u( · ) is increasing, concave, and bounded above, u(x + ll - u(x) is positive and decreasing, and converges to zero.
Since
u•(x + 1) - u•(x) = k(u(x + I) - u(x)) + (v(x + 1} - v(x}), u•(x + ll - u•(x} is negative for any sufficiently large x. not increasing around such x. that u•( ·} is im:reasing.
That is, u•( ·) is
But this is a contradiction to the assumption
Thus, if u( ·) is bounded, then there is no
non-constar.t v{ ·) such that u•(x) = ku(x} + v(x) for all x
E
[0, + co).
{c) Bv (a) and (b), it is sufficient to find u( ·) and u•( ·) such that u•( ·) is • r:1cre risk ave:-se (in the Arrow-Pratt sense) than u( ·) and u( ·) is bounded. Define u{x} = - exp(- ax) and u•(x) = - exp(- flx), whe::--e 0 < a < f3.
Bv •
Exam.:1le 6. C. 4, u( · l and u•( ·) exhibit constant absolute risk aversion with • coefficients a and {3.
Hence, u•( ·) is more risk averse than u( · ), but, s:nce
u(x) < 0 for aE x. u•( ·) is not strongly more risk averse than u( ·).
6.C.I5
TJ1.r::u2:!lo~t
-
this answer, we assume that a
;e
b, because, otherwise,
the:-e would be no uncertainty involved in the payment of the second asset.
(a)
r:-
Min {a. b}
~
1. the risky asset pays at least as high a retu:-n as the
risk!ess asset at both states, and a strictly higher return at one of them. T~en
all the v.:ea!th is invested to the risky asset.
6-17
Thus, Min {a,b} < 1 is a
necessary condition for the demand for the riskless asset to be strictly positive.
(b} If na + (1 - tt)b ~ 1, then the expected return does not exceed the payments of the riskless asset and hence the risk-averse decision maker does not demand the risky asset at all.
Thus, na + (1 - n)b > 1 is a necessary
condition for the demand for the risky asset to be strictly positive.
In the following answers, we assume that the demands for both assets are always positive.
(c) Since the prices of the two assets are equal to one, their marginal utilities must be equal. tru'(x
Thus
+ x a> + (1 - u]u'(x + x bl = uau'(x + x al + (1 - n)bu'(x + x bl. 2 2 1 1 1 1 2 2
That is, n(l -
This and x
1
+
x
2
a)u'(x + x al + (1 - wH1 - b)u'(x + x b) = 0. 1 2 2 1
= 1 constitute the first-order condition.
(d) Taking b as constant, define
then
< 0, + (l -
< 0.
Thus. by the implicit function theorem (Theorem M.E.l), dx Ida = 1
84>/Ba Bf/>1Bx
< 0. 1
(e) It follows from the condition of (b) that b > 1, that is, that a is the worse outcome of the risky asset.
Thus, if the probability n of the worse
outcome is increased, then it is anticipated that the demand for the riskless
6-18
asset is increased.
(f} Since b
> l,
Bf/>/Brr = (1 - a)u'(x + (1 - x)a) - (1 - b}u'(x + (l - x)b)
= (1 - a}u'(x because a < 1 < b.
6.C.16
+ (1 -
x)a) + (b - l)u'(x +
Thus dx/dn = -
8(/>/Bn
x)b) > 0,
(l -
> 0, as anticipated.
Throughout the answer, we assume that u( ·} is continuous, so that the
urn and the minimum are attained. (a) If the individual owns the lottery, his random wealth is (w + G, w Thus the minimal selling price R
s
+
B).
is defined by
pu(w + G) + (1 - p)u(w + B) = u(w + R ). s (b) If he buys the lotter-y at price R, his random wealth is (w -
R + G, w - R + B).
The maximal buying price Rb is defined by
pu(w - Rb +
p}u(w - Rb +
G) + (1 -
= u(w}.
B)
However. if u( · } exhibits
(c) In general, these two prices are different.
constant absolute risk aversion. then they are the same. •
two eauat1ons car. be restated as c •
w
= w + R
s
and c
In
R
w- .
~~a.-:t.
the above
= w, where c
D
w
•
are defined as in (iii) of Proposition 6.C.3.
According to the proposition,
the constant absolute risk aversion implies that w - cw = (w - Rb) - c w-R. . 0
T."l.is is equiva!er:.t to Rs = Rb.
(d) By a direct calculation,
R
is one
o~
s
= 5[(7 - 4v'3lp
2
+
(4v'3 - 6)p + 11.
the solutions to the quadratic equation •
6-19
and -
""w-R
t
3
(1 -
6.C.l7
- 10(2p
+
7p
2
- 8p + l)Rb - 25(23p
2
- 54p + 29) = 0.
According to Exercise 6.C.l2, if u( ·) exhibits constant relative risk
aversion p, then u(x) = {3x 1 assume u(x) = ax -P. argument.
1-p
+ 0
or u[x) = /3ln(x} + 0 .
In this answer, we
The case of {3ln(x) + 'r can be proven by the same
Let's first consider the portfolio problem of the individual in
period t = 1. after a realization of the random return has generated wealth
wr
level
Denoting the distribution function of the return by F[ ·). his
problem is
As
dis~ussed
in Example 6.C.2 continued (and also in Exercise 6.C.ll), we can
show that the solution does not depend on the value of w . 1
If he chooses portfolio o:
solution by a;•. at t = 1 is w
1
=
((1 - o: )R 0
0
Denote the
at t = 0, then his random wealth
Given the solution o:• at t = 1, his
problem in period t = 0 is
1-p
Since the distributions of ..x and x2 are independent and u(x) = ax· 1
• we can
rewrite the obje:tive function as
Since the firs:
int~gral
does not depend on the choice of a. , the solution is 0
This completes the proof. For the case of a utility function exhibits constant absolute risk aversions, the absolute amounts of wealth invested on the risky asset may vary over the twc pe:-iods t = 0,1, but those in period t = 1 do not depenc! or, the realization cf
a • • - .&• L
._
-
To see this, let u(x) = - f3e-P.
1.5
6-20
The individual's p:-oblem
The solution t~:-ns out to be independent of the value of w , and hence of x . 1 0 Denote the solution by a•.
If he chooses portfolio a
random wealth at t = 1 is w
1
= (w
at t = 0, then his
0
•
- o: )R + o: xl" 0 0 0
Given the solution o:• at t
= l, his problem in period t = 0 is
Since the distributions of x
1
and x
are independent and u(x) = - {3e -p, we can
2
rewrite the objective function as
Since the first integral does not depend on the choice of o: , the solution of 0
this maximization problem is the same as the solution of the p;oblem of •
•
•
max1m1zmg
But the la';te:- is the same as what the consumer would choose at t = 1 if his coefficient of absolute risk aversion is equal to Rp. Now. if R = 1, then the consumer invests a constant absolute a:::1cunt of wealth over two periods.
Thus, their proportions out of the total wealths are •
larger i!' the total wealths are smaller.
on
\\r ..
.!.
•
ana her.ce on the realization
So, the proportion
IX
/\\' now depends 1 1
Hence the proportions can no longer be
constant.
6.C.l8
(a} A cii:-ect calculation shows that the coefficient of absolute risk
aversicr: at w = 5 is 0.1.
Exercise 6.C.l2(a) shows that the coe!'ficie:1t o:
relative :-isk ave:-sion is 0.5, which is constant over w.
(b} By a direct calculation, the certainty equivalent is 9 and the probability
premium is (v'TQ - 3}/2.
6-21
(c) By a direct calculation, the certainty equivalent is 25 and the probability premium is
em -
5)/2.
For each of these two lotteries, the difference between the mean of the lottery and the certainty equivalent is equal to one.
However, the
probability premium for the first lottery is larger.
This is because
u( ·)
exhibits constant relative risk aversion and hence decreasing absolute risk •
avers1on.
6.C.l9
Foe each n, denote by f3
n
the wealth invested in risky asset n.
wealth invested in the riskless asset is then w - Eo.Sn·
The
If the individual
random consumption is x =
13 lr + l' (3 z , where z denotes the "'"'n n ""n n n n
return of asset n.
(w - l'
By
linearity of normal distributions, x is a normal distribution with mean (w -
Lr,.Snlr
+
Lni3nf.ln and variance ,S·V,S.
£[- exp (- ax)].
The expected utility from x is
But this is equal to the value, multiplied by - l, at - c: of
the moment-generating function of the normal distribution with mean (w -
t_.s k ,_ n
f3 f.l and variance '-r: n n
+ l'
(3 • V,S.
·Therefore,
By appiying the monotone transformation u -t (- li1Xlln(- ul to this utility
function, we obtain ({w - )
f3
'nn
)r + ) (3 Jl
'"'nnn
l + (,S·Vf3)a/2.
The first-orde!"' condition for a maximum of this objective function with respect to
,a
gives the optimal portfolio
,a•
= a -lV-l(J..L - re), whe:-e e is the
vector of iRr-,; whose components are all equal to one.
6.C.20
For each e:
~ 0,
let F [; ( ·) be the distribution function of the lotterv•
6-22
that pays x
+
c with probability 1/2 and x - c with probability 112.
Then,
c(F ,u) is defined as the solution to the equation £
(I/2)u(x + c) + 0/2)u(x - c) - u(c) = 0
with respect to c.
Hence, by the implicit function theorem (Theorem M.E.l),
c(F ,u) is a differentiable function of c and c (l/2lu' (x +
c) -
(l/2)u'(x - c} - u'(c{F ,u))(oc(F ,u)/8cl = 0.
c
By putting c = 0, we obtain 8c(F ,u)/8c = 0. 0
E:
Also, by further differentiating
the left-hand side of this equality with respect to c. we obtain (112)u"(x + £) + (l/2)u"(x - c) - u"(c(F ,u))(oc{F ,u)/8d c £
2
2
2
- u'(c{F ,u))(a cCF ,u}/8c ) = o. c £
Thus, by putting c = 0 and substituting 8c(F ,u)/8c = 0, we obtain 0
2 2 u"(x) - u'(c[F ,u))(o c£F ,u)/8c ) = 0.
c
c
G( · ) be thei:- distribution functions. (a) If a Be:-noulli utility function is increasing, then there exists p e (0,1) .
su:::h that the decision maker· is indifferent between the sure outcome of $2 and the lotte:-y tha-: pays $1 with probability p and $3 with probability 1 - p. Thus, the indifference line that goes through the $2-vertex must hit some poir.t en the (Sl.$3)-face (excluding the vertices) and all indifference lines must be par-allel to it.
Conversely, this condition implies that the Bernoulli
utility functio!! increasing.
By varying p vary from 0 to 1, we can identify
the area of the lotteries that are above all indifference curves going through L.
The a:-ea is shaded in the following figure:
6-23
$3
L
•
$1
$2 Figure 6.D.l(a)
Thus, G( ·) first-order stochastically dominates F( ·) if and only if L' is •
located above the segment that goes through L and is parallel to the ($1,$2)-face and also above the segment that goes through L and parallel to the ($2,$3)-face.
(b) The distribution G( · ) first-order stochastically dominates F( • ) if and only if p
1
2:
Pi
equivalent to p
3
and p
1
+ p
2
2:
Pi
+
P2·
Since the second inequality is
:s Pj· G( ·} first-order stochastically dominates F( ·) if and
only if L' is located in the shaded area in the figure below:
$3
L
$2
$1
Figure 6.D.l(b) 6-24
(Fi:-st printing
6.0.2
a
: The phrase "the mean of x unde:- F( · ), fxdF(x),
a
•
exceeds that under G( · ), JxdG(x)" should be "the mean of x unde; G( ·), JxdG(x), cannot exceed that under F( · }, JxdF(x)". the two means should be allowed.)
x and apply Definition 6.0.1.
That is, the equality of
For the first assertion, sir.tply put
u(x)
=
As for the second, let p e (0,1/2) and consider
the following two distributions: 0 F(z)
=
G(z) =
p 1
0 1
if if 0 if 2
~
z < 0, z < 2,
~ Z,
if z < 1, if 1 :5 z.
Then F(l/2} = p > 0 = G(l/2) and JxdF(x} = 20 - p) > 1 = JxdG(x).
Hence F( ·)
does not first-order stochastically dominate G( • ), but the mea."'l of F( ·) is larger than that of G( ·).
Any e1ementa..-y increase in risk from a distribution F( ·) is a mean-
6.0.3
prese:-ving spread of F( · }.
In Example 6.0.2, we saw that any mean-preserving
spread of F( ·) is second-order stochastically dominated by F( • ).
Hence the
asse::tion follows.
Let L
6.D.4
lotteries.
•
(a) Bv a cirec: ca.lc'.liation, the means of L and L' are 2 - p
-
+
p'
Pj·
3"
1
~ p
3
and 2 - Pi
"f p - p = p• T.'1Us the two lotteries have an equal mean if and only t 1
3
1
Hence they have an equal mean if and only if they are beth on a segment
that is pa:-alle! to the segment connecting the $2-vertex and the middle point of the ($1,$3)-fa::e, as depicted below:
6-25
$3
··-.....••
•••
•• •
••
••
• ••
••
L
·...... •. _
··-..... •• •
••
$2
$1
Figme 6D.4(a)
(b) If the decision-maker exhibits risk aversion, then he prefers getting $2
with probability one to the lottery yielding $1 with probability 112 and $3 with probability 1/2.
Hence the indifference lines are steeper than the
segment connecting the $2-vertex and the middle point of the ($1,$3)-face. Hence, when L and L' have an equal mean, L is preferred to L' if and only if L is located on the right of L'.
Therefore, L second-order stochastically
dominates L' if and only if L is located on the right of L', as depicted in the figure below:
$3
•
.... ... ... ••
L
l
•• •
••
•. •
L •• •
• ••
.........
•
• ••
• ••
•.
$1
$2 Figure 6.D.4(b) 6-26
(c) The distribution of L' is a mean preserving spread of that of L if and only if they are both on a segment that is parallel to the segment connecting the $2-vertex and the middle point of the ($1,$3)-face, and L' is closer to the ($1,$3)-face than L.
This is depicted below:
$3
•• ••
•• ••
L' ·-..•
••
••
·.,_ L $2
$1
Figure 6.D.4(c) (d) Inequality (6.0.1) holds if and only if Bu~.
Pj - p
3
Pi
iii:
P1 and Pi
+
(pi
+
Pz)
?!
P + 1
since L and L' are assumed to have an equal
and hence these two inequalities are equivalent to
Pi
iii:
p
1
alone.
Thus, (6.0.1) holds if and only if L is located in the right of L', as depicted belO\\':
$3
•
•• •
•• •
L' .....
••
\
·.••
•
'·.,_ L
$2
$1
Figure 6.D.4(d)
6-27
Denote by R(x,x') the expected regret associated with lottery x
6.E.l
relative to x', and similarly for the other lotteries.
A direct calculation
yields: R(x.x') = 2/3, R(x' .x) = 13/3, R(x' ,x") = (,.12 + 1)13. R(x",x') = l's/3, • R(x",x} = Cv'2 + 1}/3, R(x,x") = v'213. Thus, x' is preferred to x·, x" is preferred to x', but x is preferred to x" .
(a) Denote the probability of .state s by tr
6.E.2
5
and the expected utility
from the contingent commodity vector (x ,x l by U(x ,x l, then U£x ,x ) = 1 2
1 2
n u(x l 1
1
+
n (1- w)u(x ). 2
2
1 2
Since u[·) is concave by the assumption of risk
aversion, U[ ·) is also concave.
Thus the preference ordering on Cx ,x ) is 1 2 •
convex. (b) According to Exercise 6.C.S(a), the concavity of U( ·) implies the risk
aversion for the lotteries on (xl'x l. 2
(c) By the additive separability of U( ·) and Exercise 3.G.4(c}, both x
1
and x
are normal goods.
6.E.3
Since g•(s) = 1 + a:(g(s) - 1) for every s, we have ~(s)
>
g(s)
If g(s) <
1~
g•(s) = g(s) if g(s) = 1; g•(s) < g(s) if g(s) > 1. Thus G*(x} :s G(xl for every x < 1 and c•(x) G( ·) and
~ G(x)
for every x > 1.
c•( · l are continuous from the right, we have
c•(l) ~ G(ll.
Since Hence
property (6.D.2} holds and thus G•( ·) second-order stochastically dominates G( · l weakly.
(If g(s)
:;c
1 for some s, t}_len G•(x) < G(x) for some x < 1 and
6-28
2
G•(x) > G(x} for some x > 1. stochastically dominates
Hence, in this case, G•( ·) second-order strictly.)
G( ·)
We shall first prove the uniqueness of the utility function on money up
6.F.l
A
Suppose that two utility .function u( · l and u( ·} satisfy
to origin and scale.
the condition of the theorem.
Since the state preferences
-
...
...
~
-s
are represented
by both J(n u{x l + (3 )dF ·ex ) and J(n u(x ) + f3 )dF (x ), by apolying s 5 s ss s 5 s ss . Proposition 6.B.2 to the set of all the lotteries in some state s. we know that n u( • ) + f3
s
...
s
-
~
and n u( ·) + 13
s
s
are the same up to origin and scale.
Hence
so are u( · ) and u( • ).
It remains to verify the uniqueness of subjective probability.
...
that both 1 n (Ju(x )dF (x )) and ) x L-ss s ss '"'ss preference relation on !f..
(Ju(x
Suppose
)dF (x )) represents the same s ss
-u( ·)
Now that we have shown that u( ·) and
are the
same up to c:-igin and scale, without loss of generality, we can assume that u( ·)
= u( · J.
We can normalize u( ·) so that u(O)
=0
and u(l)
= 1.
Note here
that if a cistribution function F ( ·) puts probability p on 1 and probability s s 1 - p
s
on 0, then the expected utility is p .
Thus, by choosing p
s
•
s
suitably
for ea:::h s. any point in [ O,l}s_ can be represented in the foim
.
'
A
Hence, if (n:, ... ,n l 5
~ (n
-
, ... ,ns), then there would exist CF ..... F l e 5 1 1
!f.
S
and (Fi·· .. ,F l e ~ such that 1 n (Ju (x )dF (x )) > ) n (Ju (x )dF' (x )),
L-ss
ss
ss
L-ss
A
[ n
ss
This
contraC.i::~s -
ss
ss
A
(Ju (x )dF (x ))
ss
ss
<) n
'"'ss
(Ju (x ldF' (x )).
ss
ss
the assumption that they represent the same p:-eference. A
6-29
Thus
6.F.3
(a} If P = {n}, then UW(H) = n and
u8 (H)
=1 -
n.
Hence they are
determined from the expected utility nu(lOOO) + (1 - n)u(O).
> UW(Hl if and only if 0.49 > n.
Moreover, Uw(R)
But this is equivalent to 0.51 < 1 - n,
which is, in turn, equivalent to U (R) < U (H). 8 8 (b) We have Uw(R) > UW(H) if and only if 0.49 > Min P.
We have
u8 £R) > u8 (H)
if and only if 0.51 > Min{l - n: n e P}, which is equivalent to 0.49 < Max P.
Hence Min P < 0.49 < Max P if and only if UW(R) > UW(H) and U (R) > 8
•
6-30
u8 £HL
CHAPTER 7
Pia u 1
7.C.I
Grand Central
100 100
Empire State
0 0
0 0
100 100
Figure 7.C.1 7.0.1
Player L has M xM xM x · · • xMN strategies in this game. a 1 2 3
7.0.2
H
Player2
T
H
-1 • 1
1 • -1
T
1 • -1
-1 • 1
Player 1
Figure 7.D.2 7.£.1
(a) In order to specify a stratev for player 1, we need to determine
his moves in all of the three information sets in which he moves. typical strategy for player 1 can be written as a triple.
Thus a
The set of
strategies for player 1 are: s
1
= ( (L, x, x), (L, x, yl, (L, y, x), (L, y, y), (M, x, x), (M, x, y), (M, y, x), (M, y, y), (R, x, x), (R, x, y), (R, y, x), (R, y, y) }
1 - 1
If player 1 uses strategy (L, x, y), he plays L at the root of the game, x in his information set following his move M (we refer to this information set as "Information Set 2") and y in his information set following his move R We refer to this information set as "Informatilln set 3"). Similarly player 2's strategy specify her move at her information set (we refer tho this information set as "Information Set 1"). Thus,
s2
= {(l),(r)}.
(b) A behavior strategy for player 1 consists of a randomization of his possible moves at each information set in which he has to move.
Suppose that
at the root, player 1 plays L, M, and R with probabilities of p , Pz and p 1 3 respectively (p +p +p =1); at information set 2, player 1 plays x, y with 1 2 3 probabilities of q
-
and q
1
2
l'tPo
respectively (q +q =1); at information set 3, 1 2
player 1 plays x, y with probabilities of s Assume that player 2 plays l and r
and s
1
2
respectively.
<:
s,::q...
g~=r-,
with probabilities cr(l) and cr(r)
respectively (cr(l)+cr(r)=l). Thus, if player 1 is using the above behavioral strategy and player 2 is using this mixed strategy, the probability that we reach each terminal node will be: Pr(T ) = p ; PrCT ) 1 0 1
= p2
cr(l) q ;. PdT l = p cr(l) q ; 2 2 1 2
PrCT l = p cr(r) q ; Pr(T ) = p cr(L) r ; 6 3 4 2 2 2 Pr(T ) = p
8
3
Pr(T ) = p cr(r) q ; 2 1 3
PdT l = p cr(r) r ; 7 1 3
cr(r) r . 2
Now the following mixed strategy for player 1 is realization equivalent to the above behavior strategy: (L, x, x) with probability pl' (M, x, x) with probability p q , 2 1 (M, y, x) with probability p q , (R, x, x) with probability p r , 3 1 2 2 (R, x, y) with probability p r .
3 2
(Note: p + p q l + Pzq + p r + p r z = P + 3 1 3 1 2 1 2
Pz =PI+ P2 1 + PJ 1::: 11
If player 1 is using the above mixed strategy and player 2 is using the mixed strategy cr, the probability that we reach each terminal node will be the
7 - 2
same as shown before for the behavior strategy.
Therefore, the above mixed
strategy is realization equivalent to the behavior strategy.
(c) Suppose that player 1 uses the following mixed strategy: (L, x, x) with probability p (L, y, x) with probability p
1 3
••
(L, x. y) with probab111ty p , 2
:
(L, y, y) with ·probability p , 4
,
(M, x, x) with probability Ps
•
CM. y,
7
•• (M, y, y) with probability p 8 ,
9
••
x) with probability p
(R, x,
x) with probability p
CR. y,
x) with probability p
[pj ~ 0 for all i and
r
11
x, y) with probability p ,
(M,
(R,
6
x, I} with probability p 10,
(R, y,
... ;
TYPO
Shol.l.td be.
_.!)
1 _ __, with probability P7;,:-'---p,...;1.:...
tr,
the probability that we reach each
TYPO
pi = 1)
shou.ld. be PID
If Player 2 uses the mixed strategy terminal node will be: PrCT ) = p 0
1
+ p + p + p • Pr(T l
2
4
3
1
Pr(T > = Cp + p )tr(l), Pr(T > = (p + p ) tr(r), Pr(T > 1 8 5 2 3 6 4
= Cp5
= Cp7
+ p ) tr(l), 6 + p ) tr(r),
8
Pr(T ) = (p + p ) tr(L}, Pr(T ) == (pll + p l tr(L), Pr(T l = (p + p l tr(r), 9 5 10 6 7 9 10 12
PrCT > = Cp + p J cr(r). 11 12 8 The following behavioral strategy for player 1 is realization equivalent: At the root of the game, player 1 plays L,
~·
R with probabilities of (p
+
1
P + P + P l. Cp + p + p + p l and Cp + p + p + p J respectively; at 4 5 1 6 3 2 8 9 12 10 11 information set 2, player 1 plays x, y with probabilities of (p
5
+ p ll(p
6
+
5
p + P + P > and Cp + p )/(p + p + p + p ) respectively; at information set 3, 8 7 8 7 6 5 6 8 7
player 1 plays x, y with probabilities of (p (p
11
(d)
+ p
12
l/(p
9
+ p
10
+
p
11
+
p
12
>
9
+
p
10
)/(p
9
+ p
10
+
p
11
+ P
12
>
and
respectively.
Note that if player 1 reaches his (only) information set after player 2
moves, he will not remember whether he chose M or R.
Thus, the game is not of
perfect recall. The result of part (b) still holds: there exists a mixed strategy for player 1 which is realization equivalent to any behavior strategy.
7 - 3
Suppose
player 1 uses the following behavior strategy: At information set 1, player 1 plays L, M, R with probabilities of p , p 1 2 and p
respectively; at information set 2, player 1 plays x, y with
3
probabilities of q
1
and q
2
respectively.
If player 2 is using the mixed
strategy tr, then the probability that we reach each terminal node will be: Pr(T ) = p , PrCT ) = p trU) q , Pr(T > = p _cr(l) q , Pr(T ) = Pz cr(r) q , 0 2 3 1 1 2 2 1 2 1 PrCT ) = p cr(r) q , PrCT ) = p tr(l) q , PrCT ):::; p cr(l) q , PrCT l = p 2 4 2 5 3 1 6 3 2 7 3 cr(r) q , Pr(T ) = p tr(r) q . 8 3 1 2 The following mixed strategy for player 1 is realization equivalent: (L, x) with probability p , (M, x) with probability p q , 1 2 1 (M, y) with probability p q , (R, x) with probability p q , (M, y) with 2 2 3 1 probability p q . 3 2 However, there does not always exist
behavior strategy that is
<-
realization equivalent to a mixed strategy.
Consider the following example.
Player 1 uses the mixed strategy playing (M, x) and (R, y) both with probability 1/2.
Player 2 uses the pure strategy (l).
Suppose there exist a
behavior strategy for player 1 which is realization equivalent to the mixed strategy: at the root of the game, player 1 plays L, M, R with probabilities of pl' p
2
and p
respectively; at his information set after player 2 moves,
3
player 1 plays x, y with probabilities of q
1
and q
2
respectively.
The mixed
strategy generates the following distribution over the terminal nodes:
= Pr(T 6 ) = 1/2 Pr(T ) = Pr(T l = PdT ) = PdT l = PrCT ) = Pr(T l 4 3 5 0 2 7 Pr(T ) 1
= Pr(T ) 8
=0
The behavior
strategy generates: Pr(T ) 3
= PrCT 4 l = PdT7 ) = Pr(T8 ) = 0
Pr(T ) = p , Pr(T l = p q , Pr(T l 2 1 1 2 1 0
= p 2q 2 ,
Pr(T ) = p q , Pr(T l = p q . 3 2 6 5 3 1
in order for these distributions to be equivalent, we need: Pr(T ) = p q 2 1 1 112 .. p
2
and q
1
$
o, Pr(T l = p q 2 2 2
=o
,. q
7 - 4
2
= o since p
2
$
o, PrCT
6
>
= = p q 3 2
=
1/2 which cannot hold since q
2
= 0, a contradiction.
There exists no behavior
strategy that is realization equivalent to the above mixed strategy. In a game that is not of perfect recall the following holds: -
for any behavior strate&Y ther? exists a mixed strate&Y that is realization equivalent,
-
not• for all mixed strategies does there exist a realization equivalent behavior strategy.
[Note: for a general proof of these results refer to
F'udenberg/Tirole (1991) Game Theory. MIT press, p. 87)
7 - 5
CHAPTER 8
8.8.1
Finn i chooses h. to maximize cx.I .h. + ·1 J J
~m .h.) J J
2 - w.(h.) . 1
1
The
F.O.C. is: « + ti( .TT.h.) - 2w.h. = 0. The best response function for firm i is J~ 1 J 1 1 .
= (a: + ~
therefore: h.
1
= 0,
strate&Y iff f3
( .TT. h
JJ 2
Jill J
1
wi
.
Therefore firm i has a strictly dominant
i.e., if the best response function
on the action of the other firms.
~
If
of i is not dependent
= 0, firm i's strictly dominant
Cl
stratev is h. =
2w. ·
1
1
1 ' 2 8.8.2 (a) Suppose s. e S. and s. e S. are two weakly dominant strategies for 1
player i. and
u.(s~. I 1
s .)
l!::
-1
This implies that s .) -1
u.(s!, s .)
!:
1
1
1
-1
2
1
1
I
-1
I
1
-1
1
l!::
u.(s!,s .) V s!e S . and V s . e S . , I
1
-1
1
I
-1
V s!e S. a:td : s . e S .. In particular, 1
2
-1
1
u.(s~,s .) 1 1 -1
u.(s., s .) and u.(s., s .)
1. 2 u.(s., s .) = u.(s., s .) 1
1
1
-I
2:
1
-1
1
u.(s., s .) 1
I
-1
V s . E S ..
-1
-1
-1
-1
u.(s~. 1 I
Therefore,
V s . e S .. -1
-1
(b)
L
Player 2
R
u
1•4
2,5
D
1 •2
2,3
Player 1
Figure 8.B.2
Both of player l's strategies (U) and (0) are weakly dominant. player 2 prefers that player 1 uses strategy (U).
8-1
However,
8.8.3
Suppose not.
Assume bidder i bids b. > v .. 1
1
Then if some other bidder
bids something larger than b , bidder i is just as well of as if he would have 1 bid v .. If all other players bid lower than v.. then bidder i obtains the 1
1
object and pays the amount of the second highest bid.
If the second highest
results in the same payoff for player i as if he bid v ..
bid is bj< v., this 1
1
However. suppose that the second highest bid of the other is b
l
vi"
Then, by
bidding b. bidder i will win the object and obtain a negative payoff.
By
1
bidding v. he will not win the object and obtain a payoff of zero. Therefore. 1
bidding b. > v. is weakly dominated by bidding v .. J
1
I
Suppose bidder i bids b. < v.. I
Then if all other bidders bid something
I
-
smaller than b., bidder i is just as well of as if he would have bid v.. I
He
1
will win the object and pay the the second highest bid.
If some other player
bids higher than v .• then bidder i does not win the object regardless whether I
he bids b. or v.. I
I
However. suppose that nobody bids higher than v. and the I
highest bid of the other players is b. with b.< b.< v.. J I J I
Then by bidding b •
1
bidder i will not win the object. therefore getting a payoff of 0.
By bidding
v., he would win the object, pay b . < v ., and thus obtain a payoff of v. - b . J
I
> 0.
1
1
Therefore. bidding b. < v. is weakly dominated by bidding v.. 1
1
I
J
This
argument implies that bidding v. is a weakly dominant strategy. 1
8.8.4
Call the set of strategies for player i that remain after N rounds of
deletion of strictly dominated strategies
t':.
dominated strategy given the strategies
~ -1.
there exists a strategy given
s~1 e
'f':,1
I
Suppose s. e I
t': 1
is a strictly
of the other players Therefore,
which is not a strictly dominated strategy
~ -1., which strictly dominates s I.. Suppose further that s.1 will not be
deleted in the N+l round.
8-2
Since s. was strictly dominated by s! given 1
1
strictly dominated by assumptions).
si
given
~:~
~ -1. ,
s;; l.. ..N-1.• and
it will still be
s! E 1
~+l (with the given l.. i
Thus the strategy s. will be deleted in the next round. 1
(Note: If s. is only weakly dominated by s! given 1
1
~+~ -1
be weakly dominated given
since
~+~ -1
~-1.•
then s. may no longer 1
may no longer include the strategies
of the opponent relative to which some other strategy of player i will strictly be better.
Thus the order of deletion does matter for the set of
strategies surviving a process of iterated deletion of weakly dominated strategies).
8.8.5
(a) Suppose, player j produces qj'
Player i's best response can be
calculated by maximizing (this is symmetric for both players): max [a - b(q. ·· qj) - c)q. 1
1
which yields the F.O.C: [a - b(2q. + q .) - c) 1 J q. a-c J b.(q.)= 2" 2b 1 J
= 0, so the best response is:
-
Now, since q dominated by q q q
1
2
2:
2
1
~
0, q
::s (a-c)/2b (all other strategies would
2
= (a-c) / 2b).
Therefore, since q
(a-c)/2b - (a-c)/4b = (a-c)/4b.
Thus, since q
::s (a-c)/2b - (a-c)/Bb = 3(a-c)/8b.
obtain: q
= (a-c)/2b
- q/2.
dominated strategies, q
1
2 1
be strictly
::s (a-c)/2b, we have that 2:
(a-c) / 4b, then
Continuing in this fashion we will
Thus, after successive elimination of strictly
= q
2
= (a-c)/3b.
(b) Suppose, player j produces qj and player h produces qh.
Player i's best
response can be calculated by maximizing [a - b(q. + q. + qh) - c)q., which l J I yields the F.O.C [a - b(2q. + q. + qh) - c) 1 J
= 0,
implying the best response:
b. (q .,qh) = (a-c)/2b - (q . + qh)/2. I J J
Now, since
q , q 3 2
~
strictly be dominated by q
0, q
1
1
::s (a-c}/2b (all other strategies would
= (a-c)/2b).
Thus, since q
8-3
1
::s (a-c)/2b and
similarly q
3
:s
(a-c)/2b, we have q
2
!:
(a-c)/2b - [2(a-c)/2b)l2 = 0.
Therefore, successive elimination of strictly dominated strategies, implies that q , q , q 1 2 3
0 and q • q • q s (a-c)/2b. 1 2 3
!:
However, a unique prediction
cannot be obtained.
8.8.6
Suppose s! is strictly dominated by the strategy CT!. I
l
Suppose further
that CTi is a mixed strategy .in which sj is played with strictly positive probability cr.(s!)>O. l
strategy
cr~,
We claim that CT. is strictly dominated by the mixed
1
1
which is equivalent to CT. except that instead of playing s! with
l
1
l
probability cr.(s!} it plays cr! with probability CT.(s!). l
l
1
u.(CT., cr .) 1
l
This follows since:
I
=I: CT (s.) u.(s., CT .)
1
-1
=s . :r:i!s.• l
<
=
1
1
I
-1
CT.(s.) u.(s., CT .) + CT.(s!) u.(s!, CT .) 1
1
1
•
1
-•
1
1
1
1
-1
cr.(s.l u.(s., CT .) + :-i(s!) u.(cr!, cr .)
1: •
s. i!s. l
8.8.7
l
1
1
u.(cr~. l 1
1
1
1
-1
1
1
CT .)
for all CT . e A(S .).
-·
-1
1
-1
-·
Suppose in negation that cr. is a strictly dominant mixed strategy of I
· and suppose, s., l ... , s N. are t h e pure strategies . player 1, t h at are p led ay I
I
with positive probability in the mixed strategy CT..
Since CT. is a strictly
1
dominant strategy: u.(CT., s .) > u.(s!, s .) 1
1
-1
1
1
-1
I
\:1 s!e S. and \:1 s . I
I
-1
E
S .. In -1
•
particular, u.(CT.,s .) > u.(s~.s .) \:1 j = 1, ... , N. This implies that I
N
1
-1
I
.
I
-1
.
u.(CT.,s .) > .I: [CT.(s~) u.(s~.s .)) = u.(CT.,s .) , a contradiction. 1 1 -1 J= l 1 1 I -1 1 1 -1
S.C.l
1
Notice that the elimination of strategies that are never a
best-response is more demanding than strictly dominated strategy elimination. Thus, in every round of elimination, the deletion of never a best response deletes more strategies than the deletion of strictly dominated strategies.
8-4
Therefore, if the elimination of strictly dominated strategies yields a unique prediction in a game, then the elimination of strategies that are never a best-response cannot yield more than one strategy.
Since a rationalizable
strategy always exist, the elimination of strategies that are never a best-response will then also yield a unique prediction. If the unique rationalizable strategy is not the unique prediction after
elimination of strictly dominated strategies, then there exist a round of elimination in which this unique rationalizable strategy was strictly dominated.
However, if this strategy was strictly dominated it was also
a best-response. rationalizable.
8.C.2
n~ver
This contradicts the assumption that the strategy is Therefore, both procedures must yield the same prediction.
Call the set of strategies for pl"lyer i that remain after N rounds of
deletion of never best-response strategies best-response to any strategy in deleted in the N+l round.
w£:.1
Suppose s. is never a 1
~ -1.. Suppose further that s.1 will not be
Since s. was never a best response to a strategy in 1
.
-N+l
., it will clear 1y not b e a best-response to a strategy m J.. -1
..N
. s= J.. -1
••
-1
Thus this strategy will be deleted in the next round.
8.C.3
Suppose that
s
1
is a pure strategy of player 1 that is never a best
response for any mixed strategy of player 2. not strictly dominated. tr.
1
Suppose in negation that s
1
is
Construct the following correspondence for any
e t.(S.) for i = 1, 2: 1
A
gl(sl, ,.2) ~ gl(,.l' The first part of this correspondence is the best response function for player 1 and therefore satisfies all the conditions of the Kakutani fixed point theorem.
The second part of the correspondence is the set of mixed
8-5
strategies of player 2, for which s
1
is not strictly dominated (it is a
non-empty set since s is not a strictly dominated strategy. i.e., it is not 1
strictly dominated by cr l.
Therefore, the second part of the correspondence
1
is convex valued and upper hemicontinuous due to the usual assumptions.
Thus,
by Kakutani's theorem there exists a fixed point (cri ,cr2 ) of this correspondence such that g (s , cr2) a: g (cri, cr2l from the second part of the 1 1
correspondence, and g
Ccrt• cr2)
1
1
!:
g (cr , cr2,) 1 1
g (s , cr2) a: g Ccr , cr2,) for all cr 1 1 1 1 1 that
s
1
E ~cs
1 ),
for all cr
1
e
~(S ) ..
1
Therefore,
which contradicts the assumption
is a pure strategy of player 1 that is never a best response for any
mixed strategy of player 2.
Therefore, if s
1
is a pure strategy of player 1
that is never a best response for any mixed strategy of player 2, then s
1
is
strictly dominated by s011e mixed strategy of player 1.
8.C.4
(First Printing En-ata: a typo appears in the lower left box of the
payoff matrix.
Player l's payoff should be
1t
+ 4c and not 11 + 4£.]
For the continuation of this answer, a strategy for player 2 is to play u with probability a and D with probability 1-a, and for player 3 is to play l with probability f3 and r with probability 1-(3.
Denote by P A the
expe.~ted
payoff of
player 1 when action A e {L,M,R} is taken given a and (3. Direct calculation .
and simple algebra yield:
p M = n + ( 3cx ; 3{3-
3a{3 - 1 ) TJ
PL = n + (2(3 - l)c PR =
n +
(1 - 2(3)c
(a} To show that M is never a best response to any pair of strategies of players 2 and 3, (cx,(3), we have three cases: Case
t.
{3 > 112
apll Note that in this case
aa
= 1J[3/2 - 3(3) < 0.
8- 6
Thus the highest payoff for
and his payoff will be p (a.=O) M
player I if he plays M is obtained when a. = 0, =
1t
3
+ 11[~ - ll <
1f
3
+ 4c:(~ - 1) <
4d2t3 - I]
1t +
= PL.
Further note that PL
is independent of a., so that these inequalities hold for all a..
Therefore, M
cannot be a best respopse in this case. Case
~
13<
CIPM Now, B« a. = 1,
1/2
> 0, the highest payoff for player 1 if he plays M is obtained when
and his payoff is PM(«=ll =
n + 11Ii -
~
+
~
1f
+
1J{~
+
-13] < " + 4£[1 - 2131 = PR.
~
- 313 - I] =
n +
1J(~
-
~] <
Further note that PR is
independent of a., so that these inequalities hold for all o:.
Therefore, M
cannot be a best response in this case. Case~
13= V2
In this case PM = n - ~ <
1t
= PR = PL.
This concludes that M can never be a
best response.
(b) Suppose in negation that there exists a mixed strategy, in which player 1 plays R with probability 7 and L with probability
1-r.
that strictly dominates
M. Case
.!:. 7 :s 112.
If 13 = 0 and o:= 1 then PM = n + TJ/2 > n. payoff of
1l -
4dl-27) :s n.
The mixed strategy will give a
Therefore, M cannot be a strictly dominated by
the mixed strategy in this case. Case 2: '1 > 1/2.
-
If 13
= 1 ~nd
o:= 0 then PM
=n
payoff of n - 4d27-ll :s n.
+ lJ/2 >
n.
The mixed strategy will give a
Therefore, M cannot be a strictly dominated by
the mixed strategy in this case. This implies a contradiction, so that M cannot be strictly dominated.
!c)
Suppose players correlate in the following way: Players 2 and 3 play
8-7
(U, r) with probability 1/2 and (0, l) with probability 1/2.
Any mixed
strategy for player 1 involving only L and R will give him a payoff of n. However, playing M will yield him a payoff of n + 1)12. best-response to the above
8.0.1
correlat~d
Thus M is a
strategy of player 2 and 3.
We know already from section B.C that a and b are not rationalizable 4 4
strategies.
Thus, these strategies cannot
in a mixed strategy Nash equilibrium. strategy equilibrium In which a positive probability.
1
~e
played with positive probability
Suppose that there exists a mixed
and a
3
are both played with a strictly
Then the expected payoff from playing either one of
them has to be equal (see exercise 8.0.2).
This implies that the probability
that player 2 plays b has to be equal to the probability that he plays bJ" 1 Now, suppose that player 2 plays b. and b_ with probability « and b ~
1
probability 1-2«. a a
1 2
or a
3
2
with
The expected payoff for player 1 obtained by playing either
equals: 7« + (1-2«)2.
The expected payoff for player 1 when playing
equals: 5c + So: + (1-2«)3 = lOa: + (1-2«)3 > 7« + (1-2«)2.
mixed strategy equilibrium a probability since playing a
2
1
and
~
Therefore, in a
cannot both be played with positive
would give the player a larger payoff.
Suppose, there exists a mixed .strategy equilibrium in which player 1 plays a
1
and a
2
with strictly positive probability.
Clearly, player 2's best
response to this strategy of player 1 does not involve playing b
3
with
strictly positive probability (given the strategy of player 1, playing b strictly better for player 2). and b
2
2
The payoff for player 1 from playing a
yields: 5(3 + (1-(3)3 > (1-(3)2.
mixed strategy equilibrium a
is
Thus player 2 will play b with probability f3 1
with probability 1-(3.
equals: (l-tJ)2, playing a
2
1
and a
positive probability since playing a
2
2
1
Therefore, in a
cannot both be played with strictly is always better.
8-8
Similarly, it can be shown that there exists no mixed strategy equilibrium in which a
2
and a
are both played with strictly positive
3
probability. Therefore, player 1 always plays a Player 2 will then play his best resp.mse b . 2
2
in a Nash equilibrium.
Thus Ca , b ) being played with 2 2
certainty Is the unique mixed strategy equilibrium.
We will show that
8.D.2
any
Nash equilibrium (NE) must be in S
00 ,
the set.
of strategies which survive iterated strict dominance. Since it is assumed that this set contains one element, this will prove the required result. Let (si,sz·····sjl be a (mixed) NE and suppose in negation that it does not survive iterated strict dominance.
Let i be the player whose strategy is
first ruled out in the iterative process (say in the kth round). Therefore, there exists
and a. such that
fT. 1
1
k-1 u.(cr.,r. .) ,.. u.(a.,s .) 'tJ s .e S . , and a. 1
1
-1
is played with positive probability s!(a.). 1
1
1
1
-1
-1
-1
1
Since k is the first round at
which any of the NE strategies, Csi,s2·····sjl. are ruled out, we must have k-1 that s•.e S . . -1
derived from playing u.(s~
Hence, u.(fT'.,s•.) > u.(a.,s•.). Let the strategy
-1
fT •• 1
1
s~
1
-1
1
1
-1
s~ 1
be
execpt that any probability of playing a. is replaced by
1
1
We thus have that:
,s• ) = u.(s!,s• ) + 1 1 -1
1 1 -1
s~(a.) 11
· [u.(cr. ,s• ) - u.(a.,s• JI > 1 1 -1 1 1 -1
u.(s~.s• ) 11-
1
which contradicts the assumption that (si,s2·····si) is a NE.
First of all, notice that the first auction bid is a simultaneous move
8.D.3
game where a strategy for a player consists of a bid. Player 1, and b
( i)
2
Let b be the bid of 1
be the bid of Player 2.
lf b > b , Player 1 gets the object and pays b for it; 2 1 1 Player 2 does not get the object.
8-9
Thus, in this case:
.
u lb ,b l=v -b (l's valuation of the object 1 1 2 1 1 minus what he has to pay for it].
If b = b , each player gets the object with probability 1 2 '1 1 (vl-bJ) 0 = ul(bl + b2) = 2(vl - bl)+ 2 .
(iii)
~:
z·
Similarly:
u (b , b ) 2 1 2
=
(v2 - b2) 2
Therefore, we have for i,j e {1,2}, i 0,
J
I
1
2 (vi - bi) • bl (v.- b.),
(a)
j:
< b.
b.
1
:;e
•
= b2
b.> b. 1 J
We claim that no strategy for player 1 is strictly dominated.
negation that b
1
Suppose in
is strictly dominated by bl. i.e., for any b : u (bl ,b l > 2 2 1
u Cb ,b ). Take b2 =max {b ,bl} + l,then: b2 > b , and b2 > b}· 1 1 2 1 1 u (hi ,b2l = u (b ,b2) = 0, a contradiction. 1 1 1 1 is strictly dominated.
Hence,
Therefore, no strategy for player
Similarly, one can prove that no strategy for player
2 is strictly dominated, and thus no strategies are strictly dominated.
(b)
We now claim that any strategy b for Player 1 such that b > v , is 1 1 1
weakly dominated by v r Note that, if b > v l: 1
(i)
If b
2
=0 ul(vl' bl=v-v 2 1 1
u 1(bl, b2) (ii)
Jf b2
= v 1:
= v1 -
bl < 0 = u (vl, b2)
1 - v ) = 0 u1(vl, b 2 ) = -(v 2 1 1
8-10
(iii)
u l (vI' b 2 )
=0
ui (bi' b2 ) = vi - b1 < 0 • u 1(vi, b 2 } •
If vi < b2 • bi:
(iv)
u (vi , b l = 0 2 1 u (bl'b ) = 1/2(vi- b l < 0 = u (vi,b l 2 1 2 1 1 u (v , b ) = 0 1 1 2
ui(bi, b ) 2 Thus, in all cases, u (vi' b ) 1 2
2
= ui(v , b 2 ) 1 cases strict
u Cb , b ), and in_ some 1 1 2
!:
Thus, b > v is weakly dominated by vr
inequality holds. strategy b
=0
1
1
for Player 2 such that b
> v is weakly dominated by v • 2
2
weakly
We claim that, in this case, bi = 1
Now suppose vi > 2. dominates bl = 0.
2
Similarly, any
Observe that:
(i)
If b
2
= 0:
ui(l' 0) = vi -1 u
Co. 1
0)
1
= 2 vi
Since v > 2, u (l,O) - u (0,0) = (v -l) 1 1 1 1 (ii)
If b
= 1:
2
u C0,1) = 1
(iii)
If b
Thus, in all cases ui(l' b ) 2
!:
2
(v
1
- 1) > 0
0
u 0,. b ) = u (0,b l 1 1 2 2
> 1:
2
I
u Cl, 1) = 1
1 > 0
= 0.
u (0,b ), with strict inequality in some cases. 2 1
Finally, suppose that v e {1,2} . We claim that, in this case, 1
b = v - 1 weakly dominates b = v : 1
1
1
(i)
If b
(ij)
If b
(iii)
If b
2 2 2
1
< v - 1:
u Cv -1,b ) ::: 1 > 0 = u Cv ,b ) 1 1 1 1 2 2
= v -1: 1
uCv -1, b l = l/2> 0 1 2
> v -1:
u(v -l, b ) ::: 0 = u Cv ,b ) 1 1 1 2 2
1
1
= ui (vi,b2 l inequality in (i)
and (ii).
Similarly, it can be shown that: -if v
2
-if v2
> 2, b :!!::
2
= 1 weakly dominates bl = 0
1, b2::: vz-1 weakly dominates b' z= v2
8-11
(c)
Define the best response correspondence for Player 1 as the set of
maximizers of l's utility, given the strategy for 2.
We already have the
expression for u Cb , b ); in order to find out the best response 1 1 2 ce~rrespondence,
all we have to do is maximize this function.
Denoting this
best response by R (b ), direct maximization of u(b , b } yields: 1 2 1 2 if b
( (b2+ 1),
If
{b2' b2+1}.
if {b2}' {O,l.2, ... v }, if 1 {0. 1 • 2 •...• b2-1 ) if
< v -2
1 b = v -2 2 1 b = v -1 2 1 b v1 2 b > v 2 1 2
=
Similarly, if R Cb ) is the best response correspondence for Player 2: 2 1 {bl + R Cb ) = 2 1
< v2 - 2
•
if
b
{bl' b1+1}.
if
{bl}
if
b = v - 2 1 2 b v2 - 1 1 bl = v2
1}
,
(0,1, ... , v }, if 2 {0,1, ... bl-1}. if
1
=
bl > v2
A Nash equilibrium is a pair (bi,b2l where b~ e R Cb2)• and b2 e R Cbi)· It 2 1
can be verified that a NE always exists, and, for sufficiently large of v and v 1
2
Nash
values
equilibria are not unique.
More explicitly, it can be shown that, for v , v 1 2
2 the following are Nash
equilibria:
(i) (ii) (iii)
(iv)
If v = v : (vl' v ), Cv -1, v -l), (v -2, v -2) 1 2 2 1 2 1 2 If v = v +l: (v -1, v -2), Cv -2, v -2) 1 2 1 1 1 1 If v = v +1: (v -2, v -1), Cv -2, v -2l 2 2 2 1 2 2 If v > v +1: (v -x, v -x-ll, with 1 1 1 1 2
x
:=s
v - v
:=s
1
2
Note: if v = v +2: (v -2, v -2) is also a NE. 1 2 1 1 (v)
> v +l: (v -x-l, v -x), with 1 1 2 2 2
If v
8-12
:=s
x
:=s
v
2
- v
1
Note: if v
2
= v +2: Cv -2, v -2) is also a NE. 1 2 2
Thus, generally, uniqueness of NE does not hold, although existence does.
(a) If player i demands y
8.0.4 x
~
0 is payoff equivalent.
~
100, then any strategy of player j with
Therefore, there exists no strictly dominated
strategy.
(b)
Any strategy demanding more than $100 is weakly dominated,
Case
t.
player 2 demands y
payoff equivalent. Case
~
!:
100. Then any strategy of player 1 with x
~
0 is
player 2 demands 0 ::s y < 100. Then, player 1 could
demand x = 100 - y and would obtain a payoff of 100 - y.
Demanding x > 100
will give player 1 a payoff of 0. Therefore, any strategy demanding more than $100 is weakly dominated.
(c) Any pair (x, 100 - x) with 100
~
x
!:
0 is a pure strategy Nash equilibrium
of this game. Proof: Suppose, player 1 demands x with 100
!:
y = 100 - x, his payoff will equal 100 - x
0.
2:
x
!:
0.
If player 2 demands
If player 2 demands y > 100 -
x, the demands sum to more than $100 and both players get 0.
If player 2
demands 0 ::s y < 100 - x, he will obtain his demand and therefore be· worse off .
than if he would have demanded 100 - x.
Thus, if player 1 demands x, player
2's best response is to demand y = 100 - x.
Similarly if player 2 demands 100
- x, player 1's best response is to demand x.
8.0.5
(a)
Vendor 2.
Let x
1
be the location of Vendor 1 and x
2
be the location of
Thus, we can associate a strategy for Player i with x. e [0,1]. 1
First, let us find out the payoff function for each of the vendors.
Since the
price of the ice cream is regulated, we can identify the profit of each vendor
8-13
Suppose that x < x • In this case, 2 1 xl + x2 all consumers located to the left of (below) will purchase from 2 XI + X2 Vendor 1, while all customers located to the right of will buy ice 2 with the number of customers s/he gets.
cream from Vendor 2. ul (xi. x2)
Thus: xl + x2
-
xl + X 2]) ( ., length of [0, 2
2
XI + x2 U2(Xl' x 2 > = I 2
We can derive a similar result for x
= 21.
XI + x2 ,1]) ( • length of [ 2
2
<x : 1
• • Thus, summar1zmg:
1
-2 ' 1 - xl+ x2 ' xl
< x2
1 2
=X
"2
u2(xl' x2)
-
--
• xl
2
xl + x2 2
' xl > x2
It is straightforward to check that x.l = x firm can do better by deviating). x
1
= x2
< 1/2.
2
= 1/2 constitutes a NE (no
To show uniqueness, suppose first that
Then any firm can do better by moving by
since it will sell almost 1 - x it can-be shown that x
1
= x
2
1
£
> 0 to the right,
> 1/2 units rather than l/2 units.
> 1/2 does not constitute
8-14
'i
Similarly
NE. Suppose now that
x
1
< x 2 . Then firm 1 can do better by moving to x 2
this could not have been a NE.
£,
with
> 0, therefore
£
Similarly It can be shown that x
1
>x
2
does
not constitute a NE.
(b)
Suppose that an equilibrium (xi, x2, xjl exists.
xi • x2 = xj.
Then each firm will sell 1/3.
Suppose, first, that
But any firm can increase its
sales by moving to the right (~f xi = x2 = xj < 1/2) or. the left (if ·xi = ~ = xj
!:
1/2), a contradiction.
let's say xi = x2. xi + c.
If xi = x2 < xj. then firm 3 can do better by moving to
If xi = x2 > xj. then firm 3 can do better by moving to xi -
contradiction. points.
Suppose that two firms locate at the same point,
£,
Finally, suppose that all 3 firms are located at different
But then the firm that is located the farthest on the right will be
able to increase its sales by moving to the left by c > 0, a contradiction. Thus, there exists no pure strategy NE in this game.
Case !: u > w and 1 > y.
8.0.6
In this case player 1 always plays his dominant strategy a . 1
Player 2 will
play his best response to this strategy, i.e. if v > m he will play b , if 1 v < m he will play b Case
~:
2
and otherwise he will be indifferent.
u < w and 1 < y.
In this case player 1 always plays his dominant strategy a . 2
Player 2 will
play his best response to this strategy, i.e. if x > z he will play b , if 1 x < z he will play b
2
and otherwise he will be indifferent.
""C"'"a..=s.: :.e J,: v > m and x > z. In this case player 2 always plays his dominant strategy br
Player 2 will
play his best response to this strategy, i.e. if u > w he will play a , if 1 u < w he will play a Case 4: ==-
2
and otherwise he will be indifferent.
v < m and x < z.
8-15
a
In this case player 2 always plays his dominant strategy b . 2
Player 2 will
play his best response to this strategy, i.e. if 1 > y he will play a , if 1 1
C'3.se 5: ---
and otherwise he will be indifferent.
2
all other cases.
Suppose player 2 plays b
with probability « and. b
1
with probability 1-o:.
2
Player 1's best response will be a mixed strategy If he obtains the same payoff from playing either of his strategies: y - 1 ClU
+ (1-o:)l =
ClW
+ (1-o:)y,
Similarly suppose player 1 plays a 1-[3.
1
Cl
= --------~--u+y-1-w .
with probability f3 and a
2
with probability
Player 2's best response will be a mixed strategy if he obtains the same
payoff from playing either of his strategies: Z
-
X
f3=v+z-x-m· Player 1 playing a
(a ) with probability [3 (1-[3) and player 2 playing b (b ) 2 1 2
1
with probability « (1-o:) as defined above is a mixed strategy Nash equilibrium.
S.D. 7 (a)
Suppose w. = u. (CT!, CT•.). The following is true by definition: -1
I
1
v . = m i n ( max u • (CT. , ,. • ) ) -1
CT • -1
1
I
-1
-1
2:
min [u.(CT!', CT • ) ) = CT.
CT • I
I
-1
1
U • (CT!', 1 1
tr•.) -1
-I
We have:
Suppose (CT!, CT•.) is a mixed strategy Nash equilibrium.
(b)
I
u.(CT~, CT•.) 1
I
-1
= w.. -1
-1
=max u.(CT., CT•.) and u .(CT!, CT•.) =max u .(CT!, CT.). I
CT.
1
-1
-1
1
-1
CT •
I
-1
I
-1
Since
- I
this is a zero-sum game we can rewrite the second equality: - u. (CT! ,CT•.) = max [ - u. (CT! ,CT • ) I or, 1
I
-1
I
CT •
I
-J
• • )- -m.In
U • ( CT. ,CT • 1 1 -1
..
-I
This yields: max u. {CT., CT•.) CT. I
1
I
-1
= u.I {CT~,I
-
.
CT.
• U. (CT. 1 I
8-16
-1
-I
CT•.) - m1n u. ( CT. , CT • ). -1 -1 1 I CT. -I
,CT • ) •
Now,
= u 1(cr!,cr• 1),
min [max u.(cr.,cr .)] :s max u.(cr.,cr•.) 1 1 -1 1 1 -1 cr cr. CT. 1 - 1 I
- m1n u.1(cr!1 • •
u. (cr!, cr• .) -1 1 1
CT
CT
-
•) < -1
1
-
max [min u. (cr. 1 1 CT cr. •
-1•
•
and also
CT- i
Taking these two
)],
-1
1
inequalities together we get: v. =min [max u.(cr., cr.)] :s u.(cr!, cr•.) :s max [min u.(cr. -1 1 1 -1 1 1 -1 1 1 CT
, -1
CT. 1
CTi
But we know from (a) above that v.
-1
v. = u . (cr!, cr•.)
-1
1
I
-1
w..
!:
CT
, CT
• -1
=
• )] -1
W, -1
Therefore, we must have:
-1
= w-1.. (cr~.
(c) Suppose (cr!, cr•.) and 1
-1
1
cr'.) are mixed strategy Nash equilibria.
We
-1
must therefore have that:
= -u -1. (cr~1 ,cr•.) -1
(i) u.(cr• ,cr•.) !: u. (cr~ ,cr•.)
1 -1
1
1
1 -1
= u.1 (cr~1 ,cr'-1.)
-u . (cr~ ,cr' .) -1 1 -1
:!::
(ii)u. (cr! ,cr•.) = -u . (cr! ,cr•.) :s -u . (cr! ,cr' . ) = u. (cr! ,cr' . ) :s u. (cr~ ,cr' . ) 1 1 -1 -1 1 -1 -1 1 -1 . 1 1 -1 1 1 -1
(these inequalities follow from the properties of NE and from the zero-sum property). From part (b)
we know that
u.(cr~,cr'
1
1
.)
-1
= u.(cr!, 1 1
cr•.) = v. = w .. -1 -1 -1
This, together with (i) and (ii) above yield:
v. = u. (cr~ ,cr' . ) :s u. (cr~ ,cr•.) :s u. (cr! ,cr•.) :s u. (cr! ,cr' . ) :s u. (cr~ ,cr' . ) = v. -1
1
1
-1
1
Therefore, u. (cr~ ,cr•.) 1 1 -1
1
-1
1
1
= u.1(cr!1 ,cr•.) -1
-1
1
1 -1
u . (cr~ ,cr . ) -1 1 -1 1
zero-sum).
u. (cr. , cr•.) 1 1 -1
'V cr.. 1
'V cr . , so u . (cr~ ,cr•.) = u . (cr~ ,cr' . ) -1 -1 1 -1 . -1 1 -1
=u
that u . (cr~, cr•.) -1
:!::.
-·
. (cr~ , cr' . ) since u. {cr~ ,cr•.) -·
1
This implies that
-1
1
(cr~
1
1
-1
-1
= u. (cr! ,cr' . ) = u. (cr~ ,cr' . ) = v .. 1 1 -1 1 1 -1 -1
Since (cr!, cr•.) is an equilibrium we have: u. (cr! ,cr•.) 1 -1 1 1 -1 u. (cr~, cr•.) = u. (cr!, cr•.) 1 1 -1 1 1 -1
1
1
-1
!:
'V cr., so
u. (cr. ,cr•.) 1 1 -1
1
Similarly, u . (cr~ ,cr'.) -1
it: !:
1
u . (cr~ ,cr . ) -1
1
1
1
!:
'V cr . (Note,
-1
= u. (cr~ ,cr' . )
-1
-1
and the game is
-1
,cr•.) is a mixed strategy Nash equilibrium. -1
Similarly it can be shown that (cr!, cr' .) is a mixed strategy NE. 1
8.0.8
-1
Let (cr. ,cr .) be a mixed strategy Nash equilibrium, and suppose in 1
-1
negation that cr i assigns strictly positive probability to the pure strategies s ~ and 1
s~, 1
i.e. cr. is not degenerate. 1
This implies that
8-17
s~1
and
s~1
are each a
best response to
o:s~I
+
fT" • -1
(1-a:)s~I e S .•I
1 2 u.( o:s. + (1-a:)s .• 1
I
1
1 2 and u.(s.,CT .) = u.(s .• CT .). l
I •
-1
I
I
-1
By the convexity of S., I
and since u. is strictly quasiconvex we have that I
fT" .) -1
1 2 > u.(s., ,. .) = u.(s.,CT .) for all« e (0, I
I
-1
I
I
-1
1).
1 2 This contradicts the fact that s. and s. are each best response to CT .. I
I
-1
Therefore, any mixed strategy NE of this game must be degenerate.
8.1;).9 (a)
Playing L or R is quite risky, since we do not know what player 1
will be playing.
The risk of obtaining a payoff of -49 is very large compared
to the payoff of 1 if player 2 played L, and the risk of obtaining a payoff of -100 is very large compared to the payoff of 2 if player
~
played LL or R.
Therefore, it seems "reasonable" to play M.
(b)
The two pure Nash equilibria of this game are (U,LL) and (O,R).
To
check for mixed strategy Nash equilibrium, player 1 must mix between U (with probability p) and D (with probability 1-p).
Player 2 then has 11 possible
mixing combinations: {LL,U, {LL,M}, {LL,R}, {L,M}, {L,R}, {M,R}, {M,L,R}, {LL,M,R}, (LL,L,R}. {LL.L,M}, and {LL,L,M.R}. We will show that only the first
combination. (LL,L}, is part of a mixed strategy NE. For player 2 to mix between LL and L (with probabilities q c;:1.d 1-q p·(2)+(1-p)·(-l00) = p·(l)+{l-p)·(-49) which
respecti.vely), we must have that
The utility of player 2 from each strategy is then: u CLU =
2
Then, for player 1 to mix between U and 0, we must have: q =
~·
and
q·(IOO)+(l-q)·(-100) = q·(-100)+(1-q)·(IOO) which yields
u (U) = u CD> = 0. Therefore, p = 1 1
'2i
and q =
strategy NE. For the rest of the answer we call this "NE•".
~
is a mixed We now show that
no other mixing combination of player 2 can be part of a mixed strategy NE. (i} If player 2 mixes with the combination (LL,M}, we must have
p =
~~
, which gives utilities
u (LU = ui(M) = 0, and
2
8-18
1
= Si > 0, so
this cannot be part of a mixed strategy NE . .
(ii) If player 2 mixes with the combination {LL,R), we must have
p
= 4. which gives
utilities
u CLU 2
= u 2 (R) = - 49.
and
u (M) 2
= o.
so this
cannot be part of a mixed strategy NE. (iii) If player 2 mixes with one of the combinations {L,M}, {L,R}. {M.R).
{M,L,R), then player 1 will have 0 as a strict best response, which in turn has R as player 2's strict best response. (iv) If player 2 mixes with one of the combinations {LL,M,R}, {LL,L,R}, or (LL,L,M,R), then the analysis of (ii) above implies that this cannot be part of a mixed strategy NE. (v) If player 2 mixes with the combination {LL,L,M} then the analysis of (i) above implies that this cannot be part of a mixed strategy NE.
(c)
The choice in part a) is not part of auy NE described above.
to see that strategy M is rationalizable: If plyer 1
play~
p =
It is easy
1
2 then M is
the unique best response of player 2.
(d)
If preplay communication is possible, the players can agree to play one
of the pure strategy NE, which are payoff equivalent and Pareto dominant for both players. Therefore, player 2 will play either LL or R depending on the agreed upon equilibrium.
8.E.l
There are four pure strategies contingent on the type of player:
AA: Attack if either weak or strong type, AN: Attack if strong and Not Attack if weak, NA: Not Attack if strong and Attack if weak, NN: Never attack. The expected payoff of each pair of strategies can be easily computed and are
8-19
given in Figure B.E.l:
Player 2
AN
AA AA
NA
M 3M s+w -w ----2 4 2 -4 4 2 4
M s+w M s+w M s+w
-4 2 • -4 2
AN
M s+w M --4 2 • -2 4
Player 1 NA
-w 3M s+w 2 •4 4
- --
I
M-s
M-s I
4
'
M s --2 4
I
M 2
4
M-w
O,M
4
0
M 10
I
'
I
NN
NN
M --2 4
M-w
I
•
I
4
M-w M-w 4 4 I
0
M 1 2
-
M 210
M 210
0,0
Figure 8.E.l Any NE of this normal form game is a Baysian NE of the original game. Case 1: M > w >
-
S1
and w > M/2 > s
From the above payoffs we can see that (AA AN) and (AN~AA) are both pure 1
strategy Bayesian Nash equilibria. Case 2: M > w > s, and M/2 < s
-
From the above payoffs we can see that (AA,NN) and (NN,AA) are both pure strategy Bayesian Nash equilibria. Case 3: w > M > s, and M/2 < s
-
From the above payoffs we can see that (AN,AN), (AA,NN) and (NN,AA) are pure strategy Bayesian Nash equilibria. Case i: w > M > s, and M/2 > s From the above payoffs we can see that (AA,AN), (AN,AA) and (AN,AN) are pure strategy Bayesian Nash equilibria.
8-20
8.£.2
(a) Suppose that all the bidders, use the bidding function b(v). that
is if their valuation is v they bid b(v). The expected payoff for a bidder whose valuation is v. is given by: 1
Cv -bCvi))·Pr(b(v ) > b(vj)} + O·Pr{bCv ) < b(vj)}. (Note that we ignore a tie 1 1 1 since it is a zero probability event given that b(v.) is a monotonic linear 1
function.)
Since both players use the same monotonic linear bidding function
then Pr(b(v.) > b(v .)} = Pr(v. > v.} = v./ 1 J 1 J 1
-
uniformly distributed on [O,v].
v (since the
valuations are
b(v) will in fact be the equilibrium bidding
function if it is not better for a player to pretend that his valuation is different.
To check this let us solve a bidders problem whose valuation is v.
1
and who has to decide whether he wants to pretend to have a different
-
valuation v'. The bidder maximizes: (v. - b(v))·(v I v), and the F"OC is:
-
(v. - b(v))lv 1
1
- = 0.
-
b' (v) vi·•
b(v) is an equilibrium bidding function if
it is optimal for the bidder not to pretend to have a different valuation, that is, if v
= v.1
is the optimal solution to· the above FOC, i.e., If
(v. - b(v. ))lv - b' (v.) v ./v = 0. 1
1
1
This is a differential equation that has to
1
be satisfied by the bidding function b(v) in order to be an bidding function.
equilibriu~
The solution to this differential equation is b(v) = v/2.
Thus a bidder whose valuation is v will bid vl2 (a monotonic linear function).
(b)
We can proceed as above by assuming that all bidders use the same bidding
function b(v). ( v.) 1
1-1 1 v. -
Now, Pr{b(v.) > b(v.)"' j 1 J
Proceeding as in· a) above,
~ i}
= = Pr{v.> 1
v."' j J
~ i}
=
we get the following differential
equation: (I -1 )( v.
- b( v. ) )( v.) 1 1 1
1-2 /v
1v
b'(v.) (v.) 1- 1
differential equation is b(v) =
I
1-1
1
1
· v.
As I ...
= 0. The solution to this m,
b(v) =
1-1
1
.
• v ... v, 1.e., as
the number of players goes to infinity each player will bid his valuation.
8-21
8.E.3
= H or
A firm of type I
L, will maximize its expected profit, taken
as given that the other firm will supply qH or qL depending whether this firm is of type H or L.
A type i e {H,L} firm 1 will maximize: 1
2
1 c.)q.]
1
Max (1-IJ)[{a - b(q.+ qH) - c.).:(.) + IJ.Ha 1 1 1 1
1
1
qi
The FOC yields:{l-IJ)[a -
2 2 b(2q~+ qH ) - c.) + IJ[a - b(2q~+ qL ) - c.) 1 1 1 1
'l'b .
. Bayes1an ' Nas h equ1 1 r1um: qH1 a symmetr1c
=
2 qH
= QH
-
an d
QLl
= 0.
In
2 = QL = qL ·
Plugging this into the F.O.C we get the following two eqations: (1-IJ}[a - 3b qH - cH] + IJ[a ~ b(2qH+ qL) - cH) = 0, {1-IJ)[a - b(qH+ 2qL) - cL] + IJ[a - 3b qL - cL) • 0. Therefore, we obtain that qH = [a - cH +;eeL - cH>]
·it; ,
qL = [a - cL + l;IJ(cH - cL))
8.F.l
·it; .
For the proof of Proposition 8.F.1, we refer to: Selten, R. (1975) "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games," InternationaL JournaL of Game Theory.
Another good source is Section 8.4 in Fudenberg & Tirole, (1991) Game Theory, MIT press.
8.F.2
For the solution of this question we refer to: van Damme. E. (1983). Refinements of the Nash Equili.brium Concept. Berlin: Springer-Verlag (pp. 28-31).
8-22
8.F.3
For the proof of this statement, we refer to: Selten, R. (1975) "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games," International Journal of Game Theory.
Another good source Is Section 8.4 in Fudenberg & Tirole, (1991) Game Theory, MIT press.
8-23
CHAPTER 9
9.8.1
There are 5 subgames, each "ne beginning at a different node of the
game (this includes the whole game itself).
9.8.2
(a)
Clearly if
fT" o
r E'
is a Nash equilibrium of
and
rE
is the only
•
proper subgame of
r E'
then 0
r E"
Thus, by definition
(b)
Assume in negation that
t1'
induces a NE in every proper subgame of the game
fT" o
is a subgame perfect NE of 0
t1'
r E"
is a subgame perfect equilibrium
of
r E'
but
it does not induce a subgame perfect Nash equilibrium in every proper subgame of
r E"
Then there exists a proper subgame (say, HE) of
restriction of
t1' o
to
nE
is not a SPNE.
rE
This implies that there exists a
proper subgame (say, nE) of nE in which the restriction of NE. ~
in which the
fT" o
to ~ is not a
Since nE is a proper subgame of nE and ITE is a proper subgame of is also a proper subgame of
proper subgame of
9.8.3
rE
r E'
Therefore,
0
t1'
r E'
then
does not induce a NE in a
- contradiction.
Let player l's pure strategy be s e
player 3's be s = (x,y,z) where x,y,z e{l,r}, x is what 3 does after player 1 3 played L, y is what 3 does after 1 played R and 2 played a, and z is what 3 does after 1 played R and 2 played b. The pure strategy SPNE identified in the example is (R,a,(r,r,l)), which is easily seen to be a NE. Three other NE which are not SPNE but yield the same outcome are (R,a,(l,r,l)), (R,a,(l,r,r)) and (R,a,(r,r,r)).
For each of these NE, player 3 is not choosing a rational
move for some of his nodes. If player 3 is reached, he will always do the best thing for himself, therefore if 1 plays L, 3 will play r. To support this as a
9 - l
NE we need strategies for 2 and 3 that give player 1 less than -1 in the subgame starting from player 2's node, but recall that these strategies need not be subgame perfect. Therefore, (L,b,(r,r,r)) would be another NE, in which player 3 is again not
~cting
sequentially rational. There can be no NE with
player 2 being reached and then choosing b, since after this move 3 will choose l, giving 2 a payoff of -1. Player 2 will then prefer to deviate and play a, resulting in a higher payoff no matter what 3 will do then.
Proposition 9.8.1 claims that in any game, in which no player has the
9.8.4
same payoffs at any two terminal nodes, there exists a unique SPNE.
Now
suppose one of the players has the same payoff at one of the terminal nodes. This means that he will be indifferent between two actions that lead to either one of them.
However, since
~he
game is zero-sum, the other player will also
be indifferent between the payoffs resulting from these two actions.
Thus, in
a finite zero-sum game of perfect information, there may exists many different (in terms of the strategies played) SPNE (because of potential indifference of the player between different terminal nodes), but all of them will yield the same payoffs for the players, i.e. there are unique SPNE payoffs.
9.8.5
Note: for parts (a} and (b), m. denotes the number of strategies that .
1
player i has. For the remainder of the question, m. denotes the move of player 1
i. This is done to be consistent with the question.
(a)
Since the game is simultaneous, each player i has m. pure strategies. 1
If we allow for mixed strategies then each player has a continuum of m.
•
strategies: l:.= { (pl, ... p 1
(b)
m.
1
)
E
IR
1
+
Since player 1 moves first, he cannot make his strategies contingent on
9 - 2
any history, thus he still has m (pure) strategies. 1
Player 2 can, however,
condition her play on player I's move, thus allowing her to specify one of m 2 ml actions for each of player l's m plays. Therefore, she has 1m ) (pure) 1 2 strategies. There is, of course, a contintJum of mixed strategies. (c) Assume in negation that for all Cm ,m ) and (m} ,m2) where either m :;e mi or 1 2 1
m2-= m2
we have that
cf»i(m ,m2 1 "' ct»1Cm}.m2l for both i=l,2. Then, due to lack 1 •
of indifference (the negation condition) player 2 will have a unique best response at each of her nodes. By backward induction, and by lack of indifference, player 1 will have a unique best response to the SPNE strategy of player 2, which contradicts multiple SPNE. (d) Since player 2 has no indifference, she will have a unique best response after she is reached. Let (mi,mz) be thf" NE for the game in (a) yielding a payoff of
Jt
1
to player 1. Clearly, player 1 playing mi and player 2 playing m2
at each of her nodes is a NE in the extensive form game. However, this is not necessarily a SPNE since player 2 is not necessarily playing a best response at each of her nodes. Note that in any SPNE player 2 will play m2, after player 1 plays
mi.
therefore player 1 can promise himself a payoff of
1tr
Given
player "2's unique SPNE strategy in the sequential game, player J can therefore do at least as well as the NE
Cmi,mz)
in any SPNE.
This conclusion would not
necessarily hold for NE of the sequential game - low payoffs for player 1 could be sustained in a NE with incredible threats by player 2. (e) (l} Consider the normal and extensive form versions of a game depicted in Figure 9.B.S{e.ll:
9 - 3
Player 2
u
L
R
0,1
1,1
Player 1 t-----+---~ D
0,0
Player 2 1,0
1 1
0 0
1
0
Figure 9.B.5(e.l) In this game condition (ii) holds for some strate&Y ·pairs. It is easy to see (by backward induction) that any path in the extensive form game can be supported by a SPNE. Therefore, if we consider the NE (U,R) in the normal form version, then the conclusion of part (d) does not hold. (2)
Consider the normal and extensive form versions of the "matching
pennies" game depicted in figure 9.B.S(e.2):
Player 2
u Player
L
R
1,0
0,1
I~---+---~
Player 2 D
0,1
1,0
1 0
0 1
0 1
1 0
Figure 9.B.5(e.2) The unique mixed strategy NE in the normal for game gives each player an expected payoff of 112.
lt is easily seen, however, that the only two SPNE in
the extensive form game, (U,(R,L)) and (L,(R,L)), give player 1 a payoff of 0. Again, the conclusion of part (d) does not hold.
9 - 4
9.8.6
To find the mixed strategy equilibrium of the post-entry subgame,
suppose that firm E plays Small with probability x and Large with probability 1-x; firm I plays Small with probahility y and Large with probability 1-y. For firm I to be indifferent between playing Small and Large we need: -6x - 1(1-x) • lx - 30-x), or x = 2/9. For firm E to be indifferent between -6y - 10-y) = ly - 3(1-y), or y = 2/9.
playing Small and Large we need:
Thus in the mixed strategy equilibrium of the post-entry game firms E and I play Small with probability 2/9.
This gives both firms a payoff of -19/9,
which will cause firm E to choose not to enter. Therefore, the following strategies constitute a SPNE: firm E plays Out at the first node and randomizes between Small and Large in the second node (with probabilities 2/9 and 7/9 respectively).
Firm l plays Small. with probability 2/9 and Large with
probability 7/9 given that firm E entered.
9.8.7 (v,
Consider the last period of the game. and the offered player will accept.
0)
The offering player will offer
In the second to last period, the
offering player will offer the other player a share that will make him indifferent between accepting now and rejecting and obtaining v in the next period.
With the given cost c for making an offer, the offered player in the
second to last period will accept any offer that gives him at least v - c. Thus, the offering player in the second to last period offers ( c, v-el. Similarly, we can show that the offering player in the third to last period offers (v, 0). If T is odd, then the player 1 will be the last player to make an offer. Thus, he will offer (v,
0)
in every period in which he makes offers and accept
an offer if he obtains at least v-c.
Player 2 will offer (c, v-c) in every
9 - 5
period in which she makes offers and accept if she obtains a non-negative payoff. The result is that in the first period player 1 will offer (v, 0) and player 2 will accept. If T is even, then the player 2 will be the last player to make an offer. Thus she will offer (0, v) in· every period in which she makes offers and accept an offer if she obtains at least v-c.
Player 1 will offer (c, v-c) in
every period in which he makes offers and accept .if he obtains a non-negative payoff. The result is that in the first period player 1 will offer (c, v-c) and player 2 will accept. The argument above holds for any finite game, but if T = m, then there can be many SPNE of this game.
For an analysis of this case we refer to:
Rubinstein, A. (1982) "Perfect Equilibria in a Bargaining Model,"
Econometrica, 50:97:109.
· 9.8.8
Take all the proper subgames that do not strictly contain another
proper subgame. finite.
Since the whole game is finite, all proper subgames must be
Thus, by Proposition 8.0.2, there exists a mixed strategy NE in each
of these subgames.
Now construct a new finite game with terminal nodes at the
root of each of the previous subgames, and
associate the payoff for every
player for each such terminal node by the payoff obtained from playing one of the mixed strategy NE in the subgame that followed in the original game.
We
can again look for all the proper subgames that do not strictly contain another proper subgame.
Repeating the above process we can find the mixed
strategy Nash equilibria in these subgames. · Since the game is finite, repeating the above procedure will end after a finite number of rounds. have therefore constructed a SPNE of the game.
We
In every subgame players will
play the strategies that constitute one of the mixed strategy Nash equilibria
9 - 6
in this subgame.
9.8.9
The pure strategy NE of the one-shot game are (a ,b ) and (a ,b ). 2 2 3 3
Thus any SPNE involves playing either of these in the second period. Thus, playing either of these strategies in both periods constitutes a SPNE. Additionally, the players could use them in any combination in the two periods.
This results in the following two classes of SPNE (a total of four
SPNE): and player 2 plays b. in both periods, i e {2,3}.
1) Player 1 plays a.
1
1
2) Player 1 plays a. in the first period and a . in the second period; player 2 J
1
plays b. in the first period and b. in the second period, i,j e {1,2} and i:ltj. J
1
However, there exist more SPNE in (or 2) can
~unish
thi~
.;arne.
the other playe-:-- by playing a
The reason is that player 1 3
(b ) in the second period, if 3
the other player did not cooperate in the first period.
[Note that this can
only happen because there are more than one NE in the second stage}. This gives rise to two more classes of SPNE: 3) Player 1's strategy: Play ai, i if player 2 played b
E
{1,2,3} in period 1; Play a
2
in period 2
in period 1, otherwise play a . 1 3
Player 2's strategy: Play b
1
in period 1.
Play b
2
in period 2 if player 1
played ai in period 1, otherwise play b . 3 4) Player 2's strategy: Play bi' i if player 1 played a
E
{1,2,3} in period 1; Play b
2
in period 2
in period I, otherwise play b . 1 3
Player l's strategy: Play a
1
in period 1.
Play a
2
in period 2 if player 2
played bi in period 1, otherwise play a . 3 Too check that each of the 6 SPNE described by these classes in indeed a SPNE, note that by deviating a player loos.es 4 in the second period (no
9 - 7
discounting) and no player can gain more than 3 in any of the described strategy profiles.
9.8.10
The simultaneous move game following the choice of both players to
accommodate has the following Nash equilibria: both (U,L) and (D,r) if x and only (U,L) if x < 0.
i!::
0,
This implies that if x < 0, then the only
continuation equilibrium payoffs after both firms accommodate is (3,1), and therefore the whole
game will have the same SPNE as discussed in Example
9.8.2, except for the fact that the players will play CU. L) after they both accommodated. If x
2:
0. the SPNE discussed in Example 9.8.3, with the addition that the
players will play (U,l) after they both accommodated, will also constitute a SPNE in the modified game. However, if x
0, there will be additional SPNE.
2:
Now (D,r) is also a
Nash· equilibrium in the subgame after both players accommodated.
In this case
we can "replace" the simultaneous move game after the entry of firm E by the game in figure 9. 8.10:
Firm I Ace. Fight Ace.
x,3
-2,-1
1,-2
-3 -1 '
FirmE Fight
Figure 9.B.l0 Suppose first that x
i!::
1.
Then player E playing In, both players
accommodating, and then playing (O,r) constitutes another SPNE.
9 - 8
Now, suppose 0 s x < 1.
For the given value of x, this subgame no longer
has a pure strategy equilibrium. subgame is as follows:
The mixed strategy Nash equilibrium of this
Firm E plays Accommodate with probability US and
Fight with probability 4/5.
Firm I !"lays Accommodate with probability 1/(2-x)
and Fight with probability (1-x)/(2-x).
Firm E's payoff in the subgame after 0 s x < 2/3 is negative. This
its entry is therefore (3x-2)/(2-x), which for analysis implies the following additional SPNE:
If 0 s x < 2/3, firm E will play Out at the first node, given that when firm E plays In, the firms play the mixed strategy NE described above in the reduced continuation game. If 213 < x < l, firm E will play In at the first node, given that when firm E plays In, the firms play the mixed strategy NE described above in the reduced continuation game. If x = 213, firm E will randomize between playing In and Out at the first node, given that when firm E plays In, the firms play the mixed strategy NE described above in the reduced continuation game.
9.8.11
Direct calculation show that it is not worth for firm A to stay in the
market for t > 20, even if it has a monopoly position. not benefit by staying in the market if t > 25. the last period in which A could still be in.
Similarly, firm B will
Consider period 20.
This is
If 8 continues to stay in, it
will obtain monopoly profits during periods 21 through 25 of: (51-2·21) + (51-2·22) + (5!-2·23) + (51-2·24) + (51-2·25) = 25. If firm A were to stay in at period 20, then 8 will obtain its duopoly profits of 10.5 - 20 = -9.5 in period.
The net profit for firm 8 of staying in at
period 19 is therefore 25 - 9.5 = 15.5 > 0.
Thus, at period 20 firm B will
stay in no matter if firm A is in the market or not.
9 - 9
This implies that if
firm A stays in at period 20 it will make a loss of (lOS - 10 · 20) = - 95, which ln turn implies that firm A will stay out from period 19 on. The above argument can be applied backward and we will get the (unique) SPNE of this game.
However, the following reasoning applies: The analysis
above shows that with rational play firm 2 will exit after firm 1. Therefore, as far as firm 1 is concerned, it will never make monopoly profits. It follows that we only need calculate the last period in which firm 1 makes positive duopoly profits and this will be the last period in which both firms enter the market. In later periods, firm 2 will be in the market alone (until period 25). The unique SPNE is: For t = 1,2, ... , 10 12, ... , 25 A is out an B stays in; for t
9.8.12
i!::
both firms stay in; in t = 11,
26 both firms are out.
The unique SPNE strategies are as follows:
The offering player i offers (
ca .-a.a .l
o-a. > J
o-a.a.l I J
·v
J
'
'J·v].
ct-a.a.> 1 J
accepts if and only if he is given a share of at least
The offered player j
ca .-o.o .) 1
J · v. To show
J
o-a.a .l I J that these strategies form a SPNE, suppose that the offered player j does not
0-o. l accept the offer.
1
Then he will offer - - --·v in the next period and the
o-a.a .) 1 J
current offered player i will accept (if j offers less then i won't accept
given the strategies). Player j's payoff will be
a• J
Therefore he did not benefit by deviating. deviates and uses a different strategy. offering more than
c 1-a.a I
I
I
o-a.I aJ.>
v=
J
1
J ·v.
c1-o.a .) 1 J
Suppose the offering player i
He clearly does not benefit by
ca .-a.a .) 1 J
ca .-a.a .l
o-a.1 >
J • v to player j.
.) J
9 - 10
Suppose then that he offers less.
ca.-a.a .) 1 1
Player j will not accept, and offer - -- --=J:... • v to player i in the next
0-a.a .l 1 J Player i will then accept and obtain a payoff of
period.
a.2 · 1
c1-a i )
c1-a . > · v < ___,J"-- • v. o-a .a.) c1-a. a .) 1
J
1
Thus he will choose not to deviating from the
J
above strategy.
9.8.13
In the exercise with the cost of delay c being equal for both
players, there will generally be many SPNE of this game. For an analysis of this case we refer to: Rubinstein, A. (1982) "Perfect Equilibria in a Bargaining Model,"
Econometrica, 50:97-109 (pp. 107-108).
9.8.14
(a)
The extensive form ot the game is depicted in figure 9.B.l4(a)
below (the letter "d" is used instead of
a).
Simple backward induction leads
to the unique SPNE which is shown by arrows in the figure: Firm E enters at t=O, and always plays In thereafter. Firm I plays Accommodate for all t=l,2,3.
(b) The extensive form of the modified game is depicted in figure 9.B.l4(b). Using Backward induction, firm I will always play Accommodate in period t=3, and therefore if t=3 is reached, firm E will play ln. This causes firm I to choose Fight in t=2 since this causes firm E to exit the market, and due to condition A.2 we have that for firm 1: Z
+
ay
+
a2X = Z
+
cS(y
+ ox)
>
Z
+
a(l
+
a)z = Z
+
az
+
a2Z
•
This causes firm E to choose Out in t=2. Working backward we get that at t=l, firm I chooses Accommodate and firm E chooses ln. E at t=O depends on the value of k. enter, and if k < 1 then E will enter.
However, the choice of firm
If k > 1 then firm E will choose not to For k = 1 both are part of the
9 - 11
(unique) continuation SPNE, so there are two SPNE in this case.
T
Enter
t=O
l
T I I t=l
I I
i
T
I I t=2
I I
-1- k 2 y+dx+d
i
T I I t=3
1+d -k 2 +dz+d
1- d -k
+dy+d
-1-d -k
-1 +d -k y+dz+d\
2
y+dy+d
2
I I
Ft.i 2
2
2
2
2 l+d+d-k -1+d-d-k 1- d+d- -1-d-d-k 2 2 2 2 2 +dy+d z z+ y+ "i y+dz+d 2 +dz+d y +dy+d +dy+d
1- d + d -k 1- d- d -k
Figure 9.B.l4(a)
9 - 12
2
T
t=O
l
T I I t=l
I I
'
E Out
T I I t=2
I I
-1- k 2 y+dx+d
'
-
E Out
T I I t=3
I I
1- d -k
l+d -k 2 z+dz+d x
2
z+dy+d x
i 2
2
1+d+d-k 1+d-d-k z+dz+cfz z+dz+d)
Figure 9.B.l4(b)
9 - 13
9.C.I
First, if tr is a NE then (i) and (ii) must hold.
If (i) weren't
satisfied, then some player's information set is reached with positive probability in which he is not playing a best response to his opponents' strategies, contradicting the fact that tr is a NE.
If (ii) were not satisfied
then some player's information set is reached with positive probability in which his beliefs are not correct given his, and his opponents' strategies. Therefore. this cannot be an equilibrium. Second, if (i) and Cii) are satisfied, then tr is clearly a NE since at each information set which is reached with positive probability, sequential rationality implies that each player is playing a best response to his opponents' strategies, with correct beliefs at each such information set.
Let tr f" be the probability that firm I fights after entry. let
9.C.2
firm I's belief that ln tr , cr
1
1
be
was E's strategy if entry has occurred and let tr , 0 1
denote the probabilities with which firm E actually chooses Out, In , 1
2
and In
~-t
2
respectively.
2 As in example 9.C.3, we cannot have ,.,. < 3 (same analysis). We could, however, 1 2 have JJ. > j : In this case firm I will play "Fight" with probabi.iity 1, and 1 having 7<0 implies that firm E will choose "Out", which supports any beliefs in the information set of firm I. This concludes that one class of weak perfect Bayesian equilibria is with
~~~ ~·
firm E playing "Out" and firm l
playing "Fight". The second class of weak PBE is where 9.C.3 with
trF = 11(7+2), and
~~=
(tr ,trl'tr ) =
0
2
§.
and the analysis is as in example
(0,~.~).
This, however, is an
equilibrium only if 1 is "large enough" in the interval (-1,0). When 1 is close to -1, then trF is close to l, and the expected payment for firm E of the
9 - 14
strategy
Ccr ,cr ,cr ) • 0 1 2
(0,~.~) Is close to -1, therefore E would choose "Out"
with probability 1. If, on the other hand, 7 is close to 0, then crF is close to ~· and the expected payment for firm E of the strategy
(0,~,~)
(cr ,cr ,cr > = 0 1 2
is close to 1, and E would be happy with the mixed strategy described.
The set of 7's for which the equilibrium of example 9.C.3 will still be an equilibrium can be calculated.
9.C.3
(a) For given values of
~
and a, there exists the following weak
perfect Bayesian equilibrium (WPBE): Case
1: ;\[(1-a)vH
+ avL] + (1-;\)avL
!:
VL.
The seller will demand a price of (1-a)vH + a.vL in the first period and vL in the second period. at most vL.
A buyer of type L will buy the good if the price is
A buyer of type H will buy the good in the first period if the
price is at most (1-a)vH + avL and in the second period if the price is at most vH. To show that these strategies constitute an equilibrium suppose the high type buyer deviates and does not buy in the first period.
Then he will buy
the good in the second period and obtain a payoff of acvH - vL).
Buying In
the first period gives him a payoff of VH- (1-a)vH- avL= acvH - vL). does not benefit by deviating.
Thus he
A player of type L does not benefit by
deviating and accepting in the first period since he will obtain a negative payoff.
The seller will not benefit by demanding a higher price in the first
period since nobody will buy at that price (both types will then buy in the second period).
Suppose he offers a lower price.
If this price is higher
than vL' then only the high type will buy, and the seller will obtain a lower price without increasing his sales volume.
Suppose he offers the good for vL'
then both types will buy the good and the seller will make a profit of vL.
9 - 15
If
he uses the above strategy he will make a profit of (1-A)cSvL
!:
vL'
A((l-a)vH+ avLl +
therefore he will not deviate.
Case ~: MU-a)vH + avL) + (h\)avL < VL. The seller will offer the good for vL in every period, and both types buy the good in period 1. Clearly, both types of buyers will not benefit by deviating, since they obtain the- good for the "lowest possible price" vL' and the analysis of Case 1 shows that the seller will not deviate with the assumption of case 2.
(b)
Suppose the buyer offers the good for the price x in the first period and
y in the second period (clearly, x,y e [v.v]). Then there exists a unique
-
type v• who is indifferent between buying the good in the first period or in the second period,
v• -
X
(x -
= a(v• - y), or •·• =
ay)
Given that all
•
c1-a > types v
!:
v• will buy the good in period 1, in equilibrium we must have that
the price y was set to maximize the second period profits: y • Pr{v!:y I v
y - v 1 -
v•-
- v -
The FOC implies that the optimum will be attained at y into the expression for v•, we obtain: v•
= (1
~
- y)
y(v•
•
v• v• . Plugging this 2
--
o/2l ·
The expressions for y
and v• obtained imply that if the seller charges x in the first period he will charge
2
~ a in the second period,
all buyers of type v
will buy in the first period, and all types with ( buy in the second period.
rr
=[ v = [
v-
~
~ 8 /Z)
!: 2:
v• = { 1 _ 8 /l) v
!:
~ a
2
will
The seller's profits will thus be:
.,..,..-x---:::-=,.... ] . x (1 - a/2l
(1
1
X
+ 0.
X
2 -
cV2) ] . X + 0 . [ 2
~-x--::-7:::> a. (1 - cS/2) [
~
_
2
x_ .., ] 0
2
o]
The optimal selling price for the first period can be obtained by maximizing
9 - 16
Summing up, the weak perfect Bayesian equilibrium of this game. is as follows: The seller offers the good for v(2 - a) and for y • _ a 6 8
v(2 - a> X = 8 - 6cS
2
in the first period
-2
in period 2. All buyers of type v e [V(
4 -
will buy in period 1, and all buyers of type v e .l
- cS)
Ja •
vc2 _ - aa> , vc2 _-JcSa> ] 8
6
4
-) v
will
buy in period 2.
9.C.4
[First Printing Errata: in part (c) it should read "Now allow Ms. P,
after ... "] For an analysis of this problem see: Nalebuff, B. (1987) "Credible Pretrial Negotiation," Rand Journal
of Economics, 18(2):198-210.
9.C.S
For an analysis of this problem see: Nalebuff, B. (1987) "Credible Pretrial Negotiation." Rand Journal
of Economics, 18(2):198-210.
9.C.6
Example 9.C.3: Since the example restricts attention to 1' > 0, there is
a unique Weak Perfect Bayesian Equilibrium (WPBE). Since every Sequential Equilibrium (SE) is a WPBE, this is the only candidate for a SE. sequences of strategies be: For firm I, (CT~,cr~) = ( . n n n 121 farm E. (CTo· CT 1' CT 2) = (ii, 3 - 2n
11
• 3 - 2n ) •
2
1 + 1'
Let the
1 1 -) and for n ' n '
--
where n = 1,2, ....
Clearly,
these strategies converge to the WPBE strategies described in the textbook. The sequence of strategies for firm E generates a unique sequence of beliefs that firm E played In as follows: 1 2
n J.ll
=
1 3 2n 4n - 3 ~2--;1;--';'""1--:-1- = 6n - 6 + 3 2n 3 2n
-
-
--
Thus, the WPBE described is a SE.
9 - 17
n
2
CD
3"
u (s,).)
p
~n-cp
for
s~~n-cd
s
for
s~~n-cd
-~n-c
=
d
-s
for
s~~n-cd
for
s~~n-a
Therefore, Ms. P's expected payoff for an offer s will be given by (recall that beliefs for Ms. P are determined b)
1
(s-cd)/n
E~up(s.~)
=
f(~)):
(s-cd)/n
I (~n-cp)f(~)d~ I sf().)d~ I (~n-cp)f(~)d~ =
+
0
(s-cd)/n
+ s(l - Fr:cdH
0
Ms. P will choose s to maximize this expression, and the FOC (assuming an interior solution) yields the pure strategy of Ms. P who solves:
(i)
(b)
Implicit differentiation of (i) above yields
(ii) -
which will determine the change in the optimal value of s given changes in any of the parameters.
For example, if
~
were uniformly distributed on the
interval [0,1] then the expression in (ii) reduces to s = n - c
p
so that we
ds ds have-= 1, dc = -1. and dn p
(c)
For an analysis of this part see: Nalebuff, B. (1987) "Credible Pretrial Negotiation," Rand Journal of £conomi.cs, 18(2): 198-210.
9.C.S
For an analysis of this problem first see exercise 9. C. 4 above (for a
guideline on how to solve such a problem), and then consult: Nalebuff, B. (1987) "Credible Pretrial Negotiation," Rand JournaL
9 - 18
of EconomLcs. 18(2): 198-210.
9.C.6
Example 9.C.3: Since the example restricts attention to 7 > 0, there is
a unique Weak Perfect Bayesian Equilibrium (WPBE). Since every Sequential Equilibrium (SE) is a WPBE. this is the only candidate for a SE.
Let the
1 sequences of strategies be: For firm I. (crFn"tr'An) • c - .!. , .!.). and for 2 + 7 .n n . n n n 121 11 ftrm E. (t1' , crl" t1' ) = (n. n , j - Zn ) , where n • 1,2..... Clearly. 3 2 0 2 these strategies converge to the WPBE strategies described in the textbook. The sequence of strategies for firm £ generates a unique sequence of beliefs that firm E played In as follows: 1 2 n Ill
1
3-2rl
4n - 3
n
2
- 6
CD
J"
= ..2-~1..::..._.......;:;1~--:1- = 6n 3-2ri+3- -2n
Thus, the WPBE d"!scribed is a SE. Example 9.C.4: The WPBE described In the textbook cannot be a SE since the only beliefs consistent with any mixed strategy of player 1 is 2's information set.
(~.~)
in player
Thus the unique SE must have player 2 believe that if
his information set is reached then each node is as likely as the other. causes him to play "r" if his information set is reached (since
~·10
+
This
~· 2 >
5), which in turn induces player 1 to play "y". This is the uniq\.e '::[ (which is also a WPBE). Example 9.C.S: The WPBE described in the text can be supported as a SE. Let a strategy of firm E be (tr'
0
,t1' F"tr'A)'
the probabilities of playing (Out),
(ln,Fight). and (In,Accommodate) respectively. (c:r f'tr' a),
Let a strategy of firm I be
the probabilities of playing Fight, and Accommodate respectively (if
reached).
Consider the following sequences of str·ategies:For firm E,
Z1 ),
. n n 1 and for fu·m I, (tr' f'tr' a) = ( 1 -
n , .!.n ).
n
These sequences converge to the strategies described, and furthermore generate
9 - 19
the belief sequence:
IT - n-1 n
n-1 2 n
n
1 +2 2 n n
CIO
1.
This supports the described strategies as a SE. Another SE (and WPBE) is where firm E plays Cln,Accommodate) and firm 1 plays Accommodate. This is supported by the sequences of strategies: For firm
.!._
' 2n '
9.C.7
(a)
1 -
n
1 n 1 - !). and for firm I, (CT f'CT a) = (n' n ' n
!)
The extensive form game is given in figure 9.C. 7(a).
R
Player 2 D 4 2
1
5
1
1
2 2
Figure 9.C.7(a)
The set of pure strategies for player 1 is S
= (B,T),
1
and for player 2 is
(DD,DU,UD,UU} where playing AB means playing A at node v
s 2=
and B at node v . 3 2
By backward induction it is easy to see that the unique SPNE is (B,DU).
There
are two more classes of NE: (i) Player 1 plays T, and player 2 plays UU with probability p and DU with probability 1-p, with p
?:
~
,
and (ii) Player 1
plays B, and player 2 plays DU with probability p and DD with probability 1-p, . h p
Wlt
~
J1 .
9 - 20
(b)
In this case the extensive form game, and its equivalent normal form
game, are depicted in figure 9. C. 7(b).
Player 2
T
Player
D
U
5,1
2,2
I~----~----~
B
4,2
1,1 4 2
5
1 1
2 2
1
Figure 9.C.7(b) Since playing T is a strictly dominant strategy for player 1, we have a unique NE: (T,U).
(c)
The modified game is depicted in figure 9.C. 7(c). I
1-p
....- ..- ..-
-4
2
1 1
4
2
_....,- '--=' -
-
2 Io
-
-
-
..........
-----2
Ir
u 1
1
........ ........
........ ........
u
D
s 1
2 2
s 1
2 2
Figure 9.C.7(c) Ik denotes player 2's information set after she observes k e {B,T}, r is the probability she assigns to the event that player 1 played B after she finds herself in information set 1 . and similarly q is the probability she assigns 8 to the event that player 1 played B after she finds herself in information set
9 - 21
lr·
Let s e [0,1] denote the probability that player 1 plays B.
We
can
three possible situations in a WPBE: First, player 1 playing s = 1;
have
second, player 1 playing s = 0; and third, player 1 playing s e (0,1). 1 playing s
=1
cannot be part of a WPPE.
Player
Indeed, if this were the case we
must have q = r = 1, which implies that player 2 will always play D. But given that 2 always plays D, player 1 will prefer to deviate and play T. player 1 playing s = 0 is part of a WPBE.
Second,
Indeed, if this is the case we must
have q = r = 0, which implies that player 2 will always play U, and given that 2 always plays U, player 1 will prefer to to play T.
Thus, player 1 playing T
and player 2 playing U in each of her information sets is a WPBE. To consider the possibility of a WPBE with s e (0,1), we first note that this will induce a unique pair of probability beliefs q and r derived by Bayes rule.
In particular, in such an eC'•Jilibrium we must have: r
=
s·p -o-(1::--_-s..... )(,....,l--:....._p.,....)_+_s_·_p ' and
Simple algebra shows that if and only if
> 1 r = < 2'
> s = p
<
q
=
s(l-p) -s;-:(1~--p""')_+......_p'(l=----s')
if and only if
> 1 > , and that s q ( 1 - p)
<(
2
This observation allows us to concentrate on 4 cases
as follows:
> p and s > ( 1-p): In this case we must have
(i) s
q >
~
and r >
4· This
implies that player 2 will always play D, which in turn implies that player l's best response is s = 0. (ii) s
Therefore there cannot be a WPBE iii this case.
< p and s < (1-p): In this case we must have
q <
4and r < 4· This
implies that player 2 will always play U, which in turn implies that player l's best response is s = 0.
This coincides with the pure strategy WPBE
described earlier. (iii) (1-p)
r >
4·
< s < p: (which implies p >
4>
In this case we must have q <
1
2
This implies that player 2 will play U in information set IT and will
play D in information set 1 . Player l's best response will now depend on p. 8
9 - 22
and
Playing B will give 1 an expected payoff of 4p + 1(1-p), and playing T will give him
2p + 5(1-p).
If
p •
32
then player 1 will have a unique best However, if p =
response which rules out such WPBE.
~ then we have a mixed
strateu WPBE as follows: player 1 plays B with probability
s e (
! , ~ ),
and player 2 will play U in information set IT and will play D in Information .
set 1 . 8 (iv} p < s < ( 1-p): (which implies p < case (iff) above.
If
p "'
i)
This case is symmetric to
J1 then player 1 will have a unique best response
which rules out such WPBE.
However, if p =
~ the we have a mixed strategy
WPBE as follows: player I plays B with probability
s e (
1
2 , 3 3 ),
and player
2 will play D in information set IT and will play U in information set 1 . 8 To conclude, there exists a unique pure strategy WPBE as described earlier, and if p is randomly drawn from the interval (0,1) then the pure strategy WPBE is the unique WPBE with probability l. p
=;
However, if p =
j
or
then in addition there exists a mixed strategy WPBE as described in
cases (iii) and (iv) above.
9.0.1
In a similar manner to the proof of proposition 9.8.1, let N be the
maximal number of moves from a terminal node to the root of the tree. statement is clearly true for N
= 1 because
The
only one player will have to play
at this node, and since no two terminal nodes give him the same payoffs, he will have a unique optimal strategy that is not weakly dominated. that the result is true for N
=n
Now suppose
-1, and consider the case where N = n.
Let
the set of final decision nodes (those that are before the terminal nodes) be
v0 .
Since for every player no two terminal nodes yield the same payoffs, then 0
at each node vk e V node.
there is a unique optimal strategy for the player at that
Let tr k be this unique strategy.
We can therefore associate with each
9- 23
0
node vk e V with N
= n-1,
a payoff to all players given crk.
These payoffs induce a game .
and by our hypothesis this game has a unique prediction obtained
through iterated removal of weakly dominated strategies.
Playing the optimal
strategy in the n"th node given the unique prediction of the N = n-1 game will give us the unique prediction obtained through iterated removal of weakly dominated strategies.
9 - 24
CHAPTER 10
10.8.1.
(a) Suppose that a feasible allocation (x,y) is strongly Pareto
efficient, and take any allocation Cx' ,y') for which u (x') > u Cx ) for all 1
J
1
1
i. Then, by strong Pareto efficiency of (x,y), (x',y') cannot be feasible.
This, (x,y) must also be weakly Pareto efficient. (b) [First printing errata: omit the assumption of inferiority, it plays no
role given that X
1
= X 2 = R+
for all t.]
Suppose that a feasible allocation (x,y) is not strongly Pareto efficient, i.e. that there exists a feasible allocation (x',y') for whichu (x') !: u Cx ) I I I 1 L . for all i, and u (x') > u (x ) for some k. Since X = R and preferences are k
k
k
k
•
I
strongly monotone, we must also have u (x') > u (x ) k
k
k
k
2:
u (O),and therefore k
x' "' 0. Therefore, we must have x' > 0 for at least one commodity s. k
ks
Let
x" = x'i I for i "' k, l "' s·, II x"
Is
x" kl
= x'is = x'k I
x' ~ = x' ks
ks
+ lj(l-1)
£
for i'ltk;
for l "' s; - e.
Intuitively, we are redistributing a small amount c of commodity s from consumer k to all the other consumers, so that they become better off than under the original allocation (x,y). Observe that as long as 0 < e < (x", y') will be a feasible allocation. I
5
By strong monotonicity of
preferences,· we must have u (x") > u (x') I
x~ ,
I
!:
I
u Cx ) for i. I
I
;t
k, for any
£
> 0.
By continuity of u ( · ), we must have u (x") > u (x ) for c > 0 small enough. k
Therefore, we can find an
k
£
k
k
k
> 0 small enough such that the corresponding
10-1
allocation (x", y') is feasible and makes every consumer strictly better off. Therefore, the original allocation (x,y) is not weakly Pareto efficient. Together with part (a) above, this implies that the notions of strong and weak Pareto efficiency are equivalent.
(c) [First printing errata: omit the assumption of interiority, it plays no role given that X = X I
Let 1
= 2•
= 1•
L
X1
= IR for all t.]
2
+
= X 2 = IR +
(no production). Then (x
1
,
x
2
)
1
u (x ) 1
11
1
0, u 2 (x2 )
= x 2,
w > 0, Y
=-
IR +
= (wj2, wj2) is a weakly Pareto efficient
allocation, but it is not strongly Pareto efficient. Since consumer l's preferences are not strongly monotone, we cannot reallocate utility from consumer 1 to consumer 2 to make every weak Pareto improvement a strong Pareto improvement, as we did in part (b).
10.8.2.
Suppose that the allocation
vector p
•
e IR
L
(X
• ... , X,• 1 I
and price
I
-
constitute a competitive equilibrium. Then
•
•
•
•
1
I
1
J
the allocation (x , ... , x , y , ... , y ) and price vector a.p
•
(a. > 0) also constitute a competitive equilibrium:
(i) Profit maximization: solves
Max p Y e y
•
•
solves
J
j
Max a.p • ·y J y e y J
a. Max
Y j e YJ
J
(ii) Utility maximization: Consumer i's new budget constraint is •
•
a.p · x
I
J
~ a.p · w
L e lj
+
I
J =I
•
•
•
(a.p · y ) ~ p · x j
i
~
• p ·w
J I
+ "e L. lj
J =I
which is consumer i's old budget constraint. Therefore,
x
I
solves
Max
x e X I
u (x) s. t. the new budget constraint I
I
I
10-2
~
•
p ·y
J
x solves
Max . u I (x) s.t. the old budget constraint. I
I
X
1
EX
1
I
•
(iii) Market clearing: t"' l.. XII 1=1
=wI +
J • t"' y l.. lJ j=l
for each l = 1, •••• L - does not
depend on prices at all.
lO.C.l.
(a) The consumer solves L'
L
I:
Max
log x
1=1
rpx
s. t.
1
1•1
l
l
:s w.
The first-order condition for the Lagrangean of this program can be written as
x • A/P , l=l, ... ,L, where A > 0. Substituting in the budget constraint, we 1
I
find A
= wfL,
therefore the demand function can be written as x (p,w) = Lw. 1
The wealth effect is
(b) As
L ---+
10.C.2.
(a)
m,
ax (p,w)jaw l
the wealth effect
p
1
=
ax1(p,w)jaw
-+ 0.
The consumer solves s. t. p x + m s w . The
Max a + (:J ln x + m
m
(x,m)
first-order condition (assuming interior solution) yields x(p) = fJjp. The firin solves Max pq - trq. q~O
The firm's first-order condition (assuming interior solution) is p = tr.
(b) From the two first-order conditions and the consumer's budget constraint,
the competitive equilibrium is p
10.C.3.
•
•
::: ,., x
•
= (:Jjtr, m • = w
m
- (:J.
(a) Assuming interior solution, the first-order condition is
c'(q ) = A J J
> 0 for all j.
10-3
J
L c'
(b) C'(q) =
J
J=l
•
J
(q ) dq fdq •
d(
J
J
J
• • r ;\ dq fdq = ;\ r q > fdq = ;\ dqfdq = ;\. J J J J
•
=1
=l
•
Therefore, C'(q) = c'(q ) for all j. j
j
(c) Each firm j solves Max pq - c (q ). q j j j J
The first-order condition (assuming interior solution) is c'(q ) = p. J j If p = C'(q), then we have c'(q ) = C'(q) for every j. If c'( ·) is strictly . J J J •
increasing for all j, we must have q • q j
• j
(q) -
the solution to the central
authority's program in part (a) for total output q. Therefore, J
J
Lq
Lq
=
J=l l
•
(q)
J=l
= q. In other words, if the market price is C'(q), then the
j
industry produces q. Therefore, C'( ·) is the inverse of the industry supply function.
(a) The central authority's problem can be written as
lO.C.4.
I
Max
EXI
s. t.
(X , • • , X )i!:.O I
::s
X.
I= 1
I
Assuming interior solution, the first-order condition is
•) = ;•ex I I
A > 0 for all i. I
(b) 7'(x)
= L ;·I
•
•
i
I
(x ) dx fdx
I
•
L i\
=
1=1
dx 1/' ldx = i\ d(
1:1
I
•
Lx I
) /''dx
= ;\
dxldx /'
= ;\.
1=1
•
Therefore, r'(x) = ¢'(x ) for aJI i. .i
I
(c) Each consumer solves Max X
A.
¥'1
(x ) - Px. I
i
I
The first-order condition (assuming interior solution) is
;·ex) = I I
P.
If P = r'(x), then we have ;'(x ) = 7'(x) for every i. If ;'( ·) is strictly .
I
I
decreasing for all i, we must have x
I
I
= x • (x) - the solution to the central I
authority's program in part (a) above for total consumption x. Therefore, I
I
•
L x 1 = Ex 1
1=1
(x)
= x. In other words, if the market price is 7'(x), then the
1=1
10-4
aggregate demand is x. Therefore, 7'{ ·) is the inverse of the aggregate demand function.
lO.C.S.
The system of equations (lO.C.4)-(lO.C.6) here takes the following
form:
•
9'>'(x ) 1
= p
I
•
•
+
t,
i = 1....... I,
•
•
j = 1, •••• J.
c'(q ) = p ' ~ J. J •
rx
I =1
=rq.J
I
J =1
•
•
•
These equations describe the equilibrium (x • q , p ) as an implicit function of t. Differentiating with respect to t, we get 9'>"(x • ) x •'(t)
= p •'(t) + 1,
i = 1, ••. , I,
c"(q • ) q • '(t) ~ J J J
= p • '(t),
J = 1, ••.• J'
1
I
I
• r x 1' =
1=1
• r q '(t).
J=1
J
• • • This system of linear equations should be solved for (x '(t), q '(t), p '(t). I
•
J
•
-
This can be easily done, for e.:ample, by expressing dx /dt and dq /dt from the 1
J
first two sets of equations and substituting into the third equation.
We
obtain I
•
(p '(t) + 1)
r {¢;'(x:>r
1
= p
•
]
'(t)
r {c"(q J =1
I =1
J
•
-1
)J .
J
•
From here we can express p '(t): I
• p '(t) =
[ r¢;'(x:>r
1
I =I
I
[ 1=1
•
J
f¢~'(x ~ n-1- [ fc:'(q ~JJ-1 I
I
J=l
J
J
Compare to the expression on page 324 of the textbook.
10.C.6.
(a) If the specific tax t is levied on the consumer, then he pays p+t
for every unit of the good, and the demand at market price p becomes x(p+t).
10-5
The equilibrium market price p
c
is determined from equalizing demand and
supply: x(p
c
+ t)
c
= q(p ).
On the other hand, if the specific tax t is levied on the producer, then he collects p-t from every unit of the good sold, and the supply at market price p becomes q(p-t). The equilibrium market price pp is determined from
equalizing demand and supply: x(pp)
= q(pp
- t).
It is easy to see that p solves the first equation if and only if p+t solves the second one. Therefore, pp = pc + t, which is the ultimate cost of the good to consumers in both cases. The amount purchased in both cases Is x(pP) = x(pc + t).
(b) If the ad valorem tax
Y
is levied on the consumer, then he pays
(1+-r)p for every unit of the good, and the demand at market price p becomes x((l+-r)p). The equilibrium market price pc is determined from equalizin-g demand and supply: c
c
x((l+-r)p ) = q(p ).
(1)
On the other hand, If the ad valorem tax T is levied on the producer, collects (1-T)p from then he pays (l+T)p for every unit of the good sold, and the supply at market price p becomes q((l-T)p). The equilibrium market price pP is determined from equalizing demand and supply: p
p
x(p ) = q((l--r)p ).
Consider the excess demand function for this case: z(p) = x(p) - q((l-T)p). Since x( ·) is non-increasing and q( ·) is non-decreasing, z(p) must be non-increasing. From (1) we have
10-6
(2)
c
z((I+-r)p )
= x{(l+T)pc) - q((I-T 2 )pc)l!:
c
!: x((l+T)p ) -
c
q(p )
= 0,
taking into account that q( • ) is non-decreasing and using (I). Therefore, z((I+y)pc)
!:
0 and z(pP)
= 0.
Since z( ·) is non-increasing, this
implies that (l+T)pc s pp. In words, levying the ad valorem tax on consumers leads to a lower cost on consumers than levying the same tax on producers. (In the same way it can be shown that levying the ad valorem tax on
con~umers
leads to a higher price for producers than levying the same tax to producers). If q( • ) is strictly increasing, the argument can be strengthened to obtain the strict inequality: Cl+T)pc < pp. On the other hand, when the supply
-
is perfectly inelastic, i.e. q(p) = q = const, then (1) and (2) combined yield c
x((I+T)p ) •
-
p
q • x(p ), and therefore p
p
c
=
.
Cl+T)p • Here both taxes result m
the same cost to consumers. However, the producers still bear a higher burden p
when the tax is levied directly on them: (1--r)p = (1--:){l+T)p
c
c
Therefore, the two taxes are still not fully equivalent. The intuition behind these results is simple: with a tax, there is always a wedge between the "consumer price" and the "producer price''
Le ;ying an
ad valor_em tax on the producer price, therefore, results in a higher tax burden (and a higher tax revenue) than levying the same percentage tax no the lower consumer price.
lO.C.7.
To compute the price received by producers, we can use equation
(lO.C.S)
In
•
the textbook: x'( p •)
p:(O) =-
x'(p•)- q'(p.)
£
£-1
--
Aep.
Aep. £-1
A£p•
10-7
- arp.1-1
= -
£
Acp.
-
ocrp!
-
ex(p.)
£
=- -£X....,.(P-.-)r----7-q~(r-p-.....) = -
£
-
7
•
.
(We have multiplied both the numerator and the denominator by P. and used the fact that p • is an equilibrium price, therefore x(p •) = q(p • ). ) The price paid
.
by consumers is p• + t, and its derivative with respect to t at t=O is p'(O) + 1 = -
c c - 7
£ -
7'
From these expressions we can see that when 7 • 0 (supply is perfectly inelastic) or c -+ -
(demand is perfectly elastic), the price paid by
consumers is unchanged, and the price received by producers decreases by the amount of the tax. On the other hand, when c = 0 (demand is perfectly inelastic) or 7 -+
m
(supply is perfectly elastic), the price received by
producers is unchanged and the price paid by consumers increases by the amount of the tax. Computing the elasticity of the equilibrium price with respect to changes in k=l+-r is left as an exercise.
IO.C.S.
(a) Each firm's profit can be written as n(q, o:)
= p(o:) q - c(q, o:).
The first-order condition of the firm's profit-maximization problem can be written as· p(o:.) = c (q, o:.).
(FOC)
q
Denoting the solution of the firms' problem by q.(o:), we can write the firms' reduced-form profits rr (o:.)
•
= rr(q • (o:.),
o:.l.
Using the Envelope Theorem (in simple words, taking into account that Bn(q, o:ljBq lq=q_(a)
= 0),
we can write
n;(o:) = Brr(q .Co:), o:ljao: = p'(o:) q.(o:) - c «Cq.(o:), o:).
By assumption c ( · ) :s 0, and it is the first term, Cl
10-8
p'(cz) q.(a:), which may present pi·oblems. To determine how the market clearing price depends on «, consider the following system of two equations: p'(cz)
= cqq (q • (a:),
x'(p(cx)) p'(cz)
q•'(cz) + c
«)
q«
(q (a:), a:). •
= Jq;(a)
The first equation is obtained by differentiating (FOC), the second by differentiating the market clearing condition x(p(cz)) • Jq.(a) (both with respect to a:). Eliminating q;(a) from the system. we can solve for p'(et): p'(a)
= c
1
/U - [ q«
c
qq
x'(p(a)))
(for convenience we have omitted arguments in the derivatives of
c( · )).
Now we
can rewrite (•) as 1
- J- c
qq
x'(p(a))) - c . Cl
(b) The last expression can be rewritten as n'(a) •
•
[c
qo:
q (o:) - c •
0:
+ c
1
0:
J- c
Remember that by assumption c and x'
::5
qq
1
qq
x'(p(a))Jjlt - J- c
qq
x'(p(a))].
> 0 (costs are strictly convex in q),
0. Therefore, the denominator of the fraction is always positive,
and sign n.'(o:) = sign [c Since by assumption c
ex
qO:
q • (o:)
> 0, c
qq
-
.c
Ot
+ c
a
J
> 0, and x'
-I
c x'(p(a))}. qq
::5
0, the last term must be
non-positive, and sign n;(o:)
::5
sign { c
qa
q (a.) - c ] = • a
Thus, in order for profits to be increasing in « for any demand function, is sufficient to have
ajaq
[ca(q,a.ljqJ ::5 0 at q
= q.(a.).
On the other hand, if this condition is not satisfied, we can take a demand function with I x'(p(a)) I sufficiently small. For such a demand function,
10-9
the last term in the right-hand side of
c••)
will be very small,
and we will
have sign n;(a) = sign [c
qa
q (o:) - c •
a
1>
0,
i.e. profits increase in a.
(c) If a is the price of factor input k, then by Shepard's lemma (Proposition S.C.2 (vi)) c the condition c
qa
Gt
is the conditional demand for factor k. Then
:s 0 means that the conditional demand for k is
non-increasing in output. i.e. that k is an inferior factor.
10.C.9.
The first-order condition of the firms' profit-maximization problem
can be written as p(w) = c (w, q). The market clearing condition can be q
written as x(p(w)) = Jq.Cw). Differentiating both equations with respect to w, we obtain the following system of two equations: p'(w)
= c
qq
(w, q.(w)) q;(w) + c
x'(p(w)) p'(w)
Eliminating
q~(w)
wq
(w, q.Cw)).
= Jq:(w)
from the system, we can solve for p'(w):
p'(w)
= c
w
I
(1 - J
-1
c
qq
x'(p(w)))
(for convenience we have omitted arguments in the derivatives of c( · )). Observe that the· second-order condition for the firms' profit-maximization problem is c
qq
2::
0, and that in a partial equilibrium context we must have
x' ::s 0. It is then clear from the last expression for the sign of p'(w) coincides with the sign of c
p'(w) that
•
If w is the price of factor input k, then by Shepard's lemma (Proposition S.C.2 (vi)) c c
wq
q
is the conditional demand for factor k. Then the condition
::s 0 means that the conditional demand for k is non-increasing in output.
i.e. that k is an inferior factor. Therefore, p'(w) ::s 0 if and only if k is an
10-10
inferior factor.
IO.C.IO.
The first-order condition for the firms' profit-maximization problem
can be written as p = c'(q). The market clearing condition can be written as x(p) = Jq. Substituting the functional forms for ·c'( ·} and x( • ), we obtain the following system of equations:
= {Jq.,,
p
a.p
£
= Jq.
Taking logs on both sides, we obtain a system of linear equations in log p and log q. The solution of this system is log p = (log {J + 71 log a - 71 log Jlj(l - £71), log q = (c log (3 + log a - log Jlj(l - C1J). From here we can compute the elasticities: Blog pfBlog a = 71/0 - £71). Blog pfolog (3 = I/0 - £71) Blog qfolog a
= 1/0 -
£71),
Blog qjBlog (3 = cj(l - c71). We see that increasing the elasticity of demand I c I reduces the sensitivity of p and q to shifts in o:, reduces the sensitivity of p to shifts in (3, but
increases the sensitivity of q to shifts in (3. Similarly, increasing the inverse elasticity of supply 71 reduces the sensitivity of p and q to shifts in (3, reduces the sensitivity of q to shifts in o:, but increases the sensitivity
of p to shifts in o:.
lO.C.Il.
Let us start with a situation where both firms' parameters are equal
to a, and then increase a
1
and reduce a. marginally in such a way that the 2
competitive equilibrium price does not change.
10-11
Firm's j's profit can be written as
= pq - c(q, a).
n(q, a.)
J
J
The first-order condition of the firm's profit-maximization problem can be written as p(a. ) • c (q, a ). J I J
(FOC)
Denoting the solution of the firms' problem by q.(a.), we can write the firms• reduced-form profits w.(a. ) • w(q (a. ), a ). J • J J Using the Envelope Theorem (in simple words, taking into account
Iq=q.(a.J ) •
Bn(q, a.ljaq
J
0), we can write
n'(a. ) = Bn(q (a), a: l/Ba. = c (q (a.), a). •J •J J J 2•J J
If a
I
- a
= t.a.,
2
then
-=
n.Ca ) - n (a ) a n'(a.) t.a 1
•
2
•
c (q ·ca.), a.) b.a.. 2
•
(i) Suppose in negation that
10.0.1.
(x, .. , x , q, ... , q ) and (x', .. , x', q', ...• q') are both solutions to 1
I
J
I
(10.0.2), and that x
x;•
= 1/2
1
- x
k
+ 1/2
xi
I
J
I
for some k.
k
x;
Take
for every i = l, ... , I,
q" = 1/2 q. + I/2 q' for every j = l, ... , J. J
J
J
. . ( 10 . D. 2) , an Cl ear 1y, ( x " , .. , x " , q " , ... , q " ) sa t.as f'aes th e cons t ramt 1
I
I
since both (x, .. , x, q I
I
J
, ... , 1
q ) and (x', .• , x', q', ... , q') do. As cost J
I
I
1
J
functions are convex, we must have
c (q") J
j
~
1/2 c (q ) + I/2 c (q') for every j = 1, ... , J. J
j
j
j
As utility unctions ¢ 's are strictly convex, we must have I
¢ (x;·>
~ 1/2 ¢ (x
1/Jk(xk")
> 1/2
1
1
1
)
+ 1/Z q,icx;> for every i. - k, and
1/Jk(xk) +
1/2 t/Jk(x~).
Adding up all the inequalities, we obtain
10-12
J
I
I: tP 1(x") 1
I: c
-
I
J =1 J
1 =1
1/2 ( l>t~ (x')
>
(q")
J
I =1 1
J
I: c
-
J =I
1
l
1/2 [ r f/dx 1
+
l =1
Cq')] + J J
J
r c (q )]. J•l J J
) I
which contradicts the assumption that (x , •. , x , q , ... , q ) and (x', .. , 1
I
J
1
1
x', q', ... , q') are both solutions to (10.0.2). I
I
J
Therefore, Individual consumption levels x 's at a solution to (10.0.2) are l
uniquely determined. I
J
I: q J
( ii) The optimal aggregate production level is
I: x 1 ,
=
and it is
l =1
J=1
uniquely determines since all the x 's are uniquely determined. I
(iii) Now suppose that the cost functions c ( ·) are strictly convex. J Suppose in negation that (x , .. , x , q , ... , q ) and (x', .. , . x', q', ... , 1
J
11
1
q~
q;) are both solutions to (10.0.2), and that qk •
x " = 1/2 x 1
q" = 1/2 q J
+ 1/2
I j
x' for every i =
11
for some k.
Take
1, •.. , I,
1
+ 1/2 q~ for every j = 1, ... , J. J
Clearly, (x", .. , x", q", ... , q") satisfies the constraint in (10.0.2), 1
I
1
J
since both (x , •. , x , q , ... , q ) and (x', •. , x', q', ... , q') do. As cost 1
I
1
1
J
I
1
J
functions are strictly convex, we must have
c (q") ::s 1/2 c .J
j
(q ) +
J
j
1/2 c (q') for every J. J
c (q") < 1/2 c (q ) + 1/2 c k
k
k
k
k
k, and
#
j
(q'). k
As utility unctions ¢ 's are convex, we must have 1
; (x") z:: 1
1
1/2 ¢ (x ) + 1/2 ¢ (x') for every i = 1, ... , I. I
I
1
1
Adding up all the inequalities, we obtain
r ¢ (x") I =1 '
1
-
r c (q") j =1
J
I
J
I
>
1/2 (
J J
r ¢ (x') I =1 I
+ 1/2 (
1
1
r ¢ (x
l =1
I
I: c
j
(q')] + =1 J J J
I
) -
r c (q J=1 j
J
>J.
which contradicts the assumption that (x , .. , x , q , ... , q ) and (x', .. , I
10-13
I
1
J
1
x', q', ... , q') are both solutions to (10.0.2). I
1
J
Therefore, individual production levels q 's at a solution to
'
(10.0.2) are uniquely determined.
10.D.2.
The program (10.0.2) for the economy in Exercise 10.C.2 can be
written as follows: f/>(x) - c(x) + w
Max
m
x!:O
The first-order condition yields x
= « +
• ""' f3jtr.
x -
(3 In
tr
x + w
m
This is the same consumption (and
production) level as that obtained in Exercise IO.C.2.
• • I • • J (i) Suppose in negation that (x , m } , {z , q } is a solution I I 1=1 J J J=l
10.0.3.
to (10.0.6), but not a Pareto optimal allocation. This means that there is an allocation {x', m'J I
I
1
1=1
which is feasible (i.e. satisfies (21), {zJ', q' / J j=1
,
(2m), (3)), such that m' + ; I
I
(x') I
~
m • + ¢ (x • ) for all i = 1, ••• , I, and
m' + ¢ (x') > m k
k
I
I
k
• k
I
•
+ f/> (x ) for some k. k
k
lI I=1'
If k = 1, this means that the allocation {x', m' 1
(z'
j'
J
q'} J J=l
satisfies all the constraints in (10.0.6) and yields a higher value of the 1 . . f unction . o bJectlve t han { x • , m •} , i i 1=1
, q •}J , w h'1c h contra d.1cts the J j j=l
{ z•
assumption that the latter allocation is a solution to (10.0.6). Suppose, therefore, that k > 1. Let A = {m' + f/> (x')J k
allocation (x", m"} I
i
1 1=1
,
fz:', J
q"/ J j=1
k
k
• {m k
+
•
¢ (x )]. Take an k
k
which has some numeraire redistributed
from consumer k to consumer 1: m" = m'1 + A • 1 m" = m'k - A• k 1
and which otherwise coincides with the allocation { x' m' J 1'
10-14
I 1=1
, {z
, j'
q '}J J J=l'
It is easy to s~e that the new allocation { x .. , m .. }1 l
1
1=1
, { z'',
J
satisfies all the constraints in (10.0.6} and yields a higher value of the objective function than { x •, m •l 1
I 1•1
, {z •,J
q• /
J J=1
, which contradicts the
assumption that the latter allocation is a solution to (10.0.6). • • I m J , {z J' I 1=1
• {x , 1
(ii) Let
• J
q/
J=l
be a Pareto optimal allocation and let
-
u = m • + '(x• ) for i = 2, ... , I. We will prove that (x • , m • }I • (z• , I I I l l 1 1•1 J • J
q }
J J=l
solves the program (10.0.6) with these utility levels.
First of all, it easy to see that {x • , m • }I , (z • , q • }J I
I 1=1
J
satisfies
J J=l
all the constraints in (10.0.6). Suppose in negation that {x
• 1
,
• I , 1 1 1
m J=
•
• J
{zJ, q/J= 1 does not solve the program, i.e. that there is an
allocation {x', m'J 1
1
I 1=1
, (z', j
q•/ which J J= 1
satisfies all the constraint and
yields a higher value of the objective function. Since (x'
1'
m'}
I
I
{z' 1 q'} J J
1=1'
satisfies (21), (2m), (3), it is feasible. The
J=l
j
constraint ( 1) implies that m' + I
•
I
I 1=1
I
I
i
Since {x', m' l
,
•
m + ' ( x ) for every i = 1, ... , I.
t/) (x') ~
1
(z', q' / J
j
I
yields a higher value of the objective
j=l
function, we have m'. + ¢ (x') > m I
1
1
Therefore, {x', m') •
I
• I
m ) I
1=1
J
1 1=1 • J
I
(x •
•
•
, (z , q ) j
j
+
I
•
'(x ). 1
, {z'
I
q'}
J'
J
j J=1
is a Pareto improvement over
.
j=l
, which contradicts the assumption that the latter
allocation is Pareto optimal.
10.0.4.
The Lagrangean for he program (10.0.6) can be written as J
I
L = m 1+ '
1
(x ) + 1
L 0: IX 1
+
I =1 I
+
L '1.
I
l=Z
fm.I
r ~Jq J J =I
-
I
+ ,"' i (x I ) - u 11 -IJ{Lx I I =1
10-15
J
-rq j =1
1J
I
J
I: m1
- v{
, 1
13
J.l., v,
, 1
t
KJ,
I:z
.._
j =1
I =1
where «
J
- w 1 + [K{Z m J =1 J
J
c(q)],
J
j
J
are non-negative Lagrange multipliers.
1
The first-order conditions with respect to original variables are: BLjBm
1
•
BLjam = 1
1 - v == 0 • v = 1;
r1
-
BL/Bz . J
=- v
8L/Bx 1
= «1 +
v = + K
for L
=0
J
;;ex 1 )
-
JJ
J
!!::
2 • 1 = v for all t 1
K
J.L
= 0 for all i.;
J
J
!!::
2.
= v • 1 for all j;
•
K c~(q
BL/Bq • 13 + J.1. J
o
) = f3 + J.l. - c'(q ) = 0 for all j. J
J
j
Besides, there are first-order conditions with respect to dual variables and complementary slackness conditions. Complementary slackness conditions say that o: x I
1
=0
for all i and f3 q
J J
= 0·
for all j. Using this, (•) and (.. )
become equivalent to conditions (10.0.3) and (10.0.4) in the textbook respectively.
The first-order condition with respect to J.L yields condition
(10. 0.5) in the textbook. The remaining conditions serve to pin dow the variables
lO.E.l. tariff
n
I
and z , which are not present in program (10.0.2); j
(a) To determine how the domestic market price p depends on the T,
observe that the demand x(p), the domestic supply is J q (p), and d
the foreign supply is J q (p f
f
T),
d
where q ( ·) and q ( ·) are the supply f
d
functions of the domestic and foreign firms correspondingly.
The market
clearing condition can then be written as x(p) = J q (p) + J q (p - "T).
r r
d d
Differentiating this condition with respect to
T
and solving for p'(-r), we
obtain J q'(p) p
'( >I T
r r
-r=O = J q'(p) + J q'(p) - x'(p) d
d
•
r r
As c ( · ) are strictly convex, we have c"( ·) > 0, and therefore f
f
q'( ·) < m. The expression (•) then implies p'(-r)
r
10-16
< 1.
Domestic welfare is the sum of consumer surplus, domestic profits, and tax revenue (which can be, for example, distributed lump-sum to domestic consumers): DW(T)
= CS(p(T))
+ J
n (p(T))
c1c1
+ TJ
q (p(T)-T).
rr
We will differentiate this expression at T=O, taking into account that CS'(p) = - x(p) (Roy's identity) and
n~(p)
= qcl(p)
(Hotelling's lemma):
=
DW'(TlfT=O = -x(p)p'(O) + Jclqiplp'(O) + J,q,(p)
= f-x(p) + J clqd(p)]p'(O) + Jrqr(p) • = - J,q,(p) p'(O) + J,q,(p) • = J rqr(p) (1 - p'(O)}.
Since we have established that p'(O) < 1, we have DW'(O) > 0. i.e. the imposition of a small tariff raises domestic welfare. (b) Now we have c"( ·) = 0, and therefore q'( ·) = r r
111.
From (•) we see that
p'(T) = l, and from the result of part (a) above, DW'(O) = J q (p) [1 - r'(O)] = 0. Thus, there is no first-order effect on
r r
domestic welfare from a small tariff, and we need to look at the second derivative to determine how domestic welfare· is affected by a small tariff. However, instead of computing the second derivative of DW(-r), one can answer the question with a simple observation. Observe that since foreign firms produce at constant returns to scale, they always have zero profits, and domestic welfare equals total welfare. Imposition of a tariff distorts the global economy away from the competitive equilibrium, and therefore lowers total welfare, .which is equal to domestic welfare.
10.£.2. A
CS(p) =
Substitute s = x(p):
rJ,.. cA>p ·
A
{P(s) - p] ds
0
((x(~))
=
J,.,
[P(x(p)) -
P(OJ
10-17
~]
dx(p)
=
1\
-
(p - A p] dx(p)
= [p
1\ - p] x(p)
1\
'p J p
a.
-
x(p)dp
=
p
Cll
Cll
J"
x(p)dp.
(The second line carries out integration by parts.}
lO.E.J.
(a) [First Qrinting errata: The first sentence should read: "Show
that if t' = 0 and t' 1
I
=t1
c
1
t I for all l "' 1, then tax vector t' raises
+
the same amount of good 1 as does tax vector t." ] Given constant returns to scale, in equillbrium a firm should receive c
1
units of numeraire for each unit of good l. Under the tax vector t, a
consumer pays c units of numeraire directly to the firm for each unit of I
good L. For this transaction, the consumer has to pay the government
t units of numeraire as a tax on good 1, plus c t units of numeraire as a I
.
I I
tax on his transfer of numeraire to the firm. Thus, the total tax consumers pay on each unit of good l is t
1
c + t . This is the same amount as under the I
I
tax vector t' , and we know that t' = 0. Therefore, for each tax scheme there 1
~eaves
is an equivalent tax scheme which
numeraire untaxed.
(b) The government's problem is L
Max L CS I (c I + t I ) (t • • • • t ) 1=2 2 L L
L. t I x 1(c I
s. t.
+ t) I
1=2
>
R,
where x (p ) is consumer demand in market l, and CS (p ) is consumer surplus I
I
I
I
in market l given price p . The Lagrangean of this program is
C
L =
L cs.(cl
L + tl) + A[
1=2
where ;\
~
L tlxl(cl
+ tl) -
R},
1=2
0 is the Lagrange multiplier with the constraint.
Differentiating
with respect to t and using Roy's identity (CS'(p ) :: - x (p ) ) , we obtain I
I
the following first-order condition:
10-18
I
I
1
(for brevity we have substituted p = c + t ). • 1 I I
From here we can express t 1:
tl • u-~)/~ x.(pl)/x;cpl).
Denoting the elasticity of demand for good l by .: • p x;Cp )jx (p ), the last 1 1 1 1 1 expression can be rewritten as
This result is known as the tnverse elastictty rule: optimal rates of taxation are inversely proportional to demand elasticities. (c) From the inverse elasticity rule, optimal tax rates on two goods should be equal if demand elasticities for the two goods are equal; and markets with less elastic demands should be taxed more heavily than markets with more elastic demands. Using the argument in part (a), if t = Ct , 0 .... 0) is a tax scheme which 1
= (0,
taxes only numeraire, then t'
t c ... , t c ) is an equivalent tax scheme 1 2
1 L
leaving numeraire untaxed. Such a scheme sets equal tax rates on all (non-numeraire) goods, which is only optimal \'/hen all markets have equally elastic demands.
lO.F.t:
(i)
Suppose that p > c'(O). Using Taylor expansion of ..:·(q! near
zero, for a small c > 0 we have c(d = c(O) + c'(O)c + o(c) = c'(O)c + o(c). Therefore, n(p)
= Max pq - c(q)
2:
pc - {c'(O)c + o(c)] = {p-c'(O)Jc + o(c) > 0
q'l:O
for c > 0 sufficiently small.
(ii)
Suppose that p s c'(O). Since c( ·) is convex, we have c(c) s
cfq
c(q) + (1-c/ql c(O)
for every c,q > 0. Therefore,
10-19
= cfq
c(q)
c(q)jq
and consequently c(q)
2:
!:
c(c)/e
-+ c'(O)
c'(O)q for all q
!:
as c -+ 0,
O.But then profits
at any output q can be bounded: pq - c(q) s pq - c'(O)q = (p-c'(O)]q s 0.
Therefore, n(p) = Max pq - c(q) :s 0. q!:O
(i) Profit maximization ~ p • c'(q)
10.F.2. (a)
(ii) Market clearing (iii) Free entry
"*
A - Bp
=«
+ 2.f3q.
= Jq.
~ pq - c(q) = pq - K - o:q - (Jq = 0. 2
The solution to this system of equations is
q. =
/kf;.
p. =
Cl
+ 2
J• = (A -
Aggregate output is p
vk;,
o:B)~ - 2 a.(:3. • • • Q = J q = A - o:B
•
and q
•
and· q
•
are increasing in A.
•
r-:-'
- 2a.(3-1Kj(3.
are independent of A. This is not surprising, because
• are entirely determined by technological parameters - q • is the • • optimal scale of production, and p is the minimum average cost. J and p
Q
(b) In the short run the number of firms J • is fixed, p and q are determined by profit maximization (i) and market clearing (ii). Since we are interested in market responses to short-run changes in demand, we can differentiate (i) and (ii) treating p and q as functions of A:
= 2(3q'(A),
p'(A)
J • q'(A).
1 - Bp'(A)
We can now solve for p'(A) and q'(Al: p'(A)
= lj(J
•
+ 2(3),
10-20
q'(A) =
213/<J•
+
213).
rxB)~ rx8)~ -
As A Increases, J• = (A -
= (A
When A -+ m, J•
-
- 2 CC/3 increases, and p'(A) and q'(A) fall. 2 a.f3 -+ m, and p'(A) -+ O.The intuition is
simple: in a large market the equilibrium number of firms is large, each firm's production needs to change only slightly to accommodate a short-run shift in demand, and the market price becomes insensitive to short-run demand shifts.
lO.F.J.
[First printing errata: You need to assume that taxes are small,
which is necessary for a definite comparison. Also, the condition f/>' in the third line of the exercise should instead be· ;• and J • denote
respec~ively
1 ( ·)
1
( ·)
<0
> 0.1 Let p •, q •,
the market price, each firm's production, and the
number of producing firms at the initial equilibrium.
These variables
should satisfy conditions (i)-(iii) on p.335 in the textbook. Introduction of an ad valorem tax demand function x(p) with x(p(l+-r)). (iii) intact.
T
can be represented by replacing the
This change leaves conditions (i) and
Therefore, the new long-run market price and firm output have
to satisfy (i) and (iii), and thus have to coincide with p
•
•
and q .
The new
long-run equilibrium number of firms J is determined from the modified (ii): •
A
•
x(p 0+-r)) = Jq
Differentiating this expression with respect to
T
and substituting
T
= 0, we
obtain
•
•
J' (0) = p x'(p ljq ,..
•
.
(1)
•"'""' The resulting tax revenue is R(T) = -rp q J.
of this function around ~
~
T
= 0 can be computed as
~
R(-r) ::: R' (0) T + R' I (0) •
•
A
The second-order Taylor expansion
T:
2
/2 =
""'
= P q .n-rJ' (-rl + J(-rll 1-r=o -r + (-rJ'
10-21
2 (-r> + ZJ' (T)) IT=O T /2 ~
"""
I
1•
•
•
•
•
•
•t
A
• p q J T + p q
J' (T)T.
(2)
Introduction of a per firm tax T, on the other hand, modifies the free 0
entry condition (iii), so that now it can be written as p q The profit maximization condition (i) yields p
0
0
c(q0 )
-
-
T = 0.
= c' (q0 ). Combining with the
previous equation and substituting our cost function we can write ,, (q 0 )
= ('(q0 ) + K + T)1, /q 0 .
.
0
This equation determines q . Differentiating it with respect to T, we obtain
qo' (T) The number of firms J
0
(¢'' (qo)qofl.
•
can be determined from (ii) and the profit maximization
condition:
Differentiating this equation with respect to T and evaluating at T • 0, we obtain
I I
= [x'C¢'Cq 0 )) ¢' '(q 0 ) - J OJ q 0 1 •
•
•
(T) q 0 •
= [x' (p )¢" (q ) - J 1/C¢" (q )q
The tax revenue from the per firm tax is R
0
•2
T=o
(3)
). 0
= T J (T).
(T)
The second-order
Taylor expansion of this function around T = 0 can be computed as 0
R (T) z R
0 '
(0) T + R
1
2
(0) T /2 =
(T) + J (T))
'
= J•T
'
0
0
= (TJ
0
+
IT=O
T + (TJ
0
1 '
(T) +
I
1
2
2f (T)) T=O T /2 =
.f' (O)T 2
Since at the initial equilibrium the two taxes raise the same revenue, we must
• •
have T = Tp q .
Substituting in the last Taylor expansion, we obtain 0
=p
R (T)
•
•
•
•
•
2
q J T + (p q ) J
0 I
(O)T
2
(4)
Comparing (2) and (4), we see that the first-order terms in and the second-order difference can be computed using A
0
•
•
•
•
R (T) - R(T) ::: p q [p q J •
•
= p q [ p
• X
1
•
(p ljq
•
•
A
0 '
•
(0) -
J' (O)]T •
2
(1)
•
•
•
- p J /(¢' '(q )q ) - p x' (q ljq }T
10-22
2
coincide,
and (3):
=
•
T
=
.. - p
•zT z J •I•' Cq• ) < 0. I
Therefore. a small ad valorem tax raises more revenue than the corresponding per firm tax.
10.F.4.
[First printing errata: you should assume that c(w, q) is a twice
continuously differentiable function, and that most efficient scale is uniquely defined. I (i) The long-run equilibrium price and output of each firm are given by the
following conditions: Profit maximization .. p = c (w, q), q
Free entry
"*
pq - c(w, q) = 0.
Let f(w,q) = c(w,qlfq be a firm's average cost. These equations Imply that 2
f (w,q) = c (w,qlfq - c<w,qljq = (p - pljq = 0. Intuitively, this is the q
q
first-order condition for a firm's operating at the minimum average cost in a long-run equilibrium; Now the above system can be rewritten as
-
- = f(w,q(w)) p(w)
(1)
-
f (w,q(w)) = 0,
(2)
q
-
- are equilibrium price and firm output as a function of w. where p(w) and q(w) To find out how equilibrium price depends on w, differentiate
(l)
with respect
to the input price vector w and use (2):
-
Vp(w) =
-
'1'J
w
-
f(w,q(w)) + f (w,q(w)) q
'1'J
-
w
q(w) =
'1'J
-
w
f(w,q(w))
(•)
Now, using the definition of f(w,q(w)) and Shepard's Lemma (Proposition 5.C.2 in the textbook), we obtain Vp(w) = where z(w,
q)
'I'J
f(w,q(w))
w
= Vwc(w,q(w))jq(w)
= z(w,q(w))jq ~ 0,
is a firm's input demand function. Therefore, the long-run
equilibrium price is non-increasing in factor prices.
10-23
(ii) Take any a:
> 0, and substitute a:w instead of w in equations
-
-
p(a:w) = l(a:w,q(a:w))
(1 ),
(2):
(1')
-
f (a:w,q(atw)) = 0.
(2')
q
Using the definition of f(w,q) and homogeneity of c(w,q) in w (Proposition 5.C.2 in the textbook), we obtain f(a:w,q)
= c(a.w,q)jq = a:
= a:
c(w,q)jq
f(w,q),
I (a:w,q) = « I (w,q). q
q
-
Therefore, equation (2') can be rewritten as f (w,q(a.w)) = 0, which together
-
q
-
with equation (2) implies that q(a.w) = q(w). Using this fact, the above derivations, and (1), equation (1') can be rewritten as p(a:w)
=
a. f(w,q(a:w))
= a.
.
f(w,q(w))
= a.
p(w),
-
which means that p(w) is homogeneous of degree one in w.
(iii) Differentiating (•) with respect to the input price vector
w and using
(2), we obtain 2-
2 w
-
-
-
D p(w) = D f(w,q(w)) + ajaq [V f(w,q(w))] 'I'J q(w) = w
= D2 f(w,q(w)) + w
'1'J
w
f (w,q(w))
w q
'I'J q(w) w
= D2 f(w,q(w)). w
(The order of differentiation has been changed, which we can do due to the twice continuous differentiability of
f( ·, ·).)
Now, using the definition
of f(w,q(w)), we obtain 2-
2
-
2
-
-
D p(w) = D f(w,q(w)) = D c(w,q(w))jq(w). w
w
According to Proposition 5.C.2 in the textbook, c(w,q) is a concave function 2
of w, and thus D c(w,q(w)) is a negative semidefinite matrix. Therefore, w
2
D p(w) is negative semidefinite, too, which implies that p(w) is a concave
10-24
function of w.
(a) Let q (p) be a firm's long-run choice of output as a function of
JO.F.S.
L
its price, with p • being the initial price and q • = q Cp • ) the initial output. L
Let z (p) k
p, with z
= z k <w.qL (p)) • k
be the optimal choice of factor k when output price is
= z k (p • ). Denote by q L (pI z k ) the firm's short-run choice of output ·
as a function of p, the output price, and zk the fixed input of factor Jt. Let 1t
s
(pI z
k
) denote the short-run profits as a function of output price p and
factor It's input z . k
expressed as
KL (p)
Then the long-run profit function can be
= ns(p I zk(p)). Since the firm can always keep its
short-run demand of factor k in the long run, we must have n (p) L
!:
n Cplz•) S
k
for all p. Moreover, we know that the initial situation was a long-run equilibrium, which implies that n (p • ) = n (pI z (p •)) L s k
•
n (pI z • ). Therefore, S
k
the function h(p) = n (p) - n (pI z • ) achieves its minimum value of zero at L
S
k
•
p = p . For this to hold, the following second-order condition has to be
satisfied: h"(p • ) = n"(p • ) - n"(p • I z • ) L s k
2:
0.
By Hotelling's lemma, q (p) = n'(p) and q (p) = n'(plz • ). Ther ~fer~. the last L l s S k inequality can be rewritten as •
•
q'(p ) L
!:
•
q'(p ). S
(b) We will hold the number of firms J fixed both in the long run and in the short run. (If free entry occurs in the long run, then the equilibrium price will go back to its original level which is the minimum average cost, and the question becomes trivial.) If each firm's supply function is q( • ),
the market-clearing condition can be written as x(p,cx) = Jq(p(cx)l.
10-25
Differentiating with respect to a, we obtain
x (p,a) + x (p,a)p'(a) = Jq'(p(a))p'(a) P
a
From here we can express p'(cx.):
• = xu
p'(a }
•
/
[Jq'(p ) - xp},
where a • is the initial value of a, p
x
p
• x
•
p
(p ,
a •).
•
•
p(a ),
x
a
• x
a
• a •),
(p ,
From the exercise's assumptions, both the numerator and
denominator of the fraction are positive. Denoting by ps(a) and pL (a) the short-run and long-run price responses as functions of ex respectively and using (•), we can write
•
p~(a ),
i.e. the long-run response of the market price to a marginal increase in a is smaller than the short-run response. The aggregate consumption in the market is Q(a)
= x(p(o:),a).
Differentiating
with respect to a and using the result from part (a), we obtain
•
Q'(o: ) =
X
a
+ X
•
P
p'(a )
Denoting by p (a) and p (o:) the short-run and long-run aggregate S
L
consumption as functions of a respectively and using (•) and x'(p) < 0, we obtain Q'(o: • ) =x +X p, (a: • ) p s 0: s
~X
ex
p'(o: • ) =
+X P
L
•
Q~(a. ),
i.e. the long-run response of the aggregate consumption to a marginal increase in a is larger than the short-run response.
IO.F.6.
First we derive the long-run cost function for the Cobb-Douglas
technology:
c(q) =
wz +wz
Min (z
1 I
,Z ) I 2
s. t. z
0: I
1-0:
z2
2 2
i!!:
q.
10-26
The first-order condition for this program can be written as
w.Jw2 = cxz/U-o:)zl" Together with the binding constraint in the program, this gives us factor demands:
z z
1
2
[o:w
(q) • (q)
=
I
(1-o:)w
{(1-o:)w
1-Cl
1
1
q.
fo:w ~ q.
1
2
•
Therefore, we can write c(q) • w z +w z 1 1
A = w [o:w /(1-o:)w 1 zl I
i-a.
2 2
= Aq. where
+ w [(1-o:)w 'o:w 2
1/ '
2
1« = w« w (l-«) lo:«(l-4 ) 1-«. 2
I '
- A)q
= 0.
1
The long-run equilibrium is given by (i) Profit maximization .. p = c'(q) • A. Market clearing
(ii) (iii)
"*
a - bp = Jq.
= (p
Free entry -. pq - c(q) = pq - Aq
Observe that since technology exhibits constant return to scale, profit maximization (condition (i)) automatically implies zero profits (condition (iii). Also, the number of firms and the scale of every firm are
indeterminate. Condition (i) gives the equilibrium price p • = A, and condition (ii)
gives the aggregate output Q• = J •q • = a - bp • • a - bA . In the short run capital z is fixed at z •, and the number of firms J is 1
• fixed at J . From f(z . z ) = 1 2
= q 1/tl-Cl) fz •Cl/(1-a) . I •
c(q I z l 1
I
.
q we get the short-run demand for labor •
. IS . 1/!1-al
The short-run cost funct1on •
= w 11 z +
s
•
22
I
w z Cq I z l
= w 1z
•
w q
+
2
•a/(1-al
fz 1
The first-order condition for the firm's profit maximization is
Now, z • can be computed as the long-run demand for capital: 1
z
•
1
= z I (q
•
) =
1-Cl
[o:w /(1-a)w 1 zl
I
and the first-order condition can be rewritten as p =
wo: w c1 -« l / a a( 1-a )1-o:. · ( q/ q • )Cl/(1-al 1
2
10-27
•
q ,
.
From here we can solve for the short-run supply function of a firm: s( )
•( .IA)U-«l/«
q P = q Pt
·
The short-run supply function of the industry is Qs(p)
lO.G.l.
= J•qs(p)
=
J•q•(pjA)(I-4)/Cl = Q•(pjA)U-Cl)/«.
(i) Firm j solves
M
Max
L p mqmj
t.t
q E IR J +
•
c}qlj, ... ,
-
qMJ ).
m=l
The first-order conditions for all firms are p
• m
:S
• ljaq , acj (q •IJ •••• , qMJ mj m
= I.
> 0,
with equality
(1)
.. , M, j = 1, •• , J.
The firms' first-order conditions give. JM equations. Consumer i solves M XE I
Max IRM • me IR +
s. t. m + I
~
I:
m + 1
1/J(x I
m=l
11'
M
J
•
c.J+[B LPX.::S rn ma 1 lj m= 1 J =I
... , X
(
Ml
)
•
• •
L pmqm1
- c/qlJ'
... '
m=l
q • )). Mj
The first-order conditions for all consumers are
• , ..., x • >fax a;cx Mi mi I II
::s p • , with equality if x m
• > 0, mi
(2)
m = 1, .. , M, i = 1, .. , I.
The consumers' first-order conditions give IM equations. The market-clearing conditions are I
•
'x L mi 1=1
J
•
= 'L q mj , m = 1, .. , M.
(3)
j=l
(By Lemma 10. 8.1, we do not need to check that the market for numeraire
clears.) Thus, we have the total of JM + IM + M = (J + I + l)M equations and (J + I + l)M unknowns - prices, output vectors, and consumption vectors.
Since the equations (1), (2), and (3) do not contain consumers' wealth levels
10-28
w1, the set of solutions to these equations is independent of consumer wealth. (ii) Consider the following welfare-maximization problem: I
L • I ex11 , ...,
Max
1=1
1
• L X ml
s. t.
J
x
J
-
1=1
• L q mJ
Ml .
l -
L c J(q1 J ,
••• ,
q
J =I
MJ
) + w
• 0, m = 1, •• , M.
J=l
By examining the first-order conditions to (•), it is easy to see that an allocation (x•, •• , x •, q •, .. , q • ) solves (•) if and only (p • X •' 1 I 1 I I
•
.• , x •, q •, •. , q • ) solves (1)-(3) for some price vector p •. I
1
I
(For conciseness, every variable here· has M components corresponding to the non-numeraire goods.) Suppose in negation that ep, x , .. , x , q , ... , q ) and I
I
J
1
ep• , x'1, .. , x', q', ... , q') are both solutions to (1) - (3), and therefore I
e•>.
to
I
J
and that x
x" I
q" J
Cl ear l y,
*
k
- 1/2 x 1 + - 1/2 q J +
·
for some k. Take
k
1/2 x; for every i = 1,
... ,
I,
Ijz q' for every
...,
J.
j • 1,
J
-
ex l" ,
since both
x
.. , x " , q " , ... , q "l sat'1sf'1es th e const ram · t m · I
(x , .. , l
c•>.
J
I
x, q , ... , q ) and (x', .. , x', q', ... , q') do. As cost 1 I I I I J 1
functions are convex, we must have c (q") j
j
:5
1/2 c (q ) + 1/2 c (q') for every J. = 1.... , J. J
j
j
j
As utility unctions ¢. 's are strictly convex, we must have I
~
¢ (x") I
I
1/2 ; 1ex 1 l + 1/2 ; 1ex:> for every i
*
k, and
¢ (x ") > 1/2 ¢ (x ) + 1/2 ¢ (x'). k
k
k
k
k
k
Adding up all the inequalities, we obtain
r ¢ (x") - r c Jeq:·>
I :1
I .
J
I
I
I
. 1 J=
>
1/2 [
J
r ¢ (x')
....
1
I
I
+ 112
-
r cJ
(q')] •
j=l J
J
c r ¢ I (x I > - r c (q I =1
which contradicts the assumption that
J
exI , .., x I ,
10-29
J =1 J
H. J
q , ... , q ) and (x ', .. , 1 l J
x', q', ... , q') are both solutions to (•). I
1
J
Therefore, individual consumption levels x 's at a solution to (•), and thus I
at a solution to (1)-(3), are uniquely determined. I
J
r qJ = r x
(iii} The optimal aggregate production levels are
I =1
j:1
I
• and they
are uniquely determines since all the x 's are uniquely determined. I
(iv) Now suppose that the cost functions c'( ·) are strictly convex. J Suppose in negation that (x , .. , x , q , .•• , q ) and (x', •• , x ', q ', 1
I
1
1
J
1
••• ,
1
q') are both solutions to (1)-(3), and therefore to (•), and that q J
k
"' q'
k
some k. Take
x;•
= 1/2
x
q" = 1/2 q J
x;
+ 1/2
1
for every i = 1, .•. , I,.
+ 1/2 q' for every j = 1, ... , J.
J
.
J
Clearly, (x", .. , x", q", ... , q") satisfies the constraint in (10.0.2), 1
I
J
1
since both (x, .. , x, q, ... , q ) and (x', .. , x', q', ... , q') do. As cost 1
1
J
I
I
I
J
1
functions are strictly convex, we must have
c (q") J
J
:s 1/2
c
(q ) +
J
j
1/2 c (q') for every J. J
k, and
#
j
ck(q~') < 1/2 ck(qk) + 1/2 ck(q/ As utility unctions ¢. 's are convex, we must have I
¢.(x~') ~ II
1/2 tf>.(x.) + 1/2 ¢.(x') for every i = 1, ... , I. II
•
I)
Adding up all the inequalities, we obtain
r ¢i(x;') I =I
-
r c.(q:'l j =1 J
J
I
J
I
>
1/2 {
J
r ¢ (x') I =1 I
+ 1/2 {
I
I
L t/J I (x I )
-
r c (q:ll J=1 J
-
J
[c
J
(q )],
J =I J
I =I
+
j
which contradicts the assumption that (x , · .. , x , q , ... , q l and (x', .. , I
I
I
J
x', q', ... , q') are both solutions to (•). I
I
J
Therefore, individual production levels q 's at a solution to I
(•). and thus at a solution to (1)-(3). are uniquely determined.
10-30
I
for
IO.G.2.
The first-order conditions (l) in Exercise IO.G.1 give for output of
good l:
p• c;/q•JJ) :S
1
, with equality if
q• J
> 0, j = 1, ... , J.
1
(1)
The first-order conditions (2) in Exercise 10.G.2 give for consumption of good l: •• (x 11
•)
s p • • with equality if
11
X
I
• 11
> 0, L = 1, •• , I.
(2)
The market clearing condition for good l is I
• Jell
I:
J
•
-I: qlJ.
(3)
J =I
1=1
(1)-(3) is a system of J+I+l equations with J+I+l unknowns -
•
(p , x I
•,
.. , x • • q • , ... q • ). These equations do not contain prices and
11
II
11
1J
quantities in the other markets.
From these equations, the price and
quantities in market l can be determined independently of other markets. Total welfare can be computed as I
S(x , .. , x , q , ;., q ) I
I
J
I
I
L" t/Jii(xli)
= l
J ~
-
L. j =I
1=1
= S
(X
1
11
,
.. , X
II
,
=
L t/J I (xI )
1=1
c (q lj
Ij
I
)J
+
l
J
L t/J -1.1 (x -1,1 ) - I: c -1 J(q -1
q , .. , q ) + 5 IJ
11
-1
•
•
1=1
J =I
(X
-1,1
, .. , X
-1,1
,
q
-1,1
•J
)}
, .. , q
= -I,J
).
The first term describes the surplus in market l, and the second term describes the surplus in all the other markets. The first term takes the form I
5 (x , .. , x I
II
II
, q , .. , q ) = 11
lJ
L t/J II Cx 11 )
1=1
J
-
L c lj (q I j ),
j=l
which is exactly equation (lO.E.l) in the textbook. The welfare analysis .
of market l can proceed in the same way as in Section 10. E, without consideration of other markets. If all goods are separable, every market l can be solved independently of others using equations (1)-(3). Moreover, total surplus can be written as
10-31
I
I
L
=
L' (x )
q , .. , q ) =
S(x , •. , X ,
I
J
l=l l
I
-
I
j=l L
J
L { L ' 11 (X II }
L C lj (q lJ )]
-
=
J •I
1•1 1=1
Lc
(q )
J
=
j
L 5 I(X ll ,
•• , X
l =I
II
,
q , .. , q ). IJ
U
Therefore, welfare analysis of every market can be performed independently of other markets.
IO.G.J.
•
•
2
3
(a) Let p , p
th~
be
market prices of goods 2 and 3, ·Jet x
•, x • 2
3
be
consumption of these goods by each consumer (since all consumers are identical. the will consume the same amounts). and let q •• q • be production of these goods by the firm. Assuming (x• • x • ) 2
with taxes t , t 1
2
3
2
3
» 0, the market equilibrium
on goods 1 and 2 is a solution to the following
equations, which are a modification of conditions (1)-(3) of Exercise IO.G.l:
•
P2 =c 2 +t· 2'
(1.1)
•= P3
c
(1.2)
• = P2
• • B'*'(x ,x >fax
3
+
t . 3
2
Y'
1
3
2
(2.1)
:
(2.2) Ix
•
Ix
•
(3.1}
2
3
•
= q.
(3.2)
3
Equations ( l.l) and ( 1. 2) pin down equilibrium prices, and equations (2.1), (2.2) pin down consumption of goods 2 and 3. Equations (3.1) and (3.2) pin down aggregate production levels. Since production exhibits constant returns to scale, aggregate production can be allocated across firms in an arbitrary fashion. Assuming that ·tax revenue is redistributed lump-sum across consumers, .
aggregate surplus can be computed as •
•
2
3
S = I¢(x ,x ) - (c q
•
•
f
•
•
•
•
+ c q ) = ltp(x ,x ) - c Ix - c Ix. 3 3 22 33 2 3 2 2
10-32
•
Let x (pz,p 2
3
and
)
•
.tC (p ,p ) 3 2 3
be consumer demand functions, i.e. the solutions
to (2.2), (2.3). Then aggregate surplus can be expressed as a function of taxes:
•
S(t ,t )
c +t ),x (c +t , c +t )) -
2 3
3
3
3
2
2
•
3
3
•
- c x (c +t , c +t )) - c x (c +t , c +t ) )] 2222
33
3322
33
t 3 = 0 and differentiate with respect to t 2, using (2.1); (2.2);
When we set
(1.1), (1.2), we obtain
BSCt ,O)jBt 2
2
= I[(p• - c )8x•(c +t , c 1/Bp + (p • - c )Bx•(c +t , c >/Bt J •
• +t = tlaxz(c 2
2
2
,
2
7.22
3
2
3
3
322
3
2
c 3 ljap2•
Thus. the marginal welfare change is the same as the marginal change in the area between supply and demand curves for good 2 (the derivative of expression (10.E.6) in the textbook), holding the price of good 3 constant at c . 3
(b) When we differentiate (•) with respect to t , using (2.1), 2
(2.2), (1.1), (1.2), we obtain as(t ,t >fat 23
2
= I[t ax
•(c
+t • c +t lf8p
2222
33
•
2
+ t 8x (c +t , c 3322
3
+ t )j8p ]. 3
2
The first term in the expression is the change in the area between supply and demand curves for good 2 (the. derivative of expression (lO.E.6) in the textbook), holding the price of good 3 constant at c +t . The sign of the 3
3
•
second term is the sign of ax (p ,p ljap . This derivative is positive when 3
2
3
2
goods 2 and 3 are substitutes, then the area measure would overstate the welfare loss from taxation. When goods 2 and 3 are complements, this derivative is negative, and the area measure would understate the welfare loss from taxation.
lO.G.4.
Differentiating (•) in Exercise 10.G.3 with respect to t
obtain
10-33
2
and t , we 3
•
•
8S(t ,t lfBt = I[t Bx Cc +t , ' +t )fBp + t Bx Cc +t , 3 3 2 2 2 2 2 2 3 3 2 2 2 3 BS(t
,t lfat
23
3
c +t )fapi; 3
3
= I(t ax• (c +t , c +t >fap + t ax•Cc +t , c +t >fap ]. 2222
33
3
3322
33
3
Now, S(t ,t ) - S(O,O) can be expressed, for example, as 2 3
S(t ,t ) - S(O,O) = S(t ,0) - S(O,O) + S(t ,t ) 23t 2 t 23 2
=
-
S(t ,0) 2
=
3
J
J
~
0
BSCs,Olfap2 ds +
BS(t ,slfBp ds • 2 3
2
=
J
I sax ·cc +s, c >fap ds 2
2
3
+
2
F 3
+
JIlt
ax·cc +t ,
2222
c3+slfap3
+ sax•cc +t , c +slfBp 322
3
3
1 ds.
0
lO.G.S.
(a) Let p
.. 2
,
p
·•
be the market prices of goods 2 and 3, let x
3
2
,
x
. 3
be
consumption of these goods by each consumer (since all consumers are
•
•
2j
3J
identical, the will consume the same amounts), and let q , q
•
•
2
3
of these goods by firm j. Assuming (x , x ) » 0, with taxes t , t 1
2
be production
the market equilibrium
on goods 1 and 2 is a solution to the following
equations, which are a modification of conditions (1 )-(3) of Exercise 10. G.l:
• = P2 • = P3
•
t ·
c'(q ) + 2 2
•
c'(q ) 3
3
( 1.1)
2'
+
t 3'·
(1.2)
(2.1)
(2.2)
Ix
• = q; • 2
• Ix 3
(3.1)
2
• = q . 3
(3.2)
Eliminating prices and outputs from the equations, we obtain
• •
a;cx ,x >fax 2
3
2
= c'2 (Ix •2 )
+
t2
a;cx • ,x• >fax = c'(Ix • ) + t 2
3
3
3
3
(l) (2)
3
Now, we differentiate these two equations with respect to t , 2
10-34
treating x • and x • as functions of t : 2
3
2
"" x'2 + "" x'3 = I c'' x'2 + 1 "'22 "'22 2 ~2 3x' + ' 2
x' = Ic" x'3 3
333
(To· avoid cumbersome expressions, we have defined ¢
•
•
c" • c"(lx ) x' = dx ldt ). I
I
I '
I
1/'
= 8¢Cx•,x• )/Bx Bx ,
II
2
3
I
1
•
2
From the above two equations we can express
x,3 • .,.. ""7:JfI["'.,.. 232
+ I c""" 2 .,..33 +
""22.,.. ""33 .,..
-
x;: Ic"' 3 22
Differentiating equation (1.2), we now obtain: •
2
dp-3'ldt 2 • Jc"x' • lc''• I[• - ' 3 3 3 23'' Z3
'
22 33
+ Ic"' 2
+
33
2
Ic"• - I c"c"]. 3 22 2 3
2
Since u( • ) is concave, D u( ·) must be negative semidefinite, which implies, in particular, that ~
22
it follows that ~
22
s 0, ~33 :s 0, '
2 23
- '
(/1
22 33
:s 0. Since
the cost functions are strictly convex, we must have c" > 0, c" > 0. These 2
3
inequalities together imply that the denominator of the above expression for
• dpjdt
2
sign of '
• ldt is negative the the sign of dpis always negative. Therefore, . y 2 23
•
When '
23
< 0, goods 1 and 2 are substitutes, and an increase in
t raises p • . Conversely, when 2 3 increase in t
2
.
4>
> 0, goods 1 and 2 are complements, and an
23
reduces p •. 3
(b) [First printing errata: Assume that c(q , q ) = c q 2
3
2 2
+c (q ). J 3
3
Assuming that tax revenue is redistributed lump-sum across consumers, aggregate surplus can be computed as
• •
S = lf/J(x ,x ) - (c q 2
3
•
22
• Let x (p ,p ) and 2
2
•
• •
3
2
+ c (q )) = ltf>(x ,x ) - c Ix 3
3
2
• 2
- c (Ix • ). 3
3
be consumer demand functions, i.e. the solutions
3
to (2.2), (2.3). Then aggregate surplus can be expressed as a function of taxes:
•
•
3
3
•
p ),x {c +t , p )) -
- c x • (c +t , p • ))] 2222
3
2
c
2
3
• (lx • (c +t , p3))J 3 322
10-35
Differentiating the above expression with respect to t
2
and
using (2.1), (2.2), (1.1), and (1.2), we obtain as(t ,t )jat 23
2
= It ax•(c +t , p • ljap + It ax •(c +t , p •)jap + 2
222
+ [It
2
3
2
•
ax •Cc +t ,p ) lap 2
2
2
3/'
3
3
322
+ It
3
•
3
(•)
2
ax (c +t ,p •>jap 1 dp-• ldt . 3 2 2 3 3 y 2
Now, in Exercise 10.G.3 we calculate this derivative assuming that p • is fixed 3
at c
3
+ t , i.e. dp-• ldt = 0. This is fine when technology· exhibits constant 3 y 2
returns to scale, because then the equilibrium price of good 3 is unaffected by t . 2
But in the present case, an increase In t
substitutes and lowers p
3
2
raises p • when goods are 3
when goods are complements (as shown in part (a)).
How would ignoring this effect bias are calculation of the welfare loss? We always know that we know from part
ax•3 ( ·, ·lf8p3 < 0. With complements, 8x•2 ( •• ·lj8p3 < 0, and
(~)
• that dpjdt
2
< 0. Thus,· ignoring using the formula in
Exercise lO.G.J (b) ignoring the correction terms in (•) would result in an overstatement of the welfare loss. With substitutes, Bx• ( ·, ·lj8p 2
•
know from part (a) that dpjdt
2
If t
3
> 0, and we
> 0. Thus, the first correction term is is
negative and the second correction term is positive. unclear.
3
The net effect is
= 0, the formula in Exercise lO.G.J (b) would be overstating
the welfare loss.
But if t
3
is large and t
2
is close to zero, the formula in
Exercise lO.G.3 (b) would be understating the welfare loss (which would actually be a gain).
10-36
CHAPTER 11
ll.B.l
Note first that the technologies are CRS so that producers will
either choose no production or will choose producing at full capacity .. Assume that the prices of apples and honey in the market are such that producing both products is efficient.
Since technologies are CRS, maximum profits are
equivalent to minimum costs, and generically Jones will choose only one of the possible technologies. and only if
In particular, he will choose artificial production if
H(pm+ w) < H(bpb+ kw), or, if and only if pm < bpb + (k - l)w .
Now, artificial production is socially optimal if and only if the total cost of producing both capacities is cheaper under artificial production of honey.
i.e., if and only if Hb H(p + w) + Aw < H(bpb+ kw) + (A - c)w , m
or, if and only if p < bpb+ (k m
So, if
*
~c :s:
~)w c
.
l, Jones' choice of honey production will be socially efficient. If
> 1 then for
bpb + (k -
~)w < pm <
bpb + (k - I)w Jones will choose artificial
production but natural production is socially optimal (for other values of p
m
Jones' choice will be socially optimal). Clearly, k and pb have no effect on the efficiency of Jones' choice. changes iri the value of p
m
will cause Jones' choice to be efficient or
inefficient as described above.
The values of b and c will determine the
degree of the externality: as b increases, or as c decreases, the externality becomes stronger and thus the possibility of inefficiency is larger (the interval in which p
m
will cause an inefficient choice becomes larger).
Smith's costs of producing A apples when Jones uses artificial honey
11 - 1
production is Hb
(A - -)w.
c
Aw, while his costs when Jones uses natural honey pro:luction is Hb Therefore, Smith will be willing to pay up to -·w in order to c
bribe Jones to switch to natural honey. It is obvious that if both farms were owned by the same owner then full efficiency would be reached since the owner would minimize total costs. A government can achieve such efficiency by subsidizing bees so that when it is . socially optimal to produce naturally, the subsidized price of bees will cause Jones to do so.
(a)
11.8.2
This can be financed by a lump sum tax on Smith.
The problem becomes: Max h,T
Letting
~
fl> (h) + w 1
1
- T
denote the Kuhn-Tucker multiplier, and assuming an interior
solution, the FOCs are:
= 0
(l)
(2)
aq, ch. w +T) 2 2
-1 + ;\ • __...:=---.8:--w___;;~-
=0
•
which together yield, 0
0
0
0
aq, Ch •w +T )lah 2 2 •
B,P (h ,w +T )/Bw 2 2
(b)
Consider consumer 2's problem:
Letting h(ph, w ) denote his demand for h, the FOC satisfies: 2 aq,2(h(ph,w2).w2- ph ·h(ph,w2)) 8h
Differentiating this expression with respect to w
11 - 2
2
we get:
82¢2 8h(ph • w2) 8h
2
•
= 0 .
Under very general conditions (which we usually assume) we have that the cross 82¢ 82¢ 2 2 partials are equivalent, i.e., that w h = Bh w , which changes the above to 8 8 28 2 (we discard the subscript 2 since only consumer 2 is relevant):
Bh(ph, w)
(•) 8w
-- 82¢
82¢ ph. 2 aw 2
-
. Bh 2
a2;
-
Bh8w
2
2 a ; 2 Ph • 8h8w + ph· 2 8w2 82¢
•
, and substituting this into the
From the FOC
expression (•) above will express the wealth effect in the required terms. 8¢ /Bh 2 0 From part (a) above we have that ¢1 (h ) = - a; ;aw · 2 2 .
(c)
this with respect to w
(
..
)
Bh aw
0
2
2
and rearranging terms we get: 8¢2 82¢ 2 • 2
B¢2 82¢ 2 • aw
Bh aw2
8¢2 aw
We need to show that
Differentiating
c••)
2
2 ·¢"(ho) 1
+
2
BhBw
8¢2 82¢2 • aw2
Bh
2
2
-
a;2 a ;2 •
Bh Bh8w
is positive (for a positive externality.
2 To
proceed, we define consumer 2' s indirect utility function in the usual way, denoted by:
Now, consumer 2's demand for the externality, h(ph,w l, can be obtained using 2
11 - 3
Roy's identity:
and differentiating this with respect to w
-----=-
we get:
2
a2 .;2 t8ph aw2
2
8 .; 2 + z' 2 ca.; taw ) 8w 2 2 2 8.; taph 2
•
It is given that consumer 2's demand for the externality is normal, so that 8h(ph,w )18w > 0, which together with (3) above (after multiplying all terms 2 2 2 • In (3) by ca.; taw ) > 0) implies that: 2 2
2
(4)
a.;2 8 .;2 •
2 aph _aw 2
a2.;
>
2
8Pjl2 •
•
8ph8w2 aw2
Since the externality is a normal good, it is not a Giffen good, which implies
aph
or In inverse terms, Bh
< 0.
Multiplying both sides of (4) by
this we get:
---·
(5)
8ph •
•
From consumer theory (chapter 3) we know that the following ider.:ities relate the indirect utility function with the utility function:
-
•
•
---· aphaw
2
=
•
'
Bh
•
Substituting these identities into (5) above we obtain: 82¢ Bt/>2 a¢2 a2¢ 2 2• • < 2 ' 8h8w aw ah aw2 2 which implies that the numerator of the expression given in the model
0, and
11 - 4
c••) > 0.
is negative.
It is
Furthermore, strict
82¢ 2 quasiconcavity of ¢ Ch, w l implies that < 0, and 2 2 2 Bw 2
<0.
All these
inequalities cause the. denominator of (.. ) to be negative as well, thus we 0
have shown that
Bh
aw2
> 0 for a positive externality.
The reverse can be shown
for the case of a negative externality using the same methods.
With a Pigovian tax, bargaining will change the level of the
11.8.3
externality.
To see this consider the situation where an optimal tax is
chosen such that consumer 1 chooses the efficient level h (w.l.o.g.) that h is a negative externality.
0
•
and assume
An infinitesimal reduction in h
has a zero first-order effect on agent 1, and a positive first-order effect on agent 2.
Therefore, there are gains from reduction of h below h A
agents can bargain to obtain some h < h
0
0
,
and the
which will be the final choice of h
given the Pigovian tax and the ability to bargain. On the other hand, this argument will not hold for quotas. At the point h an infinitesimal reduction in the level of h by 9'>]. (h
0
)
and will increase 2's utility by
will -9'>2(h
0
,
reduce consumer l's utility 0
),
which cancel out given
the optimal choice of h
0
•
Therefore there are no gains from trade and the optimal
level will not be bargained down (the· opposite holds for a positive externality).
11.8.4
Assume that ¢ Ch,e) is such that given any level of h chosen by 2
consumer l, there is a unique optimal level of e that is consumer 2's "best response".
That is, there exists a function e•(h).
Since e will be optimally
•
chosen by 2, and it has no external effect on consumer 1, there is no reason to tax or subsidize this activity - it is not an externality.
It may,
however, still be optimal to impose some tax/subsidy on h, even though
consumer 2 can choose an e in response.
ll.B.S
(b)
(a)
The firm solves
(1)
p
(2)
0 s
$
Max p · q - c(q,h), and the FOCs are: qlh
Bc(q• h•) aq t
with equality if q• ) 0
1
1
1
with .equality if h• > 0 .
Since the consumer's utility function is quasilinear with respect to
money, the utility posibility frontier is a linear line with slope -1, and therefore we can find the Pareto optimal level of h and q
by maximizing the
sum of the profit function and the utility function without wealth, i.e., Max q,h
p·q - c(q h) + cJ>(h) , 1
which yields the FOCs, 0
(3)
(4)
(c)
0
acCq lh ) p s aq
, with equality if q
8c(q 0 ,h0 )
t/>' (h) s ---:::~Bh
I
0
with equality if h
> 0 ,
0
>0 .
Let t denote the tax rate on output, the firm solves: Max q,h
p·q - dq,h) - t·q ,
which yields the FOes, (5)
p s
(6)
0 s
Bc(q, h)
Bq
+ t
Bc(q,h) , with equality if h ah
is imposea on h, say T, the firm solves: p·q- c(q,h) - T·h ,
11 - 6
I
>0.
This gives the same level of h = h• as in part (a) above.
Max q,h
> 0
, with equality if q
If, however, a tax
wh1ch yields
t~e
F'OCs, <
-
(7)' p
8c(q,h) aq
8h
=-
T
with equality if q
> 0
1
with equality if h
>0 ,
ac(ql h)
(8) and if we set
1
9'>1 (h
0
then this gives the level h
)
= h0
1
as in part (b)
above, and efficiency is restored.
(d)
Let t denote the tax rate on output, the firm solves: Max q
p·q - c(q,o:q) - t•q ,
which yields the F'OC, 8c(q, o:q) p 8q <
(9)
, with equality if q
and if we set t = - o:·9'>1 (h
0
)
0
then the pair (q ,h
0
)
> 0 •
wiJI solve this F'OC as in
part (b) above, and efficiency is restored.
ll.C.l
.Following the analysis of he textbook, it can easily be shown that
the subsidies which will lead to an efficient equilibrium are s. I
ll.C.2
0 = t'. .(j'>'.(q ). LJ""l J
The analysis of the textbook regarding private provision of a public
good yields two FOCs. (numbered as in the textbook): 9'>~
I
(x•)
~
p•
with equality if x! > 0 I
with equality
if q > 0
(ll.C.3) ,
(ll.C.4)
If we have a subsidy s for the firm then (ll.C.4) becomes: p• + s
~
Let k • Argmax {9'>~ (x •
1
0
c' (q) )},
with equality if q > 0 .
i.e., k is the index of the consumer with the highest
I
marginal utility at the optimal level x
0
(there will generically be only one .
such consumer, if there are more then simple modifications will do). subsidy in the following way: s = L;ek9'>jCx
0
11 - 7
o· o
).
Set the
The pair (x ,p }, where
p
0
= f/>k(x
0
),
will constitute a competitive equilibrium where consumer· k will
purchase exactly x. = x
0
1
,
and all other consumers will have f/>'.(x 0 ) < p 0 so J
they will not purchase any additional amount of the public good. firm will produce q
0
=x
0
Clearly, the
since it faces a gross price of p + s = ~fl>i (x 0 ). 0
First, we refer to the solution of exercise 10.E.3 for the basic
U.C.J
.
setup and the results of the Ramsey taxation problem without a public good. Second, We modify the setup of this exercise so that the consumers have identical utility functions so that we can view the problem as if we had a representative consumer. The more general case is a straightforward extension, yet the algebra is messier. are not affected.
Furthermore, the qualitative results
Let each consumer's utility function be given by: L
uCx 0 ,xl' ... ,xL) = x 1 +
L ¢ 1Cxl,x 0 ).
l=2
The government budget constraint becomes (the cost is divided by the number
of consumers since we are looking t.t a representative consumer): c(x )
L
0
E tlxL = - -
1=2
I
The Lagrangian of the problem becomes: L L
= xl
+
L
f/>l (xl,xO)
1=2
which yields the FOCs (1)
= l - ;\,l = 0,
(2)
-
11 - 8
-
(3)
(4) Note that the first, second and fourth of these equations are the equivalent to those in the regular Ramsey taxation problem, and the conclusions of that model apply.
The difference is that here the level of taxes to be generated
is endogenous and is chosen to maximize the utility of the representative consumer.
The condition for this is in (3) above, which states that the
marginal social benefit from a marginal unit of the public good must equal the marginal cost of that marginal unit.
11.0.1
Suppose that h
= h2 1
= ... hi
=h•
is a symmetric Nash equilibrium of
The it must be that for every i, h• solves
the simultaneous move game.
h.
;
Max h.
-
1
1
1-lh. Thi+ I
1
h~1 ··2
which yields the FOC 1
h.
;'
1
1
•
1-lh. yht I
1 I-lh. yhi+ I - hi - h. 1 2 l -h + 1-lh" I i I
·r
For h• to be a NE this should hold for h. = h", 1
which yields
h" =
0
- h· = 0,
be a symmetric Pareto optimum. 0
of each student i will be 0
0.
(o: fl·~'(l) )·
Now, let h = h = • · · h • h 1 2 1
h
•
1. e.,
-
ui =
The utility
2
;(1) - (h ) , which immediately implies that 2
= 0 is the Pareto optimal outcome.
The intuition is simple; by studying,
every student imposes a negative externality on others, and the competitive outcome has too much studying.
ll - 9
11.0.2
For
th~
optim~l
Pareto
which yields the FOCs
=ah--.
+
Max
{h.}
r.!_ (t/>.(h.,I:jh .) 1 L11 1 J
+ w.), 1
1
a4,.•
a;.1
r
s 0 with equality if h ~ > 0
a[.h. ·Lk-i
I
J J
I
for all 1=1, ... 1.
outcome we solve
On the other hand, in a competitive equilibrium each
individual solves
Max
f/>.(h.,h.+I: .... h.) + w. , which yields the FOC (for I
h.
I
J
J~l
I
I
I
eveFy i)
8¢. 1 Bh + 1
Bf/>. I .h. :s 0 J J
ar
with equality if hi > 0.
Bt/>k .h. for all k), then if J J
at
externality is negative (i.e,
h~
1
necessarily have h! > h~ (due to the concavity of f/>). I
> 0 and the
If, e.g.•
> 0 .we will
To restore the Pareto
I
optimal outcome in a competitive equilibrium, we must set an individual tax 0
for each i of
11.0.3
0
Bt/>k ( hk I I: .h . ) t. = h J J . 1 k~i j j
r
at
In a competitive equilibrium a price-taking firm j solves Max q. J
where Q = qj + Lk:;ejqk .
n.(q.,q .) = p·q.- c(q.,Q) J J -J J J
1
The FOC for this program is
p- c (q. Q) - c (q.,Q) q J 0 J 1
c
0,
and the SOC is satisfied by the assumptions in the exercise, - c
(q .,Q) - 2c cq . 1 Q) - c cq .,Q) < 0 . qq J q0 J 00 J
Therefore, this program has a unique solution q•, independent of j.
o•
= Jq•, then a competitive equilibrium ( 1)
_ P - cq
o•
•
will be determined
Let
by
co· r· o•l • c0 co· r· o•l .
which has a solution for p > 0 because cq > 0, c
o•
o•
0
< 0, and cq + Jc
together imply that cq(T,Q•) + c Cy-,Q•) > 0.
0
In contrast, the program for maximizing total surplus is
Q
Max S .: ql''''lqJ
J p(x)dx - \ c(q .,Q) , . .L J 0 J=1
J
11 - 10
0
>0
and the FOCs for this program are
:s
=0
= p(Q) - c (q .,Q) q. q J J
'
and we can check that the assumptions ensure that the SOC is satisfied. q
0
Let
denote the solution to this program (again, independent of j), and let
0
Q
0
= Jq .
The (2)
(Again, by cq > 0, c CQ
0
< 0, and cq + Jc
< 0, (2) implies that
and
}o[cq(~,Q)
+
p(Q
c 0 (~,Q)]
0
> 0, a solution will exist.)
0
0 )
< cqcf-.0°)
j(cqq(~,Q)
=
0
assumption), then we must have that Q
+
+
co(~
0
,Q
0
) •
(J+l)cqQ(~,Q)
> o•.
Since
Also, since p' (Q)~O. +
Jc 00 c~,Q))
> 0 (by
This is intuitive since firms
ignore the positive externality that they create, and we have an under-production competitive equilibrium.
To restore efficiency the
government can subsidize production with a subsidy of Firm j's FOC will then be 0
easy to see that Q = Q
11.0.4
p -
and p
0
0
(J-l)c (~ ,Q
=
0
s
o
Qo
= -(J-l)c0 c1 .o ).
= cq(qj,Q) + c (qj,Q), and it is 0 Qo 0 p(Q ) will cause q. = to solve this FOC. J J )
For the Pareto optimal outcome we solve
Max \ __: ¢.(h , ... ,h ) + \ .: n .(h.) ,
s n'.(h ~) with equality if h~ > 0
which yields the FOCs
for all j=l, ... J.
J
J
J
On the other hand, in a competitive equilibrium each
firm maximizes profits individually, and we get the FOC shown in condition (11.0.1) in the textbook.
To restore the Pareto optimal outcome in a
competitive equilibrium, we must set an individual tax for each j of
-
t. = J
\ I -L
•
Each firm will face the same tax rate if and
i=l
11 - 11
1 only if we have \ l.i=l
U.D.S
(a)
for all j,k.
(First Printing Errata: the assumption that /(0)=0 should be added.]
This is a model of free entry so fishermen will send out boats as
long as there are positive profits from doing so.
Therefore, the equilibrium
•
number of boats, b•, will be reached when
This condition is that average revenue equals average cost. (We ignore integer problems, but if we are to give the integer equilibrium number then it is b• such that
(b)
- r
2:
- r < 0 .)
0 and
To characterize the optimal number of boats we must solve for maximum
total surplus, i.e.,
Maxb p·f(b) - r·b , the FOC is
0
p · f' (b ) - r
:::5
0, which
is necessary and sufficient since the SOC, p · f"(b) < 0, is satisfied. Therefore, the condition for the optimal number of boats is f' (b
0
)
= ~·
•
1. e.
•
that marginal revenue equals marginal (and in this case average) cost. Assuming that f(O) = 0 ensures that b
(c)
:::5
b• (equality only at 0).
To restore efficiency we need the equilibrium condition satisfied at b
i.e., we need the tax level to satisfy
(d)
0
f(b•) b•
=
r+t p'
or t =
0
,
- r.
Clearly, if owned by a single individual, the problem to be solved is
exactly that solved in part (b) above, which results in b
11.0.6
(a}
0
.
First, if the firm decides to go off and generate any level of
the externality, absent of an agreement, it solves
2 Maxh p(h) = a + 13h -,.,.h •
the (necessary and sufficient) FOC is t3 - 2J.lh• = 0, or h• = ~ . This yields 2 J.l the firm profits of rr(h•) = a + ~J.l , which is the firm's reservation profits. '
A coalition of 81 consumers making a take-it-or-leave-it offer to the firm
11 - 12
firm, and we assume that the coalition splits T equally among its members. can therefore
m~ximize
the utility of a representative member, subject to the
firm accepting the offer Max h,T
i· (
'1 - 1Jh)
s. t. n(h) + T Clearly,
T -81
n(h•) .
2:
at the optimum the constraint will bind, thus for any h that solves
~2
T = n(h•) - n(h) = « + 4;l - (o:
the program,
~2 + f3h - J.Lh ) = 4,! - f3h 2
2
+ J.lh .
Substituting this into the objective function we obtain the (necessary and sufficient) FOC, -
T+81t3- -
= 0 , which yields
h o (8) =
~
- 871 21-L
•
and
2 2
To(e) = e T/
41J
•
We can now determine the subgame perfect Nash equilibrium level of 8 by requiring that no (marginal) individual outside the coalition wishes to enter 0
0
it, given that the values h (8) and T (8) will be offered, and the firm will accept the offer .. Given 8, the utility of an outsider to the coalition is: uout =
f·
71ho(8)) .
(7 -
If this individual joins the coalition then
e
increases by
~ which reduces the
equilibrium externality level, but this individual now has to share the 0
T (e +
transfer T and pay
~~
--~_I_ <
e
+
1 -J · 1
Thus an outsider will be
•
I
indifferent between staying out and entering the coalition if: 1
oo
·r (7 - 11h (e >J and substituting for h
0
( ·)
1
= I" <1
- 11h
oo
+
1
(80+
_,.71 2 I
yll - -.....,4,_1-L-=1- - •
from above we get,
· h g1ves · · f ract1on · wh 1c us t h e equ1'I'1b rmm
e0
1
= I ' i.e. • the equilibrium
coalition consists of only one individual (the second individual may be
11 - 13
We
coalition consists of only one individual (the second individual may be. indifferent, therefore may enter).
(b)
0
Note that h (1) (the externality generated when 8 •
optimal level of externality. I ·= 1, we have 8 t h a t f or a II I
U.D. 7
!:
0
= 1.
1)
is the socially
From the analysis in part (a) above, when
When I = 2, we have 8
2 , we have that 8 °e
[First Printing Errata:
1 2 ,
0
1
<2 ,
E
Generally, we have
1}.
0 so t h at I'tm _. 111e • 1
o•
(i) The question should begin with "A
continuum of individuals can build ... "
(ii) Part (c) should continue with
"... and allowing transfers between individuals."] (a)
Assume in negation that only one neighborhood is occupied.
First
assume it is B. and consider the most prestigious individual with Since
e8 = i
(1 + 1)(1 +
a =
1.
, then this individual's utility from staying in neighborhood
1 2.) -
utility would be
c
8
= 3 - c
8
s 3.
(1 + 1)(1 + 1) -
B Is
If he would move to neighborhood A his
c A = 4 - c A > 3, so all individuals in
neighborhood B cannot be an equilibrium.
Now assume that only A is occupied
and again consider the most prestigious individual with 9 = 1. from staying in neighborhood A is (1 + 1H1 + utility from moving to neighborhood B is
~)
- cA
=3
(1 + 1)(1 + 1) -
c
His utility
- c A' and his 8
=4
- c
8
> 3 -. cA
so all individuals in neighborhood A cannot be an equilibrium - contradiction.
(b)
Let an equilibrium be a pair
(El A'e ),
8
- -
where eis {8 : type 8 locates in
neighborhood i}, and let e A'eB be the average prestige levels associated with such an equilibrium.
Claim: 9 A must take on the form
A
A
[8,1), for some 9.
Proof: Assume 9' prefers A to B: (l + 8' )(1 + SA) - cA> (1 + 8' HI + SB) - cB. Rearranging gives us: (1 + 9')
CA - CB !:
-eA
11 - 14
-
- eB
,
which implies that all types
locates in which neighborhood, and it is calculated by solving:
.
which yields,
.e
,.
- cA
= o + em
+
... e -) 2
= 2(c A - c l -I. 8
.... (c)
Starting at the equilibrium with e as given above, if a small group of
individuals from the lower end of neighborhood A move to neighborh
the average prestige in both neighborhoods will rise.
In particular, if
.
,.
for some c > 0 the segment [9,9+£] moved from A to B, the average prestige in
So, in both neighborhoods, an individual
both neighborhoods would rise by ;.
.
of type 9 who did not move will have a positive change in utility of (l+e)~. For a type
a individual who moved from A to B, there will be a negative change ... ,.
in .utility equal to (1 + 9)(1 + ; + (1
+ 9)(
c-1 2
) + (c A- c l. 8
¥) - c8
- ((I + 8)(1 +
~
+ ;
) - cAl
=
We denote the total benefit from such a change as B,
and the total cost as C, so that we have
e
1
= I 0+9)~
B(c)
de +
O+e>; de ,
9+c
0
C(c) =
I...
9+e £-1 ,. [(1+9)( 2 )
Ie
and we can evaluate the effect of such a change when e = 0: dB(c)
de
·= e= 0
9. 1 Jl 1 ... e 0+9l-z de + ,. O+el de - O+e+cl-z 2 9+c 0
I
2 9+e 1 1 =-+-+- + - 2 4 2 4 2
9 9
dccc> de
e+c
=
Ie ,.
-
2
(9+£)
2
4
,.
e
3
- (1+9+c)2 = 4
t
" e-1 ) [(1+9+e)( (1+9)2 d9 + 2 1
A
9+e
-
..
9
- -
2
+
(9+c) 4
2
""2
9 -- + 4
[Note that the last equality is true since from the· conclusion of part (b)
11 - 15
above we have that 1 + 8 = ~(cA -c8 )J,
We conclude that a marginal Increase in
c at c = 0 will lead to higher benefits than costs, and therefore we can find an
> 0 and transfers from the individuals who don't move to those that move
£
that will make all individuals better off .
ll.E.1
(a)
. The optimal quantity h
which yields the FOC functional form we have so 1ve: h..
2:
is determined by solving
:s 0, and substituting the
..
....
1 - ch + E[71] +
7 + 13 • h equa 1'1ty f or h.. b , Wit c +
l!:
fJ - bh + E[a) :s 0, from which we
0. .
(b)
Given a tax t, the firm will maximize profits and will choose h according
to the
roc an~~· 91
- t = 0, which yields the firm's "reaction" function to a
b - t . · e , as h(t,e) -- 8 + 13 tax t, given Max
The opt'1ma 1 tax t. . WI·11 b e given · by
E[¢(h(t,e},T!ll + E[n(h(t,e},e)] ,
t
. h . ld h FOC h w IC yle s t e
.
Smce
roc,
E[a¢(h(t,9),7J),Bh(t,8)] E[an(h(t,8),8),Bh(t,9)] = O ah at + ah at .
ah ( t a ) 1 . ah { t a ) at' ,= - b IS a constant, we can cancel out the at' from the and substituting the functional form of our functions the 7 - c·
E(8)+{3-t b
+ E[ 71] +
f3
H.••~
becomes:
E[ I b .Elel+f3-t b + e = 0.
from which we solve:
(c)
The choices of h and t are depicted in figure ll.E.l(a).
The intersection of the expected marginal profits and marginal utility curves solves for h and t as shown.
Consider a realization of 8 and 71 that gives
rise to curves intersecting at the point x as shown.
The optimal level of the
... externality is, therefore, h•.
If we use quantity regulation h, the dead
weight loss {or just "loss") is the triangle xuv, while if we use tax regulation t, the firm will choose h(t,e), and the loss is the triangle xyz.
11 - 16
We need to compare the expected difference in losses to determine which policy is better.
-E[~] /
/
/
/
/ /
X
~T
,..
t
I
'
I
I
'
II
I
i'--1
I".,
I
I
I
I
...
h
h* h(t,6)
''
h
E[ C)n 1
Figure ll.E.l (a)
C1h
Before we proceed, we will introduce a non standard, yet helpful way of calculating the area of a triangle. Consider the triangle in figure ll.E.l(b). The area is normally calculated using the formula A = -ied. We can divide the e edge e into e and e , and we can then write d = ~ , where b is the slope of 2 1 We can also write e = 1 2 1 e substitute into the above to get: A = 2 · b + c .
the top edge of the triangle.
Figure ll.E.l(b) 11 - 17
b~c · e
, which we can then
Getting b~ck to Figure 11. E.l( a), we can apply this to calculate the area of the triangle xuv: it is the edge uv squared, divided by twice the sum of the slopes of both marginal curves.
The height of the edge uv is
..
...
an ch, 8 l _ (- a~ ( h, 11) ) = ({3 _ b. 1' + 13 + 8 > + (,.. _ c. 1' + {3 +1J} = 9 + 1J. ah 8h c + b c + b
so that the loss from quantity regulation is,
L - (8 + 1J)
2
h- 2(b +'c)·
To calculate the area of the loss from taxation, we need the height of the edge yz. which is calculated by
...
...
....
an(h(t,9),8) arp(h( t, 8), 71) - ---:8:-:-h-'--ah
= - (1' - c . 8 + bf3 -t + 11 > _ (f3 _ b·8 + f3 -t +e) b = -,.. +
..
Ece
...
...
+ {3 - t) - 11 - t
..
= dh(t,8) - h) -
Tj,
...
...
where the last equality follows because we can rewrite t : c · h - 1' (follows by
... simple algebra). we get that
...
We need to find h(t,8) - h, and again using simple algebra
h(~. 8)
-
h=
~·
which gives us the loss from taxation,
(~ b
- Tj)2
Lt = 2( b + c)
.
We can now calculate the expected difference in losses,
• which implies that the optimal choice between quantity and tax regulation depends on the sign of (b - c).
When this term is positive, the economy is
more sensitive to changes in the firm, and therefore taxation is better since it changes the level of the externality marginal profits.
on the firm's realized
The reverse is true when the term is negative.
...
ll.E.2
dep~nding
...
The optimal quantities (hl'h ) are determined by solving 2 Max
...
...
11 - 18
which yields the F'OCs A
A
:!ii
...
...
... Substituting the functional form we get h
... h
i!!:
0,
2
=
... h2
i!!:
7 + [3 2c + b , with equality for
0. Similarly, given a pair of taxes Ct
,!2)•
1
and will choose hi according to the FOC
each firm will maximize profits
im.(h. ,8.) 1
a~.
1
-
t = 0, which yields each
1
firm's "reaction" function to a tax t, given e , as hi(t,e > 1 1
9.+
13 -
t
= ---=-b-1
.
... The optimal taxes t. will be given by 1
EI41Ch Ct ,e )+h Ct ,e >. 11) J + E[n Ch (t ,e ),e )] + E[rr Ch Ct ,e ),e )), 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1
~ax...
t 1, t2 Which can be calculated in a similar way as in exercise ll.E.l. When comp~ring between the two instruments things become a little more cumbersome, and for an analysis we refer the reader to Section V in:
.
M. Weitzman (1974) "Prices vs. Quantities," Review of Economic Studies 41:477-91.
It is, however, worthwhile to convey the intuition for the effects of a-
12
. If
the marginal profits of each firm are perfectly correlated, then it is as if we have one producer and the analysis of exercise ll.E.l follows through. As
a-
12
falls (they become less correlated) then it is more likely that taxes,
which yield negatively correlated output decisions when the shocks are negatively correlated, will be more efficient than quantity quotas.
ll.E.3
Assume that there exists another weakly dominant strategy that is not
truth telling, i.e., there exists c' such that. the consumer announces ...
c{c' ) = c"
...
~
c'.
If c" > c' then when c" > b .> c', the consumer is not
11 - 19
-
playing a best response to b, and any c > b is strictly better for him. Similarly, if c" < c' then when c" < b < c', the consumer is not playing a
... best response to b, and any c < b is strictly better for him - a contradiction to there being a weakly dominant strategy which is not truth telling.
We can write the scheme for the consumer as t(b,c) + T(b), where
ll.E.4
t(b,c) is the scheme described in the textbook. Given b, the consumer will ... want to maximize his utility, u(b,c) = t(b,c) T(b) - c. But this is -equivalent to maximizing t(b,c) - c, the same as in the textbook. Therefore, adding T(b) changes nothing (as does adding T(c) to the firm). ...
+
...
ll.E.S
Consider the scheme where every agent i announces a benefit b.. 1
call an agent i "pivotal to build" if
\~
b.l!:: K and \'. .b.< K. That is, if 1 L1= 1 LJ*' J
agent i's announcement causes the project to be built. agent i "pivotal to not build" if
L~=l bil!::
We
\~
b
Similarly, we call an
b. < K and \'. . .l!:: K. Ll=1 I LJ#I J
(Note that if
K then there can be more than one pivotal agent to build but no
pivotal agents not to build, and similarly, but reversed, for
K.)
L=l bil!:: K then the project is built, and each I
The scheme works as follows.
b.< \~ L1= 1 1
If
...
agent pays a tax t. in the following way: I
t.= I
0 if agent i is not pivotal to build, A
t.= K - \' .. b. > 0 if agent i is pivotal to build. I LJ*I J I
A
If L=l bi < K then the project is not built, and each agent pays a tax ti in the following way: t.= 0 if agent i is not pivotal not to build, I
A
t.= \' . . b. - K > 0 if agent i is pivotal not to build. I LJ*I J We need to show that this scheme chooses the choice of building in a Pareto optimal way, and that truth-telling is a weakly dominant strategy (we will
11 - 20
deal with balancing the budget at the end). A
A
Case 1: \ ..... b. < K and b.+ \.,..b. - LJ .. l J 1 LJ~1 J If agent i
We show this for all cases:
!:
K.
. announces some b. such that the project Is not built,
he is
1
...
not pivotal in any way and his utility is zero. If, however, he announces b. 1 ,. such that the project is built (in particular b = bi ), then he is pivotal to 1
.b.)
build and his utility will be u.:= b. - CK - \j . 1 1 L *1 J
!:
0, so that In this case
truth telling is a best response.
. .b. < K and b.+ \ ..... b. < K. ...
Case ~ \. LJ*1 J
1
LJ .. 1 J A
If agent i announces some b. such that the project is not built Cln 1
particular b ), he is not pivotal in any way and his. utility is zero. 1
If,
...
however, he announces b. such that the project is built then he is pivotal to 1
...
build and his utility will be u.= b. - (K - \ ..... b.) < 0, so that in this 1 1 LJ .. 1 J case truth telling is a best response. A
A
Case 3: \ . . b. > K and b.+ \ . . bj > K. - LJ*1 J 1 LJ*1
... If agent i announces some b. such that the project Is built (in 1
particular b.= b.), then he is not pivotal in any way and his utility i.s b .. 1
1
1
If, however, he announces b. such that the project is not built, then he is 1
...
pivotal to not build and his utility will be u.= 0 - (\' . . b. - K) < b., so . 1 LJ"'1 J 1 that in this case truth telling is a best response. A
A
Case 4: \. .b. > K and b.+ \' .. b. < K. - LJ"'l J 1 LJ"'1 J I( agent i announces some b. such that the project is built, then he is I
not pivotal in any way and his utility is b.. I
If. however, he announces b.
1
such that the project is not built (in particular b.= b.), then he is pivotal I
b. -
to not build and his utility will be u.= 0 - (l .. 1 LJ"'1 J
1
K) > b., so that in 1
this case truth telling is a best response. We conclude from the cases above that· truth telling is a weakly dominant strategy, so that in equilibrium we will have all agents telling the truth.
11 - 21
This, tugether with the building decision rule build if and only i.f
·
~~ b.~ K L•=l 1 •
implies that the Pareto optimal decision will always be carried out.
Finally,
it is clear that if we add a constant to the tax paid by agents, then truth telling is still a weakly dominant strategy, and we can add such a constant that on average will balance the budget.
ll.E.6
The analysis is almost identical to that of exercise 11.0.6, where A
A
each agent i announces a vector (b.(l), ... ,b.(N)), project n is built if and 1
only if
}.:!=l bi(n)
1
~ K(n), and the taxes are similarly defined·.
It can then be
shown that truth telling is a weakly dominant strategy for each project n, independent of the other projects, so that truth telling is a weakly dominant strategy for all projects.
The balance budgeting will also be done by adding
constants to the taxes.
ll.E.7
We can describe the variable total tax as a. function T(h)
which promises that T' (h)
=-
-
B¢(h,7J) •
8h
Thus, when the firm maximizes its
net profits, n(h,8) - T(h), then for any realization of chosen according to the FOC
= -tf>(h, 1)),
8n(h,9) 8h
=-
8¢( h, Ti)
Pareto optimality.
11 - 22
8h
e
we will have h
, which is the -:ondition for
CHAPTER 12
12.8.1
(a) The monopolists maximizes x(p)p - c(x(p}), which leads to the
following first order
m m m m m x'(p )·p + x(p ) = c'(x(p ))·x'(p ).
condition:
[pm- c' ( x(pm)) 1 ___..;;1_ m , where the elasticity of
Rearranging this we get
p
m
= - x' (pm)· pm
E(p ·) •
x(pm)
(b)
If c' (x(p
m
[pm c' ( x(pm)) 1 [pm- 01 )) > 0 always, then """"'----__.;......-.:,__;..;....;. < = 1. p
1/~:(p
m
p
m
Therefore
m ) < 1, or c(p m ) > 1.
12.8.2 t h e Foe
(a}
The monopolist's problem is to maximize o:p -c(p - c), which yields
« (1 - c )p -c + co:cp -c-1 .
= 0 , w h'1ch
. g1ves us p m =
-cc < 0 , so t h e 1 - c
problem is not well defined.
(b)
Using the monopolists FOC, pm=
Consumer surplus given price p: S(p)
-~cc, 1 =
we get qm =
1-c o:p SPx(t)dt = ( c-l ) . Cll
o:( 1 -~cc)-c· The deadweight
welfare loss can be calculated by subtracting from the consumer surplus under competitive pricing, the consumer surplus plus profits under monopoly pricing: o:c 1-c o: DL = ( c -1) ( c -1 l
b-
1-c
-cc
c)
( -c )-c] 1-c -cc -cc -c a. 1 - ( - c )a: = c 1 - c ( 1 - c) 1-c [ c -1 1-c
(c) After some messy algebra one can show that the deadweight welfare loss is decreasing in c (i.e. BDL(c);ac < 0).
As c increases the demand becomes more
elastic without changing the size of the market, which forces the monopolist to price closer and closer to the competitive price.
In the limit as c ...
the deadweight welfare loss goes to zero and the monopolist charges the competitive price.
12 - 1
111,
Assume the partial derivatives satisfy x < 0, x < 0, c > 0, and 1 11 1
12.8.3 c
11
> 0 (these are standard assumptions).
The monopolist's problem is Max
x(p,8)p - c(x(p,B),f/>). which yields the F'OC,
Differentiating with respect to 8 gives us: •
- 1)
and under our assumptions above we will have Bp/88 > 0 if x
> 0 and x
2
12
> 0.
Differentiating the FOC with respect to f/> gives us:
and under our assumptions above we will have 8p/B¢ > 0 if c
Suppose c
12.8.4
costs c , c 1
(p
1
2
1
> c , and pl' p 2
respectively.
- c )x(p ) 1 1
(p
2:
2
2
p
2
;S
1
> c
> 0.
are the monopoly prices given marginal
By revealed preference of the monopolist we have
- c )x(p ) and (p - c )x(p ) 2 1 2 2 2
these two inequalities and rearranging yields (c and since c
12
it must be that x(p ) 2 2
2:
2:
(p - c )x(p ). 1 2 1
Adding
- c HxCp ) - x(pl )) z:: 0, 2 2
1
x(p/
Therefore we must have that
p , or the monopoly price is increasing in c. 1 Suppose in general, that p , p 1
2
and q , q 1
2
are the monopoly prices and
quantities given cost of c(ql'¢ ), c(q ,fl> ) respectively [that is, qi= x(pi)]. 1 2 2 By revealed preference we must have that, p q - c(ql'¢ )!: p q - c(q .¢ l and 2 1 2 2 1 1 1 P q - cCq .t1> > 2 2 2 2
2::
P q - c(q ,¢ ). 1 1 1 2
Adding these two inequalities and
rearranging yields, [dq ,¢ ) - c(q ,¢ )J - [c(q ,¢ ) - c(q ,¢ )) 1 2 2 2 1 1 2 1 ¢2 ql 2 a 8qB¢ c(q,¢) d d~ > 0 q ., - .
12 - 2
2::
0, or
2
8 c(q,t/)) So, If 8q81/)
Thus, the monopoly price is increasing in ¢.
This is the condition we needed
in exercise I2.B.3 to get 8p/8¢ > 0.
12.B.S
(a)
Since each consumer desires at most one unit, the monopolist will
not gain by introducing a quantity discount.
Since the monopolist is unable
to discern any particular consumer's preferences he cannot price discriminate. Thus it is optimal for the monopolist to simply charge a price per unit.
(b)
Since the monopolist is unable to discern any particular consumer's
preferences he cannot price discriminate.
However, suppose the monopolists
charges different prices for different quantities bought. for q units by p(q).
Denote the price
Since resale is costless and the resale market is
competitive, the unit price of the good in the resale market must equals Min {p(q)/q}. q
q•
This implies that consumers will only buy a quantity equal to
= Argmin {p(q)/q} . q
and then they resell each unit for p(q•)tq•.
monopolist is just as well off by charging a price per unit of
12. 8.6
Thus, the
p(q•).
Let t be the tax/subsidy per unit of output: The monopolist maximizes
p(q) - c(q) - tq, and the FOC is p' (q)q + p(q) - c' (q) - t = 0.
Since we want
an efficient outcome, the monopolist must choose q so that p(q) = c' (q), which implies that we need
t = p' (q)q < 0, which in turn implies that the
government subsidizes the monopoly.
12.8.7
(a)
therefore
(b)
In a competitive equilibrium we must have p xm • a -
em c,
and x
w
= a -
ew c.
If the monopolist will serve both markets it solves, Max (a -
e m p)(p - c)
+ (a -
12 - 3
e w"p)(p - c)
m
-= p
w
= c, and
The FOC yields
p• =
c 2
: 8 m
+-.
However, since the women are willing to pay
higher prices than men for the same quantities (9m> ew) then it may be better to charge a price above :
.
This cannot be captured in the program above
m since it causes negative quantities in the men's demand, so we have to compare the solution p• above to the solution of +
Max (a - ewp)(p - c) which yields
c
2 . Figure 12.B. 1 shows the two cases.
p
p Separate Demands
Aggregate Demand
a -6w
a -6w
\ \
a -9m
_\ \
I
1
I
c ...........
a
aem-e...
q
em
........._MR
2a q
a
Figure 12.B.7 Due to the kink in the demand curve, the marginal revenue curve MR has a jump at that point (the dashed line).
When equating the marginal cost. c, to MR,
we could have one or two solutions.
When we have one solution, it means that
c cuts MR only to the left of the discontinuity, or only to the right. cuts MR only to the left, then the optimal price is above a/9
m
If c
and we only
serve the women. If c cuts MR only to the right, then the optimal price is below a/9
and we serve both markets.
m
When we have two solutions, then c cuts
MR both to the left and to the right of the discontinuity, and this is shown in the figure by points x and y.
When the monopolist moves from point x to y
A
(i.e., from p to p•) it loses profits equal to the triangle A, and gains
12 - 4
profits equal to the triangle B.
The costs and benefits of such a move will
... determine if p is optimal (only the women are served) or if p• is (both markets are served).
(c)
Given
quantity X, we maximize consumer surplus by solving, Max
qm Jqw p (x)dx + p (x)dx J0 m 0 w
s.t. qm +~=X and Jetting ). denote the Lagrange multiplier we get the FOCs
pm(~)
-
qw)
p
w
(q ) w
= ).,
(a which yield p (q ) •
with the constraint we solve for q
(d)
or,
m m
m
qm)
am
(a -
= ). and
aw
•
Together
and q .
w
The discriminatory monopolist solves
which FOC's yield quantities of q under price
m
p
= (a
m
=
~=--
29
These prices yield
m
-c9 )/2, and q m
•
w
= (a -c9 )/2. w
Note that aggregate output
discrimination is equal to the aggregate output without· price
e )c m w ---=--(9
discrimination if both markets were served (Q = a -
+
2
).
From part
(c), we know that the welfare maximizing distribution for a given output is to .
set p (qm) = p (qw), which is the case without price discrimin~tion when both m w markets are served. Notice that under price discrimination p (qm) """ p (q w) if
m
9
m
'*
w
e w , which implies that welfare is lower under ·price discrimination. ·
If,
however, without price discrimination only the women were served, then by allowing the monopolist to discriminate it will open the men's market (assuming that c < ~ ) without changing the price in the women's market. em This means that there is added surplus from the men's market and welfare under price discrimination is higher than welfare when price discrimination is
12 -
s
forbidden.
12.8.8
(a)
The monopolists intertemporal problem is
a-c m and the FOCs yield q7 • G = b-m > 0 (by the exercise's assumptions). 2 2 (b)
•
A benevolent social planner maximizes total surplus (assuming no
discounting of the future, we just add up both periods' consumer surplus, to both periods' firm's profits, and subtract both periods' costs),
- cq - (c-mq )q 1 1 2 and the FOCs are, (i)
(a -
bq ) + mq = c , 2 1
(ii) (a - bq )
2
which yield qi = q2 = ~=~ > 0. see t h at q m . < q •.. 1
1
=c
- mq
1
,
Comparing this with the monopoly quantities we
The way we wrote down the FOCs show that in fact there is a
sense of "price equals marginal cost".
Recall that price is margir."l surplus,
and the left hand side of both FOCs is exactly the effective marginal surplus from· each period's good: In the first period, aside from marginal consumer surplus, given q , any additional unit of q reduces marginal cost next period 1 2 by mq . The right hand side is the effective marginal cost in each period. 2 (c)
As we have seen, the social planner would want to produce more in every
period.
By increasing the output in the first period above q~. welfare in the 1
first period will be higher, and this will lead to a lower second period marginal cost.
This lower second period marginal cost will induce the
monopolist to produce more in the second period and will therefore further
12 - 6
increase welfare.
12.8.9
(a)
The monopolist will solve
q, 1 p(q) · q - c(I) • q - I,
Max which yields the FOCs are
m
(i)
(b)
c(l ) •
The social planner will maximize total surplus, max q, I
Jqp(x)dx - c(I}q - I , 0
and the FOCs are,
The monopolist produces less output than is socially optimal, qm < q•, and price is abov-:! marginal cost.
Given this, equations (ii) and (iv) imply that
-c' (I•) < -c' (lm), which in turn implies that 1• > Im.
Note, however,
that given the output level of the monopolist, he chooses the optimal level of investment.
(c)
Given a level I set by the government, the monopolist will set q to
maximize its profits, i.e., it will set q to equate MR = MC.
Therefore, the
governments problem is to maximize social surplus subject to the monopolist's behavior.
That is,
J q
max
q,
I
0
p(x) dx - c(l)q -
s. t. p' (q) • q +
p(q)
I
= c(l)
The Lagrangian is L = Jqp(x)dx - c(l)q - I - :>.[p' (q)q + p(q) - c(l)), which 0
yields the
FOCs ,
12 - 7
p(q)
c(l)
..
- ;\[p"(q)q + 2p' (q)] = 0 , ....... (vi) -c' (l)(q - ;\) = 1 •
(v)
Note that (vi) implies that the chosen investment level I is not optimal given ,.
....
the quantity q, and since ;\ > 0 we will have I greater than optimal given q. '
The Intuition is that in this second-best environment, the social planer
chooses an investment level larger than optimal for the given level of output •
in order to induce the monopolist to produce more.
12.8.10
Let p(x,q) be the inverse demand function for a given quality
level q.
Since x(p,q) is decreasing in p and increasing in q, then p(x,q)
is decreasing in x and increasing in q, and the monopolist solves, Max and the
roes
p(x,q)x
x,q
-
c(x,q),
are, (i)
mm mmm mm p(x ,q ) + p (x ,q )x = c (x ,q ) , 1 1
( 1.1.)
Pz
(
X
m
,qm) X m = c2 ( X m .qm) .
Given a quantity set by the monopolist, xm, if the social planner could set the quality q it would maximizes social surplus, m
Max and the
roc
q
[. p(s,q)ds
- c(x
m
,q) ,
0
is,
r
m
so that generally we will have q•
:;e
p(s,q•)ds = c2(xm,q•) •
0
qm, i.e., the monopolist will generally
not choose quality optimally given his quantity.
The intuition is that given
the quantity picked by the monopolist, the social planner cares about the marginal change in quality over the whole market (i.e., the average marginal change [J~ p (x,s) dx]/x ) while the monopolist only cares about the marginal 2 value of quality to the marginal consumer (i.e. p Cq,s) ). 2 linear demand in quality, the two will coincide.
12 - 8
Note that for
Following the arguments in the textbook, we cannot have a Nash
12.C.l
•
Equilibrium (NE) with p.> c for all i.
Note also that p
1
1
= p
2
= ...
= pJ
= c
is a NE - none of the ·firms can gain by raising its price and it will lose money if it lowers its price.
Extending the argument given in the textbook
one can show that if at least two firms charge p • c and all the other firms charge at least c we have a NE.
Therefore there will
be multiple
Nash
equilibrium, all yielding the competitive price.
12.C.2
(NE).
Assume in negation that there is a mixed strategy Nash equilibrium It must be characterized by firm 1 mixing between prices
. h p 2 Wit 1
1 2 1 .l. · d 1 · 1y, an d f'1rm 2 m1xmg · · b etween pro b a b 1 1t1es q an q = - q respectiVe 1 1 1 1 2 1 2 . . h pro b a b'l' . pr1ces p l an d p w1t 1 1t1es q an d q = 1 - q respec t'1veJy. I n any 2 2 2 2 2 1
NE both firms cannot lose money so we cannot have that for some i both p. < c I
2
and p. < c. I
1
2
1
1
If for some i both p. > c and p. > c, then j=i can gain by .
. settlng p.1 = p.2 = M"m { p.I , p.2} - c. - J J I 1
some i both
p~I >
c and
p~I >
c.
Therefore, we cannot have that for
The only remaining case is that for both
1 2 players we have (w.l.o.g.) p. > c and p. < c. I
setting
p~ = p~ = p~ J .
J
- c.
I
But then firm j can gain by
This argument contradicts the existence of a mixed
1
strategy NE, therefore only a pure strategy NE can exist, and Proposition 12.C.l proves that p
12.C.3
(a)
1
= p2
= c is the unique pure strategy NE.
Let n e IN satisfy (n-l)fl s c < nl1.
We have to show that both
firms charging nf1 is a pure strategy Nash equilibrium (NE)of this game.
Note
first that both firms get a strictly positive profit by using this strategy. Neither firm wants to raise its price - if one of them did so, its sales would be zero and it would make zero profit.
Neither firm wants to lower its price
- if one of them did lower its price to at least (n-llA, then it would make
12 - 9
neptive or at most zero profit. strategy equilibrium.
Therefore, both firms charging M · is a pure
It also follows from the above argument that, given
that the other firm charges M, charging M is the unique best response. Hence it is not weakly dominated by any other strategy.
(b) This follows form (a) above and from the fact that and M
12.C.4
(a)
.. c as lJ.
First, there cannot be an equilibrium in which both p
strictly above c
2
1
and p
(by the same argument as for the symmetric case).
-+
2
0.
are
Second,
firm 2 does not charge less than c , since it would otherwise make a negative 2 profit.
Third, firm 1 can guarantee itself a profit as close as possible to
£.
(c -c ) q(c ), by charging c 2 1 2 2
Since the price charged cannot exceed c , 2
this profit is also the highest that firm 1 can obtain. that at common price c
Unless one assumes ·
firm 1 gets all the demand there is no equilibrium -
2
firm 1 will want to choose
£
Such an
One can define the t-:quilibrium as the limit, so
p = c 1
2
£
does not exist.
as close as possible, but different from, 0.
and firm l's profit is Cc -c ) qCc l. 2 1 2
When the monopoly price of firm
1 is lower than c , then it will charge this price without worrying about firm 2
2's threat (so in this case there exist an equilibrium)
e IN satisfy (n-l)!J.
(b) Let n
:::5
c
2
< n!J..
Firm 1 charging (n-l)!J. and firm 2
charging n!J. constitute a pure strategy NE of this game, which does not involve the play of weakly dominated strategies.
As lJ. .. 0, (n-l)!J. ... c , so the 2
equilibrium approaches the NE of part (a) above. Note that, firm 1 charging (n-x)!J. and firm 2 charging (n-x+ll!J., with x such that c
l2.C.S
1
:5
(n-x-l)!J.
:::5
c , also constitutes NE. 2
The first q agents bidding v and agent i with i > q q+ 1
bidding v.
I+ 1
(the agent with the lowest valuation can bid anything strictly
12 - 10
lower than his 7aluation) is a pure strategy Nash equilibrium not involving weakly dominated strategies.
Clearly, the first q players cannot do better.
Bidding higher will only reduce their payoff.
If they bid lower, they will
lose the object and therefore not make a positive payoff (if they bid v
q+ 2
they wlll have 507. chance of loosing).
Players with i > q cannot do better by
bidding lower, they will still not win the object and obtain a payofr of 0.· By bidding higher, they will only be able to have a chance to win the object if they bid at least v
. q+ 1
However, if they win the object they will have a
negative payoff, since v. - v I
changing their bid.
q+ 1
s 0 for all i > q.
Thus they do not gain by
Clearly, this is a competitive equilibrium since the
demand at this price is q, which equals the fixed supply. Suppose there exists a pure strategy NE, in which one of the buyers 1 through q, assume player j s q, did not receive a unit.
Then one of the other
buyers, assume player i > q, must have received a unit.
Since player i could
have bid 0 and therefore obtain a payoff of 0, he has to get a positive payoff in the NE, which implies that his bid was at most v.. 1
not get the object his payoff was 0.
Since player j did
However, he could have bid slightly
above v. and obtain a strictly positive payoff of v. - v. (since j > i), a 1 J I Contradiction, since player j did not use his best-response.
Thus in every
pure strategy NE, buyers 1 through q receive a unit.
12.C.6
Each firm j maximizes [a - b(q .+ qk) - cl· q ., and the FOC is J J
a - 2bq. - bqk - c = 0 (when q. > 0), which yields the best response function J J b}qk) = Max{O,(a - c - bqk)/(Zb)}.
To calculate the Nash equilibrium we have
two equations (both BR functions) with two variables since we set b J'qk) = qj to get the equilibrium.
12.C.7
Straightforward algebra yields the results.
Using equation 12.C.6 and plugging in the linear demand curve we
12 - 11
obtain
= [J/(J+l)] ·[(a-c)/b),
which yields Q•
and p
= (a+Jc)/(J+1).
get the monopoly quantity (a-c)/2b and price (a+c)/2. and the quantity increases as J increases. goes
to
= 1,
For J
we
Clearly the price falls
In addition, as J
~ m,
the price
the competitive price c and the quantity goes to the competitively
supplied quantity (a-c)/b.
12.C.8 (a)
Firm j maximizes max p(q + Q ) q - c(q ), q 2:0 J -J J J J
The set of q 's which solve this maximization problem clearly depends J only on Q , and we can denote this set by b(Q ). -J -j (b)
Consider the following example: there are 10 consumers of which 5 value
the good at $10 and the other 5 value the good at $5, where each
con~umer
only wishes to buy one unit of the good. There are two firms with zero marginal (and fixed) costs.
If q = 0 then both q = 5 and q = 10 are best 2
I
1
response quantities for firm 1 which yield the (monopoly) profi!s r:f $50.
(c)
We will use a revealed preference argument.
When the total output of
other firms is Q, firm j prefers to produce q rather than producing Q + q - Q. (This quantity has been chosen to keep total output the same as when j responds optimally with q to other firms' producing Q.) Formally: ,..
"""'""
,..
""'
""'
,..
""'
p(Q + q)q - «:;(q) =:: p(Q +(q + Q - Q))·(q + Q - Q)- c(q + Q - Q)
= p(Q + q). (q
=
Q - Q) - c(Q + q - Q).
+
Similarly, when the total output of other firms is Q, firm j prefers to A
A
produce q rather than producing Q + q - Q. Formally: ... p(Q + q)q - c(q)
:!:.
p(Q +(Q + q - Q))·(Q + q- Q) - c(Q + q - Q)
12 - 12
A
A
-"'
A
A
A
= p(Q + q). (Q + q - Q) - c(Q + q - Q). Adding up the two inequallties and rearranging terms, we obtain
...
...
,. (Q - Q){p(Q + q) - p(Q + q)]
!: A
A
A
A
{c(q) - c(q + Q - Q)} - {c(q + Q - Q) - c(q)]. ,. .... Suppose in negation that Q + q < Q + q. Since p( • ) is a decreasing function .... and Q > Q, this implies that the left-hand side of (•) is negative. To analyze
the right-hand side, observe that our negation assUmption implies that
...
...
q + Q - Q > q.
Using this and the convexity of c( • ), it is easy to see
that the right-hand side of (•) Is positive. This contradicts the inequality ,..
....
in (•). Therefore, we must have Q + q
2:
q.
Q +
Take a point (Q,q), with q e b(Q). Take an increasing sequence Q corresponding sequence q
k
..., Q, and a
e b(Q ). By the result above, we have
k
k
•
Q
k
+ q
k
s Q + q. The ref ore, q
k
s Q + q - Q •
liminf q .
k
k
But then
s Q + q - lim Q • q. k
Mathematically, this implies that the infimum of b(Q) is upper semicontinuous in Q. In words, b( · ) can jump only upward. When b( · ) is a function, the above conclusion implies that b(Q) + Q is increasing in Q. lf b( ·) is in addition differentiable at Q, we must have b' (Q) + 1
(d)
!:
0, i.e. b' (Q)
!:
-1.
Since price is decreasing in total quantity and by convexity of c( ·)
-
average costs are increasing for each firm, there must exist a quantity q such that it is never a best response to produces
q,
i.e., b(Q) e [O,ql for all Q.
Using this fact, and the upper semicontinuity of the infimum of b(Q) established above, we can prove existence of a fixed point which will be a symmetric Cournot equilibrium i.e., that q• e b((J-l)q•).
This is formally
established in the Lemma presented in: Roberts, J. and H. Sonnenschein (1976) "On the Existence of Cournot
12 - 13
Equilibrium without Concave Profit Func:ions," JET 13:112-117.
(e)
Consider the following example: there are 10 consumers which all value
the good at $10 and each wishes to buy only one unit of the good. two firms with zero marginal (and fixed) costs. (q ,q 1
2
)
= (x,10
- x) where 0 s x
There are
Any pair of quantities
10 will be a Cournot equilibrium .
:5
•
(f) We will show that if the demand curve p(q) is strictly concave, the
symmetric Nash equilibrium is the unique Nash equilibrium In pure strategies. (Strict concavity p(q) is equivalent to strict concavity of the demand function x( ·) = p -l( · ). ) The proof utilizes the following claims, which follow from the strict concavity of p(q): .
Claim 1: The best-response correspondence b(Q ) is single-valued, i.e. -J b( · ) is a function. Proof: Firm j's profit maximization program formulated in part (a) is strictly convex, therefore, its solution is unique.
Claim 2: b(Q) is non-increasing, i.e. best-response curves are downward-sloping. Proof: Suppose that Q
:5 Q,
and let q = b(Q), ~
= b(Q).
Using a revealed
preference argument, we obtain 1\
1\
1\
p(q+Q)q - c(q) ::. p(q+Q)q - c(q), 1\
A
1\
1\
p(q+Q)q - c(q) ::. p(q+Q)q - c(q).
Adding up the inequalities and rearranging. terms, we have 1\
1\A/\
A
[ p(q+Q) - p(q+Q)]q :5 [ p(q+Q) - p(q+Q)}q
Suppose in negation that q < ~- As p( ·) is strictly concave and
... downward-sloping. and Q
:5
1\
Q, we must then have 1\
A
A
p(q+Q) - p(q+Q) > p(q+Q) - p(q+Q) ::. 0.
~ultiplying by ~, we obtain
12 - 14
A
A
~
1\
(p(q+Q) - p(q+Q)]q
>
....
A
[p(q+Q) - p(q+Q)]q
which contradic_ts (•). Therefore, we must have q
A
!: !:
[p(q+Q) ·- p(q+Q)]q,
~.
,.
Claim 3: Q < Q • b(Q) + Q < b(Q) + Q, i.e. the slope of the best-response function is strictLy greater than -1. Proof: We can use the same revealed preference argument as in part (b). The only modification is that due to uniqueness of a firm's best-response (Claim 1), the revealed preference inequalities can be written as strict. Continuing as in part (a), we obtain the claim.
Claim 4: The symmetric pure strategy Nash equilibrium is u·nique. Proof: This follows from the fact that best-response curves are downward sloping (Claim 2). Suppose in negation that there are two symmetric Nash equilibria, q and q', and that q > q'. Since q is a Nash equilibrium, we
= b((J-Z)q).
must have q
Using the fact that b{ ·) and q > q', we obtain q'
b((J-1)q) :5 b((J-2)q' ),
which !Deans that q' cannot be a Nash equilibrium - contradiction. Thus, the symmetric pure strategy Nash equilibrium is unique. (The existence of such an equilibrium has been shown in part (d)).
Using Clai"ms 1-4, we can finally prove that no asymmetric pure strategy equilibrium exists.
Suppose in negation that this is not true, i.e. that
•
•
•
1
J
1
there exists a Nash equilibrium (q •... , q ) in which q
•
certain pair of firms l, k. Let Q =
L q •J .
:oe q
• for a k
Consider the two-firm Cournot
J ,e 1 ,k
game between firms j and k where the output of all other firms is fixed at Q• .
•
•
•
•
I
J
I
k
If (q , .•. , q ) was a Nash equilibrium in the original game, then (q , q ) must constitute a Nash equilibrium in the two-firm game. The demand curve in this game is p(Q
•
+
q), which is strictly concave in q as long as p( ·) is
strictly concave. Let b( ·) be each firm's best-response function in this
12 - 15
two-firm game.
Let q
0
be the symmetric Nash equilibrium of this game (which
is unique by Claim 4), i.e. q that q
•
s q
I
0
(If we had q
•
0
= b(q
0 ).
Suppose without loss of generality
0
• •
> q , then by Claim 2, q = b(q ) s b(q ) = q
I
k
we would switch firms 1 and k). If we had q• = q
0
I
q
•
•
0
0
,
0
I
'
and
then we would also have
0
= b(q ) = b(q ) = q , which would contradict our negation assumption that
k
l
q • • q •. I
Therefore, we must have
k
q
•
I
0
.
(1)
and therefore
•
qk
•
= b(q I)
!:
0
(2).
q .
Using Claim 3, (1) implies that •
b(q I)
0
0
0
> q + b(q ) = 2q.
But using the result of part (b), (2) implies that q
•
I
+q
•
k
=q
•
k
•
+ b( q ) sq
0
0
0
+ b( q ) .. 2q •
k
which contradicts the previous inequality.
Therefore, in the Nash equilibrium
•
.... q ) every two firms should produce the same outputs. J
The last part of the proof is easily understood geometrically. Consider the space (q, q ). I
Claim 3 implies that to the left of the point (q
k
0 ,
q
0 ),
firm l's best response curve lies above the (-45°) line passing through 0
0
(q , q ), while firm k's best-response curve lies below that line. Therefore,
they can never intersect to the left of (q
12.C.9
(a)
0 ,
q
0 ).
In a Nash equilibrium (q •' q •J), each firm j maximizes its J
-
profits:
• max n (q_, q ) = (a - b(q + q • )) q 2:0 J
j
J
-J
J
-j
The objective function of this problem is concave in q .
I
j
Therefore, q = 0 J
•
solves this problem if and only if an (q , q • ljaq = a - b q . - c s 0. -J J J J -J J q .=0 J
If this inequality holds and firm j does not produce in this equilibrium, the
12 - 16
rival firm (- j) maximizes as a monopolist against the linear demand firm, and its optimal output is given by q
•
= (a - c )/(2b). Substituting in the above
-J
-J
inequality, we see that firm j may not produce in a Nash equilibrium if and only lf c ~ (a+c )/2. Since we always have a > c , the necessary condition J ~ -J for firm j not producing is c > c , i.e. only firm 1 may not produce in a j -J Nash equilibrium. Thus, if c
~ (a + c
1
2
)jz,
the output pair (0, (a - c )/(Zb) ) constitutes a Nash equilibrium . 2
Now, let us solve for an equilibrium where both firms produce. First-order conditions for the two firms' maximization problems can be written as
a - bq
•
a - bq
•
z 1
- 2b q • - c = 0, 1
- 2b q
1
•
- c
2
2
= 0.
The solution to this system of equations is given by
• -ql • -q2
(a + c (a + c
- 2c lj(3b),
2
1
(••)
- 2c lj(Jb).
I
2
• Whenever inequality (•) is satisfied, the first expression
• lD (
..
•
) produces ql
< 0, which means that there is no equilibrium where both firms produce positive amounts. The only Nash equilibrium in this case has firm 1 produce zero. On the other hand, whenever (•) is violated, firm 1 has to produce a positive amount in equilibrium, and the equilibrium outputs are given by (.. ). (b) As (••) shows, each firm's output increases as its own cost decreases, and as its rival's cost increases. Equilibrium profits are given by
•
•
2
1
•
•
2
2
+ q ) q ll
• =
2
p(q
• I
+ q ) q
- c q
•
1 1
= (a - 2c + c l/(9b), I
2
- c q • =- (a - 2c + c lj(9b). 2
2
2
1
Each firm's profit increases as its own cost decreases, and as its rival's cost increases.
(c) In Nash equilibrium, each firm solves
12 - 17
p(g + Q
max
q !:0
J
• -J
) q
J
- cJqJ,
J
I: qk.•
•
where Q_J =
Without loss of generality, assume that all the J firms we
k"'J consider produce positive quantities in equilibrium. (Otherwise, we would focus on the set of producing firms.) Differentiating the objective function with respect to q , we obtain the following first-order condition: J . p - cJ
= - p'(Ql q J
= (-
p'(QlQ/pl p · (q
jO> = U/d
pal
where Q : q J + Q•_J is the total industry output, p = p(Q) is the market price, aJ = q
jO
is the market share of firm j, and c is the elasticity of market
demand curve. Industry profits can be written as the sum of individual firms' profits: J
J
TT=I:n J =1 j
J
=I: (p
- c ) q = J
J=l
J
r (p J=1
J
a q .l/r: J
= I: pa2 J
J=l
J
(Q/cl
= H· (pQjr:l,
where H is the Herfindahl index. Therefore, TTj(pQ) = Hjr:.
12.C.l0
(a) Each firm i chooses its output q
-
p(Q . + q.)q. - a.c(q.). -I
1
1
1
1
I
!:
0 to maximize its profits
Assuming a positive solution, the first-order
condition for this maximization problem is
-
p(Q) = a.c' (q.) - p' (Q) q. 1
1
-
(FOC),
1
where Q is the total output. Since c( · ) is increasing and convex, the right-hand side of (FOC) is increasing in a. and decreasing in q. . 1
1
Since
(FOC) holds for every firm, we must have q. > q. whenever a. > a .. 1 J J 1 Subtracting (FOC) for firm i from (FOC) for firm j gives:
-
-
o:.c'(q.)- o:.c'(q.) = J J 1 1
-p'(Q) (q.- q.l. 1 J
lf a . > a. then q. > q ., and the last equality implies that J 1 1 J
J
-
a .c' (q .) > a. c' (q.). Therefore, marginal costs across firms are. not equalized, which
J
I
1
implies that the aggregate output is not produced efficiently.
(b) The welfare loss in this case is equal to the loss of consumer surplus due
12 - 18
to non-competitive pricing plus the higher production cost due to productive inefficiency.
The latter loss has not been considered in section IO.E. -
(c) The Cournot equilibrium output and price level obtained in Exercise 12.C.9
are q = (a-2c +c l/Jb, I
I
2
~ = fa-2c 2+c1 l/3b, p
• (a+c +c l/3. 1
2
Total profits of the two firms can now be computed as (p-c )~
2
(p -c )q + 1
1
= (2a 2 + Sc2 + Sc2 1 2
-
2a(c +c ) - 8c c l I 9b. 1 2 1 2
I
Consumer surplus can be computed as q, +q2
J
0
2 q1 +q2 p(q)dq = (aq - bq /2) = (a-c -c )(Sa+c +c l/18b. 0
1 _2
1
2
Adding up total profits and consumer surplus, and differentiating with respect to c , we obtain 1
asjac 1
= (9c - 9c I
This derivative is positive when c > c 1
2
2
+ (4/9la, i.e. when firm l's costs
are substantially higher when firm 2's costs. reduces social welfare.
- 4a)j9b.
In this case a decrease in c
I
The reason is that when c slightly falls, firm 1 l
steals more business from firm 2, which raises production inefficiency.
When
c is substantially larger than c , this effect actually dominates the I
2
increase in consumer surplus due to a lower price .
12.C.Il
p
1
Suppose firm 2 charges p • = p(q + q ). 2 1 2
(a}
s p(q
1
If firm 1 charges
+ q ), it will sell at its full capacity, and its profit will be 2
equal to Cp - c) q . Therefore, firm 1 will not find it optimal to charge a 1
1
price lower than p(q
1
+ q ). 2
Suppose instead that firm 1 charges a price above pCq
1
+ q ).
2
Given the
described mode of rationing, firm l's residual demand will be x(p ) - q . l
12 - 19
1
Therefore, firm I acts as a monopolist with the residual demand given by the inverse demand function p(q
1
+ q
2
l, and its profit maximization problem can be
written as Max [p(x + q ) - c) x • 2 1 1 XI ~Ql Given that firm 2 sells q , the optimal output for firm I fs b(q ). 2 2
However,
since by assumption b(q ) exceeds firm l's. capacity QI, producing b(q l is not 2 2 possible.
Given the assumptions of the exercise, firm l's objective function
is concave in xi' therefore it is optimal for firm 1 to produce the closest feasible amount to b(q ), which is ql' 2
This is achieved by firm I's charging
p(qi + q2). Therefore, it is optimal for firm 1 to charge p(q 2 does so. p(q
1
1
+ q ), given that firm
2
By a symmetric argument, firm 2 also finds it optimal to charge
+ q ), given that firm 1 does so.
2
Therefore, p
1
= p
2
= p(qi + q l 2
constitutes a Nash equilibrium of this game.
(b)
[First printing errata: you need to assume that p(q + q ) > c.] Observe 1 2
that given that p(q + q ) > c, each firm can make a positive profit by 1 2 regardless of its rival's strategy by offering a very low price.
Therefore,
in a NE both firms must make positive sales. We will look for a NE Cp , p L 1 2
Suppose first that the firms charge
different prices, i.e. that p. < p .. Since firm j makes positive sales, firm i J
I
must sell at full capacity.
Then firm i can slightly increase p. while still 1
selling at full capacity, which would raise. firm i's profit. Therefore, the firms cannot charge different prices in a pure-strategy NE, and we can restrict attention to equilibria where p
= p . 1 2
There are
three cases to consider:
(i)
If pJ = p
2
> p(q
1
+ q ), then at least one firm sells below capacity,
2
12 - 20
and this firm would be better off by slightly lowering its price and either selling at full
c~pacity
(ii) If instead p
1
or stealing all customers from the rival.
_=
Pz < p(q
+ q ), then both firms sell at full 2
1
capacity. Then each firm could gain by increasing its price to p(q
+ q ), 1 2
which would still enable it to sell at full capacity.
(iii) Finally, suppose that p
1
= Pz = p(q
firm i's raising its price above p!q
1
1
+ q ) and qi > b(qj). Consider 2
+ q ). As shown in part (a), 2
firm i's
best response would be to sell qi > b(qj}' which is now feasible (in contrast to (a)).
For this purpose, firm i should set a price higher than p(q
1
+ q ), 2
therefore, this case does not describe an equilibrium.
As all possible cases have been exhausted, there is no pure strategy NE.
J2.C.l2
(q. , q . ) < 0, i = I, 2. I J
(a) Assume that
The first-order condition fo:· firm i's profit maximization problem is
IT~
(b. (q. ),q.) = 0. Differentiating with respect to q . gives: ab. (q .>faq. = I J J 1 J J J i i Therefore, the sign of 8b. (q. ljaq. coincides IT12 (bi(qj)'qj) I ITll (b.1 (q J.l,q.). J 1 J J •
with the sign of IT~
2
(b. (q .),q .). I
J
J
In words, firm i's best-response function •
is increasing (decreasing) provided that
I
rr 12
is positive (negative).
(b) In the Cournot model i IT (q.,q.) I J
Then
=
p(q.+q.)q. 1 J 1
- c(q.). 1
(q.,q.) = p"(q.+q.)q. + p'(q.+q.), which is negative if p(·) is lJ
lJI
downward sloping and not too convex.
IJ
Therefore, the "normal" slope of best
response functions in the Cournot model is negative.
I2.C.l3
For definiteness, suppose that it is firm that is playing its best
12 - 21
response to p , and suppose in negation that some consumers strictly prefer 2
not to purchase. textbook.
This situation is depicted in Figure 12. C.6(a) in the
Consider a small reduction in firm 1's price p. 1
As long as firm 1
does not start to compete with firm 2, its sales are given by the consumer x who is Indifferent between buying from firm 1 and not buying at all: p + tx I
= v,
which gives x(p ) = (v-p lft. 1
1
Firm
1's
profit
can
then be written as
n (p ) = (p - c)x(p ) = (p - c)(v - p >ft. 1
Differentiating
with
1
I
respect
1
to
p
, · 1
1
we
1
obtain
n' (p ) = (v+c-2p lft. I
1
I
By assumption, some consumers strictly prefer not to buy rather than buying from firm 1, so at least this should be true for the consumer living at point z = 1, which means that p > v - t.
Substituting this inequality in the
I
expression for n' (p ), we obtain n' (p ) < (c+2t-v)jt, which is negative by I
assumption.
I
1
I
Therefore, by raising p slightly firm 1 could raise its profit, 1
.
which contradicts the assumption that firm 1 is playing its best response.
12.C.14.
There can be three types of equilibria (p
•• p •J) 1
in the model:
Type 1: The consumer who is indifferent between buying from the two firms strictly prefers buying to not buying. The indifferent consumer is located at distance x from firm i, where p
x =
(t -
•
• p I
v - pi - xt
+
• p l/2t.
=v
J
-
•
+ xt
I
= p •J
+ (1-x)t, and therefore
The utility of this consumer from buying from firm i is
tf2 - (p •1 + p •J l/2. This consumer strictly prefers buying
when
•
•
I
J
v > (p. + P.l/2 + t/2
(1)
In a type 1 equilibrium, the firms compete for a positive measure of consumers and each firm's demand is affected by the other firm's price. Type 2: The consumer who is indifferent between buying from the two firms
12 - 22
strictly prefers not buying to buying. Given the derivation above, this happens If and only if
v <
• p l/2
+
+
j
t/2
(2)
In a type 2 equilibrium, the sets of consumers buying from each firm are separate, and each firm's demand is unaffected by the other firm;s price. Type 3: The consumer who is indifferent between buying from the two firms is .
also indifferent between buying and not buying.Given the derivation above, this happens if and only if
v = (p• + p • )/2 + tf2
(3)
J
I
In a type 3 equilibrium, if a firm lowers its price slightly, it starts stealing customers from the rival firm. On the other hand, if a firm raises its price slightly, the lost customers do not buy from the rival firm - they do not buy at all.
(a)-(b) In a type p
• I
1
equilibrium, if firm i changes
it~
price slightly from
to p , its demand will be determined by the consumer x who is indifferent I
between buying from the two firms: p
I
+ xt
=p• J
+ (1-x)t.
Therefore, x_(p_, p • ) = (t+p •-p )M/2t. Firm i chooses p to maximize profits: I
J
J
I
I
I
• Max (p - c)x (p , p.) P.I
I
I
I
•
=
(p_ - c)(t + pJ - p_)Mj2t. I
J
I
The first-order condition for this program is
. (t + p
• J
- 2p + c)M/Zt I
= 0.
The first-order condition for firm j is symmetric. Combining the two
•
•
first-order conditions, we obtain pi = p J = c + t. Using (1), we see that this will indeed constitute a type 1 equilibrium if and only if v > c + 3/Z t. (c) In a type Z equilibrium, if firm i changes its price slightly from p
12 - 23
• to
p
, 1
its demand will be determ_ined by the consumer at the distance x < 1/2 from
the firm who is indifferent between buying from firm i and not buying at all: p + xt I
= v.
•
Therefore, x (p , p ) = (v-p )Mjt. Firm i chooses p I
I
l
j
Max Cp 1 pl
clx (p
-
1
•
, p ) 1
(p
1
to maximize profits:
- c)(v - p
I
lM/t.
1
The first-order condition for this program is· (V - 2p
+ c)M/t
1
= 0.
From here firm Cs equilibrium price p • can be determined: I
•
= (v
+
c)/2.
p J = (v +
c)/2.
p
1
From a symmetric derivation for firm j,
•
Using (2), we see that this will indeed constitute a type 2 equilibrium if and only if v < c + t.
(d) In a type 3 equilibrium, if firm i raises its price slightly from p
•
= v - t/2 to p , its demand will be determined by the consumer who is I
indifferent between buying from firm i and not buying at all: p
I
+
xt
= v. •
•
p ) = (v-p )M/t for p ~ p . Since firm i J 1 I I
Therefore,
•
chooses P. to maximize profits, a slight increase in P. from P • should not I
I
1
increase profits:
p•I ) 1 p =p • 1
= Cv
1
= ajap i
- 2p •
1
(p
I
+ c)Mjt ~ 0,
•
1. e .•
v :=s 2p • - c
( i.l)
1
A symmetric derivation for firm j yields v :=s 2p
• J
(j.l)
- c
Also in a type 3 equilibrium, if firm i lowers its price slightly from p
12 - 24
• 1
to
p , Its demand will be determir.ed by the consumer who is indifferent between l
buying from firm t and buying from firm j: p + xt = p
.= - .
.
l
Therefore, x (p 1
, 1
x (p , pJ) -=
pJ)
J
+ (1-x)t.
(t+pJ-p )M/2t for
1
1
•
1
:s p •. 1
Since firm i chooses p to maximize profits, a slight decrease in p 1
1
from p • 1
should not increase profits:
= BI''ap 1
= (t
(p -c)(t+p l
+ p •J - 2p•I + c) M /J?t -
!:
•-p )M/2t IP1=p·I J I
0,
i.e .•
•
t
•
+ pj - 2p1 + c ~ 0
(i. 2)
A symmetric derivation for firm j yields
t + p • - 2p • + c J
I
~
(j.2)
0
Adding (i.l) and (j.l) and using (3), one obtains v (j.2) and using (3), one obtains v
::$
2:
c + t. Adding (i.Z) and
c + 3tj2. Therefore, type 3 equilibria
can only exist when
c + t :s v
::$
c +
3tj2.
On the other hand, when (•) holds, p
• I
=p •j =
v - t/2 satisfies (3). (i.l),
(j.l), (i.2), (j.2), and is therefore a symmetric type 3 equilibrium. When the
inequalities in (•) hold strictly, there also exists a continuum of asymmetric equilibria:
•
-
v
-
t/2 + c,
• = pj
v
-
t/2 - c,
PI
where c
•
IS
small enough
•
1n
absolute value.
(e) In a type l equilibrium (cases (a)-(b)), a reduction in t makes competition more intense, prices and profits fall. At the same time, in a symmetric type 3 equilibrium, a small reduction in t increases prices and
12 - 25
profits.
Howev.er, when
t falls substantial.'.y, a type 3 equilibrium is no
longer possible, and the game switches to a type 1 equilibrium.
12.C.l5.
The consumer who Is indifferent between buying from firm l and firm
j is located at distance
x from firm l, where P + tx
2
= PJ
1
2
+ t(l-x) •
From here we obtain the demand for firm i's product: X (p ,p ) I 1 J
=
[lj2 + (p - p )/2tJ M. 1 J
(Observe that the demand is the same as for linear transportation costs. The equilibrium is therefore also going to be the same.)
• • We are looking for an equilibrium (p ,p ). In equilibrium, firm I J solves (p
1
- c)
x 1(p I.p •J )
= (p I
- c) (1'2 + (p /'
• j
- p >/2t] M I
The first-order condition for this maximization program is (t + p
• J
- 2p
• 1
+ c] Mj2t
= 0.
A symmetric derivation for firm j yields
•
•
It + pi - 2p J + c) Mj2t = 0.
= p •J
•
Solving these two equations, one obtains pi
12.C.16.
= c + t.
If the circumference of the circle is 1, then the firms are located
1/J apart from each other. We will be looking for a symmetric equilibrium
where all firms charge p • . Suppose that firm i. deviates and charges p . Then I
the consumer who is indifferent between buying from firm i. and firm i+l will be located at the distance x from firm i., where x solves p
1
+ tx
2
=p
•
+
t{IjJ -
2
x] .
Observe that since consumers on both sides of firm i buy its product, the
• demand for the product is going to be x (p .p ) = 2xM. Using the equation 1
I
12 - 26
above, we find
~oh·e.'!>
In eq1Jillbrium. rirm i
Max (p - c) I
PI
• .lt)P,·P l •
fl/J •
"' Cp, - d
•
• (p
- p JJjt/M. I
The fint-order- condition for this maximization p!'ogram is
11/J
+
• (p
- 'Zp
• p ,
value p • p
As
t
J
• p
0 or J · } 0. we have
fall or the
numb~r
! •
1
+- c)Jjt}M "' 0
• we can solve for- p : 2
= c • tfJ '
---.. c. Intuitively. <ar.
the
transpr:>rtation costs
or firm<.> grows without bound. the industry be"Comes
extremely competitive and price go-es down to marginal cost.
12.C.t7.
For now, assume that prices arc sut:h that both rirms sell positive_. . I •
•
•
amounts, i.e. tr.at ther-e exists a consumer who ls indifferent between bOJying
from the two firms. This
eo~sumer i~
located
where
Bt
the distance x from fiz·m 1,
•
p
=
tx
+
p
+-
tll-r).
' 2 from here we obtaln the demand:. fo: the two firms' products:
x 1 lp I .pZ ) -= [i t•· ~ •
We are looking fer ~ax
P,
If
0\ll
t
(p
I
• • equilibrium (p ,p/ In 1
P)/2ll M. ._
-
•
I
= !p - c: } x (p .p ) "' (p - c ) IL/2 1
I
I
I
I
'I!
i
f-
{fJ
"the fint-order condition for this maximization program is
It • A
srmm~tr""lc
p
• 2
-
• 2&> I
+
c I J Mf2t ..,
-detivaUMJ ror· firm j yidds
12 -
27
~()ives
(!quilibrluru, firm!
o.
• 2
- p
1f2tl M
1
Solvin& the&e two equations, one obtains
• p : (2c • c l/J • t, I I Z
• p2
c,l/3 •
.. t2cz •
t.
Mter- substituting these expression• into the expres'Slons for profits, we obtain
• I( l
t
:!.
•z
I
•
t2
cz
1
C'
1
Jf c
• It ... 2 z
-
2
•
- C'2
I
•
Jr
An inc:<ease in each film's costs increases lts equilibrium price and reduces its equilibrium profit. No\\' we have to check when firm 2 Cthe higher-cost flrml makes no said. The equilibrium demand
ror rirm 2's product is
• • • x fp ,p l • 11/Z • lp 2 1 2 1
• p >j2t/
•
2
AI = ll/2 + (cl - c )/6t} Af. 2
- c • < 6t, this demand Is positive, both firms make positive sales, 2 • and our equilibrium analysi$ Is correct. When c - c: ~ 6t, in equ\Ubrium
When c
2
I
firm 2 makes no sales. Since fkm 2 can ofref" any price greater than c
2
•
equilibrium fir·m J prices so that p • t • c , I.e. the consumer Jiving next I
t<'l
2
firm 2 ls indirferent between buying from the two firms.
(a) Firm 1 chooses q
12.C.I8.
produce b (q l, l
I
wh~re
oZ(q I )
I
knowing that given its choice, firm 2 will
2's
i'!> firm
best•respon~e
function. Firm ts
pr¢£rllm, then·forl!!, Is
The
(irs~-order
condi1fon (u:· this program Ciln be written aS 1
n h~ ,o Cq H = -
n'2lq1.b lq H 21
1121
(sin<'e
1
nZ
quantities
(q ,b (q )) < 0 a"d b' (q I
2
I
simultanc:o\~sly,
2
I
i < 0).
r Zt
b'tq
<
I( the firms instead choo<>e
the rirst-order· condition be:omcs l
rr 1
cq ,b (q n • I
2
12. -
o
L
o.
in
Since
1
n11
(q ,b (q )) < 0, this implies that the Stackelberg leader picks a 1
2
1
larger quantity in equilibrium than in the Cournot game.
Since the best
response function of firm 2 is downward-sloping with a slope larger than (see Exercise 12.C.8(c)), this implies that the follower picks a smaller quantity and aggregate output increases (and therefore price decreases) . •
Since the leader could have chosen the Cournot quantity, we know that his profits as a Stackelberg leader are higher.
The follower produces less and
obtains a lower price than in the Cournot outcome, which implies that his profits are lower.
(b) See Figure 12.C.l8.
N denotes the Nash equilibrium outcome, while S
denotes the equilibrium of the Stackelberg game.
s
Figure 12.C.l8
12 - 29
l2.C.l9. See Exercise 8.8.5
•
• • ),
In a Nash equilibrium (q .q
12.C.20.
1
2
firm J solves
•
Max p(q + q )q - c(q ). q c:O J k J J J
Assuming positive production levels, the first-order conditions for the two firms can be written as + p(q• + q • ) =
p' (q •
1
2
1
c' (q•1 ) •
p' (q• + q • ) q • + p(q• + q • ) = c' (q• ) •. 1
2
2
2
1
2
Adding these two equalities yields
• •
• •
• •
•
1
1
1
1
p' (q +q )(q +q l/2 + p(q +q ) = (c' (q ) + 2
2
2
c' (q • H/2. 2
Observe that since c( ·) is convex, c' ( ·) is increasing, and therefore (c' (q • ) + c' (q• ))/2 ::s (c' (q•+q • ) + c' (q•+q• ))/2 • c' (q•+q •). 1
2
1
2
1
2
1
2
Combining this with the previous equality, we obtain
• •
• • l/2 1 2
p' (q +q )(q +q 1
2
• •
+ p(q +q ) 1
2
::s c' (q •+q • ). 1
2
Since p' ( ·) < 0, (•) implies that
> c' (q •+q • ), 1
2
i.e. the Cournot equilibrium price is higher than the competitive price at the same aggregate output level. Next we show that q •
•
1
2
p(q +q
}
• + q • > qm, i.e., that the equilibrium duopoly price 1 2 m
is strictly less than the monopoly price p(q }. The argument is in
two parts. First, we argue that q
•
1
+ q
increasing its quantity to
•
2
a J
il::
m
q . To see this, suppose that q
= qm - q •• firm k
m
> q
•
1
• 2
+ q • By
J would (weakly) increase the
joint profit of the two firms (the firms' joint profit then equals the monopoly profit level, its largest possible level).
In addition, because
aggregate quantity increases, price must fall, and so firm k is strictly worse off.
This implies that firm J is strictly better off. and so firm j would
12 - 30
have a profitable deviation if qm > q: + q •.
We conclude that we must have
2
q
• 1
+q
• 2
m itq.
Second, condition (•) implies that we cannot have q • + q • = qm because then 1
. p' (q m)q m/2 + p(qm)
:S
2
c' (qm),
which is incompatible with the monopoly first-order condition U2.B.4) in the textbook. Thus, we must have q
12.D.l.
•
1
+ q
•
2
> q m ..
(a) Monopoly profit in period t is max 1t x(p) (p-c) p
= 1t
max x(p) (p-c) p
= 1t
rrn.
•
If a firm deviates at t = -r, it can obtain 1'trfl in that period, and it will get zero forever after.
If it does not deviate, its payoff is
c:o
1T
L
(7l5) t
Ifl/2
= 1/U-715)"
1-r Tim/2.
t=O
Therefore, deviation is not profitable if and only ifl/(1-.,.a) 1't Tf112 it 1't
Tfl,
(b) If a firm deviates, it can obtain nm in that period, and it will get zero forever after.
If it does not deviate, its payoff is
c:o
r (ral t
nm12 = 1/0-rol nm12.
t=O
Therefore, deviation is not profitable if and only if
K-1
(c) If a firm deviates, it can obtain
L ot nm
=
(1-oK)/(1-a) rrm
in the next
t=O
K periods, and it will get zero forever after.
If it does not deviate, its
payoff is c:o
Lot TT m/2 = 110-ol Tim/2. t=O
Therefore, deviation is not profitable if and only if
12 - 31
12..D.2..
n• >
Let
0 be the firms' equilibrium joint profits, which in
equilibrium are split equally among firms. The best deviation for a firm is to undercut the rivals by a small c, in which case it can steal all the demand and obtain almost as much as
n•
in that period.
In the following punishment
•
phase, the deviator will get zero forever after. If the firm does not deviate, CD
L
its payoff is
15t TT· ;J.
t=O
Therefore, deviation is not profitable if and only if or a
2:
lju-ant/J
2:
rrn.
CJ-l)jJ.
Since (J-1)/J increases in J, as J increases, a has to increase in order for collusion to be still sustainable. Therefore,as the number of firms increases, it is harder to sustain a collusive outcome.
12.0.3.
(3.)
Example 12.C.l in the textbook solves for the static Nash
equilibrium in the game. The equilibrium yields the Cournot outcome in which each firm makes a profit of (a-c)79b.
The maximum gain from deviation can be
obtained by playing the best response to qm /2 = (a-c)/4b, which is 3(a-c)/8b. 2 This maximum gain is 9/64 (a-cl /b. Monopoly profit can be calculated to be 2
(a-c) /4b. The payoff from deviating is . 9(a-cl 2 /64b + t
The payoff from not
r
ot
2 2 2 (a-cl /9b = 9(a-cl /64b + 15/(1-15) (a-c) /9b.
=1
deviating is Cl) t 2 L o (a-c) /Sb
=
.
2
1/0-15) (a-c) /Sb.
t=O
Therefore, deviation is not profitable if and only if 110-o) (a-c) 2 /Sb
!:
2 2 9(a-cl /64b + o/Cl-o) (a-cl /9b, or o
!:
9/17.
(b) The maximum gain from deviation can be obtained by playing the best
12 - 32
response to q. which is (a-c)/2b - q/2.
This maximum gain is b{(a-c)/2b -
q/2)2. The payoff from
deviating is
b[{a-c}/2b -q/2) 2 +
c:o
r
at (a-c) 219b
=
t=O
= b[(a-c)/2b -q/2) 2 + C5/(l-o)(a-cl 2/9b. The payoff
from not deviating is 2
1/(1-cl) (a-c) /4b.
Therefore, deviation is not profitable if and only if
I/O-a>
(a-c) 2/4b = b{(a-c)/2b -q/2) 2+
cl/(1-o}
(a-c) 2 /9b.
Thus,
2 2 2 2 C5(q) = [(a-c) /4b - b{(a-c}/2b - q/2) 1/[(a-c) /9b - b[(a-c)/2b - q/2] }, which
12.0.4.
o
E
a decreasing, differentiable function of q.
is
(a) The most profitable price that can be sustained for
llf2, ll is the rr.::mopoly price. As Exercise 12.8.4 shows,
the
monopoly price is increasing in the cost of production c.
(b) Let p (c) be the optimal monopoly price as a function of cost, and n (c) m
m
•
be the resulting monopoly profit. Let c and c
l!!:
c
•
• be the marginal cost in period one
be the marginal cost starting from period two.
•
When c = c , the
monopoly price and profits can be sustained in period 1 (see Proposition
•
12.0.1). When c is raised above c , the gain from deviating in period one stays the same, but the future payoff from complying falls.
Therefore, only
lower profits can be supported in period one. Formally, the highest profits in period 1 can be supported by the strategies which result in the best collusive equilibrium after compliance and revert to the Bertrand punishment after a deviation. The best collusive
12 - 33
•
equilibrium starting from period 2 brings the firms n (c) in every period. The m
highest supportable joint profits
•
n /2
!>
n
•
cS/0-o)
+
oj(l-o)
Observe that by assumption
o/(1-o) nm(c)
Case (i):
2:
• n
•
in period 1 therefore have to satisfy
n Cc)j2, i.e. n
•
m
2:
!>
o/0-o)
n (c). m
lj2. We need to consider two cases:
n (c • ) - a small cost increase. Then monopoly m
profits can still be sustained in period 1, and the most profitable price will still be p (c • ). The highest sustainable price may be higher than p Cc• ), but m
m
an increase in c reduces it. Case (ii):
o/0-o)
n (c) < n (c • ) - a large cost increase. Then monopoly m
m
profits in period 1 can no longer be sustained. (As an extreme case, imagine
c=CD, in which case no profits can be sustained in period 1.) Therefore, the monopoly price can no longer be sustained in period 1, nor any higher price. The most profitable price will now be the highest sustainable price, and it will be lower than p Cc •). m
12.0.5.
Let
nH
and
nL
be the profits the firms collect in high and low states
respectively: ltH
n
=
(pH - c) x(pH),
= (p L L
c) cxx(p ). L
In order .for the strategies described to constitute an equilibrium, a firm should not find it profitable to deviate in either high or low states.
If a
firm deviates in a high state, it can undercut the rival by a small c and collect n . The deviator will be punished by a reversion to Bertrand H
competition and will collect zero profits thereafter. If the firm does not deviate, it collects 'fr/2 in the current period, and its expected present value of future profits is
lj2 (o
+
2
o
+ ••• )(~nH(pH) + (1-~lnL. (pL)) =
12 - 34
= 1/2
o/0-o)
(AJrH(pH) + (1-A)JrL (pL)).
Therefore, the incentive constraint in the high state can be written as (IC ) H
Similarly, if a firm deviates in a low state, it can collect and will collect zero profits thereafter. collects
nJ2 in the current period,
profits is
o/0-o)
(An
H
nH
in this period
If the firm does not deviate, it
and its expected present value of future
+ (1-A)Jr ). Therefore, the incentive constraint in the L
low state can be written as
incentive constraints (IC ) and (IC ), we obtain H
L
1 ~
o/0-o)
(A + Cl-A)cx),
ex ::
o/0-o)
(A + (1-A)ex).
Since ex < 1, the first constraint clearly implies the second one. Expressing o from the first constraint, we obtain
o It is easy to see that
£e
~ ~
0/2.
= IjO+A
1). When
+ (1-A)ex).
o ::: o,
there is a SPNE in which both
m . f 1rms set pH = p L = p • (b) From the above derivation it is clear that when
o < o,
-
.
charging p
m
in all
periods in both states would violate (IC ). H
To find out which prices are sustainable, we rewrite · (JC ) as o(l-A) 1t H 1l
:5. H
Observe that
1t
L
s
nm
L
= (prn
-
1 -
L
o - AO
•
rn
c) cxx(p ). Therefore, in order for a price pH to
be sustainable in high demand periods, it should satisfy(p - c) x(p ) s o(l-A) H H -:-----::---'"'""::"""':::--- ( p m - c ) cxx( p m ) • 1 -
o-
Ao
Observe that a price higher than pm is not sustainable in the high periods,
12 - 35
because the deviator would h~ve the option of undercutting by charging pm. The maximum price sustainable in the high periods is therefore the smallest root of the following equation: Ml-~)
c) x(pH) = 1 -
(pH -
~a
a-
(p
m
m
- c) ax(p ) .
Suppose that this root is p • , and the corresponding profit in the high periods H
is 1rH•
• = (pH-
c)
• Since the right-hand side of (•) is .increasing in x(pH).
and we know that
a
<
a,
-
we can write
•
a(1-~) 'lrm
n: z: ~ - a - ~a • H
-
m L
-
=
n~/o: > n~
Since the profit pair (n , n ) is known to satisfy CICH) and easy to see that (IC ) is also satisfied. L
a,
.
1(
H
>
~
1(
L
,
.lt .lS
In words, it is possible to sustain
the highest possible profit n •H in .the high demand periods and at the. same time the monopoly profit n~ in the low demand periods. This is equilibrium outcome is optimal for the firms. Observe that in this equilibrium p
H
= p • < pm = p , H
L
i.e. the price in high demand periods is lower than the price in low demand periods.
Intuitively, collusion is harder to sustain in high demand periods,
because the gains from deviation are higher, and the future expected benefit of cooperation is the same as in the low demand periods.
(c)
Now a < lf2.
must ·have
~n
H
Suppose in negation that either n
+ (1-~ )n
L
H
> 0 or n > 0. Then we L
> 0. (lCHl yields
n <
~n
<
~Jr
H
H
+
(1-~)n .
+
(1-~)Jr
•
L
Similarly, (ICL) yields 1t
L
~
Multiplying the first inequality by
H
L
.
and the second one by
1-~
and adding up,
we obtain ~n
H
+
(1-~)n
<
L
~n
which is false. Therefore, we have n
H
H
+
(1-~)n ,
= n = L
12 - 36
L
o.
and consequently p
H
= p
L
= c.
12.£.1.
In a two-stage model the most efficient firm is not necessarily
active.
Take the following example: p(q)
cost K = 1/17.
=1 -
q, c = 1/2, c = 2/5, entry 1
2
= 3/4
Firm one entering and charging p
and firm 2 not entering
is a SPNE of the two stage game (As shown in Exercise 12.C.4, if both firms enter, the second stage equilibrium price is 1/2, and firm 2 will make a •
profit of 1/20 which is not high enough to compensate for the fixed cost.) However, firm 2 is more efficient than firm 1.
12.£.2. = p(Jq ) q - c(q ). J j J
1t j
drr. JjdJ = [ p(Jq J) - c' (q J )J dq /dJ
[p(Jq ) - c' (q )J j J
> 0
by
(A3l,
+ p' (Jq J) q J d(Jq j )/dJ < 0, since dq /dJ J
<. 0
CA2), p' ( ·) < 0, and
by
d(Jq J )/dJ > 0 by (AI}.
l2.E.3.
The socially optimal number of firms J max WCJ) = I J
Example 12.E.l:
0
solves
JqJ
p(sl ds
0
- J cCq ) - J K, J
qJ= Ca-c)/o 1/(J+l), p(Q) =a- bQ, c(Q) = cQ.
The first-order condition to the welfare maximization problem yields
The equilibrium number of entrants in Example 12.E.l J
As K
~
0, both J
0
and J
•
•
2
= {(a-c) /bK}
1/2
•
lS
- 1.
tend to infinity, but J
0
at a higher rate.
As for
the welfare loss from excess entry, it is W(J 0
where Q = J q 0
•
0 )
-
o, Q J •
W(J )
=I
Q0
o.
p(s)
ds
- c(Q
0
- Q ) - (J •
•
0 -
J ) K,
•
= J q •. J
Observe that this expression in bounded by K(J• - J
0
),
since the social loss
of excess entry arises from replicating the fixed cost K.
12 - 37
It is easy to see
that I i m
K(J
•
- J
0 )
= 0, which i.nplies that the welfare loss goes to zero as
K~O
K goes to zero. Example IZ.E.2: Since two firms in this Example already bring price down to marginal cost, the socially optimal number of firms J · If J
0
0
is either one or two.
= 1, free entry results jn the socially optimal number of firms.
= 2, free entry yields one firm less than socially optimal.
If J 0
The welfare loss
•
in this case is W(J 0 )
-
W(J•) =
! Q0
p(s) ds
- c(Q 0
Q•) - (J
-
0
-
J•) K,
o.
where Q0
= J 0 q J o = (a-c)/b.
we must have J
12.£.4.
0
Q
•
= J • q J • = q m = (a-c)/
2b.
However, as K --+ 0,
= 2, and therefore, there is no :welfare loss from free entry.
Since the firms form a cartel no matter how many firms enter the
market, industry output and price will be unchanged by the number of firms in the industry. increase.
Only the aggregate fixed costs increase as the number of firms
Therefore, the optimal number of firms in this industry is one .
•
If the planner cannot control entry, the equilibrium number of firms will be J • = n m/K. In terms of welfare this means that free entry leads to a complete dissipation of monopoly profits, without any benefit to consumers.
12.E.S.
We assume that entering firms do not choose their locations.
No
matter how many firms enter, they spread evenly around the circle. According to the solution of Exercise 12.C.l6,
if
J
firms
have
entered, the second-stage equilibrium price is p
•
2
= c + tfJ.
Since each firm in equilibrium serves I jJ customers, its profits are •
(p
- c)
3
I/J = ti/J .
•
The equilibr{um number of firms in the two-stage entry game is determined by
12 - 38
equalizing these second-stage profits to the
fi~:ed
J• = Cti/Kl
113
cost of entry K. We obtain
.
By assumption of ExerCise 12.C.l6, consumers' valuation for the product is high enough so that they always buy.
In this situation the social planner
does not care about the market price.
Aggregate production/consumption is I,
regardless of the price and of the number of firms producing.
The number of
firms affects only the aggregate fixed costs anc! the aggregate transportation costs.
The social planner's decision will be determined by the tradeoff
between the two. To compute the transportation costs, observe that if a consumer Jives at a distance x :::s lj(2J) to the right from firm j, his transportation costs are 2
tx .
The density of consumers on the circle is I.
Therefore, the aggregate
transportation costs of consumers with x e (O,lfC2J)] is
.tx2 I
1/2J
J
dx
= ti/(24J
3
).
0
The aggregate transportation costs are obtained by multiplying this result by
2J, which gives It/02J
2
).
The social planner's program can now be written as •
Min tij(l2J 2 ) + JK. J
The first-order condition is ti/(6J
3
)
= K, from where we obtain the sociaJJy
optimal number of firms
Observe
t~at
the socially optimal number of firms here is smaller than the
free-entry equilibrium number. The "business stealing" effect here dominates the tendency to insufficient entry resulting from non-appropriation of social surplus when products are differentiated. (See also Mankiw and Whinston, 1986.)
12 - 39
12.£.6.
Suppose there is a SPNE of the two-stage entry game where· J firms
enter, and each active firm produces qJ.
We want to show that this will also
be a Nash equilibrium outcome of the one-stage entry game.
To show this, we
need to consider all possible deviations in the one-stage entry game.
There
are three kinds of deviations: an active firm may change its output, an active firm may choose to stay inactive, and an inactive firm may choose to become active.
(i)
...
We will show that
all
these deviations are unprofitable:
If an active firm could profitably change its output, then this would not
be a Nash equilibrium outcome of the second stage of the two-stage game. Therefore, this would not be a SPNE outcome of the two-stage game, which contradicts our assumption.
(ii)
If an active firm preferred to stay inactive in the one-stage game, this
would mean that the firm's equilibrium profits are negative.
But then this
would not be a SPNE of the two-stage game.
(iii)
q.
Suppose that an inactive firm prefers to enter and produce some quantity Then entering and producing q would alsp be a profitable deviation in the
two-stage game.
Indeed, the J rival firms would respond to the entry by
choosing their outputs to be q textbook, q
J+l
J+l
.
By condition (A2) in Section 12.E of the
z:: q . In words, in the two-stage game the active firms would J
respond to another entrant by reducing their outputs.
But this means that
when the deviator produces q, the market price and consequently the deviator's profits are higher in the two stage game than in the one-stage game. Therefore, the
origin~l
outcome would not be a SPNE of the two-stage game.
(Exercise: show that Cournot competition satisfies condition (A2).) The reverse statement is not true, which can be seen from the following example: p(q) = 1 - q, K = 1/16 and c = 1/4.
12 - 40
As follows from the analysis in
Example 12.E.l in the textbook, the equilibrium of the two-stage game has two firms active, each firm producing 1/4.
However, in the one-stage game on firm
active and producing 3/8 is a NE (The best any other firm can do given that one firm produces 3/8 is to produce 3/16 which gives this firm a profit of 9/256 < 1/16, i.e. not high enough to cover the entry cost.)
Suppose that there is an equilibrium of the one-stage game where J .firms are active.
Using the notation of Section 12.£, this implies that n
(otherwise a firm would prefer to stay inactive).
J
2:
0
But then the two-stage
equilibrium number of firms is at least J (adopting the convention that an indifferent firm enters).
In other words, we cannot have less firms active in
the two-stage game then in the one-stage game.
12.£.7.
0
Suppose in negation there is an equilibrium in which J q
*
0
x(c·).
lf J• q o > x(c •), then the industry price is below c • and all firms make negative profits, which cannot constitute a NE. J oqo
Suppose that instead
< x(c • ), i.e. the industry price p is higher than c •.
Then a firm could
enter the market charging c •+,;, producing q•, and making a positive· profit of • •
cq .
0
Therefore any equilibrium has J q
J o sueh that Joqo
12.F.l.
= x (c •) ,
0
•
= x(c ).Since there exist an integer
such an equih'brium exists.
(First printing errata: Assume that
p( ·)
and c( ·) are twice
continuously differentiable.} . Assume for simplicity that c(O) = 0, and restrict attention to cr. demand function for the market of size cr. is X(p)
= cr.x(p),
it
1.
If the
then the inverse
demand function for this market is P(Q) = p(Qjcr.). Firm j's best response is obtained by solving Max p((Q + q ) lcr.)q- c(q ). -1 J /' J J q !:0 j
12 - 41
Since setting q J = 0 is always . an option, any solution to the problem . satisfies p((Q + q lja.)q- c(q ) ~ 0. J
-1
have c(q )jq J
J
:s
J
Therefore, at any solution q we must J
J
p((Q + q >ja.l :s p(O). Note that c(O) = 0 and c' ' ( ·) > 0 -1 J
imply that c(q)jq is strictly increasing in q. q
J
:s
qJ, where qJ solves
c(q
Therefore, we must have
l(q = p(O). J J
The Kuhn-Tucker conditions for (•) can be written as
= 0 if q > 0.
-
-
Denote the solution to (•) as b(Q I a.).
Define Q
~
J
~
= c'(O).
so that p(Q ) ~
Then from the Kuhn-Tucker conditions we know that b(Q
Ia.) = 0 if Q -J -J
-J < a.Q- •
i!:
> 0 if Q
-J
a.Q
-J
Now, let us focus on the case where Q
-
-J
< a.Q • Substituting q = b(Q Ia.) in J
-J
-J
the Kuhn-Tucker condition and differentiating with respect to Q , we obtain
-J
q,co 8b(Q la.>/80 = -J -J
.lex>= -J
where q,(Q
q,(Q
p''((b(Q
-J
-J
Iex)
Ia.> -J
+ p' ((b(Q
-J
•
I a.)+Q >fa.>fa. - c' '(b(Q I a.)ja.) -J
-J
la.)+Q .)ja.) b(Q .la.>jo/ +p'((b(Q -J
-J
-J
la.l+Q lfa.lja.. -j
We know that p' ( · l :s 0 and c' ' ( ·) :s 0. Hence, the result will be proved once we show that q,(Q
-J
Ia.)
< 0 for a. large enough.
Note -that Kuhn-Tucker conditions imply that p((b(Q_J I a.)+Q_}fa.l
i!:
c' (b(Q_J I a.lfa.l
Hence, we must have (b(Q I a.)+Q l/a. :s -J
-J
Q-J.
i!:
c' (0) = c'(O}
Define
p' = Max {p'(Q) I Q e {0, Q_JJJ < 0, p' = Max {p" (Qll Q E {0, Q-J ]}. 2 have 4>(Q Ia.) :s I p' I q fa. + p' fa. - p' fa. -J j 1
Then we
1
< 0 for ex large enough,
which proves the result.
12.F.2.
(a) Aggregate demand is X(p) = Ix(p) = I(a - bpl.
12 - 42
Inverting this
function, we obtain the inverse demand function: P(x) = A - Bx, where A
= afb
and B = lfbl. (b) Two-Stage Entry Model: Example 12.E.l in the textbook solves for the equilibrium in the two-stage "
.
=A
entry model with Cournot competition and linear demand P(x)
- Bx. The
equilibrium number of firms J • (ignoring the integer problem) is given by J
•
+ I = (A-c)/CBK)
1/2
= (a
- bcHI/bK)
1/2
.
The equilibrium output per firm is then determined in Example 12.E.l (Exerc;ise 12.C. 7): •
qJ• = (A-cl/B(J +1) = CibK)
1/2
•
•
The equilibrium aggregate output is J qJ•' and the equilibrium price is equal to
•
•
= A - B(J •+l)q • J J 1/2 . ( = c + KI Ib) -+ c
P(J q •} = A - BJ q • J
= c + BqJ•
as I
~ m.
Therefore, consumer (per capita) surplus increases to the competitive level as I
~
m.
One-Stage Entry Model: As shown in Exercise 12.E.6, the number of active firms in an equilibrium of the one-stage game cannot exceed J •• the equilibrium number of firms in the two-stage game obtained above. What is the minimum number of active firms in the one-stage game?
Suppose that J ·firms are active
in an equilibrium of the game, each firm producing qJ.
If an inactive firm
decides to deviate and enter. its optimal output choice q will solve Max (A - B(JqJ + q) - c)q q
The first-order condition yields q
•
= (A - BJqJ - clj2B.
Example 12.E.l, we obtain q • = (A - c)/2B(J+l). profitable if
12 - 43
Substituting qJ from
The deviation will not be
•
K ~ (A - B(JqJ + q ) - c)q
•
2
2
= (A-c) /4B(J+l) .
This gives a lower bound on the number of active firms in equilibrium of the one-stage game: J + 1 ~
J
0
+ 1
= (A-c)/2(BK)1/
2
=
(a 0
(we have substituted A and B from part (a)). When J
1
bcHI/bK) /2/2 firms are active in the ,.
market, each firm will produce qJo = (A-c) B(Jo+1)
I
· I (Jo+1) = 2(IbK) u2. = I(a-bc)
The corresponding upper bound on the equilibrium market price is
P(JqJo)
=A =c +
BJoqJo BqJo
=A
=c
I
+ 2(K Ib)
1/2
I
=A
- BCJo+1 >qJo + BqJo
=
- B(A-c) B + BqJo
---+ c as I ---+ m.
Therefore. consumer (per capita) surplus in any equilibrium of the one-stage entry model converges to the competitive level as I ---+ m.
l2.F.3.
For a fixed cr., we can use the result of exercise 12.E.3.
12.F.4.
[First printing errata: The demand function for firm j as a function -l+C
of the price vector is given by x}pl = cr.(7-f3P/P
.
-
), where p • ~pjJ, and
c > 0.] First we solve for the second-stage Nash equilibrium.
Each active firm j
solves -I+C
Max p x (p) = p cr.[r - f3P/P p J J J .
1
J
-
Differentiating with respect to p and taking into account that p depends on J
p , we obtain the following first-order condition: J -1•£
'l - 2 f3 P/P
2
-2•£
+ f3(l+c) p/(Jp
-
) = 0.
-
To find a symmetric equilibrium, substitute p = p and solve for p. This gives J
P=
[~ (2 _ 1~c)]uc.
The equilibrjum second-period profit for each firm,
with J active firms, •
12 - 44
is therefore -
nJ = p
ah -
Observe that as J
~
-
-1+£
f3P/P co, n
J
~
1 = o:r r1
~
n
•
l+c 1/C l+c l+c (2 - J )] (1 - J lfC2 - J ).
2(3 1/C = -21 o:r [1 . r
.
When ex (the s1ze of the
•
market) or (3 (the substitution parameter) are large enough, we have n
•
> K.
Therefore, in equilibrium of the two-stage game infinitely many firms will enter. However, the competitive limit (Proposition 12.F.l) will not be
> 0, while the marginal cost is
achieved: the price will converge to zero.
12.G~t.
The Cournot Duopoly
-
The equilibrium for this model (q •,q• ) has been obtained in Exercise 1
2
12.C. 9(a). Assuming that both firms produce, the equilibrium is q
q
• 1
= (a + c
•= 2
(a +
2
-
2c l/3b, 1
c - 2c lf3b. 1
2
•
From here we see that 8qj8c = 2/3b > 0. When firm I's cost goes up, firm 2's 1
equilibrium quantity goes up. Firm l's equilibrium profit is
• •I 1 2
n (q ,q 1
c ) 1
= (a
- b(q
• 1
•
•
2
1
+ q ))q
-
cq •. 1
The strategic effect on firm 1 is
Intuitively, an increase in firm l's cost makes firm 2 more aggressive, which strategically hurts firm 1. The Linear Citv Model
-
-
Assuming that prices are such that both firms produce, the equilibrium (p •,p• ) 1
2
has been obtained in Exercise 12.C.l7:
p • = (2c 2
2
+ c )/3 + t . 1
•
From here we see that BpjBc = 1/3 > 0. When firm l's cost goes up, firm 2's 1
12 - 45
equilibrium price goes up. Firm
l's
equilibrium profit is
• •I 1 2
n (p .p 1
c ) 1
= (p•1
-
c ) {1/2 + Cp • - p • )j2t} M 1
2
1
The strategic effect on firm 1 is Cp• - c 1
1
J aq-• lac > ?!
1
o.
Intuitively, an increase in firm l's cost makes firm 2 less aggressive, which is strategically beneficial for firm 1.
12.AA.l.
[First printing errata:The statement in the exercise is incorrect.
Here is a counterexample: L
I
R
u
-1, 9
1 -1.10
D
I -10,-1 I
0, 0
The only NE of this game is (D,R). However·, for some a we can sustain (U,L.l ir a SPNE using Nash reverc;ion strategies. If player 1 deviates and plays D, he will los_e 9 immediately and gain 1 forever after; he will prefer not do this provided that 9
2:
aj(l-a), i.e. a ::s 0. 9. If player 2 deviates and plays R,
she will gain 1 immediately and lose 9 forever after; she will prefer not to do this provided that 1 ::s 9ajCl-a), i.e. a
2:
0.1. Using Lemma 12.AA.l, we see
that (U,Ll can be sustained in a SPNE using Nash reversion strategies if and only if 0.1 ::s
a
::s
o. 9.
Intuitively, (U,U can be sustained in a SPNE using Nash reversion, because player 1, whose payoff is below its Nash equilibrium level, is playing his short-run best response. Thus, any deviation brings short-run losses to player 1, and If he is sufficiently impatient, he will choose not to deviate despite a permanent future gain. On the other hand, player 2 who receives a higher than NE payoff,
wil~
not deviate if he is sufficiently patient.
The correct statement which is the closest to the statement in the
12- 46
textbook is as follows: No pair of actions q such that
•
•
1
2
n (q , q ) < n (q • q ) for all i = 1.2 1
1
2
I
can
be
sustained
as
a
stationary SPNE outcome path using Nash reversion.] To prove this statement, observe that by assumption (q • , q • ) is the 1
2
unique NE, and therefore (q , q ) is not a NE. Thus, one of the players has a 2
1
profitable short-run deviation. Suppose for definiteness that it is player 1; and that his profitable short-run deviation is q', i.e. 1
1t
(q'. q ) >
1
1
1t
2
1
(q • q ). If player 1 deviates in this way, the game will 1
2
revert to the Nash equilibrium, and player 1 will receive
•
1t (q •
l
1
•
q ) forever 2
after. Since we know that n (q •, q • ) > n (q , q ), by deviating player 1 can 1
1
2
1
1
2
increase his payoff in very period of the game.
Therefore, (q , q ) cannot be 1
2
sustained in a SPNE using Nash reversion.
12.AA.2.
The first-order Taylor expansion of n (q ,q ) in the neighborhood of I
J
I
• •
(q ,q ) is 1 J 1t
• •
I
(q ,q ) "' n (q ,q ) + I
J
1
I
J
• •
•
• •
11
11J
•
+ an (q ,q )jaq (q -q ) + an (q ,q Jfaq (q -q ). 1
I
I
j
JJ
j
Since (q •,q • ) is a Nash equilibrium, we must have an (q •,q • >jaq = 0. 1 1 I l J l Therefore,
"' n
(q
•,q• )
lij
+
an (q •,q• >faq (q - q • ), i = 1,2. lij
j
• • Since by assumption an (q ,q Jfaq * 0 for i I l J J • • (q ,q ) close enough to (q ,q· ) such that 1t (q 12
12
J
J
(•)
= 1,2, there exists a point ,q )
112
< n (q •,q• ) for i = l, 2. 112
Consider a strategy profile where the players are to play an outcome path
• •
involving (q ,q ) in period one and (q ,q ) in every period thereafter, and 1
2
1
2
where the outcome path is restarted after a deviation. gives player i the payoff v'
I
(1-o)
v' :' I
= n1(q1,q2 )
(1-o)
as required by the proposition.
n
(q
This strategy profile
• •
+ oj(l-15) 1t (q ,q ). We have 112
,q ) + 0 n (q • ,q • ) < n (q • ,q • ),
112
112
112
It only remains to show that the described
12 - 47
strategy profile constitutes a SPNE. To do this, it will be useful to estimate ~ (q ) (using the notation of J
1
For this purpose, we can write ~
Appendix A in the textbook).
.
.. • ..
.
=
(q ) 1 J
•
..
n (b (qJ), qJ), and differentiate with respect to qJ at the point q / 1
1
"Jr~(qJ) = anl(ql,
qJ>faql b~(qJ) + an1(q1, qJ>jaqf
Since q • is a best-response to q , we must have 8fr (q •, q •>jaq = 1
.
.
"
Therefore, n' Cq ) 1
J
.
=
1
around q •:
< (1-15)
2
11
1
J
1
a.
Since Cq , q ) can be chosen to be as 1
2
"
we can use the Taylor expansion of n Cq ) 1
..
"n Cq ) = n (q , 1J
1
8fr Cq ,q >Jaq . 1 1 J J
close as necessary to (q•,q• ), J
J
..
J
.
q ) + an Cq ,q )/aq (q - q ) ~~~~ n (q ,q ) < J 11J J J J 11J
• • =
(q ,q ) + 15 1t (q ,q ) 112 112
1(
(l-15)
c••)
v'. 1
(We have made use of (•).) Now we can finally check that the strategies above constitute a SPNE by checking all possible deviations after two types of
• •
histories: those which prescribe playing (q ,q ) forever, and those which 1
2
prescribe playing (q ,q ) in the next period (this occurs in period one or 1
2
immediately after a deviation).
(i) As has been discussed in Appendix A,
the strategies prescribing playing
• •
(q ,q ) forever constitute a SPNE, and therefore no player can profitably 1
2
deviate.
(ii) Suppose that player
i deviates after a history which prescribes playing
(q ,q ) in the next period. If- this deviation continues for T periods (we also 1
2
allow for T = m), player j will keep playing q for T+l periods, and player i J
will get at most (1 + ••• +
aT-1 )
1\ J(
(q ) + 0 1 J
T
v' < 1
(1-a T )v'
which is what player i would get if he did not deviate.
1
T
+ 15 v' = v' I
1'
(We have made use of
Therefore, both kinds of deviations are unprofitable, and the strategies
12 - 48
constitute a SPNI::.
12.88.1.
Figure 12.88.1 describes the case where entry deterrence is possible
but not inevitable, and where point S lies to the right of point Z.
This
figure is a small modification of Figure 12.88.8 in the textbook. Geometrically it is clear that k
Jr
i:
q (S) > q (Z) = k . 1 1 z
Since entry is not
.
blockaded, point Z is to the right of the incumbent's optimal monopoly point. In words, k
z
is higher than the monopoly output, but lower than k . Therefore, Jr
by the argument on p.427 of the book, the incumbent prefers to deter entry.
1\(
Figure 12.BB.l 12.88.2.
Observe that when accommodating, the incumbent ends up producing
less then when deterring entry.
In order for the incumbent to be indifferent
between these two outcomes, the equilibrium price in case of accommodation must be higher than in case of deterrence. fixed cost of production is duplicated.
Also, in case of accommodation the
Therefore social welfare is higher if
the incumbent chooses deterrence. As we have seen in Section 12.E of the textbook, an entrant may decide to
12 - 49
enter even when it lowers social welfare.
This may happen because of the
"business stealing" effect - by entering, the entrant incurs a negative externality on other firms in the market.
It is thus not surprising that
entry deterrence may sometimes raise social welfare.
12.BB.3. enter.
If firm I enters at both ends of the city, then firm E will not . Indeed, if firm E entered in this situation at one of the ends, it
would engage in Bertrand competition with firm J's plant existing at this location. it would collect zero profits and would not be able to cover the fixed cost of entry. Now consider the situation where firm I has entered at one end of the city. If firm E enters at the other end, the analysis of Example 12. C. 2 suggests that it will collect tM/2 - F in net profits.
Entering on top of firm I is
clearly dominated by not entering at all, in which case firm E gets zero. Therefore, firm E will enter at the opposite end if and only if tM/2 > F. Now Jet us go back one stage and examine firm I's decision.
F'irm
I has three choices:
(i) F'irm I can enter at both ends and become a monopoly.
choice: it can either charge a price
~t
which the midpoint consumer buys or
charge a price at which the consumer does not buy. first option, it will charge the
Then firm I has a
~aximum
If firm firm I chooses the
possible price v-tj2.
Firm I may
consider to switch to the second option by raising Its price and losing some customers.
Suppose that it charges some p ·> v-tj2.
Then the demand at each
end will be M(v-plft. F'irm l's profits will then be n(p) = 2 (p - c) M(v-p)jt. Differentiating with respect to p, we obtain
n' (p) = 2M(v + c - 2p) < 2M(v + c - 2v +
t)
= 2M(c + t -
v)
< 0,
since by assumption v > c + 3t. Therefore, firm I will choose to charge v-tj2,
12 - 50
and its net prpfit will be M(v - t/2 - :) - 2F. (ii) Firm I can enter at one end.
opposite end.
Then if tM/2 > F, firm E will enter at the
The resulting competition will bring the net profit of tM/2 - F
On the other hand, if tM/2 > F, firm E will not enter at
to firm E.
all.
Then firm I again a choice: it can either charge a price at which the most distant consumer buys or charge a price at which the consumer does not buy. If firm firm I chooses the first option, it wiU charge the maximum possible price v-t.
Firm I may consider to switch to the second option by raising Its
price and losing some customers.
Suppose that it charges some p > v-t.
Then
the demand will be M(v-plft. Firm l's profits will then be n(p) = (p - c) M(v-plft. Differentiating with respect to p, we obtain
n' (p)
= M(v
+ c - 2p) < M(v +
c - 2v + 2t) = M(c + 2t -
since by assumption v > c + 3t. Therefore, firm I will
choose
v)
< 0,
to charge v-t,
and its net profit will be M(v - t - c) - F.
(iii)
lf firm one does not enter at all, its net profit is zero. All these cases can be summarized as follows:
If F < Mt/2. firm I will never choose not to enter, since it always gets positive profits from entering. If it enters at both ends, it gets M(v - t/2 c) - F, while if it enters at one end, it gets Mt/2 - F.
Since by assumption
v > c + 3t, firm l will choose to enter at both ends and deter firm E's entry. If F > Mtj2, firm E's entry is not a threat.
Firm I decides whether it wants
.
to be a monopolist with two plants, a monopolist with one plant, or not to enter at all.
With two plants firm I will get M(v - t/2 - c) - 2F·, while with
one plant firm I will get M(v - t - c) - F. The difference in profits is Mt/2
- F < 0, therefore firm I would rather enter with one plant than with two. the other hand, if F > M(v.- t - c), firm I will not enter at all. We can now summarize how the equilibrium outcome depends on F:
12 - 51
On
If F < Mt/2 then firm I enters at both ends, firm E does not enter. If Mt/2 < F < M(v - t - c) then firm I enters at one end, firm E does not enter. If M(v - t - c) < F then neither firm enters. There is a simple intuitive explanation for why firm E never enters in this game.
The reason is that whenever firm E could enter and collect a
positive profit, firm I could preempt by entering at the same site and collecting at least as high a profit. In general, we can expect firm I to collect strictly higher profits by preemption because then it does not have to compete with itself.
This is called the efficiency effect.
For a more
detailed discussion, see Jean Tirole's "The Theory of Industrial Organization", 1988, section 8.6.2. This effect does not work, however, when firm I can only enter with one plant. Now when F < Mt/2. firm I will only able to enter at one end of the city, and •
firm E will enter at the other end.
Preemption becomes infeasible .
•
12 - 52
CHAPTER 13
13.8.1
The three functions are graphed in figure 13.8.l(a).
This graph
also depicts the situation in figure 13.8.1 in the text.
.
e
..
Pareto
Optimum
r(8~
e
Complete Market Failure
I I
I
Competitive e uilibriwn
..
.
•
...., /
..
[9:9~9]
J .
r(8) 1
j ./
/
/
..
[9: 9~9] I I I
I
-e
9
9 Figure 13.B.l(a)
-
9 Figure 13.B.l(b)
e
Figure 13.8.l(b) dt!picts the situation in figure 13.8.2 in the text . .
Figure 13.8.1Cc) depicts the situation in figure 13.8.3 in the text .
. 6
lMultiple competitive '
1
r
equilibria
),r(S)
, ......_
_.., jE[ e: e~ 81 I I
e
Figure 13.B.l(c) A
13.8.2 But E (9
If w = 9, only workers of type 9
I
9 :!:
-91 < 9- = w,
~
9 will accept the wage w and work.
which implies that firms demand less worker than
there are in supply. · This implies that the market does not clear. If w > 9,
13 - 1
•
only workers of type a
~
h
•
> 9, with r(9 )
a
work (since r() is an increasing function).
= w,
will accept the wage w and
• • I 8~8]<8
But E [a
which implies that firms demand less worker than there are in supply.
This
implies that the market does not clear. Thus, to obtain market clearing firms have to offer a wage w < 9, which
a
implies that some workers of type 9 <
will not work, and there wi 11 be
•
underemployment in the competitive equilibrium (in an equilibrium with perfect
a
information all workers of type
13.8.3
a
~
will work).
{a) Suppose firms offer a wage of w.
:s w, will accept the wage and work.
Then all workers of type decreasing.
a
it
• a
All workers of type 9, with r(S)
Suppose there exists a
will work, since r(a)
• a
• := r(e ) =
with w
• r(e )
= w.
and r() is
Thus, the more capable workers are the ones who will work at any
given work. (b) firms can offer the wage w = will work.
-a,
and since
r(9)
> 9
no workers of type
from part (a), no worker of any type will work.
-9
Therefore, the
competitive equilibrium is Pareto efficient, i.e. nobody will work.
{c) If w = E
[8
I
e
e.
i!:
-eJ
only workers of type
> 8
a a will it
a
i!!
> e.
with
But
= w, which implies that firms demand more workers than there
are in supply, and the market will not clear.
. a
accept the wage w and work.
. r(8 ) = w,
decreasing function).
If w <
a,
only workers of type
will accept the wage w and work (since r() is a
But E Ia
• • 8::!::8}>8
I
= w • which implies that firms
demand more worker-s than there are in supply, and the market will not clear. Thus, to obtain market clearing, firms have to offer a wage w implies that some workers of type
a <9
> 9,
which
will accept the job, and there is
over employment in the competitive equilibrium
e
:?::
a
will work).
13 - 2
13.B.4
We can think of the true valuation of the good, y,
as a state of
nature s e S where S is the set of all states of nature, and both agents 1 (say the seller) and 2 (say the buyer) have a common prior that is common knowledge.
Let
H.(s)
be the set of states that agent i believes is possible
1
given the true state s (this wiii depend on the signal observed).
We can
define the following two events: T a { s e S 1 T
2
a { s e S
I I
E [ y
I
H (s)nT 1 2
1 :s. p}
E [ y
I
H (s)nT 2 1
1 2: p }
That is, T. is the event that agent i wiJJ say "trade" given that he knows 1
that event H.(s) has occurred, and that he ·believes agent j will say "trade". 1
'
Assume that there exists an equilibrium where both agents say "trade". Then, each agent i knows that event T. occurred, and since this is an equilibrium .
1
then each agent i believes with probabil!ty 1 that agent j knows that event T. J occurred. Therefore, each agent believes with probability l that event T=T{'T
2
occurred.
The seller
(1)
then prefers to trade if and only if
E[y IT)::!:p, and the buyer (2) prefers to trade if and only if
Ely IT):i!:p.
Assuming a generic distribution of values both can hold with probability zero, therefore the set T must occur with ·probability zero.
13.B.S
(a)
When r(e)=r
for all
e,
and E[9):i!:r>e, then the firms will make
-
zero profits (a necessary condition for a competitive equilibrium) in two cases: Either w•=E[e) or w•=e.
-
In this case, if w•>E[e) then firms are losing
money, while if w•e[r,E(9)) then e•=[e,e] and firms make positive profits.
-
Finally if w•e(e,r) then no one will accept employment, and together with the
-
assumption that in this case firms believe that any worker who might accept employment is a
e
-
type worker, we must have the offered wage to be
13 - 3
e.
-
If, hcwever,
-
e~r.
then when w•=e firms will make positive profits since all
-
workers will work, and E(9)>w•, so the only competitive equilibrium is w•=E(e).
On the other hand, if r>E(e) then when w•=E(e) no worker will wish
to be employed, and due to the assumption on firm beliefs, we must have w•=e.
-
(b)
In the equilibrium with w•=E(9), both firms have zero profit, there is
full employment, and all workers have a utility of E(9)>r. If, however, w•=e
-
then again both firms have zero profits but workers have a utility of only r, this is Pareto dominated by the full employment equilibrium.
(c)
The analysis of SPNE is as in proposition 13.B.l which applies {with
straightforward modifications due to a constant r() function).
When E(9)=r,
then both competitive equilibria are SPNE. ·This follows because no firm can offer a wage we(e,r] and make positive profits (because if w<E(9)=r no worker
-
will accept employment) and a wage w>r will cause losses.
(d)
Clearly, when
-
E(a):!:r>e and when
e~r
-
then the highest wage competitive .
equilibrium has full employment, and is therefore Pareto optimal.
For
the case where r>£(9) a simpler version of proposition 13.8.2 can be applied.
13.B.6
For a similar analysis we refer to:
Wilson, C. (1980) "The Nature of Equilibrium in Markets with Adverse Selection," The Bell Journal of Economics, 11:108-30.
13.8.7
-e -u
First assume there is a Pareto improving market intervention ( w , w )
that reduces employment with respect to a competitive equilibrium with wage w•.
-e
Clearly, we cannot have w < w• since then those workers who are employed
-u
are worse off. Similarly we cannot have w < 0.
0.
-e
We can then reduce both w
-u
and w
by
£
-e
-e
> 0 such that w -
£
-u
> w• and w - c
> 0 will still hold, the same groups of agents will be employed/unemployed,
13 - 4
-u
Now assume that w > w• and w >
-e -u
and the government now has a surplus (it broke even with (w , w ) since it must have a balanced budget). The government can then distribute this surplus by •
-
raising only w , and this would still be a Pareto u
improvement relative to the
-
-
competitive equilibrium. This can be done as long as w > w•, and since it was e
-e
-u
employment reducing we must have w - w• < w - 0, implying that we can reduce
-
both w
e
-
and w "'
u
-=
until w
A
e
. w•. The result is a Pareto improving intervention of
A
.. ..
"'
the form (w ,w ) with w = w• and w > 0. e u e u
The converse is trivial: if (w , w )
..
is a Pareto improving market intervention of the form w • w• e
e•
we must have less employment since the marginal type
e u .. and w > 0, then u
who decided to work
had w• = r(e•) and now can be unemployed and receive r(e•) +
wu> w• so he will
decide to be unemployed, and by continuity a positive mass of types will decide so as well.
-e -u
Second, assume there is a Pareto improving market intervention (w , w ) that increases employment with respect to a competitive equilibrium with wage
w•.
-
Clearly, we. cannot have we < w• since then those workers who are employed
-
are worse off. Similarly we cannot have w < 0.
->
·
u
""' We can then reduce both w
w - c > 0. u w
u
e
-u
-
Now assume that w - c > w• and
and w
e
-
by c > 0 such that w > w• and e
0 will still hold, the same groups of agents will be employed/unemployed,
and as before, the government can distribute the generated surplus by raising
-
only w , and this would still be a Pareto improvement relative to the e
-
competitive equilibrium. This can be done as long as w > 0, and since it was
-
-
u
•
employment increasing we must have we- w• > w u- 0, implying that we can reduce.
-
both w
e
-
and w ,.
..
u
-=
until w
u
.
0. The result is a Pareto improving intervention of
the form (w ,w ) with w > w• and w e u e u
= 0.
As before, the converse is trivial.
Finally, these facts give a simple proof of proposition 13.8.2. by contradiction: If there were a Pareto improving intervention, it must be either employment increasing or reducing.
.. ..
two forms: (i) (w ,w ) e u
= (w•
Therefore, it must be of one of the
.. ..
+ c5,0), or (ii) (w
·e
13 - 5
,w u ) = (w•
,c).
If it is of
hav~
form (i), then a firm could
deviated, proposed w• +
a , 2
and by the
conditions given. would have made positive profits - a contradiction to w• . being a SPNE. If it is of form (ii) then the government cannot balance its budget since r( · ) is strictly increasing, therefore the best of the formerly employed types will choose to be unemployed, and by paying w• to the remaining employed workers the government must lose money, in addition to the unemployment benefit it is paying out, a contradiction to (w• ,c) being a feasible intervention.
13.8.8
Assume that we have a model similar to the one studied in· section 13.B
as displayed in figure 13.8.8 (let r(a•) = w•) .
•! '
/ //
/
E[ el r( 9)
~ w]
/
.
A
/ .·
/
~.··-='4t.os~=_ _ _ _ _ _ _ _ _ _ .,
r(e) r(Er)
w
r(S)
Figure 13.B.8 In our new model, since all workers of type 8 e [e•, 8] don't work, they get to consume the product x.
Since the government observes the level of
consumption of x, and prior to intervention the economy is in a competitive equilibrium, the government is able to identify the type of all indiv-iduals for which 8 > 8•. (Because their consumption is a function of 8, x(8), which is known to the government, and x(9) is invertible because it is increasing.) In the competitive equilibrium of the model of section 13.8., firms break even, and workers' payoffs are:
13 - 6
- if
a :s a•, uC e) = w•
- if
e > a•, uCa>
Consider the following scheme:
e > a•
=
r(el
suppose the government offers a worker of type
[which has been identified) wage of r(9) + c, where c is small enough
so that rCa) + c < e for all a e (9•, 9]. this offer, and improve their payoff by
All of these workers will accept £.
At the same time, r(a) ...
< .e for
£
all a > 9•, and the firm makes positive profits out of hiring these highly skilled workers.
These profits are equal to:
-
e n•
=
[a - r(a} - c) eca}da
Since the firm was breaking even with wage w• for workers with 9 <
a•,
we
must have:
9
-
If the firm raises the wages of all workers with
e < a•
by d (to w• + d), it
will incur a loss of:
a• d lC9) da = d Pr (a :s a•)
en•
If d < 2 . Pr (a :s
e•) , this loss will be smaller than
~he
gain from the highly
skilled workers because: d Pr(a
Thus, the firm is now making positive profits, thereby having raised its
13 - 7
utility level.
All workers of type e
level (from w• to w• + d).
:S
a• have also raised their utility
Additionally, all workers of type a
raised their utility level from r(9)
to
r(a) + c.
2::
a• have
Thus, this change is
Pareto improving.
13.8.9
In the model described there are three candidates for a highest wage
competitive equilibrium (recall firms should get zero profits): Case
~
w•
= aH
and only high types are employed.
This implies that we must
have aH < r(eL) or else low types would ask for employment and firms would lose money. (This case is fully Pareto optimal.} Case
~
r(6H}
2::
w• < Min{ r(9H) , r(9L) } and no one is employed.
This implies that
r(aL), since if r(9H) < r(9L) were the case, a firm who would deviate
to w = r(9H) would make positive profits and this would not be an equilibrium. Case
£
both
w• = ;\aH + 0-A)aL and both types are employed.
r(eH) ::s w• and
r(9L)
:S
This implies that
w•.
Consider case 3 which is not fully Pareto optimal (only the high type should be employed) and assume that r(aH) < r(eL) < w•.
Let w = £be the u
unemployment benefit where c < Min{ r(9L) - r(eH)' eH - r(9H) }, and let w• remain .the employment wage.
Only low types will choose not to work, the
government will lose (1-A)c and gain Cl-A)(w• - eL) due to the l~w types relocation, and for small enough c this is beneficial and yield a Pareto improvement. Turning back to Exercise 13.8.7, we can say that a competitive equilibrium involving full employment is constrained Pareto optimal, if there is no employment reducing, Pareto improving intervention of the form (w•.c). a
13.8.10
As in proposition 13.8.2, we still have that for all (w , w ), the set e u
... ...
of workers accepting employment is of the form [9,9).
13 - 8
This follows since r( ·)
is still an increasing function.
Therefore, the details of the proal· in the
textbook go through.
13.C.l
Suppose some workers, whose types do not equal to a, do not submit
-
to the test. test a
•
Call the worker with the highest type that did not submit to the
> a Finns will now offer all the workers that did not submit to the
-
test a wage justified for the mean worker that did not submit to the
~est,
i.e. they will offer a wage equal to the e,xpected value of a for workers that •
did not submit to the test. of worker a
• which
the test.
• a ,
This expected value will be lower than the type
while if worker
• a
submits to the test he will receive a wage of
is higher than the wage he would receive if he would not submit to Therefore all workers (except for the lowest type worker) submit to
the test in the unique SPNE of this game.
It follows that firms offer no more
than a to workers who do not submit to the test (they loose money otherwise).
-
13.C.2
~-t=l
For simplicity, assume
results).
(this does not change the qualitative
The competitive equilibrium with perfect information is given in
the following Claim: Claim
.
1: At the competitive equilibrium with perfect information, type aH
(resp. aL) gets education level
~~
(eL' eL) = eL).
eH
(resp. ez.), where
~
(eH' aH) = eH. (resp.
The wage for aH Cresp. eL) is given by wH• = eH + eHeH
(resp. wL• = eL + aLeL). Proof gf Claim !:
If a worker of quality 9 gets education level e, then his
marginal productivity is a + firms are competitive.
ae,
and his wage will be equal to a +
ae,
since
Workers of type a will thus choose their level of
education to maximize their utility, given this wage level: max
w - c(e, 9) = a + ee - c(e, a)
e
13 - 9
and the FOC is: Remark:
~ (e•, 9) = e. Q.E.D.
The competitive equilibrium with perfect information is Pareto efficient.
As you may expect, both the separating equilibrium and the pooling equilibrium look similar as in the original model in which education does not affect productivity.
That is:
- the equilibrium contracts provide the firms with zero
profits.
- the low productivity type will obtain the optimal level of education •
in a separating equilibrium. The good news is that the high productivity type may also obtain the optimal level of education, i.e the above competitive equilibrium with perfect information may emerge as a separating equilibrium. In Figure 13.C.2(a) the competitive equilibrium with perfect information is sustained as a separating equilibrium:
WA I
I I 8u.
Figure 13.C.2(a)
e
In Figure 13.C.2(b) below, the outcome of the competitive equilibrium cannot be attained as an equilibrium with imperfect information.
Type eL would like
to pretend to be of type eH' if he is offered (eL_. eL + eLeL •) and (eH •. eH + 9HeH • ).
Thus, the incentive constraint for the low productivity type is not
satisfied.
13 - 10
w
Figure 13.C.2(b)
e
With a similar argument as in the text we can show: Claim
~:
The separating equilibrium of this model is as follows:
(et: eL + eL eL •) and (eH •, aH + eHeH •), with eH • e [e', e"].
This
equilibrium is depicted in Figure 13.C.2(c).
WA I• I
w
I I
~
.,
•
""
•
...
•
"'
...
'"'
..,
.,
•
•
;..I.:;..._..;...:;!.
I I
.
w' ~ •••...•••
II I
I. •
;·.
ll
/·•
: llHII ,
j:
I.
~
I
'
.
•
I
1-.
6L;
(e ~, "(")
• '
e'
e"
e
Figure 13.C.2(c) Claim
~:
·The pooling equilibrium of this model is as follows:
(e•, 0-h)(9L + eL e•) + h(9H + eHe•)), with e• e [0, e' ). depicted in Figure l3.C.2(c).
13 - 11
This equilibr
W£ I
I I
'' Poolq Break· SV$11 hDe '
·~~
e
e
ell~__<e_l_·._vr_>____________~··
e Figure 13.C.2(d) 13.C.3
We basically need to find an example that demonstrates the
situation in Figure 13.C.ll in the textbook. type be A =
Let the probability of a high
5
6• let eH = 4 and eL= 0, and let the utility functions· of both
types be: 1
for e :s and
9 wE - e + 20
2 1
for e -> -2
The separating equilibrium is (wL,eL) = (0,0), and (wH,eH) = (4,1), and the utilities of both types are uL (wL,eL) = 0 ,
and
uH(wH,eH) = 3.45.
This
equilibrium is depicted in Figure I3.C.3(aJ on the next page. Notice the linear line between the points (w,e)=(3; ,0) and (w.e)=(4,I).
As
we move the wage of the L type up from 0, his indifference curve will shift
. up.
In order to make zero profits we must give the H type a wage equal to the
wage determined by the intersection between the L type's new indifference curve and the linear line described above (this is easily shown algebraically and depends on the values of .\, eH and eL). Therefore, to give an example as required we must have the H type's indifference curve through the separating equilibrium point first go down below this linear line, and then cut it and reach the w-axis above this linear line.
This is how such an example can be
generated. A central authority who will offer (wL,eL) = (0.5,0), and (wH,eH) = (3. 9,0. 88), will maintain zero profits since
~ · 3. ~
= 3
1
3
= E(e), and
the utilities of both types are uL (wL,eL) = 0.5 , Note that since E(9) =
and
uH(wH,eH) = 3.47.
3~ < 3.45 = uH(4,1), we would not get a Pareto
improvement by banning the signal since the H type would be worse off.
I
0._-------------+~~
1
e
0.88 1
Figure 13.C.3(a)
Figure 13.C.3(b)
•
13 - 13
e
13.C.4
[First Printing Errata: the question should end with "Derive the
(nuique) separating perfect Bayesian equilibrium.") The firms will offer a wage w(e) as a function of the observed level of education.
Given a wage schedule w( · ), an agent of type 9• chooses e to
maximize his utility
2 • • 2e• w(e) - e /9 , and the FOC is: w'(e ) = e• . Since
the firms are competitive we must have zero profits, w(e•) = 9•, in equilibrium. Combining this with the FOC gives us the differential equation: w(e•)·w'(e•) = 2e•, which implies that w(e} = v'2·e .
This wage function,
together with each type choosing e that satisfies the FOC above, that is, e(9} = 9 / v'2, is the unique separating PBE. There also exist many (in fact, a continuum of) pooling equilibria
...
-
(w•,e•) of the form:
..
w• = £(9), and e e [O,e), where e is calculated by
equating: u(w•,el9) = u(9,0I9).
-
13.C.S
- -
(a) The consumer will buy the product if the expected value of the
product is higher than the price, i.e., if h vH + (1-h) Vi_
2::
p.
(b) Suppose there exists a separating equilibrium in which the high quality producers spends A on advertising and only the high quality product will be bought (in a separating equilibrium consumers know the quality of a product, so low quality products will not be bought since p > v~).
This
implies that the low quality producer makes no profit and the high quality producer makes a non-negative profit, IIH = p - cH - A
2::
0.
However, a low quality producer can make a positive profit buy spending A on advertising, since the consumer will then mistake him for a high quality producer and buy the good from him. equal
IIL = p - cL - A > p - cH- A
The low quality producer's 2::
profit will
0. Therefore, no separating equilibrium
can exist.
13 - 14
(Note, that the banks will not be compensated for the risk that they assume since they are risk-neutral): h(pGR + (1-pG)O) + Cl-h)(p R + (1-p )0) = I + r, 8 8 or, R = (l+r) I (hpG + (1-A.)p ). 8 An ·entrepreneur of type i will pursue a project if: (p. (TI-R) + (1-p. )0) 1
for i e {G,B}, or if II
2::
i!:.
I
Cl+r)/[).pG + (I-).)p ]. Since, by assumption II 8
0
2::
O+r)/pG and II :s (l+r)/p , the entrepreneurs wlll pursue a project if h is 8 large enough.
In other words, if the fraction of good projects (i\.) Is large
enough, then the banks will set a lower interest rate (R) and thus. the entrepreneurs will undertake the project, i.e. II (b)
(i)
2::
. R = U+r)/[ApG + U-A)p ]. 8
The entrepreneur's expected payoff from a project of type ie{G,B} is:
p.[ll-(1-x)RJ + (1-p.)O- (l+p)x = p.(ll-R)- x[(l+p)- p.R). I
l
I
,
l
(ii) In a separating equilibrium the banks will know the type of the
project.
Thus, the banks will offer an entrepreneur that pursues a good
project an interest rate of R a bad project an interest rate
=Cl+r)/pG s a~·
II and an entrepreneur that pursues
R = O+rl.(p
8
i!::
n.
Therefore, in a
separating equilibrium no bad projects will be pursued. Therefore, the minimum level of x, that allows the entrepreneur with the good project to signal his type, has to give an entrepreneur with a bad project a negative expected payoff if he contributes this level x of internal funds and obtains financing from the bank at an interest rate of R = O+r)/pG. That is, p (1I-R) - x((l+p) - p R) = 0. 8 8
Substituting in the
equilibrium level of R we get: p [1I-(l+r)/pG] - x[(l+p) - p 0+r)/pG] 8 8
= 0,
or,
x = [(li-(l+r)/pG))/[(l+p)/pB - (l+r)/pG] < 1. . It follows that as pG increases and r decreases, x increases. increase, x increases.
As II or pB
Since an entrepreneur with a bad project will obtain a
zero expected payoff by pursuing the project, an entrepreneur with a good · project will obtain a positive payoff . .
The separating equilibrium is then: Entrepreneurs with bad projects will
13 - 15
t·ontribute x
=0
and accept a bank's offer if R :s
n.
Entrepreneurs with good
projects will contribute x = [(n-(l+r)/pG))/[(l+p)/p - O+r)/pG) and accept a 8 bank's offer if R :s (l+r)/pG. U+r)/p
8
The banks will offer an interest rate of R =
if the entrepreneur contribute x = 0 and R = Cl+r)/pG if he
contributed x
= [(JI-(l+r)/pG)]/((l+p)/pB
- O+r)/pG].
(iii) The entrepreneurs with the bad project will not be better off, and
may be worse off in the separating equilibrium of ·part (ii). ~.
For large enough
all projects will be financed in the equilibrium of part (i) and the
entrepreneurs with the bad project will make a strictly positive expected profit.
In the separating equilibrium of part (ii), however, entrepreneurs •
with bad projects will always obtain a payoff of zero.
For small
~.
entrepreneurs with bad projects will obtain a payoff of zero in both equilibria. For small i\., entrepreneurs with good projects will be better off in the separating equilibrium, since they obtain a positive payoff in the separating equilibrium and a zero payoff in the equilibrium of part (i) (since the projects will not be financed). As i\. becomes larger, projects will also be funded in the equilibrium of part (i).
Now, the entrepreneur will have to pay a higher interest rate R on
the bank loan in the equilibrium of p"art (i).
In the separating equilibrium,
however, the entrepreneur has to contribute his own funds, which is costly to him since he is liquidity constraint.
Thus as
~
becomes large enough, the
entrepreneur will be better of in the equilibrium of part (i).
JJ.D.l
This is the screening analog of the signaling model of Exercise
13.C.2.
The competitive equilibrium with perfect information is given in the
following Claim: Claim 1:
At the competitive equilibrium with perfect information, type eH
13 - 16
de (resp. aL) has a task level of tH (resp. til, where dt (tf/' aHl :: aH. (resp.
*
ctz. aL) = aL .. The wage for aH (resp. aL) is given by WH. = aH + aHtH
(resp. wL • = eL + eL til . Proof Q.( Claim !: produces
a
competitive.
A worker of quality
a
performing a task level of t,
+ et, and his wage will be equal to 9 + 9t, since firms are
Workers of type 9 will thus choose their task level to maximize
their utility, given this wage level: max w - c(t, 9) ·
the Foe is: Remark:
de
crt Ct•. a) = a.
=9
+ at - c(t, al , and
Q.E.D.
The competitive equilibrium with perfect information is Pareto efficient.
As one may expect, the equilibrium looks similar as in the original model in which the task level is unproductive.
That is:
- the equilibrium contracts provide the firms with zero
profits.
- there exists no pooling equilibrium - the low productivity type will provide the optima 1 task level
• m a
separating equilibrium. The good news is that the high productivity type may also obtain the optimal task level, i.e the above competitive equilibrium with perfect information may emerge as a separating equilibrium. In Figure 13.D.l(a), the competitive equilibrium with perfect information is sustained as a separating equilibrium:
W£ I'
Figure 13.D.l(a)
t
In Figure I3.0.l(b), the outcome of the competitive equilibrium cannot be
13 - 17
attained as an. equilibrium with imperfect information:
w
8a
t
Figure 13.D.l(b)
Type aL would like to pretend to be of type eH. if he is offered (tL_, 9L + aLtL•) and (tH•, eH + 9HtH•).
Thus, the incentive constraint for
the low productivity type is not satisfied.
By a similar argument as in the tey+ we r.an show: Claim
~:
The separating equilibrium of this model exists and is as follows: - CeL_ •.eL + aLeL•) and (eH•. aH + eHeH•). the indifference curve of the high productivity type is above the pooling break-even line. - no cross-subsidizing contract, that will be accepted by both types, exists that gives a firm a positive profit.
This equilibrium is shown in Figure 13. D.l(c): W
£
a ..
~eH+
o · ))eL
t
·Figure 13.D.l(c) Claim _i:
No equilibrium exists in the followir.g -case: The indifference
13 - 18
curve of the high productivity type break-even line.
iS
at some points below the pooling
This equilibrium is shown in Figure 13. D.Hd ):
W'
8• l8H+ (1 ~ ))6t
Pooling Blak·
even line
e Figure 13.D.l(d) 13.0.2
(a) Once M and R are given, the wealth levels of an insured individual
in ·the two states, denoted by (w , w ), are given by: 1 2
(wl' w l = (W - M, W - L - M + R). 2
Therefore, we can think of the insurance contract as specifying the wealth levels
(w
, w ) in the two states. 1 2
The premium M and the repayment R can be
obtained by the following equations: M
=W -
w
1
and R = w
2
+
L -
wr
(b) This game is analogous to the screening game studied in this chapter. Therefore there exists no pooling equilibrium, and the existence of a separating equilibrium is not always assured. If there exists a separating equilibrium, then the high risk types are completely insured, i.e.
w~
The low risk types will not be completely insured, in fact w7 > wi. For a proof of these results, we refer to the original paper by Rothschild, M. and Stiglitz, J. 0977) "Equilibrium in Competitive Insurance Markets with Adverse Selection," QJE, p. 629 - 649 and to Laffont (1989) The
Economics of Uncertainty and Information, MIT press, Chapter 8.
13.0.3
(a) Assume for simplicity that JJ=l, which does not change the
qualitative results.
The high (respectively, low) type workers output is
13 - 19
aH(l+T) (respectively, aL (l+T) ). output level the worker's type.
A firm can n(IW deduce from the observed Therefore, any firm will propose the
fallowing contract: - aHO+T) if observed output is aH(l+T) - aL (l+T) if observed output is aL U+T). wil~
If aL (l+T)
2::
c, both types
accept the offer and work.
If aH(l+T)
2::
c > 9L O+T), then only high type worker will accept the contract.
If c > aH(l+T), none of the workers will accept the offer. (b) Denote by (wG' w
8
>
the contract offered by a firm, in which it pays wG
(w ) if the output realization is good (bad). 8
The firm chooses its contract,
(c) The problem only differs in respect to which workers will accept the offer given in part b): A worker of high type accepts, if pH u(wG) + (1-pH) uCw ) 8
i!::
A worker of low type accepts, if pLu(wG) + (1-pL) uCw ) 8
u(c). .
13.D.4
2:
u(c).
It can be shown that under conditions (i) and (ii), Proposition
13.0.2 still remains true.
However, with these added conditions The model
differs from the model of section 13.0. with respect to the existence of equilibria.
In this model 'the existence of an equilibrium is guar_anteed if no
firm can offer a pooling contract, that attracts both types of workers, and makes a positive profit.
Thus, the existence of an equilibrium is guaranteed
in Figure 13.0. 7(a} in the textbook.
In Figure 13.D. 7(b), however, there
exists no equilibrium, as is the case for the original model.
The difference
for this model is for the situation displayed in Figure 13.0.8 of the textbook.
Contrary to the model discussed in the text, firms cannot offer
multiple contracts and thus cannot cross-subsidize between the workers, and an equilibrium will exist.
13 - 20
For a proof of these results, we refer to the original paper by Rothschild, M. and Stiglitz, J.(l977) "Equilibrium in Competitive Insurance Markets with Adverse Selectiont" QJE, p. 629 - 649 and to Laffont (1989) The
Economics of Uncertainty and Information, MIT press, Chapter 8.
13.AA.I
An example appears in Section V of:
Cho, 1-K.. and D.M. Kreps (1987) "Signaling Games and Stable Equilibria," QJE, 102:179-221.
In their paper, Cho and Kreps use the notion of "Riley outcome" to refer to the textbook notion of "best separating equilibrium".
The example itself
appears under the sub-title "Case B: More than two types," on pages 212-13.
13 - 21
CHAPTER 14
14.8.1
The answer is yes, and the argument is supplied in footnote 8,
immediately after Lemma 14.8.1 in the textbook.
14.8.2
Now, the program cannot be split into a ·minimization program and
afterward a maximization program since the principal is risk averse over n - w(n).
Therefore, letting u( ·) denote the principal's utility function,
the program becomes:
Max w(n)
s.t.
u(n - w(n))f(nleH)dn
(i)
(ii)
-
u -
v(w(n))f(nl eL)dn -
where constraint (i) is the participation constraint and (ii) is the incentive constraint (assuming that eH is the desirable action). Letting 7 and ll be the Kuhn-Tucker multipliers for (i), and· (ii) respectively, the Kuhn-Tucker FOC is:
which in turn yields: w(n)) v'(w(n))
u' (n -
1 -
•
Note that in this case the incentive constraint may not bind, i.e., we may have ll = 0. The reason is that due to optimal risk sharing it may be optimal for the agent to have enough risk such that (ii) does not bind.
14 - 1
14.8.3
[First Printing Errata: part (c) should end with "What effect do
changes in t/> and v (a)
2
have?")
Direct calculation ·gives: Eu(w(n),e)
= E[a
+ ~nle1 - f/>VAR[a + ~nfeJ - g(e)
=a+
=a (b)
~Elnlel- t/>f32 VAR[nle12 2
+ {3.e - f/>~ v
g(e)
- g(e) •
Optimal risk sharing will result in a fixed wage for the agent, and
maximizing the principal's profits ensures that this wage will exactly compensate the agent for his effort.
Therefore, the first-best contract with
observable (and verifiable) effort is the solution to: M~
I
E[x e) - g(e) ,
and tl;le FOC (which is necessary and sufficient) gives g' (e•) us the optimal effort level e•, and
(c)
= 1,
which gives
w = g(e•) is the wage.
As in section 14.8 of the textbook, the principal's problem can be
divided into two steps.
First, for a given effort level e' , the optimal
individually rational and incentive compatible compensation scheme (using the result from part (a) .above) is given by: •
Min
a
+ ~e'
a,fJ s. t.
. (i)
2 2
a + ~e' - t/>~ v
- g(e' ) il!:: 0
The incentive constraint (ii) implies that given e', f3e - g(e) should reach a maximum at e'. the FOC: (i)
The condition of the question allow us to replace (ii) with
g' (e'} = 13, which uniquely determines
~
given e'. .
We also know that
will bind, so f3 = g' (e') implies: a = g(e') + tJ>g' (e' )2v 2 - g' (e' )e'. 14 - 2
We
can now find the optimal compensation scheme by solving:
2 2
M~x · Elnlel- E[g(e) + (/>g'(e) tr
- g'(e)e + g'(elnlel,
which reduces to:
Given the conditions of the question, this is a concave program, so the FOC
2 which is necessary and sufficient yields: 1 - g' (e) - 2f1Kr g• (e)g"(e) = 0, which gives us: g' (e)
=
1
This implies that 0 < 13 < 1, which is
1 + 2t/#2 g' (e) •
"economically" reasonable because the incentives are not fully aligned with As t/i increases, the agent is more averse
profits due to optimal risk sharing.
to risk (through variance) and therefore 13 will be lower, i.e., lower incentives.
14.8 .. 4
The same happens as tr
2
increases.
[First Printing Errata: in the hint of part (b) it should read "vL
and v " and not "v and v "] H
1
2
(a) Lei pi!!: f(nHiei} so that p
= ~· 1
When effort is
observable, the principal will pay exactly g(e) for effort level e, so that:
2
1
25
3
3
9
ll:(e):: -·10 + - · 0 - -
1
>3 •
1 + .!..o = -·10 2 2
1 n(e ) = -·10 +
3
and therefore e
3
~·0 3
is optimal with a wage of
1 .
(b)
Let a contract specify a pair (vH,vL) where vkE v(wk).
For- e
implementable, three conditions must hold: (i) (ii)
·1
1
8
2
- > -·v -·v + -·v - 5-3 H 2 H 2 L
14 - 3
+
1
5
-·v 3 L - -3'
2
to be
calculations show that (ii) implies vH
2::
8
S
vH :s
~
+ vL , and (iii) implies
+ vL , and clearly both cannot be satisfied simultaneously.
be implementable we must have both (ii) and (ii) satisfied.
... ) 1 ( 111 -·v
2
1
-·v 2 L
+
H
i!::
1 -·v 3
For e
2
to
Rewriting both:
2
+ -·v H 3 L
•
or,
Both can be satisfied if and only if (c)
J.Ptrst prmtlns
err~a;
\oRe •Rg a£· "'ais part cf the tttteJtiaR shewfd 7
Pead: 91 Wfiaf effet'tS dO ehaug!!i (fi
We established in (b) above that e
f 2
and ~
2
rr- st.,.,f)./?.
ug r\
ft&Ve?".l "
cannot be implemented. To implement e
get the first best contract by paying w
:f!rrf/llltJ
=
116
9 . and the principal"s
expected profit is:
n(e3 ) To implement e
1
1 2 16 -·10 + -· 0 - 3 3 9
=
=
14 - 9"
we must have both individual rationality and incentive
constraints satisfied. Consider the individual rationality constraint, and the incentive constraint with respect to e : 3 (i)
~·v 3
.. ) -·v 2 (11 3
H
H
+ +
5 - - = 0, 3 L 3 1
-·v I
5
1
2 .
4
--=-·v +-·v - 3 • -·v 3 L 3 3 H 3 L
which together give: (vH,vL) = (2,1), or in terms of wages:(wH,wl) It is easy to check that incentives with respect to e
2
= (4,1).
are satisfied. The
principal's expected profit is: 1 33 1 2 2 n(e) = -·10 + -·0 - -·4 - -·1 = 3 9' 1 3 3 3
and therefore e
1
is optimal with the compensation scheme (wH' wL)
14 - 4
= (4,1).
3
we
(d)
When effort is observable, the principal will pay exactly g(e) for effort
level e, so that: n(e l = x·10 + (1-x)·O - 8 1 1
>2 ,
1
= -·10 + -·0 2 2 11
(e3 ) = ..!..10 3 e ...
+
~·0 3
~9 = ~9 < 2
-
•
·;s olt-""'...
I
""'
•
-c.,
'
.,
_..,H
"j
and therefore as x -> 1, -e.y is &::ptimal with a v.•ase cf w • !. When effort is not observable, the principal can still implement e above with n(e ) 3
= ~·
To implement e 1
1
2
8
-·v 2 H
(ii)
1 . 8 1 - -5-> x·vH + (1-x)·VL -·v + -·v 2 L 2 H
1 (iii) -·v 2
-·v - -5 2 L
1
H
+ -·v
2
L
8 5
~
0
1 3
'
satisfied.
v'8 ,
- - z: -·v
H
Supposing that (i) and (iii) bind, we compute that (vH,vL) = terms of wages:(wH,wL) = (
as in (c)
we must have (similar to (b) aboveh
(i)
+
3
(~.;), or in
144 16 , >. and it is easy to verify that. (ii) is 25 25
We therefore get:
To implement e the principal can use a scheme so that as x is arbitrarily 1 close to 1, the c'?st of implementing e will be arbitrarily close to 8, the 1 wage which will exactly compensate the agent for his efforts. take some number a > 0, and let x = 1-c where c -> 0. compensation values (vH,vL) ·= ([1 +
aJ·v'S,!I
+ ~ -
~ -J·v'S>. c
To see this,
Set the utility It is easy to
check that for all c, given e this scheme gives the agent an expected utility 1 of 0.
Furthermore, as c -> 0 we have that vL -> -111, so that the incentive
constraints will easily be satisfied, and as c -> 0 we can make c5 arbitrarily small so that both the individual rationality and incentive constraints are satisfied, and the expected wage that the principal will pay will approach 8, resulting in
n(e ) 4 1
2.
therefore, as c -> 0 the principal will prefer to
14 - 5
implement e , a higher level than the first best level of effort. 1
Assume this is not true, i.e., that the optimal incentive scheme is
14.8.5
as shown in Figure 14.B.S(a) below:
w(rr)
w{TT)
' I I
rr
TT
Figure 14.B.5(a) Then, if any profit n
E
Figure 14.B.5(b)
(n .n ) was realized, the agent will dispose of any 1 2
excess profits above n , and receive wCn 1
n e Cn ,n
1 2
>
>
1
> w(n).
Thus, for all realizations
>.
But this can be achieved
the principal will end up paying w(n
1
by the scheme depicted in Figure 14.B.5(b) above - a contradiction.
14.8.6
(a)
The principal's problem becomes:
- C-
R Min w ( R, C)
R •
s.t.
C
- - R C
(i)
R
- C- R C w(R,C)fR(Riec)fc(Ciec) dC dR- g(eC)
(ij)
R
-
C
14 - 6
- C-
R
R C
- -
.
where constraint (i) is the participation constraint and (ii) is the incentive constraint (assuming that eC is the desirable action).
Letting 7 and ll be the
Kuhn-Tucker multipliers for (i) and (ii) respectively, the FOC can be worked out as is done in section 14.8 and further algebra yields: 1
-v~,(r-w-;(=R-=,C~)~) = 7 + fl"
1 -
fR
f c
fR(Riec)
f
clcl ec)
increases, and the concavity of v( ·) implies that w( ·, ·) should decrease.
This makes sense because we want to suppress the
incentives of the agent to choose the revenue enhancing effort. . fc
should decrease.
increases. and the concavity of
v( ·)
Similarly, as
implies that
This makes sense because we want to strengthen the
incentives of the agent to choose the cost reducing effort. {b)
ln this case the principal can no longer use R as a variable in the
compensation scheme.
The intuition is straightforward: no compensation scheme
that induces the agent to choose ec will have
=~ >
0, and if : ; < 0 for some
values of R, the manager will dispose of some revenues. Bw
have BR = 0 for all values of R.
Therefore we must
Thus, the optimal scheme will be w(C) with
Bw < 0. (The FOC will be as in condition 14.8.10 of the textbook.) BC (c)
In this case no compensation scheme can induce the manager to exert
effort level eC"
14.8.7
That is, only eR is implementable.
For an analysis of this problem we refer to:
14 - 7
"Repeat~d
Rogerson, W. (1985}
Moral Hazard," Econometrica, 53:69-76.
It is worth mentioning that the conclusion of this model is counter-intuitive: The optimal compensation scheme is history dependent.
The reason is that in
addition to supplying the agent with correct incentives, the compensation scheme also serves as a consumption smoothing device for the agent.
For an analysis of this problem we refer to:
14.8.8
Dye, R. (1986) "Optimal Monitoring policies in agencies," The Rand
Journal of Economics, 17:339-50.
Let a e
14.C.l and
1 2
E~=1h.=l. 1
1
;\> 0,
The revelation principle still holds, so we can restrict
N ourselves to a menu of contracts of the form {(wl,e.)}. 1' Program 14.C.S in I= I
the textbook now becomes:
h.[ n(e.) - w.]
Max {(w. ,e.)} 1
1
1
I
I
s. t. (i)
w. - g( e., a. l I
1
1
Oil w. - g(e.,a.l 1
1
l!:
i!::
I
v
-1 -
(u)
Vi
w. - g(e .,a.) 'f/ i and V j=i J J J
There are N individual rationality (IR) constraints given by
(i)
above, and
(N-l)N incentive compatibility (IC) constraints given by (ii) above.
However,
an analog of Lemma 14.C.l implies that only one IR constraint, that of type
a
, binds. 1
To see this note that the IC constraint ensuring that type
not choose (w
e2 will
,e >. the IR constraint of type a1, and the assumption that
1 1
gee(·.·) < 0, together imply that:
and this can be repeated inductively.
Now that we have only one IR
constraint, Lemma l4.C.2 holds as before, and we have: w 14 - 8
1
- gCe
,e ) = v -l(u). 1 1
The analog to Lemma 14.C.3 is that two type of inequalities hold: e. :s e! ,
(I)
1
and,
I
for all i < N.
e. ::s e!
The analog to Lemma 14.C.4 is that
1
1
The proof of this
can be seen with a graphical argument as in the textbook extended to N types, or in a similar (yet more cumbersome) way to that of Appendix B in the textbook.
Moreover, we will only have the "downward" IC constraints binding.
i.e., (ii) above can be replaced with: (ii') w. - g(e.,e.) a: w. 1
1
1- 1
1
- g(e.
,e.) 'f/ i > 1 .
1- 1
1
To see this, note first that for all i > 2 wee can drop the IC constraints with respect to all j < i - 1 because: (1)
w.J- 1 - g(e.1- I.e.1- l)
(2)
w. - g(e.,e.) a: w. 1
1
il!::
w.1-2- g(e.1- 2,e.1- 1) '
- g(e. ,e.) , J- 1 1- 1 1
1
summing (1) and (2) we get that: w. - g(e.
1- 1
1
,e.
1- 1
) - g(e.,e.) a: w. I
1- 2
1
- g(e.
1-2
,9.
1- 1
l - g(e.
,e.).
1- 1
I
Rewriting this, and using the fact that gee(·,·) < 0 we get: w. - g(e.,e.) :r:. w. 1
I
I
- g(e.
1- 2
1- 2
,e.
1- 1
) + [g(e.
,9.
1- 1
> -
1- 1
g(e.
,e.)
1- 1
I
> w.1- 2 - g(e.1- 2,9.J- 1) > w.1-
2
- g(e.
1-2
,e.) . 1
To show that all the "upward binding IC constraints are not binding we can follow a similar manner as suggested at the end of Appendix B in the textbook.
14.C.2
The manager's utility function becomes u(w,e,e) = w - g(e,9).
It is
easy to show (similar to the textbook analysis where the st.ate 9 is observable) that the first best effort level must satisfy n' (e!) = g (e!,9.) 1
14 - 9
e
1
1
for i e {H,L).
Furthermore, when a is observable the manager has his
individual rationality constraint binding, that is:
or,
-
AWH + (1 - A)wL "" u + ;\g(eH,aH) +
(1 -
A)g(eL,9L) .
This, in turn, gives the owner expected profits equal to: '
En "" Mn:CeH) - wH] + (1 - AHnCeL_> - wL)]
= An:(eH) +
(1 -
AlnCeL_> -
[u
+ Ag(eH,eH) + (1 - A)g(eL,eL)).
(Note that due to risk uncertainty for both the manager and the owner, any pair (wH, wL) that satisfies the manager's individual rationality constraint above with equality,- will give the owner the same expected profits as above.) Now suppose that 9 is not observable, and the owner offers the manager the following compensation scheme: w(n) =
1t -
a, where
-
a "" An(eH) + (1 - A)n(et_>- wH] - [u + Ag(eH,aH) + (1 - ;\)g(eL_.eL)]. A manager of type i who faces this scheme will choose a level of· effort that maximizes w(n(e)) - g(e,a.), which is (recall that a is a constant): 1
Max e
n(e) - a - g(e,a.) I
and the. FOC is just the first best condition: n' (e!l = g (e!,a.). We only need 1 e 1 1 to check that the O\Yner makes the same profit as in the first best scenario, and that the manager will choose to participate.
First, since the owner will
pay w(n) = n - u for any realization n, he is left with a profit of a:, which is exactly his expected utility in the first best scenario.
Second, the
manager's expected utility if he accepts the contract is:
This is shown in the (w,n:) space in Figure 14.C.2(a), and in the (w,e) space in Figure 14.C.2(b).
14 - 10
u H= TI(e~) - Ol -g(~ ,E\J)
w(n) TI(etf)-Ol
W u L = n(eL) -at -g(et ,Sr_) n(efj)-at
-- - - -- ----
Uft
n<ev-&
' n(et) -Ol I
I
n(et)
n(etj)
------
I
UL
n
'
I
I
'
etf
et
-Ol
-Ol
Figure 14.C.2(a)
Figure 14.C.2(b)
The revelation mechanism which yields the same outcome is just the two points which result from the compensation scheme above.
14.C.3
(a) Assuming the same conditions on
v( • )
That is, given a as above,
and
g( · , • )
the program for
the optimal contract under full observability is:
Letting 'l denote the Kuhn-Tucker multiplier, and noting that as in the textbook analysis the assumptions imply that the FOCs must hold with equalities, we get the following four FOCs:
=
(i)
-). + 7(Av' (wH)]
(ii)
-(1 - ).) + 7((1 - ).)v'
0 ,
(iii) ).w' Cejil - 'lAge(eH,aH) (iv}
From
(i)
(I -
).)w' (el_) -
(wL)] = 0 ,
= 0 ,
7(1 - ).)ge(el_,BL)
= 0 .
and (ii) we ·get that wL = wH. This makes sense because wages and
14 - 11
e
'l[~v'
(i)
-h +
(ii)
-(1 - h) +
CwH)] = 0 • -r[(l - ;\)v' (wl,ll
(iii) h1t' (eH) - -r~ge(eH,eH)
(iv)
(1 -
=0
,
= 0 ,
hln' (el_) - 'lCl - h)geCeL_,eL) = 0 .
From (i) and (ii) we get that wl_
= wH.
This makes sense because wages and
disutility of effort are separable now, and the risk aversion is only on monetary income so that an optimal contract will have both wage levels equal. Using this we can rewrite (iii) and (iv), for states H and L respectively, as: n'(e!) = 1
v
'(
1
,rge(e!,e.).
w.
1
1
1
This again makes sense because having wL_ = w;{ implies that eL, < eH. (b)
This contract is clearly not feasible when
is that wi. =
wH = w•.
a
is unobservable.
The reason
and eL_< = eH' will cause the manager to choose the L
pair if the H state occurs, i.e., the H type will misrepresent.
14.C.4
For an analysis of this problem we refer to:
Hart, 0. (1983) "Optimal Labor Contracts under Asymmetric Information: An Introduction," The Review of Economic Studies, 50:3-35.
14.C.5
[First Printing Errata: the first line of the question should read
"..• in Section 14.C would not change if... ") The objective function in U4.C.l). and (14.C.S) in the textbook changes to:
and the first best effort levels are determined by: i
E
{H,L}, and the condition
nH(e)
2::
11:~ (e!) 1 1
= g e (e!,e.) 1 1
n:i._ (e) > 0 ensures that eH > eL_ , and
this promises that the analysis in section I3.C would not change. however • we have
0 < '1tH(e) <
for
If.
n:i._ (e), it may be that eH < eL_ • and in this
case the analysis of section 13.C would no longer apply. (Note that the
14 - 12
-
uL= u,
w
lsoprofit n L
I I I
e~
et
e
Figure 14.C.5(a) We present a brief graphical presentation of two possibilities for .optimal contracts when eH < eL."
If the menu {(wH;eH),(wl,eL)} is incentive compatible
we must have eH > eL (see Figure 14.C.4 in the textbook). Now start from the
-
situation (wL,eL)=(wl_.el_) where wl_ is chosen such that uL = u. Becausel > eH' .
the best contract for the H type will have eH = eL = eL since this has the least distortion of the H type's effort level. pooling contract with (w,e)=(wl_,eL_J.
This means we will have a
This, however, cannot be optimal because
a first order reduction of w and of e along the indifference curve of the L
-
type, uL= u, will cause no first order change for the L type yet will cause a first order benefit for the H type both with respect to a lower wage, and a better effort choice (closer to eH) . not too large, we will end up at
a
Therefore, if the probability of aH is A
A
pooling contract of the type (w,e), where
eH s e :s eL_. and w is chosen so that the L type's individual rationality constraint is bonding. 14.C.S(b).
An illustration of this situation is given in Figure
If, on the other hand,
~H
is large enough, then it will
be worthwhile to further distort the L type contract so that we get eH= eH' _and eL <· eH.
An illustration of this is given in Figure l4.C.S(c).
14 - 13
-
-
w
.w
w --
-~ WL
-
I I
1
I
'
e
eL
Figure 14.C.5(b) 14.C.6
I
I
I
'
e•H
et
e
Figure 14.C.5(c)
This is a problem of monopolistic screening, and the firm will make
positive profits in contrast to the situation in competitive screening described in exercise 13.0.1.
Assuming that a worker of type 9. has a I
marginal product of 9.(1 + t) (recall that we took 1J. = I which does not I
change the qualitative results), then the monopolist's problem is:
s. t.
WL - dtl,9L) ~ 0
(j) (ij)
wH - c(tH,eH)
i!:
0
(iii) wH - c(tH,9H) i!: wL - cCtL,9H) (iv)
wL - c(tL,eL)
i!:
wH - c(tH,eL)
As we can expect, (i) together with (iii) and with the assumption that ct Ct,9) < 0 imply that (ii) is redundant. 9 14.C will apply directly.
satisfying tH
= tH
All the conclusions of section
That is. we will have the optimal contract
(the first best value), tL < tl_. u(wL, tL,9L)
u(wH, tH,eH) > 0. In summary, a monopolistic screening model an equilibrium, it is never first best in the
sens~
that tH =
~H
=0
and
will always have and tL < tl_,
and the firm makes positive profits. In contrast, the competitive screening 14 - 14
mode) of exercise 13.0.1 may not always have an equilibrium, we may have a first best separa.ting equilibrium, if we have a separating equilibrium that is not first best then tH > tH
I4.C.7
and tL = tl_. and the firm makes no profits.
For an analysis ·of this problem we refer to:
Tirole, J. (1988) The Theory of Industrial Organization, MIT Press, Section 3.3 (pp 142-152) and section 3.5.1 (pp 153-154).
14.C.8
(a)
The indifference curves of the two types and the firm's isoprofit
curve are depicted in Figure 14.C.8.
w
lsoprofit A
A
A
I I I I
A
v A
w
-
--
v
er
v -9s
A
p
p
Figure 14.C.8 The formal problem that Air Shangri-La would want to solve is: Max {PT. WT,PB' WB}
s.t.
(i}
aTT + WT :s v
Cii)
a8P
8
+
w8
(iii) 9 TT + WT
(iv) (v)
e P + 8 8
w8
:s v :s 9 TB + WB
:s e PT + WT
P ,w ,PT,WT 8 8
14 - 15
8
i!::
0
(b)
In the program above, constraint (i) and (iv), together with
imply that constraint ( ii) is redundant, so it is never binding. (i)
e8 < aT'
Constraint
must therefore bind for if it would not, we can reduce PT and P
and all the remaining constraints will still be satisfied.
by c > 0,
8
This implies that
tourists will be indifferent between buying and not buying a ticket.
(c)
Assum~
that {(PT,WT},(P
8
,w8 n is
an optimal, incentive compatible
contract, and assume in negation that increase P by : 8
w8 >
0.
Now reduce
w8
by c > 0, and
so that the B type's utility does not change, and the firm
B
earns higher profits from the B type.
We need to check that the T type will
not choose this new compensation package.
Indeed,
c
+ - ) + (W e8 B
contradicting that {(PT,WT),CP
8
,w8 )}
contract. Therefore, we must have
is an optimal, incentive compatible
w8 = 0.
If, in an optimal contract, the
business travelers were not indifferent between (PT,WT) and CP slightly raise P (ii)
.w8 J,
we could
and all the constraints would remain satisfied (recall that
is redundant), and the firm would earn higher profits from the business
types.
(d)
8
8
Therefore, in an optimal contract we must have the business types
The trade of that the firm faces is: By lowering PT and increasing
WT so as to keep the tourists indifferent between buying a ticket or· not, the firm can increase PB (recall that
w8
= 0).
From parts (b) and (c) above, we
can conclude that if the .firm raises WT by c and lowers PT by ~T so that the tourists remain indifferent between buying or not, then to keep the business types indifferent betwe.en their package and the new tourist package, c(9T-9B) it can increase P by 9 9 . 8 T B
Since this trade off is linear, it is true no
matter where we are' in the .(p, W) space, and therefore it will be profitable if
14 - 16
and only if:
• or, 1 - h
Note that this is independent of the cost c, because this is a revenue trade off (the co_sts are the same for both types).
Therefore, the price
discrimination scheme can take on two forms: (1)
If
).
, then only the high types will be served and the
1 - ). <
scheme will be (this assumes that c < .!_. e . 8
(2)
If l
~
>
9
T
).
- 98 98
, then both types will be served and the scheme will
v be a pooling scheme with (this assumes that c < -)· e . T •
Since the direction of the inequality in the condition above determines the •
type of ·scheme, scheme:
i~
is easy to see how changes in h, aT' and e will affect the 8
If the proportion of 8 types is large enough (h small enough) the
firm will choose to serve only the 8 types. If the B types suffer less from prices (9
8
is smaller) then the firm is more likely to serve only them.
If
the T types suffer less from prices (9T is smaller) then the firm is more likely to serve them as well as the 8 types.
Changes in the cost c are
discussed in part (e) below.
(e)
v As long as c < -
e8
•
, and we are in case
(1)
as described in part (d)
above, the firm will decide to serve only the business types.
14 - 17
If, however, we
v v < c <, then the_ scheme described in (d) aT eB
are in case (2) above, and
above cannot be optimal because the firm is losing money.
In such a case, the
firm will choose the scheme described in case (l) of part (d), and serve only the business types.
14.C.9
(a)
the firm will choose not to operate at all.
If c
The monopolist will offer the individual a policy that fully
insures him (optimal risk sharing) and keeps the individual at the same level of expected utility.
That is, if we define
the optimal insurance policy has c
(b)
1
= c
2
u
E
eu(W - L) + (1 - e)u(W), thtm
= u -l(u}.
The monopolist will offer an optimal screening contract of the form
L L
H H
1 2
1 2
Hc ,c l,Cc ,c )} that solves: Max
s. t.
}.((1 -
(i) (ii)
(1 - .;\}[(1
H
H
(1 -
&H)u(c ) + eHu(c ) 1 2
(I -
L L eL)u(c ) + aLuCc ) 1
2
-
~
u
iii';
u
(iii) (1 -
(iv) This is again a standard monopolistic screening problem and the standard analysis applies.
Solving this program (again, (ii) and (iii) will be
redundant) will have the H type fully insured, and the L type not fully insured.
A graphical analysis is given using Figure l4.C. 9 below.
-
Both
-
types start at the point A, with utility levels uH and uL respectively.
If
the monopolist would try to insure both types by offering the points B and C to H and L respectively, the (unobservable) H type would choose policy C instead of B, that is, the points B and C cannot be part of an incentive compatible contract.
14 - 18
-
UL
•
- UH UH
W-L
-.. . -------/
~-
-~~::---== A,
w
Ct
Figure 14.C.9 If the monopolist offers points A and B, then the H types would choose B (since they are indifferent),
th~y
would be fully insured, and the risk
.
neutral monopolist would make profits from their choice.
The L types,
.
however, would prefer the point A to B and no profits would be made from them. If the proportion of High types is large enough, then the monopolist will find it profitable to slightly insure the L type at the cost of raising the utility •
of the H type.
This means that the optimal contract will look like the two
points D and E for the H and L types· respectively.
The mathematical analysis
is straightforward, and results in a situation common to monopolistic screening (or hidden information):
The H type will be at the first best
insurance level and his participation constraint is not binding.
The L type
will be under insured (second best distortion so that screening is possible and profitable), his participation constraint will bind and his incentive compatibility constraint will not. (Note, that the pair of points A,B may be optimal if the proportion of L types is small.
This is parallel to the hidden
information case where the H type is at the first-best observable point with
14 - 19
no surplus, and the L type has eL = O). (c)
The difference is who gets the surplus.
In chapter 13 we discussed
competitive markets, so that the insurer was left with zero profits and the individuals had utility levels above their reservation utility.
Here, the
monopolist makes positive profits and at least the L type has no surplus.
14.AA.l
For an analysis of this problem we refer to Proposition 5 in:
Milgram, P. (1981) "Good News and Bad News: Representation Theorems and Applications," Bell Journal of Economics, 12:380-91.
14.AA.2
Sufficient conditions for the first order approach to be valid will
promise that the agent's op,timization program yields a unique solution.
Given
a compensation scheme (wH,wL), where wi is the compensation when profits ni -are o!Jserved, the agent maximizes: Max e
f(nHje)·v(wH) + (1 - f(nHje)}·v(wL) - g(e) .
The FOC will be sufficient if the SOC is satisfied, i.e., if
So, if v(wH) - v(wL) > 0, and f ee(nH Ie) < 0, then the first order approach will be valid.
The first inequality is implies by MLRP, and the second is
guaranteed by concavity of the density function.
For more on this we refer
to: Rogerson, W. (1985) "The first-order Approach to Principal-Agent Problems," Econometrica, 53:1357-69 .
... 14.AA.3
--
The program to be solved is:
14 - 20
s.t.
(i) (ii)
wH - g(eW8H) ~ 0
(iii)
wH - g(eH,aH) ~ wL - g(eL,eH)
(iv)
wL - g(eL,eL)
i!::
wH - g(eH.eL)
and we have already established in the textbook that (ii) is redundant (Lemma 14.C.l}.
We proceed with two straightforward claims for the program without ,.
constraint (ii): Claim
~
) ~ \
...
'• . ... '·
...'
Constraint (i} must bind at a solution.
.,:.' ~~·.
Proof:
Assume not.
Then reduce both wL and wH by c > 0 such that' (I) Is
still satisfied.
This will not change the remaining two constraints
and it will increase the objective function, a contradiction to •
being at an optimum. a
-
Claim .2:
Constraint (iii) must bind at a solution .
Proof:
Assume not. satisfied.
Then reduce wH by c > 0 such that (iii) is still
This will not affect constraint (i), constraint (iv)
will only be further relaxed, and it will increase the objective function, a contradiction to being at an optimum. c We now proceed to solve the "modified program" which is the original program without constraints (ii) and (iv), and with constraints (i) and Ciiil binding. We will then proceed to show that at the solution to the modified program, constraint (ivl will be satisfied.
s.t.
(i)
The modified program is therefore:
wL - g(eL,eLl =
(iii) wH - g(eH,aHl
o
= wL
- g(eL,aH)
Letting 7 and J.l be the Lagrange multipliers for: constraints (i) and (iii)
14 - 21
respectively. Then, assuming an interior solution, we get the FOCs:
From (3)
~e
(1)
hn' (eH) - IJ.ge(eH,eH) = 0
(2)
(1 -
(3)
- ). + fl
(4)
- (1 - ).) + 'I - fl
).)n' CeL) - .,.ge(eL,eL) + llEe(eL,eH) = 0
=
0
=0 -
have that ll = h,· and plugging this into (4) we get that 'I = 1.
Substituting for ll and h in (1) and (2) we get-the well known conditions:
n' (eH) = ge(eH,eH) ' ).
n' (eL) = 8 e(eL' 9 L1 + 1 - ). · [ge(eL' 9 H1 - 6 e(eL' 9 H)) · wL and wH are then computed using the two binding constraints. A
A
A
A
solution to the modified program by {(wH,eH),(wL,eL)}. that this solution satisfies constraint (iv).
Denote the
We are left to show
We proceed with tow claims
regarding the modified program.
..
... Claim ;t_ Proof:
At the solution to the modified program: n(eH) - wH
.. Assume not.
....
~ · n(eL)
- wL.
...
Then the firm can offer the pair (wL,eL)
t~
both types.
The L type will clearly accept it, and since we have shown that (iii) is binding then the H type will be indifferent between this
.. ..
pair and his original pair (wH,eH).
This, however, will raise the
profits earned from the H type while leaving the profits earned from A
A
"""'
"""'
the L type unchanged, a contradiction to {(wH,eH),(wL,eL)} being a solution to the modified program. c ~C!!:la~im~
i:. Constraint (iv) must be satisfied at a solution to the modified program.
Proof:
. ..
Then, the firm can offer (wH,eH) to both types, the H type will clearly ac.cept as he did before, and the low type will prefer this
14 - 22
-
to (wL,eL) given our negation assumption. A
... ...
Furthermore, (wH,eH) must
A
A
A
satisfy (i) because (wL,eL) did and the L type prefers (wH,eH) to A
...
A
(wL,eL).
..
But now the firm is earning profits of 1t(eH) - wH from
both types, which is (from Claim 3) at least as good as the profits
.. .
earned from the L type through (wL,eL).
The assumptions on Jr( ·) and
gC • , • ) guarantee a unique solution to the modified program, a contradiction.
Therefore, constraint (iv) must be satisfied at a
solution to the modified program.
14 - 23
CHAPTER 15 •
15.B.I
(a) Tne budget constraint implies that
for each i = .1.2.
Suppose that the above weak inequality
held wiih strict
:!:
•
inequality <, then there would be (x .,x .J e B.(p} such that Cx .,x .> >-. 11 2 l 1 11 2 l 1 •
(xli(p).x ;Cp}}, because the preference of 2• nonsatiated.
fer ea:n
•
1
consumer is locally
•••
Tn.is would contradict the fact that (xli(p),x i(p)} is the 2
demand at p.
•
eac~
We must thus have
= 1,2.
Suz,•ming over i = 1,2, we obtain •
-t..)
= 1
(b) If the market fo:- good 1 clears at p•, then Iixli (p•) -
that By Pi > 0, tt-.Js imclies •
0 by the equality of (a).
L::x2 i(p*)
-
-w
2
= 0.
0 and her:.ce
Hence the market for good 2 clears at p• as well and p•
is a Wa!rasian equilibrium price vector.
15.B.2
As we saw in Exal'ltple lS.B.l, the offer curves of the twc consu!:le:rs •
oc1(p) = (o:p·t..)/Pl' OC2(p) =
(1 -
cx)p·t..)/Pz),
{~p·t..)2/p1, (1 -
,g)p ·w2/p2).
de!Ha::ld f cr good 2 at p is thus
\ cf' Exe:-cise 15.8.1, equating this to \ J B_.,•. '·o ec uili b:- i un1 price ratio •
..... ,_ • •
Ll.o._.~
(1 -
-w
2
= t..)
21
a'"21 + 13""22 a}w + (1 11
+ w
22
~Jw
12
gives the ·
By substituting
into the cffer cu:ves, we obtain the equilibrium allocations:
equal tc
15-1
OC, (p•) is
..
1 - (X ), (1 - a:lw + (1 - f:3}w 11 12 •
and oc (p•) is equal to 2 1 -
(wl2w22 + (l - a.)wllw22
f3
(1 -
•
It is eas.-• to check that
a{pVp_2J/aw11 < o, aoc cp•J/aw 11
11
aoc21 cp•J/aw11
>
•
o, - a:H 1
(1
> 0,
=
• •
• ana,
_
•
,..-.. 51.........
-w
0 and
2
•
IS
constant,
15.8.3 Let ;:* be a Walrasian equilibr-ium price vector and x! be the demand 1 of consume::- i (i = 1,2). Since the pr-eference of consumer 1 is locally
. . . ncr:.-satratec, tne
upper contcur sets
<x
1
E
102 "'+: x
•} r· b ~ x 1es on or a eve the 1 1 1 ..
strictly above the budget line.
•
.&.
Symmetrically, the upper contour sets {x
2
E
lies on or below the budget line and the st.-ict uppe::- contour •
x
2
>-
2
x2}
. ,., .. lies strictly above the budget 11........
sets
····o 1,. yy
I
>-
2
fx ••
Henc~
x2} do not inte!"'"sect; the other
2
E IR : x +. 2
1
I
•
--- -
.
Wa1-·:-::; c:! a~ e,... d 11· b ...r·um allocation ._,• ~.
x• = (xi ,xi) is Pareto
fignre below.
15-2
the two
Hence the •. T oo~o!ma •. •
See the
p*
0:
•
x*
•
•
•
•
Figme 15.B3 1S.B.4
(a) Here is an example of a:1 offer c::..-ve (of consumer 1) with the
gross su.bstit'..zte -.-"'pe,..ty· . . Tne dotted curve :.s the indifference curve of •""".. "" consn!!H~r
l that goes through cu .
......__ ___.______________ 0: •• •• ••
•• • •• •• •
•
•
•
'
•
••
.......... •••
•
• ••
Figure 15.B.4(a}
(b) As we discusse-d in the text, at C..
-"ons··~e-• ~,;. ..... .;, :a
• ....... .,.t..__~se""· _.. and, conversely, an.y i::.:ersection of the offer curt,·es
at an allocation di;'ferent from w co;-;:-esponds
. ... Ver
,. •· •.t.J
. tn-·
.C:.L
. ·-· :.. .. : -
:Li
equilibriUl'll.
•
•
m oraer to
offe:- curves intersect only one: (not counting the
intersection a: the initial endowwentsJ, it is t:::.erefore sufficie::;.t to show •
that there is only one equilibriu.."'l pr1ce ratio
15-3
So, let p•
•
•
oc12
< OCl!(pi,l) +
oc12(pi,l)
equilibrium price vector.
=
-""r
Hence (pl'l) is not an
Symmetrically, if p < Pi, then there {P 1' 1) is 1
not an equilibrimn pr-ice vector either.
Thus the offer curves intersect only
once. •
(c) Here is an exa.-nple of a normal offer curve that does not satisfy the •
•
gross suostltute property. •
0:
•
(,J
•
o.
Figrue 15.B.4(c)
• •
(d)
Here is a:: exarnple of a preference, an initial
correspcndi:lg offe;- curve that is not normal.
0:
•
•
I (&)
• •
Ot
L_--------~==~·======~;~ Figrue 15.B.4(d.l)
15-4
As for the second statement, we consider the case in which the price of •
com.1nodity 1 inc;:-eases.
(The case in which the pr-ice of commodity 1 increases
can be symmetrically proved.)
> Pr
Assu."lle that ocli(pi,p
Let (pl'p J be the initial price vector and Pi
2
l > ocli{p1,p 2 l and oc2 iCpi,p2 l < oc21 (p1,p2 >. 2
It is sufficient to prove that if commodity 1 is not inferior, then co::runodity •
•
(That is,
Pi0Cli(p 1,p 2 l + p oc i(pl'p ) < Piwli + p 2w il. 2 2 2 2
Since the relative price of
corr...-rnodity 2 has decreased, this a.TJ.d
oc2 i
that commodity 2 must be inferior-.
This completes the proof .
< oc i(p ,p ) together imply 2
1 2
•
•
Assume
(e)
•
the offer curve of consumer- 1 is normal cna tna-: of consu"J."e:"' •
the gross substitute property.
If the initial endov. ."ments
constitute a Walrasian equilibrium (and preferences :=:re strictly convex), +'
... ne two o:fer curves intersect only at the initial endowments, because
• •r-e•· are '-·•
f
•
• th 1n • e upper conLour
.-1
ccnta~ne~
+
se~s
01.:-
w anc• 1
~
2
So S'Jooose that • •
.
. . . , _,.,l*\_ • . , .... ;-•s t,...._ . ,,_. "l"" . ·~ ... 1-··x·· e -au_ ....... , ........ .:. . do not constitute a Walrasian • '··-·&;.. c:. . . '
A
1.. ...... -
'
we
,
• ............ 'tl"l'
... .:. ... ""
.
&-1·-s· need to establish the following assertion: •. •. • ? p and p' are equilibr-ium price vecto:-s, the:-., fer- eve:-y i = .L,-
eve:-v l = i, 2, we have
-
•
In fact, suooose that we have (OCl.(p) - "'t·)(OCl.(o') - wl .) • • 1 I • ~
•
•
1
~
0.
Since the
initial endow ,.. e::ts de not constitute an equilibrium, the abc•:e weak ineaualitv . -
i:s satisfied with st:-ict inecualitv. . -
and OC c. ( o' } '-l •
•
w.. r:rus: be t!
1A,_.; c..--
-a-: riD1 - • ....
• •• ... ,. r"QS:-:r' • ..,.J.Y-
•
and the other- must be negative.
. weaK
. axJ. om c:
cr.a;" ,_._e_, .. -.;. -.;. s.a...,_, OCi(p) and OCi(p') must be outside the other budget • •
constraints, as shown in the figure below:
•
•
15-5
~
I
(h
•
OCi{p) •
•
OC;(p')
(&)
•
•
...•
01 •
. ~
F1grue b.B.4(e)
"
I
.
1:1 !he Ed.s;zewor~h box, however, this imolies that the offer- curve of the other -
ccns:.Jmer does not satisfy the weak axiom.
Tnis is a contradiction.
Hence, •
o~.·e~~ f ~-• • ~
'
D
a;""!~
1·
·•-
w•
t
~!·s·
-
•••'-'
L
have
relabelling the· indexes of the commodities if necessary, ·,•,;e can ass...une that
oc11 (p)
- w
11
> 0 for every equilibrium price vector p.
)'.;ow, let p• = {pj.I) be an equilibrium price vector such that if p = £p ,1} is 1 •
-
•
c.ny ctner-
<
.. 1 • •
•
eq:-.ll!lCrlw~
____ ..
,.... •• -,. . • .;. ...... " ·"'·!. c:. ~ .i. ...•t '
P.' . l
orice vector, then o• < p .. . . 1 l
For any p
p~
with
•
we have either •
or
Sin~e OC
•
11 --
ccnstramt. applies.
(p~) -
c.>
1 r:us
if the second case applies, •ne"' ............
11
the second case is actually impossible and the first case
Or:. the othe!" hand, by the gross substitute property, we haYe ,.,,.. .. w = H ..-··-1
•
•
•
• • •• • r ~ "..,.lU'ffi·
p lS net a:-. --.• ···-a. At'""!~.... ·,
.. p:-ice vector.
Thus Thus the Walrasian equilibrium price
15-6
'Vee'tor p• is unique.
Therefore, the two offer curves intersect only once, •
except for the initial endowments.
(f) Here is
an examole in which two normal offer curves intersect several
-
times. •
•
OC:---
• •
c.>
-
•
I..,
Figure 15.B.4(f)
15.B.S
First, we can derive from the quasilinearity that
-Q
oc21 (pl'p 2 )
r
=
The budget constraint the:1 implies Hence the consumer 1 is as given in Example 15.B.2.
offe~
curve of
We can similarly show that the
offer cur-ve of conS'..tmer 2 is also as given in the
ex~"''lple.
Rearranging the
equali:y of the total demand of the second good and its tot:a! supply and s:.rbsti tuting r = 2
8/9
- 2
l/9
. , we obtam
•
-- (28/9
-
21/9),\D. /
Tnen p /p = l/2, 1, 2 are solutions of this equation, 1 2 • -,,..e P .o. ..,...,
15.B.6
0
- 1 -
2
- 1l.
hence equilibrium
• ra-l"S .. L V ..
We cc..'1 obtain the offer curves from the first-order conditions of the
utility maximization problem:
15-7
-1/3 •
P2
-1/3
12137 2/3 (( )Pl
2/3 (12/37)pl
Set p
= 1 and Ytrite q = p
2
1/3 1
+ p2
, then q
2
-1
2 q + 12/37
-
•
q
5
12q
3
oc11 (p)
+
oc
(12/37)c• •
+
(12/37)q
+ C37/12lq
3
+ q
2
2
+ 1
; 12137
~5~------~--~~~3~~--.
q
We can check that
-l/3 ). • Pz
12
+ (12/37 ..:+ 37/ 12)q
Cp) =
+
q
1 if and only if
-7q 2 + 37q - 12 = (q - I)(4q - 3)(3q - 4) = 0. - ..,
Thus q = 1, 3/4, 4/3.
Hence the equilibrium price ratios a:-e given by ...
•
lS.B.i
We shall orove that the set of Pareto ootima! allocation looks like a • •
curve in the Edgeworth box.
More precisely, we show that
~~e::-e
is a
c:1e-to-o:1e, co::ltinuous mapping from a (non-degenerated) bounded closed in:e::-·;al of ~ into the Edgeworth whose image is equal to the Pareto set. is sufficient for the first assertion of the exercise.
•
-
1•2 • let
-
u.( • ·) be a utili:v • function of consumer i.
It is continuous, s;::-:r-.glv .
•
•
--
-. n. . .... Fo::- e __
This
mcnctcne, and s:rict1y quasiconcave.
'!:
-w}
~
-w
is a (no:.1-degenerated) ·closed bounded interval.
-
X
1
) ?:
compac:ness of {x : 0 1 •
•
SO!UtlCn.
0. :S
Denote it by [o ,o J. 0 1
This maximization problem is feasible ar.d, by the x
1
~ ~. u (~ - x } a: o}, there is at leas~ one
2
1
We shall now prove that the strict quasiconcavity of the u.( ·) 1
impli~s
that such a solution is unique.
Let x and 1
xi
be distinct solutions,
We can assume \lr·ithout loss of Then, by continuity and strong
15-8
city, ther-e exists u
2
(w
~
unique A E {0,1] such that u 1
1 1
2
A(W
u (x l = u £xi> , we have
-
(A(W
•
- X )) = 1
-w
- x ) ~ 1
Hence -
Thus u
(w
2
X )
1
+
Ther-efor-e (l/2}x
-
feasible and attains a higher utility than x
1
and
xi,
0/2)xi) is
+
1
a contradiction.
•
2
ther-e must be a unique solution, which we denote by tp,(o) e F. . +
J.
..
•
mappmg rp:
=
-w}
Thus
Deiine the
bv -
It is now sufficient to prove that tp{ • ) is one-to-one and continuous,
and its image is equal to the Pareto set. follows f;om its const;uction. u
2
(w
l
with
As for being one-to-one, note that·
-w
- cp (15) from consumer 2 to 1 would in._. ease the utility 1
level of consnw.er l, a contradiction. c:!ntinuity.
It thus remain:= to verify the
For- this, it suffices to prove that tpl ( ·) is continuous.
•
oe a seoue:.ce m •
Note first that, by continuity, u •
u,(x,) ~ u Cc; £c5)).
1 1
.1
.L
n, we ca"l find
~
2
:r-
(c.;
x' ' 1
-
such that u
o, we obtain x
For the
se~ond
-
2
·m T
(w
( o~n) 1
By strong monotonicity, fer any sufficiently large
- X.) I
Let
,., · 1'! prove t'nat 1'f ne sna
•
h~nce
the Pareto set
!c
= o; otherwise, by strong monotonicity, a small transfer
- rp.(o)}
c~llinear
The equality
2
(w
-
n}
XI
~n
~ a
.
Thus
ana
= tp Co). 1 1
assertion of the exercise, it is sufficient to prove that
the prefe:-ences of the consumers are homothetic and the Pareto set ever
C'..lts the diagonal in the interior- of the Edgeworth box, then the Pareto set ~ust coincide with the diagonal.
optimal a!lo::ation on the diagonal. p:-eference (De;'inition 3.B.6),
•
(c5w,
Let (~w. (1 -
cS)u}
e (0,1)) a Pareto
By the definition of a (1 -
o)w)
15-9
homo~hetic
is Pareto optimal for every o
E
By str-ong monotonicity, (O,w) and Cw.O) are also Pareto optimal.
({).1).
•
Hence every allocation on the diagonal is Pareto optimal.
By our- previous
r-esult (the existence of the one-to-one, continuous mapping rp{ •) ), the diagonal exhausts all Pareto optimal allocations.
Hence the Pareto set is
equal to the . diagonal.
•
Suppose that the pr-eference of consu."'tler- i {i = l,2i is represented
15.B.S
•
•
by a utility function u.: IR l
2
+
~ tR of the quasi-linear form u.(x.l = x 1• 1
~.{x .J,
wher-e rp.:
~
strictly
~anatone
and str-ictly concave.
1
an
2l
1
irr~~ediate
+
-+ IR is continuous.
21
= xZi
and i.
•
In fact. its strict monotonicitv is •
(0,1).
E
l
We shall first prove that ¢:( ·} is
consequence of that of the pr-efer-ence.
conca•Ti::y, le: x
1
+
As for- the
.. SL.!lCt. -~
By definition,
Hence, by the st::-ict com•exity of the prefer-ence,
.> 11 2
u.(.\(¢.Cx • .) - rp.(O)} !
2l
1
+ (1 - i.)(rp.(x
1
> .\:J.(CI.(x • .) !'l 1
-~..(0), .,..1
2
x
2
.J + 1
¢.(0)), i.x . + (1 - i.)x ' .) 1 21 21
.> 21
(1 - .\)u.(A...(x 1.,..1
-~..(0), .,..1
x • .>. 2l •
This is
e~uivalent •
to
¢.(.\x . + (l - i.Jx • .) > .\lf>.Cx .> + (1 - .\)¢.(x • .). 1 . 21 1 2l 1 2 i. 21 •
•
•
Thus o.( · j is str1=tlV concave. . 1 • •
is continuous and str-ictly concave.
Hence there exists a unique maximum x2
1
b order to ve;ifv the assertion of. the exercise, it is nov; sufficient •
.
.
'l'n "'- pro,'!"a ....
i:: the
• •"'=~
x IR
""'·--
i:J~:erio:-
++
is a nanwasteful feasible allocaticr:
of the Edgeworth box and x
optimal. In fact, let x be as such. fer e•;e;y >..
2
E. (0,1),
21
= x21'
then x is not Pareto
Then, by the strict concavf.ty cf q;( • ),
we have
15-10
E
Far" each i = .1,2, define ~i.i
E
IR as
- ¢i((l- A)X2i + AxZi) + ~i(x2i) + (l/2}(q.>((l - A)x21 + Axzll - q;(x2I))'
then oA.i + oi.
Since xli > 0, we have xli +
continuity.
sufficiently small i. e (0,1). u.(x . 1
11
f)(·), and o~.i
= 0 by the definition of 2
+
o~.l
-+ 0 as i. -+ 0 by
> 0 fc:r each i for a.'ly
Moreover,
o.,.n.!.,
+
o.,.n.l
+ lj>.((l 1
•
> x . 1l
+
-
¢.Cx J = u.(x .• x .l. 1 21 1 11 2 l
Thus x is not a Pareto optimal allocation.
The figu:-e below depicts the set
of the Pareto optimal allocations in the interior- of the Edgeworth box. [It is wcr-thwbile to note that if the
c~ns1.::npticns
sets were taken to be
(- co, c=} x :R .as in .De:inition 3.B. 7, then the Edgeworth "box" would have an
infinite length in the horizontal direction a::d the assertion of this exer-cise could more easily be proved, without the interior-ity assumption or an explicit Use
0~ .I
with x
x• 21
1 : .... f 't•-.-· ... ·n· e u+' ... " .. .:.. ~""'"'""'"ens .
-
.&. ..:. "" "
21
;=
The reason is that, the:.1, for any allocation x
xZl' there wouid always exist a feasible allocation x' with x2 = 1 -
. p a.-eto su;:enor . .a ... ~ lS to x.
t~-·
nonnega:i\•ity
•
This is r.ot always true when we have the
consrrai::~s.J
•
···-···
Figrue l5.B.8 •
15-11
In connection with the discussion of Chapter 10, this fact implies that .
in the absence of wealth effects, a Pareto optimal allocation (in the inter-ior of the
Edg~wo:.-th
determined.
box) of the non-numeraire commodities are uniGue!y
The differences in the consumers' utility levels at diffe:-ent
Pareto optima! allocations (this time including -the numer-aire} can all be •
generated simpiy by redistributing the numeraire . •
--
•
The cffe:- curves of the two consumers are rather t;:-ivial when the
15.B.9
prices of the two commodities are both positive. E
IR
2 ++
:
X
When one of the two
pee:
com.inodi~i~s
has zer-o price, the offer curves are given by {(x
OC (o , u. ) =
a:·p·o
oc (p , p. ) 13 c b
'r \,w
-
ar-e
.xb pa: a:
p_8
):
pee:
---·- ....
~A~; t""~Ad •
~
0}
~ ~w
if p
D •
if p ~
= 0 and pb > 0,
if •0 p > 0 and pb -- 0. } pee:
, 20):
{(O,x. ): xb oa: a:
•
-·Lnese
{(x
pee:
,0): x
0}
if
p
0
• 0 •
-
0 and pb > 0,
> 0 and !-'b -
-
(Note that 4 < 20
in the following figures.
•
0.
112
< 5.)
•
oc.
0::.
~+~--------------------~:~------~ • •• • •• • ••• •• ••• ••• • •• • •• ••
I•• I
'
I I
•
•• ••
I
• •
I
•• • ••
• •'
••• •• • • •
20 Fig:rure 15.B.9.1
15-12
OCz
I
l
•
•
OCz •
•
• •
•
o. Thus. when w
co:.
oc.
••
l5.B.9.2
= 30, all equilibria are on the boundary of the Edgeworth box
•
..
..
• -d ~J..oa eo.,:n;..,,.l···m a price ratio and allocations are gl'/en by p /~ = 0 and '"'""'"".....:. • ................ ~ p 0 1/2 ...... {((30 - x ,0}, (x , 20 )): 20 •\" ner: u = 5, tte interior ~ xpf3 ~ 30}. pet •08 . P.8 •
equilib::-i'J..:-n is the inte:--section on the above figure .
Tne bou."'lda:-y equilibria
{({S,x. have the p::-ke ratio o. and allocations /p = 0 • 0 p
co:
X.
oa:
:S
20}.
N
Q •A .. _
}, (0, 20 - x.
00:
)}: 5
~
•
. t •• c.~. when ,.,--
w decreases from 30 to 5, Al:;:>hanse's utility level pa:
in::reases a::.d Betatrix's utility level decreases, rega:-d!ess of the choice of •
equilibria to
b~
compared.
= 30, Pe:-rie!" is too This is because, when w pa:
abundant rela-::ive to Br-ie and its equilibrium price is zero, implyir.g that . . ,, . 1 • - .... Be:at.-lx essen:lC.!lY cc:::.sumes tne tota enaowment Oi ...ne economv. •
w·nen
wpee:
-
5, Pe:-rie:- is sca::-ce enough to have positive price a..."'ld Alphanse can afford •
posi:ive cons-:.1::npti-:ms of both goods.
The price of B;-ie can e•;en be driven
down to zero, in whicr. case he essentially consur.1.es the total endo,-.·ment of the econcm•v .
lS.B.lO
(a) S;.:;ppose that the preferences of the two consumer-s a!"'e ouasilinea...-
\vith respect to commodity 2 and that
t..)
21
= 0.
Suppose for a moment that
w2 1 =
0, so that the:-e is no increase in the endowment of consume::- 1 for corr. .Inodity
15-13
•
Here is an example in •which an increase in the
endo~ent
of consumer 1 for •
•
com.!nodity 1 may lead a decrease in his utility.
ili • ••
•'
••
•• • •
•• •'
••
•• ••
•• •• •• •• ••
•
•
..
•
O'z
••' • •
• ••
••
'
•• • ••• • • •'•
•
•
• '••
•
•
Figwre 15.B.IO(a) . _,..0 S
. ,_ ,.., .'I .. _ ... 1... -- e ..• U.:. ... l o, "''··a•: allocations depend continuously en the initial e::1doVI.'IDents,
we ca."'!. still have a decrease
ill
positive, but sufficiently small.
..., '·' the utility of of cons:.1mer- 1 w,n·--"'21 L
We can thus ta.<e
u2
1
•
lS
» w2 as asked for in
the exe:-::ise.
Ir: this exa.T.ple, the small
t in the initia 1 endowrr.e:.t leads to a
substantial dec:-ea.se in the relative price of cozn:,!odity 1 . •
Sin:~
the wealth
•
• •
•
of cons1.1mer- l comes exclusively from commcdity 1. his real ,._,.. eC..ltr. tnen decreases, cespite the increase in his endowment.
Hence his utilitv ~
This fact is often discussed in the theory of a quar..tity-setting
decreases.
monopoly: It is no: in the monopolist's best interest to supply a!! it could potentially do, because an increase in supply lea.cs a decrease ir. price. •
(b) Let (p,x} be an equilibrium of the original e::cowrnents
be an
. .. ~
....liD (1'• ; ! ~ :: ,
""' '!""" -~........
.1"-ffi·.
In order to
of the new endowments
a~c!v • • •
the result
of Exe.-cise l5.B.8, assume that beth x and x' belong to the interior- of the Bv • the first fundamental
and x' ar-e Pareto optimal.
theore~'""
-
•
of welfare e::on Cfill cs both x t
Thus by the result oz Exe:-cise 15.B.8, \\-'e have
15-14
- 'Xi1 and xi2 = xi2·
H.-.nce, by the definition of qnasilinearity (Definition •
3.B.7}, we ca.-. assume without loss of generality that p = p'.
monotonicity of the preferences, p
»
Hence, by
0.
Since p•xi = p·wi > p·w
have p·wi > p·w . 1
""i
~
By the strong
c.Jl and
""i
;e
w , we 1
= p·x 1 , we obtain xi >- x . 1 1 1
(c) Here is an ·exa:nple of the transfer paradox: • •
•
•
x'
w'
•
Ot
Figure 15.B.IO(c)
In this
exampr~.
a ;:ositive t::-ansfer is made
f:~m
cons-..:..-ne.r 2 to consume::- 1.
If the price ra:io wer-e kept to be at the orig!nal level ?{P • then there 2 •
•
would be an exces.;; cie•!ia."ld for commodity 1. •
•
cnange to recover- a.; ec;uilibrium. • •
Thus the crice r.:;;::o needs to •
In this exa-:1ple,
but this
de:::rease induces a r.egative wealth effect on consu..-ner l because he is the net supplier of c:::nmoci::y 1.
(d)
.
......... Followin£ t ........ ·--·•
\..,
we shall prove that
original enc!·:Jw"..1ent w. • . t.,e h tnrougn
... c::-1gtna~
Hence his equilibri''""i. ccnsU!!:ptions goes dov.'"Il from
tt~:-e
• •,..er --e C:• C -- . equilibria at the
In the figure of (c), c:-aw the budget line that goes 'l'b ' ... owment w andt'ne new ecp.u.1 r;.um x •.
e!'~ .4
must be stee?er tha.; t.hat of p' because w « 1
Hence the demand of
consumer 1 o:;: this budget line must be in the north--west of x'.
15-15
This budget line
Of course,
his offer curve with w must go through this demand, as well as the originQ.l 1
equilibr-ium allocation x.
Hence, as the relative price of commodity 1
incr-eases, his offer curve must cut the contract curve from above a: x. Sym."Iletrically, as the relative price of commodity 1 increases, the cf:e!' curve of consumer 2 must cut the contract curve from below
a~
Tnis is
x.
illustrated in the following figure: •
•
• ••
•
•
-----::::::::- ---------------~ Curve Offer Cm ve of 1
Offer CY: 11e of 2 0
0;
I
T
Figure lS.B.lO(d)
0
•
that these two offe:- curves must go through tt.e initial
It will ther. be easy to convince yourself that, in whateve!' way
··-
•' T ,, ····:. Vo
-
.... ....
•
-s extend the offer curves, they n:ust intersect at at least t1-..·o other --· ;... • ... .o.i:. • • •
·~
Hence ther-e are at least three equilibria with the original endov.-me::::.:.
15.C.l
(a)
Tnis is a simple cor.sequence of two simple fac::, both
are already mentioned in the text: First, a Walrasian
which
is Fa.-eto
Se::onci, under the stiict convexity assumptions, :.:.1e.re is a ;:..::lque
1 oo~im.. L. • • C..;, ..
.,. .. e ..... o P .....,;,
equiiioriu.z~
c:~
1 o~ti-a ~ -•·.&. ~
aPor-a•ion •• - L.. ..
(b) Here is a:.:. exa.'Tlple in which the siope of the excess cema""ld fur:.::::cn may •
change its sign. •
wa!;!e levels w
-
1
H ..... -· -, given p = 1, the equilibrium wage
and
l~vel
-·
is \\·• ::--..,... ~"'~
,.,. , ) < z 1........2 ). we have 0 < ..:..- , r••• !
with w.
\,
.1.
~
15-16
at
Hence
z C·) must have a positive slope somewhere between w and w , and a negative 1 1 2 slope somewhere between w
and
2
w•.
q
Xl(Wl, WI+ Wtl. + 1t(Wi))
••
•
•• •• •• •• •• ••• •• ••• •• •• •• ••• •• • ••• •
•••
0
L - yt(wl)
••
••• •
••
'•' ••• •• • •• •• •• ••• ••• • •• ••
•• ••' ••• •• •• ••• ••• •• •••
•
•
•• •
-
•• ••' •• •• •• ••• ••• •
-L - )'2{w:1)
•
L
Figure 15.C.l(b) ·
To prove that the slope of the excess demand function is negative in a
c:
neighbo&hooci
the equilibrium wage level
and denote the wealth level bv v = •
wL
w•.
assume its differentiability
+ n(w), then
He!"'e, by (vi) of Proposition S.C.!, -
'v.·•l 1 = ·1' V
X ( •.,.
1·'"'
w•L·
•
... -( ..••)) . ••"" .
Thus the sum of the first two teems is equal
to the diagonal element corresponding to labor of the Slutsky matrix of this He::1ce it is negative by Propositions 3.G.2 and 3.G.3.
By (vii) of •
Proposition 5. C.l, Yj{w•) < 0.
Hence z'(w•) < 0. 1
•
(c) Suppose
the two consu.111ers are endowed with the same amount of labor . •
Then, at a:1y ...,.·age level, the total wealth of t..i}e economy is split equa!ly betweer, then.
. ... . . moal1 ~cation ...
Here is
a.."l.
- Exa.'!!ole
o~
example of multiple equiiibria.
4. C.l.
•
• • •
•
15-17
It is a
q
.-··
•
•• •
-·
--· ........ -· . -·-· .
.
•
•
.•• ......
•
••
• • • •• ••• ••• • ••• ••• • •• • •• • •• •••
••
•
..
...... -----· ...•••
0
•
•
L
Figure 15.C.l(c.l) If the fi:-m ope::-ates under constant returns to scale, then the!"e is a U."lique equi!ib.riu."n allocation.
To prove this, note firs: that the p::-ofit of •
Thus, to find an eouilibrium, we •
the firm must be ze::-o at any equilibrium. can assume 'that n(w) = 0 for- e'ler-y w.
Since, in addi-cion, the individuals a:-e
endowed with labor- a!one, they are always net supplie:-s of labor-.
1 :1us
their
off-· curves look as follows. •
• •
•
•
q
••• • •• •
••• ••
••• •• ••• ••• •• ••• •• •• ••
'
•
•
•
••
• •• •• ••
0
L Figure 15.C.l(c.2)
L
Hence the total offe:- curve, that is, the sum of the tw..o ofier cur:Yes, also looks as abo\'e, anC. the equilibr-ii.L"'l is described as an L'1tersection of the
15-18
total offer curves and the boundary of the production set, according to the profit maximization requirement of an equilibrium.
Since the total offer
curve can cross the bO\L"1dary only once (otherwise, the labo!" demand function would be multi-valued), an equilibrium must be tmique . • •
It is sufficient to calculate the Pareto optimal production levels,
IS.C.2
tion problem
·which is a solution to 'L.'1e
1/2 ( uz • 1 - z) = ln •
By the first-order condition,
:. !1e equilib:-iu!n consnmption is (1/3
-z /z2
1
+ ln (1 - z).
2
The equilibrium profit is 1/(2 · 31/ }.
112
,2/3).
For a..."'ly allocation z = (z ,z ) in the Edgeworth box, we have 1 2
(a)
< z
z
z = 1/3.. If we fix the output price to be one,
112 /2. 'then the equilibrium wage is 3
15.0.1
V2·'
1z if and only if z lies below the diagonal. 22 12
Hence the assertion
!- :. l'lOWS.
(b) For this
first recall that the differentiab!lity of the cost •
•·-:..,_.,.:on ' _ ... w "'· -
O••L'
0
--
.. ~,ui~·c.-l~=tr..,.. _... ---"" •
of a .(w) at ev_ • .,. wl is c.(·) kr eouivalentiy, the uniqueness J . J •
~,....
... ._
•
~avu.,~
-
.....Q
that f.(·) is strictly quasiconcave . J
Tnis implies that •
:b.e ma...-ginal rates of substitution changes strictly monotonical!y along the C
.•... a.... •
. • ·,.... an~ z . a .•c z .. ·r • J J 1
--·,..,s+t•u·• o~ a•'- z ::>~\..1 u •
t..
•
'-
a."ld thus (together with the homogeneity of de51 ·ee zero)
-
f o.,...
~
:""'
.,_ •• ., w.; il _,
'-•
..
J
/ z' / • z2j zlj ... 2j zlj'
then the marginal rates of
and z'. are differ-ent. J
Now suppose that a ray from the origin of firm 1 and the Pareto set of fa:::tor allocations intersect at z = (z ,z ) (which is not the origin) and let 1 2 z:' =
(zi,zzl be
a~o~her
point on the ray.
(The case ir.. which a ray starts
f:-orn the origin of firm 2 can be similarly proved.)
I':. is sufficient to prove
•
:l'.at z' is nc: Pa:-eto
o~-::imal.
By definition, z /z = z2/zi 2 1 11 15-19
;e
-z 1zr 2
-
0
ir-·J'r·,.s .. - .. that t:.i.e marginal rates of substitution of fi& inequality z
/z 22
of firm 2 at z
2
oe z_2 /zi
12
2
2
1 at z
111
Zi
are tle same.
of substitution of firm 1 at z
1
By Pareto optimalit:_.:, the marginal rate
and that of fir· m 2 at z
a:_ the sa..-ne .
2
z
The:-efore, the marginal rate of substitution of fi.: "' 1 at
-
•
2 at z_2 are different.
The
implies that the ma:g:..nal rat#== of substitution
and z_2 are different.
•
1
and
•
Hence z' is not Pareto optimal.
•
and that of 0
N:::e that this result
·-
is equivalent to saying that· the factor intensities at differ
t
Pareto
optimal allocations are different. Let's now show the (strict) mon.otonicity of ti::.e facto:- L>ltensities and of the supporting relative factor prices along the Pareto s--· sufficien: to p;:-ove that of the fo1mer. bounded, closed (non-degenerated)
int~rval
[c5
,c\l 0
·exis~
a
a co::.:buous,
a,1d
cox ..,.:_:lse image
0
monotor:~
We want to prove that it is (strictly)
not, then the intermediate value theorem would irm::iv 0
one-to-one.
•
t..~at
In fact. if
i: is not
That is, two different Paretc optimal allccat!c-:: would ha:;e the
•
same
-t ac:or 1n:ens1ty. . .
paragraph.
But this contradic:s the res::lt in tt: .:Jrecedin::: ~
Hence 1,!1( • ) is strictly monoto;:e,
imply~.::g
t.,'1e (;;-_-ict l
n:cnotonicity of t_i-).e factor intensi:y along the Pare:o se:.
15.0.2
Let z = Cz ,z ) and z' = (zi,zi_) be two feas:ble fac:.::- allocations. 1 2
Let >.. e !O,li.
We want to prove that the consump:ion vect::-
il.{fl(zl},f2{z2)) + = •
0
lS ln
'\I:' ( ( 1\•
z
)
1 1
+
(
(1-
>..)(fl(zi),f2(z2))
")I:'(') i - 1\ ~ z ,
1 1
prcduction possibility se:.
'\I:' 1\•
(
z
2 2
't
Clearly,
15-20
+ (1 - ,,,,,r ( z ')'1
2
2
•
COlnCJC.eS
by letting ljl(cS) be the factor btensity at c5 e [o ,o 1, the!l 1./J( ·) is 0 1
continuous.
.
•
Define a ms.;::> 1/J[- i from io ,c5 l 0 1
with the Pa.reto set of factor- allocations. ~
0
lt lS
By Exerc!se 1S.B.7, :he:-e
one-to-one maP q;-{·) from [c5 ,c5 l into the Edgewort.&'1 0 1
ir.to
-
•
•
and, by concavity, f C~z
1
1
+ (1 -
A.)zi)
l!::
Ar Cz l + (1- ~)f Czil, 1 1 1
f CA.z + (1 - A.)z2l ~ A.f cz ) + (1 - i\.)f cz2l. 2 2 2 2 2
Hence the proof is completed.
•
15.0.3
We shall give two proofs.
•
The first one uses Figure 15.D.6(a}.
The
•
second one is more formal along the line of the ·Proof of Proposition IS.D.l. In both proofs, we consider the case in which the price of geed 1 increases. The case in which the pr-ice of good 2 increases can be treated simila.:-ly. For the first proof, suppose that the price of good 1 increases from p
1
Let w• = (wi,wz) be the equiiibrimu factor price vector-
to APl' where i\ > 1.
We want to shov.· that
w~• J
•
> A.wi.
Of course, both w• and i\.w• are on t.."le salne ray.
Since p
2
coes not
change, both w• and w•• are on the same unit-cost curve {w: c Cwl = p } cf 2 2 By the equilibrium condition a11c the
firm 2, which is dov.'11ward-sloping.
homogeneous c! degree one of c ( · ), both 1 Th~
w••
and
~w·
are on the sa!:ne. unit-cost
w••
positions of w•, A.w•, and •
figure below: •
•
••
• • •
• • • ••
• • •
•••
• • • • •
• • •
""'.•.· • • •
• • •
I • • ••
•
• • •
•
• • •
••
••
•
• • •• •
• • •
••
•
•
At) I
-
01 •
0
W;
Figure 15.D.3
15-21
a:-e
::iepi~!.ed 1n •
•
tne
w•• > 1
we must have .
For the second proof, note from the Proof of Proposition lS.D.l that dw = Ca
22
cw•); I A I )dpr
1
Thus,
dwl/wi - dpl/p1
-
wi
(w• )/I A I
( (a
22 a22(w•)
-
- 1/pl)dpl
1
V'/i
1 a (w•)a cw• J + a2l (w• }a22(w•)( w2/\\'i) 22 11
-
IA I
•
cpl > 0.• •
-
15.0.4
(a) T."l.e utili-.:y function of consumer i
= 1,2) is denoted. b\' u.( , · }.
(i
~
•
The
producti~n
•
fu!lction of firm j (j = 1.2) is denoted by f.(·), which is J
assumed tc be homogeneous of degree one.
A price vector of the two •
consu1nption gocds is denoted by p = im::'.l!.s bv w .
of the two .
the consmm:rticr. vector of consu.":ler i •
~
D"i X. E -
-2 fir-m j b V Z. E ~ • -
~-.
+
z =
+
H:e:;: an equilibrium is defined as a vector (p•, w• ,x• ,z•) such that
fc::- shc::-t. 1. (L'tili::y
J
1
..,2
~i?xin:ization)
xi
For each i,
•
•
•
soives the constraint maxlmrzatlon
.' prco1.en:
~ w~. 1 1
Ma.x u.(x.) s.t. p•·x. 1
1
2. (Profit ;\·fa xi-J.:.ization} For each j,
z~
J
max;m'z-·r·o.., solves the constraint .... -· ~ c=.:.. •• •
--- --
• • -. "'r,l'"'!'n, P ...
p~f.(z.}-
J J
J
w•·z. s.t. z. J
3. (Market Cie= ... i;;g) [ixf = (f (zil.f (z2)} ar.C: 2 1
Of cou.:.-se, •
~
c. • • \.,
= (1,1). p~
J
=
2
·~al.u L.
x•• z••1 •
"'
•
(b) Suopose ;:;ow ••
f (z•) :s •1 1
+
I z• . z •.. I 1 1 -J
~
Llj
!R-.
can be replaced by the first-order condition
-....
(p•• t w••
J
E
?
~
•
Assume without loss of ge::er-ality that
f 1\-l '~··' ,..,.4 f (z•) 2 2 I
..._. • ._.
~
.,.;.aw•;p• f (z•' w• f ("'•' f 2 (z••) Note ••• ~ 1 2 2 2)' 2 1 ~1'' 2 • • utili:v maximization and the market •
15-22
,
clearlngo
can be
Acc;:ording to Exercise lS.Dol(b), the input allocations z• and
depict~d
z••
as follows:
•
'
.... ·· .•
.......··
•
.... ····
••o
...····
o•"
0
••••
• 0
~·
_ 0
z• 0:
•
Figure 1S.D.4(b.l} •
= p•• = 1) because of c• 1 ., • I
is, the •orice of inout 2 •
•
cannot inc:-ease even in the absolute value:
0
0
0
w•
•
0
0
0
• • •
•
• • ••
••
••
0
•
•
•
0
••• ••
•
• • ••
0
•
••
0
•
0
Fi2ure 15.D.4(b.2)
-
the f.{·} are :>t:-ictly C£Uasiconcave, this implies that z• J = , P •1 = c•• .1 .,
15-23
= z ...
Hence, bv•
fc) Suppose now that we have two equilibria (p•,w•,x•,z•) and
(p••,w••,x••,z••).
Assume without loss of generality that
•
Pi = pi•
= 1.
We
only show that it is not inconsistent to have fl{zil < fl czr•), f 2(z2l >
=J ~ (z•) w•• ll' 1
w• f 2 (z••) 2' 1
the sa.rne time. •
wi•
and
wz
= f
cz••) 11.
w•/p• 12
= f 2 (z•) 2'
and w••/p••
2
2
= f 2 (z••J 2
In fact, again, we know from .the unit cost curves that
w2•.
>
wi
at
<
But, as we saw in the proof of Exercise 15.0.3, the profit •
maximization a.'ld the factot- market clear-ing implies that w2•/w2 <
P,i./Pz·
that
.-'
•
•
Denote the initial factor allocation by z = (zl'z l and the new factor 2
15.0.5
allocatio~
-z
by z' = (zi,z2), after the endowment of input 1 increases from
1
_
!\ate en Figure 15.0.7 that z. and z'. are proper-tiona! for each j =1,2, J J •
Since the endowu:er.t of inout is • fixed at
lev·el of
-
-zl - zi 1
Th.us,
<
-z
1
-
z11 ,
\..,. a·=.... ··1·a·1· ... ••c;:o the left hand side by z ....
·~
•
~
0
··I!!!
•
oecause cf z < 11
15.0.6
-z
By proportio:::al!ty,
this implies
11
-z'
> 1 -
that is,
-z
and the right ha::..d side by
J, we obtain zi{z 11 1
.H..... -··c.._,
1'
-z
1
(and
By the homogeneity of degree •
. .
. eauilibrium conditions for- w• a-(a) \\'riting w• = (\ w , w }, tne •• \,.oi. • 1 2
... -
... t .... . __ ( \.1~• . -· q•' a -J . - - .... !
L.
c c
:
•
(w•) i1::
pl' and c (w•) = p whenever qi > 0; 1 1
(w•) 2:
p , and c cw•) = p whenever q2 > 0; 2 2 2
1
2
au(·,.,.·•Jqi + al2(v:•)q2 =
-zl; -
(b) S:Jppose first that an equilibrium w and (qi ,q2) satisfies
> 0; By the fa=tor
•
•
-· ... ...... -
• • ·~·e--,.+y· • • • • ' 0 .... • '
= p2 and
Gz
> 0.
condition, (the inverse of) the slope a (w)/a. (w} 11 21 15-24
•
•
the optimal input vector of firm 1 must be greater the slope a
Df
of the optimal input vector of firm 2.
G.i
q2
> 0,
-z/z-
> 0, the slope
two slopes.
22
Cw)
By the market clearing condition and
of the endowment vector must be between these
2
This is equivalent to saying that
diversi:ication cone. •
12
!w)/a
-z
belongs to the
•
•
Suppose conversely that
-z
belongs to the diversification cone.
0, then the !Harket clearing condition impiies that
cone.
-
Sirnilar ly.
- q
'I" 1.
•
2
and hence
-z
does not belong to the dive:-sificatior,
• 0 h :;:: , t en z o.oes not belong to the di·;e:-sification
•
•
cone e1t:1e:-. .-:. -c:. 1'• ( c ) 'iN•.=• c_•. ~
'I"\ ~
1:"•
. o,. e t!l- asse:-tion of this q•.Jestian by showbg ~
,.
~hat
if the
• • . . 1' Un1!.-f1C !C.:' isoqua..I.ts inte:-sect more than once, then the fac:::- 1nte:ls1tv . •
is r.ot satisfieC.
........
•
• • ne st. a terne:1: 1n t ..... •n·n· .. ! .. •
•
I 1"'~.0 ,._.__
two pc •. -
·!.. .,
• !'"!
L.
if they intersect more than once,
~
. :-.aT.... •
•
)
ec:.""" .s ....quan. proportional to each --
,
!""\
1
....
slopes of the isoqua:1:s at these points are identical.
.
-
•'na~ t.O.
•
tne
- firm
• ..---· eac!l z .. , •
---~
~J
•
-
. po1n .. j at
a.;. -
such tha: the
The slct:e of •
a.,.; bv d P.nr-'!" - ....J'·-- •
is en! •v one z J. such that 2
I
~!""P
As the :::•a a: is •
~
.. .,1· ~ _,-1 0 ...p:;::1' s~·v ... ~ ~"!!In~ Oi ........ • ~ .L-• W'lr.oi -~• r. •
•Ll._a,...e "'" _..
- ......
o~he~ ~~d
. ' ( . ,... -. te-:m.c.o:.. c:.r. .. pe:-h:ps c:mecessa.rily !ongj. ·it can be skiooed. •• .~
.. . --
...,.,,
The next paragraph is devoted to a •croof of
.. (z .. ,z .i aes 2
on the
J
lJ
we can sutn:i!·ess z . s.[·}.) from the augment of the slope function • .. 2J J 2 2 Le: v e ~ and v' e IR be two different intersectior:.s of the two
!soqu~:;!,
++
•
++
De::ote b.;•
1SOqUa:1~5.
c
the region of the unit-dollar isoqt.:E.."lt of firm l
.
. then C is a comoact .COU!lC.eC . . b Y V ana• v , .m::!USlve, • •
.1 ... ......... . . . a··. ., -'!"
,.,:.;,
~~· ... :.-~
•
.. g .. n .. -a' ••··
-· -· -- -: •
-. . '!" 1: •1""1· · - · -
.. ,. S
-
A.,... A
~·
-
~
C;
e
, • • n, b """'• ., .l!.~ -.:,..,a-s.o.,...1 0'"'"' e a ...,.., - - ;..., •"- cf the two isoquants on C, 1et tv J
~I. LaC,.;.:, 't'
. ,_.,.p.,...
secuence of •
•
•
·•··-· se .... ons
1• .&.n•ArC:•-t 1nr " ' - " _ .... .........
.1n c •
We car1 assume
-v
-v
• ·~~.···~au· •
. -·· o 1
~
••
1~
~
•
- '1
15-25
•
, ,
• -. "'0 C--
con ..·e;ges to a point e c . Then lS n v - 'i'2 2 • na•;e we Sin~~ s (·· ) - lir~: sl(v 1} • i' '1 n •• VI
• •r-.-• Ll.C::.
,...,.
-
-
I
loss of
an
-
s2(vl).
Hence
•
blnt is ve:-ified for this case of infinitely many intersections on C.
Jf
•
there are only finitely many intersections on C, pick up two consecutive intersections.
To simplify notation. let v and v' be as such aad v < vi. 1
Sin=e one of the two isoquants is above the other every where between v and v'
So sa-ooose not , ..
s Cv) or s Cv'} = s (v'}, then there is nothing to prove. 2 1 2
•
•
-
.
•
of fir-m 2 is above that of finn 1 everywhere between v and v'. case can be proved similarly.) For each z
0.
(The other
Then s Cv > < s
2
2
1
1
e [v , vi), let A(z ) !:: 1 be such that, if (z ,z ) lies on 11 11 21 11 1
the iscquant of firm !, then i\(z Hz ,z > lies on that of' fir=. 2. 11 11 21
Ncte
that i\(v > = i\Cvj> = 1 and i\(z lz e [v ,vi1 for every z e [·:rviJ: 1 1 11 11 11
}.;ow
tt.en g~·;.) < 0 a!1d
> 0.
g(,-;) ••
By the continuity of g( ·) (which is implied by the continuous
• • •n ... ,..-...... t diffe:r-entiabilitv of the f.(·)} and the intermediate 1;alue ... ,..__,. -•••• .o.l........ -e m"s-L
•
&.
J
.. .:......
But this imclies tha: s. Cz ,} = • 1•
!
s £i\(z
2
11
)z
11
i\(zl.!Hz ,z 11
). that
:> 2
•
15,
•
if lies on the isoqua..."'lt of fi.:-rn 1, then 11 21 •
lies on thc.t of firm 2 and the slopes
are
!SOOUa."'ltS •
•
the same o!'l. those two points.
The Hint is thus proved .
•
By the hint,
Now suppose that ther-e are more than one inte=-sections.
ther-e are zi and A > 0 such that zi lies on the unit-dollar isoqua."'1t of firm l, i\zi lies on the tmit-dol!ar isoquant of fir-m 2, and the slopes at those •
oc1~ts
•
a.:-e the same .
is such the.: w /w is equal 2
•,...,. ·a .. .... . ..... _ slope, then zi attains the minimum of w·z •
•
the m '., 'm·,..,..,.. of w·z -··--·.
I.Ll.O..
one, this implies that p
zi 1
1
1
on {z.: f. (z,) :::. l/p } and i
By
2
.L
1
.1.
.• rh .~..omcgene1 •Y o.~.
d eg:ree
attains the minimum of w·z en 1
•
s:nce
p i\zi a:-e p:-opcrtiona!, the factor intensity condition is 2
15-26
t
satisfied.
-w
As for the graphical construction of the diversification cone, if
vector, then, for each. j, w supports {z .: f .(z.) J J J
equilibrium input 1/p .} at (1/p .Ha
J
J
.(w),a
1J
is an
.(w)).
Moreover
2J
-w • {{1/p .Ha .(wl,a J
1J
.(w)))
2J
~
= 1.
Hence we obtain the following figure.
0
'
• • • • ••
..:· ( l/p2}{aiztwJ,az(wJ)
• • • • •
••
' • •
0
•
••
• • •
• • •
•
• • •
..-·---·
• • •
•
(1/pt)(ati(w),mi(w))
••
-
•
..-
•"
0
Zl
Figure 15 D.6(c)
(d) As o.,•:e saw in the discussion pre=eding P:-opcs:r.ion 15.0.2 (Rybcszynski
-.
.- l ~ the Th .. o....-••-'
:::a ~~ 1~ intensity condition implies that there exists .:.x ---..•
fact·:=~:"'
F. c.-c~o.,... .. 1· ............. o,... )J· ""'- ·--""' ..
v.· = (•"1• .. w 2 )
sue"~-
~..
-.u .....:. .,
tn· a-..., "'or -nv· -otal 1",.,it:-1 e.,,.:o .. , ...,.. <= 1 <=.I... .. ...... "'c::. ......... Y'fl~-···-,
the facto:- pr-ice vector- of any equilib:-ium invol•ing positive p:-cduction of • • •s e ...... :both goods ,;. ""';...C. i
0
....
:..·..,
'tt·
n .
By (b), the totai ini:ial endowment vector
-z
gives
rise to an (unique) equiUbdum that i:wolv:s pcsi:ive prod'Jction of bot.:.'i goods
0l!
. anc
cnly if
belo.,v t:t.e cone,
.. . ~na:.
-z
belongs to the dive:-sific.ation cone of -
.
:hen the economv s:::ecializes - .
1s,
factor price vector w• is
- - anc
If, on the
0
hand,
lies above the cone, that is,
specializes in
prod~~:icn
-Z/Zz -
of gocd 2 2..L""!d
,._
.!.~
t.Joo_ _ - ....
-.;-._~.·.:. 1.-o·--~·l·ng t"V
-
::!:
t:..~
= (1/f' 7 l·'.lus"!"-c.-~A...; ·~ ... .~ .. .__
•• -· z 1.._es T.:-
A
ir: prcducti::m cf geed 1 and the equilibri·•
-z
-w.
a
12
.......
15-27
and
:"'! liD .,...
A
22
Cw), then the
eqt:iiibriu:n factor price
(z))z
pi"ctT•-,e.
(.,v)/a
.... 0 .. __ .
c!"
Tnese a.;-e
w**
.
• • •
•
••
zz
• • •
z
• • •
• • • • • • •
•
w• z Zl
0
15.0.7
It is straightforward either from (15.0.1) and {15.0.2) or from
( ...'SDS' • • J
tha~ -..:..& ...
__ ...;
_1/2/2-j./2 ::, z. ,
C.r.e-
l
15.0.8
Filzme 15.D.6(d) -
..._.. ~··
2
--
It is easy to check that the production of good 1 is relatively fc.-::~or-
intensive in
We
1 than in the production of good 2.
ca~
m·~r-e
thus anolv .-. .
gr-aphical appa.:-atus obtained in Exercise 15.0.6 to answer this question.
th~
Bv-
some st:ra!g'h::for-wa:rd calculations, we can show that the unique equilibr-ium factor price \'ector w = ( w , w 1 that involves positive produc!ion of beth 1 2 213 213 goods is equal to (2 /3, 2 /3) and is independent of the total factor . . 113 endowments; a..::c tnat ali (w) = 2 , a
-
21
(w)
= 2
-2/3
, a
..
12
(w)
= 2
-213
• a
22
h•;) =
... ..
.
S.... ~c- th-e unit-dollar isoquants are equal to the (standa.-ci) isoqua.'"l.tS {z.
J
e
2 r.::: '
.' + . f ...(z .) ,
1
= 1} by p = (1,0, these results yield the foilowing fig-u.re:
~
15-28
j=2
j=I 2
1/)
·····-········; •• •• •• • ···---------·--i········----···· •• ••• •• • •• •• ••
•• •• • •• •
2-'Z./i
0
2 l/J
Zl
Figme 15.D.8 He.."l.ce the total factor endowments ·~ 11
•
·~
1
-2/3
113
-
-z -
gives rise to positive production of both 113/2-213 h . < 2 • t,.at 1s, 1/2 <
< z11 z2 1ne e::PJ!libriur:l level q• = oroauctlon (qi,q2} must then satisfy • •
goods
anc
u 2
on~y
•
· .n_ . - A'·1C ~~a-•
~
L *""o ~,._
I2
•
-CJ
- -
< 2.
=
2 x 2 It" a ..,,.. . as defined in the Proof of Proposition 15.0.1 and ..... lX
·' T 15 . ltS . •.., tZ"a."lSCOSe . •
Solving this, we obtain the equilibrium factor
allocation:
-
z• = 11
lf
-z
eco~orny
=
(213H2z
(l/3}(2z
1
1
iies beio?• the diversification cone, that is,
specializes in pr-oduction of good 1.
-z/z-
2
i1'!
2, then the
The eouilibrium factor •
ve"'·"r - L-.. .... ""' ..; s
Thus
v.•• =
-
S TW:. .. a.!U-L .. .':'T'
•!CC:.!J. - •I ~,
A .... 1'"" •
good 2.
I
"
•
• ••
l,
- -
.:s 1/2, then the economy specializes in p:roducticr. of
price vector w• is given by Tr..e ec:uilibriu.,: factor •
15.D.9
ec•.!ilibrium of this twc•
15-29
cmmtry model, where p•
E
IR:+ is the international price vector of the two 2
4 consumption goods, wA• e IR is the factor price vector in country A, zA• E !R ++ ++ 2 is the facto!"' allocation in country A, q• e tR is the cutout levels of the A ++ •
++ is the aggregate demand for
two consnmption goods in country
the two cons:!.-nption goods of country A, and similarly for
cou.~try
B.
Bv the
-
xB are proportional. By the · c ..' earl.::tg con d.,.. 1.1on, x.A_ + xB = q_A + qB and this su1n is proportional By the budget constraints, p•·(x~ - q.A_) = p• · ("B
ass11mptions on the utility functions, x.A_ and ..... ,. arket...
to
•
Hence, in orcer to verify this theorem, it is sufficient to prove that
We shall now prove this inequality.
Let A =
-z
21
/z
22
> 0 and consider an
a11xiliary ccu:m·y C, which is endowed with the total facroor allocation
i?.~.. ....
;\z
2
e
It is easy to see that, faced with the international p!"'i-=e vectcr- p• of cons~'~ption
goods, the factor price vector
w_B
would clea.- the input
l'!larkets in co'.:..'1try C; the corresponding factor allocation would be equal to ;l.zB a.'"lc ti:e ::::.:tput levels to A.qB.
Country A is endowed with the sa:z•• e amount
of factor 2 as country C and a larger amount of factor 1 than co'!.lmry C. Hence (as
r:.~~:her
cf country A nor C specializes) the Rybcszyuski Tneorem is
applicable t: comparison between countries A and C, implying that and
15-30
qiA
> A.qiB
CHAPTER 16
By Proposition 3.C.l, there exists a continuous utility function ui:
16.C.l ~
X.!
Since X.1 is nonempty, closed, and bounded. Theorem
IR that re:Jresents >-.. . -1
M.F.2 imolies that the function u.( ·) has a maximizer, denoted bv • 1 •
Hence u.(x!) == u.(x. ), that is, x! >-. x. for every x. e X.. 1
1
1
1
1 -1
1
1
1
x~ E 1
X l..
Tnus >-. is
-·
•
•
globally (and hence locally) satiated.
1 -1
w .. " Suppose that there is x.
~
x~.
We shall p!"ove by contradiction the property "If x. >-.
16.C.2
l
1
any x:
X. sufficientlv close to
E
1
l
-
then p · x.
1
L
X. such that x. >-. x! and p · x. < w ..
E
1
1 -1
1
x.1 satisfies p · x~1 < w.l
1
!
~,d,
Tnen,
bv the local -
nonsaUation, there must exist at least one such x: which moreover satisfies 1
X.'
>-.
1
t
l
.!la:..•
'I
-
1
X~
1
•1A •r-~sl···T!"'it" L..&L.. a.:,, &...&. .. .. y' -
B1•
X •.
x· >- x• ..
1
..
But this contradicts the ass"mntioi:i. •
••
1
1
is max.i:11al for >-. in set {x. e X.: p·x. s w.}. -I
1
1
1
1
•
(a) Fer tne fi.:-st assertion, if we had two different two global
16.C.3
~nd
satiation ooi:nts x. -
l
x:, then the convexity of X. would imnlv that (l/2)x.
•
1
.
1
·-
.
•
(l/2)x: E X. a;-ad t.ne strict convexity (1/2)x. of >-. would imoly that . .....1 •
+
1
(112lx: >-. 1
global
1
1
1
1
+
• x. -. X ..
Tnis would contradict the hypothesis that x. a.'"la x:• a.:-e
•
Hence every i can have at most one global satiation
!
!
•
•
satra.-~!On
1
po:..uts .
•
•
1
•
..... -·
n: n'!: Pu
For the second assertion, let x.
1
• • • --·a-•c .... ,...,_.,...,. S c:.~~ ~. "'' ,.. .......... ·....
X. be different from the global
E
1
Then there exists x: e X. such that x: >-. x .. 1
-·l
O'X.. + (.!. -
a:)x. >-. 1
1
X ..
1
1
•
•
•
··~·c~ \'T '"',;. ~ .;. • t
as
\Ve
a.'"l
1
!
Hence x.l is loca!lv nonsa:iated. -
• • (b) L e: (x • ,y • ,p,' b e a pr1ce eqm"l'b 1 rlllm shew that if
1
...,.,.. Tnen. f ~·
Wl"th TT"" •• ans1&- e.,.... s
(
• ""l' ... , v.:• ) . 1
We shall
x. > alic-cation (x,y) Pareto dominates (x•, •v•), then ~.o· Lr l
~.w .• !.1 l
saw i.."l the Proof of Proposition 16.C.l, implies that (x,y} is not
16-1
•
feasible.
In fact, if x! is a global l
tiation point for
then
•• • ll. IS
•
•
unique by (a) and hence X. =
x~. 1
1
= p·x!1 = w1..
implying that p·x.
1
a global satiation point, then >-. is locally nonsatiated at x! by (a}. -1
~
p·x.
1
1
1
1
l
1
1
> p·x!1 =
is not
1
Thus
l
ln addition, if x. >-. x!, then p·x.
p·x! = w ..
x~
If
Since
w .• l
r.o·x. > '.w .. 1' ! .L.1 1
ther-e is at least one such i, the smnmation over i yields
For the first assertion, it is sufficient to prove Liat if x is net
16.C.4
•••
Pareto ootimal relative to the u.( · )'s, then it is not Pareto optimal relative • 1
•
•
•
to the U.( · )'s eltner.
So suppose that x' Pareto dor.tinates x relative to tne
1
u.( · )'s, then u.(x.) 11
1
~
u.(x.) for every i and u.(x.) > u.{x.) fa::- some i. 11
11
Tnus U.(x) > U.(x) for every i. 1
l
11
•
:""'·. Thus x' Pareto dominates x relative to -·~
U.( • )'s toe. 1
This resu!t dces not mean that a conununity of altruist.:: can use competitive rr:.a;kets to attain Pareto optima.
In fact, suppose that, f..r.. a.-:
Edgev.·c:--th tcx :pure exchange economy, consu.rner 1
O'-\'"DS
•
c-! the eccnomy and his utility function U ( · ) pu:.s n1ucn 1
.
•
e.~.phasis 0 -"
pleasure function u ( ·) of consumer 2. 2
--"~>A
:. .;, .:. ._
1
allocation, a!
;:,;~ich
1"!1c.:--~
(Consumer
. . .. . e::Ju·:' ---:urn
•
other- hand, c::es not so much care about u ( · ). )
t~:a:
most of the
Tr.en a Wa!:-as!an
.
... .......
consumer 1 consumes most of the total endowments, is not tr~-:s:e~
Both of thel'Il prefer a
from ccnsurr:e::-- 1 to 2 to attain a Pareto ootil'Ilal allocation (relative to •
.
.
) . . ( ) .u. · s at ;,vn1cn 1
-. . ...
•• ,;-.a
-·-
consumer 2 consumes most of the total endoY..'!Ilents .
Tne above a.:.-gument does not depend on the concavity.
16.0.1
Sup-c·:.se that if •
x.1
then p · x.
x~, 1 1
>-.
1
~
p · x~. 1
··n • to. a~.
n
X. 1
>-.
X. l l
Thus, by n
•'-o- everv n. •
~ c::,
.
By the transitivity,
we obtain p·x.
1
~
p·x!. l
n
1 -1
1
•
in X. convergmg to x. 1
n
X. 1
>-.
1
•
s~.:.::l
1
•
X .• 1 l
n H -!1-- p X.1 a. ,...._
Hence x! is expencitu::-e
16-2
By ~-·-·-
Let x. >-. x'!'.
1
local nonsatia.tion, there exists a sequence {x?}
•
.
-· . ..... .;. .. ...
"""' •
i!:: -
•
. .
• X ..
-y·-
1
..J... a...
m
p
{x. E X.: x. >. x!}.
16.0.2
l
1
1 -1
1
1he implication of the other direction of obvious. .
Here is an exa.;nple of the failure of Proposition 16.0.1, in which the
preference is not loca.!iy nonsatiated and a •thick" indifference curve of the preference is drawn.
Any point in the intersection of the interior of t...l}.e
production set and the thick indifference curve is Pareto optimal, but can be •
suppo.
~.ed
neither- as a pr-ice equilibrium with transfers nor as a price
-
quasiequil!brium with tr-a.:."'lsfers.
•
•
X2
•
0
XI
Figure 16.D.2
•
-
16.D.3
Let (x•.y•,p) be a quasiequilibrium with transfers (wl' ... ,w }. 1
Supp=se that y = (-r. .... , •Jl e Y x ... x YJ satisfies ~ .v. 'l ~-J 1 + P ·().v. -J- J
+
-w)
> 0.
-w)
= Ll '· w.l > 0.
the L
p·fl:ljl
~ p·(Llj) by profit maximizatioi'!, we ha•::
Hence w. > 0 for some i. •
0, then
1
By Proposition 16.0.2 !a!ld
this c:;nsu.-ner mus: then be maximizing her pr-eferences Li {x, e
-
!R : p · x. ~ w.}. + 1 1
saw in the
SL;,ce
+
-w »
By t._l;e strong monotonicity, this implies that p »
s~a!l-let:er
0.
As we
paragraph on page 556, this implies that (x•,y• ,p) be
'"b riW:l . . . tr-ans;1:" ers. an equ111 w&tn
16-3
Denote the identic;o.l prcferenee toy ~ an(! lel V - ht e ll!: x !:
16-D-4
Write e • !1,11 e
~-
We sh.all first show that if
c.1 -
Ae 11 Co V for any II. > 0, th.en the
symmetric alloca-tion in which' every consumer gets equilibrium.
In taet, if
Col -
ror
;I.e If 01 V
1o1
?tel s
•
P;~. · x ror every x
C!
01 V.
2 such that p · e ,. 1 a.nd p, -> p e Jt2 .
Jt
+
Hence
Co \'.
"
t.l
Then p- w
..
e ll!+
:!i
p · x for every x e
2 Slnce "' e R , p- w > 0 and h.em:e w Is
is cost minimizing at p.
utility maximizing.
11
Moreo'ler, we can assume th.at there
11
01\l
~ '11.2 with P;~. ,. 0 such that
By the stroJ1i monotonlcity, p
and we can thus assume that p • e "' 1. exists p
Is a Wal!'"i1.$lan
any II. > 0, then, by the separating
hyperplane theorem (Theorem M.G.:J), there exists p:\ pi\· (w
w}.
++
Thus the symmetric allocation in which every
consut~~er
aets w i$ a Walrasian equilibrium. Therefore, If the symmetric allocation in ·Which every consumer gets
c.1
is ·
not a Walril.$lan equilibrium, then there exists A > 0 such that w - )le e Co V. Then there exist a poslti'l'c Integer N. lin > 0, and xn
.. that l' =1 and ""nn I
~
Ae
c.1 -
= t' 1o1
x .
Write x' "' x n
~nn
n
10
+ :I.e, then t' Jl x' -. w and ~nn
x' >- "' by the transitivity and the strong monotonicity. n
enough, then there exist nonneg11tlve integers r (Jl
n
r/r )x' >- w, by the continuity of >. n
n
or
n
Now, If r is large
such that t" r .. r and "11n
(That Is. for each n, the fraction
n
r /r approximates p. • l
n
V (n • 1, ... ,Nl such.
The allocation in which, for each n, r
n
consumars (out
rl get the consumption bundle (II r/r lx' is feasibJe, because n
'r r 111 r /r )x • ,. rfr' 11 lC • ) "' r~o~.
"'nn n e~·ery
16.E.l
n n
Col
n
Hence the symmetrie allocation In which
~nn
consumer gets
n
Is not Pareto optimal.
(a) Suppose that consumer I cares only ab
2 cares on!)' about consumer 2.
u=
(1.1 C!
r:
U' ., {u e U:
Then ul s ul(wl) and u2 s u2(;2n. 1.1
1
a:
1
~.< (0) llf!d
16·4
Uz
~ u (0J).
2
(Recall ti:Lat the disposal tecl:mologies are assumed to be available).
Tr..ese
sets are illustrated ln the following figure.
0
UL
Figure t6.E..I(a.)
1 1
-
2 2
It is easy to see that the unique Pareto opti:nut:!. is (u (w J.u t~o~ JJ
ji
U'.
(b) We can assume without Joss of generality that u.(O) ,. 0 for every i.
The
l
strong
mor.oto~icity
impiies that u.lx.) :;o 0 for every x. ~
l
l
E
L R , and hence U' c +
UnR L .
•
We shall prove that if u e U' and u is not a Pareto optimum, ttum u belongs to the interior of U.
Let x be a feasible
So let u be as such.
Since u is not a
cxmr;wnption allocation such that u. = u.(x.l for ever)l i. l
l
l
Pareto optimum, there exists u' e U' such that u'
;to
u an
rea:oible consumption allocation such tt-.at u: = u !x:) ror t:vtJry i. l 1 I for which uj > ur Pick up
t •
Since u < U'. u ( RL ;;nd beflce u.(x:) > 0. +
for which x;. > 0.
L
Denote by et co ft +
is one and the other components "-l'e' ~ero.
u. for any sufric:iently small rc > 0. l
uk{xk •
1
xj •
t~ue
Pick up i "Jbu:; x: " 0.
I
I
vec:tor who-.._, tth component
By the continuity, uilxi - •:o::tl >
!ly the strong monotonicity,
~ 1 ell > u~ ~ ulc for any k " i.
which consumer i gets
I
Let x' be a
The consumption allocation in
Ect and an~· other 1c • i gets ~ •
slmply· a r-edistrib\>tie>n or x'. and hence: feaslbh:.
o;
1-l"t
Sine" all consumers'
is
utility leYels are increased by this redistribution from u, u belongs .to the •
interior of t:. •
Tnerefore, if u belongs to U' and the boundary of U, then u 1s a Pareto optimu.'ll. • •
·(c) Let (x•,p) be a quasiequilibrium with transfers {w , ... ,w ). 1 1 allocation such that u.(x.) > 1
1 .
u.(x~) 1
l
for every i.
that x is not feasible.
By cor.dition fer every i.
•
.
.·-
l
-
~
W E ~
L ++
tne consilmcnon sets ar-e • S
. . ·~ m - -
--·· .. •
S t- • .;...a . . l.;. • 5
1
.'
O'V•c::i.,.. ._" ....
,
-
Hence w. > 0 for some i.
p ·W
~L. +
wa.:1.t to
1
Prooosition 16.0.2 implies that p· x. > • 1
we obtain p.
r.x.
1 1
sho·.~
> p. w.
•
·~
1
•
1
. "'ne....
c·,
~
-'
C·'"'"'"S"' - """.. - t
p:-od~:::i~n
11 •. f""''o a .. oca .1...-n.
u.(x'.'J 1
l
16.F.l
= 1\y +
that
(1 - ::\)y',
the summation is· taken component-wise.
By the convexity of
sets and the consumption sets, (x",y") is a feasible
•
!o.!cr-eove:-, the concavity of the utility functions implies that
~ A!..!.{X.) ...• (1• -
• •
••
A.)u.(x~)
1
1
for each i.
•
Su:ppose first that an allocation (x,y) is a solution to problem
(!6. F.l).
If it is
no~
Pareto optimal, then there exists another feasible
allocatior. (x' ,y') that Pareto dominates (x,y). (x' ,y') satisfies constraints (2) and (3). it also sa.tis:ies constraint (1). the
s~ch
We
In fact, let
x" = 1\x + (1 - 1\)x' and -y"
the
r-or
VI. 1
Hence x is not feasible.
that ther-e exists another feasible allocation (x",y")
u.(x':J ~ ::\u.(x.l + (l - 1\)u.{x:) for each i. l . ,
..
Since
Let (x,y) anc (x' ,y') be two feasible allocations and 1\ e [0,1].
16.E.2
VY ' •
l
0 (by the profit maximization of the free disposal
•
• :~rn
It is sufficient to show
(ii) of a quasiequilibrium (Definition 16.0.1), p·x. ~ w.
Since p
technologies) ar;,d
Let x be an
hypcthes~s
Because of the feasibility,
The Pareto dominance implies that
If u (xi) > u Cx ), then this contradic:s 1 1 1
tha: (x,y} is a solution.
If u Cxi) = u (x l, then, by 1 1 1 16-6
Pa~etc
•
dominance, there is a consumer i
¢
1 such that
u.{x~) 1
1
~
> u.(x.) l
1
-u .. · l
By the
strong monotonicity, a small transfer from consumer i to consumer 1, such as
ex:,1 increases the utility level of consumer 1 and, by the continuity of u.( · ), stili satisfies the constraints.
But, again, this yields a
1
hy~othesis
contradiction to the
that (x,y) is a solution.
Hence (x,y) must be
•
•
Pareto optL.-nal. • •
Suppose conversely that (x,y) is Pareto optimal. for each i
~ l,
If we put u. = u.(x.l 1
1
1
then (x,y) is a solution to problem .(16.F.l) by the definition
of Pareto ootima!itv. • •
•
with the ncr...;.egati-;ity constraints xli
~
0.
By augmenting the domains of all •
the u.( · J and F.(·) in the obvious way to include all the variables of (x,y), J
1
and then by applying Theorem M.K.2, we obtain the first-order necessary n- a•·lf"'r.s. e -.:.; ·'- - ...
••
•
eu/ax.H = lle - vll for every t;
o
= - <\(cu/axti) + f.lt - vli for every l and i * 1; •
for every t and j. ' . Rearran.gir:g these terms and by eliminating the v li. by the complementary 0 =- llD
+ "1.(oF./oyn.) J J 'J
•
•
slack.'"less condition vlixli = 0, we obtain (l6.F.2) and (16.F.3l.
Since 13. = 1, Vu.(x.) » 0, and VF .(y.) » 0, we have 13. > 0 and 7. >
16.F.3
!
0 for a!! i and j.
cu. ax •. ~
;~
l
~1
= ;-x c..!.. a ,. 2 • •
~~::---=:.:...~, •
1
J
1
Thus, by (16.F.2), Ill/~ i
J
1
8u.,/8x 11 . , 1
'1
OU, ,18X 6 , . , 1 c. 1
1
-
...... -e (1"'0 • F . 4.'· i established. •u•-••!,.; . I .S aF .1ay D. lltl"' j J ~J -=-==-~-=-- = eF.Iavn•· f.lt .1"1 j J •' J
By (16.F.3),
Hence 06.F.S) is established.
Condition (16.F.6) is implied by the abmre
oF .,;ay • .• J <.J
aF .,;ayl, ., J
equations as well.
16-7
J
=
1-lf/"1 j'
1-Lt ,I"1 j'
J
•
multipliers associated with constraints (1) and (2), respectively, and define
o1
= 1.
Tnen the firs-: order conditions of this problem is the same as
(16.F.2), which imnlies (16.F.4), as we saw in the answer to Exe:-cise 16.F.3 . • •
•
multipliers associated with constraints (1) and (2}, respectively .
Then the
•
•••
first-crde::- conditions of t.r.'lis problem are
Hence. fer everv j az:-::: t ~
8F.18vl. J J j 8F .18y . 1J J
J
J ~
J 1, we have
8 -
-
.,/c> 0 . , J • 'J
oF .,/Bv ., J
• 1J
• which imolies •
(16.F.S}. •
Finaily, fc;:- (16.:.9}, denote by p c:: 0 the multiplier associated with ~'-e L,· • -
•
.. a •.....,..
co,-:~ L._ ..o;;
..
-·
l
.....
ne:: the first-order conditions of the problem are
.Su/ox
obtai:!.
the
1
~neorem
Applyi:lg the EnYe!op
ou/Bxl
= p a..•·1d
8u/8xt
= -
p(Bf /8yl).
(Tneore."'Il M.L.l) to (16.F.7) a:1c (16.F.8), we
= lle f:-:- every l. c:: 1 and
8/ /By t = -
~ 2.
c-t for e•tery l
Thus
fi:-~-crde:_- cond.::~cns
for (l6.F.9.) can be written as IJ./IJ.l = u-l. But 8F ./8y l. for every i and and crt = aFJ . J : ::r -:·.-e:-y j. This . . .18 v .
CU/8X •:
here p. ./Ill = ~ ,·/ _ t. l cu. ox.. l
J
~!
o#
1J
•
imolies (16.f.6). ••
I6.G.l
The Pare:o o:::imality and the strict monotonicity of the u. ( ·) imolv ... l . -
.•. cond 1 don
(~··""
111 J.
The ?ar-eto optimality also implies conditions (!6.F.2l 2.!1d
sL.~ce 0~
(16.F . 3} a:1C,
-
= 1,
'iiu.(x~) l
»
1
> 0, a.-:d r; > 0 fer all t, i, and j.
... .. . (.' o e ... . C:v -n...,, 1 • · • t.Oi. ••• 0 ~
.& .....
. h r • • c ... _ '""-
.1, d a•, -• .. n._ w . ~·
A
r
l
•
•
quas1concavrty
-- u.,• :-)
~~
p·w
»
0, we have
p.~
'
> 0, o. 1
Hence, by putting p = (J.L , ... ,1lr_), we 1
first-order condition
=
-
0, and "VF .(y~) J J
vu.(x~) 1 1
= (l/!3.)p. 1
For each •
•
+ "[.p·y. by condition (iii), anc tne
J
J
implies that condition (ii) is also satisfied.
16-8
l6.G.2
Let the input be gcod 1 and the output be good 2.
By applying the
implicit function theorem (Tneo:-em M.£.1) to the equation F .(y.} = 0, we know J J 8F ./8y . 1J J that the marginal productivity of the input is equal to By oF Jay . · 2J J 8F./ay . 1J p J (16.G.l), this is equal to P/Pz· Hence p = oF Jay . 2· 1 2J J
16.G.3
Let goods 1 anc 2 be :he inputs and good 3 be the output. •
production set Y of the :::!:
1s
•
g1ven by Y = {y
0}, wher-e F: (- IRl x (- iR) x IR
1 ne
IR is defined by
2
3
domain of F( · ) is not the whole 1R , but this
.. .~"---:-. _ , , . , .. ~ ..
constant retu;-ns to scale,
F(y)
IR : y I
E
1
• • . --e • •Nill turn out to be J, .. '
3
- 21-1/p((- y )p+ (- y )p)l/p
F( •..,,J
and p > 1 is close tc !.
~
The
b".lt,
Note that this production sets exhibits for any q > 0, the set {y e Y: y
0
3
~
q} is not
y1
Figure 16.G.3.1
N-:::te aiso that, as
p -;. 0,
the boundary {y e Y: y
3
= q} becomes flatter and
3
flatter sc as to cci::cice with {(y ,y ,q) e IR : y :s 0, y :s 0, 1 + Y = 1 2 1 1 2 2 •
- q•'. f'..:."":c::on
or
1/3 113 1/3 d th the consumer is u(x) - x x x an e 1 3 2
16-9
iuitial endowment is
-w =
It is easy to show via the first-order
(3,3,0).
conditions that, for every p > l, the production and consumption plans and the price vector
x• = (2,2,2), y• = (- 1, - 1, 2), and p = (1,1,1) constitute a marginal cost pricing equilibrium.
Moreover, if p is
sufficiently close to 1, then it is the unique one. which can be shown as follows: Let (x,y) be a:."1y marginal cost pricing equilibrium allocation. 3
first that x e 1R +..;. • y
1
< 0 and y
2
Hence y
1
> - 3, y
•
2
> - 3, and y
3
> 0.
r\ote
If, furthermore,
< 0, then, by the equality of the consmner's marginal rate of
substitution at x and the finu's marginal rate of technologicai substitution ••
Although cne of y and r may we!! be 1 2
at y, zero, by letting p
~
!, the marginal rate of technological substitution ca.."l be
•
made as much close tc l as needed a•·1d thus we can have neithe::- y 0.
•
This can be illustrated by an Edgeworth box of
~
-,.-
.,....,
••
•
:--------,_
•• • •••
••
•
...
•
•
••
• • •
••
•
•
•
,• •
_..
-·
l-llill
. ........··· ,.
•
• • •
allocations:
•• ••
··...... •••
l!lOUt •
1
= 0 no:- y
•• • • • •
•••
••
••
.
•
··..••
j I
'
Indifference Curve
···..... !Isoquant Curve ~··------------------~·J___ Consumer IT ••
Fi!ZUl'e !6.G.32 - .
--·· ...,c H-. • .. - a·~ c.:..;.. J
..
-~-::r•n ' !.i....... 0 .;.. .;. .. C:...i.i
,...-c:· - .., ... .... pricing equilibriw '!, we cust have
'!: J 1
..
< 0 a.-d \' < ' ~L.
"T":"' Then, as we C.i.s=ussed, y = y given this equality, ' ~ . . "".., -.a.•· 1 2 . . th,e equa:r:y 01 the cor.sumer' s marginal rate of substitution between the
0.
output and the inpu:s and the marginal produ=tivity implies that y
16-10
3
= x
3
= 2
2
-
and y
= y
1
= - 1.
2
Hence the above marginal cost pricing equilibrium is the
•
omy one. It is now clear from the first figure that, given p = 0,1,1). in order
to produce y
3
= 2, it is not cost minimizing (and in fact, cost maximizing) to
use the L-lput combination (yl'y ) = (- 1, - ll. 2
The cost can be decreased by
moving towards concentrating on only one of the two inputs.
Suppose that the ·set of all feasible allocations is nonempty a.'ld
16.G.4
compact. Then there exists an allocation (x,y) at which the utility of the single consumer is maximized. (16.F.2) and (16.F.3).
Hence it is Pareto optimal and satisfies
Hence, by Proposition 16.G.l (as proved in Exercise
l6.G.l), there exists a price vector p such that (x,y,p) is a. marginal
cos~
. . pncmg
., . "" . eqUhh.. l'"lU.'"n.
16.G.5
(a) Let (Le,Te} and {Lf,Tf) be an optimal allocation of the endowti.lents
•
a:ta
2 e• = (min {L ,T }) and e e
•tet
Hence L
•
..-
= T
e
a.'ld L
u(e•,r•J = L (l - L e
+
"' -112
e
>
r•
Then e• > 0 and Hence e• =
Lf
.
r· >
0.
= 1 - L . e
= 2/3.
By the first-order conditions, L
e
He:1ce T
e
= 2/3, e• = 4/9, and f• = 1/3.
(b) Let (pL,pT'pe,pf) be a supporting price vector of the Pareto optimal
allocation iden:ified in (a). profit
co:-~diticn
As a nonnalization, put pL = 1.
The zero
and tne cost minimization condition of the food production
i.:npiies that pT = 1 a.:.·::! Pr = 2.
Under (pL,pT) = (1,1), the marginal cos• of
education at the input levels (L, T) with L = T is equal to 1/L. i::"" ..gir.al
cost at the octimum is 3/2 an.d hence p = 3/2. . e
the education sector is (2/3 subtracted
fro~
+
Thus the deficit of
2/3) - (3/2)(4/9) = 2/3, which is to be
the consumer's budget as a lump-sum tax.
wealth is 4/3 a.'1d his utility
Thns the
He!lce his
ne~
2 2 on probiem is to maximize el/ rl/
16-11
Thus
subject to (3/2)e and
r·
+
2f :s 4/3. The solution to this problem is in fact e•
= 4/9
= 1/3.
(c) As calculated in the ~-;.swer to (b), the lump sum tax is 2/3.
(d) Let e e [0,1] be a planl'1ed level of education and s-uppose tha-= it is
produced bv the input combination L = T = e · e e
1/2
Ut can be s.hc·w-n that this
•
equaiity a!;;rays holds at a!l Pareto optimal allocations.) as we sav: in the answer to
(b),
Let pL = 1, then,
the zero profit condition and the cost
minimization condition of the food production implies that PT =
,.. c:.na - . Pf
= 2.
Hence, lf the government follows the marginal cost pdcing rule, then it sets 0 .e
= lle
112
from the
a:1ci the deficit is 2e
lando~-ne:-
112
- ele
112
= e
112
.
Tnis is t;'1e transfer
wr.en the planned level of education is e.
(e) As V.'e saw in the answer to (d), at any marginal cost prici:;g equilibrium with a plan."'led level e of education, the equilibrium price vect=::-
scalar
mult.ip~e
th~
vector,
cfi (pL,pT'pe,pf) = (1,1,1/e
112
!~Just
be (a
Faced with t...'"lis price
,2).
112 den: a.• C. of t.he laborow:ler is (e 12, 114) and the cema."ld of the
( e landowne:- is 2 - e
1/2. T
1 - el/2
l. -
2
J
4
l/2
).
Hence the aggr-egate de:r.and for .
1/2 1/2
food is --":'_,--- . The inputs used for educatlcn are (L , T ) = (e ,e ) e e 112 and thus the food s'..lpply is 1 - e . Hence we must have e = 4/9. Tnis is •
the planned level of education compatible with a marginal cos: pricing e~•
'1''~1
•
u1 orl u":.• .
We sh.a!! now prove that the marginal cost pricing equili":r-ium is Pare:o cptima! by sho...,ving that the production possibility frontier is below the
Scitovsky co!ltour (page 120), as iilustrated in the figo..u-e below.
16-12
1
·-······-·
4/9
••• •
Scitovslcy Contour
••
I
•
1/3
r
1
Figun: 16.G.S(e.l)
Note first that the production possibility frontier is giver: by the equation f = 1 - e
112
, or equivalently, e = (1 -
n
2
.
On the other hand,
since the ag3 • egate dema.:1d at the marginal cost pricing equilibrium is (f•,e•) = (1/3,4/9} and both the laborowner and the landowner have the identical,
1 homothetic utility function u(e,f) = e /2fl/2, the Scitovsky contour is given 112 112 112 112 by e r = (4/9) , or equivalently, e (113)
= 4/27f.
It is sufficient
to show that "the Scito.,·sky contour touches on the production possibility •
frontier onJ. ~· at (f•,e•) ~
ever:,·-where else.
= (V3,4/9)
and the former is above the latter
For this, it suffices to prove that
e·:er-y f e [0,1] with f
;~~:
1/3.
(1 -
But this follows from (1 -
depicted in the figur-e below.
16-13
n
2
< 4/27f for
2
< 4/27, as
n r
e ············--·····-;:;~.
412i
•
•• ••
•• • ••• •• • • •• •• ••• • ••• •
-
•• ••
•
0
113
1
f
•
Figure 16.G.5(e.2)
n n)} b . A . . ~LI iRLJ ( , ' { Le: ,x ,y n e a sequence 1n converging m · '"' x to x,y J,
16.AA.I
By definition, then, x~ ~ x. and y~ ~ y .. L 1 J J
•
Since every X. ana ever-y 1
'\.' 1
. j
.
lS
closed, and x~ e X. and y~ e Y. for every n , we have x. E X. ;:nd y. e y ,. l 1 J J 1 l J J . r,.x.n = u - + L .v., n we h ave ~.x. = w - + L .y .. Hence (x,y ) E A. S tnce -l
1
j"
J
J J
1
We showed in Exercise 16.AA.l that the set A is closed.
16.AA.2
s·J.fficient to prove that it is bounded.
-
•
•
It is th..:s
First observe that if ~(xn, vn)}
-
'
-
be
an a:-bitrarj· sequence in A such that the sequence {yn} is •
.
...... -,.... se..,, W"o•e "" ___ .. .,,. .,_....... \'( x n v·n)} is also bounded. . n
lyll
(I -
< s fo' every l a~d n.
lk + s
.... Lue X ... 1
.. !"":
-+-
Tn~s
-wt
To show this, let s > 0 sa:isfy n
n
-
n
Then, by [.xt. = Yt + t.Jt' we have - r < xli < 1 1
for every l and i, where r > 0 provides a lcwer bou:ld for
•
t;:e secuence •
n
{x~ .} ,1
n
is bounded, and hence so is the whole
T'\
{( X ,1i"')\; .
-
n
Now suooose that A is not bounded, then there exists an u.t1bounded • • n n {(x , -. .,. '}} n .-
secue~::e
. .A. -
·~l
~
According to the observation of the previous
•• •1-1 .. ~ n p;:r-ag~ ·apn, 0 • takir.g a subsequence if necessary, we can assu..-ne ... Jat "L· X. - wll •
-
l
T
,..o y e IR, .
16-14
1
Since Y is
convex and 0 closed, y
E
n
Y, (lilly ll}y •
n
=
n
n
•
n
(1 - 1/lly 11)0 + (1/Uy U)y
E Y.
Since Y is •
Y.
E
-
By the equality
+ "'·
1 __;n:____
we also have _ '\'. 11-'-i.x i - wll
-"'l
ri 1
n
- - - - C [ . x1 t .1 U'"'.x~ wll Ll 1
-
and hence
-w)
>
t
-ri -
-w
ll[.x~ ·-
wll
.L.l
-7
Since the right-hand side converges to zero as n
1
l
~
=.
1
we obtain y e
L
"'""' + . But
since y e Y and !lyll = 1. this is a contradiction. He:-e are
fc~.;r
exa.'Tlples to show that each of the four assumptions is
indispensable to gua:-antee the boundedness.
Note that each example violates
only one cf the fo:.L" cor:.ditions.
Tne first exa:-nple shows that assumption
(i) is indispensable:
Fig;ure 16.AA '.1
-
-1 .n "'"- s _____ ..., eX"''nple shows that •
Q
,..
~- '"""! ....
(ii) is indispensable:
16-15
y.
------. l
•
n cr.x. - w)
Figure 16.AA.2.2 The third exaz.uples shows that (iii) is indispensable.
0
Figure I6.AA.2.3
The last exa.-::.ple s~o\'liS that (iv) is indispensable.
16-16
.4
16.A...\.3
We shall first establish the following letmna
Suppose that, for each j
{v~}
= 1,2,
is a sequence in IRL and v. 1s a J n J
n J ll\"
n 1
+
r
vz!l _,
as n _,
c::!
•
.
n
n
J
J
0.
J
Then
c::!.
To prove this, ncte first that, since Cv + v )·vj > 0 for each j, there exist 2 1 n n n n n a positive in~eger N such that ((l/llv u>v + (l/llv Hlv J·vj > 0 for every n > 2 1 1 2 0 N . 0
Tnen.,
f~r
any s > 0, there exists a positive integer N > N
0
such that.
fer every r. > r-;,
HO/IIY~ll )v~ + ( i/Hz:~JI )v~lf > n
llv .II > 2s/llv
1
No\\. •cick
1
n > N.
~'1"~· •
+
(l/2}:tv + v 11; 1 2
v 11 for each j. 2
n n Assume for a moment that llv 11 ::s 11v 11. 2 1
Then
= llv7!1·ll0/llv~ll)v~ + 0/llv~ll)v~ + (1/llv~ll - 1/llv~ll)v~U. But here II :' ;.. / u'' \:):11 'II )··• n• '"
l
= H( l / I! v~ II )v~ +
2(1/Ji,.·~JI
+
-
1/llv~ll )((1/llv~ll)v~
+
+ ('/II J. • .nli 1 .. -
16-17
Hence
We can establish this inequality in the same way for the case of Thus llv
n
n
+ v 11 ~=as n? 1 2
CIO.
We shall now prove the assertion of this exercise by contradiction. Suppose
tha.~
the assertion is false.
Then there exist sequences
n
and {v·2 } n n lly 11 1
~
= as
n n O/ny nJy 1 1
y~
+
y~
+
n n y and U/lly 11)y 2 2 1
~
-w
~ =;
n
il!:
Thus we also have II y u 2
y ; 2
0.
By Exercise l6.A..6,..2, the sequence {y~ n
~
~
paragraph, we must have y
= as 1
+ y
n ~
2
+
Ill,
= 0.
y~}n in Y = Y1
+
Y must be bounded. 2
and, by the lemma in the preceding Since lly 11 = 11y 11 1 2
z:
1 and the
irreversibility asstirnption holds for Y, in order to derive a contradiction, it remains to show that y e Y and y 1
n 0/ II Y 1
+
n Y II )( y 1
2
1
2
1
e Y.
+ y ) e Y for every n.
2
Since Y is convex and 0 e Y, we have Moreover, by taking a subsequence if
n 1 n 1 . . necessary, we can assume that the sequence {(l/Uy + y 11Hy + y )} m Y 1s 2 1 1 2
convergent.
Our second application of the lemma (let v~ = y~ and v~ = Since Y is closed, y
We can similarly show that y
2
e Y.
16-18
1
e Y.
CHAPTER 17
17 .B.l
Y . 1
~
Suppose first that Yi
•
By p
l!::
0, p·y
~
1
0, p · Yi
0 for every y
= 0,
and p
v• e • 1
0.
?!:
e Y and hence p·yi 1 1
Suppose conversely that Yi e Y and p·yi 1
2:
p·y
1
l!::
.p·y
for every y
1
for every y
1
Since
Yr
E
1
•
needed by taking yll very small.
But this is a contradiction to p ·
for ever-y y
Hence p· y
1
e Yl"
Thus p =:: 0.
1
~
0 for all y
1
E
Yi
l!::
P ·v -1
Yl" Finally,
si:!ce 0 e Yi' p· Yi = 0.
17.B.2.
We shall prove property (v) by contradiction.
exists a sequence {p
0
}
0
such that pn
n
»
Suppose that there
0, pn ~ p, p :~; 0, Pt = 0, and the
n
sequence {Max{z (p ), ... ,zL(p )}}n in IR does not diverge to infinity.
Suppose
1
that consumer i satisfies p · w. > 0. 1
We first observe that, by the strong
monotonicity, the demand of consumer i at p is not well defined, that is, there is no x! e iRL such that p · x! = p · w. and x! +
1
p ·xi :s p · wi.
1
1
1
~... 1
x. for every x. e IRL with 1
+
1
This is because, by simply adding a positive amount o: good t,
whose price is zero, consumer i can get better off.
L
Since the consumotion sets lR • {Max{z (pn), ... ,zL(p
0
1
{Max{z .(p
0
11
), .... zL.(p
}}} 0
1 .
0
)}}
+
are bounded below and the sequence
in iR does not diverge to infinity, the sequence n
in IR, which consists of the excess demands of
0
consumer i along {p }n' does not diverge to infinity either.
Hence, by taking
a subsequence if necessary, we can assume that the sequence {z.(p 1
convergent.
De:1ote its limit by z! and define 1
o·x .. = p·w .. • • 1
1
x~1 =
17-1
1
)}
•
n
IS
z! + w., then x! e IRL and 1
1
We shall now derive x! >-. x. for every x. 1 -1
0
1
E
+
1
IRL with p · x. :s p · w., +
1
e
1
which is a contradiction to the observation in the first parag1·aph. ~
let x. e 1
L +
and p·x. :s p·w.. 1
For each n, let ).
1
n
n
= o •<J./p·w. •
1
1
i!:
In fact,
0; this is
.,L an d p n · C"'1\ nx. ) :s p r. ·c..: .. l ' d e f.mea. b ecause p·w. > 0 . Then "'nx. e"' we. 1\,
1
+
1
1
1
z. (pn·) + w. >-. ). nx. by the definition of the excess demand func:ion. 1
1 -1
1
z·.(pn) ~ z~ and A.n ~ 1, z.(pn) + w. ~ x~ and A.nx_ ~ x .. l
1
>-., we obtain
-1
17.8.3
1
x~ >-. 1 -l
(a) Let {pn}
1
1
l
Tnus
Since
By the continuity of
1
x .. 1
n
be a sequence such that pn >> 0, p
n
~
p and pi > 0.
Bv •
property (v) of Proposition 17.8.2, it is sufficient to prove that the sequence {zg 0, P'"'/Pt
"'A· · 1 1arge n. SU1TIC1ent.y proof.
s·mce o
i!:
0 is well defined
n t h"1s camp 1etes tne · :s xli (p n .p n ·wi ) :s p n ·wi 1 Pt•
[Step 4 of the proof of Proposition 17.C.l contains a simila:- pr-oof of
this fact.
J
l/2 112 (b) We conside!"'" the case cf I = 1, u Cx ) = x + x and w = 21 + x31 ' 1 11 1 1 (1,2,1}.
For each n, define p n
0
= (l/n,l/2n,l).
Then
2
n
x Cp ,p ·w > = (n + 3/2 - l/4n, 1, l/4n ) 1 1
and hence n
z(p ) = (n + 112 - l/4n, - l, 114n
- 1).
2 remains in net supply.
Hence, although
17.8.4
2
We shall first prove that the profit functions n .( ·) and the supply J
Let r > 0 provide an upper bound for the Y.
fu."lctions y .( ·) are well defined. J
(that is, y •. < r for every y. e Y. and t). ~J
J
J
17-2
Let p » 0 and
-y.
J
J
e Y ., and J
consider the subset {y. e Y .: p · v. J .J J
p • y- .) of Y .. J J
:!:
Then this subset is bounded
because it is included in :s r for everv t>. •
Thus it is compact.
y~
hence
J
e Y. such that J . J
a·y~
Hence there is
is a profit-maximizing production vector.
p·~~ ~
JJ
o·v.
and r·v~ ~ •• J ,. 'J'
Its unioueness is shown in •
Hence both n .( ·) and y .( ·) are well defined functions.
Prooosition 5. C.l. •
J
J
The homogeneitv of the n .( · ) and the y .( ·) is also sholh-n in Prooosition • J J S.C.!.
Let p• » 0 and let C be a compact subset of {p E IRL: p » 0} such that p• e Int L C. iR
Let
-y. J
some o e C} of Y .. -
satisfy
p · yj
p·yj
l!:.
d for all p
l!:.
then p·yj l!:.
Y .: p · y.
E
J
l!:.
J
J
In fact, let b e l!:.
c for all p e C.
Then, for every y. J
E
for ~
L ... ~
-·
Y. and o e C, if J .
b and hence
l!:.
E
IR satisfy b/pt
E
C.
E
•
b/pt - 11c~(pk/pt)Ykj
Therefore, for every p •
C, and c
E
•
y lj
J
J
b for all p
d > 0 satisfy pk/pl
p·yj'
Y. and consider the subset {y.
We now show that this is bounded.
J
~
E
p · y.
l!:.
c - Lk~ipk/pt)r
l!:.
c - : L - l}dr.
C,
•
{v. e Y.: p·y. -J J J
p·y.} c {y. e Y.: c- (L -l)dr :s y 9 • ~ r for every l}. J J J <.J
l!:.
Give:1 this boundedness, we can now prove the continuity of the y .( · ) and J then.(·). The subset •
J
Y.(C) = {y. E Y.:
J
•
1s compact.
J
..
Let y .: C J
J
-7 Y .(C)
J
c- (L- l)rd
~-y •.
<.J
:: r fer every l },
be the correspondence that associates p with
..
the maximizers of p · y. over Y .(C) and w .: C -7 IR be the corresponding profit J J J .... • function. By the theorem of the maximum (Theorem M.K.6), y / ·) 1s an upper
... hemicontinucus correspondence and Y.(C) J
:::>
{y. e Y.: p·y. J
J
J
l!:.
11' .( • )
J
is a continuous function.
..
But since
..
p·y.} for every p e C, y.(p) = y.(p) and n.(p) = J J J J 17-3
•
If .(J)J
J
p•.
Thus both y .(·I and 11" .{ • l are continuous r~nctions at J J Since tl\e choice p• » 0 hu. ~n arbitrarily chosen, this shows that for e\·cry p E C.
both y .( ·) und J
If.[· I
are continuous functions everywhere on (p
J
ll
RL: p» O}.
'
Propertie:l (il and 1m or Proposition 17.8.2 for the production inclusive "gsrcr;ate eltcess demand functinn are immediate con$equences of the homogeneity and the continuity we established in the preceding pllragraphs. For property iiiil. simply note that
rilp·wi .. [jElij'lj(p)) • LtP"I.Ii • L/"Y/Pl
= 0.
'"[.p•IJ. + [.n.(p}- t'.p·t..l.- r.n.(p) I I JJ '-i I JJ
> 0 provide an upper bo\lnd for the Y. and a lower
A"l for (iv). Let :-
J
bound for the xi. then, for every t and p, zt{p) > - rl - }1r.>li ·- rJ. Finally, we sh&U establish (vl.
0, pn
"*
>c.~. "i I
ye
+
p, ·p
I#
o,
=0
and Pt
RL , we have ++
Let (p")
for some
E-P'"'· I I
t.
+ [.nj[p)
J
0
be a 5«1uencc s:ucb \hat pn »
Since there exists a
:~ p·!I:.w. I 1
+
y)
y 4i
Y such that
> o. !AithoiJgh we do
not h~e p >> 0, RJ(p) can still be defined as the supremlJtll of tile profits p- y j .for yj
E
YJ' because Yj Is bounde-d above.) Since
rr"'1 •
rjlliP'
thc7e Is a consumer i such tbilt p·w
1
= r;'P""'; +
.. rJeitjtpn.
t.e .. TIJ(p) :.1 IJ
> 0,
For such. i, we can
show that t.la.x(zlitp"J, ... ,zLi(pn)) ~ ao, as in Exercise 11.B.Z. -
n
Since
n
;o:l{p l > zll(p ) - r!l - I) - [ILiti - rJ
for every
17.8.5
t, we also have
-
n
-
n
Max(;o: tp ), ... ,zL(p )} 1
~
•·
By the Shepbsrd's Lemma !Proposition S.C.Zivlll. for each j and p,
Vr:}p) represents the input dem;,.nd of rim• j at p for unit output. runction a
J'
(p e Rl..: p
»
0} -+ Rl. by
17-4
Define a
If good t ls the output of firm j;
lr ,eood l belongs to th.e domain of c.(·J; J
otherwise. Then a pair (p,a:} formed by a prke vector p »
0 and a vector « IC
activity levals constitute an equilibrium if and only if they solve
a
z(pi - [ .« _.., .(p) = J J J
and
p·a.(p) J
17.B.6 Let (ow·• ,p) be an i. let
=
a.(p·a.(pJ) = O·fcr all j.
cquflib~ium
J
J
in the sense of this exercise.
For each
e Y solve the maximization problem of consumer i together wlth v!.
y~ 1
1
Oeflne y•
Xf
o.
~
tl
LtYi.
then y• e Y because Y is a convex cone.
vl + Yi E Xi'
(xl .... ,xp.
Wrlte x• =
For each i. define
We shaH show that (x•,y•.p) is a
Walr··aslan equtubrium.
F[rst of all, sln.ce L1 ~.v! • t".w., we have : £.1 l Denote e
(1, ...• 1)
m
L e R .
1.x~
4..j
l
= t'.w. L.l I
+ )'•.
Note that, by the monoton£city or
preferences, p·e > 0; otherwise. the consU!'ncrs can always get better off simply by adding e to
. .
vi
(without changing the production plans yl), whose
.pnce 1s nonpos1t1ve. We shall prove that y~. I
then p·v.
1
a
0.
Thus
p·y~ ~ l v~ + 1
0 for every L
v. satisfies the 1
Define v. • - {p·y! /p·e)e + l
budge~
I
Hence if
constraint.
this vector is chosen and no produ-=tion is carried cut (that is, y.
:.!
J
OJ. then
the resulting consumption is v! +- v. • (v~ • y~J - (p·y• tp·e)e"" x!- [p·y!/p·e)e. 1
l
1
1
I
1
1
Since (v!.y!l is maximal for >- •• we must have x~ >-. x! - (p· y~/p·e)e. I
1
monotonic£ty, p• Y(
.. 1
;c
0.
Thus p·y•
I
~
-1
I
I
By the
0.
We shaU next prove that p · y .:s 0 for every y
17-S
E
Y.
let y
E
Y.
Define v
~
(p·y/p·e)e - y, then p·v = 0. Alsc, we ha•:e
y~ + 1
Thus, v! + v satisfies the budget constraint. 1
y E Y because Y is a convex cone.
v anc the p:-oduction plan y! + 1
•
IS
Hence if the choice
v~ + 1
y are combined, then the resulting consumption
· •
(v~ 1
•
Since
(v~,y~) 1 1
+ v} + (y! + y) = 1
•
x~ + 1
is maximal for >-., we must have -1
monotonicity, p· y
:!:i
{p·y/p·e)e .
x~ >-. x~ + 1 -1 1
(p·y/p·e}e.
By the
0.
The profit maximization condition of a Walrasian. equilibrium is therefo:-e '-'. . d .
To establish the utility maximization condition, note that.
estau~1sne
•
since p · y :s 0 for every y e Y, {v. + 1
L
y. e IRL: p·v. :s p·w. andy. e Y} = {x. e 1R: p·x. 1
1
1
1
1
L
e IR : p · x.1
Hence p · x ~ :s p • w. and x! is maximal for >-. in {x. 1
1
1
-1
1
1
~
:!:i
p. w.}. 1
p·w.}. 1
A possiole interpretation of this implication is that if a single convex, •
constant retu!"'ns to scale (in fact, additivity suffices) production set Y is accessible to every consumer, then the profit maximization is a consequence of their utility maximization. ass~ption
This .result thus gives a justific<'.tior: for the
of profit maximization (to the extent that utility maximization is
jus~ified).
•
17.C.l
Let p e b., q' e f(p), q" e f(p), and A e (0,1].
We consider two
cases, p e Inter-ior b. and p e Boundary b., separ:ately. Let p e Interior b., then, for every q e a, z(p)·((l- 1\)q' + .i\q") = U- .i\)z(p)·q' + .i\z(p)·q" ~
Cl - A)z(p) · q + .i\z(p) · q = z{p) · q.
Hence (1 - .i\}q' + Aq"
E
f(p). •
Let p e Boundary
a.
If pl > 0, then
17-6
ql = ql •
= 0 and hence (1 - 1\)ql +
A.qt = 0.
Hence (! -
~)q' +
A.q" e f{p).
For each positive integer n, we let tJ.
17 .C.2
n
= {p e IJ.:
1/n}. •
Step 1:. For each n, there exists rn n
p e t. .
>0
nL
n
such that z(pl c (- r , r ]
•
for e11ery
This is an immediate consequence of property (i) and the compactness
An c. u . &-
F'or each n, define
Step 2: Construction of the fi.xed-poi.nt correspondences.
n
nL
r , r I
~
f..
n
n
nL x [- r , r ] by .
n
rcp,z) = {q e IJ.n: z·q:: z·q' for every q' e tJ. } x z(p}. •
Step 3: The fixed-point correspondences are convex-valued and upper hemicontinuous.
This can be proved as in Step 4 of the proof of Proposition •
17.C.l, by property (i), ·and the convex-valuedness of z( ·).
Step 4: For each n,
rc ·)
•
This follows from Step 3 and
has a fixed point.
Kakutani's fixed-point theorem.. Denote it by (pn,zn), then pn·z every p e l!n and zn e z(p
0
).
By (iii), pn·z
0
::
p·zn for •
0
=
0 and hence p · zn :s 0 for every
•
p e IJ.n.
Define ·
rf
L n = {z e IR : p·z :s 0 for every p e tJ. and zl :: - s for every l},
1 then nn :> rf+ , zn e
each
rf
rf,·
and hence z
0
e
is bounded; to show this, let z e
rf
for every N and n
~ ~d.
il!:
N.
Moreover,
for every t., qt = is equal to 1 - (L - l)/n
and the other coordinates are equal to 1/n.
Then
l l l (1 - (L - 1)/n)zl = q~t = Lk;elqk(- ~) :s CI:k;etqk)s = (L - l)s/n.
Hence
(L - l)s/n 1 .:. "(I.. - 1 )/n
17-7
•
L - 1 s. n - L + 1
ihus the sequence {z3 ) is bounded. Step 5: The sequence {pn}
ln Interior ll..
vector p• e 11. Interior ll.
Since
lL
n
in t. hAS a StJOsequence conver-gtng to _, prlce vector
is compact, it has a subsequence e<:JL'\\'erging to a price
n By !vl and tl:u: boundedness of li }, p• must belong to
Therefore, by taking a subsequenoe if neoessary,
that pn • p• e Interior A and Step ti: Tile ltmiC
17.C.3
lly (l), z• E
can assume
~ z• e f!L.
p• i.s a Walras!_,n equUlbrium price vector.
for all n, tl:tc upper bounds that z• ,; 0.
·i'
w~
L ~
n -
1 + 1
s for t!ll!
rr"
Since z•
o&
n"
obtained in Step 4 imply
By livl, z• = 0.
:z(p"~,
[Firil Printine: Errata: In the definition of an equilibrium with
taxes, the budget set a,!p,p·loli + 11/tli:t.ittiptxli) should be writll:n as Bi(p,p·ui + Clllll:tllptxt/
That Is, first, t!le rell!vant consumptions are
the !!Qul!ibrium ones; second, the indexes for the consumers in the sum of the
tax revenues should be something othe:r than I, because it is already belna used.
Also, the difficulty lewel should be C.)
(aJ For example, if we take
t
11
s
t
21
= t
12
= 0 and t
22
equilibrium in an Edgewortn box looks like the following
17-8
> 0, then an fil'l~-
Budget Line for Cons11mer l to) •
Budget Line for CoJtSwner 2
x•
Figure 17.C.3(a) As we .can see in tl\is figure, the N!lative prlce.: Ute consumers
an~
filced with
can be dlHerent, and hence an equilibrium with the uxes need not be a Pareto optimum,.
(b)
lElrst adnting errasa: The phrase "Ap?lY Proposition 17.C.I" should
really be •Apply the idea or Proposition 11.C.l. "I
For each I, define -r.: IRL +
I
Ti(xl) = (xt/(1 • tli), ... , XL.{(l + tLi)) for every x. e RL. +
-~
L For e11ch i, define anotn.,r preference :.-! on IR+ by the -I
following rule: For eaeh x. e IRL .and y. I
!:;I Tl(yil'
+
I
RL, x.
o&
+
>-~ y.I
if 11nd only if
I -1
Dehote by xr(p,w1J the Walrasian demand function or
!;j
T
tx l 1 1
and by
t
xi (p, w 1 the demand function of C(lnsumer i under the budget constraint
1
B (p, w J. Then 1 1
x~(p,w.J = dx~(p.w.ll l
I
I
I
L
for every x. e IR • I
p•x1 ":!i "'r
For eac!l
p
» 0
and r
Z:
•
I
T
Cx 1 1 1
fl
B.(p,w.) if and only if I
0, define
t t zi(p,r) "' x cp, p·wi + r) - wI•• 1
17-9
I
Th ls ls tltc excess dem11nd function of .;ansumer i uncle<" the bud~~:<:>t <:onst<"aint with the
tiiX
rates t~i and rebate r.
Define zt(p,rl = !1z~lp,r). N.,te thilt if p is
3£grcgate excess demand function.
This Is th~
an equilibrium price
vector and r is tile associated tax r~:bate for each ;onsumer, tben zt(p,r) "' 0. We shAll now show that, converselr. If a price vector p ;md thl!! associate tax
rebate r sati:or!es zt(p,r) • 0, then they constitute an equilibrium wltn t;,J(I!!S.
In
f~t.
t r 1x.(p,
then,
l
I:titliptx~i(p,
p·~o~. + r) • I
[.w .. l
I
Morcov~r.
P'"1 • rl t
t
'" l:tiU • tti)ptxll(p, p•ul + r) - I:liPCtt(p, p·wi + r).
lly the budi;et constraints, the first term is equal to p·li1"'i) + Jr.
The
second term ls equal to t
t
Lttplzti{p,rl + wll) • p · z (p,r) + p · lliwi l = p· lliwi). Hence Ltittlptx: cp, p·w + rl • Jr.
1
1
Thus p Is an equilibrium price vector
and r is the associated rebate ror each consumer if and only if ztlp,r) = 0.
T6/ prove
the existence of an equilibrium alottg the line of Proposition
17.C.l, note rlrst that the
a~:gregate I!!XCC'SS demand functian zt( ·) with taxes
does not satisfy Walras· law.
(That Is, p·7hp.r) "'0.)
So we modify zt(•)
to satisfy Watras' law: Detine
w t t L 2 (p,r) '" z {p,r) - (p·z (p,rl/p·plp e R , w
then p· z (p,r) = 0 for every (p,rl.
Just like the original exces.: demand
function z t l ·l. thi:; mod if"ied exc~ dem-and function z w ( • I is cant lnuous (In both p and rl.
-
w
MorcoYer, for each. fixed r a: 0, z ( • ,rl satisfies properties
(lv) and (-.) of Proposition l-,,11.2.
r(lr this, since Zt{ • ,r) Htisfles th~se
pro~rties, it Is suffi;lent to prove that p·zt(p,rl is bounded above. But this follows rrom
17·10
s [.p·x~lp, p•w. • r) • p·([.w.l + Ir l
:!;
I
I
t •
1Mu(L wti' 1
l
I
I •...• LH + Jr.
Note also that if a price veet<:..- p and a rebate r satisfies zw(p,rl .. 0 ac~a t
p·z (p,r) = 0, then they constitute an c:quilibdum wlttt taxes. w
In fact.
the~
t
t
z (p,r) = z (p,r) and hence z Cp,r) = 0.
To construct the correspondence to which the fixed-point the'Orero is to be applied, define IJ. =
Let
-t and let
r- > 0
= Maxitli: L " l, ... ,L an
s:atlsfy
utu .. illr - (ito • i llp· ri:iwi 1 > o. We shall now prove tllat p·zt(p,rl > 0 for c:very r z. r. t
In fact,
.
= ~iptlxulp, p·c.'! + rl- r.~til =
t
EuPrti{p, P'"'i • rl - EtiPflti
- -1 ~ u .. tl I:dt + •
- -1
(I • t)
t
ttt>Prtilp. p·wi •
1
(p·(~w l + lrl- p·(~wil
-
-
- (Jill + t))r - (tl!l • -tllp· 1I: w J > o. 1 1 Now detine a correspondences /: t.
w
J(p.r)"' {q
E
IJ.: z (p,r)•q
f(p,r1 '" (q
o& t.:
i:
l<
[0,
rl -) ~ by
w
z (p.rl·q' f'or all q'
p•q = 0) If p
E
E
.0.) if p e Interior
lloundary A.
Note tllat we cannot have p e f(p,rl for p e lloundary ll.
"'
lllso, since z (·I is
continuous lin booth p and rl and z"'! · ,rl satisfies properties (iiil, Civl and (Yl of Proposition li.B.2. we can prove just as in the proof
or
l7.C.I that f( ·) is
(The proof for
conve~e-valued
and upper hemicontinu'OuS.
Proposition
upper hemicontinuity can be made easier by elltablishing this property with respect to p and r so::para.tely.l
t.,
Define tt..::n another correspondence g: ll x
17-11
Note that we cannot have r•
~
g{p.rl because p·zt (p,r) > 0.
g{p,Ol either. because p·zt(p,OJ = -
LtittiPt"~i(p,
p•w ) < 0. 1
We cannot nave 0 It is easy to
check that gC • ) is convex-valued and upper hcmlcontinuous. We shall now
pro~e
that the a filced point or the prodll(;t eorrospondence
cr whicb maps
(p,r)
In fact, if [p•,r•)
ll x
1o.r:a
-i
~.
"
1o.r1.
to f(p,rl >< g(p.rl, constitutes an equilibriiJm with taxes. E
f[p•,r•)
Hence p" >> = 0.
x g):
l(
g(p•,r•), tt:en p• e flp",r"'l and r• e
o and r• e (O,rl.
Hence, as we argued berore, zt(p•,r•) = 0 and p• is an equilibrium price
vector and r• Is the associated rebate. Finally, the pl'Qduct correspondence (f " llemlcantinuous because so are [I·) and gl· l. point is guaranteed.
g){·
l is conVe')(•valued and upper
Thus tnc exlstenec of a ri.Xed
This completes tbc proof of the existence
or
an
equillbri,IJJn. (c) !first Printing terata: On the rleht·lland side of the equation defining •
B.Cp.T.), pl. should be rcphoeed by pl. I t
I
In this ease, a budget set has a.
t
kink ilt. a consumption x e 1
R~
with lCtl = wli if tt > 0.
which-0 < t i < 1. 0 < t i < I, and T > 0. 1 1 2
(If
tl
~
Here is an example in
1. then lh!i consumers
wlll not increase his disposal wealth by selling t!ieir lnltla.l endowments of commodity l.
In the case of L = 2. therefore. the budget line will then have
nonnegative slope at x 1 with "ti > "'tr)
17·12
IE
......... ···---·
0
XII
Figure 17.C.J(c)
(d1 Denote· the buylni tax rate for good
for good f by t~
t
by
t~
!!:
0 and tl1e selling tax
ra~e
The budget set e !p,Til is then equal to 1 L b {xi e R 0 : p·(xi- t.~i) + LtttptMaxfxli - wti' 0} i!:
0.
$
+ EtttptMax{"'ti - xli' 01 .s
T/
(c) The existence of an equilibrium for the modification in (c) can be guaranteed by applying the proof of Proposition !'7.1111.2.
The key fact in this
application is that the g;n-:u:·-theol'etlc approach to tllc existence of an equilibrium allows the budget s1:ts to t:.e different among consumers and to hue kinks.
The details, such as che<:kln,p, convex•va!uc:dncss and upper
hemlcontinulty, are almost identical as tM proof, so we omit them liere.
l'l.C.4
(a) According to the rule.
p·t.~
1
-
(I/Z){p·w
2
-
O/Zllp·..,
Hence, by
~o~
p-~
1
1
2
- (!12lp·(w
~he con~umcrs'
1
+
w ))" 2
after•tax wealths aJ'e
p·(~J/4lw
1
• (l/4l
- (ll2lp·l"" .. w Jl = p•((l/4lw • 1 2 1
• (1,21 and w
2
(JI4)~o~ ).
2
.., (2,11, tl:.c ;,fter-tax wealths are {S/4)p .. 1
17-lJ
(7/4)p
and (7/4)p
2
(5/4)p . 2
+
1
•
(b) Because of the after--taxes wealths in (a), the after-tax aggregate excess demand function is equal to the standard aggregate excess demand function of the economy in which the initial endowment of consumer ! is equal to {3/4}w +
(l/4Jw
and that of consumer 2 is U/4lw + (3/4)w . 2 1 2
1
It thus satisfies the
conditions in Proposition 17. C.l. •
17.C.S
(a) Let x. e IR 1
L
, IRL and x. e .
+
+
1
>-~
x! does not hold.
1 -1
I
. ' x.' , there exists a y: e Y1. w1tn 1
>-~
Then, by the definition of
Suppose that x.
-1
+
1
' e IRL sue h v. • 1 +
L that for eve::-v y. e Y. with x. + y. e IR , we do not have x. + y. >-. x: + • 1 1 1 1 + 1 1 -! l
By the completeness of the original preference >-., we then have -1
-.
J1: 0
r
By the definition of >-!, we then have x'
0
1
-1
:;,L , x. e IRL, an d x... e IRL. L et x. e ~, 1 + 1 + 1 + I
Let
•
e Y ..
y~· 1
>-~ -1
x.. 1
>-~ -1
Tnus
Suppose th a t x.
• >-.
x.1
-1
• 1
X. 1
d an
x.1 -l >-. x .. 1 I
•
e Y.1 with x:1
Tnen, by the definition of >-! , there exists a y!
1
1
>-. -1
is complete. I
! -:
X~ + V~
y~. 1
1
II
+
• ~L . y. e ~~"' sucn that x~ + y: >-. -x~ + .y~. arid there also exists a v. e Y. such 1 + 1 1 -1 1 1 • 1 1
that
x.
+
1
• v. + y .. >-. x! • 1 ... 1 1
By the transitivity of the original preference >-.,
1
-1
this means that x. + y. >-.
1 ....1
1
_,
By the definition of r •• we have x.
+ y"..
x...1
1
>-~ 1 -1
I
•
x'.'. 1
•
>-~ lS ""1
Hence
• •
trans1t1ve.
L (b) Let X. e IR , x: e IRL, and A e (0,1]. 1 + 1 +
• d Suppose th a~ x. >-. x. an x. >-. x .. 0
1 -1
1
Then, by the definition of >-! , there exists a y! e Y. such that
1
-1
• • _L d • x. + y. e ~~ an x. +
1
1
+
+
1
1
y. >-. x. + y., and there also exists a y': e Y. such that 1 -1
1
1
1
1 -1
1
1
1
-1
+ v~) + (1 - AlCX~ + y~) "1
.
1
1
= (Ax: + (1 - .:\)x':) + (;\y: + (1 1
1
1
;\)y~) >-. 1 -1
x. + y .. 1
1
•
• •• •
1
By the convexity of the originai
preference >-., this means that .:\(x~ 1
1
I
x'.' + y'.' e IRL and x'.' + y': >-. x. + y .. 1
1
•
•
Let y. e Y .. 1
II
1 -1
•
1
•
17-14
•
Since and :>..v: + • 1
(l -
•
1
implies tha-:. i\x: + (1 1
>-~
:>..)y': e Y. by the convexity of Y., the definition of 1 1 :>.. )x~ >-~ 1 -1
x .. 1
Hence
-1
•
>-~
Is convex.
-1
(c) Assuoe that, among the L goods, the first M goods are the marketed goods
and the last L - M goods are the non-marketed household goods.
We say that
constitute an equilibrium if they constitute a standard Walrasian equilibrium
By the definitior:: of the
>-~.
-1
this means that the consumers maximize prefe:-ences by
using the household technology Y. and the marketed goods 1
x~ 1
as inputs.
This
is an instance in which a non-Walrasian model can be reduced to a Walrasian one of Sections B and C after some modification, so that the results in these sections are also applicable to non-Walrasian settings .
•
17.C.6
We are going to draw the images of the aggregate excess demand
initial endovnnent has been displaced to the origin.
Any equilibrium then
col:"'responds to a price vector at which the graph intersects the origin.
(Note
however that, u,1Lke the offe:- curves of consumers, the image of an aggregate excess demanc f'u:1ction may intersect the origin more than once.) Here is an example to show that condition (i) is indispensable: •
•
•
•
17-15
a
.••
••
••
••• •• •
••.
•.
••
••
0
Figure 17.C.6.1
Here is an example to show that condition (iii) is indispensable:
0
Figure 17.C.6.2 He:-e is an example to show that condition (iv) is indispensable. that we have taken the example so that as p
17-16
n
~ (0,1),
~
co.
Note
0
Figure 17.C.6.3 Here is an example to show that condition (v) is indispensable:
0
Condition {ii) is not included in the list because it is dispensable. That is, even if (ii) is not satisfied, the existence of an equilibrium price vector is guaranteed as long as the given function z: IRL
++
other four conditions on the restricted domain RL
++
17-17
"
~
~
IRL satisfies the
of normalized price
vectors.
17.0.1
•
From the first-order conditions, for any normalized price vector p = •
•
•
It is then easy to check that
equilibrium price vector.
z
11
(1,1) +
z U,ll =
- p -
0.
12
Also, the derivative
1
Hence p
az 1(1,1}/op1 is
= (1,!)
is an·
equal to
1
---....,.......,.,..,.....---.-.. . . . .--~~-:--=---- (2
+ 211(1-p} + 2 -1/0-p))
•
Hence, by the index theorem (Proposition 17.0.2) (or the intermediate value theorem, as L = 2), there must exist at least two other equiiibria . •
17.0.2
Since rank
Df(v)
= M, we can assume, by relabeling the indexes of the •
unknowns, that the Jacobian matrix with respect to the first M unknowns •
at 1 (v)/o\11
•
•
•
is
no:-~-sing,Jla:-.
•
• •
•
at 1 (v)/ovM I
• •
Thus, by the implicit function theorem, there are open
neighborhoods A' and B' of
Cv , ... ,vM) 1
and
(vM+l''"'vN)'
and a continuously
differentiable function 1J: B' -+ A' such that for any (vl' ...• vM) e A' and • •
{vM+l, ... ,vN) e B', we have (vl' ... 'vM} = 1J(VM+l'''''vN) if and only if f(v , ... ,vM,vM+l' ... 'vN) = 0. 1
-
Hence, in the neighborhood A' x B' of v the of (N - M) parameters
solution set can be parameterized by
•
•
•
•
17-18
•
17.0.3
e IR
L-1
the excess deman.d for the first L - 1 •
commodities of ccnsumer 1 with initial endowment vector w .
Tnen D z{p;w) = wl
1
D
A
wl
z.(p;w ) e IR
(L-l)xL
1
l.
.
Denote •
•
the partial derivative of his demand function for the first L - 1 commodities •
•
with respe:::t to wealth, and by IL the (L - 1) x L matr1x 1
0
0
•
•
•
•
•
0
•
•
1
•
0
The!l
Thus, for
-z(p;w)v
D
-v
ev~ry
-v.
= -
e IR
L-1
v
• let v =
L
e IR , then p·v = 0 and hence
•
Thus the range of matrix D z(p;w) is equal
"'1
•
Thus
"'t
its rank is L - 1.
17.0.4
Denote the upper-left (L -: 1) x (L - 1) submatrix of the Slutsky
matrix S.(p, w.) e IRLxL (p. 33) of consumer i by S.(p,w.). 1
1
1
A
-
•
prove that if z(p;w) = 0 and v e IR IRL x ... x IR
L-1
We shall first
1
, then there exists (tl' ... ,t l e 1
([rSi(p,p·w ))v.
•
J..
such that [iti = 0 and Dz(p;w)(v,t , ... ,t l = 1 1
Let p, w, and ; be as such. A
1
Write e = (0, ... ,0,1) e IR;.
For each i,
L
-
define t. = (v·z.(p;w.))e e IR . 1
1
1
Moreover,
-z.(p;w. )v- +
-z.(p;w.)t.
Dz.(p;w.)(v,t.) = D D 1 1 1 p 1 1 w. 1 1 1 1 ,. .. T"" .. .. .. = S.(p,p·w.)v - (ox.(p,p·w.)/ow.)z.Cp;w.) v + (v·z.(p;w.))(8x.(p,p·w.)/ow.) 1
1
1
1
111
1
11
1
1
= S.(p,p·w.)v. 1
1
•
.A
A
A
A
Hence Dz(p;w)(v,t , ... ,t J = ~Dzi(p,p·wi)(v,ti} = 1 1
cr.s.(p,p. w. ))v. 1 1
l
The appropriate differentiability condition (Appendix A of Chapter 3, p . •
17-19
95} says that Si(p,p·wi} has rank L - 1 and negative semidefinite
(Pr-opositions 3.G.2(ii} and 3.G.3).
Since S.(p,p·w.)o = 0 and p·S.(p,p·w.} =
0, S.(p,p·w.} is negative definite on T 1
1
sum rfsi(p,p·wi}.
l •
1
p
= {v
IRL: p·v = 0}.
E
By Theorem M.D.4(iii}, the (L- 1) .
~
1
1
X
1
Hence so is the
(L - 1) matrix
I
So it has rank L - 1.
1
Therefore, the result in the previous paragraph, -
""
-
..
A.
Dz(p;w}(v,t , ... ,t l = [.Dz.(p;w.Hv,t.) = ('"".S.(o, o • w.) )v, L1 1 • . l 1 1 1 1 1 1
..
implies that the matrix Dz(p;w) has rank L - 1 (while the total endowment is fixed). Hence, by the transversality theorem, for almost all w
IRLI with I:.w. =
E
++
1 1
-
L-1 L-1 w, the r.1apping z{ • ;w}: lR ... IR has the property that rank D z(p;w} = L - 1 ++ p
for every p with z(p,w) = 0.
Hence, each equilibrium is locally isolated.
By
•
property (v) of Proposition Proposition 17.B.2, we know that there is c
E
•
(0,1)
such that if p is an equilibrium price vector, then p •
E
L
[c,l/c] .
Since
the set (E:,l/c]L is compact and each equilibrium price vector is locally •
isolated, Theorem M.F.3(ii)· implies .that there are only finitely many equilibrium price vectors.
Since each equilibrium price vector p uniquely
• •
determines an equilibrium aliocation, the number of equilibria is finite.
17.0.5
For given initial endowment vectors
(pl. 1)
·2 IR
E
•
pr1ce vector p = constitute an equilibrium if and
++
only .:\.Vu.(x.) - Cp ,1) 1 1 1 1
=0
for each i
= 1,2;
xll + x12 - wll - w12 = O; P1Cxli - wli) + Cx ~ w i> 2 21
=0
for each i
= 1,2.
The left hand sides of this system of equations define a continuously • •
17-20
•
differentiable mapping from IR
4
x IR
++
2
x IR
++
++
!R
X
4 +
4
wnere ••~.ne •
into 1?. ·
Denote the mapping by 1/J{ • ).
unk."'lowr:.s are (xl'xz'"'rA. ,p ,w ,w ). 2 1 1 2
T.'1en the
above equilibrium condition can be rephrased as the equaticr:.
It is then sufficient to •creve that the row rank of this Jacobian is 7, that is, for any (vl' v .s. t , t l e 2 1 2 2 2 IR X !R X IR X ~ X IR, if
T T (v ,v ,s, t 1
2
then Cv ,v ,s,t ,t > = 0. 1 2 1 2
,t )D.p(xl'Xz'Al,;\ ,p ,w ,w ) = 0, 1 2 2 1 1 2
•
In fact, suppose that the above equation h.o!ds.
Then, by the partial derivative of 1/J( ·) with respect to w., we know that
•
1
s(- 1, 0) -
t.p
T
1
= 0.
By the partial derivative of 1/J( ·)
Thus s = t. = 0. 1
with respect to ;\,, we know that Vu.(x.lv. -- 0 . By the partial derivative of 1
1
1
1
2
•
1/J( • ) with respect to x. and by s = t. = 0, we know that v. ·:1..0 u.(x.) = 0 and . 1 1 1 1 1 1 2
hence v. ·A..D u.(x.)v. = 0. .
1
1
l
1
1
By the differentiable strict quasiconcavity
(that is, if vu Cx ,x )v = 0 and v 1 1 2 1 1 Vu.(x.}v. • 0, we have v. = 0. 1
1
l
1
.
•
• •
17 .D.6
The system of equations that defines an equilibriuzn in this context
is the same as in the answer to Exercise 17.0.5, except that the first equation •
•
is now replaced by
With this replacement, we can continue to use the same symbol 1/J( •) to denote this new system of equations that defines an equilibrium in this context. •
the same way as before, we can prove that for any •
17-21
In
IR
X
R
X
?., if
then (v ,v ,s,t ,t ) = 0. 1 2
(We need to assume the differentiable strict
1 2
quasiccncavity with respect to the own consumption, that is, if • •
=
o ar..c
I
2
The assertion that, generically,
there are only finitely many equilibria follows from this, just as we saw in •
Exe:-cise 17.0.5.
17.0.7
{a) Assume that, for each island n (n = l, ... ,N), there are L . n
commodities and I
n
consumers.
Since there is no communication among the N
islands, an array (xl' ... ,xN) of the consumption allocations for the overall economy, where x
n
L I e !R n n is an allocation of island n, is an equilibrium· of •
the overall economy if and only if, for each n, xn is an equilibrium allocation of island n.
Since each island has three equilibriu.'ll allocations, N
the number of the eauilibrium allocations in the overall economy is 3 . • •
I\:ote that this result does not imply that the number of equilibrium price
N . vectors is 3 . In fact, .there is a continuum of equilibrium price vectors for .
L
.
the over-all economy, because, if ~ e IR n is an equilibrium price vector of island n, then, for any positive numbers
> 0, ... ,
~N
> 0, the array
an equilibrium price vector of the overall economy.
(A. q , ... ,;\NqN) is
1 1
(b)
~"!
•
[First printin,lt:
strictly convex.
· The consumers' preferences should be assumed to be
This implies that there is no equilibrium of the overall
economy generated by the replication, which is discussed in Section I.] L
n
= L and I
n
= I for each n.
Let
be the initial endowments of the L
commodities in each island.
17-22 .
•
vectors of each identical island such that q
11
= q
price of c·::Jmmodity 1 is normalized to be one.)
12
== q
13
= 1.
(T.."lat is, t,he
We shall prove tha: the
equilibrium price vectors of the overall economy are (scalar multiples of) (ql, ... ,ql}. Cq2, ... ,q l, Cq , ... ,q l e 1R 2 3 • 3 •
•
equilibrium price vectors.
LN
.
It is clear that they are-
~LN . '1' b . ) ( Suppose t hat p1' ... ,pN e ~.... 1s an equ1 1 num
p:-ice vector of the overall economy and p
11
= 1.
commodity 1 in island 1 is normalized to be one.)
= 1, ... ,N,
and n'
= 1, ... ,N
•
(That is, the price of If there are l = 1, ... ,L, n
such that pln < pln .. then the demand for
commodity t. endowed in island n' is zero, which cannot happen at equilibrium . •
·Hence, for any
t., n, and n', we have pln = Ptn' that is, p 1 = ... = pN. L
Denote the=. bv p e IR . •
Since the consur.ners' preferences are strictly convex,
•
the aggregate demand vector of each island for the L commodities at p, which •
is denoted by x(p) e IRL.
•
Then the aggregate demand for commodity t of the
overall economy is equal to Nxip) and the aggregate supply from tt1e N islands is Nwt
-w.
Hence we must have Nxipl = Nwl for every t, or equivalently, x(p) =
Thus, by p
= 1. p e {q ,q ,q } and the number of the normalized 11 1 2 3 •
•
•
equilibrium price vectors is three . •
•
•
17 .0.8
Let the consumer's utility function be u(x) =
[£a:l = 1, and his initial endowment vector be w e
with c:C. > 0 and Then his excess demand
function is given by •
z(p) = x(p,p•w) - w = (a: p-w/p , ... ,«r_p·w/pL) - w. 1
By the chain rule, for every p e IR
L ++
1
, l < L, and k < L,
where w = p · w, the wealth of the consumer at price vector p. •
17-23
Moreover,
a!ld
cxip, w)/cw
= a.lpt
2 - a.,_w/pl.
if t. = k,
0
otherwise,
Hence the (L - 1) x (L -
1)
matrix
Dz(pl
is the sum
of the following two matrices: the first one is the diagonal r:1a:rix whose lth · the second one is equal to
•
• ••
•
for every t < L,
Dz[p)p
al/pl •
=
and hence
-
•
•
•
•
This equality means that if we add to the first column of of
Dz(p)
•
Dz(p}
the tth column
multiplied by PIPl for all I. < L, then we obtain the vector on the 0
right hand side, which we denote by v.
Dz(p)
Denote by Z the matrix obtained from
by replacing its first column by v, then det Z = det
Dz(p).
We now
•
obtain another new matrix Z' by adding, for each l = 2, ... ,L - l, the vector
det Z' = det
z,
and Z' is the sum of the following two matrices: the first one
is the diagonal matrix, whose first diagonal is zero and the lth diagonal is 2 equal to - al.w/pl for each t = 2, ... , L - 1; the second one has the property •
that its first column is equal to v and all other columns are equal to zero. Thus
17-24
L-1
det Z' =
2 a w /p )) . (-
( rr (l=2 I.
l
Tne (normalized) equilibrium price vector is
•
Plugging this into the above equality, we obtain
z• = (-
det
1)
L-1
..
L 2 . L-'"1 ( II "'lat) · ('\./wL} . . l=l
By cet Dz(p) = det Z', the index of the equilibrium is + 1. •
•
By differ-entiating both sides of z(;\p) = z(p) with respect to :1. > 0
17 .£.1
•
and then evaluating at :\.. = l, we obtain (17.£.1) . • •
By differentiating both sides of p·z(p) = 0 with respect to p, we obtain
(17.E.2). • •
17.E.2
By z .(p) = x.(p,p·w.) for all p and the chain rule, 1
1
1
.
-
--
Dz.(p) = D x.(p,p·w.) + D p
1
for ever-y
-p.
1
w.
1
--
--
x.(p,p·w.)w. 1
1
--
1
T
-
1
--
--
T
Since D x.(p,p·w.) = S.(p,p·w.) - D x.(p,p·w.)x.(p,p·w.) , we p 1 1 1 1 w. 1 1 1 1 . 1 •
•
have •
-
--
Dz .(p) = S .(p,p·w .) 1
1
1
-
·
--
--
D x.(p,p·w.lx.(p,p·r.>.) w. l 1 1 1
T
1
--
--
--
--
T
+ D x Cp,p·w.)r.>. w. 1 1 1 1
T
= S.(p,p·w.l - D x.(p,p·w.)z.(p,p·w.) . 1 1 . w. 1 1 1 1 1
Hence, by sununing over i and replacing in place of p to emphasize that D
by p, we obtain (17.E.3).
-x.(p,p·w.)
p 1
-(p,p·w.);
respect to prices evaluated at
-p
1
1
.
(We used
•
is the partial derivative with
that is, D
p
does not operate for
-p ·w .. ) 1
l I. Let p, A, a , e , and x. be ·as defined in the proof of Proposition •
17.E.3
l
17-25
•
-p
in
-p
17.E.2.
For each
o. for a::ly k = • K
£_
1
(b~,1 ... ,b~) 1
for each i, the family
~
define u.: IRL 1.;.
Ll
{b~·y. 11
-
b~·x.: 11
continuous. concave, and strictly monotone. 1
l
{b~·x. 1 1
-
b~·x.: 1 1
t
Note that.
So, for each i,
is linearly independent.
IR by u.(y.) =.min
u.(x.) = min
++
1
•
t. = l, ... ,U.
Then u. ( · ) is 1
Note also that
= l, ... ,L} = 0,
l P = I:eCL - 112l qlibr -1
The second equality means that the price vector p can be spanned by the "gradient ve::tors"
b~1
of u.( ·) with strictly positive coefficients 1
• -1 (L - 1/2)-~qli, ... , (L - 1/2} qLi.
L Hence the upper contour set {y. e ~ : 1 +
u.(y.) =:: u.(x.)} is strictlv supported by p at x.. 1
l
1
1
1
~
The utility maximization
condition is thus satisfied at x. under p, implying that z.(p) = x. - w. = 1
· i T p.(a) . 1
1
l
1
. l i t t. i Moreover, smce b.· (x. + e) - b. ·x. = b. ·e = 1 for all i., the .
11
11
wealth expansion path of consumer i under p
1
be a straight line parallel
•
•
•
1
to e (locally around x.). l
contour set {v.
~ 1
S.(p,p·w.) = 0. 1 . .1
1
Hence D x.(p,p·w.) = (1/p.)e. w. 1 1 1
Since the upper
1
IRL: u.(y.L =:: u.(x.)} is strictly supported by p at x., + 1: 1' 1· 1 1 • • l 1
E
Hence Dz.(p) = - D x.(p.p·w.)z.(p) = ea. 1 w. 1 1 1 1
•
17.E.4
Here is the offer curve of a function z( ·) that is continuous,
•
homogeneous of degree zero, satisfies Walras' law, and cannot be generated from a rational preference.
The last property follows from the fact that the •
weak axiom of revealed preference is not satisfied at p and p'. ha.~d.
On the other
it is easy to check that if the offer curve does not go through the
initial endowment point as in Figure 17.E.2, then the weak axiom always holds .
•
•
17-26
p
z(p') p'
0
Figure 17.E.4 •
17.E.5
We shall prove the assertion by contradiction.
Suppose that there is
an economy that generates the given function z( · }, whose initial endowment satisfies 0 2
sets IR . +
~
-w
1
.:: 1 and 0 ::s
-w
2
-w
.:: 1, and whose consumers have the consumotion •
Let x.( ·) be the demand function of consumer i, z.( · l be his excess
for every i.
1
1
According to Figure 17.E.3, ~zli(p)
must thus have zli(p)
=-
wli' or xli(p,p·wi)
=0
similarly show that- x .Cp' ,p' ·w.) = 0 for every i. 1 21 depicted as follows.
17-27
=-
1 .:: -
-w = 1
for every i.
I:iwli"
We can
These demands can be
We
p'
•
•
p
•
•
•
'
0
Figure 17.E.S
Thi.s is a violation to the weak axiom of revealed preference.
Hence there is
no economy that generates z( • ). • •
•
L L Let p e IR , p' e IR , llpll ++ ++
17 .E:.6
= llp'U = 1,
and p
:;e
p'.
Since p · z(p) =
p' · z(p') = 0, it is sufficient to prove that we have either p · z(p'} "> 0 or p' · z(p}
> 0.
So suppose that p. a: 1
p. - p:(p·p') > p. - p: 1
l
1
0.
1
Since p·p' < llpllllp'll = 1, p·z(p') =
p~. 1
~ p~.
If p.
1
1
we can similarly show that p' ·z(p)
•
•
> 0.
• •
= llp'll
= 1 and p is
directly revealed preferred to p' by consumer i, then pi < Pi·
This follows
We first note the following observation: If llpll
17.E.7
from the fact that p·z(p') = p. 1
p~(p·p') :::= 1
0 and 0 < p·p' < llpllllp'll = 1.
Now suppose that p is indirectly revealed preferred to p'.
. .. ex1s~s
r·1mte .
a
Then there
n . h . 1 N h th 1 N • c am p , .•• ,p sue at p = p , p = p , an d p IS directly
revealed preferred to p
n+1
for all n
~
N - 1.
Hence, by the above
•
n < o b servatton, p. .
1
p~+l for all n ::s N - 1. Thus p. < 1 1
17-28
p~. 1
Hence, again by the
above observation, p' cannot be directly revealed preferred to p.
The strong
axiom is therefore satisfied.
17.F.l •
Let
s 1..
X., 1
-X,
-w.
and
be as defined just before (17. F.2} and de!"ine •
1•
c
r.1
=
L:
D=
1
1
1
• p ·w.1
Then, by (17.f.2), Dz(p) =
x)[x. 1 p·w
1 [x. 1 p·w. 1
p ·w.
p·w .
•
p·w
p ·w.
x][w. 1 p·w 1
[x. I
-
1
r.s. 1 1
o ·w. •
p·w
sufficient to prove that dp· (C - D)dp
xz(p)
T
p·w
- (C - D}.
1
-T xJ '
, -T
WJ
•
It is thus
0 for any dp with dp·z(p} = 0 .
:!::
•
•
If the initial endowments w. are proportional among themselves, then 1 p. w. 1 • w = 0 and hence D = 0. Since C is always positive semidefinite, p·w
w. 1
our assertion follows.
•
p. w.
1 -
If the demands x. a.-e proportional among themselves, then x. - - - x = 1 1 p·w 0 and hence C = D = 0.
17.F.2
The proof is thus completed.
By construction, A
•
•
•
A
-
A
,.,
-
q·z(q'l = Cpizu(a:p1,a:p2l + Pzz21Ca:p1,a:p2ll + a:Cp3z32CPj·P4) + p4z4zCPj·P4D
-
,..,
....
....
..
""
= (plzll(pl,p2) + PzZ2l(pl'p2)) + a:(p3z32(pj.P4l + p4z42(p3,p4))
-p z
3 32
-p z
4 42
q·z(q') is negative.
Cp_J,p4) < 0.
Hence if et > 0 is sufficiently large, then
Similarly,
-
•
""'
....
-
""'
""'
q'·z(q) = a:(plzll(pi,pi) + PzZzl(Pi·Pz)) + (pJz32(a:p3,etp4) + P4,Z42(etp3,a:p4)) .... ,.. ... "" = o;(plzll(pi,pi) + PzZ2l(Pi·Pzll + (p_jz32(p3,p4) + P4,Z42(p3,p4))
..
....
..
"""""'
""'
"""""
Note that p z Cpi,p2_) + p z Cpi,p2_) < 0 and p_jz Cp ,p ) + P4,z Cp ,p l does 1 11 42 3 4 32 3 4 2 21 not depend on a: >. 0.
Hence if a: •> 0 is sufficiently large, then q · .z(q') is
17-29
•
negative.
Define
17 .F .3
•
The aggregate excess demand function is then
Hence, if we let p
= (4,1,8,4)
and p' •
= (1,4,4,8),
•
•
then
z(p) = (- 1. - 1, 1/2, 114) and z(p') = (- 1, - 1, 114, 1/2) and hence p' · z{p} = p · z(p'} = - 1.
Thus the weak axiom of revealed preference
is not satisfied.
If the individual excess demand functions satisfies the (weak) gross
17 .F.4
substitution propez-ty, then so does the aggregate excess demand function (p. 612).
lt is thus sufficient to show that the excess demand function of every
consume:r has this property. 1
w. e u:i- with w. 1
1
+
=0
So suppose that consumer i has initial endowment
and spends an· his wealth on good
t.
Dencte by
et
the
. _L . II • d" . d vector m ~~ wnose (.tn coor mate 1s one an all the other coordinates are +
zero.
Then h!s excess demand function z.( ·) is given by 1
w.. 1
Thus azk!(p)/opm = 0 for every k
;e
l and m;
azli(p}/apk = wk/Pt ~ 0 for every k
:;e
l;
Hence it has the weak gross substitution property.
17 .F.S
For the g:-oss substitute property. with any initial endowments, recall
17-30
-8xki(p,p·wi}/8wi
!:
0 and
-oxki(p,p·wi)/opl
> 0 for every k and l with k
t
z
(The assumption in (d) that all u.t.( ·) are identical is not necessary for this result.)
Hence 8zki(p}l8pl = 8xki(p,p·wi)/8pl + raxki(p,p·wi}/8wi)wli > 0.
We
1
- [xtiuli(xti)/u.li(xti)J < 1.
1
1
< p < 1,
1
a:lix~i'
In fact, if uli(xli) =
th~n
then •
and
p-2
l)x l i .
Thus - [x 0 .u;.(xn.l/u~.(xD.}] = 1 - p < 1. t.l t.l (. t t.1 t.l
li.F.6
erl"'ata: The hypothesis made do not seem to be
[First
sufficient for the desired property.
We proceed to give a proof unde::- the
additionai hypothesis that the utility functions of the consumers for tr.e two consumption goods are Cobb-Douglas.) the utility function of consumer i.
.1 .DV •
W. E I
~2 ;~
+
•
.
For each
.
1
= 1, ... ,I, let u.: IR 1
t.
+
~
!R. be
Denote the initial endowment of consumer
For each l = 1,2,
consumption good
2
fun:::tion of
By the Cobb-Douglas assumption, we have
a. 1-a: . 1 l u.(x.) = u.(x .,x .) = x .x . , 1 1 1 11 2 1 11 2 t •
whe;e cxi e (0,1} a:-td t3e 2
•
vectcr of c-::msu::nption good prices and q = Cq ,q l e IR + be a vector of input 1 2 •
prtces.
Denote the demand function of consumer i by x.(p,w.l = 1
1
demand function a.'ld the cost fu:-tction for unit output by ziq} = (zlt(ql.z iq))
2
by Example 3.0.1.
17-31
;!:
0.
Then,
x.(p,w.) = (cx.w./p , (1 - cx.)w.lp l. 1
By Example 5. C.1,
1
1
1
1
1
2
1
•
- [3 ))
- 13t)lf3t)
t
1-13 ((1 -
f3t
13e (q1/q2) ).
13 t 13t 1-13t f3l)lf3l) )q1 q2 .
Let's now propose an equilibrium concept for the inducec!. economy, which will later shown to be equivalent to the standard equilibrium concept for the original production economy. 2 f . d . IR . . . x + o mput vectors an a pr1ce vector q e
:e 2 ~"+
f or mputs . . const1tute an
indiJced-econou.y equilibr-ium if (i} For every i, v7
(ii)
't"".v~ ~ L,l 1
= LC'ti(c (q),c (q),q ·wi)zl(q). 1 2
'"".w .. Ll 1
Condition (i) can be motivated from the standard equilibriu!"!l concept in the following
\\ray:
If the input price vector is equal to to q, then the
weal~h
of
•
•
consumer i is a· tha! the output • w.. 1 and the zero profit condition imolies • price ve::tcr must be equal to· Cc (q),c (q)}. 1 2 vector is xitc (ql,c (q),q •w/ 1 2
Then his consumotion demand •
The vector of inp•JtS necessary to produce
this consumption demand is LC'ti(c (q),c (q),q·wi}zt(q). 2 1
By condition Oil,
an induced-economy equilibrium is defined to be a point where the scm of these • mpu ~ vectors is less than or equal to the aggregate resource vect.oi [.w .. •
1 I
2
Suppose now that the consumption vectors (xi, .... xi) e !R+ x ... x ·
• ·
d
( •
•}
!R
2
2
~+'
vectors p and q constitute a
standa:-d Walrasian equiiibrium.
For each i, define vi
= I:extiziql.
We shall
an induced-economy equilibrium. satisfied.
Since p = (c (q),c (q)}, condition (il is 2 1
By the market-clearing condition for inputs of the original
17-32
equilibrium,
:: [.w .. 1 1
Hence condition (ii l is satisfied. 2
~ + for inputs
and q e
Suppose
For each i, define
constitute an induced-economy equilibrium.
x~ 1
=
•
xi(c (qJ,c (q),q ·wi). 1 2
For each
Let p = (c (q).c Cq)). 1 2
t,
define
zl
= (Iixli(c (q),c Cq),q · ui)}zt(ql. 1 2
We shall now prove that (xi·····xjl. input demands
2 x IR , and price vectors p and q constitute a standard Walrasian + •
equilibrium.
First, the profit maximization conditions are satisfied because
p = Cc Cq},c (qj} ai:d z£ is proportional to ziq}. 2 1
condition fa! lows from p = Cc (q),c (q)) and xj 2 1
The utili!y maximization
= xiCc 1(q),c 2 Cq),~ · "'/
As
fer the market-clearing condition for the inputs, •
r~z~ (. l!
= [ t... (t.xo.(cl(q),c2(ql,q·w.})zt(q) l c.: 1 =
•
Li 1[lxti(cl(q),c2(q),q·wi)zt(q))
C[ixti(c 1(qJ.c 2 Cq),q · "'i) lziq ),
~o~
r
= Iivi == [iwi"
• the out;;uts, since z£ =
c~.x~)z.(q). Ll ! t
W·e
hav!"-Jet..r ( z•) -
•
By this equivalence, it is sufficient to prove that there is a unique
Since it is given by t" .. inequality
reduced-ecor:.omy equilibrium.
[.[l)xl).(c.(q),c Cql.q·w.)zl)(q) - [.w. :: 0, lc.Ul lc. 11 2 it i:; sufficient to prove that for each i, the input demand function
has t!:le gr-oss substitute property.
In fact, by the explicit expressions for
Lexti (c 1(q),c2( ql. q · wi lz1e' q) 1-(3
= (la(r Hf3{(1 - f3 ll 1 1
1
.
+
((1 - ai)/,.- Hf3 /0 2 2•
17-33
It is then easy to show by a straightforward calculation that the gross substitute property holds.
In
•
. .,. . . . fact, 1. we cer me
•
1-{3
{31
0 2i = ((a:/r1HO - f3l)/f31) the:~.
o1i
+
o2 i
= 1 and the input demand function is equal to the standard 6
1·1
6
2= . demand fu=tc:ticn derived from the Cobb-Douglas "utility" f unct1on v 1 • v . ' f or • l 21
•
the inputs.
Thus the gross substitution property follows fr-om this fact and
Example l7.F.2. Note finally that the above proof shows that the uniqueness of •
equilib:-ium for- this setup can also be obtained from Exercise 17.F.9. this~
supp~se
To see
2 . . . x !R and an . +
that
input price vector q constitute a standard Walrasian equilibrium.
Jus-: as
•
above, if we define
Yi
= Ltxlizl(q) for each i, then q
·vi
~ q · wi and
Thus no trade is required to attain the allocation if the technologies o. the f l(zl) are freely available.
17 .F. 7
Suppose that z(p) = and z(p')
:;e
0.
By the homogeneity of degree zero,
B·;• Walras' + p'z (p'' 0 la w, p'z(p'J 11 22 J==.
The gross substitute property implies that if
p > Pi· the:: 0 = z {p) < z (p'} and hence p z (p') > 1 1 1 1 1
piz1(p').
Pi· then 0 = z (p) > z (p') and hence p z (p') > piz (p'). 1
1
1 1
either case, we have •
17-34
1
Also, if p < 1
Therefore, in
17.F.8
Let p and p' be equilib;-ium price vectors and ;\
;\p + (1 - ;\)p'.
Define p" =
[0,1].
E
We shall prove that z(p") = 0 by contradiction.
then, by (l7.F.3), p·z(p") > 0 and p'·z(p") > 0.
Suppose not,
Hence p"·z(p"} =·;\p·z(p")
+
•
•
(1 - A)p' · z(p")
17.F.9
> 0, a contradiction to Walras' law.
[First
· The production set Y should be convex.
Let (x , ... ,x J be
not necessary to assume that preferences are continuous.] an eq•..1ilibrium allocation such that xi - wi e Y. an eC!uilibriu:n. •
We shall prove that x. = 1
x~ 1
It is 1
1
Let (p' ,xi, ... ,xil also be
for every
•
Bv x. - w. e Y, 6e
1.
-
1
1
a.ssumpticn of constant returns to scale, and the profit maximiza::on conditicr. of an equilibrium, we have p' ·x. :s p' ·w. for every i. 1
1
Thus, by t:te utility
maximization condition of an equilibrium, x: >-. x. for every i. 1 -1
1
Since Y is
convex, the consumotion allocation • •
Moreover, bv the strict convexity of the >-., if x. ,.: • . -1 1
is feasible.
X.'
••
I
•• L.nen
{1/Zix.. + C:/2lx:1 >-.• X .. On the other hand, since prefer-ences a.:-e strongiy • 1 ' monoto::~e, t!'le fir-st welfare theorem holds, implying that there is r.o i for whom (1/Z)x.
1
17 .F.lO
+
(l/2lx: >-. x..
Let p be an
1
1
1
We must thus have x. = 1
e~uilibrium
e price vector and do •
x:L ~
L
for eve:-y i.
By the definition,
.
cdp) - z(p}).
c;:>·Dz!plcp = But by (17.F.3l, {(p + cdpi - p) · (z(p
+
cdp) - z(p)) = (- p) · z(p
+
cdp)
Hence +
cdp) - p) · (z(p
17-35
+
cdp} - z(p)} .:: 0
~
0.
and thus dp · Dz(p}dp :s 0.
17.F.ll
Let p be as in the exercise, fix an l, and denote by Z the (L - 1} x
(L - 1) matrix obtained from Dz(p) by deleting the lth row and column. • ~
is negative semidefinite and, by Theorem M.D.4(i), det Z ~ 0.
Denote bv• ~
(L - 1} x (L - 1) identity matrix. L-1 IZ /(0) = (- 1)
I and f(a:)
~ m
Define /: IR as a:
~ m
degree L - 1 and the coefficient for a:Lthat f(O} > 0.
("'1 .... - Z)v·
we must have a:
•
(because f( · } is a polynomial of is one}.
It is sufficient to show
~
0.
So let f(a) = 0, then there exists v e IRL-l such that = 0.
Hence
Since det
-Z
v · Zv
= .... "'.
s·mce
Z 1s . nega:1ve . .. ...., "d e.;·tnhe, .• S--·-1
;e 0, we actually have a: < 0.
p be as in the exercise, fix an t, and denote by
L~t
~
1R by /(ex} = I cd - Z 1. then
•
17 .F .12
-Z
the (L - !) x
(L - ll mat:-ix obtained from Dz(p) by deleting the lth row and column.
each k
= l,
-I the
Fo-:- this, it is sufficient to show that, for- a.'1y a e IR, if
f(a) = 0, t!ien a < 0. l!v'ol = 1 a'"'.·c' ,;
1
~
Ther;
•
let Z. be the row vector of K
Dz.K(o} of Dz(p). .
-Z that
For
corresponds to the kth rc\,..
L-1 Der.ote by p e IR ++ the vector obtained from p by deieting ~
the ith entry.
We shall prove that p is has the property in the definition of
a dominant c!iagcnal (Definition M.D.2), that is,
-sign pattern Zkp < 0.
But, by the definition and again the gross substitution
•
s1gr. patter-n,
--
Zkp = pk&zk(p}/opk + Lro:;et,kpmazk{p)/opm
=- lpkazk(p)/c3pkl
+ Lm~tklpmc3zk(p}/apml.
Thus 1pkozk(t:>}/opk I > ~;e£,k I pmazk(p}/opm 1.
17-36
-z
17.F.l3
[First printing
that is, p < - 1.1
: The coefficient p should be less than - l,
Let l = 1 be the home good of country 1 and l = 2 be the
home good of country 2. •
•
•
Then, by the first-order condition cf utility •
•
rna Xlmlzauon, if their demands at a price vector p are in the interior
e=
then their individual excess demand functions for
1 are giv!:!n by
•
z (p) = - (- p)l/(1-p)(p /p )p/(1-p) 11 1 2
( ) = (-
z12 P
P
)1/(1-p)(
I
P1 P2
)-1/(1-p)
•
I
·
Hence the aggregate excess demand function for t = 1 at p zl(pl'l} = ( - P )
1/(l-p)(
·
P1
-11(1-p)
- Pl
p/(1-p)
).
He:~.ce
- pl 1 nus,
by p <
p/(1-p})
.
- ....' •
:.. 1 " (.1, 1) /cp " = (- p)ll(l-p)( az 1 1 1 - p
-
p 1 ) > 0. - p
Thus the index at p = (1,1) is negative . •
17 .F.l4
--
(a) For any two different l and •
> 0.
(b)
Hence any two inputs are complementary at anv v . •
[First printinsz err-ata: Tne statement is not exactly true.
Just as il'!
the case of perfectly substitutable inputs (Exercise S.C.10}, we can find only 0, if we do not impose any further
assumption. = L - 2.
To ottain a strict complement, we should assume that rank
We s:-tould also assume that v »
s:::ale, Of(v)v = f(v).
0.]
o
By the constant returns to
By differentiating both sides with respect to v and
2 rearranging the derivatives, we obtain D t(v)v = 0.
17-37
2
Thus, for every l,
2 By the concavtty, 8 f(v)l8v t :s 0 . •
2
Thus there
f(v)
must exist an l'
:;e
l
:: 0, that is,
:: 0. ·we have thus obtained a weak complement.
2
strict complement from rank D f(v) = L - 2, suppose that there
To obtain a •
1s
•
no stnct
complement. = 0 for every t.'.
one and the
cthe~
2 D f(v)e = 0. 2
~
D f(v)
So, denote by e the vector of IRL whose lth coordinate is 2
Then e·D f(v) = 0.
coordinates are zero.
By the
•
symm~try.
Since e and v are not proportional, this implies that rank
L - 3, violating our assumption.
Hence there must exist at least one
strict com;:,le:nent . • (c) This is the same as Exercise 5.C.7.
(d) The above result says that if goods l and t' are complements, then the input demand for l will decrease as the price of good €' increases. direc!ion of change is opposite to that required by the gross prcpe1ty.
This
subs~itute
Hence the presence of complements makes the gross substitute •
property less likely to hold for production inclusive excess demand functions (17.8.3}.
17.F.15 by
~.
Denote the consumer's preference by >-, his initial endowment vector
-
and the
(ag~egate)
production set by Y.
An allocation (x• ••v•} and a
•
pr-oduce:" price vector q constitute a distorted equilibrium if (i}
q·y
~
q · y• for every y
E
x• is maximal for >- in {x -
(iii} x• = w + y•.
Y. E
IRL: +
·
Note that we are assuming that when the consumer sells 'his initial endowments, •
17-38
he receives the producer prices. The uniqueness of of the equilibrium can be proved as follows.
First,
since the production sector is of the Leontief type, by the Nonsubstitution Theorem (Proposition S.AA.2), we can assume that there is only one elementary activity for each producible good.
Then, by Exercise S.AA.2(b}, the
equilibrium producer price vector is unique up to scalar multiplication (assuming that every producible good is actually produced by means of Y at equilibrium).
Let q be an equilibrium price vector.
Suppose that there are
two differe:J.t equilibrium allocations (x,y) and (x' ,y' ). ~
Then x
= x'.
Since
is str-ictly cor.vex, this means that the two co&responding budget sets are '&'1. T
d ...... e,.,.. en,1"....
r · x'.
Tnus r · x
= r · x'.
Assume without loss of generality that r · x >
Ther:., by the normality condition, x > x'.
feasibility condition (iii).
Thus q·y > q·y'.
profit maximization condition (i) for y'.
Hence y > y' by the
But this contradicts the
The equilibrium allocation must
therefore be unioue . • Here is an example which shows that the normality condition is indispensable. t
In the example, L = 2, good 1 is the p:iimary factor,
> 0, and w = 0. 2 2
17-39
t
1
= 0,
Wealth Expansion Path {pt, (1 + u)pz)
0
Y+{w} Figure 17.F.15
Since good 2 is net nor-mal. the wealth expansion path for p can intersect the .I b our..'"'ary o.&'
17.F.l6.
more ·h ~· an once.
'V 1
= p ...... m
(a) Let
'0.
&;.
&H
.. a ..... m -- r.:••
~o •
'
is, g
~>1.}
m
(p'}
> g
m
Then, by the SGS property, a.g (p')
m
(p)
for any m
= n.
+
p
m
> a.g {pl
m
+
p , that
m
Hence g( ·} satisfies the GS property.
(b) An example tha: satisfies the GS property but not the SGS property ca:1 be ccnst:ru:::ted as foEows.
Define h: [- 1, 1]
~
IR by h(p} = p sin(n:/2p } if p
Define g: [0,2] ~ IR by g(p) = h(p - ll - p.
0 and h{O) = 0.
2
2
=
1\"ote that, as
required at the beginning of this exercise, we have g(O) = 1 > 0 anc g(2) = - 1 < 0.
The function g( ·l trivially satisfies the GS property, as i: does
not impose a.r:.y cons:raint for functions defined on a subset of iR. prove that it does not satisfy the SGS property. .!t 1s .
. . 11 r eren:1ao~e.
d"'~-
'I
We shall
For this, we now prove that
It is sufficient to prove that h( ·) is differ-entiable.
If p :c: 0, then, by the definition, h( ·) is differentiable at p and .
2
2
2
3
h'(p) = 2p·sm(n/2p) + p cos(n/2p )·(- n/p)
17-40
The differentiabiiity at p = 0 follows from limp-+O I h(p) - h(O)I /I p - o I ~ limp-+0 I pI = o. Thus
is also differentiable and g'(p) = h'(p - 1} - 1 for every p e ·
g( ·)
[0,21. We can now show that, for any a: > 0, the function a:g(p) + p is not increasing.
In fact, for every p e (0,1), h"(p)
2
2p - (1t/p)cos(Tr/2p l.
!:!;:
Hence, for every {3 > 0, there exists p e (0,1) such that h'(p) < - (3. for- every
a.
> 0, there exists p e (1,2) such that a:g'(p)
+
1 = a:(h'(p - 1) - 1) - 1 < 0.
Thus the function cxg(p) + p is not increasing.
Hence the SGS property does
[:\ote however that there exists an a: > 0 such
not hold.
(0.2] fo:- all p e [0,2].
tha~
o:g(p) + p e
To establish this, define D = {p e [0,2]: h{p - 11 ::
0}, ther.. 0 E D a:-:.d D is compact.
Consider the function p/(p - h(p - lll
Tnis is continuous and p/(p - h(p - 1)) > 0 for evei"'y p e D.
defined on D.
Let ex > 0 be smaller than the
So the minimum is attained, which Js positive. min!.rr:.um,
Thus,
th~n
we can show that a:g(p) + p e {0.2] for aU p e [0,21.1
As for the continuouslv differentiable case, write q = (r, ... ,:-} e • Let g: [O,r]f\; g(q) «
0.
~
iRN be continuously differentiable and satisfy g{O) »
~~++ . 0 and
We remark the following two facts:
-·-
For e1i"' ... Y For everv •
!:) •
e [O,r]N and every n, if p
:J E •
N [O,r J and every n, if p
n n
-
0, then g (p) > 0. n
r, then g (p) < 0. n
If p e {O,q}, then these facts follow directly from the assumption.
If not,
they are immediate consequences of the GS property.
g (p} :: 0}. n
Then; by the above facts, C c {p e [O,r]N: p < r} and D c {p e n n n
17-41
N
[O,r] : p
n
> 0}.
Thus, by the continuous differentiability (or, actually,
just by the continuity), the following two minima, Min{(r - p }/g (p): p e C } n n n and Min{- p lg (pl: p e D ), exist and are positive. n n n than those two minima. have 0 ::s ag (p} n
+
Let K
n
> 0 be smaller !\
Then, for every a: e (O,K ) and every p e [O,r]" , we n
::s r.
o
·n
I
For each n, define L = Max{lc3g (p)/8p 1: p e [O,rl }. n n n
This is well
defined because each Bg (p}/8p is a continuous function defined on the n n N
compact set (O,rl .
Then. for each ex e (0,1/L ), n
e[a:g (p) + p }lap = cx(ag (p)/Bp ) r. n n n n
+ 1 ::!:: -
a:K + 1 > 0.
Note also that o[cxg (p) + p ]lap = cx(8g (p)/8p ) > 0 for any m m m n m n
:;e
n by the GS
property. Now let K = Min{K , ... ,KN.liL , ... ,11LN} and a: e (O,K), then cxg(pl + p e 1 1 [O.rl!'\ and o[ag (p) + p ]/8p
m
m
n
> 0 for every p e [O,r)N, n, and m.
Hence g( ·)
has the SGS proper-ty. (c) Pick an o: > 0 for which the SGS property is met. 'O L
t
,i\ r· .. I
Then the function h:
'O ,;...; defined by h(p} = cxg(p) + p satisfies the conditions of the t ,r J
Ta:-ski fixec
po~n:
such that p = h(p).
theorem (Theorem M. 1.3).
Hence there exists p e [0. r l
N
By the construction of h( · ), this implies that p is an
equilibrium. A graphical illustration is given below:
g(p)
•
•I
:
- - - - - --·
~.;..--
0
Figure 17.F.l6(c) 17-42
r
•
p
(d) Define g; [0,4]
2
~
[0,4]
2
by
g (p) = - (p - 1Hp - 2Hp - 3) - P 1 1 1 1 1 g (p) = - (p
2
2
- 1Hp
2
- 2Hp
2
- 3) + p
+ Pz·
1
-
Pz·
then g( 0, 0 J = ( 6, 6 ), g( 4, 4 I ::;; (- 6, - 6), g( ·) is continuously differentiable and satisfies the GS property. Moreover,
_, g -coJ
Hence, by {b), it satisfies the SGS property.
::> {(1,1),(2,2),(3,3)}.
(e) Suppose that g(p) ::;; g(p') ::;; 0.
such that p"
1!::
p, p"
1!::
We shall now prove that there exists p"
p', and g(p") = 0.
(The existence of p" for which p" ::; Define q e
p, p" ::; p', and g(p") = 0 can be symmetrically proved. I •
qn = Ma.x{pn,p~} for each n. o:. > 0. + p
h~p)
=p
•
= ag(p)
+
s
+
1!::
cxg(q) + q
1!::
o:.g(p') + p'
p is in fa::t well defined.
= p'
.
+ q i!::
1!::
q
By the Tarski fixec point tl:eorem,
1
1!::
q
The proof is thus completed.
· max (f) We shal! prove the existence of p • symmetrically pr-ov"ac!..) Define
cxg(p)
Hence ag(sl + s
there exists p" e !q ,rl x ... x [qN,rl such that h(p") = p", :hat is, p'' and g(p"l = 0.
by
Assume that g( ·) satisfies the SGS property with
Then, for any s e [q ,rl x ... x [qN,rl, o:.g(s) + s ::: ag(q) 1 and o:.g(s)
[o.d''
•
(The existence of
mm
D
•
car. be = 0}.
For each n, let qn Clearly, if g(p)
thus sufficient to show that g(q) = 0.
= 0,
then q
1!::
p.
It is
For this, we need the followir.g le::nma: N
N
{pm} in [O,r] converges to a p e [O,r]" with Suppose that a secuence p . m m N converges to 2 e [0, r r . then :::: p fer al: m, and the sequence {g(p )}
m
m
b. fact, by the
- .g n (p)
Thus zn - gn(p) :::: 0, implying that z :::: g(p).
17-43
!$
(p
n
-
pm)/"' ... f or every n an d m. n
For each m, by applying (e) (N - 1)-times, there exists pm e [O,r]r-; such
m that g(pm} = 0 and ·q - 1/m s p s q for each n. n n n above lemma, g(q)
l!::
0 and hence o:g(q) + q
!:
q.
Thus p
m
.. q.
Thus, by the
Thus, by using the same fixed
.
point argument as in (e), we can show that there exists s
By the definition of q, s s q and hence s = q.
0.
!:
q such that g(s)
=
Therefore g(q) = 0.
(g) Analogously to the index theorem, it is sufficient to prove that, for •
eve::-y p e [O,r)n, if g(p) = 0, then (- l)Nsgnl Dg(p} I = 1.
On fact, the •
boundar-y behavior of g( ·) should require some attention to establish this sufficiency.
But we shall neglect this.
Showing the above equaUty on the
sign wili turn out to be sufficient for the application in (g}.) ~
iR by /(?.) = I ?.I - Dg(p) I, where I is the N x N identity matrix.
(- 11 • an~
N
I Dg(p)! and [().)
~
lXI
as A
~
lXI
r~. . f i\N . ) t ••e cce • lCtent or1s one . h
•
(because /( ·) is a
poiynorr:ia~
Then f(O} = of degree N
It is sufficient to show that f(O) > 0.
-.
For this, it is sufficient to show that, for any f. e IR. if i\ 0.
Define f: IR
l!::
0, ther.. f(i\) ;::
So let :\ ::: 0, then (f.I - Dg(p))y = f.v - Dg(p)v >> 0 since Dg(p}v
the GS sign par.tern, this means that ?.I - Dg(p) has a positiYe
« 0.
dominan~
•
diag::r:a~.
By Th.eor-em M.D.S(i), f(f.)
~
0.
(h) It is sufficient to show that, under the given assumptions, Dg{pl has a negative dominant diagonal. derivative of the demands for the first N + 1 goods with respect to p:'\+1" The:1, by apply:ng (l7.E.ll to the usual excess demand system for the good, we obtain Dg(pip = - Bg(p)/8pN+l' - ag(p)/cpN+! « 0. the equilib::-ium is
Hen~e
~,;nique
~
... 1
By the gross substitution proper1:y,
Dg(p) has the negative dominant diagonal.
by (g).
17-44
Hence
8'1
-
11l++
Let w e
17.F.l7
be the identical initial endowment vector of the
.
consumers and let p e IR
2 ++
be a price vector.
The demands for- good 1
constitute a uniform distribution on [0. p·w/p 1 and hence the average demand 1 is p · w/2pr
Thus the average excess demand for good 1 is p · w/2p
Similarly, the average excess demand for good 2 is p·w/2p z(p) = (p•w/2p
2
1
- w .
- wl"
Thus
2
- w , p·w/2p - w ). 1 1 2 2
This can be generated by a consumer whose utility function is u(x) (Cobb Douglas} and initial endowment vector is w.
-x
Le~
17.G.l
1
be the demand function of the first consumer for the first
( ·)
L - 1 good, with the price of the last commodity normalized to be one.
-z (p;w 1
•
1
J= x
-(p,p·w l
1
1
-
Thus
-z (p;w -- )
D_ 1 wl
--
where p =
(pl' ... ,pL-l}
Thus, for eve:-y v e lR
-
-wl"
- -
{D ... z (p;w )v e IR 1 1 wl
(1,0, ... ,0} e
L-1
~L-l,
=
...
wl
-T
the (L - 1) x (L - 1l identitv matrix. -
. with p·v = 0, we have D_ z (p;w )V" = 1 1 wl
--
1
- e)
wl
-
by the strict normality assumption.
=
-xl(p,-p1(p·Dw 1
1
1
Thus
-0. w - ) • 1
-
1)
- - Thus
-D ... z (p;w le 1
1
E {v e IR
"'t Hence {D ... z
\. 0
have
-
-(p;w
-
Moreover, w::-iting e =
:
-p·D_ z- (p;w -- le = p· {D xl(p,- - - 1
-- -
-
we
+
1
....
and
L-1
Then
L-1 v e Jv e IR :
and rank D~ •
"'1
17-45
"'t
·: p·v = 0}.
L-~
--
z 1(p;w 1) = L - 1.
•
-Dpz(p;q)
Since rank
= L - 1 (which is implied by the differentiability of
this and (17.G.l} imply that rank Dp(q) = L- 1.
p(·)},
l7.G.2
Let
-A
be any (L - 1) x (L - 1)
· L-1 (1,0, .•• ,0) e IR .
and
-- p·v < 1.
-p · v
= 1.
+
..., . negative definite matrix. -nnte
•
Since e · Ae < 0, there exists
-
-
-
-v e
Define v =
where p = (p , ... ,pL-l). 1
IR
(v,
L-1 ++
e ==
•
such that e·Av < 0 - -
1 - p·v-) e IR
L ++
, then
Now let the first consumer's preference be represented by ui (x ) = 1
-
Ccnsidel"
L-consumer economy such that the aggregate excess demand at
ail
-
p is
equal to - z 1 (p;~.} and the (L - 1} x (L - 1) submatrix of the Jacobian l
A
~-1
of the agg:-egate excess demand function is equal to A
- D
p
-z. (p:w -- }; 1
!
•
the
existence of sud: an economy is guaranteed by Proposition 17.E.2, as a?;:>lied
Then p
in the proof cf Preposition 17. G.l.
is an equilibl"ium pdce ve:::o:- of •
the (L + lh:onsumer economy, which now includes the firs:
consur:~er
as well.
•
Fer this (L .;. !}-consumer economy, we have 0 f';
d e.
...
.e.
~ra
p
;(p;~.l 1
=
A.-
1 ,
which is negative
BY (17.G.l), •
Do(w . ,)
=
•
-- -
-
- -
+
T
"'ulp
= - Avp .
.
--pe·Av>
Bv - our choice of v
Thus 0.
-· --
--
H... n~"' op(w. )/owl.
17 .G.3
1
l
-- T
1
> 0.
[First orinting e:-rata: On the third line of this exercise, "the SOS • .
property" should be "the SGS property". 1 We s h a ll prove t nat
-p max
::: p
ma.x
.
•
•
(The other- inequality
::: p
mm
can be symmetrically proved. )
17-46
For this, it
is sufficient to show that there exists p .
~
smce g(p function
max
max
~
p
max
such that g(p} = 0.
But
) = 0, we can apply the fixed point argument to the
)
ii!:
g(p
o:g(p}
+
p on the domain [p~ax,r] x ... x [p~ax,r], just as we did in •
(e) of Exercise 17.F.l6.
Thus there exists p
ii!:
pmax such that gtpl = 0 .
This
•
•
•
result is illustrated in the following picture for the case of N = 1. •
•
r •
0
p
g(p)
Figure 17.G.3 •
•
l7.H.l
Let
t
e :R . +
= {t = l, ... ,L: 1/J(p(t)) = zip(t)}/pt(tl} and N
The:1.
= {l, ... ,L}\.M. ·r 0 l•
Define M
•
a:td oniy if t. e N.
Then, by Dz(p(t))p(t) = 0,
dz(p(t)}/dt = Dz{p(t))(dp(t)/dt) = Dz(p(t))z(p(t)}
= - Dz(p(t))(!Jt(p(t))p(t) - z(p(t))).
Hence, by the differentiai version of the gross substitution property. dzl(p{t} )/d-:.
<
0 for every
l e M.
Therefore, for every
t.
e M,
< 0.
Thus l,!l(p(t)J is decreasing through time . •
'17-47
(b) The assertion was already established a~ Proposition 17.H.l.
Here we
shall give an alternative proof based on the fact that the function 1/J( ·) is a Lyapunov function. Let c
= inf
{l/l(p(t}): t
~ 0},
then t/J(p(t))
-7
-
c because 1/1( ·) is -
By property (v) of Proposition 17.B.2 and the fact that
decreasing.
is constant over t, we can suppose, by taking a subsequence if necessary, that p(t)
-7
p• for some p• e
decreasing at p•.
IR:+.
Then t/J(p•) = c and hence 1/J( ·} is not -
Hence p• must be an equilibrium price vector.
(So, in
fact, c = 0.}
17 .H.2
(a) A Walr-asian price dynamics
is given by
n:( ··)
dtt(t)/dt = z(n(t)) - p -l(n(t)), for every t.
Or, mor-e generally, it has the property that, for every t
the sign of dn(t)/dt is equal to that of
z(~(t)) -
p -l(n(t)).
~
0,
It is
interpreted as the t:rajectory of prices (perhaps in fictional time) adjusted •
by the auctionee;, who increases the price whenever there is an excess demand. A Marshallia.., quantity dynamics (:( · l is given by
for eve:-y t.
Or, more generally, it has the property that, for eve:-y t
the sign of dC(t}/dt is equal to that of z
-1
(C{t)) - p(C(t}).
~
0,
It mav be •
•
interpreted as the trajectory of quantities adjusted by the producer, who increases the p;-oduction level whenever the consumers' willingness tc pay (demand price} exceeds the marginal cost of production (supply price}. 1
(b) Let f(p) = z(p) - p
-l.
(p), then dn(t)/dt = f(n(t)l and the system is
Walrasian s~able if and only if f'(l) < 0.
_I
Also, let g(z) = z •(z) - p(z) and
17-48
d~(t}/dt
= g(<(t}) and the system if Walrasian if and only if g'(l} < 0.
By
(p -l)'(l) = 1/p'(l) and (z -l)"(l) ::: 1/z'(l), we have
f' (I) =
z'(l} - 1/p' (1),
g'(l} = 1/z'(l) - p'(l).
Since the technology is nearly of the constant return type, p'(l} is very close to zero.
Thus f'(l) < 0.
only if 1/z'(l) < p'(l).
On the other hand, we have g'(l) < 0 if and
Cin particular, if z'(l)
< 0, that is, the slope of
the excess demand function is negative, then this condition is satisfied.) This is depicted in the figures below:
z
I
p{·)
f·· -- •••• ....... ••• ---- --·
•'' ••
z(·)
• ••' •• •• •
0
1 Figure 17.H.2(b.l)
17-49
p
•
z
p{·)
1 ········ ..••.....••..•. /
.•• ••• •• •• ••
l•
z(·)
•• • •
0
p
1 Figure 17.H.2(b.2)
(c) In the following general price and quantity dynamics, the price Tr{t) is consid~red
as the price that the consumers have to pay and the quanti-:y c';(t J
is considered as the quantity that is produced (by the producers).
Then the
•
dynamics are given by •
dtr(t)/dt = z(n(t)) c;Ct), • dc';(t)/dt = n(t) - p(<(t}} . •
Tha: is, p:-ices are adjusted according to the difference
be~ween
:he
•
consumers' dema.TJ.d and the output level; quantities produced are adjusted according to the difierence between the price and the marginal cost of production. To dra\r,.· a phase diagram and argue about dynamic trajectories, let's define f(p,z) = z(p) - z and g(p,z) = p - p(z).
Then the Jacobian of this
system at (p,z) = (1,1) is given by df(l,l)/dp
df(l,l)/dz •
dg(l,ll/dp
dg(l,l)/dz
z'(l)
-
•
1 •
•
17-50
- 1
- p'(l}
Hence the trace of this Jacobian is equal to z'(l) - p'(l) and the determinant In the typical case of z'(l) < 0, then it is
is equal to 1 - z'(l)p'(l). possibl~
quite
that z'(l) - p'(l) < 0 and 1 - z'(l)p'(l) > 0.
This is
equivalent to saying that the characteristic roots of the Jacobian are either two negative rea! n:.Jmbers or two conjugate complex numbers with negative real •
•
•
parts.
But this is a necessary and sufficient condition for the system to be
locally stable.
If, furthermore,
(z'(l) - p'Clll
2
•
- 4(1 - z'(l)p'(l)) = (z'(l) + p'(lll
2
- 4 <
then the chara-::teristic roots are in fact two complex numbers. dynamic trajectories spiral around equilibrium.
o.
Hence the
This is illustrated in the
following figure. •
z
p{·)
•
1 ..............-......... . •
•• •• •• •
•• • •• •• •
z(·)
•• ••
0
1 Figure 17.H.2(c)
p
•
(d) The system in (c) is locally stable if and only if z'(l) - p'(l) < 0 and 1 - z'(l)p'(l) > 0. o~ly
As we saw in (b), the Marshallian stability obtains if and
if 1/z' ( 11 < p' ( ll.
If we take p' (l) to be 0, then th!!se two conditions
are equivaient to z'(l) < 0.
Hence the system in (c) is locally stable if and •
17-51
•
•
oniy if the equilibrium is Marshallian stable.
(e) The simplest price and quantity dynamics in this limit case is dn(t)/dt = 1 - l;(t), •
-
•
d~(t)/dt
= n(t) - L
Hence df(l,l}/dp
dfU,l)/dz
0
- 1
= dg(l,l)/dp
dg(l,I)/dz
•
1
Hence the character-istic roots of the Jacobian is ± i.
0
Thus the dynamic
tr-ajecto:-ies are s?irals that keep the same distance from the equilibrium (l.U, as decicted below . •
p{·)
z
•
1 - - - - - - - - - - - - - - - - z(·)
•
•
0
1 Figure 17.H.2(e)
p
A modified q•Jantity dynamics is given by d~(tl/dt
wher-e 9 > 0.
= n:(tl - 1 + eCl - l;;(t)),
If we define h(p,zl = p - 1 + e(l - z}. then dt!;(t)/dt =
•
. 17-52
df(l,l)/dp
df(l,l)/dz dh(l,l)/dz
dh(l,l)/dp
-
0
- 1 •
1
- e
Hence the cha!"acteristic roots of this Jacobian is - 9/2 ± (1 -
2
e /4)i.
Hence
the modified dynamics is stable and the dynamic trajectories are spirals . • •
converging to (1,1}.
17 .H.3
He!'"'e is an example of tatonnement trajectories for L = 3 in which
there is a
singl~
equilibrium that is locally totally unstable.
•
It is impossible
~o
make such a single equilibrium a saddle.
The reason is
that a saddle has index - 1 (as pointed out in Figure 17.H.2) and, according to the
ind~x
theo:-e::!l, if an equilibrium is a saddle, then there must be at
least twa ethers.
17.!.1
Suppose that we are initially given I types of consumers (>.... ,w.) (i = -1
1
1, ... ,1) and a positive, rational distribution (ttl' .... ,tt ) for the I types, 1 •
with [.tt. = 1. 1 1
For each i, let m. and n. be positive integers such that tt. = 1
1
17-53
I
n./m.. 1
1
Let K be a common multiplier of the m.. .
l
We shall now show that the
initially given distribution (trl' ..• ,n ) can be generated by (any replication 1 Note first that Kn. is a positive integer
of) an economy with K consumers. and [.Kn. = K. 1
1
1
We can thus define a K
= (>-.,w.) if )<.Kn -1
!
'1n
+ 1 ::s k :s )::s.Kir • m '1n 1 m
1
Then the proportion of the
consumers k for which (>-k•'wk•) = (>-.,w.) is Kn./K = n .. -
-1
1
1
1
Any replication of
this K-consume:- economy thus results in the same distribution as the initially •
g1ven one.
17.1.2
Suppose the the input is measured by the horizontal axis and the
output is measur-ed by the vertical axis.
Then the production set Y of the
. f . 2 pro d uctlon unctlon q = v is depicted in the following figure: •
•
...............
v1
0
-v y
Figme 17.1.2
As we can see in the picture (and also in the equality q/v = v}. Y is additive and y• = {y = (yl'y } 2
E
2
IR : if y
2
> 0, then y < 0}. 1
The nonconvexity of Y is large in the sense that marginal productivity of the input is increasing on any part of the boundary of Y, however far away it
17-54
is from the origin. •
Hence, however big the number of the consumers is (and whatever the positive price of the output is), the firm gets an arbitrary large profit by supplying an arbitrarily large amount.
Hence there is no profit-maximizing
•
production plan and there is no useful sense in which an equilibrium, even •
•
neany, ex1sts. •
17 .1.3
(a) T:"le convexity assumptions on the consumption sets and the
preferences ar-e not satisfied. •
(b) Let (p ,p , w) and (vl'v ) be the price of the high-quality good, the 1 2 2 price of the the low-quality good, the wage level, the input demand of the production of the high-quality good, and the input demand of the pr-oduction of the low-quality good at an equilibrium (assuming that it exists).
We want to
•
t~at
prove
v > 0 and v 1
2
> 0.
We shall fir-st show that p and p
2
1
> 0, p
2
> 0, and w > 0.
The poshivity of p
1
fellows from the utility maximization condition of an equilibri'..1m.
Suppose now that w
~
0.
Then the profit maximization condition implies that •
the supply of the
low-qua~~tY
good r.:.Jst be one.
However, the wealth of the
poor- does not exceed zero, implying that the demand for the low-quality good is zero.
The feasibility condition of an equilibrium (together with p
2
> 0)
is th':.1s viola-ced. Hence w > 0. Since w > 0, the rich, as well as the poor, get positive wealth. the demand for the high-quality good must be positive. the low-quality good, since f2(v ) 2
~ co
as v
2
that of v . 2
17-55
Hence
Hence v > 0. 1
-+ 0, the positivity of p
2
As for imolies •
(c) We assume that each agent has one unit of labor and normalize the total population of the economy to be one, so that the populations of the rich and of the poo::- are both l/2. capita
(This means that consumptions are measured in per
Let lp ,p ,wl and (v ,v l be the equilibrium valu!!!s as in (b). 1 2 1 2
t~l"'m.)
Since 0 < v < 1 and 0 < v
We ca:! assume that w = 1.
p t
2
.
cv > = 1. 2 2
Hence v
2
1
= (f3p l
1/(1-~)
2
.
, f
cv l 2 2
.
= (~p l
2
.
< 1, p 2 1
f3/(l-f3)
/0- )
•
= w = 1 and .
, and the prof1t 1/(1-
1/(1-
Hence the (per capita) wealth of the rich is
\\o"e shall now prove by contradiction that p poo:-
d~
th~s
have p
d~mand
not
2
~
1/2.
!ow-quality good.
2
= 1/2.
If p
> 1/2, then the
2
the low-quality good at all, contradicting v_ Suppose that p
2
2
> 0.
< 1/2, then the poor concentrate on the
Since their wealth comes exclusively from labor,
equal to 1/2 (in per capita term).
We must
i~
is
Thus the demand for the low-quality good
•
and the rich's (per capita) wealth is 1/2 + (1 - {3)/2 = 1 - /3/2.
By p 1 = 1. this •
is the (pe:- ca?hal demand for the high-quality good; by t Cv l = v , this
1 1
1
= 1 +
13<2
2113
- l)/2 > 1, a contradiction to the feasibility condition.
We have shown that p
2
= 112 if an equilibrium exists.
that this in fact constitutes an equilibrium.
We shall now prove
Note first that \"
2
==
This and the feasibility Also, the profit from the production of the low-quality good is
17-56
•
((3(3/(1-(3} - (311{1-(3})(112)11(1-{3)
and the (per capita) ·wealth of the rich is 112 + ((3(3/(1-{3} - 131/(1-(3))(1/2)1/0-(3).
This is also equal to their (per capita) demand for the high quality good. The r.tost important part of this exercise lies in the consumption of the poor, who has the role of clearing the output markets because of the multiplicity of More specifically, since p = I, p = 112, and each agent is 2 1
thei:- demands.
endowed with one unit of labor, the demand of each of the poor consists of one unit o: t!ie high-quality good and two units of the low-quality good.
We can
thus p!'escdoe the fractions of the poor who choose one of these two demand vectcrs to clear the
ma::k~ts
in the following way:
Tne fractior:. (,8/2)/3/{l-/3) of the poor consumes two units of the lew-quality geed.
The (per capita) consumption for it is thus
.2· ({3/2)/3/(l-(3}. (1/2) = ((3/2)(3/(1-/3) .
The fract1on 1 - ($/2) 1-.igh-quality good.
/3/(I-(3)
of the poor consumes one unit of the
The (per capita) consumption for it is thus •
It is easy to check that the feasibility condition is satisfied for labor and the two goods. To sum1narize, ther-e exists a unique equilibrium of this economy, in
....
"'hi "" "' c"-·
(pl'p ,wl = (1.112,11; 2 v. = 1 - (f.V2)1/(l-(3) and
.
1 eac!'l of the rich consumes 1 + ({313/( -/3) - (31l(l-f3} }( 1/2 1/3/(l-(3} units of
the high-c;uaiity good; .
the fracuon ({3/2)
/31(1-/3)
· of the poor CQnsumes two units of the
17-57
low-quality good; and .
the fract10n I - (/312)
{3/(1-13)
of the poor consumes one unit of the
high-quality good. • •
17 .AA.l
Let x and x' be Pareto optimal allocations such that
(u Cx ). ... ,u (x ll and (u Cxil, ... ,u (xjll are proportional to (s , ... ,s )
1 1
o.
»
1 1
1
1
1
1
By Pareto optimality, we must have Cu Cx ), ... ,u Cx )l = 1 1 1 1 Define another allocation x" by x': = (l/2lx. + (l/2)x: 1
for each i. if
-.... x'1..
X.
1
I
This is feasible and, by the strict convexity of the prefere:1ces, •
then u. {x:·) > u.(x.) = u.(x:). 1
1
1
1
1
1
Thus, by Pareto optimality, we
must have x. = x: for every i, that is, x = x'. 1
l7.AA.2
1
1
The proof is thus completed.
(a) It is sufficient to prove that the mappings x:
I
IR- ar-e well de:ined and continuous. +
.u.
~
_u 1t1 ·+
d an p:
By assuming that the u.( ·) are 1
co!'ltir.uousiy differentiable and noticing that for every s, there exists an for which. p(s} =
~u.(x.(s)) !
1
(if x( ·) is well defined}, it is then sTJfficient
to prove that x{ · ) is well defined and continuous. (u. (x l. ... , u.(x.l). 1
1
I
1
Write u(x) =
•
L
We shall first s!iow the well-definedness.
-x
there exis!s a feasible allocation
By [.w. >> 0 and u.(O} = 0, 1 1
1
such that u(x) » 0. Hence (bv• free
disposal). for each s e t:., there exists a feasible allocation x such that u(x) is propo:-tio:1al to s.
Since the set of all feasible allocations is compact
(Proposition 16.AA.l:, for each s e t:., there exists a feasible allocation x such that u(x) = As and. if x' is another feasible allocation for which u(x') •
= A's, then A
:!: :\'.
The strong monotonicity then implies that x is Pareto
optimal (which can be proved by the argument in the answer to Exe!'cise
17-58
16.E.l(b)J.
Moreover, by the strict convexity of the preferences, there is no
other feasible allocation x' for which u(x) = u(x').
So, by taking x(s) to be
this x. we know that x( · ) is well defined. We
n
Let {s }n be a .sequence in .
shaH next prove that x( ·) is continuous. •
converging to an s e D..
By the compactness of the set of feasible
a.llcca:ions, we can assume that the sequence {x(sn)} allocation x. U', where
converges to a feasible
n
Also, for each n, u(x(sn)) belongs to the boundary of U and to
t: was defined in Section 16.E and U' in Appendix A to Chapter 16.
By Proposition 16.AA.2, U is closed and U' is bounded. n
that the sequence {u(x(s ))}
n
conve.rges to a point u on the boundary of U, By the continuitY of the u.( ·}, u = u(xl. ~ 1
which is included in U itself. Hence u e U'.
Therefore, by Exercise 16.E.l(b), x is Pareto optimal. n
-
that u is proportional to s, as u(x(s }) :: u(x) » a!location
-x
Hence we can assume
Thus x = x(sl,
establishing the continuity of x( • ).
•
(b) For each s, '.g.(s) = p(s) · ('.w. - '.x.(s)) = 0 because '.x.(s) =
(c) If s. l
.£...1 1 .
= 0, then u.(x.(s)) = 0. !
•
1
1
L..
1
Hence p(s) · x.(s) = 0. 1
1
w. >> 0, p(s} ·w. > 0.
.L.l
1
'.w .. L.l 1
Thus, by the strong monotonicity and
1
Par-eto optimality, x.(s) = 0.
~ote
0, where the feasible
was introduced in the preceding paragraph.
.L.l 1
•
On the other hand, by
Hence g.(s) = p(s) · (w. - x.(s)} > 0. 1
1
1
(d) Our ex:stence proof here will proceed in the same way as the proof of Pr-oposition l7.C.l.
Namely, we shall first construct a mapping to which the
fixeC. poi:tt theorem is applied and then show that, by the construction, a fixed point corresponds to an equilibrium . •
s. + rna x
Define a mapping /: tJ. •
~
D. by f.(s) = 1
17-59
1
{g. ( s), 0} 1
for each i and
each s e D., then this is a well defined continuous mapping. 1
1
1
= s.
then f(s)
1
e
5
In fact, then, g.(sl > 0 bv (c) and
D. and s. = 0 for some i, then f(s) ~ s.
hence f.(s} = g.(s) > 0 = s..
Note that if
1
-
»
Note also that if s e D., s
0, and g(s}
= o,
In fact, then, there exist i and k such that g.(s) < 0 and
•
1
gk(s) > 0 by (b), and hence 1 + l:hmax{gh(s),O} >· 1 and si + max{gi(s),O} = s., implying that f.(s) < s.. 1
1
g(s•) = 0.
Therefore if s• is a fixed point of f( • }, then
1
Brouwer's Fixed Point Theorem {Theorem M.I.ll guarantees the
existence of such an s•.
l7.AA.3
For each A e D., consider a maximization problem:
LI r.A.U. (x.) s. t. [.x.
Max
1111
IC
XE:n
11
::5
-W.
+
By the c:lmpactness of the set of feasible allocations (Proposition 16.AA.l}, a
solution always exists.
Suppose that whenever x is a solution to this •
preble=, we have x. e RL 1
+-+
for every i with A. > 0. 1
Then x. = 0 for every i 1
•
sucn •
that i\.'Vu.(x.) = p for every i with A. > 0. 1
1
1
1
We can thus associate with every ;\ a solution x(i\) to the above •
m.axi::nization Problem and the unique p(;\) e IRL such that .:U7u.(x.(A)) = p(;\) •
for every i wi:h
+
i\~
•
> 0.
Then define a mapping h: D.
h(;\} = (p(;\)·(w
1
~
IR
1
I
1
by
- x (A)), ... , p(;\)·(wi - x (;\)). 1 1 1
Then (p(i\),x(;\)) is an equiiibrium if and only if h(;\) = 0.
17 .BB.l
Her-e is a g:raphical example in which a Walrasian quasi equilibrium •
with
s~rictly
positive prices is not an equilibrium, and the given four
conditions are satisfied.
Here I = 1,
x1
=
17-60
(1,1), and the preference is depicted in the figure by means of the
Note that the indifference curve at w is tangent to the 1
indifference curves. boundary of Xr
X21
2
Ca>l :
•• •
••• •• ••
1 2 Figure 17.BB.l
0
e:~.cbwn:ent w
The initial
1
Xll
and its supporting price vector p = (1,1) constitute
the unique Walt·asian quasiequilibrium.
But it is not a Wah·asian equilibrium
because, ir.. the budget set of p = (1,1), he could afford (0,2) e
wr
strictly bette:- tha.: ;~
·••
t"n .. t•x"'" be,...-us.. • - "• -a - •
t'n
'
x1,
which is
This example does not contradict any result given this example, there is no
-x.
1
e X. such that w. 1
1
~
x.
1
and w. = x .. l
17.BB.2
l
We shaE prove the assertion by contradiction.
. ' IRL . Wltn p e . anc p
...
$
Walrasian ecn:ilibrium. •
Suppose that (p,x)
0 is a Walrasian quasiequilibrium, but it is not a
Then, by Proposition 17.88.1 and the assumption that
the consumptio::1 sets are nonnegative orthants, there exists a consumer h such that p · "'n = 0.
Also, by Iiwi »
0, p · wk > 0 for some k.
So define H = {h:
p · wh = 0} an.d K = {k: p · wk > O}. then H and K constitute a nontrivial
17-61
partition of the consumers.
An application of the indecomposability condition
implies that the consumers in K desires some of the commodities OTW"!led by the consumers in J, say, commodity
t.
preference-maximizing for every k
Again, by Proposition 17.88.1, xk is E
K.
Since (some of) the consumers in K has
strongly monotone preferences for t, we must then have pt. > 0. for some h
E
Hence p · wh > 0
H. a contradiction. •
17 .BB.3
In order to argue the existence of an equilibrium, first of all. we
assume that w.
1
» 0 for both i; without this, only the existence of a
Wal:-asian quasieqcilibriurn is guaranteed (Figure lS.B.lO(a)).
We also assume
that the initial endowments (w ,w l do not constitute an equilibrium; if they 1 2 do, there is nothing further to be proved.
A final additional assumption is
that the preferences are representable by differentiable utility functions. This assumption enables us to discuss tangency of offer curves and •
indifference curves, although it is not strictly necessary; an analogous arg•Jment is available for continuous, quasiconcave utility function as well, •
but it requires some knowledge on nonsmooth analysis, which we omit here. Given these assumctions, the indifference curves of the two consumers • that go though the initial endowment point must intersect.
Since the offer
curve are tangent to those indifference curves (p.518), they also intersect. This fact is illustrated in the figure below:
•
17-62
•
~--:-.----~ '-----------~ Oz
OC2 OCt··....
••
••
•• ••
•
•
• • •
•
Figure 17.BB3
It is thus sufficient to prove that the offer curves intersect once again at a point not equal to the initial endowments (p. 519}.
In other words, we need
to show that they do not get stuck within the Edgeworth Box . •
Since the Edgeworth Box is bounded, in order to show that the offer •
curves cannot get stuck within it, it is sufficient to show that offer curves extend indefinitely.
Formally, this can be established by proving the '
•
•
Here, by saying that OC.(p)
following t:t-.ree assertions for both consumers i.
1
•
is well defined, we. mean that there is the unique utility maximizer under p.
n
2.
Suppose that a sequence {p } in IR
oc.• (p
n
2
•
) is well defined for every n, and OC.(p) 1s not . 1
•
n
with p
converges to a p e
~
0,
The:'!
n
Max {OCli(p ),OC i(p )} 2 3. In
If
D .
2
e - IR ... , then OC.(p) is not well defined. !
fac~. take the set of normalized price vectors to be S
Denote by D. the intersection of S and the set {p e !
defined}.
tFl:
= {p e
2
~ :
llpll = 1}.
OC.(p) is well 1
Sta::-ting from any strictly positive _price vector on S, move a price
17-63
vector p along S towards (- 1, 0) or (0, - 1) until it gets out of D..
By
1
Assertion 3, D. does not intersect with - IR 1
outside D.. I
2
and hence p will eventually be
+
By Assertion l, S\.D. is closed and hence, for each direction 1
towards (-1, OJ or (0, - 1), there is the "first" vector p• to be oUlside D .. .1 •
•
•
Then. by Assertion 2, Max{OCli(p),OC iCp)} -+ 2
as p -+ p•.
lXI
Hence the offer
cur-ve extends indefinitely. Suppose that the set {p e
We shall prove Assertion 1 by contradiction. 2
IR : OC; (p) is well defined} is not open.
Then there exist a sequence {pn} in
•
S'\D
1
conver-ging to a p e Dr
By "'i
»
0, the correspondence that associates
2 . . b d { IR ·eacn p to tne u get set x. e : p·x. 1 + 1
t
•
.
p·w., 1s contmuous.
~
1
Hence there
. . IR2 sueh t h at p n ·x.n -< p n ·CJ. an d x.r. .. OC . (p ). ex1sts a sequem::e { x.n} m 1n + 1 1 1 1
. h { n} 1 now constr-uct anot er sequence y. s h a'
.m IR2 sucn . t h at p n · y.n
1n
>-.
-:
x~. and lly~ - x~ll = 1. 1
1
+
1
~
We
n n o ·w .. y.
•
1
1
In fact, since. OC.(pn) is not well defined, there
1
1
An 2 n An n -n n -n n must exist a y. e iR such that p ·y. :s p ·w., y. >-. x., and llv. - x.ll > 1; +
I
1
~ pn·c.;! and llxi - x~ll :s
1
1
1
1
"1
I
1}, Which ·always exists by the compactr..ess, is the
2
utility maximize:-- on the overall budget set {x. e iR: p·x. :s p·w.}. 1
n
v. = • 1
1
~-:-n----
n lly.- x.ll 1
n
n
then p ·y. :s 1
-n
y.
1
+ (1 -
+
1
1
1
n
) --=-~---X.,
n
r.
1
I
lly. - x.'ll
1
1
n n nb h . ,... · w., v . >-. x . y t e stn ct convexity, and lly~ t"' 1 "1 1 1
1
n Hence the sequence {y.}
n . d . ..-l y. 1s as es1re ... 1
s'.lbsequence if
1
ne~essar-y,
is bounded.
n
Define
- x~ll = l. 1
Thus
By taking a
we can assume that it is convergent.
Denote the
limit by yi' then it is in the budget set under p and satisfies llyi - OCi(plll •
= 1 and v. >-. OC.(pl by the continuity. .. 1 -:
(112}y.
I
+
1
By the strict convexity, we obtain
(1/Z)OC.(p) >-. OC.(o), which is a contradiction. 1
1
1 •
•
Assertion 2 can also be proved by contradiction. •
17-64
Suppose that ther-e is a
sequence {pn} in D. converging to a p e S'\D. such that the sequence 1
1
{Max{OC .(pn).OC .(pn)}} does not diverge to infinity. 11 21 n
Then, by taking a
subsequence if necessary, we can assume that OC.(pn) converges to a x. e 1
1
Then p ·xi :s p · wi and, by the continuity of the budget corresponde!}ce an~ of 1 -1
n
OC.(p }.
1
1
1
This is a contradiction.
1
In order to pr-ove Assertion 3, it is sufficient to prove that if p 2
~ ++ •
-
1
wei~
OC.(p) is 1
2
2 ++
defined.
{x. e iR : 1 +
By p e - !R2
++'
So suppose that p
+
X. l
iR
E -
++
and
2
>-. OC.(p)} c {x. e lR : p ·X. 1
-·
+
1
1
~
p·wJ. 1
2
•
tne set {x. e IR : p·x. =: p·w.} is compact, and hence so is {x. e
x.1 >-. OC.(p}}. -1 1
+
is equal to - IR •
2
By the utility maximization,
2
iR :
E
then OC.(p) is not well defined, because of Assertion 1 and the fact
that the clos'..lre of - !R
2
-
+
1
1
1
1
By the continuity, then, there exists a maximum on this
2
set, which is in fa:::t a global maximum on the whole IR . +
•
But this contradicts
Assertion 3 thus follows.
the local nonsatiation.
•
It fcliows directly from Assertion 3 that at equilibr-ium at least one •
•
•
pr-ice must be pcsitive.
17.88.4
The:-t
S.mce p · (" L·x.• 11
p·y7.
Thus, by (ii. p·yi•
p·y
•
1
r 11 .w. - L"' .y.·> JJ
for all y
1
E
Yr
=
o
~-X~ Ll
1
= ".w. L; 1
+
v'• .I
· ·oy· (1u .... 1, •c · y1•• agam
Hence (x•,yi•·yz, ... ,yj.Pl
is a Wa1rasian quasiequilibrium. •
•
17.88.5
Suocose that x. • '
lCX
= cxx. + (1 - a)x~ and we have both X. >-. X. l
1
•
X~ 1
>-.
X. . 10: 1
Bv the continuity and the local nonsatiation, there .
17-65
1
1
. . ex1s ... s
10:
an
and x~· E 1
1
1
X. and X. 1 1« 1
X."
X. such that x. >-.
1
>
>-.
1
II
>-.
>-.
X. 1 1
By the convexity,
X. . .1 1«
cx)x~
x. = a:x. + (1 1a: 1
1
>-. X"• • -1 1
which is a contradiction. •
•
Let zj e
17.88.6
y·:
and
J
-y/x,y,p),
e
J
a: e 10,1), y.
J
JO:
Then p·v. = a:p·z. + (1- a:)p·z'. ·Ja: J J
Y ..
J
Hence y. e y .(x,y,p). JCX J Let q e p(x,y,p), D..
z'. e
-y .(x,y,p),
Thus
-y .{x,y,p)
il:
= a:z. +
a:)z'.,
(l -
J
J
= p·y"J
a:p·y·: + (1 - a:)p. y': J J
••
is convex.
J q' e p(x,y,p), a: e (0,1), p
ex
= cxq
+ (l -
a:}q', and p" e
Then
cr.x. 11
L·Y .}·p JJ a:
[.w. 11
= cx([.x. - [.w. - [.y J·q + 1 l l 1 J J ~
a:([.x. - [.w. - [ .y.) · p" + (1 - a:)([.x. - [.w. - [ .y.) · p" ll
L..: 1
ex
JJ·
11
= ('.x.-
Hen-:;e p
a:H[.x. - [.w. - [ .y .l· c;' 11 11 JJ
(l -
'.w.L..l 1
JJ
11
'.yJ·p". L..J J
-p(x,y,p).
e
11
Thus
-p(x,y,p)
•
•
convex.
IS
To prove that · · t n d n ( n n n) h em!c::~r.ttn'.10us, 1e. z . -') z . an z . e y. x ,y ,p . J J J J
-Y/X,y,p).
n
A
n
Let y'. e Y .. then p ·zJ. J J
Hence z. e v.(x,v,o). J -J - • To prove that p(·) p(xn,yr:,pnl.
n
i!:
-y .( . ) J
•
1s
We need to s!:.ow that z.I e
..
As n -') m, we obtain p · z.
p ·y'.. J
J
is upper hemicontinuous, let qn -') q and q
We need to show that q e p(x,y,p). l
As n
~ ~.
1
-
n n r.w.- r.y.)·q 1 1
J J
, n (L...x.
~
l
1
-
\L..w.- L...v.)·p. \n, l 1
J. J
we obtain •
o::.x. - [.w. - [ .y .> •q Hence q e
-p,x,y,p).
11
11
JJ
~
C[.x. - [.w. - [ .y .) · p'.
17-66
11
n
Let p' e D., then
•
n ([.x.
upper
11
JJ
E
• ~ p·y ..
J
We assume that there are I consumers, indexed by i = 1, ... ,I, with
17.BB.8
•
preferences >-. on the consumption set IRL and initial endowments w. e IR +
-1
1
of +
labor. •
(a) We say that a vector (x•, v• ,p) in
X
. ••
•
X
ts
an
•
equilibrium if For every firm j in sector 1.,
( i)
•
for everv v n • •
<.J
(iil For ever i.
:?!
p·x~
and if x. >-.
1
1
• then p·x. > pL•w. 1 1 1
+
X. •
(iii) For
t < L,
[.x~.
1 <.1
1
=
and
r.c.,J.v~. <.J
= [.w .. 1 1
•
•
•
(b) Ccn.C.i:ion
(!}
of Proposition 17.BB.2 is clearly satisfied.
We assume that
w; > 0 for all i, which guarantees that every Walrasian quasiequilibrium is
•
•
actualiy an equilibrium. Let
-w
=
~.w. L.l .1
> 0 be the total
amount cf good I.
< {3 I.
0
::5
w·e can show that condition (iii) is met as follows: endo~ment
of labor.
Then the maximum producible
is attained by using input ~.>/ J t for each of the J I. firms
1} ir.. sector
(by
l, which yields the production level _Pt.
J · 7 •w t t
f3t
_
• (w/J ) l
•
Hence the set of feasible allocations is bounded, and thus com;:,act. • _P t•S i. 7n'W
c.
·JD
1-13I.
(,
.
L
A
for every t, and defme X. = {x. e !R : 1
-
-w
- 1
::5
yLI.j
::5
0, 0
+
1
::5
yUj
::5
Xn. ::5 <.1
r} and
r,
and zklj = 0 for. any k ~ t, L.} (Note that
•
ts not a subset of Y lj' unlike the 17-67
-Y j
in the p:--oof of.
Let r >
But we will construct a fixed point correspondence so
Proposition 17.88.2.
that, at a fixed point, the production plans are in the Ylj") Define the wealth functions w.( •) and the best responses corresponde:1ces
-x.( ·) 1
and
-p( ·)
1
just .as in the proof of Proposition 17.88.2.
The best response
correspondences are convex valued and upper hemicontinuous. As for the production sector, condition (ii) is not immediately satisfied by the current model because of the externalities.
However, for each industr-y
l, once we fix the value of the coefficient al by determining the labor inputs of the fir-I:l.s in the sector, then the _production set of each firm satisfies (iiI.
Thus we can define the best response correspondence
sector l by taking
-Ye}x,y,p)
-y i} ·}
firm j in
to be the set of profit maximizers on
-y t.} · ) of
~
L/ujl (-
~l zLlj)
is convex valued and upper hemicontinuous.
}.
Thus there exists a
fixeC. point (x•,y•,p•) with Ytj e Ylj' implying that it corresponds to a Wah·asian equ!librium. • •
(c) Given using
a:~
\'I/ JR.
0
i:-..put level v1.' the maximum output level of good l is attained by
u:lits for each of the Jl. firms (by 0 <
~l ~
ll, yielding the
aggregate p!'"ocuction level
Hence the agg" egate production set of sector l is given by this aggregate p!"'oduction func:ion
-f t' ·).
It thus exhibits increasing, constant, and
decreasing returns to scale, respectively if and only if pl = 1, and pi. + f3 i. < l.
~ ~t
> l, pl + {3l
\\r'hile there exists an equilibrium in the cases of
constant and decreasing returns to scale, there is no equilibrium in of increasing returns to scale, because
unbou~dedly
17-68
th~
case
large profits can be made
at any (positive) output prices . •
{d) If the exte:-nally of sector t. is internalized and the aggregate production set exhibits increasing returns to scale (that is, pl. + (3t. > 1), then the sector can attain an arbitrarily large profit.
Hence a Walrasian equilibrium
does not exist.
(e) We assume that p
1
> 0.
Then, by
a1 = 1,
the individual production
function exhibits constant returns to scale and the aggregate production function exhibits increasing returns to scale.
Because of the quasilinearity,
there exists a normative representative consumer (Example 4. D. Zl and his (direct) utility function (Exercise 4.0.4) is also quasilinear. uCx ) + x . 1 2
Let it be
We. assume that u( ·) is differentiable and strictly concave..
Let
v• > 0 be the equilibrium level of labor input and v .. > 0 be the socially optimal !eve: of labor input.
They can be graphically illustrated as follows .
•
•
•
•
X2
'' ''
•• •• •
•• '' ' '' •
•
•• •• •• ''
..
w - v** w - v*
• XI
0 Figure 17.BB.8 We shall now analytically prove that v• .< v••.
17-69
Note first that the
-1 equilibrium price of good 1 (with labor being the nume:-aire) is equal to 0:1 -1 -p • 1 Thus the budget constraint of the representative consumer at the 1"1 v 1 .
equilibrium is given by -1 l
•
v•
1
•
-p
1
x
~
1
-w.
-
p1+1
-
Hence, for every v :s v•, the consumption vector (7 v • , w - v), which is 1
made available by using the labor input v, satisfies this constraint, because -1
;
1
-p
v•
p +l
1
1
(a' v
+w-v=w+
)
1
- 1) :S
-W-
Thus
(w
+
- v) :s
By the strict concavity of u( · ), the demand is unique and hence, for any v <
(w -
+
We ml.!st thus have v••
~
v•_
< u(-r1v•
v)
p +1 1
) +
To show that this weak inequality actually holds
with equality, note that the first-order necessary condition for •
•
•
max1m1Zat10n at. the equi!Jbrium
•
1s•
tn~
utility
•
g1ven by
•
p1+1
u' (a' v•
)
1
-
-1
cxl
-
-1
1"1
• v
-p
1 •
On the othe;-- hand, the first-order necessary condition for- the so::Ea.l optimum is given by -1 l
By p
1
l
-p v..
-1
•
1
(p
1
+ 1)
.
> 0, these two first-order necessary conditions are not satist'ied at the
same labor input_
17.88.9
Hence v• < v••.
\\i'e shall only outline how the existence argument for the
two-player-game approach should checking
con~..-exity
proce~d.
The details of the proofs, s\.!ch as
and upper hemicontinuity, are exactly the same as in the
17-70
•
proof of Proposition 17.88.2, so we do not provide them here.
-y .:
First, for ea::h j, define a correspondence be the set of the profit maximizers over
-Y. J
1::. ~
J
J
bv taking -
-y.(p)
to
J This is a convex valued,
under p.
Associated with
upper hemicontinuous correspondence.
-Y .
•
-y .( · ). J
consider the .
•
profit function n .: 1::. J
~
IR +
by letting
1t
.(p} be the maximized profit.
This is
J
a continuous function. For each i, define the wealth function w.: 1::. 1
".6 . .n .(p). Lj 1J
J
1
1
1
I -1
1
1
2:
= [.p·w. + [.1r .(pl. Define then the l 1 J J ~ X. by letting x. e x.(pl if and only
Moreover, [.w.(p)
-x.: 1
w.(p} anci x. >-. x: for all
"!!:
1
J
individual demand correspondence p . X.
IR by w.(pl = p·w. +
Tnis is continuous (as so are the n .( · )) and satisfies w.(p)
p ·xi for ever-y p e fl.
l' f
~
1
1
1::.
Xi
1
l
1
• e Xi satisfying p·x.I < w.(p). 1
•
Then it
is convex valued and uooer hemicontinuous. • •
De,;·1ne Z = [.X. - <[.w.} - [ .Y .• then it is a convex, com:>act subset of • 1 1
L R .
J J
De:ine a correspondence z: 1::.
then it is
-p:
1 1
c~nvex
4
Z by z(pl = [.x.(o) - {[.w.} 11.
valued and upper hemicontinuous.
Z .. t. by letting
-p
~:
Define a cor-respondence
e p(z) jf and only if p·z:!: q·z for every q e t::.. then
it is convex valued and upper hemicontinuous.
corresponder..ce
11
L .y .(p), .JJ
t. x Z ~ 1::. x Z by ~(p,z) =
Finally, define a
-p(z)
x
-z(p).
the:t it is convex
valued ar..d u::mer hemicontinuous. • • By Kak:.:.ta:ti's fixed point theorem, 'It(· I has a fixed point (p•,z•).
and z• =
[ ...• " ..
-
j" J
• •) 1s • a f ree d"1sposalI T .... .. e:1 ( x • , ... ,x • ,y • , ... ,yJ,p 1 1 1 •
quasiequilibriuo for the truncated economy.
It is thus a true
quasiequilibr-ium of the original, untruncated economy . •
•
17-71
[.x~ 1 1
Then
- [.w.
1 1
CHAPTER .18
18.8.1
Let (x•,y•,p) be an equilibrium.
Suppose that there is a coalition
s •
L Then, for each i e S, there exist x. e IR+ and y e y 1
that can improve upon x•.
•
such that x. >-. x~ a11d 1
1
1
L·1E 5 x.1
L·1E5 w..1
= y +
condition of the equilibrium, p·x. > p·w.. 1
By the preference maximization
Hence p·y
1
""'L·lE 5 Cp·x.1
- p·w.} > 0. 1
0
Since Y is a convex cone, p · y• = 0 and hence p · y > p · y•. profit maximization condition.
This contradicts the
Hence x• must have the core property.
0
18.8.2
Define a three-consumer exchange economy by u.(x.) = 1
- 2, for i = 1,2,3
1
w = w 1
2
= (l,l), WJ = (0,4).
:-\'ote that cor:sumers i and 2 have the same type.
We shall show that an
0
allccaticr. x• = (xi,x2,xj) defined by
x• = (2 - 2 1
112
• 21, 0
•
1 2 2 = (8 / + 1 - 10'1/ •
•
21.
1 2 2 x! = (10 / - 1 - 21/ • 2) . .)
•
is a core allocation and does not have the equal treatment property.
In
fact, the quasilinearity implies that, for each feasible allocation x and for each coalition S, if "'· su.(x.} LlE 1 1
;l!:
L-lE 5 w11.
+
ttscr.1E5w2 1J#S)
1/2
- 2{#s>.
then S cannot improve upon x; this can be shown by applying the ar-gument in the answer to Exercise 15.8.8 to coalition S. hand side by v(S).
Denote the value on the right-
(This is the same notation as in Appendix A; by the •
quasilinearity, the ex:::hange economy give rise to a TU-game \-'.) v({i}}
= v{{l,2}) = 0
for every i,
18-1
Then
v( {1,3}i
= v( {2,3}) =
v({l,2,3}) = 18
112
1 10 /2 - 3, •
- 4.
On the other hand, •
Hence it is now easy to check that x• has the core property and xil <
Xiz·
that is, x• does not have the equal treatment property. •
It is impossible to construct an example of this effect only with two In fact, the initial endowment allocation (w ,w ) is the only 1 1
consumers.
allocation i:l the cere. core anc (xl'x J 2
;e
To see this, let Cx ,x l be an allocation in the 1 2
(wl'w 1, then x 1
1
convexity, (l/2Jx + (1121x >- wr 2 1 1
~
1
w and x 1
2
~
"'r 1
But since Cl/2lx
Hence, by the strict
+ (l/21x
1
= w , this 1 2
is a ccn':.radict ion.
l8.B.3
. (x • ,p } e IRLHN ' . Suppose that ther-e exists a Wa!rastan equt·t·1b rtum , +
such that the nth and the mth individuals of type h consume diffe:-ent .Th en .
'1 • · • • b unc.es, t:1a.t lS, xhm :;:: xhrf
p· Ul/2)x.• nm = (ll21p·xhm ~
+ (1/21 +
xh• ) n
0/2)p·xhn
(l/2lp·wh + (l/2lp·wh = p·wh
and, by the str!c-:. convexity, we must have either (ll2}x~
+ (l/2lx~
nm
un
>-h xh•
m
or (112lxh• •
m
+
(l/2lxh•
>-h xh• _. n ••
This is a contradiction to the utility maximization condition of an equilibrium.
18.8.4
Thus x• must have the equal treatment property.
Write v = x
1
- wl'
By Taylor's fonnula (or simply the definition of
the directional partial derivative), there exists a function rp{ ·) such that
18-2
uh(xh + cxv) - uh(~) and I cp(cx) I /ex -+ 0 as ·ex ~ 0.
= cxVuh(xh)·v
cp(ex)
+
Since Vuh Cxh) • v is a positive constant
(independent of a:). uh(xh + cxv) - uh(xh) > 0 for any sufficiently small ex > 0.
18.8.5
(a) [First printing errata: To be rigorous, the phrase in the
parentheses "i.e., every individual consumes the same number of shoes of each kind" should be "i.e., every individual consumes the same number of shoes of .
each kind, possibly except for one individual, who consumes one more left shoes than right shoes".
This is because there is one more left shoes endowed
with the economy and extra left shoes do not decrease utility.}
It is easy to
see that if all pairs are matched in a feasible allocation, then it is Pareto optimal.
It is aiso easy to see that if some pair is not matched in a
feasible allocation, then we can find a Pareto superior allocation by combi:~ing
the unmatched pair.
The statement of this question is simpiy the
combination of these two facts . •
(b)
\~"e
• •
•
claim that ar. feasible allocation has the core prope:-ty if and only
if, at the allocation, each owner of right shoes receives a pair of shoes . •
To prove this, first let x be a feasible allocation at which each owner of right shoes receives a pair of shoes.
Suppose that there is a blocking
coalition S, which contains J owners of right shoes.
This means that the
whole coalitior! S owns J right shoes and, at x, it receives J pairs of shoes.
By the definition of a blocking coalition, every member of S must receive at least as many as pairs as before, and some must receive more. must thus receives mor-e than J pairs at the blocking allocation.
The whole S However,
•
this is not feasible because the whole S owns only J right shoes. •
•
is no coalition that blocks x and x is in the core.
18-3
Thus there
Next, let x be a Pareto optimal allocation in which some owne:- of right receiv~s
shoes
no pair.
Since there are I + 1 owners of the left shoe and
only I right shoes are available in the economy, some 0\\-"ner of the left shoe receives none of the right shoes.
Thus these two consumers' utilitv • levels at •
x are both zero.
But if they form a coalition, then at least one of them ca:1
get a pair (and the other nothing).
Hence the coalition can block x, implying
that x does not have the core property.· Thus, if x is in the core, then each •
owner of the right shoe receives a pair of the shoes.
(c) We claim that (x,p) is an equilibrium if and only if p is a positive •
multiple of (pR,pL) = (1,0) and, at x, each owner of right shoes receives a pair of shoes. Let's first suppose that (x,p) is an equilibrium.
By the utility
.
maximization condition, p
:!::"
•
reason is as follows.
If p
0 and p
»
:;e
However, we cannot have c• >> 0.
0.
The
0, then each consumer can afford cnlv a shoe . • •
Thus the agg:-egate demand consists of at most 21 shoes (including both the •
right and the left ones).
•
Hence at least a shoe is left unsold in the market,
wh!ch contradicts the m
= 0.
We must thus have either pR = 0 or pL
If pR
equilib:-iu~ ~ogether
=0
and the
with p »
0.
(and hence pL > 0}. every
owner of the left shoe demands a pair, which, again, contradicts the feasibility condition because there are I + 1 owners of left shoes. thus have pL = 0 ar.d pR > 0.
We must
Then each owner of the right shoe re::eives a
It is easy to see that if a price vector p is a positive multipie of •
(pR,pL) = (1,0) and, at a feasible allocation x, each owner of the right shoe •
receives a pair- of the shoes, then (x,p) is an equilibrium. •
18-4
(The remaining
left shoe can be assigned to any consumer.)
The proof is thus comoleted . •
(d) The core and the set of equilibrium allocations coincide, regardless of the value of I. •
•
• •
We shall prove the stated properties of an effective budget set by
IS.C.l
showing that it can be expressed as a parallel shift of the epigraph of some
L-1 (Let D be a subset of R and f: D
strictly conca...-e function.
epigraph of f( ·) is defined to be the subset {(z,zl) e D x IR: zl !RL, which we denote by epi f.
IR, then the
4
~
f(z)} of
It follows immediately from the definition of
str-ict concavity that if D is convex and f( ·) is strictly concave, then its epigraph has an upper boundary containing no straight line. l proof can be r:n.cre specifically explained as follows.
Our strategy of
Fix a consumer i.
Let
a_i = (alk'at-:clt.
1
.a~,
iC=l
(-
lXI,
> 0 and [k .a~k > 0 for every l < L. <.i< =1 (.. ~k L. -..at'·._, l ~ ~ bv ., ""
•
•
-·
•
For each l < L, define hD:
•
is strictly increasing and strictly convex. •
Define f: t: ~ IR by f(z) = strictlv• concave.
Write
Lt.
Define E = {z e U: Lt
"'u}·
then E is convex and closed, because the function Lt
u
c..
.
•
ts
convex
and
E c
-
a: •
X •••
X
-
lXI
•
•
Define then D = {z e E:
zt
0!:: -
wli for every l < L}, then D is convex and 18-5
•
closed.
We shall now prove that: L
The effecti11e budget set {x. e IR : +
1
e A.} is eaual to (epi (/ID) 1 •
- w. ::: g(a.;p(a.,a .}} for some a.
X.
1
1
+ {w.}) 1'1 (~
L-1
1
1
1
-l
1
x IR ), where g( · ), p( · ), and +
A. are as explained in Example 18. C.3, and f I D denotes the r.estrict,ion of 1
• •
the function f( ·) onto D c U. -
2
To ease the notation, for each l ~ L, define pt: IR + -+ iR by
CI:k~ialk) + bl
2 ( .o 'n• b") • e IR, . £. t:. +
This function assigns to each bid bt. of consumer i
_at trading post l the market clearing price. -
= pl(bt)
-1
""
b£ - b£ for each t < L, and gL(b)
(b , ... ,bL-l). 1
. 2(L-l) L Defme g: IR+ -+ IR by gt(b) -
= Lt
- bf}, where b
=
This function assigns to bids b of consumer i the transaction (In fact, for each t < L, gibl is uniquely ·
bundle that he ends up with. determined by bt.)
With this notation, we have
L
~
{x. e iR : x. - w. +
1
1
l
L
= {z e !R : - w. ::s z 1
g(a.;p(a.,a .) for some a. e A.} 1
1
-1
1
•
1
g(b) for some b e A.} + {w.}.
~
.
• •
1
•
1
Al:hough our for-mulation allows the consumers to put both the good (on the offe::- side} anc the money (on the bid side) simultaneousiy at a trading • •
post, it turns cut that they can enjoy exactly the same excess demands by concentrating on one side at each post. exists c e A. such that 1
clcl
Formally, for each b e A., there .
1
= 0 for every t. < L and g(b) = g(c). To see 2
this, for any given b e A. and any l < L, define c e IR (L-ll as follows: +
1
If
-g(b) = 0, the ct. = 0; -geCol > 0, then - (0, p,_lblg,_(b}); c,_ -gt(bl < 0, then -- (- g,_(b), 0). ~
If If
ct.
•
It follows immediately from the definition that Ct• •
18-6
_<
bt' and ct'' -< b"l' 1'mply·ng 1
that c e A ..
With a bit of calculation, we can show that
1
•
-
We shall next show that D =
-g(b)
-·
=
-g(cl.
e A.}. 1
First
gf(b) = 0 if bl = 0;
-gl(b)
•
=
-
> 0;
Hence - wli :s
-gt(b)
-
•
< [k~ialk.
Moreover
if bl = 0; C[k;r:iatk)bl ( rk~ i at k } .._ b
< 0 if bl > 0;
i"
e D.
Thus Con·•e:-sely, let z e D, then define b
as follows: •
If
zi. = 0, then bt = 0;
1.,-
Zc -c.
> 0 • tnen · bl = •
•
•
•
If zt. < 0, then bl = (- zl' 0). ,_ .•
Th 1 -··
0 ,,
(.
= - zt :s uli and
L:eb£
Hence b eA •.. •
= [t.hl(zl) :s wLi.
We
have thus
. L-1 proved tha": 1) = {(g!(b), ... ,gL-l(b)) e IR : be Ai}.
It is easy to show that
-gL (b)
-
~
= f(gl (b), ...• gL-1 (b)) for every b e A.1
Hence x
IR: Zr_ = f(z)} =
{g{b)
L
e IR : b e A.}. 1
By taking the free disposal (comprehensive) hull in D x IR, we obtain
{(z,zL) e D x iR: (zL,z) :s (z',f(z')) for some z' e D}
= {(z,zL) e D x IR: (zL,z) :s
-g(b)
for some b e A.}. .
1
Since /( · l is strictly decreasing, the left-hand side is equal to epi (f i D}. By the definition of D, the right-hand side is equal to
18-7
{(z,zL) e D x IR: there exists b e Ai such that zL
-gL (b}
!0
By adding the initial endowment w. to both sides, we know that epi (fiDl 1
{!.J.} 1
+
is equal to
-gL (b}
{(z,zL) e D x IR: there exists b e Ai such that zL ~
..
{w.}. 1
. ' Finally, by taking the mtersect1on (epi (fiDl + {w.}) n (iRL-
1
1
1 "th IRLWI x IR +' we obtain
x lR) +
L
= {(zL,z) e lR : - wi ~ z ~ g(b) for some b e A.} + {w.}. 1
1
' • Tlne proo,& ' 1s tnus camp l•d e ..e .
Wnen L = 3 and w. >> 0, the effective budget set can be depicted as 1
follows. X3i
Xii
Xli
Figure 18.C.l
Let
18.0.1
(x~ .x~l l
w·r1te
1 ""1
"
z. = x.• - w..
both i.
1
1
be a feasible allocation such that x! >-. w. for both i.
1
1
It is sufficient to prove that w. + z. >-. w. - z. for .
Suppose not, then w. - z. >-. w. + 1
1
1
1
18-8
1
z.I for some i.
1 -1
1
1
Then, by the strict
•
convexlty, (l/2Hw. - z.) + (l/2Hw. + z.) = 1
1
1
This contradicts the assumption that
•1
X.
1
>-.
-1
W ..
1
CJ.
>-. w. + z. =
111
1
x~. 1
We must thus have w. + z. >-. 1
1
-r
w. - z. for both i. 1
1
18.£.1
The homogeneity of degree zero can be proved by noticing that, for
ever-y ;\ > 0, any allocation (xi•· ...
"iJl
for the H types solves the
maximization probiem (18.£.2) with a profile (1J , ... ,~) if and or:ly if it 1 does so .,,tith the profile (;\j..t , ... ,;\~). 1
H H As for the concavity, let ll e IR , j..t' e IR , and t e [O,ll. Let x = + + . . h d • ( • , l ~LH . b e a so l ut1on w1t 1-l an x = x .... ,xH e • oe a 1
solution with IJ'.
e IR
LH
by x" = ( tj..l. /IJ:· )x + ((1 - t )IJ.' /u." )x' for each h, then x" is feasible h n n h h·n h
with.ll"·
Mcreove:-, since uh(·) is concave,
Hence, by multiplying ll:' and taking the summation over h, we obtain n
I:hllhuh(xh) ~ tv(J!) + By the definition of (1 - t)V(J.t').
18.£.2
Hence
V(IJ") ::: I:h~uh (xh).
v( • ),
v( ·)
(1 -
t)V(IJ').
Thus V(J.l") ~ tV(J.l}
+
is concave.
Because of the strong monotonicity of the u. ( ·) and the interior n
solution, we can repiace the inequality constraints (i) of (l8.E.2) by the equalities and neglect (i.i).
Since 8[LkJ\'ic(Xk)]/8j..th
= uh(xh) and
8[[kpk(xk - wk)J/cj..lh = - (wh - xh), the envelop theorem (Theorem M.L.l}
impl!es (18.E.S).
18-9
18.E.3
[First printing errata: In order to obtain an example of the desired
prope::-ty, we should not require the value function v( ·} to be differentiable. If v( • } is not differentiable, we should define the marginal contribution of
type h at
-~
is defined as
-
-
-
-
-
Y(lll'""''~-l'f.!h'~+l''""'f.!H) -
lim
-
Y(IJ.)
-
-
~-'h <~-th'~-+IJ.h
•
That is. it is defined as the partial derivative of v( • ) with respect to 1-th from the left. 1 We consider an example with two types, h = 1;2.
Let !.Jl =
(2,0,01 and w = (0,2,01. 2
w~th
x . = x .. 1n
u.n ( · I is con:::ave
2n
and not differentiable at any xh with x h = x h. 1 2
-1J. =
Let fcilo~ting
By the symmetry and the L-shaped preferences, the
(l,lL
constitute the class of the equilibria:
...1-' --
xi
p = (1. 1. p
- p l .and· 1 2
1
+
p
2
= 112 and p
x2 =: o.. 1.
p
3
= 1;
- p l. 1 2
Note thc.t the kinks of the identical L-shaped prefere:1ces admi: multiple supporting price vectors at (Xih'xZh) = (1,1) (h = 1,2). and the consumption allocations for the numeraire depend on the choice of the equilibrium p::-ice vectors.
The utility levels at the equilibrium allocations are given by 1 + ... !-'•• - p •
2
and
Let's new calculate the values v(loll for J.L near
-1-1
= (l,ll.
then the L-shaped preferences and the strict concavity of the
If 0
squa~e
imply that the followi:lg constitute the class of the solution to probler:1 ( 18. E. 2):
18-10
~ ~~
::s 1,
root
x21 ~ 21l{C~1 + 1), x22
?:
21J./(Il1 + 1), x21 + x22 :s 2,
x31 + x32 = 0 · + 1)
=
vz, 1l
The marginal contribution of each type at
-ll
+ 1).
Likewise, if 0 :s J.L < 2
= (1,1) can be calculated as
follows: v(IJ. ,1) 1 ll 1
Hence, for examcle, if p •
l
v(/-( ,1) - v( 1,1)
v(1,1)
2
1
= p
ll
2
2
-
1
= 3/2.
= 1/2, then both types get less than their"
marginal contribution at the equilibrium.
18.AA.l
s 1•... ,SN
AJlY partition
par-tition with
an
(in the standard sense) is a generalized
= 1 for every n.
(1,2,3,4,5}, N = 4,
s1
= {1,2,3},
s2
Here is another kind of example: I =
= {3,4,5},
s3
= {1,2},
s4
= {4,5}, and on
= 1/2 for all n.
Suppose first that a TU ganie (l,v) has a nonempty core. outcome u the
on (1
_r
E ~"(
b
e
tn
.
1ts core.
Let
s 1•... ,SN
be a gene:-alized pa::-titi::m and
::s n :s N} be the corresponding weights.
N [ CS v( S ) ::s , n n
n=~
.
N
Let a utili:v -
Then
I 0 ) a X. = [ x.( [ an C[.IE s x.l = 1 1 1 n {(i,n):ieS } n i=l {n: S 3i} n= 1 n n n I = [ x. = v(I). .1= 1 1
L
r
Hence (I,v) is balanced. Suppose conversely that (1, v) is balanced.
To show the nonemptiness of
its core, we consider- a maximization problem whose solution is guaranteed to exist by the balancedness, then apply ttie duality theorem of linear programming (Theorem M.M.l) to derive the existence of a solution to the dual
18-11
minimization problem, and show that this existence implies the nonemptiness
of the core. Denote by !I the set of all coalitions (that is. the set of ali nonempty subsets of I l.
Consider the following maximization problem:
Es~i 0 S -
s.t.
- Ls~ioS
1'
~
1 for all i.
+ 1' ~ -
1 for all i.
Note t!'lat the objective function is linear with coefficients v(S) (S e !I} and - v(l), and the left-hand side of the constraints can be wr-itten by means of
!
. the 2I x 2 matr1x such that, after assigning a number m ( 1 :s m
:s
2
1
- 1} to
each coalition: The (l,m) entry of the matrix is 1 if l I •
~
I
I, m < 2 , and player t belongs
•
to coaL1t1on m;
1 Tne (i,:n} entry of the matrix is - 1 if t > I. m < 2 , and player t - I belongs to coalition m;
Tne (£, m)
en~:-y
of the matrix is 0 if t :s I, m <
and player £ does
not belong to coalition m;
1 The (t,rr.} ent::-y of the matrix is 0 if l > I, m < 2 • and player l - I does not be!ong to coalition m;
Tne (f,2 J entry of the matrix is - 1 if l ~ I; 1
. ( " 21 1 er:.t:-y of the matrix is 1 if l > I. T :le '· ' Define ((oS)Se.:f'3'•) by
c3$
= 0 for all S
that ((oS)Se.:f'-:r•) is a solution.
:;e
rs:/)s -
l
We now show
It is easy to see that it satisfies that
constraints and Lse.1'v(S}os - v(I)r• = v(l). constraints, t!'ler.
I, oi = 1, and ,... = 0.
Let ((c5S)Se!f'l"l
?:
= 1 for all i and hence the (1/(l
constitute balam:irig weights for the generalized partition !f.
18-12
o satisfy the +
r})c5s
Thus, by the
balancedness, I:sv(S)(l/(1 + 7))as - v(l) v(l).
Hence ((oS}SE!f'-r•) is a solution.
~
0, that is, •
Thus, by the Duality Theorem, the
minimization problem: Min(( V. } , (Z. )}>O ~
1
1
L·IE 1y.l
5. •&. ..
-
L·IE 1z.l
[.IE sY.I -
r.lE sz.1
+ [.
z.
lE 1 I
has a solution
((y~l.(z~}) 2: 1 I
the core of (l,v).
0.
ii!:
v(S) for all S E :1,
2: -
vCI),
By the constraints,
(y~ 1
z~). 1 1E 1
-
belongs to
So the core is nonempty.
This can be proved by Exercise 18.AA.l and the fact that S and T
18.AA.2
constitute a generalized partition with coefficients as = oT = l. diie~t
p;oof is also available.
But a more
Namely, suppose that v(S) + v{TI > v(I}.
u be a utilitv Olltcome such that ~. u. L..IE 5 1 -
~
= [. Su. ~ [. Tu.
So u is not feasible for the g:-and
lE
lE
1
t
i!::
v(S} + v(T} > v(I}.
v(S) and •
coalitior. a:-1d hence does not belong to the core. if v(Sl + v(TI > v(IJ.
~.
Le!
L..1E
Tu.
1
:2::
v(T), then
~- u. L.;e 1 1
Hence the core must b: empty
The contraposition of this is the statement of the
•
exerc1se.
l8.AA.3
the core.
I~
was shewn in the text that the proportional allocation belongs to To prove that it is the only one, let f(zl = cz with c > 0, then
any feasible utility Ol.!tcome u = (u , .•. ,u } satisfies [.u. = 1
proportion a:locatior: u• feasible utility such that u. < 1
ot:t::~:ne
v( {i}).
= (u~ .... ,uil
1
satisfies
1 1
ur
= cwi
c([.t.~.) 1 I
= v({i}).
and the
Thus if a
u is not equal to u•, then there must exist i e I
Hence u does not belong to the core.
Thus the
proportionai allccaticn is the only allocation that belongs to the core.
18-13
The following proof is a bit lengthy, but it is direct in the sense
l8.AA.4
that it uses neither the four axioms nor the linear combination of Exercises 18.AA.S and 18.AA.6. If #S = 1, then (l8.AA.2) trivially holds.
S.
So suppose #S
~
2 and {i,h} c
For any T c S, let JT(T) be the set of all one-to-one functions from T onto Then [ieSS\CS,v) = LneiT(S)(l/(#S)!)m(S(n,i),i)
= (1/(#S}!)Lnemslm(S(n,il,il To show the
fi:-s~
= (1/(#S)!)(#S)!v(Sl
= v(S).
eaualitv of (18.AA.2), note that • • Shi ( S'\{h}. v) = ( 1/(#S - 1J!)LneiTCS'\{h}
mc n, i}. 1
Here, for each n e IT(S\.{h}), there are exactly #S elements of IT(S) whose restriction onto S'\{h} is equal to n.
Hence
S\(5'\{h},v) = (1/(#S - ll!)LneiT(Sl(li#S)m(n:l (S\.{h}l,i) •
= (1/(#S)!)Lnerr(Slm(n I (5'\{h}l,il,
where n:[(S'\{h}) denotes the restriction of n e n(S) onto S'\{h}.
~cw
Hence
define TT = {n e IT(S): n(i) < rr(h)}, 1
IT then rr and 1
rr2
2
= {n
E
ll(S): n:(i) > n(h)},
constitute a nontrivial partition of TT.
so that, for each rr
€
Define P: TT(S)
~
TTCSI
IT(SJ, P(rr) is obtained from n by permuting i ar:.d h . •
T:"len the rest.-ictions PI TT and PI rr are one-to-one and onto mappings between 2 1
rr 1 and rr 2
(and, in fact, each one of them is the inverse of the other).
If n e
rr 1•
then S(rr,il = SC1t I (S'\{h},i) and hence m(Tr,il- m(ni(S'\{h}l,il = 0.
If n
E
1! , ther:. 2
18-14
m(n,i} -
mt'~rl
(S\{h)},i)
= v(S(n,i) v {i}) ·- v(S(n,i)) -
v((S(~r,i)\{h})
v {i}) + v(S(n:,i)\{h}).
Therefore, Sh.(S,v} - Sh.(S\{h},v) 1
1
-
Similariy, Shh(S,v) - Shh(S\{i},v)
= (1/0tSJ!)()
TT v(S(~r,h)
'-n:e:.l
v {h}) -
r' IT v(S(7I,h)) '-Tie 1
- )'-Tient v((S(lr,h)\{i}) v 1
Hence, if n e 1
rr1
and n-
2
{h}) + ) IT v(S(lr,h)\{i})). '-Tie 1
= PCn ) (and hence n
1
2
e
n2
and n = Ptn Jl, then 1
2
S(n ,iJ v {i} = S(n ,hl v {h}, 2 1
S( n:2, i) = (S(rr,.h)\{i}) v {h}, •
(S(rr ,0\{h}) u {i} = SCn ,hl,
2
1
SCn ,U\{h} = SCn ,h}\{t}. 2 1
We thus obtain S\(S,v) - S\(S\{h},v) = Shh(S,v) - Shh(S\{i},v).
IS.AA.S
Suppose that a utility outcome u of the unanimity game (I, v) of S
satisfies the efficiency, symmetry, and dummy axioms. axiom,
L·lE 1u.I
= v(S}.
By the definition of a unanimity game, if i f S, then
vCT u {i}} - v(T) = 0 for every T. [ieSui = v(S).
By the efficiency
Hence, by the dummy axiom, u. = 0. 1
Thus
To apply the symmetry axiom, note that if another game (I. v')
is identical to (I, v} except that the roles of the two players i e S and h e S are permuted, then, in fact, (I, v') is identical to (1, v ). •
•
ax1om requtres u. = u.. l n
Thus the symmetry
Hence, by ~eSui = v.(S), we obtain u. = v(S)/#S fer 1
18-15
every i e S.
18.AA.6
Let
~
be the set of all coalitions and, for each S e
~.
denote by
(l,v } the unanimity game such that, for every T e ~. v(T) = l if S c T and 5
Denote by G the vector space of all TU-games on I. I 1 I = 2 - l, dim G = 2 - 1. Define v(T} = 0 otherwise.
this is a linear transformation.
then
For the first statement, the:-efore, it is
sufficient to prove that f( ·) is onto. s•.1fficlent to show that if
By #!I
f{(o:.S)Se~)
For this, by dim G = 2 = 0, then o:.
5
1
- l, it is
= 0 for a!l S e !1.
We
shall now es:ab!ish this by induction on #S. Let S e
~
and #S = l.
!1, YT(S) = 1 if T = S and vT(S) = 0 otherwise.
LTeJ'~vT(S)
0:,.
f((~}TeJ')(S)
=
Suppose now that S e !I and, for everv T e !1, •
Thus
if #T < #S, the!l
Hence
Then, fer everv• T e
Then
= 0.
.f( (~)Te!I)(S) = L{Te!I:TcS}a,.vT(S) = ~Te~:TcS,#T=:#s}';-vT{Sl. But here. {T e !1: T c S, #T =:: #S} = {5}. 0.
Hence
f((CX,.)Te~)(S)
= cx . 5
Thus a:
This cor::pletes the proof of the first part. To show the second part, for each S e
for
,..s
iC.entified in Exercise 18.AA.S.
and
s ui
= 0 for a:1y i E S.)
(That is,
Let v = Lse~a: v
5 5
prove that the utility four axioms.
Since each u
~.
let uS be the utility outcome
u~1 =
1/#S fo• eve:-v• i e S
be a TU-game on I.
We shall
only one that satisfies the 5
satisfies the efficiency, symmetry, and dummy
axioms, the linearity axiom implies that if there exists a utility out::ome It is therefore
Our of the four, the linearity axiom is satisfied by construction.
18-16
The efficiencv •
5
=
axiom can be easily checked.
It thus remains to verify the dumrr.y and symmetry
•
ax1oms. Let i e I satisfy v(S) - v(S'\{i}) = 0 for every S e :1. induction on #S that, for every S e Y. if i e S, then a: Let #S = 1, then S
such.
v(e) = v(S)
= 0.
Thus cx
5
= {i}. = 0.
= a:5 .
Hence v(S)
We shall prove by Let- S e !I be as
= 0.
5
On the other hand, v(SJ -
Suppose now that S e !f, i e S, and, for every
«-r
T e !/, if #T < #S and i e T, then
= 0.
Then
v(Sl - v(S\.{i}) = L(Te!f:ieT,TcS}cx,-vT(S) = ~Te:l:ieT,TcS,#Ti!::#S}cx,-vT(S). But he:-e, {T e !f: i e T, T c
s.
#T ::: #S} = {S}.
Thus a:
= 0.
5
This completes
the proof of the dummy axiom. As for the symmetry condition, assume that another game (I, v') is ice~tical to (!, v) per-m•Jt~C..
except that the roles of the two players i and h a:-e
Defi:-.e
't:
!! -+ !f by T(S) = S if {i,h} c S or {i,h}
= (S'\{h})
(S\.{i}) v {h} if i e S and h E S, and T(S)
Then T( ·I is one-to-one a."ld onto, •
•
T
= T
-1
v
S = e, T(SI =
I'\
{i} if i E S anc h e S.
, {S e :1: i e S} = {S e Y: h e T(S)},
•
and #S = #-rfSJ. Moreover. for each S e !f and T e !f, S c T(TI if and only if Hence u:1 =
Ls 31.ex T,rs.I*~=S J
=
The symmetry axiom is •
thus ve::-ified.
18.AA. 7
[Firs: printinli!' errata: The difficulty level should perha;:>s be B.]
We shall fir-s: estab!ish two lemmas.
Denote by the set of all coalitions by
n For each S e !I and i E S, define rp .: (O,ll -+ IR by IPs .(t) = 5 •1 + ,1
!!'.
•w .. ~
1
i':ote tha: if S c T and i E T, then rp
.(t) :S rpT .(t) for all t.
5 ,1
'1
Lemma 1: Let S e :1 and i E S, then v(S u {i}) - v(S)
18-17
L-...-swh
,1
1
U-
...
In fact, v(S u {i}) I
Lemma 2: Let z
1
1 = I 0 Vf(rpS,i(t))widt. IR n , z • e IRn , and z • -,.. z, then 'Q'f(z')
E
+
+
•
1
1
il:
vnzl.
In fact, define ¢: [0,1} ~ IRn by f/l(t) = z + t(z' - z), then +
Vf(z') -
=
1
2
I 0 D flrp 5 )tl)(z' - z)dt
~:
o,
where the integral is taken component-wise. Now let S c: T and i E T, then, by Lemma 1. 1 v(S u {i}) - v(S) = J 'li'f(rps .(t))w.dt and 0 ,I 1 1 v(T u {i}) - v(T) = I Vf(rpT .(t))w.dt.
0
,1
l
But, by Lemma 2,
I
1
0
'17fCrp
.(t))w.dt
5 ,1
1
,1
1
Hence v(S u {i}) - v(S) -s v(T v {i}) - v(T).
The convexity is thus proved.
2 IR +
~
18.AA.8
Let n = 2 and define f:
convex.
Let I == 2 and w = 0,0) and w = (O,ll. 2 1 2
and v({l,2}) = 21/ .
l8.AA.9
IR
+
by f(z) = Cz
2 1
+
2 1/2
z l 2
. , tnen
f(·)
. 1s
Then v({l}) = v({2}) = 1
Hence, by Exercise 18.AA.2, the core is empty.
(a) It is easy to check that v(S) =
f('[.
w.), where f(z) =
lE 5 1
(b) A utility outcon:e u is in the core if and only if u.
1
2::
0, [.u. = 2, u 1 1
1
+
•
•
= 1.
saying that
18-18
This is, in turn, equivalent to u = (a:, ex, 1 - a:, 1 - ex) for some o: e [0,1]. •
(c) After the modification, if a utility outcome u is in the core, then u
u
3
+ u
4
l!:
2.
He:~ce
u
1
+
u
3
defining the core are now u u
= 0 and u = 1 3
+ u
1
+
4
u
3
=2 i.'!:
and u
1, u
1
+
2
u
= 0. 4
l!:
The other inequalities
_1, u
l!:
3
l, and u
4
2:.
Thus
l.
Hence, the welfare of consumer 1 is worse off.
(d) By the symmetry axiom, Sh (v) = Sh (v) and Sh (v)·= Sh 1v). 1 4 2 3 efficiency axiom, Sh Cv) + Sh (v) = 1. 1
3
By the
By (18.AA.3), we can obtain Sh (v) = 1
Hence Sh(v) = Cl/4,1/4,1/4,1/4).
114.
(e) Denote by v' the TV-game obtained after the modification of (c).
Define
another TU-game v" by v(S) = 1 if S = {1,3,4} and v(S) = 0 otherwise. = v
1
+
+
v".
for any i
Then v'
Mo:-eove:-, Sh Cv") = - 1/4 by (18.AA.3l and, thus, Shi(v") = 1/12 2
= 2,
by the efficiency and symmetry axioms.
axiom, Sh.{v') = Sh(v) + Sl'l(v") = (1/3,0,1/3,1/3}.
-
-
Thus, bv the iinearitv
Hence the Shapley value for
player l increases by the modification, which is in the opposite direction to . f . tnat o t!'le cere
18.AA.l0 (I,v(
C"''":C""..,~.
•
(a) Let I = 2 and the spaces Cx ,x ) are g1ven. 1 2
ll bv • x ,x 1 2
v(
CS) = - CCL:.1E x.). 1
x ,x 1 1 2
5
Define a TU-game
The proposed cost aliocation is
- Sh (l,v( ))). 2 x1,x2
This can be explicitly
calculated as:
(b) Tne marginal cost on division one is (-r/2HCx
marginal cost on division two is (-r/2)(Cx + x l 1 2
18-19
-r-1
+ x > 1 2
-r-1
+
-r-t x ·). 2
+
The
(c) [first printing errata: In (c) and (d), we should assume that 7 > 1.] re-numbering the two divisions if necessary, we can assume that a: answer depends on the values of a
and a . 1 2
1
~
By
a: . 2
The
That is, the cost allocation rule
proposed in (b} leads to a profit-maximizing choice of overhead if and only if" a. :: 2a: . To prove this, suppose first that a. ::· 2a . We shall show that the 1 2 1 2 . cx , x > = cc a / 7 ll/(7-Il , O) 1s . t h e umque . . .. a 11 ocat1on pro r·1t-max1m1Zmg 1
1
2
allocation, a.'ld led by the cost allocation rule in (b).
In fact, the forme:-
follows
The latter follows
from (1/2)(Cx (r/2}((x
1 1
.... -1
'Y-1 .7-1 X )" + x" ) = 7Xl = al, 2 1 'Y-1 "Y-1 7-1 X )" + x" ) = (7/Zlx =
+ +
Suppose next that za.
2
2
2
1
> a. > a. . 1
a.{2 ::
a: . 2
Then the allocation (xl'xZ} =
2
1 ) , Ol 1s · tne · . l/\3"· . . a II ocat1on, . b ut 1t . car:.n:.: b e (( o: / 7 ) umque pro r·1t-max1m1Zmg 1
•
led by the proposed cost allocation rule because
Suppose fina!ly that o: = 1 ·~
1;
a. ~
2
!flen iln allocation (xl'x ) is profit-maximizing 2
X• •• •
and hence such
a.'l
allocation cannot be attained by the proposed rule.
If X, =
•
x , tnen 2
He:1ce the a:Iccation cannot be attained either.
(d) Rigorously speaking, the answer depends on how to interpret
efficien~
•
decentralization choice.
In fact, we shall establish the foliowing two
statements.
18-20
-•
(i)
There is no allocation rule 1/1 ( ·) and 1/1 ( ·) with the following two 2 1 pr-ope:-ties: First, ¢ Cx ,x l + ¢ rx ,x ) = C£x + x l for all (x ,x ); 1 2 1 2 2 1 1 2 1 2 second, for every Ccx ,cx J and for every combination Cx ,x J that is 1 2 1 2 profit-maximizing under (a ,a J. the two divisions are led to choose 1 2 (xl'x J when faced with tP C·) and I/J ( ·) under (cxl'cx J in the sense of 2 2 2 1 (c).
(ii)
There exists a cost allocation rule tP ( ·) and ¢ 1
2
with the following
C ·)
t·wo properties: First, ¢ Cx ,x l + I/J Cx ,x ) = CCx 1 1 2
2
1 2
1
+
x J for all 2
(x ,x J; second, for every Ccx ,cx J and for some combinatio:'l Cx ,x ) that 1 2 1 2 1 2 is p!"ofit-maximizing under (cxl'a l. the two divisions are 2
l~d
to choose
Cx ,x J when faced with 1/1 { ·) and tP ( ·) under Ccx ,o: J in the sense cf 1 2
2
1
1 2
(c).
Thus the beca'~se,
•
~equ!re:nent
for "efficient decentralized choice" is
strong~;:-
in (i),
foro every (cx ,cx l. (i) requires all profit-maximizing combinations 1 2
Cx .x J under Ccx ,cx J to be attained as a result of decentralized choice; on 1 2
1 2
•
•
•
the other hand, (iil requires some profit-maximizing combination Cx .x J to be 1 2 attained.
As we saw in (c), there are multiple profit-max:mizing combinations
(x .x J if a:1d only if o: = cx . 1
2
1
2
The difference in the notion of decentralized
choice betweer. (il and (iii therefore arises only when cx = a: . 1
2
But, when it
comes to the existence of a cost allocation rule of the des!red p:-ope::-:ies, this seemingly subtle difference brings about two different conclusions. Let's •nrcve
•
(i l
first.
Suppose that an allocation rule 1/J ( • ) and r/J ( • J 1 2
snow that the:-e exists a combination
(x ,x
1 2
) such that either e!ji
(xl'x
1
In fact, suppose that
18-21
2
J/ox
1
=
(x .x L
Then a.p (x ,x l/ax 2 1 1 2
1 2
=0
and a.p Cx .x l/ax 2
= tP1Cx1l
We can thus write ljli(x ,x l 1 2
1 2
= 0 1
respect to x
1
1 2
= t/.12 Cx2 l.
and tJ!2 Cx1,x2 l
cost on each division depends only on its own usage.
C(x + x l for every (x ,x L 2 1 2 1
for every (x ,x ). That is, the
Hence 1/1 (x ) 1 1
+
¢ (x ) = 2 2
Now differentiate both sides twice, first with
and then with respect to
Xz·
Then we obtain > 0,
which is a contradiction.
Hence there exists. a combination
(x ,x
1 2
l such that
By re-numbering the two divisions if necessary, we can assume that a.pl(xl,x2)/oxl
= C'(xl
Let cxl = 0:2 = C'(xl
+ x2).
+
X2), then (xl.x2) is a •
profit-maxicizing combination under (a ,cx ). 1 2
-x
chaos~
1
because o¢
Cx ,x
1 1 2
Let's next prove (ii).
~
)/ox
1
But division one does not
Statement (i) is thus proved.
cxr
Define CCx )
if x
C(x + x ) - C(x l 2 1 2
if x
1
•
1
~
x , 2
< x , 1 2
It immediatelv • follows from this construction that
To verify efficient decentralized choice, let a
CCx + x l for- all (xl'x ). 1 2 2 2:.
1
0:2. •
a!lo!:ation (and the unique one if cx > a ), and 1 2
aVJ 1(£cx1/7l l
~
a.p
2
1/(7-1)
((a /7)
1
-
,ovax - a , 1
1
1/(3'-ll
,0)/ax
2
=
a
1
sense of (c). is the unique profit-maximizing allocation and chosen by the two divisions . •
18-22
Statement (ii) is thus proved.
18-23
CHAPTER 19
19 .C.l
F"o.:- each s,
by the concavity of u (·). u (A.x
s
Mu!tipiying rr
s
1 ••
s
s
5
~
Denote this value bv• k > 0. _,
1' " ·r I
a,,_ o, .. 1 J"": t""1
.,...
u
-•
t
,.
) > . x li -
•
AU(x) +
1
We must thus have k < 1. suppo~~ing
Then k •
•
'. B !-· tn:s ts
(
2 = H ence.
·rI•
w + w 21 22
-
p --
} ·ic c. -
t.., ,.., ~ ,,...
1
2
'. ••
pr:ce vector-. then •
I9.C.3
s
Let (x .x l be a (interior) Pareto optimal allocation. 1 2
-
• t·'!"
s
;\u (x) + (1- ;\)u (x' l.
1nus U(·} is concave.
H"'nc• ......... > -
~
.:\)x')
0 and summing over s, we obtain U(:\x + (1 - ;x)x'}
5
( l - i\)U(x'l.
19.C.2
s
+ (1-
•
•
•
\Vri:e x. = (x,., ... ,x .} E iRS and define l 6 1 5l
-X.
= ([ 7t .X ., •.. , ) 7t .X . } s 51 51 "s 51 51
i
E
s R .
• ~~-.ca··· .......... " I" ~.. ¥
·· of the Bernoulli utility function, •
• • ----.~o·~!""'·,..··v m........ 1.;.. ......... .;.\. ,
the
•
•
-x .
,...D •• H fb . -:1--- . •" ~
1
•
•
which implies the local nonsat!aticru,
o · x.
2: o • .
.
•
But -
.. X. • •• ~
-
:;: ·X. • !
= (L"\'
0
s·
)(\ 7t .X . ) S L..s Sl St
-
\p
~ 5
X
. S1
= \
Tl
.OX
~ 5!.
• Sl
\
L.s
.o
= 'Ls ( rr Sl' - o. 5 )x 51.. Hence ' L.s
{it .c - o )x . ~ 0.
s:·
--- .
·s
s1
,.;;_..., 'oo·" . .... ' . 5 . . . sides of this inequality by 8 •,... . d;,_..: ... -~·
•
19-1
-p,
we obtain
p
S
X
.
s:
· x 1..
[
s
(rr . - p /p)x . ~ 0.
s
51
A possible interpretation is thus that the
5!
•
equilib:-ium consumption X. is biased towards the states s .,,,..hose equilibiium 1
•
/o
•
pnce ra<.tc .-o s . is lo\.'ier than the subjective probability •
assrgns to
I
•
•
[First prmtmg errata: Every u.(-) should be
concave,
JUSt
1
•
•
tnat consu::ter
lt.
s.
sta~es
•
l9.C.4
•
~o~
This is because risk neutrality needs to be a !I owed for. j
strtct•Y co:'lcave.
- ..
b .. a,.. in~erior Arro""·-Debreu equilibrium. Then
(a) Le:
By :he eq:.:a:!::; of the subjective probabilities, u2Cx
-
12
llu~t>~
22
J
Tbs
= 1.
• •,., .. b ,.,. t .. 5 ·-· t..• 1.'-·L concavity u ( · }.
-
1
< l
ii" a,-,.1 •' ,,._.
"'"'lv -u J
·lrX I 12
>
X 22'
l ·..vr~e:: f':is p:-oOabiiity assessment for the
,...,-
- ··-·-···-·
,.. 0 '!""'! c:
~·
0
•I
1 ~- ~ ...
sta~e
•
_. -
,_ ~.,... ; ..........
... • ., ... 1 .. --.·
\ f ..
•
o
•
o
,-. •
d DC:
, •
~
•
r.: ./r.:2. a:v.: li. ~ ...,
I
'
•
I
0
'-c:.n-1 -- .\
1 !.. •• i::::.
l9.C.5
• .• . . rx• ·-----·cx•J a ... ··- ' l' ... ' I
J.;-;
••••
'
•
'' 1. '.
' '
(iii
~
t"nrou-"' ::;:• his •
•
o
-..J
., o
0
0
•
o ...
0
0
:
..,.. r -
.. _
• • !. X •.
~-
•
I
·- .... _
--. .-.'.x~ L~ t
. .1
.'
=
• 111:1!"'1 .... ~··~_...e,.··
;-;·;al-
_., .w'- "";. •J.
..o..;.;. "';.
•
---c:· .•
..., \w,, ... ·-s· • • •
(_.,
gol. ng
----· ~
::- /c~
.., • ·..
budget line.
o
- •-
~
:ilS
... Oi
.
"IL"ne''
•
-
·-"~.-· l .......
gam fiom trade because. a:: ecuilibrium.
..... ...,ce l·-t41·:-~e ~ 1.;. .i. • - .;, •
•
··•
•
:~-...,.
n~s
o
•
•
the state is larger than ---~
is lar'ge:- thar-i t!1at
c;onsumer 2 consumes more in s:a:e 2
...
r.-.~s·
cons~~es
Hence consume:' 2
•·e.-Tc-.~ 'I
---·
n• = -
an Arrow-Debreu equilibrium if
is maximal for >-. in {x. -1
'
l
E
X.: p. ••
X. I
--
... .
. ' ". -,.; "'""··· •
1
= '.w L. ... •
•
•
..,s e ~, . For each :, define +
,..... ·..-; . .... . :.-.. f"!:n ...,. ~
•
19-2
x~
'•
s R .
== (r.·w., ... ,rr·w.l E I
I
and a p.-ice vector p =
= [ n -rr·w. = rr·w. and hence
rr·x~
each i,
I
and rr·x.I
S
5
I
• •
l
1
h t .• e
By
rr·w ..
~
constitute an Arrow-Debreu equilibrium.
. • r1sK
. avers1on,
x~ 1
is in the budget set.
( rr·x., ...• n·x. ) r. 1 I -1
•
monotoruclty, x':'" = (rc·w., ... ,n·w.) >-. (rr·x., ... ,rc·x.). 1 ! 1 -· 1 1 •
•
•
~~
lw&
all s.
fc~
19.D.l
Then
a:
Ou:
,... .....-
I:.x•. 1s1
= n:·([.w.) = l'n: k = k. 11 Loss
1/ -
•
••
•
\.. ... ·-· ..
(
D...I ': • til remams true. •
I~ • ..":1
.
-
~
•
•
~~"
•
•
•
-. - -. "'r• ,..,.. or- ·-•""' b •-a,.se ..... ....
5
--·
'.;.,.
=
•
- -
(w w
1' 2
-
•'-a· .... . •
.,
-""' E !••
:~ \ t
0
~
•
I e iR
2
+.;..
Define
•
5
~.W . L.l 51
The seccnC
Si
S!
This
•
c: -. ,\ .·'-•h r. e _____ to •
:"'\ e O...
• :- • • .
;:>
liD,..'!-
·s
..
Thus we can no l:::r:ger- r. o:--:::atlz~
xi
•
be tne aggregate
++
2
L
= (:>,D) • •
=
=
=
xz
•
..
Define •o =
=
-( l/2)(w,w) e
x•J ·.·s a:1 A:-;-:J\\·-Deb;eu equilibrium.
IR
4 ++
.
~r.
vu{(l/2lwl
-·1 nen
. 1t
•
!S
Let's now f:gure
...
•
ve!ctor
~naot.· ·utent
•
e
- -
'"asv .. 0 ,. .,J. ...
-c~o. l~ e c:.
- ... .... .
.. . ·-
•• . ,,..,
•
. 0 ..
-
p·((l/2:•w
I
-
.. .. • - -
• "':-o"'- r• •' J .
-
-
..... /
_. ...
~-
.
•
•
tO C:leCK
·!--~ .. • • -
J!.lst as i :1
-
.... ·-
= ;:; .• ( \'1/'Z ,..a.; ···'1,=~x ....,\.,. • ..,. t _;•
- (p·wl/2, 1.
By
1
----
= -
-
01
p • . r.c-·· . . -··. ·"'" ......... ,., ... •
-· I .I '
'..,Jo
O'-'
• z' I •• . !""! p
.
. . .... ..... •. --·---o"
"0"'~ ·~g~':""-
•
s•, .. _ ...... .. _,
eq!.lilibrium with a pr1ce vector (q: ,q 7 J fa.- the •
..... .......
•
••
and w
-
• K
commodity equal to 1.
0
.:.
51
is not homogeneous of degree
•
19.0.2
l I.
5
•
eo·
1
Hence we can no::-:nalize
ckles not involve p.
•
r
x. by the
Hence (fi) is rr:.et.
for examo!e, •
~--~=~,..·~-··~~As ...~1-wl..i.C. ... ""•-
-w
1
+' ~ne
x~ >-. 1 -1
He:1ce
I
re"'laced by p · x . :s p · w . + z . fer everv s.
v-
•
~
;.
......
•
must
' ' ' \4;,.}
I
o:- ... ·..... .....
• 1 ,.. •l- a---· ...........
r.s~
"'-•
r •. '
1n • Po:""'•t-: _...., ... c::. ....!••· L.Y'
1
L p• X. e X. -~
:he tv:o inequaiities (i) and (iii that constitute the b;Jdge:
- ···-·-- - - ,.;-.a,. ... . ... '
• ..,,..., ........ C'"
By
x ..
As for condition (ii), •te~•
·-a--··1"1." an"' ,.,:, .. L. ., ':.:f t ' condition (i) is met. •
First, fe-r
.-;::,:.
we must have
= - (p·w)/Zp 1 = -I ....
~
vi
•
-
Ev•
Vu((l/Z)w)·w
zau( (l/2)w)/oxl . .
the contingent commodities,
•
19-3
51
zZL
= -
.z". • • ••
BY•
-··-
'!' )o, "'
fea5ibi!it'i-
= - z• 22
co:-~str-aint,
Hence
Q u ( ( 112 }w ) . "'
-
2c3u((l/2lwl/ax
151
By inducing the consumption at t = 0, we have X. =
19.0.3
1
,.,Ul~Sl
""'
. . . anc a spot pnce vector p 0
to the spot price vectors p Write p = Cp ,p .... ,p 0
1
, 5
5
e !R
1
. 'q ,... ' '- ~s •-= as - ...... \ r·····"tsj ~·
Ul+Sl
~.=.rare
·oJ
...
= (p
15
.
= (p10 •.... pLO l
•... ,pL
5
l e
~
L
e
~
L
. ·
at t
.oL(l-SJ ~-
+
• w. e I
=0
in addition
in each state s at t = 1. •
A contingent commodity pr1ce ve::tor q ::
' and the L ph.ysical goods at t = C
-
-r ...... .. S
a:~C:
n~e U .: ..... lt' •. -'
contir.ger.: ccrr:modities are traded in the markets opened at t = 0.
o
I
~o'ole~ ''O D L' m -x=,! .. iz--',..n d. L. ... w c' ••• .. ,., • • I P ~c:.
~.
Max
U.(x .. x .•... ,x .l Ul+Sl I 0 I 11 51 ( x!ii. Xl; •... xs; )eR ... . . .
-.
I
r •1 ·1 "·X
s.t.
- CZ +)
' . '"'o ' u1 n. ·
< 0 • '·' . -s ·s st - · o "'oi ·
,. ..,. ) "" ·x . s 5; \. • "
;;..J
<
•
.--· ,, __ _. ~
-i 1:-5:-gooc . . tj,t .....
x~ I
C
--
11
e ,....,:••l•!""'rt•'rn -· ... • ....... .....,
•
•
•
r,o,,... co-·l·"g""'"'" ,..i..'.. &oL.
.....
•
s
••
r:-! state s = for eacn oerioc a •
. -1
l~ll)Z
at each period and state,
....... '---
Cons ' -tir··~"'
c:.A
Radr.~r-
.......
l'.
i.
rr:ax • ,. :~:,-=.'ti,...,..., •••••--••V •
.. , .......
~
LU ... Sl
a~d
...
•
. ... . .
s !:=:
e
• a spot pnce vector p
.. -!11,
:::,t
e ·~~~:~-·· ' -· v
~
s = t. ... ,S .
fer eve:-y consumer i. consumption plan z7 = (zi i ..... zS i)
Vl
0
-
ever~·
-
= (x! .. x• ...... x:.. I e •••
5'
p z . for 1s 51
~.
----r-:~~1 ~; ... ! ......... - .....
s.
. 5
•W • +
...... a
'
l"'.ro~.=.,..,
• •
Li
-
<
. Si
-
,.,.
II
t:--.~
cons:..unption plan
(z~ .x~) 1 1
solves the abcve
o~n~•-m •
•
..,.>J .. - •
••
s = 1, ... ,S, and [.x•. :::: [.w . for I 51
l
e~.re~v
Si
•
s =
C•,l, ... ,S.
The 5taterr:e:'lt c:-
..
COffiHlOCLty
Proo~sition •
•
19. 0.1 remams the same, except that
•
·-~ lo..: ....
. . . IRL(l+Sl "' . . .. .Ot:ri ::i .~es an-...~ pnce vectors now 11e m an ... , 1n pa:-t l 11 1..
19-4
+'
c!'le
'
•
"rr:ul!i;:Jlied" contingent commodity vector is now g1ven as -. p::-ccf, the definitions of the two budget sets, I n 1-.s
. . max1mtzat1on
the above fc:-h.u!atlon cf the utility tv.·-~
bet\veer'. the
!!
0
= {{1,2}.{3,4}}, and !1
1
E-3·4
s ..-- Xl .. in ;.:)
--"'-s"~-.··a"._,.;. I \.0. • o. & u L. .. .0 •
"- .. ,:,
.
•
. -"' -
0
....... '-··•
•
end
-
at
0
:
•
•
•
•
?
.• I
.. '.'3 ....•..
o
-
"''"''-' I
'
_.
~:;::.~A-D
!
discussion at the end of Section 19. Dl are ~,"5 .........
I
. o •
• r ___ _ •D -. d..;. ;.·s
... -
.. .• -. :)
~~
•
e
...--
• ..
X
-
•
:.
-_
x~
'
.-
• ,.,_.,.._ ' .. _ vecr.o::- •::J = XI and a -...........
constitut~ :.:--~
solves t!":e maximization
•
A:-ro\v-De':.:-et:
.-·· .
• • -Ov""": ~rr P.
Max X .E .X. U.(x.l I I I
1
X <'\ D W P L~Ee~1 tE' tEi - L...t_EeA· tE tEi.
s.t.
..-··...·-- ..... .
:
.... ·-{'
-. "'-.
·~·
•
ana
are well defined bv•
' ... .. l '.x~ ~
I . \
(b) A
!
Ll
:
•
•
• "1 . .I "O · ~--· 1 .... '!' ,... -m .. ,.., DV .................. \,;l . . . . . . .Ll .... •
•
•
•
.,;
-·e~o-··
I -•
X
.::- :'::'·. H for everv tE e 11 ...
:-"'.,-
"!::..:&! ..
(
•
•
,..,
•
J
x1
,., \ \o.i_,_. . . . . .... - ..
.,.,., -·.... - se: C·l ....• o!'1P
bv A .
t:OII:to-":
.l.r .....
'"'-
• •
2{1}, 2{2}, 2(3). and 2{4}.
~
'
;:. .-. ,··, ·" 5 --: : : c . .. -_ ---.
W.
_H·3·~ E :.";
•
'!"""\ ICII
... • •
.
2
The
The admissible
Section 19. B.
"--
.,. 4 •
I
--
3, and
+ 1 =
•,-.-=• constramt, which was explained in "" ··-
.... s~··--. .... __ _,._, """-.. ....
..... -
e ,.,~,~·,_,.~l~n""'" ........ ,. •
•
~.,,;._
0
The
= {(1},{2},{3},{4}}.
and the initial
.._
•
•
p;oble:n).
modified budget sets can be proved just as before.
= {{1.2,3.4}}, !f .
C
I
(a) The given event-tree structure implies that S = 4, T
19.0.4
should be
' f o:- BR. :n . to in:::orpcrate the consumption at t = 0 (as was cone
wcdi:i~d
easilv•
B~, 1
and
a price vector
t+
19-5
•
ro~ I •
co:1tinge:1t fi;-s:-good commodities, a spot pnce
o ,.. e iR •t:...
H
for every tE e A. and, for every consumer
• •
z~
1
•
1.
consumption plan
= ((z•E.(t + 1, E'))E. UJ t 1 E.;r.
E' E) E " • c t e~' ~+ 1
e X. constitute a Radner equilibrium if '•
x~
anc
•
1
•
•
( i l F o:- every i. the consumption plans (z~.x~) max1m1zes U. (X.) with I
1
respe-:t to x. e X. 1
1
E' }}E' EJu:
1
1
~· El.,;::-..::~·1 . ·- C "".._ __
t ...• •,
-
sub iect to the budget constraint in the small-tvoe . discussi:::r: at the ~
19.0. E' l =: 0 for every tE e s3 and E' e !f ~.w
LI
\Vith E' c E,
t .. !
E' for every tE e ~ t I
......
.N o ....... t .. a·.. , sir.-:::e there is no predecessor at tE = O{L,2,3,4}, •a.
0~
this acmissible pairs becomes :s
" • .X G·.-· ,...,....•._•:.. •c..i
-- .. --
c:.:• ~.-,a.
Also.
-
•
=~~C ,·,,r.sr--,. .. -·~'·
b:.1dget
•;...eo-.=.
~...:.~---
·-
•
t"':.,
-··
__
::1~se
.
.,._... sso.S ' ·~~
:c:
•
~a1r-
•
<
-
t~=" -
E
{2{1} ?(·'\\ ... • 2{2} • 2{3\ ,,c-,~, ..
•.. ._,. ::-
b··c·-... ..... o-t..
•
oecon1es •
•
~ · .... ---" L=.. ·"~-· Lt..!
a •L
oc_ ·wa~·. . 't.. .t..l
""'
... •"zt.. ~z-· i::...l
•
•
(2 E) for E -= {{r\ {2\··. anc• · r2E 1{1.2}1. • .. · "·
+ o
z
;::- e (2 E) for Z P ... 12E l{J,4}i ' -
{{""~
U.}>
..>.' •
•.
(c) J·Js: as :r: t::e o:-ccf of Prooosition 19.0.1, it is • • t·:)
sr.cw tha: t:-te
-.
,...., ... Sa .....
-· c::. -Jioi..-. Fo:-
fo:-
•
..1..:-:-ow-Debre~
-
........
. .
eou!iibrium orice vector o = {p ... __) ... -. - A. • •
--\...!·~
. .. ... .... --·· ~
choosing equilibrium pr1ce ,.P,..•n-s "-~'-•' .
.1--,...·~;-rr~i.._~-u
~······-·
set a:--e tne
budget set and the Racine; •
a~;:::r~p:-iateiy
~!""
'
•
• • .. , \ ~ • '!",. •"'I
~.::...=.
-
ror t:1e COi:.t!:-tge:1: ii;-s: good commodities by q E(t + i, E' l = p •.. ~·. t
!
••
~H
a once vector •
1
i~
· tE
E
•
•
x. e X.•
D
....... .,.,..,.,.._v-D ..-breu .. . •
L. · -
~ :\
~
budget set under p, define the
•
19-6
1.~-J..:.. •
transaCtiO~
For
o:- the
...
contingent first-good commodities,
.
• .... !""' .....
by the following
If
= l,
If t
~
then
•
= 0 a:-.c E' = {1,2}. then
t
= (pi{l,2} · (xl{l,2}i
ztEi(t + 1• E'}
[E"e{ {l}, (2}} ql{l,2} ( 2 , E"}zl {1, 2H ( 2 . E" }}/pll {l, 2}:
+
E . = {3
If t = 0
.' ..
•
··-cu ... n .. ·-:.• - :a -· . . --.. . .-- . Z .
tr-a~sc.::t!or:;; •
•
...
1
l
E
X. is in the Radner 1
1
..
•
1
•
b:.ldzet se:. ~
•
•
... ·s~ 1..- =:.::a ~- -
b~.:dget
se:
. .........
price vectors ( p. q), then x. l·s a.ls"-· .;,.,
~v
....
1
z. of the contingent first-good commodities.
•
• • :;:J~~a·,n~l"'1
tnat x. e X. is also in the Radner i:udget set a::a:ne::
that, con..,ersely, if x.
c:-c~··
;,
then
•
., . ... ~··· S .........
to .... bv t •
~~ . ,: t
= (p'{3 4}. (xl{J 4}' wl{J "}.}}/pii(J ~• 1 •. "1 ,'+1 ........ ,
ZE.(~+!, E'}
•
•
- wl{l,2}i)}/pll{l,2}
~~-
•
•
- Ra--i-•- eq d l i bri urn pri:::e vector :;:.
\,
-
.;. .. -
0
~ = ((q_E(t + 1, E'}}E. •
.. _
'
EJ
• t+ 1
E' ;:-)-..
c_
I
tt.-~
• d ... ,... ...... P· = 1rst-goo commodities and a soot once v ,....., • • •
r-o- t •,.., ..
'
(JI
~~·
for everv tE e il, define 1J.. = (IJ.. ,..).E_ • t
IJ. ••- --
·-
•I
:r ~~ t
--
' ' I ( 1-!tE = C.:\l, 2 ;.'2.EJ/pll(l,Z} I-Ll{l, 2 }
·r 1
t
E
X. 1s I
•
1~
= 1.
t =
--
•
•
2 ' E l .. \ '2}' t = ana e .t~ ... t J.
u • t~
~ ·:l
X. l
E
2 and E e
{(J}.{~}}.
X. is also in the Arrcw-Debre!J bu::ige: I
V"''"'""'" (II .., I ---"· .-.E"'t• · t. t E e.:t,,,.
under the
~
0 •~·. •
0'
= ~;;, 2 ~ J.~(l.EI/p; 0{1 2 3 4} if u .... ~..>,.. •. ··~
• :. I l 4.!::1 r cu.:. 't 1!"'. ::!
c..
1 t: l!"-
•
It is also routine tc show t!'la:. if x. I
•
tr:e Ar-.-:::··'11'-Debreu budget set under the price vector •
19-7
•
•
then i: is a:so in tne Radner budget set under pr1ce vectors (p,q}, attained rOi
by the trar.saction z.
the contingent first-good commodities defined as in
l
the previous paragraph.
Let ...- .(o ,\v .l and v .(p ,w .1 be the indirect utility iunctio:1s l1 s 51 0! • 0 0l
l9.E.l
Suppose that, at a Radner equilibrium. we have asse: prices q = (q , ... ,qK) and spot prices p 1
,...Ul+S) ~-.
choice of -::c:isumer i. De:1o:e bv z~ e IRK and x~ e IRL(l+S) the ootima! •
.
-
1
1
w•. = p·x•. for each s = l, ... ,s.
w,-;te v.· • · x!. ' •• Oi = o • 0 U!
a ,..,d ,.,.. • •
5i
> 0 r-o- e".. !"'Y s
•
a
5!
9
"'
...,.
I
r.ssum__\
51
so that the solution is in the
•
•
•
•
• '1. ... -
";c.
The
,,..,·e~lo.... ·.. ~ I •
fi:st-o:-de:- condition is now that there exists o:. > 0 s:.1ch that 1
- .... .
k. -
= [ r. .(8;: .(p ,w•.l/ow .lr , for S Sl 1! 5 S 1 51 SK
t
, ,:\..
b• •••
-
..
~5
•.ve obtai:-:
f",
..... .........
•
(b) In
•
•
-
= \ u r i-s· s s.~
"'K
s =
for
1
1, ....
s.
•
•
•
•
s::1g!e g·:JCC case, the first-order conditions in te:-r.:s
.. ctaect
-. ··--t·.. . i
........ .:.~.....
h.JI
is that there exists a:. > 0 such tha:
~
1
c:. . = "' """ v: ~ .. \,'x• 0 .. I . . .
-
t
a:.-..,"!.
·• ""·
0
C
-
~~:~ro•
•
1!
.u • .(x•. lr
-s s'.. 1•'
......... . .:; -
.. b• •
= ) •
......... .;.
~
=
for everv k = 1, ... ,K.
1
s ..• s !'\.-
(~.L!' ...
J
,,..,.sl
by
J.! = n .u · .(x•. llu • .(x •. I for everv s = !, ... ,5. 01 01 s s• 11 51
-
..,_
\•·~~:~
c'"'• _._c.. . ·.....r c. = ' u r , . . :~ Ls· 5 S:< •
19.E.2
•
By re!ateli:1g the S states if necessary, we can assume tZ\a: 'r > r
19-8
•
2
>
For each s e (l, ... ,S - 1}, let c
.' .
. • be .. ne opt;on with strike prtce c . s ~'
s
e (r
,.r l and let asset 5 s+~ s
Let asset S be the primary asset.
Then
the return matrix is rl
- cz
•••
r2
- cz
•••
r. - c, •'
••
0
Si:"lce r
s
- c
• • •
• •
0 0
0 0
r2 CS-l
r2
' '
•'
• • •
•
rS-1
..
...
> 0 for every s < S and r
s
CS-l
•
'
-
r,
rl
-
• •
•
'
-'S-1
cS-1
0
rs
. ' > 0, this matr-ix is ncns1:1g:.t!ai.
5
Hence the asset s:;:-ucture is comclete . •
[: is sufii::ient to prove that if (x.,z.l is in the budget set of the
l9.E.3
1
I
then x. is also in the budget set cf t:-.e 1
. . ' 0 -·g•P'I::::-' "
•... .. To this effect, let (x .. z. I be as 5 """:...... .. '
···--
;,
. ......... .
... -,,a.r~·
0
. 5
~
I
'. .... • I""''
. 5'
-
~..
. ) ~ ', p 2k.r .. '-r;. ls . 1 s:.: 5!
1
. oor.atn '
Summing over s. we
,....tx ..... ' ..
•
• •
•
r-5'-'· ..a. s z ...,. r -··.. :\,
1
:\. •
.::. ~
~s-"' ""'
n·'x L;-.:• l . •
1.
0.
:s;
. z. . r A. •
• 1 ..
~-
'
, .....
-
COSSIO .. e. •
~
•
lh
...
• • .,.._15 ~.
,-~~
•
.. r ... ls &
"'
= 0.
Since unlimited
_,... __
•
Sd
• -. 5 "5
""-
........
!
··-- ....
•:-:'"'I ~ ;.:. ..... ::
-z
••
""~ ..... .. ..
" ,..
-z
• •
•
tnat the consumers can always ace
•
o~cget
,i.J
-
Then
....
•
="'
~
e ...-• -''l•'""-"''~ .Lu .. •"""••'·
..~r-r O\'w"- De t:rel.J
=-,a.-
•
1
•
...
•'
[ka...,-'k' -
•
a:2c_. .'
1,• \ ~.I
But r.ere
-) > . o, H a..,,...~ -~~---
-
constraint.
= p!s(r slcxlqJ
+
Moreover.
-'3
• . , .... a_ 1 ........ 5
r s2a:2q3 - r s3(alql
+
to an·v •
.•
= c:.r...
...
•
.... 2. 2' ~
a:2q2);
= ?!srs3(q3 - (a:lql.+ a:2q2)1. :-
•
3
lS
nonnegative and nonzero, I:kp •
19-9
15
r skzk
i!:
0
-
I 0:""
<
-
[kp isr skzk
and
> 0 for some s.
By the strict monotonicity of preferences, the
-
consumers can always increase their utility levels by adding z to their..
H-::),.,vever, this cannot happen at equilibrium.
DOr't f OilO . •
We must thus have
We can then shew tha:, bv
-z,
subt:-acting
aga1n cannot
-.·,-......,-
•
•
·. s ••· n
~
.a th .......
•
.
... -. .
(il
(J)
A
pro•;e that if a terminal-node
-·I. .. ·-··'DAf-:.~e· --· v
~
.. " ' - · · 0 ..
~
.• z
1a1
+
a·a z a:. 2
z.. . "::1
+
q. z . . = z . 1a! o 2 o:
+
q. z .• o 2 a1
z. -- . .;.. c·c z,_c!. = z' . .at .-·
+
q c z2 at..
;:!!;
~
l :.;..•
z.
~
•
(-) .:J
..... 0
+
.+
1
•
.• z ..• z ..• z ..• z .l S'.!c!-, t ha: 2a1 101 201 1C~ 2::t
0
z.~a~.
•• l
1? ... const.:m:lt :cr. X. E " • •
• budgp.r - ..
c• ••. : !":""t ;• ,.. asset trading Cz -
c.
::a
at ecuilibrium. •
s~,~·'CIIID:"":.•
.:.....-~o·,·• ,,
·-·• ::n. ... ·-. .. - ..- xl-·•
• l .
the consumers can always increase their utility levels, which • nappe~
•
19.£.5
-
••
~I
z ... 101
:.+ z~b-1' "£.
•
x __ = (..;· ..... .,. zl_:. • .j! .J1
-. •
•
~
l
>:. . =
-.
I, I J
• • •
w • . .,.
-.
I •
•• •
• 0
{j.)
•
~.
-c:. .;, • ... ·....;, •
·-
.. s ..l. .-:....
~-
•
··"' c. ..'I"DS :.... ....
----1-''"'rl
uniouelv • • determine the left-hand
-
•
SlCeS
•
oi (2) a:-tc
tr. at
1: thus remains to prove thar tne pa:r•
(ziai'zz,) s-:: de:er-mined satisfies (11. .\r .t• 'J.
•
~ ~
. ,. : \..>1. ,..,.,
co~str-a1n:
lC ~
c.·o = q . there ex!sts a unique pair ( z 1 .• z2 . ) at at
l· .._s .co 1"~ . . ,. rz~ \ 1 a .......
•
1 l
•
We can then apply (51 and (7) to unique!y dete::-mine
(6}.
• • • ·- ....•.. .:. ~ ..... ,_, .i.
_
2
..
-~
1
. . z2 .. --··---.
Sa
z
x., \'-.re can uniouely determine z b. an::! z. . so as to
sa:isf·.--
~t·sr.
1.•.
+
..... 0 -
r, -1~ ~· 1 - - ·.. ,
.
z .
0
•
From the Arrow-Deb:-eu budget
:hrough (71. we obtain 0
i\ C J.L3Zl Cl· + i\ C J.L4(zl Cl· + 2 2 Cl.) :: 0 ·
19-10
zz.. Cl ( J).
1. the left-hand side is eaual to • + A (z C
.
1Cl
+ ~"
"'t
z
.} .
2 C!
By (2). (3), J.!.z = q .• and IJ. = qc, this is equal to 4 c il.b(zl a1· + 1-12z2a1.) + Ac (zl at· + IJ.,Zz .l. ~ at
1-lz c
+ ::\ JJ.., = q , this is equal to z' . + ,..- z 2 .. c .. a ta: ·a ar
= l am:! A,
H..-··~c·-
W holds.
vector is r = (l - 0.75}r
19.E.6 Q
•
-
-
{ ~ ..
1
= (!6.4,1}.
•
Crice •
~s
"1
(b) The
rA~:: :""":""":: • - · - · ·-
=
'"!"ef'• Qr" is r \i _ ......
un.e~n""" -- •- --
((l-
•
Hence the price
lS
q = ( 1/~Jc
"1
-
•
•--•
~ •• :,.-..
1-,
•
• lS
.:.-.~:~
•••-•
•
.. 0' \ .:.0,""':", . } .
_,.~
0.75)-6.:., ( l - 0.75)·16, OJ
((1- 0.75)·64, 0, 0) = ( 16.0,0).
-
:
-
I.
.'-. ..·••
. ·-·s :.. ;, 1;,.
,;:,
-
p.-s .. ;..0 -"' •y
-·.... -c. :. .s .-
n~
•
' . ~ ..__ "! :"
1 .i. .1.
La\..!
•
•• ~ l '
• -s ... • cannot be priced by aroltrage. a :.
-·
•
•
(d) , .......
-
117-;- ...
asset
. . ,... . - ...,_. - , ... d..·- -- .., ,
•
...
- C. r5io. = 8.
'-
L • ..,A-'J . C ;;;..,_';,.,:'\,
-11.e
•
-:,;.::
,.,._.,
-
....
-..•'
r
'.J
= 0.
•
'
•
= (!,1.1) and its pnce ....
-
•
"3
• •
'" •
•
•
•
-. ·--
Thus there is an arottrage opporturu :y. •
H.
..-
~~:~ror
~
...... h,_.. d...-·-~···ar:,, {c) cannot be priced by arbitrage . • .... • .. .... a--.. !:!.::> .... ~- r t'n •
, .. A..,.,. •
.!"'I
0
-
11:11 •
sa~e
~
,.. .....
-~
...... . -,..,
• • • ,.,. ., I.;.:..., • o
is r ::: (Max{16,1}, Max{4.1>, Max{i,!}! =
s-- ........ - ,. - -_. ;-·
(f) T .;.e • - . . t.:.
.LS tne .
- - .... -
-_
···- -
• -. ., :5 •
I"":~
So its price is q = 8.
Max{O,l}) vector is r = (Max06,l}, Max(4,l}, -
. as 1n ,_J. '~~:~~'
( 1o.~.-! ~ • ...
.
So its price is q == 8. •
••
19.F.1
Let (..:-'• x•1.... , x··~ ['
L x iR x ... x
L IR be a \lo'a!rasian e C• -·-
19-11
-·
.....
-
• • • \".., • • I,: .... ;. •
_. _.
!1 ~ ; t" ~ r-t' ~ , 'I""!""'
?h .. - spct economy, which exists under the standard assumptions. ( p, ... , pI e
~
LS
1
rr · R e IRK.
• ·h ..... ( l . • a~ l. X.,z. IS In
s s
1
sk
r-: -n· S .llrr1· '~1-S':l
It is
to prove
budget set of this Radner equilibrium, tne:1
:5 u.(x~)
1
{r
•
1
51
R =
u .(x~).
=
1
1
In fact, let (x.,z.) be 1
1
Then the
1
ave:-age spo': co:-:sumption [ n x . e IRL is affordable under p, becacse
s s
:::;, · (~ rr x .
"
Ls S Sl
51
.... '
-
::s Ls ~ n (~kr ' z. . i = s '- s.: ;o
w.)}
'·' I
1
1
LJLn r ,)zki •. "' s s" .
= ~. q. zk. :;: 0. '-K K
Thus, b •..• U
'x•' l"x•
• \
• J
I
1
-.l "
.1
·-
.
\,;. . .. .. .1... \....'
·.' -::.
1 ~ '
s
~ rr u.(x
'-s s .•
' so.
•
•
•
I
'
'-
•
•
~ •..;
.:. • • "..J,;,
... · - ;
~
. ll",...~ln.~
-· ·--
-
•
• · - .. '•
o·;• •
...
.•
s
·
...
--~·
i
-- ·-
,.,,.,.~,...a·
~
s s
~
--...
u-n~
rr u.(x .:.
;, ......
51
·I is continuous and concave !f so
U.• (··
•
J. . If the returns are
I
. ' imolies that • 1
-~__ ... S c:,;;
. IS .
0~
-- .
£.,• . :'.
0~
"t
• ~~~
•
~ [
-~c:.~-
c:' -.
-·..... - ---,...-:"""1 ..... - .....,,....p
u.C[ n x .) 1 s s 5!
h- •••
.l.
1
•
l
-
U~!
.,., .. -.... ·,- -· .........
•
Bv the concavity,
. J ..
:., ..T ..
19.F.2
-
\\"A
..... 0
• • I \ ... X u.\L •L
> -
1
. ' . -........ . maximization condition of the eauiiib;:-it.:m, •
-:.. ., .. L
Is
•
U~( ·} 1
- Exercise
proof
- ......
is stric:h·•
Oj
.~a
-.
,
·!"'I-· . I - . . ·-•
o.l...~,
sigh [. beca•..;se, rec!aced by the summation •
-- ·-•
,.~~
r:,;,';,,,.~
'
I
• I
• • I .I. " ' · - •
........ ..---·· .. l e ........... . F eacn state. l9.F.3 (a) .""·--··:here ar-e two goods in .. ............. "'--"-···-· ? . ... ( ' 2'. ...... . -~ ..... 8 .. --o"ll"l ._,.:,..,_ . or function. ....... utili tv -. - ..' --·. • ,...- • . ,.. ,"". . - ... ... _ . •r- ~"1 ra,..n·. i e {1,2} and s e {1,2}, ... ·., -· .... -· :.. •
•
r '
•
._ '"' I
"' • •
•
•
·' '
•
...:.. ..... ~
••
-
~.a
•
~•
•
- · &1
....
1•
·-o-s·~~p ~.:.
•
•
--~
•
... ~·-
• ....... _ •
•
0
•
•
-·
ea
•
'!' ...
~
..~,
~--
·_.::::.
-~s
-..
"'
~
- "·"'"'SU'""'<>'" ........... . ··-·
c~
...... ..........
,...,... ......
..... ....,
jn:;,..l'- 1 ~· ~
-~: ~
•
.. - . ""'• .. •
P-~
u.,;; --
;;
.• - . I .o
• ~ _r.:..-.c:.u -
J p.':'
on
• -
• 1 a·
.
•
_2
if(
++
,
state s.
...
..... L.O.I"
•
• p.
•
We assume tna:
..,
- ..i . ) ~ ~
• •
Qu.(x.) 1
1
e IR
19-12
2 ++
and
•
..........
~1---.~~:~-
-·--· . . ..
~•
l
We claim that (p,q,xi·····x[l. combined wi~h no
a: all, is a Radner equilibrium.
[ r. u.(x .)
su::~scot-free •
This is a
• . .. _ ·u-... .... , .... Assuming that the asset structure is tne inat:-lx
asset !
i. •
def~ne c:;T =
IRSxK,
1
i
ccnst:mt'tion. E
and X~ = (x~ .... ,x~l
LS e iR for each
=
De~:r.e c •
~
•
•.
!""' D
~
•
definite fo:- e1:er-y X.
?.
E
I
and w212 = l.o.'222. sim~ly
is •
2 ++
, •
That
the uncertainty on the endowments for good 1
IS,
a swa;:>ping between the two consumers acoss the two s:a:es, and ther-e • -..
no •'n'"'o-•a:nt'' en the endowments for good 2. ....... ..... ........ 1.. '
IS
•
•
th::o~
l
So denote the total
there ts no aggregate uncertamty.
·- ..
·,.,...oJy ....... .
These
~
bv ~
-u.
-
We can use the Edge•.vorth box of size w for both states. We shall now
c~nsider
the equilibria of economies
and
u.(·) 1
-
• "fr, ere:1 t owr..ership assignments of the total endo.,o.-·ment vector- u. tn.:-ee a!
.. ··n .. - u~ioue .
•
• r;
E
ve-::cis u 11
~2
.• 2 '
•
......... . 0". .
Le: and
2!
?
""-! =-"' ·---
.-a
. . ..
••
•
a:1c ' L/?J''·' - -12 . l
•
•
'. /?) '-• - w::::>,.
....
I
I
•.
'~•....: a~• .,., ,a. r .,. ,a.
--
•
IS
...... •
•
•
1
\\"~ ;ea:-~ o loa.,. -
oi. ..; ..
-
ve-~~rs '-L'-'
i
•
A
-··· ·-·-
• 1..!""\ e 1""'1"'1-! • ........... ~
.... . _.....-·o-_ . .
'
I
-~·
•
~
a~e
•
~1
=
--·
__
•.,. .::. ..... -\. 0,.. .
..... _. ;. ;. .;. c: ;. •
•
corres~cnCing
a· .. . ·-
S -:-
• •
the mean is taken oo;e:- the
,..~,..su.,.,.,•.-
•r"l. !"'- D
0
''/2)·· \1' ._-~1
•
--=· . . . . . . 2 . and
•• . n"'
'
•
t ....·o
...... ...,_.
--:=-~~:~c:: ;:,_
., ;:;.G.
++
•
1S
L ...
--
.._,,_..., ,...n . . . "!\·-a .... ... ···-·'"" ~
x•• I J e :. ,...... is the consumption vector of consuiT,e:•
.l
tr.e C"""'--··--·10'" w ! .. .::.. ..,; ;.;. •
? ":)-
• •
..
. ;~ . 1 1:1.. ·.Ia.
•
::.i":.e u:-:ic•Je eauilibrium with these mean • • ver"""!'--s - ... v£
..
l"".' ... - ....
•
which is the same as lr.
··,. I endowment . .. ean .... tn1.1a '
. I ...
•
ec;ual to p•
lS
....- ·-
0 0 •
where
IR2
•
,.. - · .... Q
is the consumption vector of c-=r:s•..;me:- 2. •
"y~~ ,"""' ........
···-· . •' . - .'1 I • I -
--
".. .. ••
•
. F
1s a
umque equilibrium ior state 2 ( ... ; ... h
co.nsu:ne:-
•
•
.
.
-.~ns··mer l '-...,... .
•
w
•
..
• c~
,.,. ..
p2 vector of consumer 2. and •. -. . is the co:1sum:::tion •
\"~I"~
•
is the consumotion •
++
••
E
••
•
·-1··a· for state 1 (with eot!ilibrium . ".. ... . .. •
.
•
X•
•
--
L pf
a
.
0 ......• ,..:;, :
.\..-
.•=>•
Ti".ese
. . . ' to the three different ·n···-, -. .. ' .-·-.
de?i·:ted in the follo..,,·ing figure:
19-13 •
p•
p•
I u"'* •
02
.x*•·_,
•
Wli
,....,_, ,.
---··············· -·-··--·-··-·-·····
•
• '
Figure 19.F.3(a) ~'a~"' il'-0 ... -
__
;....
...
L.o~C1
•! ... -s'" • ... ~.
•
•
--'""'··
•
•
e•
•
. IS
•
•
..'" .;. d ·. .::.
• • •• • .......... - · • Cr.'
•
• of gocd the relative pr1ce •'
15,
•" v• •) and thus that the allocation ( xl1'''12
.~•T!: I~-~~-
"'""' 'l"-1""'" '-'• .... .~.~ • -.
that
>
..
•
•
•
l
•
-
•
-
'"X-.Jio<"~Pi~'
s' tr:.e
•........ _;. ...
-
,...,_._ ,-. .•
•
•
!~ -
-
~---~----
>
•
0
-·
0
probability 112
-.. ··------· ' .
,_,·en., ..
a
J..:=.
•••\.O.~L._.."
$'If l'"! ...... ,""-
hL.)
I
•
...
;..
".
0
. . ... •
..... -
•
•
.:.
0
L.~~-
0-,......i -. = ~··-· ~~ ·"' •
•
I''"'-:::: • - I ... _.
e .~"'!
o •
•
·...::..o~i.
..;.uc:.:.. ·... -.
o 0
...... •
-•
.I •
tC·
•
· I · ..
• •
•
1..02\"' 12
,.. Vl
.,. •.... ,a.
• I ••
1
, 2. . • . . - \./ }u.?'··-.~l·' "'"
+
.. 'v•• 1\
•
-. c:.-..... 1 -. ... -·... o I •
tha~
••
•
o ..
I
G.
> <:
•
llr
-ss··,..... .. that ·-:,,_
... ;....c:.'
•
0
15
h 1gner 0
small positive
,'2 .I .. ( '( • ) + ...... ·=-~\· i!
I ..
0
'\/'·--. ,1, "I'
~·
•
0
.... x•" fer sure. !1
•
o
I
•
,-
0
0
--·"-,'~!~ ._ 0.,..: ' ~ ,=:J - ,; • 0 • : .:>
"""'
•
'\",:a.
wit~
/.
~··.:
r.~~:~-cc::::=.~v
•
•
. -'
.. .---... ......... ···..... . . -
-o-s··-·-. ···.. -······.::>
that consumer 1 obtains
• X
•
o
B ••.,·
i. r~
third
•es
0
... ' ,'? . .
•
J.·m .;. .. l ..... ;....: -· .. •
tha:
J1o
...... '-· ,_
Ol
of the first
the a! location
+
'~.
highe:- for-
,.~:
- --
'
• • exoec~ea • '!..." h,,· "'- "'··•
-
utili ties of the t \.'.:o
. ,.
t n- sum
01-
. .. ... .. .'-s·
-" • • .. •
' "'""!.-
~
i -·
•
the utilities
('o •;)
~.
. -. " ... -· .. --- . •
I
""'
r ...... r
0 •
~
~ •
. -.. -.- ..____ . -··1· -1····---···~ •
•
•
•
I
.... 0 . . . . . . .
•
OOXJ
•
o
•
0 • •
~.
-o-.-"'s"'o-• •
•
.;..~··
•
'-'·
•
.
,,, e
•
•
•- -- •. .... ~ t; S .it'.J'"'y n 1
110
•
0
'
-
oJ
.;.~·
. ..., ... '.J
•. ··o -
\
0
•
.... . •
~r-ra"...
these
equilibria (all fo:--
for the two states
19-14
•
•
\\l"ltn
. • -g.ao·· . . .
S ~-~~ - . 0 ....
•
a, _ ~~
• •......... : .. ,,.,' ........... . .
...
.
-- -··-· ....
~
:-~.-.
Firs: of a!1, a:1 eqt.:flib.-ium without assets is simply a combination or- a ••,., equiiibr i~,;m for the first state and another for the second. consumotion vector •
IR
X
2
Hence the
of consumer 1, the cons'.Jmption vector
++
IF!z
consumer 2, and the price vector (p•,p•) e
........
canstitu:e a unique equilibrium for the no-asset case.
IF!2
X
+•
+~
As for the case with
t·wo Arrow secu>ities (contingent first good commodities). define
This is s:mpty the deviation of the initial endowment of consumer i for good
-,.. ...... ,..,..,
0
1n state s
~
... u
tne mean.
....
w' ' "''!-., L•• ~e - (1, 0 j - --' • 0:::
-~
--
'1 ---
--...1 ~u
c.;
1
-h t.en
•
u.I
.
w
•• z 1i +
-·
0
<>nro -.- .. ~-A-,. _:,_,.,.,.~-·~'-
Si
-
...... c:. .. . . ---.... . . ••. . ---.-s---·o-s
.
• ........... w' .
lli
+
w
res;.; I: i r. ?
-= o-
We claim that the cons:..unot. i-:::;n •
2 ,, .ne t'
pnce 0
t spa~
vee t or
-=-
~A
.... ·-
2
A-1••.
I "
0
"- '"1 ....... :.
-s ... -
--
"
--t-~ J-·l .. · - •
~.·~,-.-
• ...,._
...
·- --
. l..,r•••...,...,
r: •
~•
...., s .. '- " :. •
- ·-,..•..
..... ... ·w.o
0
l.o ... - -
('
0:
l. • • •
... , -- ··-····
. -· --·
,-~-·.
Defi::itic~
.
• • ,.. ,_, .... -
. __ " .. ·-·
T·•- rna. o
•
0
1 • :·~-
-..... . I .
• I," ' .
..... 1 ...
••
•
... , ... ,a.- ar: ...."'0"-~ -~ ................ ::, 0
.. . ..-- -· ·- "'-·· ~ ::::. """"'
-
-
...
0I
'.
0
1
... ~ !"' -
X-.
•
+~
•' -,
~ "'"' p- ::. l _ ..... · •
!/2)" (x:• ) \-1 1!
I
+
·'
-•
an~
1
.... 0
But this follows
~-.p
o:-
(ll21u (x• ) a.nc the 1 21 0
19-15
•• • .. U i...; .... \i • 0
['
r• r. ......
c.ec. .... ,s .... '!"'"
.
alioca::·~r.
equilibrium, the 0
X
0
.. fjc~~; t :..~-
•I
:..
'
o
rli..,j,i-
.-,r-·.... - '
A~
-:-
maximization condition (condition (i) c.f
equilibrium.
::::11....,_•
1 "'-··-a~-.::;.
""
Deiinition 19.E.2) follo\vs iri!mediateiv•
19.=:.2). i: is S'.Jfiiciem to sho.,,.· that the above
0
i;)-
S .....-,...:. -- :he market is complete, b}' Proposition 19.E.2(!J, i:-:
C -:""1-·-1·--·" -..~ ::. ... .. . .
;:')2
•
o
0
• .. ,...
.,
constitute a complete-ma:-ke:
both consume!"s are fully insured.
0
..... ::t!'
• • • • i ~ I I
~-~ > ' , ' I
= ( l,l) e R
•
-•
2
- •' --
p··~ - ~(,:""'\.•• .' · ' = · ..
0
1
... \ ":.:-1· f :. l-
!2i .
(z•• z••1 or cons··,.., .. 1 ·he consumption pian 1:~ 21' . " '..1···-' ' L.
and •
smce
I
first good
0! 1_! D ,-. ~ ·"" ~
= ( l/2Hw
•
...,.~ 2 .........
•
then
0
. . "'a"' - ... ....,...
2i
~lore over,
we have z•• + z•• = 0 for eac.". s - {' ?\ 112. s1 s2 "' • • ~ • ·
Sl
1
= 0 for i e
•
u . + z·~e
-·nus
z••
., ' 0. ,_:.
1
0
At the se=o:.C., campiete-market equilibrium, the utility level of consumer 1 is eqaal to Hen~e.
b•; ~
our consticct.ion, the sum of the utilities at the complete-ma;-ket equilibrium is smaller than the sum of the utilities at the incomplete-market equilibrium.
--·t rus.
(b)
.1s just a matter of fixing up the necessary notation for ali four
ea-::h s
I
. -.-
--
e ~•
{ ..,.I!\. ...J 1--:""
E
- j.,' w
r .,
•
•
ea.:h s e {1,2}, define us (·1 = u (·I and u
F~r
s:a~es.
--
-~
~Dc--1·n ... f
,j.
- - .;_
u
-
sl
(·)
'.
= u2 ( · )
and
~22'
w 42
1
1
and u
(·l
· - . . . . \.,.I '
,..._.. S,...,t, of : " l - ... ,, ...."g
-- x"I ~ ,
::.
••
-.
•• = X .u ' . 2:
i2'
12'
-·· "'32
22'
.:.1
,,.. x• x• x• l \.~~..r ;y ?:·· ,:~ ".::.;
-.
-..
.
z•• = D...... !-;.,,. ........ 31
= x•• and x• • =
z•• = z••
Z ••
F' 0:""
A Is-~ define
for the
= "'21'
• X
a
.
z•• = z•• 42
•
..
~~-l·~~:~ro•v;s :"1..-• L ...,
.;,-~ !~
.0.
. -· .... -.-. -. . ....._. . . ( . .. - ... -· . ........
(c)
= u l( · i .
•• • 1"'-l"
,....,!1!;
...
s2
' . ( . }. (·I = ...2' s2
~ :~-==-·
32
Finally, for
21'
for
and
... -:.•
oa~
.....
•
'•
z~·. I
•
•
We should assume exolicitlv that • • '""~
-~-s
.;..-
•
•
=
X~
.... _ ""'m" e ,..--..;,,'..J , .. ., I': as •
•
•
.
J.
.
•
•
•
...
.
1
At t = 0, all four states are 1nu lS trng~..! 1S~La: 1e.
-- ..-
-... . . . -.. ......... "" ·........
- •...... c.
;
!
"'~·-
! _,..... ...
• • • • • • • ..c.-•,-.... ... -~ ...... ,:, ....... !) ..:..!ll;..c:·...,.· ......
\r.-,.. ... c::-I.f'\..., .... ...,.;. ...... :;
(
I
./
o
......
-~=
-
........ o .._.. 0 -
.;
I
..
.,
·,
'
o
~
•
i •
t
-
••
."\
.. _
Sectio:-~
-
as ~P'!' ---
= ((!.2},(3,4}). and :1'2 = ({0,{2}.{3},{4}1.)
t:;e asse: s::--ucture to be introduced as follows:
"'-::l-.::.~.--;1"":_· .... - · · ... - - .... vi..
..-·....----
the
1 . •
~
. S --~ .. -·-. -· O:.!""~
•
"'SC:::P-~I..-
- - ...
,. ·o-. \•
I
•
t:-1-1 .o<:_
!11
•~•
~:-a..,.,--1 ...
---·
. -. . . c:- tr.e
e-~-
--~.
•· t ."\~
~~1.-.•,t.·D~ _._.,...,,, __ •
5 -a·~...... !)
...
B. !!
5
=
---- ·c: ,a. ... ,...., !"''; ~
..... ~
•,_? .. , _
0
=
•
a.so
l
s::e~ifv .
.J • .;..;.
the uncertainty is completely
nctatlon in the small-type discussion in
'!."""\
f.
s = !.2 and the second two sta:-es s =
~-a~A-. ... - ! )
-....
.- •
:-\.:...
...
-u
=
I
. . t.., ... ,.. ... 1"'.. o - •
-
1 d e nt {1 2' •f., .. ,,..,..~ .... s ...- .... l··:ps = an ~·e . r, "'·- ·"'' 'v.. ---'· ··-
dd a-e ·-a e • t...l
At
.;.
t
= 1 and event
two states s = 3,4 are traded.) •
•
consumptton vectors
-
'....J • ..,.,. -.'.
(..
.-
I .,- "'ner-='". .
and the spot pr-1ce •
i:c; s:atesl constitute the umque eauilibriu:n. •
19-16
~-~.~
•
x~ 1
_\ t,_ ,""\
•
·h'i. I
~ !)
no
equilibriu:-r,, the utility level of consumer 1 (at t = 01 is equai to (l/4Ju Cxi 1 + (l/4}u Cxz 1 + 0/4)u Cxj 1 + (l/4lu cx; ) 1 1 31 41 11 21 1 1 = (l/4Ju (xi > + (l/4Ju
1
1
and the
u:i~ity
(x2 1+ 1 1
(l/4)u Cxi 1 2 2
+
0/4Ju
(xz 1. 2 2
level of consumer 2 (at t = 0) is equal to 1 • (l/4Ju Cxi 12
C/4lu
2
= (!14)u
1+ (xi 1 1
(l/4}u
1 + cx2 22 2
1 + (x2 1 1
{l/4)u
1 Cxj 32 2
(ll4lu Cxi 1 2 2
+ {l/4Ju
(l/4lu
+
-
(fer 01-
. :·,o;c events {,•. 2'1 an d (3 , 4} at :r:e
-
b . ,, . '. . e.., .. t., , t-m ......
0
.,
0
•
eaua~ •
IS
...
1
the soot •
.
t •
- .. 0 ~·•.~
•
t = 1) constitute a Racine:--
:his equi!ibrium. the utility level of consumer
~,.
= !, then the
states), and the asset price vector q••:;:: \1,!)
! O;J!""
. eacn
z~·.
2
Cx2 1. 2 2
If the four Arro.,.,.· securities are available for trade at consumptio:1 vectors xi·· the portfolio plans
42
<x~ J
·, .,
I
\.
a·L : = 0)
t~
( '/4'J,, • 1 • (x ... • • J. + ~
-- -:
I I
= t'r/2i ~ .. ol'x••J l 1i 0
•
and the
.. .. . •" ievel of consumer 2 (at 0
0
•
,,.~:1":"'"~
"-
Oo/'. . "'"' f ' , ,
0
;_1/
t
= Ol is equal to
( •• ,
xl.,,
- ...
0
, .. 2, , •• , = '-~ , __ lX ··' ' . . I
0
•••
- .... •
We sav:
.! ....
.~
'
' ..... ,,c.r
•
0
. a·
-L.~.....•
...
''/2'ro a..:. iu.l (x•~,)' ! 1
> ""t . , 'x••' I : l I
'x••J -z' • 12 •
+ ''
Thus ==cth •
2e!
a.:
0
securrt1es
l9.F.4
.. , ... \
..... _ a:-: _ ..... ·---· ·o
1... da~ en ....... ·"-··-=> a,~ ~~+ •
0
I
,.. ... ,, ....... .:.-...•0
.l- -n· \," ;.
-"'
•
economy with I = 2. L -
assume the.: the:-e is no consumot ion at t = 0. • 0
...
L = :...
-~-.,, I o
o
wo;-se off after the introduction of
•• ~.ne
-o
f I' ... as c lOWS.
19-17
= 2, The
T
=
1. S = 2, a::d K = 2°
- .
U tloli.·~,·o •
.. ···-
fu,..._.--:-=r.r"s a:-.. d ··--·~ol
~·.. A
,. . e
\' 0
u11x 1J
= (114)lnx
u2 cx2 1 ::
111
(3/4)lnx
(3/4llnx
+
211
+ (l/4)lnx
112
x
+
212
+ x
121
+ x
+
122
22
t.
x
222
.
L.•i = (w1!1'w2ll'wL2l'w22l} = (l/Z,J/ 4 ,1.11.
w2 = (wll2'w212'wl22'"'222) = (i/Z,l/4 •1•11. Note that if state 2 is realized, then the two physical goods are ;:>erfe::t Moreover-, the U.( ·l are quasilinear with these goods.
substi tut=s.
1
•
firs~
The <e:urn structure of the two assets are as- follows: The one en.:: of good l in both states.
oa~·s J
•
•
•
good 2 i:'l co::1
S
-·1 nus,
'!"-~,.c::
·-C. ... - - .
. , . ... . • o·l The second asset pays one U..;.l.....
~
•
when the spot pnces ar-e p
nol = •
•
•
•
~ 5 ;:.
I"' ::11 :
•
•
I
-.·•
o-
'
• l_ ..
..._'"'{- ........... I-···· A~''l,,,..._ I • , 0
•
.......... --
---- -
Q._. A
-: •
- Pzz -
r:'"'----- ·•z•... pl2
'""
,
.\,;~
-~
•
~,,1:2.~·
1.
pll
-
V(p)
- r.•., .... .=. :\.
H...... '-.Pf"'!r"~
.-.
-
'
: ' I ,... . • .. _, J •
2 . that
,..,_ .. .
- ..
-·.-.
C•
5 ·-·c··,. •
!..l .i.
•
;.;.Y
•
:· f a-1:.C.·
... ... _
~
•
• • ...=.. • .... ,, 5 .!. .. • -
1 i:
--
.. -
~ n,~
•
1
~-( p}
•
IS
,. • .,
~omo I \., 0• • •-'--•
.
• 0- -•
Co.~
&I
-
•
•
•
•
...... · '
= 1• that is, (since tne
... ,_. t:te t ,,.-. •
•
• \\
-.
I
I
~
!""" ,..._. ,..
c:. .~:~ P"'j -
0\' •
.
Note that, unlike in the case in whi.:h t:-te
!"'l--··,a
i. L
----·=- .. I
0 .... ....
,
•
l!ia~
• "'!'!~ .. • ... 0 • •
+
I
A
~n-re
-
01
is no Radner equiiibrium in •••
•
_...,... .... , e ,...-r"'O,....,V
•
tr~lS
a ~·--· t:Jo h . equ:!lOr1u:n on :::es, two possibilities oi "./ • •
-
•
•
r
-. I 1rst that
--
c;,. :.
and, --o--'"···-· •
I
-
c::.~
~ !""?!'
0
pnces. ceoends on soot • •
_.__c:,,
and ""'~- •• --
•
•
e s .. a., ,,._,,. •"" ...... '
-
·. . c:. ..... ,
y
·-
0
•I
s~--~~:~
,
__ ,, ...... -a c:... a _::,_1,.. S ............ ~·-~
.
r ~~:~r•~-n.
• I
I• I
__ ...... =-
•
is impossible to tr-ansfer' wealth a.::-oss
---
•'
.-
·in amounts of a single physical good. the ~-·· •
~·l ·...;
•
hand.
P '• •. - :-'2:• .
-
-
•
Pzr
....- •ur~~:~ the asset s·ru L. L. • -
iS.
- ..
· ·li0:1lV
..
•
•-.o~:a-
~
•
••
Then
•
1 •
asset
Then the
t:-.e ql!astunear1ty. we· can calculate the •
•
•
•
0
as
19-lS •
--·
r- ..... n s ' ' !"'Tl jCI ~ :::::::: • -~-··-·
.,
•
•
cema:'!.cs
Hence, •' -o....,. the ma:-ket clearing conditions x ~
.~
;,
= 3/4
x 212
•.ve obtain p
+ 1/4,
11
= p
= I.
12
= p 12 at equilibrium.
cannot have pll
111
+
= l/2 + 1/2 and
which is a comrac:iictior•.
Thus we
-· Pzr 1 nen
So suppose next that o
. l! =
• • -J-' . no \,. . ec: :.n tra:-tsfer across the two states is possible, ar.a nence o = •
•
a Radner eauiiibrium pnce vector if and only if (pl: .p2! l is • ar. equilibrium of the spot economy of state l. .-~ e'-'
-
so i: is easv to check that we cannot have •
n"'....,..
-
\..lll~'Y.
= Jja.
It is a standard Cobb-Douii?las
2"•-l ... -
= 1. )
.... 1-"21
o., ..
-·
•
I
__ _
_, H .=.n.-~
This is another contradiction.
fa~t.
(Ir.
p
11
is no Radner
., .b .
• . ... I p e_U .. ll·. L,.!il. • ,
"
-/
c .... e (a) Let \,_. satisf...,· c•;z ~ • . I •• + - a a c
!9.F.S
·~
( !12.:L'
.
. .... .. e . -. • 1.:
-.
! ,-,,... ~
·~
~c ~;
I '
a
0
a
I
t.•:ttn
+
::s k/2 and
(l/21U (c l + (i/2)L'_.(c .:
(c") +
~? ~--:
e
c
a
+ cd• /2
•
k/2.
a
....
;r .. We cla!m . .. - .... .
•
- -'-··
•
•
"'~a· '"';I ,..
II:; : I .,... ""!
--· S ;-,.o!"' . . -
•
···•
.~
... .0
consutaer ..consumes c· units of the . . a
-
11:11 " • 11:11 -
••
-"·-·
1
..,
e
•
[n
fact,
•
•'-"•
I .. . I
-··
•
•
0
•• · - -
•
ne 15
.__ .. .......... 0 ..,... ·-..
---
..,
--
..
'
::.-=a.· -·..J:.- I
n~•!
"D!""'
--
l ~/?-'
~os·c:
....
• ccnsur.~er- s
•
a~d
....
·- .
. ...
-·-·' en
S •:...~............ .....
•
~
·w~~-
..
• .,_ ... delivery contmgent on any ... ~
he is able and nothing v:nen
,j;.
•
•
--o-\,..· ......'"m !-u
,., .......
• •• ~~-·:o·1o
... .
-- ' ~ c:.;.w
is equi•·alent to the following one:
-c 19-19
.
so
cr.lv if
-
part~cular
. k uruts :::
-.o~ Since a ,...onsurr.e.......... c:o,., -··5·-•
C ~~sr·----,o
... ~ •• •w-...
'
. . . ..... D,..,! 1'"a· :!"":!""'
• • .......... .....'r...... •. tne single consumption ... or g ' "' ' •
.""ic:ao· 'e also costs 112. •
""•·
--.... ·-'
a:e able and \vhich are not) are pr
.
--
o--~,..,.
goo a
•
• •
continger:t or: a:-:y particulai consumer's being able (that is. .........
..
•
•
consumct!·~r: •
-rh~
. ....
...........
SUDOCS~
•
~r·!""!s: :~,e.-s
~
I""-·. -
. I . a. O-c: ;.. . . ·::. •.
~.,; I;~~~~ IP'!"\
.,:;)
a,.. },.. of which im;olves a comolete -
• • . -,,,,, ·• . -o-,..,""'"'!.-res .... .. ........... - • • .a.
- -·· ......
:s a
•
_,., c• .. .i.,.,. • units \l.·hen disabled. a c
::::::
•
~~a~
rr:arkets. then an Arro\V- Debreu ! "::. • ~
c
•
•
••
... " •
l'~l·t•··
!':IS '""'
'
'- •
c /2 + a
s.t.
•
is a solution.
Hen=e
k/2.
Since there are sufficiently many WOrKe:-,
•
alwavs tr•.Je tnat a half of all workers are able and the
-
The budge: constratnt 0
1'L. LS
half disabled.
k/2 thus implies that the feasibilitv •
-<
+
o~her
0
0
{market-clearing) condition is met.
0
~s,-a~c
-
O(lStttve •
0
rea~tz-:.tl~n
.
·-.~r L. .. · - ..
....... 1-' . -
of
-~
0
'- ""' .... ~
~
·"''-'
1 --'
.-' .....-. 1 •. ...
-s~•--n~~
~c:..~--
,..,., to-/A• a:.. aro , :o.. .... l equilibrium
•
1 ~I
0 and an
payment
0
.. 0 -~.-.
- !""og•,..,.,.
. . . . -o·'-
·-.
...,suCo .. .. .. --~ !O
0
~·p
.............. - · .... - · .... 0 -
•• -
0
:.L~
......._ ......
.. -- . -·
.... .... ,.,... .......... .... . 0 ..
-
~
•ns··----~ ..~ • ....,.. C:.i.i:__
.oav-
0
jJ• ·-·-
0
c.
·-•=>
r.- ......
0
-
---····-oollo .
1 ..... •·1 9"'\· · -1 ---·
~,
0
available in the market and
a ·-~
I
d:sableC.
LS
~
-
re~e 1•. .
0
~'"""On'!"',
~.......
&
0:. A
•
-
an •, prom1ses
·e sorne
0
(af:er
••
orc,... possible good and makes .:. •• :.J \. ,. .:. It • 0
,...,...,_:-1-.P~
0 .... •\'
I
- . ··"'
~
0
-- .
·r •• •
0
0
.... , . . . - - . , . , . . . . , _ . .
...-,._
a
. . asset (which '"'S'''"'SS . . :-... ....
-
L
LS
. • . • of tne world) the holder c 1a1rr. .na~
••1"'....
0
ir 1.. a t .. - to
0
the consumption good ·r L. and
a
0
S:.zooose -
o:-
••.
H--r.~n.. .......... Pa.,,i
0
'!'" .....
•
An l oO ....... ..- is the ccntmgent commodity that
(b)
0
•
>
-
-.~
-r 0 .-"0"'1-. . ... ::.
the
•I
..,.,
..
a:-e
..J
. --:- . c.a·.
~ ~ S -.::,.,..,;,.!)~
~ --
•
0
.
•• •. "'
• .,
•
• ' I '
•
0
>
·..... -· r •• •
. I ! .
0
•
0
-
. -. . .
..... -~ I .
'
0
r"
-
I
0
0
E •"' ••
•
with q
...
• .I
0
0
..... .•... ...... ..... .-. .- ..... n-. ;..... ~
• •• •
.;. ....... ~
;.
0
_..__
IS
I
;
,_ -·,-;. ..
U~.l
0
-_ ..... a . ... .,
•:-1 .. ("a.-, • ol
•
\\"~.Dr"' • ,_.,,
0
_,
L - · - :... ....
, •
•
-
.. • •
I
•:--._
....
ma
....
0
.
.. . -. .
0
·-··-
-·os I~. ·~a ., -
•
,...., S
-·'
0
...,-
0
•
•
_..... ~
•
1 •
.._..I
0
•
o. .._ •
_
...·'"''"r. .....
r .........~=· ........... ... _~
+'
~.
-0 -o• .!J •
•
~
.. I• •
IS
Lone idea of perfect
.
'-,.- this condition ~
(t,ql. in the 0
•
+'
I
:"'-"!"r.el•.'"'r ~· i'.C .. ~·
,\.,...... 1 ""!··~s~r!"
a
.... Li!S, we
Tc
. . . .... ..... r.,c:a.~.
,..._ . . . =-~......,---t .'....: ..
A 0
-
disabled and
shall
0
lt
mus~
no\li
1s
C
r"\~:"""!~!":
•
0
.. : ...... r-'1
met, •n"''"' Ll - •
......
- • I\.
.;
·-
0
0
.
-... -
!.-·
\ \'" .-, :"'"' I .. 1. ...,
~
-. 0
If
·'=-· -:::-
0
-0
••
I~
t.
·~;, .....
,..._,, __
! •• ~wo
--~
c:.~~
... -
ts 0
-....
a •- : :. •
~'"""'"~ _..., ·l
••
0
-
.. 1~ t~ IS C~SSin • •
..... • _. ......... • ' I -
..
•
•
--·-
,~,..;
...
..
0
I "'
-·,· L • l· -· • ""
crave •
be preferred to
0
0
So suoocse -.
19-20 •
0
-o··---·... '-·""'·· 0
Presence or •
fai! to hold.
-·1 .-- ...
'/?~~.' \..
•
0
. . ....
,;.i,:
r-·,.~'!'"
-'--i ' - ( .. - . -..·..- .... • '
·~o·or-t·
... 0
L'
( l/2)(:' (k
claiming that he
0
•-..ac= ~·~ ..... 0
compatibility con a 1 dO!'! .
•,...
~r:s~ .;, ·...a.;..;.~...J C ....
•
••••
;. I·-
0
'
and k ->
t/2
-
- •- . ~•
•
g .. r ... -a~ .. s
'
· - · ......
--
.. n ....... and h·-.
0
.. -
de
a
(1;z;u (k
l (l/2lU (k O• I <
+
a
l/ZJU d ( t - q l.
+ (
a
~ ) = (k,Ol (no insurance at all) can be attained bv Since (''"a•'-ct -
= {0,0), - 0 'J. •
Hence
t
-
-~~-5-
Sl.!ooose now that k < t. • •
q > 0.
-
.hl'gh.o-, . .. - ccnsumpt;on level by claiming that he is disabled.
::.
a
• ."'.::> •• ., -- "S w •
. . h ot q UnitS, regardless wnetner .e is actuallv t
-· . z 1rm s
The
profit
- -·
... . . . r · • """"ga· ~,·-=1 ' ~- ... .., -· ,., a contradiction . • •
.• -a ...... . _. ... .
• ~,..···:::!)-~
•
IS
•
•
t-:: be positive ..
IS ass'..!med
........
. 1 • c:---·:
"' ·-
t
< 0,
tnat 1: ge:s a
We must thus ha•:e k =:: t.
disabled.
- q
• 11.·~~:~
then q -
.. o\· .. """'" -· '
~r-~,.. 1"
•
I. ;, h ...
-· :.c:..;..
That
q > t/2,
IS,
al ioca: ion eauilibrium • •
•
•
•
15
•
1 -"''"l~g ~;i,OJ;,." ·~
•
•
•
',_ ..
0
•-.A""\
....
•
.....,_. •
;_~IC1~
..... . -· ··-..., .. a ...,
!-• \"-
a b:.;..-er •
-
The firm's expe-:tec oroftt • •
•
•
whenever he is able to do so anc gets k
·-'"
t/2 •
-
•
~
•
• lc• • ..... ··•= &..
Hence he
•
do so and get
. . b'!e d cr net. a1sa
......
Then a buver•
•
\O,.'QI,:}ci
•.,_
• t:. t .!S msurance can
o.r
-·)
•
'-';
be attained at a comoetlt!ve •
. .... . . . . -- ..... - ........ . :- ::t =-
r~
-• ... •
·..
! ~s-
.
s . ·'oJ,., ... ~. ~-
·-
· I ( -
-· -c. .-c-. •
''"'~'T!' '/;lo·
·. .
•: 1
{ :. ,
-;
.
•
0
•
.;.
.._ .. l..,i
,,_
lo.
•
•
!.~~ ~ 1
•
-· ...
i!""\,a.
:.;,~·
\ ·-
o
•
...... ....... •_......... •
C - - -.., · · - - - - - 0
·-·
....
,.
"
-
~
(1.
J
w
··"- • •
•
• _____ •• _ .._. -,~·~,.,.'A
0
.... - q•
-
a
q -
- t/2 -
•
•
,'\;.L;.~.;,
•
I
..,.,
(a l
de;iva:~ve
is depicted 1n the fol!o•...:ir1g
19-21
>
As
This imo!ies • •
::a.~
+
The given condition on the I.
. .. --:.:.·1 .....
""
(112lU (c•}
a
v~· ~ tr:
= k/2
/2 • c ./2 -
; .. ,
l .. -
I
. .
· . . . ...,. , iir-s:-:r-rle: c:Jnditions of the utilitv• ... a···..-.· za ..to.o.·-
•
-
q, then c
...•'" e
-
•
:~'-•'
•....
•
·a<:
.
=t o:-
-- " I•
Let ( t, c) •
This implies condition Cii l.
••
-.........·-. .-- L c:.- c.- = F'(r•) a a ,. _. ... .., ....... _ -· ·-· ·--· . ·... l s, a •
•
...
t~a: ~::e
no:e
•
r :;?H' r.- ·, c c
•. +
r~
-
•
con,dition ( i i).
•
. -. . . C ~!' ~~·~Lon
•
~
-a~isr;,.s ::a .... I .. ...,.
..... "!:; : •
I
and q•
= k -
1'
k - o• and
- .....
'· t...../ ? '"'.
'~C
~
. .- .
• -~a·
""~~
(1-P~
•
• • 1 '
I
" ' I •
-
Cd
k
= Cd
C:t.
Indifference Curve
I
• • •
••
••
• • ••
••
••
• • •
• • •
••
••
• • •
• • •
• • • •
•
• • •
• • • • •
• • •
• • •
• • •
0
,~,o~
k Figure 19.F.5(b)
... ,.
(c) L J. .. - - ·
' ma .a. :-..
..
......... .... .;.::,.,... ,......
•
IS
·~~
~
that IS, k
;..
~
is not reachable at a:1v -
( _.., ~··~""'' -· J
..
................ H.....
•
we have (i )
r--=•~~:~
c..
.... '
•
····o-··~~:~r :\,...,
...,,...D
•r.:::•r~:s,
........ w. -
anc s :o:~ •
disabili~y
would claim
•
·--
........ ng • I ••• ",,.; . . . 1- 1:\.
Since
. .. :..
(•••.,
-
...,, - ""'•
r. -' 0
0
·' . .. .... -
......
I
o
I
:\,. I
•
+
2::
k/2, c
t/2, k >
c . c a
2::
d~.\..-
•
•
.0-
•
a ~.
•
LS ceot.:te~ •
'
~-A
rl"s~
;,. ~ 0 ....
• .... ~
• •
0
-
. .,_ •
..... ~'=
-
•
0
1j'r:
,.,g consumctlon bundle • •
.... '
fdlowing figure .
•
19-22
•
't
.. J IS (c• • ' c~ ,.,
a
...
t}
•
0
( ' •f\. 12 J o'"
(t,q) E
. . .. • e-:u! ttorn.!tlt under tne ne\v assum:>ticn •
.,..,s··--- - c ;.~
+
0
•
•
'
•
- --...' .•
0
a c
0
•
2 q R.
-=. ,... ... \
- •=C I ... ._ ... •
E
,"!.... 2 .
-
15 I• ..• • • 1 •
f"' .. • ~ .. ,0
(k/Z,k/2).
'
::::
Cd
Indifference C\lrve
• • •
• • •
k ••
• • •
• • •
• • • • •
••
••
••
•• ••
••
/(c..**,cd**)
. . . '.·. • . .. .. . •
. . .... . ". . . . .. .. . ........ •
• •.
• • , . 6'
....
...
. • ...-.. ... ...."!te: ... "'· ....... __ ................ ---·· -. ... ....·-·- .. --.....---......__...·........... ...... ... ..""'. ,..... .• ................ .-•·-. ............ ·., . . ... . ... .. •
.I
. . . . ., _ , ....... .
• ... • • • ··~""' • ... "=:
• .... • .. ..
-:r __......,.-~~ --· t , --....--:• - ·-.
.... . :t .-·-
• .... ..- - -":"'.- 4.\- _ .._ ........_
,
_
--~-··· ~
... ...
•
0
•
•
•
•
• •
k Figure 19.F.5(c)
The equ: I i br l~.;r:-, all cca :ion (c•• c••) is optimal relative to the allocations a · d r"" n m, .., ... tne o=.r'\... t,' - ........... •
p
~
•.. . , ......
•L.ne •
,-!:5..,..,1·~;
u ..
c.~
..
\l~
-
- ...
a c
The reason
•
15
•
•
that, smce it also cannot O:l5el"'Ve be!~ng
aliocations it can choose must
~- •
·."(
-"' C C:.oi
~
.,
) ..:: ,C.
• • -C:.;...;.;,_.., ~-, .... e •
c 12
•
a
•
+
c .12
::S
c
k/2, c
to
. ...... •
L. • • -
• • ,.. r-.. 5 1•• t •••
•
•
IS
a
•
-- -·
c:: ..:::~o ~
•
ISO.G.I
-· --. • • • ••
r~
-~:::.
... +
""-
X. I
r,...,- \\·hich there ; 'W.
••
p·
ro,..
'
(X . Sl
extsts (z .• z .l 1 at
- w .l
+
5!
-.
• · S ,,,.., ...... ... ·'
.... ,..
•
:;,;~
p. - a 5 z - . 1
"'
1
"'·
•
--n·-
c:. I
'
• ·-
a· z. • 1
....... -
S in ..... ~
.. :::!)
:!) 1'..
•
-·
•
• ..
1
•.... ,:. •~e= -.--- .., .~ t -·'-' ~ .....·! ~5::, •
•'J
1 ...
s~
I. 5
• •
a...,t·a: _........... •
0.
'Z .Ct, ',:. c::. .• •.•.•
•
:..
at
~
•i-,.,a.
• ....
I 9 . G . 2 1-· ... :.
+ v(a,q)z . ::S e.v(a,q).
. . . d r;.gn~..-nan side of the first ineaua[itv r.,... a:.1\. :"' .. -0.. !"-
a =
'L ......o ·- r _ ... ,~-"'·=.r·. + - . --··- . -. t\.
•
•
~~ ;
'
• •
~
__. ___ . -
•rrrt:~·.-,=;a· •
Since v(a,p)
z
the secane
;::>
• .Ct.l ::!i: e.v(a,ql, 1.\'he:-e a.= (a. , ••• ,a.,}
a;
·•
•
•1
1
1\.
--
..
--
":;)!\.
••
H .......-- .. a, . ..
•
cf (i9.G.ll by using the portfoli-:: z. ... a .a. I
errata: The last phrase "one cons;.Jmer
19-23
a~
--
" _I'-.
• •• -~
X. I
better off ar:.d t!le othe:- is significar.tly worse ofi" sho:Jld be rep:aced by "b otn . consumers are significantly better off". I
Suppose that the • • • , I Th.e1r ll":lt!a,
cons•.Jme:-s have the same utility function u.(x .,x .1 = x .x .. 1 11 2 1 1! 2 I •
•
It the snare
. . . l The return vector o,(" ..... L&;e lillt1a
•
~
firm is not traded, then tne consume;s consumct1on •
0!
•
•
ini~ia~
= (0, 1
(0,1).
Define a = (0,11.
asset is (l,Ol.
their
=
( !.0 I ana
e~do·..::nents
endowments and their share-holdings, wmcn are x, = ( i, c/2) and •
•
Hence thelr utility levels are u 1 (x, I = c/2
+ c/2}.
.N' a._
0.
t
•p
•I
h.a" _ , x:, 1 -
II
I
s~ar-e
c/2) as
i +
~he
.
-
consumers' initial endo.,vments.
.
-J ._.....:' .
=
... c il\2
((1 \-
. \' •'
(
~
•
( l/2.!/2 j as c
~
X~
::-~a:
-D-
•
•
~
~~a• • ;., •..J ._ ~
1
•
Jcl,
+
Zcl/(2
+
•
•
·'~\i=~ ~
I: is
. ,.cnT l!'o-:qm The ... .,. ,. o • I I:... e a I .
= •
0.
find the A:::-ow-Debreu equilibrium allocation.
t.o
\V,
~
is traded, then .'" ....e asset marke-. is
•
!('\
0 as c
~
a~d
•
•
'
•
•
.
1'"'·w o. •
}
+ Jd, (1 +
c )( l
+
2c} /(2
-
' . .J!:j ). ~
•
0 for both i =
'I •. -,
1, 2.
• •
•
•.,..
"-·
....., ........ •
-'""': • • • . : . 0 -
• ... .
C'P"""ll
!'
.,...~-
p -
~-·· ,i~ I C. r
•
..., .-- .
1
,., '!
... .
S •cr .. ·• ._ ...... _.~ ... •~· l
•
•
oe. at tne
ne~o•;
•
• ••
•
reason 1s tha:: if the sha:-e
.:.
•
•
•
•
•
~~:~=·-r"'!
,...-~.:.-
~···-·
·- -·
•
.
:.
..
. ... .
'-~··.,...a "' • • 0 •
5 "'0""'" •
o
··-
or
....,....,,
I
. ... - .. 0
•
-.................
A,...,.....
• ,
'
oi·
·-e •
I. ;, :.
1--r\~--
.. ::::. ...... .;, • -
('I'-1 • • • ;.
•
:s •
c:ons~.;mers 1 n!s
- ·
0
!'1-.:. c.;. ;. ....
::a
•
-· ...
l ."" -:: ~ .... ,;:a
.....
enna:'l·:ec •
•
•
•
,-.~··
'·
)
•
•
....... _~. •
r !"" e - ..... .::>
.. •
'.J
•
~
.... ---· . -r, ~arket becomes complete and the .... ...-...:"'\ ,_. --a--.. share and the initial asset. .... --·· ,... ____ ... ,.., .. ;-s;olf increases their utilttv ·---, ..... ---.. ·-· .. ........ - . -··-- __. . '"'-·· . --. "''- ·=-- ..... , -' , -a·"'1Pn
C "r: -~
equll!Drtum .
the associated increment in tne
.,. ,..tl"D""A-
1 •·-· .. ., ._ .., - ·
-. .... .. --· -
nnr.1•· ··~'
•
.. .;. • • - r •
:"'·A
,.-.
•
19.Ci.3
n.,..,:; S roar-. ~ ···~
....
•
.... .... .. •
are K assets, with return matrix R e
0 ~ • -~-·
-.............. -... --
"=' K •• such
•
1
•
:
~
19-24
•
. ....,-c::. .. ~
•
I..
8 ·.,·
-
•
-~ • • .. •
0
..
•
-
•
that
a
15,
Ls a., r
=
5
, for each s.
~
s~
equilib1ium of the first scenario.
I:5 a..Kq,;c .
By the no-arbitrage condition, q
a -
We shall now prove that x~.
(q, p,
··· '
I
z• l' l
X•
(z•al - 9 1l"'..... · · ·' z• 1
+
is an equilib;-ium in the second scenario. feasibilitY• condition is satisfied. ••
s:I
= {x.
r z . L.k sk k1
8~'
= (x. e
• •
!
L
s
•
+
+
K
at
·
I
1
c z .
~
·a a•
such that o· (x . - w . .
:;: L.l< ~. r St: ' zk.I for every s and q. z.
5(
:!:
51
, .'
,_,
...,
5!
s a1
ul
-
x IR such that p. (X .
a z . for every s and a·z.
there exists z. e
;;?-
ar
Define 1
~ ~
- 6 Jctl I
It is easy to check tr.at the
there exists (z.,z .) e !R
1
(z•
+
Sl
,. e
~
u
•a
•
~
i.
1
e.a: 1
0}.
I
Let x. e ••
•
8~I
and suooose that it is imolemented bv a oortfotio (z.,z .1 e "' • 0
""'
•
1
•
""' o <=. •
?
fi:s: s:ena;io, then it is routine to check that x. e 8':" ar:d i: :s I
•I !"!"!!"\ 1.a~ .=..... -. ~,..• •,..,_ • • a .... ...., - · · · - • • ' - - """"' 0
I o
...
X.• ·.·s
__
. _, ...........
.. b .,
:~-.lA~An·~~:~~ ,_. ._
•I
•
at
•
_,.....,K '1:::
' """ . ,
. ~
. I ':' 1• 5
"""!. .:. .. ';.1·-··
But .if .x.[ e
!n-=. - .......... • • • I -c!~·
-
•
S·.:p;:os~ nex: that S =-.. .- n _...,
--... . '
• .... ,., a,.... o 5
..... - \ . , . ! .. · -
•
~s
an
• • • • . . . - "'1 ... • .. ..
•
;.J •
.. -
•
E
3~I
•
8~1
X. E I
a oo:-t:ot:o z •
\..,;.
r ·',-st s-.. ..... _,na.-1o . . -
~r-. .. ...~-
o
o
It is easy to check
~
;. \,; t
;.
lZ. ,6.
1
)
I
.,
E
--
_l'-..
'J'(
I
bv a •
•
•
in the first scenario.
19-25
This show·5
t~at
l
r ...... ............. Lo.5 ,=..--.
1
!: is rout me to check that. x_ e B. and it
•
L .. , :<.
-~~:~
·-
•
the~
L
~~:~~~
-"""l - 0 • • .;. ~
.:::11.
'
I
and
•
•
..... :...,J.., ....
Sc .. n--· ~.;,
•
~
•
'-
and it is imclemented •
• 2 . • -c . . . .... sa~:5-r;e~ • .=:J ... • • • .... • ..... . . . - .. ;Define 8~ and 8 b . as JUSt ·-· ;:>, -· l 1 __ .. _K . • ..... ' s·. _,-, . . _ .... ,,.. . ' ..... i: is imolemented b\' a port!OltO Z. E ,"". 1:1 -···~ ~-. .. • • • • I • -
l
•
We shall now orove •
L.s k ·k·
I
11....-
.... .;.
e-:-1
=~a:q
"'
o
I
(q,p,x7·····xi.zi .... ,z[l be an equilibrium of
-~
-
?
to check that
c
.
.. .-., ...... . - ...,-....
1
I
p~ •I
•-
a
This sho'.·:s tha-c 8: c 8':"'
- 6. Ia in the second scenarto .
.., ..
s:•
to
/
•
•
'
• '
.., ""• • c z .... lz . - e.lo: e IRK. '"'O ... • - : LLu.. 1•
...
!
?
8~ ~ 8':"•
'•
i
,.,_., a
.....
.... ~_..;
x~1
im:~le:nented •
is
8~ c 8~.
tha: ~K ~.
1
•
(z
and it is implemented by a portfolio (z.,z .) e 2
•
aJ
e iRK.
- 9. )a. 1
S = 2 and, for both i. define U.: IR I + l/2
U .(x.) = 012 )x i 1
1
and let w = 0,0) and w = (0,1l. 2 1 1 ' , , . L. \W ... ( l/2)a ) 1 !
Let a
u 2 cw2
=
+
1 1
= (0,2) and a
:r. t'.:-.... __;,.'·'
(a) For .. n~ Lo.-
c. ,._r, e.~.ance for a •
-. = .5 - ... .
(x.
!
.
~!
v
x.•
\
!""\-
. \
I
'o
A•·,a.~~·
·"--~
___ .
.
:s ex
eJ ._. ::: ._
•
•
- ·--=........ -· ......
.....• s
~
•
= 1/2,
...
.• s:• .
..... · " ' 0
""~ •
•
~v
~25
e
!
•
. .. . . .--' r -·
•
1
'SS"~ .. •""\ '!..! J. ... "-
•
•
: •ox 151. • <
X
Z;;i -
xz S!. <
+
,
a'~o"J.
I
·· ... 1.1,x .1
• ••
to!· " i i
..,
then 'i'
(.!
--.. . =·-
0 •
= !,
- 4
1.1[ •
-
1
=
: Er"':Q
c-(2)
~.
•
u .l X -.. '• . St
•
::>.
"
.
+
x2U'
4 lnxl2i + x22!'
S!
u( . )
J
.... --. s = 2. •
..--·-. ... ......-~
information provided bvu(·}J. •
.. !:".- u?.(x. .'=--·n··+.ian l 1
l
l = 4ln3 + l.
&:...
u\ · l. l
5
and
t
I
<
-.1 . )
x. T .nus u .•r"''
.
1' 2 u( · l . Hence x. 15 not ex post optimal.
2
2:
•
-
>
-
= 2i::2 .;. 2 and u i ( x i l = 4ln4. u ., X.
•
. (X ,. ' ) l
S1
- . -..-
v~
•
• A· .. a.,......., assertion, consider the followin:=: e ·' .~.~----~ •
S c.- ..... ~r.
•
1
•• •
C''
\''
~
•
p··..:.-·· •• • ••
•
~Si
.,,
'\' •
r~-
• • ._a' ...... 0 '..J;.,j;,~~ ...
t
l
r
-
..... ·- ..
~!"''-·
•
a::-e nc:
C:. La
or a .
u2i(xl2i'x22!) = •
--.a •
2
uli(xlli,x2lil = Zlnxlli
•
i
asspr• '"ton ' if u( ·) - • L
!iT"'S~ ' ,. •
.....
c:
Ther.
(l/Zla l = I.
-
~r. 1 _v, ,
o I
= (2,0).
2
prerers return vector a
~•. • I v
-a~t·mo"U l1 ·• . . . . . . ,
2
= (l/2)2!12 = 2-1/2.
-
~· o ....
bv •
112 + ( 112 lx i 2
L) w ~
19.H.l
~ ~
2
L~:
a!
t
. .ts .1m~ 1ement~d •ov a rout me to check that X. e 8 . an d lt 1 1 -
It IS
1
19.G.4
8~1
•
•
+
[t thus remains to shew
1
1
1
oo1tfolio z. •
I
Bt.:t if x. e
IR, then
X
b'-" (z~.e.} in the first scenario. .1
I
19-26
•
t
'< ;
" I . (X ,_ • .. . . 1,
•••
••• •
~
u( · I
(b) This property is nothing but the definition of x.
•
1
c-( • ) .
(c) Tt:e ailocation x.
1s measurable with respect to u'( ·).
1
cr'(·l
•
f.::~llcws
asse1tior!
Thus the
frcm tne definition of X.
•
1
(d) By the c!efinitior.. u'(·'
'lu(s)) = [ ,(n •. lu(5llu ,.(x ,:
u.{x. l
cr'fs''
1
S
51
51
'l.
5i
Dencte by rtr. ,ilcr'(si}Ju(s)) the posterior updated by u(·) of 5 posterior (rr_,: lu'Csll a: state s.
o- ___ .. a-·..........
"'X""'~.-
•
Then, by the law of iter-ated ccnditi·::Jnal
.
"'
anotZ":~r
•
:,_
\
L
S
rr ..
;
'
l
f
I
(J' \
f
51
'(')
s 'r) .~, • • ( X (J' , . 5 S!
) -
51
\ L
5
,
((
rr
I (j . ( 5 ) ) ,,_Il s )\u. . . (...I( C" ' l. s Sl 51 s:
''') )•
J
lJ
I •
Since u' [ · l is at leas: as informative as u( ·). 'I
-:;
.. .! u '( S,'I I U\' S I',u , . ( XG''(s')) .. S 1 5 I 5 1
, 'I '.. 7T
-.. --
• r~
8 •'·/
.. defir.::ion.
~
\ L
(( I " • I I ( 1) ( C"l s I L .. ,T! .c~SJ, (j s, u. .. X S Sl 51 51 I'
~
I.
•
•••
.' . I .. , , c=l· ,.,,. _.,..., :..'. 5 • • • T \J 'o _=:J .. i -
0
•
l
•'
•
·' .""\:>
\ ) =
• •
' • \ !"(
·-... ... .-.---·i-'1 VU,;
~
•
.. ~ ·'.
•
{
• •
J r Il , • G', S· U
51
'
"
51
( Xu(, S. ) . ) = 51
"'
i
t.nts
(" ·..... • •
' -
S!
S
I
'>
•
cr(·JI. ') ( U. X. O"lS; . !
I
v~
... C o•
·- .. ,_
· l.
·r.~
JL::,.
e ;. ·
. •o( 5 l · x IS .. + x2 . .:= w.(s) for 1 51 -
s -·
e --~ ::1\.. ••
•
~ . I .. -~-
~
1
5 I
cr (I s =
,.,,
u•, s , ~.
·~
.. h--
\.l.t:: •.
~:\ p ....
.... .
. ',
•
1
~_,-·(·)
c 8. l
.
Hence Proposition 19. H.l is stili valld l:I:::ier
-· ".
5 .. ~
l9.H.3
Si
1""''-l'··
l v ·, ::::. . I.....
~-
• •
•
a: leas: as inf'crma:ive as u( · }, and p( . I and w.( · I
:-
"n-1· ~-P' ...:. ~- c:..:. ..... - - · •
'
of Proposition 19.H.l, for eacn signai
5l
••
,,
•
x _ = x . . w h enever r.- ,-.'l'·'
(.....
.
C''' c: '.
i ul 5, 1.
:.; . ( :-<: . ! I
•
:.. ;.!
r:-t • i
8~
S
t
t
. u!·l . . ,
~
I
•
s
•
•
19.H.2
\ L
s
l
olo
\L •
--
s
...-; n; ... ,·en
\.,i,..,..;.
.. -
u(s') u(s"j x .. = x ... .
if <::s·) = C"(s") = u(s), then r •I
Bv •
r u(s')) , (( Tr , • I C' '( 5 )'I I C'1' 5 )'IU , . 'X , . . 5 S I S < S l
~-
Fo:-
-·
.I •.;. -5' . :..
19-27
1
Then, l:v
-
•~o
t•
! "-
ce: •
...
•
•
0
t:-.L:I:;n
o:
a-'.)
X.
l
'
I
0"( • ) u.(x . la-(s)) 1
ll. (X . I IT( 5))'
i!:
51
1
S1
•
or- eauivaientl•;, if c-(s} = c e IR, then we can wr1te • • 0"( • ) u.(x . lc) :: u.(x .lcl I
•
oy tne
o
0
•
•
o
I
Cl
Cl
r...• Hence, by taking the exoectation ove!" c e • -''
ccnstramt.
measu:-a~!:ltv •
1
'.':e cbtain
L·~ E:.,..'{'
S}J([ • , ••• ,
1
sea-
a-( • ) .lu.(x . I c)
_ 1(
c
~1
n ) 5l
I
C1
n )ui(xci I cl. 5
S.:S!.I
H....... r bv t=:e de:i:-:i:ion of u.( · lcds)) ::: u.( ·I c),
Jo
-·
....,.
•
1
. ,{) Lce:...?1.t· -1.... , 5 1 :-
• ~-
. iX . ' L... I
•
,•
. 'J •
:
--- ... --
0 ..,
~'(s)
-.L( ) St
c
1
s
Cl
'
'
•
•
&
5
5!
5l
•
. -. ' ,..... ..,::;-
-·I ~·~=
}.
·~w_,.
- o-• ..-a-. -· .' ~
•
51
S!
51
•
fs that t3 = !
a-( • )
x. e 8. I
1
. then
X
.
Sl
=
I
l
X.
E
.
Hence 8.
I
I
u' ( · l
c 8.
I
5 1
X c: '.,
-.
rh.·5~.. !"'ourJ·ons,
····--..J-··--··'"- - . . . ,.. .... ...,_,.,..,...,_a: probability ·.. .... ·•
•
-·i i:e ...• . . ~-
•
~s·
I
i-.~•-.:.:-:.-1.~1'"1!~
•
........... _. _,
•~rn.=.n~~:~-.~~-
l; the second one is that {3 = 2 ar.d c:. = - 2.
•
•
0
•
.__,~,·-• ~LVII
•
both
of
"'"r. 5·,·
..
.
'•
C"' = -· •
-c:..~._
8 .•
a-( • )
"""e~ ,... ".,. . . . . . . . . .. _ ,f ... -·
•.
•
Sl
•
X
•
:. I/::: ,"'\ ...J-..'
1
l
:: v. = 2, then there are two possibilities:
~
•
•
r;-'(··)
I
•
=
Sl
=c-(s'J whenever a-' ( s l = a-'(s'l. x. =
•
19.H.4
[
n .lw.(x .lc) = [ n .u .(x . ) = U.(x.l.
.
sec
I c) =
. c:(·) l.;.(x. ),
•
.~
}o
=
C
... ,... "Sa.-••an on I -~ ._ -''-' - · v .... •
. . , ,... 5 . ...,·:,.
-~(
rr .)u.(x: . J s1 1 ct
u(·l rr .u .(x . )
U.(x. J.
~
:.:. ,a. ,...
u( . )
,
SEC'
L=e~\t.L, ,., ..... , 5 .1 1([
1
possibili~ies
ha-:;e
= 1 given
= 2 is l/2
(T,
...
s~.· •
the
-. .
• ~,,.,a~
.. . -. -,··a..... - - ·- . to . '. - 2 ·-•c:: also 1/2. The same argument can oe ,..,. ....... the conditional probability Ot = case . _ . ... ·- .. -.,.·- -- 2 r-nis l/2 . = 3 is also l/2. Hence. '= -·..,)· ' . ,.. expected utility function remains to be ,. nd f c·:-- ,.. ..... -.... : = L2. ·-· nr• ~-
·-:>
••
•
•
~-
. 1
Q .........
~
•
•
J
~
-
• p ....
• D..,
c:....~u.1'-w
:..
• •
-.";..,. ' I . -
....
~
.
-
•
l•
•
-1-L."~~.:. ~
r.; •
• •
•
~
~· '-'•
• • ..... ·:..J •••
..........
rf'.
~.
1
•
'.
19-28
-
...~~
•
Thus the equl:ibrium pnce of good 1 is 3/2 .
- --· •
We shall '"'·'"'n"' ..... _ ·,~P ... two cases, separately, on the value of {3 (which is •
19.H.S
.
•
f
-.
.
revealed bv m. ormatton equilibrium price function tr-.e cooled • •
•Ol
I , • •· and show
that, in both cases, consumer i with nontrivial c. has incentive tc pay a 1
s,..,.,a• ••• !.1 c3 > 0 to know the realization of c ..
Su "'t-icl·e..,t'"" l • .. ..... ~
-
We ass .. me that
1
consume:- i has no endowment of the second good.
· (Because of the
quasilinear-!ty, a:1y s;:>e:::: if !cation of the endowment of the second gccc d ""'"' -'OJ- no~..
r.,.,... a r, ·--""'
~-. ... •••-
1· -
Cas ~ -
-
~
•
1 US:On ' "0"''"' ~ ....... .o.· • ••I
....
i
-
.
...
0
s~ppose
fir-s: that consumer i does not know
. ... ..- . · ur.--· · o. ' - .... ·h··s ... .... h:s ...
• I .••. • • 1 U
•
exoectea •
•
( 1/p, p -
0
•
..
..... ~:
'
• ...... o,..,... S - - • ;.w
t
. . ''P --.:::Jo- oJt w d G-...,·.J
-
"'vp ..... . , .. c•ert ... ....
•
l• I I .....
• !)
"·
......
I
•
•
•
1
.
•
~
o
0
ex a:1 te
I
, . !!"ltl
)"
.
- c.
.... - ........ •
••• ,. ~··~· . • • . l'...
... c.) I
(l + c.llno + I •
-.-
n .
I
c. I is
!eve!
1
t: •..
•
••
•
~- '-ne 8'-"'!"'\11:11,-· ......... •...~ ........ "'--
.................._.,_ in the utility levels by rece1vmg •
• 'n---,a.~,gr,".
I: is easy to check that (l
••
•
S ·-·-···· !,.j, · - ·.. ;.'W •
I
•
£[([ + c.li:-:(l-;c.ll. • • •
'"'Or,•A-..' r-,,..__ ...... .•.-.n .... .... -·'
-···-l.o..Joo
c ..
~
0~
I
•
Case
t
.f:
.
•
~· .;,." ~
•
co~s
-
-~
I
on the value r:ot deoend •
1~ = 2
19-29 •
£:. iS 1 -!-
t'
0:
p.
c. l is ::. ~
'a i i ~ "l' Thus. bv Jensen's [ n ~~:~ ..., f........... 1
•
•
+ c. )!n(l 1
£[(1 + c.l!n(l + c.ll > 0.
... No I ~-
•
"''5
1I
I
I
•
iHS
•••
£[(1 + E::.llnCl + c.}) - lnp + p - I. ,. ' I
•
0
this:
..." ..-
•
his
}:":"\ -. 0 •
I
..., l..J,
.·- ... consumer i knows c.,1 then his utility
I
.. I C'
-
'
--
c.l/a, o - (L.+ C'.ll a.od thus his utii!!¥ leve:i is • l
.
·\~
Henc-e
level is
1 ~·.,. .._.\""'.:
•!"'l~.•
•
,~
.... ... -
u~ilitv •
He:~ce
21
1I
I
L'
b·· ' ,.
' ~ I c:,.,, . is u:(x .1!3.c;) = (l + c.llnx 1
""'t.t!
.• -
1
·ne~ a~,
1
is u.(x.l/31 = lnx . + x ..
i.
c:. ..~; ·w
.. ••.
~ •• c:..
~_
~
o:
s·.mocse fir-s: that consumer i does not know c .• • •
bv £[c.) = 0, his . I
1
ex:>ecte::! • (2/p,
0•
L:t!HtV •
___ -
•
..J.. U ""'"a~
•
-···- -
•
tnat consumer
,c.l ·"' •
__ ..... b '/
-
.
LS \r 12 +
Hence •
....
•
:"llS
1
1
x ..
•
•
2!
IS
- 2.
-r ...... ,
knows c .. then his
1
•
n1s der:1and
.'0""..
~~~ .~..........
1
•
•
nrs
I
£.I I and thus his utilitv• !e..-el
+
•
1S
1
c .lln(2
+
+ £.) - ( 2 +
l
.... a...,.. e •' .. :. ·-
•
+
•
u.(x.l 13,£.) = (2 +
c.l/o, (2 p 1 • (2
•
u.(x.ISI = 2lnx . 1 1 11
IS
•
• .. ~nnr: . S
Suo :Jose • •
•
•
his expected utility level is 2ln2 -
21
-
d ... ,.,..,, n~
-r unct1on .
•••
1
c. Hnp 1
+
level (before observing c.) I
p - 2 - £: .. •
•
•
.. d .. __ \.-,.~
•
exce:tec •
1S
c'"l
-
.,.,..,~s· ~.
c. l!n(2 1
•P"'i...... • .~.
A--~
•
•
:.;.~w_
·- ..... -.......... -... .' •
~ -. -
-. 11:11
'-·
. _ -
1
-
2tn2.
-
[t
-;::,_ ~""" I_,.... ~ :""-, .. ~X . '-'--.':." _,. . .........
-
+
----
--•
-. ~ !"""
-
•
.:-;. .... 1 - · .... ,
0-
!:'.lln(2
•
e ...
·-;,
-.··-~~:~-a=r ~·~~~·-~
..
...
•
£:.I! nO •
+
.
.
•
. .
- -···.J-···-·
+
c.ll - 2!,....·-? > I
c.ll. £[(2 ...
+
•
+ £:.I'I-'·•··~ •
+
•
•
I
•
c
c• lj -
)i<"l2 ... ••• ,
!
-
is .....-illing to .
1
·.
io o ~-I o
'
- ~. 1!1 -
- •. 0 0 cv
f;.
.
. 2
:- ?
- Jlo-
or .. ..... •
r-.-
tn~
.. ·- • .. •• ::a
(""'
·~--t•• ~ I ;, ...,· • •
lo. - '
~• c:,.
1
nontrivial
•• •
,... ,..._ .... ...._ - ' • ...... .1:1 ,.._
.. • ...... -J
. £: . ., '
•
< .\I.:r~ ~.:.t\~ , ,.. • ' ,
.
•
•
I
··~.-
IS
•
•
..... -
~
.1:1 .. "" -
•
::ct r:.::)'""e'"'o ... on the value of o. • •
:
. T
•
Thus, bv- Jensen's 1
• • ..... ~. s ' " . .... ·-- .;. ...
•
is easy to check that (2
f; .. 1
•
2.
utili tv • levels bv recetvmg c.I
-h t. •• e
...
E;.,~ ••
• •
a
• ~it
-(A """""' D -, ~
c. I I - 2lnp + p
+
.;;.,. • • ••
•
l9.H.6
.......
'"" ..... "-··"""'•
(a} Le:
•
•
)
Arrow-Debreu equilibrium.
~---
•
-............--..... ·--::::. •
,a. .""""! • ! ! • 1 ""': ~ •
u
.
~
•
I
'
,.
--.... -... ~.
•
. - - - -c. - ,..-~=Cu,;.,:,_.
r
""
~ou
,.......-
. !,..~c.:.
I •
o
,....
• ~r~~:~r. .;
--·.......... -- -.. •
•
perfect substitute for consumer l a .. -
---·'·-· .... -I/? ',.1 . '2'.......... ..? - - •• r-or .'
•
:""'!.-
""' :""": ~ ~ .~..., ~ '!" I A -
• 1 • 0
lr. I a ...
• l.~
•
•
•
. ... ... . .. ·- .. • e a c: S .. -
• •
15
•
in the mtenor,
e·:erv- s.
Thus xls 2 = l/4
-
that
-
l/4 .
l/4,
Thus X2 I s.
19-30
::: . ·"'
....
·-
..... .... c.;.,"
-
-- :-:zz - -
•
• -. v
-
.
:""'·~
"--'
-
C'-l ...,,
•
•
•
- ... .!a ~ - !""" : ~ :: !"" I "' ::. a:.a. :-.. ...... ----· ·•.::::.
.
. . ., ' -. . -
(
.
•
...
~
U .., 1. "t
(b) De::1:::o:e the •D !"" 1,... .. Oi the noncont ingent delivery of (one
q.{.
i.'!
0.
P.
~~
.. b
of) good
••
•
•
Taking (q 1 ,q.,l as g1ven, consumer i solves the following maximiza:ion
.
~
.....·
~o"r e~ ~.~.
( L/Jl[ u.(cu S
1
z 1..
. +
151
1
w
.
2 51
+ z .l.
2I
s . :. \Vhere
~'
lsl
there !s
and
S
(\\'e ar-e
0
ass:.~:ning
that
nr! a" .. r... delivery) prtce vector (a• 1.. a• J a ·-,, ::sc: __ 2 •
~SSP' c:: -·-
.....
_, v = 0 for ev pr•• •
sec:: market ooen after a state occurs.) • •
:10
A~
0
= 0,
• 'z• z• • , . } 7 constitute an equilibrium if \ : l' ----12'-,?
-
..
'
....... ""_
~ -- t": e--··
!="",...• v.
.l ';. .I
--
~-
•.:.,
-!..~
-·a·.... ,_- .. nc ecuilibrim,) allocation is Pareto - .... --.-0
•
0
l
0
.
' ' ' I
•
....... ---.--. .. --.. -. ,..., "" .... .... :·-··-- . ·-
.........
~.
-~~:~
.
0
.
0
•
~··
.·-
•.,JI
• .., . s . . To -... ··.. ; ' ..
·-a· 0
•\.i..a.tl 00 •
'-''-'•'-•••·
•
•
....
...
-
.
•
•
-
.... 0 •·~•
...- ····-·· -·=----··.-. . . . _ . ;_ .... . :. ·-··. . "'" '
~_
I
'
-
'
0
:'\:ote second that,
-
. li
... ..J.;....
.::::.
~
:..
;;,_;:. ~
.~..
.;:,
T"
.. nen •
•
"" _,. +2 ".1.'...i- / - "'1'2 !' • - .. ,.., ~a ..... -.... . ......... - • n::. :: X.,? i • = .:.
•
... ,. •__ .,:_ ...... ........... .
_,.. I
-~
_""\
0
• • •
•
,
• •
,~ , ._.
:: ~ ..........-
• •
-
(c} lf ..
.
! :""1
-.
:;l
......... _
••
- ...
\"-;}1'~?~
I. _ ...._-,. ~
.. -
U
and the asset (contingent
•
<,J
-
•1
---
1 .. ::::. !)
-----... 0
~-- . '
-.. ...... .
·-'
0
....
.
~,....c~ ·,. ,., Com ...... ;._ '-'• •
-~e
•
,..,
•
'
.. .... -•' maximization problem is +
s. t.
.,. z ..
• •• ....... . • •• •
\,_X
•
xl ......, = w.-. . "" J_ '~-
•
(unless xl22 = xl32
,ort=..
...•-
•
is Pareto optimaL
0
w~rw--.
•:-.A, ..
.
..
.--.-t'r"'~e,..
•
~
!"
0
•
'}
.-
-~
•
•
'
7
.. . 0
0- •
L
z2li },
....
.: •. •I
...
'-~,..,·--A.. . . . ' •.
-·----· . --
•
anc
asset (none on t!ngent
it is
19-31 •
,.. .. --· __ a -· ·--- . 0
•
•
,...
0.
-
,....
....
0
~
.. ...
.... ,... o.~asll inea!"" it v, if an allocation x is Pa:-e::J ...
0
1/'' .,. . = >:,_? = •
..X ~;? --
.... ,...,.. •......• .o-
'm .. 1 • ••'z:o• :o,;. the above max
•
'
.
wz S .l
{i/2}\ 2 Ju.(wl . + z12"' L-s= , 1 S1 1
s. t.
2
qi2 !2i
+
q22z22i
z22.), I
+
°·
::5
--
• -s ... • p-· -.a. •teeter (q l'q An a ::::. . ,....,._ ;q .q l and an asset allocation 1 21 12 22
-·
eo• ,.ilib.-t"·u,.,., . . .....
...;r .
r:taxim:zatio;. problems.
,\!t, .. \ (d) !f
~
L·z~·· l (. ; l
-
0 ior every f and j.
•
s:ate 1 na.s cccr..:rr-ed, then the utility
as in ki.
I·.-
•1 n·,=:.s _
s~ate •
,.,.o~•
....... "':laL
-!5
•
....... 'OJ
problem is the same
1 occurred, t h en t h e uttT 1ty max!t·:11za · · L.:on · pro b .ern
-· . Co ,,.,-ing'"n.. l....... l commodity ma •
·- -.- in the
e !""!•·.j::"'.-~~ ..................... •
- '. .!:. c.
,... I-·-··· .. .. -
oo ,......_.
maximizaL.~on
• ~·s : f\.-L.
(e) s:r.ce S ''
.
\
•' . . .
,...,.... ___
•
, ....
,
. . . . . _ _ _ .._ __
" - \..I ~ ;,
~.
fa~
states 2 and J.
. ·-
interior, the marginai r-a• ..
--~
•
l bet\'.:ee:t anv• two of the six co;:-: t mgen:
,~11:11-
::a ~ ~ .;. .;. - •
budget sets of consumer i must be as
.'.oi •
.· a '
4
I
;,
=
1
u ·-. ..., •
i._. ~
'~ "' • •
-
•
•
I •
=
•
...-
' . \X
••
•
:)0.
•
f.X.
E -'
••
•
•
... .
[.,' x~ : ~
Ln
L•'-• 5 xdc.S I·
Lc..Swttc.S1·
t,S c.S!
~
r;.
s
WD
•
•
...,~·d••!", Com .., •••w • ·"'"' --
-• ·""'·10\'"S" .. lo
\..I ..
..
•
),
c.S I
and
..... ..-~
= lX e R : ..;. I
i=!c
-.• •
I=-·••
-·1'"'"5 ..- ......• -
•- I
_c .... .... .• 8":"
•
l
! •
• •
At
- ....
- -
. ...... a -· ·-
~.
c: ...
• •
•
1
•
•
[ex t''I; ••
<
Ltw,,c.u-· w~:r c.-1
< L• :'\ :. a .•._ ,xo : l~e. Lc.,s>. c.s. ,,s> s . tt
= Xn..,· t...~l
'a
.....=. ''•lo• ... l T"........
·-·
r.
•••
/a ,....
•
•
o 1 o ,
•
o
•
o: oo .. n t:.
.w., .}. Lt:,s>~ c.s:
::!: '··
•
a;-s -..... . ..
-\.,.:u . . .0. . .o •u:. .. & •
0
I """"'
i\ •
...
•
0
... ·...... .,. "'a·f..
.. .
c a,..,-i B.• . we can conclude:
_ ...
-
• " • ..
~·
t...~.
f
and X- . [ewtli Lc.,s>l lst
••
•
.......
.l :-:
.l , .... I
~
level w . 21
0
alloca!ions of (al and (dl are L ·..., • '
w~..,;
.
,;·..J~
•
I -.I ' .
. --
~
ailocations of (a). (b). {c), a:-:::1 w'..t. .. lYl e·"'"l·--·•·m •
l ::)
,,,.
• •
•
~
-""'~-I.-~~:~ \.. ...., •• .=:J ..:. • .. a.......
-"' ·-·-' -
.J
••
•
2
-0 •
\X.•
--
•
Rc
•. l
•
•
•
x.t .I -
,
-. -
1
-<
•
n R-
I• -~ •
6
•
-
g. :""' ,a.
• •'
5
an x.
I
E
• • .,-::>. Pareto 00• \oO.···-··
d 8. such that
r;- .... _ l • -· ~
= L/4. j
= XLJ2
1
, oniv if aLoca:.ion of (b) is Pareto optima! if anc • •
19-32
.
""uz
= w!Z2 =
wi32'
(iv} The ec•.Jilibrium allocation of (c) is Pareto optima! if and o:-:ly if •
IT'r.e .Important
Thus the ir.:crnation in (c) and (d) are socially valuable.
0 ..'
•
he~e
fact
•
•
•
is that the equilibrium pnces are not changed by the 1n t.:-ccucuon
.. . ···-·· ......
l.'"'for--::~~·1·~,.,
or contingent commodities, and consumer 1 atta:ns the same Nothing as in Example 19. H.l haooens i:1 t!"lis ' .
everv• 0
1
•
examo.e. 1 •
•
20.H.7
........ _ e .... raT.a·. •
The utility function of consume:- 2 in s:ate 2
"'"1'"'-rt~e ~~
..... S"o"ln '· .... _ l.i- z;,...x .... '?
•
L-
-.----·· -
(a} As ··-··-! .... . •
c•·--,·; .. necessary calculations eas:e:--. makes the -G:::, • •;~e"rt·t .;.,; 'f' . •
·lh · •.. -
' . I .D-.Cu r '·' cera::; lD.lE:: -- ( 1, 6/(' 0 .~ •
.
utili:·:• ~= ....
:~:
s
:""'
~ ·~ u .=.!" .... ... .... .__
•
f~rtct:icns
of consumer 2 are as in C:al.
e"" \,;,
functjon
u:ii~tv •
5/(6
. . • r. :- 0 ... -.aLl~~
;.
-p
= .-.-·
(c} lf ;,.,._...,
+ c)).
.::..
(b)
. i _..... :..,. - s m• a ,ha.•
We
(e:) ;:
+
-
1
•
0
1 "'"' g. -
~
,.:.
___
._r- •:'\ ~
• :-, ~........ : ::.
• 'I"! :a ,.... .:.
ell. and hence consumer ! can a .-:--,·aih· - . - ...•
-
• tne spo.• pnces.
·::> •·,:
s ·-i.a;...,,
-· -·
\\.""..., c "' ,.,.
•
•
,... """" r; -w ..
-
• I ) = lp!\C ,p 7 ( C )\; =
.
-
•1
c '-·.
(d)
If
~
,-.---·1-A._·...,;; ......... -. ~,:,
l.
. --- -·-... .....-·---.... is no rational exoectations ..q·u L'l ; ·· .- . . . • ooservmg the soot prLces, oecause 1::
== 0, _..., ..... -.... l
•
• -. .. .- - :"'!"1 ,. ,.,
L~~.
~~
- "'""
• !'~D~D
..........
~;. ..... -
•
o:;;::
•
•
... U) ~
~ ~ ........ ~ ~
.1. ;,.t;,
-
a:
• • • ft ................ .... •••
~;J.
-
•
•
a!sc no rational expectations equilibrium
-p, ( c I .
... --. n·-,. -· ···-·
c::..
... ~
•
•
:....
•
•
~
LS
-
> 1 >
. •. ·t:•' t;-\ .r.::.. O·"'"' !9.H.J, 1 T;,::; exar:-:p!e t:as the same nature as Examole .. ~. •
.. -··-·
19-33 •
•
-
.
,-h~m ..... -
0
illustrates the .C·:>ssibiiitv of the non-existence of a rationat exr:·ectat!ons • •
equtlibrium.
•
A difference between the two is that, here, tnere
•
LS
•
unc~rtatntv
-
while, in Example l9.H.3, there is uncertaintv ir:. util:tv . functions.
In beth cases,
for excess deman::ls. r-ational
ex~ect:tions
howe~r·er,
there is uncertaintv• in t!tiiit•; fl.!ncticns •
This is the uncertainty that is essential in moce:s of equiiibrium.
19-34
-
CHAPTER 20
20.8.1
Let c be a nonzero consumption stream, then u(ct)
i.'!:
u(O) fo.- all t
and u(c J > u(O} for some t, because u( ·) is strictly increasing. t•
•
V(c) >
~a
t
"Tr.us ·
-1
u(O) = (1 - .5) u(O).
Hence u(O) + .SV(c) < V(c).
But, if c' = (O,c ,c , ... ,ct_ , ... ), then 0 1 1 t=m t V'tc' l = u(O) + ~= .s u(ct-l) =
1
Hence rtc') < V(c).
•
Hence V( ·) exhibits time impatience. •
20.8.2
Fo~
that ct =
c~
the first statement, let c and c' be two consumption st:-eams such fo:- every
t-'{c} -
::s T - 1.
t
Then
nc·) •
Hence V(c} - nc')
i.'!:
0 if and only if V(c T} - V(c'T) 2:. 0.
Hence stationarity
holds. As for ti':.e se-:ond statement, let L = 1 and define u: IR + ~ IR by u(ct} =
l/2 ct .
Let
a1
:;;
a2
= 3/4 and
o3
= 1/2.
(0,1,0,0,0, ... } a."ld c' = (0,0,1,1,0, ... ).
(1/2)
3
= 11/16, anc hence V(c} > V(c').
1 1 = 21/16, and hence V(c ) < V(c' ).
20.B.3
Let
-c',
Compare two consumptior: streams c =
Then V(c) = 3/4, V(c') = (3/4) l
1
2
+
But V(c ) = 1, V(c' ) = 3/4 + (3/4)
Hence stationarity is violated.
-
c", and c" satisfy V(c' ,c")
V(c' ,c"), ~
2:.
20-1
then
2
~
He!:lce Lr>tu-.(cT)
L.>tu-.(c-.).
:!!::
Thus, for any c' = (c0 , ... ,ct), A
~:5tu-.(cT) + ~>tuT(cT) •
Hence V(c' ,c")
2:
V(c' ,c").
?;
Lr:stuT{cT) + ~>tu-.Cc-.).
The first property is thus proved.
property car. be similarly proved.
The second
These two properties can be interpreted as
the proper-ty that the preference between two consumption streams does not depend on the consumptions on the periods where they assign the same •
consumpttons.
20.8.4.
Let u: IR
L
~
+
iR
+
be a one-period utility function and G: IRL +
~ IR be + •
the aggregator function defined in the exercise. all bounded sequences in !RL.
Define M to be the set of
We shall prove that there exists V: M -+ IR
+
+
such
1
that F(cl = G(u(c J,V(c ll for every c e M, where c T is the T -pe:-iod backward 0
shift of c for every T = 0,1,.... M
0
M.l
= {c = {c
e M: c
e M: c
t
T
= c
Define
t+ 1
for all t
= 0,1, ... },
•
e M for some T . • •
0
= 0,1, ... }.
We shall establish th.e existence of such a V( ·) by first defining it on M , 0
.. t h.en ex:e:1::11ng i: to M • and finally to the whole M. 1
'' , d e r·1ne f·. ls-110-o:l To define a ,.c v · J on l\1, o ,co > 0
~IR
+
bv ~
f(Vl = v - avo:. then it is easy to check that f(ol/(1-o:)) = 0 and
r cvJ
= 1 - aa:vo:-l
:!!::
11 1 - oa:Co 0-o:llo:-l = 1 - a: >
o.
•
increasing and onto. Hence f( · I is strictlv •
Thus so is the function V
~
/(VIlla:. tt(x,x, ... I satisfy f(V(x,x, ... ))1/o: = u(x).
Then, by the definition of f( ·), •
we ha.....-e
~i(x,x, ...
) = G(u(x),V(x,x, ... )) and, by the strict monotonicity of
20-2
L
and x' e IR +' if u(x)
f( · ), for anv x •
2:
u'(x), then V(x,x •... )
!:
•
V(x' ,x' , ... }.
We now extend V( ·) to M as follows: Let c e Mr 1
If c
T
e M . then 0
define
•
This definition does not depend on the choice of T.
·r
In fact,
1
CT
u
EiYL
0 •
By the
Moreover, by
the monotonicity of u( ·) and (where the inequality
2:
V( ·)
on M , if c e M , c' e M , and c 0
1
1
is taken coordinatewise), then V(c)
2:
!:
c'
V(c' ) .
•
To finally extend V( ·) to the whole M, for each T = 0,1, ... , define hT: M ~ M
by letting hT(c) = (c ,c , ... ,cT,O,O, ... ) for each c e M. 0 1 1 ~+l(c) 2:
any c e M and T, we have .
and
!:
..
for
•
,.(c) and hence ll(hT+l(c)} ;: V(hT{c)).
•
. L -Moreover, by letung c e sup{ct: t = 0,1, ... } e IR+' we have (c,c, ... ) e M
-'V(c,c, ... )
-
Th~:1.
V(hT(c)} for every T.
nondec:reasing and bounded above:
0
Thus the sequence (V(hT(c))T in !R + is
Hence it is convergent.
We define V(cl
t~
be the limit of the sequence. It thus remains to prove that V(c)
let c e M.
for every c e M.
By the continuity of G( • ), 1
.
1
Since
and hence
Thus
= V(c).
20-3
So
20.B.S
Let c and c' be two bounded consumption streams and let ;\ e [O,l),
then the convex combination ~c + Cl - Ale' is also a bounded consumption he:~.ce
stream and
V(Ac + (l - ;\)c') is well defined
Since u( ·) is concave, •
•
;\ lc') 2:. ;\u( c } + (1 t t
-
V(;\c + (1 - ;\)c') "" i!:
Lra
t
;\ )u( c') for every t. t
~otu(Act
+ (1 -
Hence
;\)c~}
t
t
(;\u(ct) + (1 - A)u{c~)) = ;\~o u(ctl + (1 - ;\)~o u(c~)
= ;\lt(c) + (1 - ;\)V(c'). •
Thus V( ·} 1s conca..-e. For- the cardinality of the additively separable form, consider a monotone Then the function - exp (- ~otu(ct})
transformation f(V) = - exp(- V).
represents the same preference over the bounded consumption steams, but it is not additively separ-able.
[Or we could apply Exercise 3.G.4(a) to see that
only linear (affine} transformations can preserve separability; although it dealt with finite sums, its proof can be easily modified to incorpor-ate infinite sums. J
•
• •
•
•
20.C.l
The own rate of interest of commodity l at t is defined as
20.C.2
Let (y ,yl' ... yt, .. } be a myopically profit-maximizing production 0
path and
Cy
0
,yi·····Y~ •...
) be any other production path.
Then, for every t,
Thus, by summir.g over t = O,l, ... ,T for any T, we obtain I;=o(pt·ybt + Pt+l·yat 1 i!:
=O Pt'Ybt + Pt+l-yat ·
Hence ty ,yl' ... yt, ... } is also profit-maximizing over any finite horizon. 0
20-4
20.C.3
A production path (y ,y , ... yt, ... ) is said to be weakly effLcient 0 1
if there is no other production path (yo·Yi·····Y~····) such that
Ya,t-1
y
+
« y'
bt
a,t-1
+ ....
Jbt
for eve:-y t. For a weakly efficient production plan, Proposition 20.C.! can be stated as follows: Suppose that a production path (y , y 1' ... y t'" .. ) is myopicaliy 0 profit-maximizing with respect to the nonzero, nonnegative pdce sequence
(p , p 1' ... , p t, ... ).
Suppose also that the production path and the p:-ice
0
sequence satisfy the transversality condition pt
(y ,y:, ... y ...... ) is weakly efficient. 0 . '
+1
·y
-+ 0.
t
a~
Then the path
We now prove this by cont:-adic:";icn.
Suppose that a production path (y ,yl' ... yt, ... ) is myopically profit0 respe~t
maximizing with (p !pl' ... ,pt, ... l. 0
to the nonzero, nonnegative price sequence
Suppose also that the production path and the pri-::e
sequence satisfy the transversality condition p •
• •
t+ 1
· y t -+ 0.
a
•
C.r 0 .Yi·····Y~···· l be a production path such that y a, t-.1 +
fer eve:-y t.
Since p ?
Let
<< v' .;. \' • • bt .. bt -a.t-1 v
0,
'
•
for eve:-y t.
Moreover there exists a t such that pt o ·(v + y ) < p ·(y' ·t ·a,t-1 bt t a,t-1
So
d-~fine
a sequence {nT}T in IR + by ,._T nT = ~=o(pt·(y~.t-1
+
~ +
0 and her..ce,
v.' }. ·ot
Yi,tl- pt·(ya,t-1 + Ybt))
for every T. ;.hen it is nondecreasing a.'ld there exists a T such that n:_ > 0. 1
Thus, by the transversality condition, there exists a T such that n:T > \' P T+l • ·aT"
Fix such a T, then we have ·
•
20-5
By rearranging te:-ms and recalling the convention y .. =0
pt-ybt + pt+l·yat + pT·ybT
a,- 1
= y'
a,-1
= 0
we obtain
'
=0 pt·ybt + pt+l·yat).
Hence we must have either pt·ybt + pt+l·y~t > pt·ybt + pt+l·yat for some t ~ T- 1, or pT-ybT > p ·ybT + pT+l·yaT" 1
The first possibility is a direct •
violation to the assumption that (y .yl' ... y , ... ) is myopically profit1 0 maximizing with respect to (p ,p , ... ,pt, ... ). 0
As for the second possibility,
1
note that ( y bT' 0} e Y by the possibility of truncation.
Hence no such (y ,yi·--· ,y~····}
violation to the myopic profit maximization.
20.C.4
Thus it is also a
0
(a) Let a production path (y ,y , ... ,yt, ... ) be overall profit-
0
1
maximizing with respect to Cp ,pl' ... ,pt, ... ) 0
»
0.
Then it is easy to check
that (y ,y , ... ,yt, ... l be myopically profit-maximizing with respect to 0
1
Since p
t
... 0, we have a:llptll
~
0 and hence p
, · y ..... t+.:. a" •
0.
Thus :he transversality condition is satisfied.
Hence, by Proposition
(b) Let a (bo·. mde::U prcduction path (y ,yl' ... ,yt, ... ) be myopically 0
profit-maximizing with respect to (p ,p , ... ,p , ... ) >> 0 (with LrPt < cx,J and 0 1 1 be any other (bounded) production path.
Then, by Exercise
20.C.2, t=T )'" I p , I.' -'-t=Ol t bt J
By
t~e
bcuncedness and
LP .. t
+
· t=T p · y ) > )'" (p • y' t+l at - -'-t=O t bt
<
CXI,
+
P
t+l
• V' J
}
at ·
the limits of both sides as T ..
they sa-:: isf •~·
Hence Cy ,y , ... ,yt, ... ) is overall profit maximizing . 0 1 •
20-6
ex~
exist and
20.C.S
[First prin~ing errata: The overall difficulty level of this exercise
should perhaps be B, although part (c) is harder, as already indicated.]
(a) Suppose that there is a production path (y ,y ,. .. ,yt, ... ) that 0 1 •
•
is myop_ically profit-maximizing with respect to (p ,p , ... ,pt•···) >> 0 and is 0 1 not T-efficient for some T.
for every t, and y a,t- 1 that if J"-~
=
+
y. t
o
v't' ·n ~ .en t -< T' .
•
L;_=O
Then there exists another production path
~
y'
+
a,t-1
ybt for some t.
Then ' by p t »
pt·(ya,t-1 + ybt) <
Take a T' so large
0' =0
Pt' Ya,t-1-+- ybt).
Here, the left-hand side is equal to •
arv.:! the right -hand side is equal to t=T'
~=0 (pt·ybt + pt+1·y~t) + PT'+l.Yb,T'+l'
Hence there must exist a t :s T' such that 0
•V
·.t Jbt This
contracic~s
+ P
t+l
·y
' ( p •y • + pt+l • Yat' at t bt
th.e myopic profit maximization .
(We could also apply
•
Exercise 20.C.2 to derive this contradiction.)
Hence (y ,y , ... ,yt, ... ) must 0 1
be T-efficien: for aH T.
(b) Let (y ,y l' ... ,y t'".) be a 2-efficient production plan, the:~., by the 0
q ) e IRL smoothness assumption, for each t, there exists a unique q_ = (q ++ r. bt' at L h . X IR sue. tnat II qt II = 1 and qt is an outward normal vector of Y at y t. Bv • ++ •
0 the 2-efficie::cy, for every t, there exists a ~-'t = > 0 such that c. f3 qb · · ·a, t- 1 t t
Hence, just in th.e small-type discussion at the end of Section 20.C, there
20-7
exists a price sequence (p ,p , .•. ,pt, ... ) such that Cy ,y , ... ,yt, ... ) be 0
1
0
1
myopically profit-maximizing with respect to (p ,p , ... ,pt, ... ) and (pt,pt+l) 0
1
is pr-oportional to ~ = (qbt'qat) for every t .. Hence (p ,p , ... ,pt, ... ) » 0
0.
1
Ther-efore, by (a), (y ,y , ... ,yt, ... ) is T-efficient for all T. 0 1 •
•
(c) Here is an example of general linear activity technologies for which the
conclusion of (b) fails.
Let L = 4 and define the production set Y to be the
set of the Y = (y v ) e iR b'· a
4
x IR
4
that satisfies the following weak
ineaualities: • y lb .:: 0 for every t; ylb ... max{yla'O} + max{y a,O} + 2max{y a,O} 4 2
:S
0;
•
Yzb + Y3a ::s O;
•••• 3'o •. -v· 4·a
-< 0 .
The::"! Y satisfi~s: assumptions (i) through (iv) on page 737.
gene:-al lbea:- a:::tivity model.
In fact, the elementary
It also exhibits a
a~tivities
a::-e
v 1 = (v'1' ••"', = (- 10 0 I 0 0 0·t 1I • 0 0 0 I 0} 1 0 y
-
L
•
••
•
v
= (v2,v21 = (- l', 0, 0, 0; 0, 1, 0, 0), 2
,·
= Cvj.v3l = (0, - 1, 0, 0; 0, 0, l, 0),
v
v
3 4
5
=
(v4,v~)
= (0, 0,- 1, 0; 0, 0, 0, 1),
=
o, o. o; o. o. o.
112>.
and the free c!isposal technologies. This techn:)logy can be understood as follows: Good 1 is the primary fa~ tor·,
goods 2 and 3 are intermediate goods, and good 4 is the final output.
The fi:-st e!ementa:-y activity v The second one v
2
1
is a simple storage of the primary factor.
transforms one unit of the primary input into one unit of
the first intermediate good.
The third ·one v
3
transforms one unit of the
•
fi!'st intermediate good into one unit of the second intermediate good.
20-8
The
fourth one v
4
transforms one unit of the second intermediate good into one •
unit of the final output.
The last one v
transforms one unit of the primary
5
input directly into a half unit of the final output. The crucial feature of this technology is that there is two ways to produce the final output from the primary factor.
·
The first way is to use v , 5
which yields, at the next period, a half unit of the final output out of one unit of the primary factor. v , v , and then v . 4 2 3
The second way is a round-about production:· Use
Although this takes three periods, one unit of the final
output can be produced out of one unit of the primary factor. t-.:cw consider the following two production paths (y ,y , ... ,yt, ... ) 0 1 (
. •' .. . } Yo· 1 1·····-"t···· : y~
and
= 0 for all t
i!:
3.
•
Then Ybo = (- 1. o. o, O), •
•
•
Yao + Yb: = Yal +' Yb2 = 0 •
v J
a2
3 = ( 0, 0, 0, 1/2 ), •o
+ v. .
.,.
Yt,
0
\.' J
bt
= {- 1.
-
0 for all ·t
l!:
4.
ii!:
4.
o. o, oJ.
v' • aO
v . . a2
+
v' .. a, t-.i
ybJ = (0, 0, o. 1), + y'
bt
= 0 for all t
Hence (y ,y , ... ,yt, ... l is inferior to (y ,yi, ... ,y~ •... ). 0 1
0
(y ,y , ... ,yt, ... } is not 3-efficient. 0 1 is nevertheless 2-efficient.
Thus
We shall show that (y ,y , ... ,yt, ... ) 0 1
So suppose that a production path
20-9
•
a no
y" + y" > a,t-1 bt - Ya,t-1 for all t.
We need to show that, in fact.
y"
+
a, t-1
for all t.
• • •
Suppose first that if yll
•
~
t
y~ a::
must have
y
t
and y"
2::
t'
y
and Yll t' t
•
~
yt' then It - t' I
y •.
Hence y" = y
..
+ y"
t
t
~
and y" =
Then we
Thus
t'
t
2.
- y bt - a, t-1
Ya, t-1
for all t. Suppose now that there exists a t such that y" = y
s
s
for any s E
{t, t + 1}, the:~
y"
2::
y"
+ y"
bt at
y
If
t a::
bl t+l
II
2:.
a,t+l
3 . t"ne:1 ybt .. -> 0 and hence
y
.
a,t+l
y~
= 0 =
1rnus ybll
• t+ 1
2:.
0 and hence
= 0 = .,vt+l"
Y3 = y3 .
Thus Yz = y _ and hen.ce 2 .. 0 ll hence_ Yzbl = 0 and y3 a!. = . By Y4a2 (0,0,0,112}.
•
by Yi:al + Ylb2 y .
10
....
2::
o.
must have Yll 1al
..
2::
= - 1, we a:::tually have y' 'bl = - 1. 1
. ., ·~ 0 F mahyl 11 t =
0
>
ylbl -
•
}''I
laO
have ylbO tl
-
-
~
l.
Hence
I
h " t en y la
Yibo
2:.
1
2::
1/2 1
·
= l,
then y_2b!
Hence ylbl :s - 1. 11
Thus by Ylal I
1 and hence
" > 1 y y by Thus •. • Y laO o = a· II
I
i.'!
o and
we must have ylbZ ~ - 1. '
2:
O
ylbl :s - 1. -
Tnus,
Si:1ce
1 y"1 = ...'"" r
•• > Since ylbO - y1b0 -
:s - 1.
1
If t
Hence
Bv v" - "'laO
+
1, we actually
Hence yl = y . 1
[I-: will be instructive to see how the smoothness assumption of (b) is
violated and the argument in the above answer to (b) does not go through. e Y such that z 0 = Ybol z
= y~ 0 1
+ ybl' and z
2
= y~ 1 >~ then Z is the set of three-period
20-10
•
production paths (made up by two production plans) and it is closed and convex.
Since the production path (y 1y , ... ,ytl"'') is 2-efficient 1 the 0 1
vector (yb
t
y
I
at
+ yb t I
1y
+1
t
> belong to the boundary of Z for every t. a, + 1
By the supporting hyperplane theorem (Theorem M.G.3), there is
x IR ( YbOI YaD
4
x IR
4
a nonzero •
of Z at
Yb!. Yai) and there is another nonzero supporting vector q. =
+
l
(qlllq2llq31) e IR
4
x IR
4
x IR
4
We shall now show
of Z at (ybl' Yal + yb21 y a2).
by contradiction that, unlike the case of smooth production sets; it is impossible that (q! 1q J = (q 1q l. 0 20 11 21
Suppose so.
scalars if necessary, we can assume that q
110
= q
By multiplying positive
= 1.
111
Tben, since c:.
Since q
.
= O, ql21 = c:.lll
00 '2
10
·v'' ~ 0
2
= •• I
· .. q21. v 4' + q31. v"4 <- 0 S LnC-
2 q121· = 2 ·
·••' + q
00
· ybO·
• I
2 q321 - q431 - . >
~
We have thus got 2
-
•
q
-
211
~
l1 a contradicticn.]
•
By :he first-order conditions for myopic profit maximizatior. in
20.C.6
•
Example 20.C.6, if we define wt = Ck
~k 1 ... ,k
0 1
1
~+l'Q'F
2
Ckt,lt) for each t, then
1... ) is myopic profit maximizing for
((qo,wo),(ql,wl), ... ,(qt,wtl, ... l. •
20.0.1
[first printing errata: The budget constraint of problem (20.C.31
should be (20.C.4}.]
We denote a sequence of money borrowed ar.c lent at each
period by (:n ,m , ... ,mt, ... ). 0 1 tl
•t
•
c
-:
::; p · w + m t
t
Then the sequential budget constraints a!'e t
for each . t, •
•
20-11
'
We shall now prove that these sequential budget constraints are equivalent to the single budget cor.straint of (20.0.4).
We assume that the infinite sum finite.
(Otherwise, the single budget
constraint would be strictly larger than the sequential budget constraint, •
beca'.!se the former would impose no constraint but the latter would do.) •
Suppose first that a consumption stream (c ,c , ... ,ct, ... ) satisfies the 0 1 single budget constraint.
Define a sequence (m ,m , ... ,mt, ... ) by mt = pt · ct 0 1
- pt · "\ for each t, then the budget constraint at each period is satisfied. th~
Moreover, since infinite sum ~
0.
Lm. ~~
Thus Lm ~
t
= \ p ·c ""'t t
t
= L,(p · c - p ·wt) also exists and equals .. t t t ~
0.
exists, the
LP
·c. ~t ~
LP
~t
·w
t
Suppose conversely that
satisfies the sequential budget constraints.
Since pt · ct ;;: 0, the sequence •
•-T
is nonr..egative and nondecreasing. t=T
Moreover, for every
£
>
0 and
t=T · for every sufficiently 1arge T, ~=Opt·ct ~ ~=OPt'"'t + L.;=Omt < ~Pt'"\ +c. Thus the infinite sum iirr...,_ 1
t=T
=Opt· ct =
pt · ct extsts an-. ts not greater- tnan
•
L .o• ·w t . L
~
20.0.2
To sho·•.,
(p . v• sum \' L..s=t s • bs
tha~ +
p
(ii' l implies (ii), let (yb,ya) e Y, then the infinite
..... ) + (p • yb + p · y ) exists and, by (ii' ), s ... l ·as t t+ 1 a
\"" (p ·v• + p -v•) Ls s • bs s+ 1 J as
'"" (p . y• + p ·y• ) + (p . v + p ... ) Ls::=t s bs s+l as t ·o \.+1 .ra ·
Thus Pt • 'j'.bt + p t+l . y•at
i!::.
p .y + p .v t b t+l Ja·
Hence (iil holds. To show_ the converse, let (y .y , ... ,yt .... } be a feasible production 0 1 path.
The myopic profit maximization implies that, for every T,
20-12
Since 0 1m1t
E
1~
f. . ) . mae.
Y, pt · ybt + pt+l. y;t =:. 0 for every t.
Together with w <
the
lXI,
=O pt·ybt + pt+l·y:t) = ~(pt-ybt + pt+l·y;t) exists (and it is
Hence
-
Thus
20.0.3
By the definition, a consumption stream Cc ,cl' ... ,ct, ... ) is 0
myopicaliy utility-maximizing if and only if, for every t, (ct ,ct~I) solves the following maximization problem: Max(c:' c'
•' ··I ~ •·
+ ou(c' 1 } u(c') t t+ 1
p ·c • + p
s.t.
t
t
t+l
·c •
p ·c + p ·c t+l t+r t+l - t t <
n-e~essarv ,
and
sufficient fi:-st-orde:- condition for the above maximization problem is that ther-e exists i\ • > 0 for which (20.0.5) holds.
Hence (20.0.5) is
~
•
'
•
eaui~·aient •
to
•
the myopic u~ility maximizati'on.
1-
20.0.4
o for
all T.
As shown in
Example
=0
= (1
-
o)wo
T
1(1 - ol
= oTil - olw/oTw = 1 - o.
Hence
20.0.5
- olw~;:To
t
. k• ( ~}110-o:l > = eto De f 1ne
o, then 1 - oF'(k•J
kt > k•, anc 1 - oF'(kt} < 0 if kt < k•. investment
pc~icy.
=
o, 1 - oF'(ktl > o if
Let (k ,kl' ... 'kt .... ) be an optimal 0
then the Euler equation is
20-13
for eve:-y t ::: 1.
-k
Claim ~- If
0
We shall prove that:
< k•, then kt > kt _ for every t ::: 1 and kt -+ k• as t -+ 1
aa •
•
To this end, it is convenient to first establish the following lemmas:
Lemma -2.
There is no t ::: 1 such that kt ::: k• and 1 - oF'Ckt) + g'Ckt - kt_ l > 1
o.
Lemma 1 follows directly from the Euler equation and g' (kt+l - kt) shall prove Lemma 2 by contradiction.
2:
We
0.
Suppose that there is a t ::: 1 such that
•
og'{kt+ Thus,
oy o e
1
- kt) = 1 - oF'(kt> + g'(kt - kt_
> o. 1 >
(0,1),
g'(k
t+!
> 1 - oF' (k > + g' Ck
-
t
t
- k
t-1
>
and k
t+1
- k
t
> k
- k
t
t-1
::: 0.
•
Write ~ = 1 - oF'(kt) + g'(~t -. kt_ ~ > 0.
1
g' (k
-r ::: t . • - k l > ~ for everv •
"['~~
"['
We shall now prove that
We have already shown this for T = t.
-r > t and asst:me that g'(k T ' + 1 - k T ,) > ~ for- eve!"y -r' e {t, t 0.
+ 1, ... ,
-r - 1}.
T.·lUs, b..the Euler eauation, og'(k • " T+ - k
T-
1)
> 1;.
and hence 1 - CSF'(k l >
Then,
T
- k ) - g'(kT - kT_ } > 0. T 1 1
Thus (3(k
(3-1
"['+
1
- k ) T
• >
~,
that is, k
Therefore, the It - I)-period backward shift investment policy (k ,._ 1 ,k t , ... ) is st:-ictly increasing and unbounded. • •
Hence, for any
•
sufficient! •v large T ::: t, we have k T > 1 and hence
20-14
Hence , -
T+.
Let
F(k l - k T
T+l
< F(k ) - k T
.
T
= ka: - k T
< 0.
T
This contradicts the feasibility of the optimal investment policy.
The proof
is thus completed . •
Suppose first that there is a t
We now prove Claim 1.
~
1 such that k
By taking t smaller if necessary. we can assume that if Then k
t.
< k•
t
~
k
t-1
and hence 1 - oF' (k l + g• (k t
contradiction to Lemma 1.
Thus kt
there is a t a:: 1 such that k
t
assume that if k T > k. • then
a contr-adiction to Lemma 2.
i.'!:
> k•.
t
t
t
then T
~
J < 0, a t-1
k• for every t a:: 1.
By taking
Suppose next· that
smaller if necessary •. we cc.n
Then kt > k• and hence
a:: t.
T
- k
<
Thus kt ::s k• for every t
2:
1.
We have thus
proved Clab 1. We nex: prcve Cla:m 2. k• for every t a:: 1.
contradic~ior:.
We shall first prove by
Suppose that there is a t a:: 1 such that kt
taking t srr:aEe: if ne::essary, we can assume that if k •
•
T
a:: k•, the::1
The:-t kt a:: k• > k:-l and hence 1 - oF'(kt) + g'(kt - k. 1} > 0, a Thus k
contradictior: that k •
SUC:-1
that k
0.
i.'!:
t
~
k
t
< k• for every t a:: 1.
> k. , for every
t
.. - ..
t
a:: 1.
•
" ..... mus-:- h-.,.. ••C:.,;""" Y"t-
-
A
<
t
-
1 fo- every t '
T i!:
Suppose that there is a
~
1.
t.
t
1 - oF' Ck + l
•
Hence kt > kt-l for every t a:: 1. 2::
0
-
We shall next p:-ove by
• then g' (k - k > = 0 and thus, by Le::nma 1, t- 1 t 1- 1
Bl.!t th:s c::mt::-adicts kt < k•.
t
con!;-adic~ion
.-
to Lemma 2.
<
Bv
k•.
i.'!:
•
that k
F(k ) - k t
t
We
=- ket - k.,. t
...
Thus the in\•estment policy
We have shown that it is also increasing. Hence it has a limit, which we denote by k••.
By taking t
eq"..laticn, we cotain 1 - oF'(k••) = 0, that is, k•• = k•. completed.
20-15
~ ex~
in the Euler
The proof is thus
2::
We shall prove that an interior investment policy (k ,ki·····k~, ... }
0
20.0.6
satisfies the Euler equations (20.0. 9) if and only if It solves the following maximization problem:
• • •
s.t. (kt-l'kt) e A for every t, k0 =
-k
0
, and #{t: kt ~ k;} < ex~.
But Ck ,ki,. .. ,k~····) solves the above maximization problem if and only if it
0
solves the following maximization problem for every T: Max •
e A for every
and IL
"I+ 1
= k..!
1 +.1
•
But the necessary and sufficient first-order condition for a ( inte:-ior) maximum of the above problem with terminal period T equation (20.0.9) holds forte (l, ... ,T}. (k ,ki·····k~, ... }
0
1 is that the
•
Thus the investment •ooiicv •
•
•
By the functional forms given,
v2 uCkt-l'ktl
+ (1/2l'17 1uCkt.kt+l) = - 11(2kt-l - ktl + 1/(Zkt - kt+l) =
Hence, by multiplying (2kt-l - kt)(2kt - kt+l) to both sides, we obtain - (2kt - kt+l) + (2kt-l - kt) = kt+l - 3kt + 2kt-l = 0. Hence k
Eul~r
satisfies the Euler equations (20.0.9) for every t if and
•
20.0.7
+
t+.1
= 3k .. - 2k ~
. t- 1
By substituting the given solution
into this equality, we obtain 3lk
-
ko
0 +
+ Ck (kl
1
- k Hz
t
- I)) - 2Ck
0
t
t
- k 0 )(3. 2 - 2 -
0
1)
.
k H2t+l ko + (kl 1) = kt+l" 0
20-16
+ (k
1
- k )(2t-1 - 1)) 0
o.
•
•
[By the standard technique from linear difference equations, we can show that
the abo•te solution is in fact the unique one.]
Let k and k' be two initial capital investment, and A e [0,1].
20.0.8
Let
(k ,k , ... ,kt, ... ) and {k ,ki·····k~, ... ) satisfy (lct-l'kt) e A and (k~_ ,k~) 0 1 0 1
e A for every t, k 0 = k, k 0 = k', V(k) = ~o\l(kt-l'kt)' and V(k') = \' atu(k' ,,k' ). "t
t-L
Then the trajectory
t
(A.k satisfies A.k
0
0
+ (1 -
+ (1 -
A.Jk
0
A.)k
eve~,-
•
t,
(1 -
+
because A is convex.
~otu{A.kt-l •
2:
(1 -
.Xlki•·· ., A.kt + (1 - A.)k~···.)
= A.k + (1 - A.)k' and
(A.kt-l for
A.k + , 0 1
A.)k~_ , A.kt + (1 - A.lk~)
1
+
•
By the (strict) concavity of u( ·)
A.)k~_ 1 , A.kt.
+ (1 -
A.~a\l(k~_ 1 ,k~)
e A
(1 -
+ (1 -
A.lk~l
.Xl~otu(k~_ 1 .k~) .
By the definition, •
V(A.i<
+
+ (1 - Alk'
(l -
t-1'
•
•
Hence
•
Ak. + ( 1 - A )k' ) . t
•
'
•
VC.Xk + (1 - .X)k')
2:
.XV(k) + (1 - .X)V(k').
•
We have th':.ls: established property (i).
Property (ii) follows directly
fro~
(20. D.lll.
20.D.9
For any given k, define a function f( ·) by f(z) =
V(k + z) - (u(k +
then. 'by property (ii), f(z) f"(OJ
i!:
0.
2:
z,
l/l(k)} + oV(I/J(k))),
0 for every z and f(O) = 0.
Bt:t here f'(O)
f"(O) = •
20-17
Thus f'(O) = 0 a!"ld
Thus V'(k) = 'V uCk,I/J(k)) and V"(k) 1
By giving up one unit of good l at period t, a consumer can save plt
20.E.l
units of money.
With this amount of money, he can buy Pa/PD units of (.., (., t+ 1
good l at period t + 1.
•
Since plt/pl, t+l = 1 + r, r can be inte!"preted as the •
• o f .In~eres • •... ra,.e
(a) Let (y ,y , ... , y t' ... ) be a proportional, efficient production 0 1
20.E.2
path a!"ld assume that y t = (1 + n)y tbe a vector that
-q supports
~upports
Y at y . 0
Y at v for every t. -t
1
for every t.
Then, by the constant retu:-r:s to scale,
By the small-type discussion at the e:td of
Section 20.C, there exists a price path (p ,p , ... ,pt, ... ) with respect to 0 1
But this
im;_~lies
\(q ,q/
that for every t, there exists At> 0 such that (p ,p t
This, in particular, implies that pt
0
1A . f.Jf.t , = f. 1 ~ -. t+ t
-
= ?.tq 0 ,
Pt+l
= f.tq , 1
t+A1 •
Pt
)
=
=
Denote this ratio by o:, then we have pt
(b) We shall p:-cve the following assertion: Let (y ,y .... ,yt, ... ) be a 0 1
proportional pr-oduction path such that yt = (l + n)yt-l for every t, and assume that it can be myopically supported by a price sequence t
( p ,p •... ,pt, ... ;' sue. · h t'na~• pt = a: p 0
1
.
f or every t .
0
D""-fl'ne -. =
(_1 -
"')/"'. .... ....
Then, the p:-oduction path is efficient if r > n; it is not efficient if r < n. The proof of this assertion is as follows: Suppose first that r > n. The:1 p
·v
-. 1 'a··.. \,...,.
(l
•
+
= ((1
+
-t
rl p l·(Cl 0
n)/(1 + r) < 1, pt•l' yat
~
t
t
+ n) Yao) = ((1 + n)/(1 + r)) p ·yao· 0
Since
•
0 as t
~ a~.
Thus the transversality condition
•
•
(Proposition ZO.C.ll is satisfied.
Thus the proportional path is efficient.
20-18
Suppose next that r < n.
e IRL.
(1, ... ,I)
Consider a vector ((1 + n)
-1
e, - e), whe.:-e e =
Then -1
rl ·p ) · ((1 + nl 0
~
-1
e, - e) =
((1 +
nl
-1
- (1 +
Thus, for any sufficiently small £ > 0,
•
•
•
-1
•
y' = y + c((l + n) e. - e) e Y. 0 0
Hence, if we define y' = (1 + n) t
t
Yo
= yt + c((l + nl
the constant returns to scale. then •v't e Y bv J
Ybo = Ybo
t-1
t
e, - (l + n) e),
But here,
+ c(l + nl
-1
e
•
· ana • Y
a, t-1
for every t
- y bt - a, t-1
Thus the alternative production path (y ,yi·····Y~ .... )
0
0.
i!::
+ y'
Hence the latter is not efficient . •
•
Let a production path (k ,k , ... ,kt, ... l be as in the stateme!lt of 0 1
20.E.3
this exercise, then F(kt) (q .q , ... ,~, ... ) by q 0
1
"V{\kt,lj
i!::
\7
Fik
q 1('i7/(k - c,l)) 0
- c1.
Define a output price sequence
= 1 and qt = qt_ 1V{(kt-l'll for every t 1
i!::
1.
Then
-
- c,li > 'i7{(k,l} = 1.
1
-
0
~
F(k
t
~
o, qt -+ o.
Since
Hence (20. C. 2) holds.
Thus, as expiained in
Example 20.C.6, (k ,k,, ... ,k , ... ) is efficient. 0 J. t
20.£.4
and
P-oner-tv • • •
V' 1F(k,l) -
0,
= 1.
For
property (iii), note first that V{(k,l)k + V F(k,1) = F(k,l) by the 2 homogeneity and Euler-'s formula (Theorem M.B.2). - k.
Thus w(k) = F(k,UIV{(k,ll
By differentiating both sides with respect to k, we obtain
20-19
w'(k) = -
2 V F(k,l) 1
-------::::- F( k ,l).
< 0 by the concavity, w'(k) > 00.
Since 0
0
20.E.S inputs
For a path (k ,k ,' ... ,kt····} of the N investment goods (and labor 0 1
i\
= 1 icr ever-y t), according to the notation in Example 20.Co4, the
input at period t - 1 is equal to (- k
....
equal t:::: (kt' G(kt ,.k.l, O)o ~
•
, 0, - Il and output at period t t- 1
lS
Hence the profit at period t - 1 is equal to
If a steady state k is myopically supported by price sequence (s •• qJ, then ~
~ 0
this .t:rofit is .maximized at (k t- .kt) = (k,k). 1 the r:1aximum with respect to kt_ fir-st equality
o~
(20.E.l).
The first-order condition for
is stV G(k,k) - qt-l = 0. 1 1
This proves the
The second equality is the first-order condition
for with resoe::t to k , which is q • t
t
+
s
G(k,kl v t 2
= 0. 0
0
20.E.6
If N
= l,
then (20.~.1) .impli.es that q/qt+l 0
= - v 1 GCk,kiiV 2 G~k.k}.
But the
:-ight-~and
20.E. 7
[First orin tin& errata: On the last line of this exercise, the
side is nothing but the slope of the leveL curve at (k,k).
inequality ex < F(k,k'l should be replaced by ex < G(k,k' ).} both sices of G(k, k' + ex}
= G(k,k'l
= 0, we obtain V G(k,k') = - 1. 2
= st fo:- eve:y t.
- ex with respect to o: and eva!uating at o:
Thus, by the second equality of
By the first equality, v G(k,k) 1
V G(k,kl - 1. 1
But here
The assertion thus follows.
20-20
(20.£.1~.
qt
Hence r(kl =
= 0 since V G(k,k') = - 1 for every (k,k' ). 2
< 0.
By differentiating
u2 v1GCk,k)
= 2
Hence r'(k) = V G(k,k} 1
•
Assuming that lt = (1 + n)t for every t, the per-capita surplus
20.E.8
maximization problem among the proportional production paths with rate n of growth can be written as
0
The first-order condition for a maximum is 'V {(k , 1 0
n)/(1
+
+
n) - 1 = 00 0
Thus the inte;est rate 'i7 {(k,l) - 1 is equal to n.
-( y ,y,
St!ppose that the golden rule path
20.£.9
0
••
,
-y, ... )
•
consututes an
•
equilibrium with a price sequence (p ,p , ... ,pt•··. ). 0 1
Then the golder:. rule Under the
path must be myopically supported by this price sequence.
smoothness assumption of Y, it is a unique supporting price sequence (up to a Thus, by Definition 20.E.2, it must be constar;,t.
scala:- mu!:ipii::::ation).
let
-p
So
0
=
p~
for every
t
~
-
By the equilibrium conditi.on (20.0.2), ct = y a + •
•
the sequence
(w
0
,&U
-yb
+ "' . t
Assuming that
, ... ,wt'".) of initial endowments is constant, the 1
So let
-c
= c . t
0
Si:1ce this co.:1sumption sequence is myopically utility-maximizing (Definition 20.0.2}, Exercise 20.0.2 implies that for every t there exists ;\. > 0 such L
that
;\ p = 'Ju(::} t
and
;\ p = o'i7u(c). t
But since
a
< 1, this is a comoradictio:1.
•
~ o:}.
20.F.l
Write r(w) =
Define (= X -+
~
bv • •
0
and
<:
X -+ iR by
20-21
~Cxl
Then
2
=
.;c · ) is
2
twice continuously differentiable on {x e X: x
;t
0} and ~( . ) is
twice continuously differentiable on X. For the gradients of these functions, note that
• •
•
-1/2 1 = (1/2Hx + x l 1 2 1
V'~(x)
•
'Q'~(
Since 1 /31 1
•
•
· ): X ~ IR2 is continuous, there exists a
~l
-1/2 -1 . Define c = (l/2)o: {3 > 0, then 1 1
for every x e X.
e'i7~(x)
«·
Hence 'il.;Cx) - c'Q't;;(x} »
for every c e (O,c ) and x e X. 1
«
'Q'~(x}
> 0 such that
0 for
every c e (O,c ) and x e X. 1 As fer the Hessians, note that 2
D
Thus ••
'1
o
+ v
x
+
2
2
2
~(x)
= (- l/4Hx
1
.;Cxl is negative semidefinite on IR "f:.
< xi
0'· '. +
) +x2
2
-3/2 1
1
11'
2
and negative definite on {v e iR :
Note also that, for every x e X, x' e X,
x2_ and v
1
v
+
2
~
0, then v·D
2
~(x)v
2
< v·D .;Cx')v < 0.
On the other
x l a.1ci hence, by depoting the 2 x 2 identitv matrix b .. · I. we have • • • 2
I
1<X. +
•
d[t:;{x + cv))/dc •
l
7 (x1 + x2l
-v"(x + x l Q 2 1 2 = 2{x -
+x))·(l-
2
+
= 2(x - -r(x
1
cv))/cc
2
r"1 cx 1 r"2 Cx 1
2
2
2
= 2llvll
2
+
2
+ x2)
7" (x + x2) 1 1 7" (X + x2} 2 1
(2c)v·(l-
+ x }J · v
+ x2)
l
7 (xl +
7" (X
2
x2)
+ x2l 1
7" (x + x2) 1 1 r"(x + x2)
2
)v
)v
)v
1
2 (2c)llvll .
>
o.
d [~(x + cv))/dc lc=O = v·D ~(x)v.
On the other hand, by the definition, •
2
Hence v·D ~(x)v > 0.
20-22
•
Now define S
2
= (v
e IR : II vii
2 v · D C::(x)v defined on X x S. compact, there exists /3
= 1).
Consider the function (x, v) -+
Since this function is continuous and X x 2
1
< 0 such that v·D C::{x)v
?;
s
is
13 for e\o·e:-y (x,v) e X x 1
s.
•
We shall now prove tnat there exists a c > 0 such that, for every x e X and v 2
e S, if llv -
- 2
-1/2
- 2
II < c or llv -
-1/2
2
•
-112
n<
-1/2
2
c, then v • D C::(x)v > 0.
In fact, suppose that this claim is false, then, for each positive integer n. 2-1/2
. ( n n e X x S such that llvn t h ere ex1sts a x ,v 1 - 2 2
II < 1/n or
- 2-1/2
•
-112 -112 n
taking a subsequence (and by multiplying - 1 to v ), we can assume that there 2
. X . . n d n exts:s an x e sucn tnat x -+ x an v -+ v =
- 2
contin~.:i-o;y.
2
v · o C::( xlv .:s 0.
actuall•; exists. •
-112 -1/2
Hence such a c > 0
Tnis is a contradiction.
So define
• •
•
s·
= {v
E
S: llvn -
II
i!!:.
c and II v
n
- 2
-
2
•
then
s·
Then, by the
•
2
-1/2 -1/2
- 2 - 2-112
S ' , a::1o........
2-1/2
E S'.
Thus v·D
2
~(x)v
i!!:.
c},
-1/2 2 ..
s·.
is compact, v·D C::(x)v > 0 for every x eX and v E
II
-1/2
< 0 for every x e X with x
=
0 and v
•
So let
e S'.
•
2
tne max1mum exists because for every x e X, x' e X, and v e !R , •
< x'•
1
Then let c e (0,£ }.
> 0.
X E
2
X,
2
v·(D ~(x)If v
E
s·.
then
X ;e
0,
V
2
e IR , and 2
2
V ;:
0.
£D c;;;Cx))v < v·D 1;(x)v .:s 0 .
•
•
20-23
If v E S', then
2
- cD ~(x)
2 is negative definite on the whole IR .
Hence, if c e
(0, min{c ,c }l and we define u: X ~ IR by u(x) = ~(x) - ec;(x), then u( ·} is 1 2
strongly monotone and strictly concave. •
20.F.2
A couple of possible graphs for a policy function and the graphical
The uniqueness of a steady state can also
dynamic analysis are given below.
be proved as follows: Since 11/J'(k)l < 1, d[tJ!(k) - k}/dk = 1/J'(kl - 1 < 0 and dec~easing.
hence the fur:.ction k -+ tJ!(k) - k is strongly
Since a steady
state k is cha.-acterized as t/J(k) - k = 0, this implies that there is only one steady state.
• , ,.
• ... ., ,. •
k'
•• •• ••
k'
,. • ,.
••
•• •• •• •• •• •• •• • ••
... ··--- -- ---)"' ,. •
• • • • • •
•• •• •• :,.
,. .,. ,.. ,. •
,.
,. ,. "
,. ,.
•
•
····------·····------··~
,. .,. "'
•
•• •
, ,.
,. ,.
·--7!.~
,. ..
•• •• •• •
••
6'~ ...............
•
•'
•
0
0
k
Figure 20.F.2
•
20.F.3
,. •• "'· ,. ,. •
.: ,., ,.
•
For the Ramsey-Solow technology, we have
v u(k,k") 1
= u'{F{k) - k')F'(k)
and hence •
v12 uCk,k') = -
•
u"(F(kl - k')F'(k) > 0 .
20-24 ••
,.•
k
•
As for the cost-of-adjustment model, we have
v u(k,k') = u'(F(k) - k' - rCk' - klHF'(k) 1
+ 7'(k' - k}J
·and hence v
12
uCk,k'} = - u"(F(k} - k' - 7(k' - k)Hl + 7'(k' - k))(F'(k} + r-'(k' - .k)} .,. u'(F(k) - k' - r(k' - kll7"(k' - k) > 0,
because 7"(k' - k}
~
0.
While a diagrammatic proof is given in Figure 20.F.S, a proof based on
20.F.4
the first-order conditions is also possible.
Denote by k~r(S} the solution of
-
the optimum problem when the shock is transitory and takes value e.
--
0.
Tner:.
By (20.F.3l and u(k,k') = g(k) + h(k'),
Bh(k~r(e},e}/8k'
+
-
oaV(k~r(e),O)/Bk' = o. - .. e h.,_,
Since
By differentia<::ing both sides with respect to e, we obtain
•
= 0.
Hence
dk~r (9)/d9 = -
-
Since
2
a
h(k,OJ/8k'
2
~ 0 if and only if
1 ~ 0)/-· -0 2 hHI<, Oi{ ·~Q Cw > ( 0•
20.F.S
A diagrammatic proof goes as follows.
2
Since a h(k,O)/c3k'89 > 0 and
2
a g(k,0)18k'ae > 0, as e goes up, the indifference curve going througr.
(k,V(k,O)}
becomes fla!ter and the graph of V( · ,9) becomes steeper.
20-25
Hence the
(k,V(k)),
tangency on the graph of V(- ,9) is attained in the right of
k~ > iC.
the:.. is,
Moreover, since the indifference curves are of quasilinear form (and
the graph of t-'"( • ,9) becomes steeper}, the tangency is attained in the right of •
•
•
Indifference Curve
" ," "
, . -" -, --, , -, ,,_ ~
v
, . " ~...... / ,. ,,' : , "" ,'""~~ . :.-!: ,
I
,.,. . . . -
.,.--~
•• ••
-----/
•
I
•• •• •• •• ••
"
•• •• • •• •
•• •• ••
••
••
•• •
••
••
0
•
• •• • ••
•• •• •• •• •• •• • •• •
•
kl kt Figure 20.F.5
k
V(k')
k'
•
A proof based on the first-order conditions goes as follows. •
· so.u ! .: k P·e) l tne •• or: 1
r
o~
•
Denote by
•
the optimum problem when the shock is
perman~nt •
takes value e.
The.r: ki(O} = 0.
By (20.F.3) and u(k,k') = g(k) + h{k' ),
oh(ki(e),B)/Bk' + CSBV(k~(9),9)/c3k' = 0.
Since BV(ki(e),6)1c3k'
= u 1u(ki(9),ki(e),9) = c3g(ki(e),e}/c3k,
we have
ahCki(el,e)/Bk' + oc3g(ki(e),e)/8k = o. By differentiating both sides with respect to e, we obtain
= 0•
Hence
20-26
and
--~~~------~.---~--~-------=-.
+
Since
2
a
2 2 2 h(k,Ol/ok' + aa g(k,ol/ak < o,
2 -
a h(k,Ol/ak'ae
> o. and
•
20.G.l
Suppose that a price sequence (p ,p , ... ,pt, ... ) and the consumption 0 1
streams (c .,c ., ... ,c ., ... ) (i = 1,2) constitute a Walrasian equilibrium 0 1 11 tl such that c . > 0 for every t and i. tl
every
t
1 nus,
'f f O!" every t, ct+l, > < ctl 1 1
= c
. t+l,.:.
and i.
+
c
Then pt /pt = ou.(c +1
1
•
.l/u.(c .) for-
t +1'1
1
t1
Hence ou (ct+l,l)/u (ct l = ou (ct+l, 2 )/u lct l for eve:-y t. 1 1 1 2 2 2
t+l.2
= 1 for every t, we must have c
•
t+ 1, 1
= c. 1 ana c ~...
t+ 1•2
= c
t2
.
Thus the equilibrium consumption streams is stationary and the sequence of equilibriu~
a.
p:-ices is proportional at rate •
•
20.G.2
•
The first statement is an immediate consequence of the stng!e budget •
•
•
constraint, which is represented in (20.G.3) and mentioned in footnote 28. (And the in-period marginal utilities of wealth are equal to ;\ in (20.0.6).)
Its implication on transfers of wealth is that as the timing of transfers of wealth does not matter, it can take the form of a transfer of commodities on any period.
•
(a) For ea:h 71
20.G.3
1
E
(0,1), let
c/nl) = (c0i(l)l),cli(1)l), ... ,cti(lJl), ... ) (i = 1,21
be the unique Pareto optimal (interior) consumption allocation such that 7i 1Jotu (ctll =
1
1
11 - 1l/i7otu £ct L 2
2
(Such an allocation can be characterized
as the unique solution to the problem of maximizing the weighted sum
20-27
for every t.)
The Pareto frontier of the utility possibility set can be
parameterized by
•
(b) Let .o(.,..,., I l =
sequence.
•
A Walrasian equilibrium l}l e (0,1) can be characterized by
Lrpt(TJl}·ctl(TJl) - ~pt(l}l)·wtl = 0 ·
In fa:::t, by the constraint ctl(1} 1l + ct CTJ 1) = 2
"'u
+ "'tz' we obtain
~pt(TJl)·ct2(1}1) - ~pt(TJl)·wt2 = O.
Hence the budget constraints for both consumers are met and thus the price sequence p(TJ ) and the consumption streams ci(TJ l constitute a Walrasian 1 1 equilibrium.
•
Ls as nanv
-
times differentiable as necessary for the transversality theorem (Preposition 17.0.3} to be applicable. •
We shall .take w (the initial endowment for good 101
1 at period 0 of consumer 1) as the exogenous parameter (and the other
vadables are fixed).
Define f: (0,1] x (0, + oo) -+ 1R by f(nl'w
101
l =
~pt(lJl)·wtl - ~pt(l}l)·ctl(TJl), then af(TJ1'"'101)/owl01 = plO(lJl) > 0 a:1d hence 1Jf(n ,w
1 101
J
=
0, that is. 1Jf(lJ ,w
1 101
) is of full rank.
transve:-sality theorem. for almost all w e (0, 101
{0, + oo), the equilibria are discrete.
cond!tion on f( ·I (such as f(O,w analcgcus to
Ex~rcise
101
+ oo)
,
if TJ
Hence, by the 1
e (0,1) and
Together with an appropriate boundary
l > 0 and [O,w
101
l < 0, which are
17.AA.Z(c)}, this implies that generically the
equilibria are finite.
20-28
20.G.4
Suppose that· a price sequence (po·Pr···•Pt'".) and the consumption
= Cc 0 i,cli'"'''ctr···)
streams ci
(i
(overall) Walrasian ecui!ibrium . •
= l, ... ,Il
constitute an inter-temporal
To prove footnote 32, we shall first prove
by contradiction that (ctl, ... ,ctl) = (ct'l'""''ct'I) for every t and. t', that is, the consumptior. streams are constant over time.
So suppose that there
•
consider the consump!ion streams that are the same as (cl' ... ,c ) excep: that 1
•
"t
a· ( -----.-c 0
t
+ 0
t '
•I
0
+
t.
t'
0 -
'•
Le..• yt e Y and Y... e Y be the production plans at period t and t'.
Since
L
ther~
is no:> possibility of intertemporal production, we can ta k e y .... L
v e . t'
tr.
T .n•"'""
.
........ !
r .c
. - [.w. + y and [.c . . = [.w. + y ..
1 tl
t
1 1
•
•
-
1t1
Hence
t
t' a --:'__:._t:-:,,.... ct. i ) ot + o
. t. 0
[.c •. 1 t 1
,.
a· - -.--.:-, C[.w • 1 1 o~ ... ·o·
11
t
e CL .~ an d
0
+ ytl +
t.
t'
a
Since Y is co:-n:ex, this proposed allocation is feasible.
Moreover
I
bv the -
strict Co,..,-a-·1-r ,. •
.;.,!...,
"'
. _.. ,
1"
u.• ( •
•
->
.
0~·
at ••
o·
+
a· +
...
o·
0
tl cti
+
u.(c .) + 1
tl
a ot
t' 0
+
0
t'
t'
t' at + 0 .
u.(c ,.) t 1
1
I
and the stric: inequality holds if c . tl
:;e
By multiplying at + at
ct' 1..
•
•
both sides, we obtain
20-29
to
t'
t'
a
a ---=t~-':""l,:- c a
i!:
a
t
u. (ct. ) + 1
1
t'
a
+
t
o
.. }
t 1
u.(ct,.) 1
1
and the st!"ict inequality holds if cti
~
ct'i'
Hence the proposed allocation
•
is Pareto s:Jperior to (cl' ... ,c l. 1
By Proposition 20. G.l, this contradicts •
the fact that the consumption streams Cc , ... ,c l constitutes an equilibrium.
1
1
Hence they must be constant. Given this constancy, it is routine to check that the given intertemporal equilib:-ium is an infinite, constant repetition of a one-period \\o'alrasian t
o p0
equilibrium a:1d that P. =
"
for every t.
Cor:versely, it is also easy to check that an infinite, constant repetitior: of any one-period Walrasian equilibrium constitute an intertemporal •
equilibriu~.
20.G.S
and that pt =
t
a Po for every t.
~
By the concavity,
0 and
•
2
1
I
'0'~ 2 u( • )
•
::: 0.
Thus
2
.- ('0'22u(·) + a'O'llu(·)) dk 1.'+ 1 . dk I = 2
•
a!'O'l2u(·)l
Hence dk 1 2 I t + I dk But he:-e, by the concavity, (IJ
2 12
-
2 2 2 c'0' u ( - 1 + av u c · > l 22 11
u( • ))
•
2
•
~ •
2 2 '0' u( • )'0' uC ·) and hence 11 22
2 2 2 (v22u(·} + aiJllu(·)) 2 2 2 2 = (IJ22u( . ) l + 2a'O' 22u(. )'0' 11 u(. ) 2::
i!:
2 2 2CSV' u( · )V' u( ·) 22 2
11 2 2~('0'12u(·)) .
•
By o
e (0·,1} and
dkt+l Hence I I > dk
"" 0,
20-30
1.
20.H.l
If we allow for the possibility for bubbles at an equilibrium • •
•
(cOi'cli'''''cti'.--) Ci = 1, ... ,I), then the budget constraint for each consumer i is that, for some M.
1
0,
2:
p w. + M..
+ 1
'-\:ttl
1
With the finite number- of consumers, we can sum both sides eve:- i to obtain
L·LP ct. 1 • t 1
= '-\: )' nt +
[.1 Pt"'t' + [.M .. 1'1: 1 1 1
•
On the other hand, the feasibility constraints remain the same: For- every t, ~ c . J...·1 1 t
= y t 1 a,-
+ yb
+ ~.w .. t L...1 1 t
Multiply P. to both sides, we obtain •~
Summing both sides eYer t = O,l, ... ,T, we obtain
T \'
T .).Pc.=\'
T n + ~ - >.p w . L.t=O"'"t t tL '-t=0 t
'-t=Q4-l' t tl ••
or
•
T
•
•
T
'Lt = Crrt + "·P w .) " _ >.p c . = '-1 t t1 '-t =0"'"1 t t1 0 Taking the lblit T
~ ex~,
we obtain
Now, by the possibility of truncation, (ybT'O) e Y and hence the myopic profit maxb1ization irr.plies that pTybT + pT+lyaT limT
oT y _
~-
+1 a1
0.
pT y T = 0. J-70: +1 a
1 1
::;;
0.
1
i!::
0.
Hence
Thus, in fact, we must
Hence M. = 0 for every i.
Hence no bubbles
•
can arise at equilibrium.
20.H.2
pTybT' or pT+lyaT
On the other hand, - [ .M .
have [.M. = lirr"T 1 1
2:
,p ) is (a) Note first that he slope of the offer curve at price (o. ·o a
20-31
differentiati:lg both sides of the budget constraint + p z (pb,p ) = (1 - c)p
a a
a
b
•
\r,·ith respect to pb' we obtain
zb(pb,pa) + pbVlzb(pb,pa) + paVlza(pb,pa) = 1 - c. Since
v1z.o (p.o·a ,o ) :::
0 and zb(pb,p } a
:S
v1z a (p.o ,pa )
0, this implies that
Hence condition (20.H.3) is satisfied _and the slope_ of the offer
v1zb(pb,pa) =
negative (or the curve is vertical if
O).
If
CU!"'Ve
> 0.
is
v1zb(pb,pal >
0,
then CO;'!ditio:: (20.H.3) is satisfied if and only if the slope is greater than one.
Tnerefore, the offer curve must look as in the following figures (or- a •
mixture of the two, in which case the curve has both positive and negative slooes l: •
1
1 •
•
(y, 1 : y)
.....
'' ''
0
''
''
•
···········' ,(y, 1 - y)
'
1-
''
:
€
0
I
'
1-
'' €
' I
Figure 20.H.2(a)
In either case. if we plot iterates on the offer curve from any caO
*
1 -
•
m
the same way as in Figures 20. H.l and 20. H. 4, then we can see tha: they must unavoidably leave the picture. Hence we can conclude that the steady state is the only equi!ib:-iUm.
20-32
•
(b) Let {(cb ,c t
at
)}
lXI
be equilibrium consumption streams that constitute a
t= 0
•
By the same logic as in (a), we can show under condition
Pareto optimum. (20.H.3) that if c
at
> 1 -
r
~
for some t
0, then the sequence of those·
iterates must unavoidably leave the picture.
Thus, if c
for ali
'!51--r
at
t ==:
As we sa'.\o' in (a), condition (20.H.31 also implies that '17 za(pb,pal > 0. 1
0.
Hence, by z(l,l} = C-r. 1 - 7) and cat
1 - 7, pt+l ~ pt for ali t a:: 1.
'!5
u(cbt'cat) :s u(-r, 1 - 7) for all t ~ l.
Thus
As for t = 0, we have cbO = 1 and· cao
~ 1 - 1. implying that u(cbO'caO) ~ u(l, 1 - -r).
Since {(c. ,c ot
at
)}
00
t= 0
is
Pareto optimal, we must have u(cbO'ca 1 = u(l, 1 - 7) and u(cb ,c I = 0 t at u(r, l - -rl for all
( -r, 1 -
for all t
;r)
t
==: i.'!:
1.
(c.T,C
o. at
) =
The proof is thus completed.
1.
(c) We shall pr-ove that any such equilibrium consumption st.-earn as in Figure 20.H.4 (as well as the monetary steady state) is Pareto optimal.
So !et
•
{'
•
•
•.cbt'c a~•
bubble.
)\lXI
'·-o
be su.ch a stream and M
Ther:.
(c~
~
0 be the associated possibility for a
L-
for e\'"!!!!"Y
,c• I -+ (1, 1 - 7), and, as seen on page 774, D c• ot at • · · • t a, t- 1
t <=: !.
Hence M/pt = c• -+ 1 - 7 > 0. a,t- 1
Therefore, a::cor-ding to
the itaEcized ciair:: on page 774, the stream {(cb•t,c• T )} a~
•
=- M
00 _
t -0
I ; is P are:o co ~rna .. t •
(d) A::co!'"ding to Example 17.F.2, if - v"(clc/v'(c) < 1 for all c a:: 0. then
u z.o·o (o. ,p I < 0 and u z (cb,c ) > 0. a a 1 1 a
Hence condition (20.H.31 is satisfied.
The fcilowing is a more general condition: If z (pb,p )v"(z (pb;p )) -
a
a
for aE (p ,pa}, then (20.H.3) holds. 0
a
a
Note that, since there is no initial
end-:::nvment in the second period of life, the first term of the left-hand side is nothing (pb,pa).
bu~
the same measure - v''(c)c/v'(c), evaluated at the demand under
This condition thus allows the measure to be greater tha.YJ. one, but
20-33
•
requires the difference to be bounded by the relative price pb/pa. Let's now prove that the above condition is in fact sufficient.
By the
first-order condition,
Differentiate both sides with respect to pb' then we obtain pav"(zb{pb,pa) + lHc3zb(pb,pa}/c3pb)
= ov'(zb(pb.pa)) + pbov"(zb(pb,pa))(c3za(pb,pa.)/c3pbl . Differentiate Walras' law pbzb(pb,pa) + paza(pb,pa) = 0, then we obtain zb(pb,pal • pb(c3zb(pb,pa)/c3pb) + pa(c9za(pb,pa)/c3pb) = 0 · s~lve
those two equation with respect to ozb(pb,pa)/c9pb and aza (pb,pa}/c3p , 0
then we obtain oz. (p. ,D )/opb = (- ov'(z (pb,p ))p + ov"(z (pb,p ))pbz. (pb,p ))/ll, o o·a a a a a a o a
•
. . ~w·nere
> 0. Hence (20.H.3l is •
sa~isfied
jf and oi}ly if
- ov'(z (p. ,0 )}p + ov"(z (pb,p ))pbzb(pb,p } a a a a o·a a
•
By Walras' law, this is equivalent to
- ov'(za(pb,pa))pa - ov"(za(pb,pa))paza(pb,pa)
< ov' (z a (p.o,p a ))pb + v"(zb(pb,p a ) + llpa zb(pb,p a ).
- 1 D
. a Here, the second term of the right-hand side is positive by zb(pb,pa) < 0. Hence this is implied by
20-34
•
•
- 1 -
z (pb,p )v''(z (pb,p )) a a a a v (z (pb' p ) ) a a
•
But this is the sufficient condition we proposed before.
The pr-oof is thus
completed. i'\o~e
that, by the first-order condition and Walras' law,
pb/pa = v'(zb(pb,pa) + 1)/ov'(za(pb,pa)) = - za (pb,pa)/zb(pb,pa}. Hence the righ-hand side of the above inequality can be replaced by these valu~s.
which a.:-e given in terms of excess demands and marginal utilities . •
Let c > 0 and suppose that the above inequality is satisfied for all
are some equilibria v;ith non-steady states, then these equilibria must involve the recurrence of ge:1erations facing (pb,pa) with pb/pa
~
c.
This is because
expression (20.H.3l i:olds at all (pb,p ) with pb/p > c and he:1::e there is no . a a equilibrium with r..on-steady states and p /p t
t+ 1
> c for all t.
a!so that if the inequality is satisfied for a!l (pb,pal' then the ~he m~asure
S;Jore::mm of .
- v"(c)c/v'(c) must be at most one. ' •
•
Hence, h·r ,
exarr.ple, the measure is nondecreasing with c, then - v"(c)c/v'(c}
:5
Iro~ L
1 for all
c a:1ci hence ti:e cffer curves looks like that in Figure ZO.H. 2. Further elaborations on conditions for (20.H.3)
:;, be fcund in
Jean-Michel Grandr:1ont' s ''On Endogenous Competitive Business Cycle" if! Eco!lome~rica.,
20.!.1
Sin~e
Vo!. 53, No. 5 (September, 1985), pp. 995 - 1045.
z( ·, ·} is homogeneous of degree zero, Theorem M.B.l implies that
'i7za( ·, ·) and 'i7z ( ·, ·} are homogeneous of degree - 1. 0 p'i7 za ( l,p) aP.d the f!rst equality follows.
2
1J 1z (l,p} + ~
a
v z
2 a
(l,p)p =
0.
Hence 'i7 za(l/p,11 = 2
The homogeneity also implies that
The second equality follows from this. •
•
20-35
TER 21
2l.B.l
Let 1t:{l,2, ... ,1}
(i)
I .1= la.1 =
(onto), then
> {1,2, ... ,1} be any permutation implies Sign
I .1= la.1
=Sign
I '=l«lt( i)
which in turn implies that F(al'a , ••• ,ai) = F(atr(l)'alt(Z)"' .. ,cxn:(l}) . 2 •
(ii)
Neutralitv: I
-Sign
. 1(-a.)
=
(iii)
l
I
F(cxl'cx , ... ,a l = Sign 1 2
= Sign - . 1(-a.) t=
1
-
= -F( -cx , -a , ... ,-a l . 1 1 2
Positive Responsiveness: I
Sign
l:!::
. l «.
1=
1
Assume that FCcx .... ,a l l:!:: 0, then 1 1 I ,.. 0, which implies that i=I«C o. Take C«J. , ... ,«1} 2: (cxr•····«r)
such
Then,
Sign
> 0, which in turn implies that F(«i, ... ,«il = 1.
2l.B.2
Let 1=2.
{i)
1=
1
A Dictatorship functional F(a ,a ) = cx is both neutral 1 2 1
between alternatives and positively responsive, but is not svmmetric. •
•
•
•
symmetric and positive responsive, but is not neutral between alte::-natives.
symmetric and neutral between alternatives, but is not
21.8.3
that
"'i•
Assume majority voting implements k = 1.
•
resoons1ve.
Then, there exists i• such
> 0 (if not, either all v = 0 and k = 1 is Pareto optimal and this 1
shows the resu!t, or v. ::s 0 for all i and v. < 0 for some i which implies that 1
1
k = 0 sh::::uld have been the case). Now, if the decision is reversed to k = 0, then agent i• is made worse off, which implies that k = l was indeed Pareto •
·optimal.
A symmetric argument hods for majority voting implementing k = 0.
21 - 1
•
Now consider transfers between agents.
An example can easily be
constructed to show that majority voting is not Pareto optimal. v = v = -1 and v = 5. 1 2 3
Let I
= 3•
Majority voting will implement k = 0, but if the
agents were allowed to choose k = I and agent 3 would pay 1 + e to each of the other two agents, then all agents would be strictly better off. This contrast is due to the nature of quasilinear preferences.
I.Fl such. a
•
•
case we can perform interpersonal comparisons of utility (use the "mean"). which is a cardinal property, while majority voting is only considered with ordinality (which is the "median").
Claim: If for some {x,y} c X, S c I is decisive for x over y, then
2I.C.l
•
for ant third alternative z, S is decisive for z over y. Proof: Consider a preference profile
(~ , ... ,?:
1
1 satisfying z >-. x >-.y for all
1
1
1
•
i e S, and y >-.Z >-.x for all i e 1"\S. Since S is decisive for x over v we must 1 1 ~
have
x
Fpl~
..... ~
1
y.
Since z >-ix for every i, by the Pareto property we
z F (>- , ... ,>- } x, therefore, by transitivity we must have
must have
p- 1
z F (>- , ... ,>- } y. p- 1
1
l
-1
Moreover, by pairwise independence this relation should
-1
•
•
•
hold regardless of individual ranking of x, which implies that whenever z >-.y 1
for all i e S, and v >-.Z ..
1
for all i e 1'\S, we should have
z F (>-,, ... , >-r) v, p -"
-
..
•
i.e., S is decisive for z over y.
2l.C.2
(b)
(a)
Majority voting with two alternatives.
Majority voting when the domain is single peaked (see section 21.0 for
more on this}.
(c)
Majority voting with more than two alternatives: The Condorcet paradox
illustrates how the resulting social welfare functional can have cycles when the domain is unrestricted (see example 2l.C.Z in the textbook).
21 - 2
•
(d)
The Borda count does not satisfy pairwise independence (see the
continuation of example 2l.C.l in the textbook}.
(e)
A constant social welfare functional (i.e., which is independent of the
preference profile).
(f)
•
A dictatorial social welfare functional, i.e., choose agent 1 as the •
dictator, and x F PC~ •...• ~
1
1
)
y if and only if x >- y . 1 •
Let X be the finite set of alternatives, and let there be I
2LC.3
individuals.
Consider the following social welfare functional :
x F (>- •... ,>- ) y if one of the following hold: p- 1 -1 (1)
X
>- y 1
(2) (3) •
•
and x >X -
1
y,
x -
2
2
y
y and
• •
(J)
x -i y for ail i = 1, ... ,1-1 and x >- y 1
This is a LextcaL dictatorship social welfare· functional in the sense that •
'
•
agent 1 gets to dete:-mine the stl-ict preference, but if he is indifferent then agent 2 decides, and so on, and if all are indifferent then the two alternati\·es are socially indifferent.
•
It is easy to see that for all
profiles of preferences where there is some indifference for all agents, and the preferences of the agents do not coincide then the social choice functional will not be identical to any of the individual preferences.
2l.D.l
Since X is finite, let N be the total number of elements in X, and we
preference relation >- as follows:
-
21 - 3
(ii)
For any X' c X such that x 6! X' or 1
X' then for all {x.,x .} c X', 1
J
x. >- x . if and only if j > i. 1 J (iii) For any X'
c X such that x 1e X' and xNe X' then
xN >- xi' and for all
{x.,x.} c X' such that N E {i,j}, x. >- x. if and only if j > i. 1 J 1 J Clearly, this preference relation is complete and reflexive. Furthermore, for
•
any strict subset of X it has a maximal element: "For subsets not including both x
1
and xN then xi with the smallest i is the maximal element, while for
subsets including both x
1
and xN then XN is the maximal element.
preference is clearly complete, and it is reflexive. maximal element on every strict subset of X.
•
This
Furthermore, it has a
Note, however, that it is not •
acyclic.
2l.D.2
•
The preferences generated from oligarchy are clearly quasitransitive:
If x F C>- , ... ,>- ) y andy F (>- , ... ,>- ) z, this means that for all he S (the p- 1 -1 p- 1 -1 members of the oligarchy} we have x >-h y· and y >-h z. each individual are taken from
-
~.
Since the preferences of
transitivity implies that x >-h z for all
•
.
h e S, which in turn implies that x F (>- , ... ,>- ) z. p-- 1 -1
To see that social
indifference may not be transitive, let I = {1,2, ... ,1} be the set of individuals and let S = {1,2} be the ·oligarchy. current preferences are
x >- z >- y 1 1
Further assume that the
y >-z x >-
and
2
z.
From the definition
of the oligarchic social preferences we have x F(?:; .... •?:I) y and 1 y F(>-, ... ,>- J x. so that x and y are socially indifferent, and we also have -l
-
1
•
indifferent.
However, since both x >- z 1
and
x >-
2
z, we have that
x F (>- , ... ,>- ) z, which shows that social indifference is not transitive. p- 1 -1 This can happen as long as there are more than one member in the oligarchy. They can all agree that one alternative is strictly better than some other,
21 - 4
•
but when comparing each of these two alternatives to all other alternatives • there can be disagreement.
21.0.3
Clearly, the generated social preferences are acyclic: for all
possibilities of individual preferences, agent l's most preferred alternative will be at least as good (but not necessarily better) than any other alternative.
To see that quasitransitivity may not hold, let the preferences
be x >- z >- y 1 1
and
From the definition of social preferences
we have that x F (>- , ... ,>- ) z and z F (>- , .•. ,>- ) y, but we do not have p- 1 p- 1 -1 1 #lw
x F (>- , ... ,>- ) y because individual 2 can veto this. p- 1 -1
Note also, that in
spite of the veto power of 2, x is the only maximal element in this case: we have that x F (>- , ... ,>- ) z and x F(>- , ... ,>- ) y. p- 1 -1 -1 -1
However, ·y is not a maximal
element since z F (>- , .... >- ) y, which happens because individual 2 cannot p- 1 -1 veto z preferred to y.
2l.D.4
Assume >- is single peaked with x• e [0,1) as the peak alternative.
-
Assume that y and z are two distinct alternatives, neither of them equals x•, such that y - z.
We need to .show thqt for all
~
e (O,l}, ;\y
+
(l-i\.)z >- y.
First, we must have either y < x• < z or z < x• < y (or else we could not have •
y - zl.
W.l.o.g. assume that y < x < z, and choose some A e (0,1).
Letting
x' = ;\y + (1-;\)z, we either have that x' e (y,x•], in which case x' >- y, or we have that x' e(x•,z). in which case x' >- z, which implies that x' >- y.
21.0.5
The six possible orderings are: (1) (x,y, z); (2) (x, z,y); (3 l
(4) (y,z,x); (5) (z,x,y); and (6) (z,y,x).
(y .x, z );
Recall that the preferences of the
Condorcet paradox are x >- y >- z , z >- x >- y, and y >- z >- x. 1 3 1 2 3 2
Agent l's
preferences are not single peaked for ordeJ:"ings (2) and (4), agent 2's preferences are not single peaked for orderings (1) and (6), and agent 3's
21 - 5
preferences are not single peaked for orderings (3) and (5) .
•
Let X- {x,y,z}, I - (1,2,3,4), and consider the following profile of
21.0. 6
~
preferences: x have #{1 : x
~
1
1
.~
y
1
z , z r 2 y r x, x 2
y) - #(i : y
~i X) -
~
3
~
z
#{i : z r
1
3
~
4
y, andy
y) - -{i : y
x ~
1
~
4
z.
We thus
z} - 2, which
implies that x is socially indifferent to y, andy is socially indifferent to F(~ ,
1
z, so we can write z however, that z
F(~ ,
1
which implies X F
...
,~ )
6
y andy
F(~ ,
1
...• ~ 6 ) x since #(i ; x
...• ~6) z.
(~1'
p -
-
~i
•
...• ~ 6 ) x.
It is not true,
z)- 3 and #(i : z
~i
x)- 1
Therefore, in this case majority voting
fails to generate a fully transitive social welfare functional.
2l.D.7
(a)
Since the cone spanned by (Vu (0),Vu (0),Vu (0)} is the entire 1 2 3
2
space R , we must have no linear dependence between the Vu.(O)'s, and the ~
directions of the gradients must be as drawn in Figure 2l.D. 7(al) {they cannot all point in directions that can be •bounded" by some straight line going through the origin). The dashed lines p , p , and p are the 2 3 1 •
perpendicular lines to the direction of each gradient respectively, and the indifference curve of agent i' that passes through the origin is tangent to p . 1 These indifference curves are shown in Figure 2l.D.7(a2).
~
•
/
.
I
• I •
/ /
p2/
~
•
I 'Vu_; (0)
/
p3
I
I •
\ .pl
Fi
\
/
·\
e 21.0. 7(al)
\ \
U,(O)
\pl Figure 21.0. 7(a2)
•
21 - 6
•
I
•
I
•
\
/
I
•
•
(We show convex preferences though they can be linear, as used for part (b) of •
this exercise.)
Given the directions of the gradients, we can find three
alternatives x, y, and z as shown in the figure for which x >- y >- z , 1 1 y >-
(b)
2
z >-
x, and z >- x >- y, and a Condorcet cycle exists. 3 3 2
(We restrict attention to linear preferences which is not too restrictive
since close enough to the origin, any indifference curve going through the •
origin will approach a linear line.)
•
2
Claim: Given a point x e IR , either one or two gradients 'O'u.(O) pass through 1
the circle centered at x, with radius Uxll. proof: Assume in negation that none, or all •
three gradients would pass through the circle. Then they would all be one the same side of the dashed line in Figure 21.0. 7(bl), and they would not span the entire IR
2
by their cones, a contradiction
"
•
to our assumpt1on. c
2l.D.7(bl)
Thus, we divide the proof into two cases: g~:,adients ~
Case 1:
•
•
•
Assume w.l.o.g. that Vu (0) 1
pass throl.!gh the circle as shown in Figure 21.0. 7(b2).
V'u1(0)
•
•
"
-2l.D.7(b2) 21 - 7
•
Then the angle between both gradients must be some a: < 180°.
Now bisect a: and
let y be on this bisecting line so that Ux-yh < ISxR (i.e., y is within the circle). which implies that u (y) > u Co) and u Cyl > u Co). This concludes case 1. 1 1 2 2 Case 2: One gt adient'
throuvh.
Assume w.l.o.g. that Vu (0) passes 1
through the circle as shown in Figure 21.0. 7(b3).
•
•
• •
/
/
./
/
/
'V~{O) •
2l.D.7(b3) •
•
The dashed line T is tangent to the circle, so that Vu (0) and Vu (o) must 3 2 pass be!ow the line T. By the spanning condition, we must have that either the region A, and the rema!ning gradient will be strictlv inside the region B. is in the region A, let p
1
and
Assuming w.l.o.g that vu (0) 2
Pz be the perpendicular lines to Vu (0) and 1
Vu (0) respectively, and there must be (by the spanning condition) an "area" 2 between these lines so that we can choose a point y in this area such that Vu (0)·y > 0 1
and Vu (0)·y > 0, which implies that u (y) > u CO) and 1 2 1
u Cy) > u Col. This concludes case 1. 2 2 Remark: The idea in both cases is finding two gradients with an angle less than 180
0
between •them .
21 - 8
•
21.0.8
(a)
The situation is depicted in Figure 2l.D.8(a) below:
V'uz(O) / /· /
•
/
Since the gradients form a pointed cone, assume w.l.o.g. that agent l's •
gradient is in the "middle" as shown in the figure.
Letting P. denote the 1
perpendicular line to the gradient of agent i at the origin, we show that no alternative to the left of P
1
can be preferred by a majority to the origin.
Any such alternative must be either in region A, or in 8, or in C.
The origin
is preferred to any alternative in region A by both agents 1 and 3 . •
Similarly, the or-igin is preferred to .any alternative in region C by both agents 1 and 2, and finally, the origin is preferred to any alternative in •
region A by all agents.
Therefore, any alternative preferred to the origin
will be prefer-red by agent 1 who is a directional median. (Note that alternatives in the "slice" opposite from A will be preferred to the origin by agents 1 and 3, and similarly for the other slices.)
(b)
In Figure 2LD.8(b) we show the desired situation.
Agent l has linear
indifference curves with the (fixed) gradient Vu ( ·) (the line with long 1 dashes), agent 2 has linear indifference curves with the (fixed) gradient 'i7u2(.) (the line with short dashes), and agent 3 has convex indifference
•
21 - 9
curves with the
(variable) gradient Qu ( ·) (the line. with long dashes). 3
We
have three alternatives, A.B. and C. such that A ~ B ~ C, 8 >- C ~ A, and 1 1 2 2 C ~
3
A ~J B.
'Vu3 (B) •
~ '~'
\
\
'
',-, '
\ ~\
,
- .. ,~
.,....~
,' ... .
.... % \
•
u 2 (A)
. 3 I
.. \ .. .
,
I
,...........-::''
~
'
•
\\ \\ •
A
'
\ •
•
21.0.8 •
•
•
At each point we have a different median agent, and we clearly have a Condorcet cycle. •
(c)
The result follows immediately from (a) above and from the transitivity •
of individual preferences; for all alternatives.
Assume w.l.o.g. that agent 1 is a median agent
lf x F
p
y and y F
p
x then we must have from part (a) •
that x >- y and y >- z. Assume in negation that z F x. 1 p 1 implies that z must have x F
21.0.9
(a)
~l
p
Then (a) above
x, but this contradicts the transitivity of
~ •
1
ther-efore we
z.
Assume that u(x) ~ u(y) and u' (x)
21 - 10
2:.
u' (y).
Then, since 1 and 1/J
•
are positive we have that f/lu(xl + 7u' (x)
i:
f/lu(y) + 7u' (y), i.e., the •
preference relation represented by f/lu( ·) + 7U' ( ·) is intermediate between the preferences represented by u( ·) and by u' ( • ).
For every pair (x,y) define the half space,
(b)
A(x,y)!! {{3
E
IRN: {3·(a - a ) > 0} . X y
where ax= Ch (x)., ... ,hN(x)) and ay= (h (y),. •. ·.,hN(y)). 1 1
We can thus write
•
B(x,y) = IRN
++
n A(x,y).
Claim 1: The set B(x,y) •
(c)
{{3 E
N R : u (x) 13
i:
Proof: Assume that f3 and (3' are both in the set and u , (x)
u (y)} is convex.
13
-B(x,y),
that is. u (x)
13
u , (y). Because the preferences represented by /3'' a...-e 13 13 intermediate between those represented by {3 and {3', we have u , (x) 13
so /3" .
Cla1m
E
£
il:
u (y) 13
il:
il:
-B(x,y ).
The set Ux,y) !! {{3
E
N
u , (y), 13
•
1s convex.
IR :
Proof: Symmetric to the proof of claim 1. Assuming that u ( ·) is continuous in /3, then the conclusion of part (b) above
13
is still correct as follows: from Claims 1 and 2 above, it must be that
-B(x,y}
n
•
-L(x v' 0•
I
•
1s convex.
However,
-B(x,y)
"
-L(x,y)
can be convex only if
-B(x,y) and L(x,y} are translated half spaces. Of course, if /3 is restricted •
•
N to IR then B(x,yl must also be restricted to RN . (See Fleure 2l.D.9(c)) ++ ++
)
-L{x.yl (
) Blx.vl n l.!x,yl
•0
Fig•ne 2l.D.9(c) 21 - 11
-B(x,yl
0
....... of ~
(d)
agent: Take N=2 and consider a density g( ·) that is
2 unifonn over a circle K c R with f3 • being the center of the circle (see ++ 0
Figure 21.0. 9(dl)).
It is straightforward to see that
• ~ is
a median agent.
0
0
0
21.D.9(dl) Non-existence of
Take N•2 and consider a density g( · ) that is
_g_
2 where the length of each of the uniform over an equi-triangle A c R ++
triangle's edges is a. as depicted in Figure 2l.D.9(d2)
a
2
' ' '
!
a
2 ,
0
•
The only candidate for a median
'
,
,
-. , , J3. , . •
'
'
a
2
-•
•
2l.D.9(d2) • agent would be {J shown
However, if we draw a horizontal line through
• t3
in Figure 21. D. 9( d2}.
as shown in figure
2l.D.9(d3) then it splits the triangle into two unequal parts, which implies that
• (3
cannot be a median agent.
0 0
...... :.-.-fJ
21.D~9(d3) 21 - 12
(e)
Consider the situation depicted in Figure 21.0. 9(e) below:
\
(x" ,y) •
•
From pa•·t (b), for any alternative y. B(x•.y) is a half-space.
Furthermore.
because x• is the single most preferred alternative of the median agent we must have that
• (3 e
• (3
B(x.y). Since
is a median agent, and
mass of agents in the region B(x• .y) is greater than (f)
Let X = IR
N
• g(/3 )
be the set of alternatives.
1
• 2
>
o.
then the
hence. x• defeats y.
For any x e X consider the N + l
functions given by. N
a~
L.
n=l
2 X n ' •
•
•
For (3
•
• • •
d~fine, •
This function is as in part (b) above, except that f3 is not restricted to be non-negative, and that given x. there is a constant term h (x). 0
change any of the conclusions of parts (b) and (c) above. explicitly, ·
•
21 - 13
This does not
Computing u (x)
13
•
N
+ ~2 n n=l
r
= -II
2
p
X -
U
N
L
+
~
n=l
N
r
But notice • that = -II x - ~ u2· +
n=l -II x -
~
2 n
-
~2 is an increasing transformation of n •
•
2
11 , that is, both functions represent the same preferences over X.
Therefore, this is exactly the same model as in Example 21.0.6 . •
21.0.10
[First Printing Errata: The sentence starting in the fourth line of
part (c) should read "Show that preferences over X are ... "]
As w• and
(a)
-w
become further apart, this means that the median is
•
moving away from the mean, or in other words we are moving away from equality.
(b)
Each agent's
Clearlv, '
if w.> w 1
..
p k" , t .• p_a
1
Wl'11
tw.
b e chosen to maximize (1-t}w. + 1
then t.= 0, and if I
-w
If w• >
> w. then t.= 1. 1
1
-w .then
'
than half the agents prefer f = 0 to ·any other alternative, and if
more
w• < w
then
the opposite is true.
(c)
Now, to find t. we 1
-w
-2w.+ 2tw.+ 1
w.< 1
-w.
1
2tw
-
. .
. max1m1ze
such that w.< 1
(1-t} w. + l
t(l-t}w.
The FOC is:
= 0, and the SOC is satisfied if 2w.1
Clearlv for w.> , 1
-\'v',
2
-w
we will have t.= 0. 1
2w
< 0, i.e., if
To solve for t. for all agents 1
1-
w • - -w the
roc
is necessary and sufficient and yields t.= I
2
1
w•1
which
1 1
1
-w
1w.e [0,- wl, and due to the non-negativity of t. 1 1 2 1 that t e [0, 1. Furthermore,
c
2
The difference with part (b) is that here the dead weight loss causes agents to be more averse to
21 - 14
•
taxes. As a tax is levied, it both decreases one's own initial wealth, and it decreases the median initial wealth. 2
2
To find t. we maximize (l-t)(I-t )w. + tO-t )w, which yields the FOC:
(d)
1
w>t
3(w.1
2
1
-2tw. - (w.1
1
w)
Clearly, this FOC can have multiple solutions
= 0.
•
so that more than "peak .. can exist for each individual. •
21.0.11
Consider the following example: Let I = 5, and consider a mechanism
of pairwise majority voting with the modification that agents 1 and 2 have a •
weight of two votes each, while the other agents (3,4, and 5) count once (as in regular votir.g).
That is, x F (>- , ... ,>- ) y if and only if p-1 -5
~.a..t3.
L..lll
> 0
where, 1 if i e {3,4,5}
+1 if X >-.y 1
«.= 1
• and
-1 if y >-.X
(3.= 1
1
•
•
2 if i e {1 ,2}
This is a social welfare functional because it is equivalent to a majority voting mechanism in a society with 7 agents, which consists of our 5 agents with one more agent of types 1 and 2, and by Proposition 21.0.2 this indeed gives a social ordering.
Furthermore, the social welfare functional is ••
•
•
clearly pairwise independent, and Paretian, but it is not eauivalent to • pairwise majori:y voting on our society of the 5 agents.
21.0.12 ti
+
(a)
Consider the alternative Cti,t2:tjl
~
•
0,
t2 + tj = c. The following alternative is preferred by agents 1 and 2:
(t!,t2,t)l = (O,t.2_-£,tj+ti+c).
Similarly we can find other alternatives
preferred by 1 and 3, or by 2 and 3.
(b)
With (ti,t2,tjl
~
0 such that tl = 0 for one i only, we have the same
problem as in (a) above: reduce the tax paid by each of the two agents by c, and increase tr.~ tax of agent i by 2£; this is preferred by both agents with
21 - 15
positive initial payments.
t~
If, however, two agents have
1
= 0, then this is
a "weak" Condorcet winner. this alternative will be preferred by the two agents who pay nothing to any alternative in which they pay something.
For
•
alternatives in which one of the two still pay nothing. and the third agent pays less while one of the two pays out the difference, we have social indifference - the agent who still pays nothing will be indifferent and the agent who pays less will prefer the new alternative.
•
If we ignore
•
indifference in the sense that a status quo is changed only if a majority strictly prefers the new alternative, then any
(t~,ti,tj)
with two agents
•
paying 0 will be a Condorcet winner.
21.0.13
(a)
Suppose in negation that y ~ {xr ... ,xJ}.
Since J is finite, we
can trace a line through the point y which does not contain any of the points
x3
Xs
A
x,
x3
•
•
•
y
8
x, -
A -
Xs y~,
,-
y
x4
8
x4
-
•
2l.D.l3
e21.D.l3al
Fi
Denote by A and 8 the two (strict) half planes that lie on both sides of the line traced through y.
Because J is odd, we cannot have the same number of
most preferred points, and say (w.l.o.g.) that A contains more.
Since we
have ex = ex = · · · = ex , then the mass of agents in A is larger than that in 1 2 J B.
Therefore, any alternative y' that is a small displacement of y into the
region A, will defeat y (see Figure 2l.D.13(a2)}, hence, y could not have •
21 - 16
-
•
been a Condorcet winner.
•
•
Note that if J is even this does not hold. demonstrates this for I
=4
and ex.= 1/4 for i 1
Figure 2l.D.l3(a3)
= 1.2,3,4.
x, •
y
•
•
•
Fi (b)
Consider any point y tE xh and trace the straight line that connects y •
with xh.
By the "general position" assumption. there can be at most one other Denote by A and B the two half spaces created
preferred point on this line. by
this line and by C the closed half-segment of the line created by the point y, and that does not contain xh. (See Figure 21.D.l3(bl))
..
•
A
•
y
8
• • •
c
21.0.13
We must have that the mass cf C is less than ~ (because h is dominant and C contains at most one other type). the reverse.
Also. we either have mass(A)
!:
mass(B), or
The following argument is similar to the one in part (a) above.
21 - 17
Assume w.l.o.g. that mass(A)
mass(B) and consider a point y' that is in
il:
•
region A, close to the point y, in such a way that (y' - y) creates an acute •
(but almost straight) angle with (xh - y) as depicted in Figure 2l.D.l3(b2). •
-- ---
A y ____ _ y
-
-
8
• • •
c Then, mass(A) +
'1; > mass(B)
mass(C), which implies that y' defeats y.
+
Therefore y could not have been a Condorcet winner .
.
21.0.14
(a)
Due to indifference we can view majority voting in one of two
ways: _if agent i is indifferent between alternatives x and y, he will either vote for both "x preferred to y" and for "y preferred to x", or he will abstain from voting at all.
-
'
.
Considering the first voting procedure, and given
the preferences of the five agents, x is defeated by w (2 vs. 3), y is •
defeated by w (3 vs. 4), z is defeated by x (3 vs. 4), v is defeated by z (3 vs. 4), and w is defeated by v (3 vs. 4).
(b)
The linear order
x
will satisfies this problem.
It is
easy to verify this by drawing the preferences of each agent given this order.
(c)
The following utility functions shown in Figure 2l.D.I4(c) for each agent
will give rise to the preferences over the alternatives given in the exercise (the utility functions are actually weakly concave due to the "weak" single peakedness}: •
•
21 - 18
•
----'-· • I
I
_1_
L
I
I
- -
•
I
I
-_, - .... ------- 1..
I
I
I
I
L.
I
I
-
I
I
I
I
-
I
-1 -
I
I
I
I
I
I
J
I
I
I
I
I
I
---- ----
I
-1-
-
-1-
I
- - - -' - - - -
L
I I
I.
-1-
•
I
I
I
I
-I I
1..
....
I
I
I
I
-
•
L I • I
•
0
y z v
X
w 1 X
0
X
y z v w 1
X
•
21.0.1 (d)
We construct an example which is an extension of the preferences given to
a continuum of alternatives, where the "gaps" are filled with indifference. In particular let X = {0,1], and let the agents' utility functions be giv~n by the graphs in Figure 2l.D.l4(d). For example, the utility function for 3 is: •
•
•
3
for 0 :s.' x < 0.2
4
for 0.2 :s. x < 0.4 and for 0.4 < x :s 1
5
for x = 0. 4
Similarly for the other agents.
Note that these utility functions are
•
quas1concave. •
•
•
• •
21 - 19
-
-
,_
-
-I •
-
_, ..
-
., -
-
I
I
I
I
I
I
I
I
•
•
I
I
I
I
I
I
I
I
1
I
-r
I
I
t
I
t
f
.. - - - - - - - - -
I
t
I
I
I
f
f
I
t
t
t
I
I
I
t
I
t
I
I
I
I
I
I
I
I
...
-
-
I
I
I
I
I
I
I
I
I
0 .2 .4 .6 .8
-- r I
I
I
I
I
I
I
t
1
I
I
I
I
I
--
I
I
I
I
I
I
I
•
I
' ..
~-
• I
_,_-
-·-- - ~
I
I
I
I
I
I
I
I
0 .2 .4 .6 .8
X
-- r-
P
I
I
I
I
'
I
•
I
I
I
'
I
- •- - - •- -
-
1
.. -~- - · - - ~--
I
I
I
L
I
-t-- -e- I '
I
-
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
'
I
I
--
'1
I
--
I_
-
_I
I
I
I
I
I
I
--
•
•
I
•
I
I
•
-
-
I
I
I
0 .2 .4 .6 .8
I
- .. - -·- - -· - •
I
'
•
I
•
-
I
1
I
I
•
•
-
-
I
I
I
I
I
I
I
I
t
I
I
•
I
I
I
I
I
I
I
I
I
I
I
'
I
I
I
'
•
L
' -
-
t_
I
I
I
I
I
I
I
-·-I
I
I
•
- - - - ------- - - -• • I
I
I
'r
• I
I
'
I
- - - - - - - -' - - - - - -
--
I
.J
-
-1
--
.J -
I
I
-
....
I
I
I
0
I
I
I
I
•
•
0 .2 .4 .6 .8
1
2l.D.l4(d)
21 - 20
X
~
I
I
0 .2 .4 .6 .8
X
1
-
I
I
~
....
I
I
I
I
..1
I
- - - - -----• - - - -·•
I
I
I
------
- - - - - - - - - -• - - I
X
r
•
--,.- -.- I
~--
1
I
-- -- --- ---I
--
I
I
J,
t
I
I
I
-1
- - - - ---------
---- - - - - ' - - -' - - - - - '' _,_ -- -· - -
--
-.
--·--·--
r
'
1
X
Using majority voting as described in part (a) above, we can see that. 0.2 beats any alternative in the open interval (0,0.2).
This follows because all
agents but agent 2 and 3 are indifferent between any two alternatives in (o,0.2], and agents 2 and 3 strictly prefer 0.2 to any alternative in (0,0.2). Similarly, 0.4 beats any alternative in the open interval (0.2,0.4), 0.6 beats any alternative in the open interval (0.4,0.6), and 0.8 beats any alternative
in the interval (0.6,11.
Now observe that if we label these alternatives as x =
{0,0.2,0.4,0.6,0.8}. 0, y
c
0.2, z
So, we are left with the alternatives
= 0.4,
v
-=
0.6, and w
= 0.8
then the agents' preferences are
identical to that in the previous parts of the exercise and we know that there exists no Condorcet winner in this case.
•
2l.E.l
First, we show that a unique Condorcet winner exists.
From
Proposition 21.0.1 we know that there exists a Condorcet winner associated with the media.> voter (there may generally be more tan one}.
In our case
preferences are strict and there is an odd number of individuals, so we cannot have two Condorcet winners. (Because when voting between any two alternatives •
x and
X
1
we would have a strict maj_ority for one or the other).
Second, we show that
f(~
•
..... t l that assigns the Condorcet wmner to 1 1 •
preferenc~
every
profile satisfies weak Pareto.
Suppose in negation that
.t"') l I I) d • t h ere ex1sts a pro11.e ~ , ... ,t an x = f(~i , ... 1 1
y e X such (~l
,... ,ti I,
tha~
y
>-i
y d~feats
Condorcet winne:-.
x for all i = 1, ... 1.
,ti ),
and there exists
But then with preferences
x in majority voting, contradicting that x is the
Therefore, f( • , ..• , ·) must satisfy weak Pareto.
Fir..ally, we show that !Ct , ... ,~ 1 is monotonic. 1 1
Let x = !C~}·····?::'r ).
Since I is odd and preferences are strict we must have that for all y e X, Y
~ X,
#{i : X >-~1 ~v}
> I/2.
Take some (?;~ ·····?:~) e A such that x maintains
1't s pos1"t"10n f rom (?::'I! , ... ,?: I ) t o ( ?:II , ... ,?:II) · 1 1 1
21 - 21
Then, for all y e X, y
~
x,
#{i : x >-'i y}
2l.E.2
> #{i : x
>-i
y} >1/2, which implies that x =
!Cti·····t'i).
First, the first agent is a dictator by definition 2l.E.5, so that •
the defined social choice function (SCF) is dictatorial.
Second, the SCF is
weakly Paretian because that dictator is always getting one of his most preferred alternatives, and we can therefore not make all agents strictly better off.
Finally, the SCF is monotonic since if. a chosen x maintains its
position, it will maintain its position for the dictator. Therefore, for the dictator, x's lower contour set (weakly) increases, implying that it must still be the smallest indexed alternative among the dictator's best alternatives, and it must be again chosen. argume:~t
The
for strict preferences is even simpler, and along the same
lines, since then agent one has a unique best alternative which is always chosen.
21.£.3
•
Suppose that we had z = f(?:i· .. ··?:}L
'
•
position from
from mcnotonicitv that •
f(?:~·· .. •?:[l = z.
(?;: ..... ?:: ).
1
1
But (~!·····t!l also takes {y,z} to the top from
Tnerefore we conclude that z F(?:; ..... ?:: ) y, a contradiction to 1 1
the assumption that y FC~ ..... t 1 l z. 1
2l.E.4
(i)
Ct}·····t[)
Cti•···•t'i). Since z maintains its
e A that takes {y,z} to the top from •
Consider a profile
Le~
(1) if x >-.
1
I
~
Hence, z
f(?:i·····?:'{J
':1:-
3, X = {x,y}, and define F: 1>
~ 'P
1
such that:
y fer all i then x F (>- , ... ,>- l y, p- 1 -1
(2) if (l) does net hold then y F (>- , ... ,>-I) x if and only if p- 1 #{i :
X
>-.1
y}
2::
l/2.
That is, x is socially preferred to y if it the unanimous Condorcet winner, and otherwise the Condorcet loser is socially preferred between the two
21 - 22
•
•
alternatives.
The induced f(: • ·) is vacuously pairwise independent, and •
clearly satisfies the Pareto condition.
By definition, x
: X >-'~ y}
X >-~
1
F
1
FCti•"·•?:il x,
y andy
f{?::i· .. ··?::il = x and f(?:i•·"•?:il = y.
(?:i· .. ··?:Jl to (ii)
X
Z
•
Z
>-'2 y >-'2
given by x >-" y >-" 1
1
and
< I.
which implies that
But x maintained its position from
implying that f(···) is not monotonic.
Let I = 3, X • {x.y,z}, and
>- '1 y >-'1
Cti····•?:r)
f(?:;l , ... ,?:r) = y, and
such that #{i :
Now consider
X
the domain of preferences to
I I d pre f erences { ?:lII , ... ,?: " ) are d an z >y >x, an • 3 3 1
z , Y >-"2 z
>-z
X , and Z >-j y >-j X.
Define the social
welfare functional F( • • ·) such that y F (>-' , ••• ,>-' ) z, z F {>-'1, ... ,>-'t) p.... . p- 1 - 1
z F (>-'', ... ,>-''1 y, andy F (>-" , ••• ,>-" 1 x. p- 1
-1
p- 1
.... 1
X,
F(···) is clearly Pareto, and
looking at the ranking between x and z, and between x and y, it is also pairwise independent.
However, even though y maintains its position from
that f( · · ·) is not monotonic. •
•
•
•
21 - 23
CHAPTER
22
The set of alternatives X in an exchange economy (Example 22.8.1) is
22.8.1.
I
I
r
given by X = {
X
1=1
1
r
s
w }, where
(w )
1
1=1
1
is the vector of initial
1 1=1
endowments. Observe that this set is convex. A sufficient condition for the •
UPS in this economy to be convex is that u ( ·) be concave for every i. Indeed, l
•
suppose that u'
I = (u (.x')) E 1 I 1=1
U and
X. Then by convexity of X for any (1-~)x" E
Ax' +
I u"= (u (.x")) E I I 1=1
~ E
[0, 1] we must have
X. Concavity of u ( ·) implies that 1
~u (x') + (1-~) u (x") ?: u (~x' 11 II I I
+
(1-~)x") I
for any i. Then by definition of
a UPS we have >.u' + (1-~) u" = (~u (x') + (1-~) I
22.8.2.
U, where .x', .x" E
I
1 u (x")) E U. I I 1=1
The Lagrangean for the constrained maximization problem can be
written as L = v(p , p ) + 12
~
((p -1) X (p ) + (p -2) x (p )). 1
11
2
22
The first-order conditions are obtained by differentiating the Lagrangean with respect to p
, p 1 2
and using Roy's identity foi:' the quasilinear case:
I
2
•
22.8.3.
The second-best problem can be written as A
A
max v(p , p)+ (p -1) 1 z l •
X
" 1 1
(p, p ) + (p -1) 2
2
X
" 2 I
(p •
Pz
where the last two terms represent the tax revenue which can be, for distributed among the population lump-sum. The first-order conditions (22.8.2) •
are then obtained by differentiating the objective function with respect to p and using Roy's identity for quasilinear utility:
22-1
2
•
22.8.4.
•
The one-dimensional "boundary" of the UPS in Example 22.B.4 is given •
parametrically: I
L
(v (p ) + (p -1) 1 2 2
•
We will
x1(p2), v2(p2), ••• , vl(p2)).
1=1
the "curvature" of this boundary around p
2
= 1. For
this purpose, consider the following derivatives: I
u~(p2)
I
L xa{p2)
=
L x:(p2h
+ (p2-l)
1=2
1•1
= - xk(p ) for k > 1.
u~(p )
2
•
2
aap
(We have used Roy's identity for quasilinear utility:
v(p)
=
-
X (p).) 1
Furthermore, I
I
u"(p ) 1 2
=
x'(p ) + 2 1 2
L x'1{p 2)
+ (p -l) 2
1=2
l"' x"(p )· L 1 2 ' 1=1
•
= - x' (p ) for k > 1.
u "(p ) k
2
k
2
2
=0
•
Substituting p holds
f~r
u'(O)
> 0,
1
and observing that x'(p ) < 0 (by the Law of Demand which 1
2
the quasilinear case}, we can see that
< 0,
u"(O) 1
u"(O) 1
<
0
u"(O)
•
k
>
0.
Now, take any k > 1, and consider the boundary of the UPS's projection = 0 is
on the plane (u , u ). The slope of this boundary at p u. '(D)
du k
k
-- = --du
1
2
< 0, which is not surprising. The curvature of this boundary at
2 k
2 1
•
1
this pomt
du
k
u'(O) •
du
1
-
IS
equal to du.
d dp
•
1
du
k
1 •
- -
u."(O) uj(O) - u~(O) u"(O) k
1
u'(O) 1
(u'(0))
1
3
1 •
using the signs in (•). This, the boundary is convex at p
> 0,
2
= 0, and the UPS is not concave.
•
•
22-2
•
22.C.l.
(i) U
1
c IR is synunetrlc, convex; W: ~ •
1
•
1
~ is mcreasmg,
-+)
•
•
•
syuunetric, strictly concave. Suppose that
• W(u )
= max W(u), and
• uJ
•
*
ueU
u k . Let
j
-
t = k j, L
uI' i •
• u
convexity of U, 1/2
• u +
WU/2
1/2
k
•• W(u )
•
•• u
By symmetry of U,
*
e U. By symmetry of + 1/2
•• • u ) > WCu ),
•• u
• ),
•
• = W(u ).
e U. By strict concavity of
By
•
{ · ),
which contradicts the assumption that
• u
was the
•
•
optimal utility profile in U. 1
c ~ is symmetric, strictly convex; W: IR
(ii) U
•
SYIWiletric concave. Suppose that
• u •
•• u l
•
u,
•
-
•
1/2
+ 1/2
•• u
·we · ) is
Since
•
W{I/2
•
• u
+
= max W(u), and
Let
k •
J, L
1
•• u
*
k
U. By symmetry of WC · ),
E
strict convexity of U, 1/2
• u
~ is
j
1
l
By symmetry of U,
)
1
ueU
• uf l
• W(u)
1
• u
+ 1/2
•• u
•• W(u )
=
• W(u ).
e Interior{U). Therefore,
ce e U for c > 0 sufficiently small, where e = (1, .. ,1).
increasing and concave, + l/2
•• u
+ ce)
> W(l/2
which contradicts the assumption that
• u
• u
+
1/2
•• u )
?:
• W(u },
was the
optimal utility profile in U.
22.C.2.
By
(a) According to Proposition 6.E.l (Extended Expected
Utility Theorem), if the planner's preferences over lotteries satisfy the continuity and extended independence axioms, they can be described by the utility function
22-3
U(x • ••• , x ) = u (x ) + ••• + u (x ). 1
I
1
1
I
I
Suppose that all components of the outcome except x are fixed.
If the
l
planner's preferences are non-paternalistic, then the planner's preferences over lotteries over x should coincide with individual i's preferences. 1
But
then u ( ·) must be individual i's von Neumann-Morgenstern (vNM) utility l
function. Since vNM utility functions are defined up to a liner rescaling, we • •
can conclude that the planner's preferences can be represented by a (possibly} nonsynunetric utilitarian SWF: W(v , ••• , v 1
I
J = 13 v
11
+ ••• · +
13 II v,
where v is the utility level of individual i, and 13 > 0 for every i. I
l
(b) If in addition the planner's preference relation is symmetric across individuals, then it can be represented by U(x , •.• , x ) = u(x ) + ••• + u(x ). 1
I
1
1
Therefore, the planner's preferences can be described by a utilitarian social welfare function where all individuals are ascribed identical utility functions. •
22.C.3.
[First printing errata: The definition of a Paretian social
utility function should be strengthened. It should say that, in addition, • •
U(p)
?:
U(p') whenever U (p) I
?:
•
U (p') for every i. J I
(a) Let final outcome 1 give the object to individual 1, and final outcome 2 give the object to individual 2. If p is the probability of implementing outcome 1, then U(p) = u p 1
u (1-p).
+
2
If we are indifferent
between outcome 1 for sure and outcome 2 for sure, we must have U{l)
= u1
= U(O)
= u 2.
But then U(p)
= u 1p
+ u (1-p) 1
= u1
regardless of p,
i.e. all the lotteries are socially indifferent. This means that the society •
•
22-4
does not care how expected utility is distributed between the two individuals . •
This conclusion follows from the linearity imposed by the independence axiom. For example, if U(p) were a strictly concave function, we would have UU/2) > U{O) = U(l). In this case, society would prefer to allocate the object using a fair randomization, rather than giving it to either individual with probability one. •
(b) See Figure 22.C.3(a). The vectors U and U l
2
represent the normals to the
two agents' Indifference curves. The vector U represents the normal to the social indifference curves. The social utility function is Paretian when U lies "between" U and U , i.e. the slope of social Indifference curves is l
2
•
If U does not lie
between the slopes of individual indifference curves. between U and U , we I
find two points A and B such that both individuals
2
prefer 8 to A, while society prefers A to B, i.e. the social utility function is not Paretian (see FiiUI"e 22.C.3(b).
•
•
•
0 •
22.C.3
22.C.3
(c) In this part we will restrict attention to the case where N = 3 and •
I = 2. When the two individuals have the same preferences over lotteries. then •
the only Paretian social preference over lotteries is the one which agrees with both individuals. Indeed, if a social preference relation does not agree •
• •
•
22-5
with the individual preferences, there will exist two points A and B such that •
each individual preferes B to A, while society prefers A to B (see Figure 22.C.3(c)). Therefore, in this case the Paretian condition determines the •
social preference uniquely.
•
•
B.
•
A
22.C.3 •
When the two Individuals have
preferences, any social
preference relation whose indifference curves have the slope between • •
the slopes of the two individuals' indifference curves, is Paretian (see Figure 22.C.3(a)). When the two individuals have strictly opposing preferences, any social utility function satisfies the Paretian property vacuously. (d) The social utility function can be written as U(p) • U • p, where •
I
N
U = (u , •.. , u ).
Similarly, every individual L's social utility function
can be written as U (p) I
= U I • p,
where U
I
1
= (u I ,
N
... , u ). In the case where I
N = 3 and I = 2, the geometrical analysis in parts (b)-(c) suggests that the •
social preference function U( • ) is Paretian · if and only if the vector U lies in the convex cone of U and U • i.e. U = 13 U + 13 U • with 13 .13 =:: 0, and I
~~
2
ll
22
12
> 0 for some i. We will show that this result holds generally:
22-6
I
I:
Suppose U(p) =
(3 U (p), where 1 1
1•1
t. Suppose that
for
I
I
L f31(U1{p)
U(p) - U(p') =
?:
- U (p')] 1
1=1
tt
0 for every (, and fJ >
l
U (p) - U (p') I
f3
I
0 for every i. ?:
Then we must have
0.
Suppose furthermore that U (p) - U (p') > 0 for every L. I
o
Then we must have
1
•
I •
L
U(p) - U(p') =
(3 [U (p) - U (p')]
1=1
1
1
1
it
(3k(Uk(p) - Uk(p' )]
>
0.
Therefore, U( · ) is a Paretlan social function.
Let C be the convex cone of (U, •••• U }, i.e.
Necessity:
.
l
c = ( r f3 1u I I f3 I
?:
0 for all
o.
.
-1
l
If U(.) cannot be written as
1= 1
•
where f3
I
~
0 for every i, and f3 > 0 for some i, then either (i) U lies 1
outside C, or (ii) U = 0, i.e. the social functional exhibits complete
•
•
indifference. In case (i), the Separating Hyperplane Theorem (Theorem M.G.2 in the Mathematical Appendix) says that there exist vectors c,d U·d c
E
N
IR such that
(1.1)
< u·d for every u
E
(1.2)
C.
Substituting o e C in (1.2), we get c < 0, which together with (1.1) implies
•
•
•
u. d
< 0.
(2.1)
Also, inequality (1. 2) implies that u ·d Indeed, if we·had u·d
?:
0 for every u e C.
for some u e
c.
(2.2)
then by definition of C we would have
(3u E C for any (3 ?: 0, and ((3u) • d = (3(u ·d)
< c for f3 large enough,
which contradicts (1.2).
• p
Now, take any interior probability distribution
•
22-7
.
e Int(A), and c > 0
small enough so that p
• that U(p
ed) -
+
•
•
+ £d e 6. On the one hand, inequality (2.1) implies
• U(p )
= U ·(£d) < 0.
•
• t, U 1(p
implies that for every individual U
1
On the other hand, inequality {2.2)
e C by construction of C.
+ ed) -
• U (p ) 1
= U ·(ed) 1
2::
0, since
These inequalities show that the social utility
function U( • ) is not Paretian. In case (ii), observe that by assumption •
I
I:
=0
(3 U -. U I
1=1
I
as long as {3
0 for all L and (3
it
I
1
> 0 for some L.
We can formulate the following useful lemma, which is a strengthening of the Supporting Hyperplane Theorem (Theorem M.G.3 in the Mathematical Appendix) in •
application to convex cones: I
Lemma: Let
c = {r
(3
1=1
I
I:
u I 1 1
1 1
Then there exists a vector d
1
?;
0 for i
= 1, •••• n
s;; IRN, where •
(3 U -. 0 as long as {3
1= 1
(3
•
1
2::
e
0 for all i, and (3 N
IR
such that U • d
•
1
> 0 for some t.
1
<
0
for all i.
•
Proof: We will use induction on N, the dimensionality of the space.
The
•
statement is obvious for N = 1. Suppose it is true for N-1 dimensional • spaces, and look at an N - dimensional space. Observe that (•) implies that ~
that 0
lnt(C). This allows us to use the Supporting Hyperplane Theorem,
which implies that there exists a vector d e IRN such that u · d ::s 0 for all •
u e C. Now, restrict attention to the hyperplane Hd = {u e IRNI u·d = 0},
which is an N-1 -dimensional space. The induction statement implies that there exists a vector d' e H vector
• d
= d
have U ·d > I
o;
+
d
such that U ·d < 0 for any U e H . Now, consider a 1
ed', where
£
l
d
I
> 0 is small enough.
and therefore U
Second, for any U e H
·
I
• ·d =
we have U
I
d
First, for any U II! H we I
d
U ·d + eU ·d' > 0 for £ small enough. I
• ·d
I
= U ·d 1
+ eU
1
·d' = eU ·d' > 0. Therefore, I
the statement is true for N-dimensional spaces. By induction, it must be true for any N.
•
•
•
22-8
•
•
Applying the Lemma, we see that U • d < 0 for all i.. Take any· interior probability distribution
• p
• p E
1
Int(6), and
• U (p
On the one hand, we have
+ ed e 6.
> 0 small enough so that
£
1
• U(p
all t. On the other hand, we know that
• U (p )
+ ed) -
1
• U • (p
+ £d) =
= U • (ed) < 0 for I
+ ed) = 0 •
• U(p ).
•
Therefore, the social utility function U( ·) is not Paretian. This completes •
• •
the proof.
•
•
•
The result says that any social utility function which is Paretian and satisfies the independence axiom corresponds to a weighted. utilitarian SWF • •
The coeeficients 13 are the weights which the SWF places on the utility of I
individual (. Different individuals can have different weights .
•
(a) Let >-
22.C.4.
R
represent the Rawlsian social preference, while >-
LM
•
represent the leximin social preference. Observe that
u' >- u -. min u' > min u .. R
1
1
1
I
u'r 1
On the other hand, we can have u' and u
,r 2
R
>
r u -. 1
u' >-
LM
u but u' >-
LM
u.
u, for example when
leximin is a refinement of the Rawlsian
preference.
(b) Observe that the leximin ordering coincides with the lexicographic •
ordering above the 45° line. As shown in Example 3.C.l, lexicographic preferences cannot be represented by a utility function. (c) Suppose u is a LM optimum, but not a Pareto optimum. Then there exists u' such that u'
i!:·
u, u'..e u. Let k = min {i I u
,r l
r}
> u . (Observe that the set is 1
non-empty.) We must have r
r
1
I
u' = u for any i. < k, r
u.
•
k
22-9
Thus, u• >-LM u - contradiction to u being a LM optimum.
Let u =
22.C.S.
Then we can write u
I }1-P ]1/U-PJ. Observe t h at w h en p W (u) -- (~ L u1-p)1/U-p) = U (~ L { p I I I U
> 1,
at least one tez·m in the smu is exactly one, and others may be between zero
-
•
and one. Thus, the sum In the square brackets is between 1 and I for p > 1. 1/U-p)
When p .. m, 1/{1-p) .. 0, and both 1
and I
1/U-p)
go to one. Therefore,
as p .. •·
W (u) .. u p
•
22.C.6. (a) See Figure 22.C.6(a). •
•
•
•
•
•
u •
•
•
•
(b) The utility possibility Mts for quulllnear I
u
= {u e IRI I
r
u
1•1
I
I
~
L uI ), u·
= {u E IR
1=1
1
are of the
:
I
I
r
I
u s
1=1
I
r uI
I } •
If
u·
passes the weal<
... 1
compensation test over (U, u), then by definition the exists a u'
E U'
such
•
that u'
1
i!:
-u
I
l
for every £. Adding up, we obtain I
hand, u' e U' implies
r 1=1
1•1
I
u'l s
r u'.1
I:
I
u
1
1
i!:
I: u 1.
1=1
Combining, we see that
On the other
r ul
1=1
I= 1
•
22-10
I
I
~
't"
L 1=1
-u'
l'
i.e.
-u'
-u.
is a utilitarian improvement over •
Also, the last intequality
implies that U c U', i.e. U' passes a strong compensation test over U. This argument is graphically illustrated in FiJUre 22.C.6(b). If we use a nonutllltarian SWF,
-u'
may not be an improvement over
-u.
•
•
•
22.C. 22.C.7.
[First
•differing only in distribution of the
initial endowments" should be interpreted to allow for different Initial endowments. J Take two consumers with utility functions u {x 1
1/3
n'
1
12
2/3
x ) =x11 x 21
u (x , x
21
22
)
•
•
213 1/3 =X X •
12
22
Take the total endowments for U to be {2,1), and the total endowments for U' to be (1,2). Choose then individual initial endowments to be
= {2-c,l-c), 1
w
w = (c,c), 1
w
2
= (c,c)
for U,
w = (l-c,2-£) for U'. 2
For c > 0 small enough, the competitive equilibria in the two economies will •
•
yield utility vectors u and u' located as in Figure 22. C.7 in the textbook.
22.C.8.
(a) See Figure 22.C.8(a) for a situation where Kaldor comparability
is. possible.
Kaldor comparability is not possible in Figure 22.c;:.1. in the •
• •
22-11
textbook.
•
u •
22.C.8(a) 0
(b) A relation >- is
if a >- b and b >- a cannot hold at once.
If (U',u') passes the Kaldor compensation test over (U,u), then by definition
U' passes the weak compensation test over (U,u), therefore, again by •
definition, (U,u) does not pass the Kaldor compensation test over {U', u'). Therefore, Kaldor comparability is asymmetric . •
•
(c) See Figure 22.C.8(b).
(U'.u') passes the Kaldor compensation test over •
(U,u), and (U" ,u") passes the Kaldor compensation test over (U' ,u'). But U"
does not pass the weak compensation test over (U,u), and, therefore, {U",u") does not pass the Kaldor
test over (U,u).
•
u 0
•
22.C.8(b) •
• •
22-12
22.0.1.
(a)
-u
F(u , 1
3
is always complete. since we always have either x
)
,.,_ ..,._,.,F(u • u • u ) y or y F(u • u • u ) x (or both). ~
z
1
3
1
2
3
F is Paretian:
....
-u
....
u (x)
u (y) for every i.
?:
I
-u (x)
I
....
>u
1
..
{y) for every t
1
x
•
.. x .
for every t
....
1
p
.. y F(u
u (y) > u (x) for every i I
Feu, 1 F (u , ....
....
y
l
..
y
3
1
}
y. not y
....u ....
2
-u
,
....
3
1
) x• 3
) y. •
u ) x.
,
3
,.. F(u, u ) x and 1 3 .... F (u , u , u ) x. p
F(u, u
2
1
not x
F(u • u)y• 1
3
3
F satisfies PI, since whenever -
u (x) l
= u'(x) I ....-
#1111
and u (y) 1
= u'(y), I -
-
we have u { •) 1
= u'(I ·) -
(there are no other
alternatives). •
Suppose in negation that F can be represented by a social welfare function WC • ). Choose a utility profile
(u
....
1
.... , u , u ) so that 2
3
-u (x) = u- (x) = -u (x) = 1, u- (y) = -u (y) = 1, -u (y) = 2. 1 2 3 I 2 3 .... Then by definition of F we have y F(u, u , u ) x, but not 1 2 3 -Nx FC u , u , u } y, i.e. y F (u , u , u ) x. This implies that - -
l'fJ
12
3
p1
2
3
•
W(l,l,2) > W(l,l,l). Now choose another utility profile,
(u' u', u'), so that 1 2 3 -r.t'(y} = u'(y) = u '(y) = 1, 1 2 3 1
u'(x) = u'(x) = 1, u'(x) = 2. 1 2 3 ... have x F(u', u', u'l y. This implies
Then by definition of F we (1,1,1)
=i
•
"""
1
2
3
that
W(l,l,2) - contradiction to the previous paragraph. •
(b) The Paretian condition fails to be satisfied, since we can take -
I
-
u e 'U such that u (x)
x F
p
(u , I
... ,
-u
1
I
-
(y)
1
for all i, and we still have
) y. All the other conditions of Proposition 22.0.1 are
satisfied. However, F cannot be represented by means of a social welfare •
•
22-13
•
function. Indeed, suppose that ?; is a social preference relation on utility •
vectors which represents F. Choose a utility
-u (x) 1
Since x F
p
-u
=
(u , 1
•
2
•
-u
(x) =
3
(x) = 0, and
-u ) y,
••• ,
-u (x) 1
-u
I
=
2
(x) =
-u
3
=
1
2
3
we must have (0,0,0) >- (1,1,1). On the other hand,
consider a utility profile •
-u (y)
profile u so that -u (y) = u- (y) = 1.
-u'
such that
= 1, and u (y) = u (y) = u (y) 1 2 3 F (u , ... , u ) y, which Implies
(x)
Now we must again have x
I
p
= 0. (1,1,1) >- (0,0,0).
I
Contradiction. •
(c) Consider the Borda count. •
-u
I
(y)
= u',I
and
-u
= u for every i. I I FCu, ••• , u ) y. By the
we must have x
z z
u e tl
Take a profile of utility functions
22.0.2.
F(u. "1 F(u,
••••
... ,
1
-u ) I -u ) I
I
-
Pareto property, we also have
I
F(u •••• , u ),
Hence, by the transitivity of
X•
I
Since u >- u• using the pair x,y,
(z)
1
with u (x) = u ,
1
I
we obtain
y.
•
22.0.3.
(i)
Let u'
a utility profile
~
u, where u,u•
u E 'li
such that
functional· F is Paretian,
x
F (u) p
a utility profUe u·
i
= u'
1
E
= u' >-
-
1 E IR •
I
~
= u', I
-u(y ) I
-u (y)
= u for all L. Since the
1
1
= u for all i. implies that I
-u(y)
= u.
Choose two alternatives x,y -
E
X and
-
'U such that u (x) = u', u (y) = u for all i.. Since the
functional F is Paretian, p
-u(x)
u·· » u, where u,u' -
x
-u (x)
y. Since the functional is represented by the social preference
-
F (u)
I
IR • Choose two alternatives x,y e X and
•
relation >-, we must have Cii) Let
-u (x)
E
I
-u (x) 1
I
= u'
I
1
I
> u(y } =
1
u for all t implies that
1
1
y. Since the functional is represented by the social preference
relation >-, we must have
-
-u(x) •
= u' >-
-u(y)
= u. •
• •
22-14
•
22.D.4.
F(u ,
x
1
The social welfare functional is defined by ••• ,
-u
1
> uh
-uh (x)
) y •
J
(y)
or k
= n)
,.. ~ uh (y) for j J
= 1,
•. , k and
for some k s n.
k
~
(a) Pareto:
-u (x) I
for every i. Then
-uh (x) .?: uh - (x) •
J
J
. for every k = 1, •.., n, which implies x F(u, •••, u) y. If. moreover, .
1
-u (x) > -u (y), 1
X
then we know that
1
F(u , 1
••• ,
-u
-
I
(u', ••• , u') e 'U, with 1
I
X
•
•
-
E
1
X and
-
u (x) • u'(x) and 1
F(u , ... , ui) Y* 1 - (y) for -uh (x) ?: uh
I
•
J
-
1,
.. ,
(u '
•
-uh (x) -> -uh (y) for J J x F(u', ..... u') y.
- ) ••• , u 1
1
-
u (y) I
k and
J
J
•
which implies
) y.
Pairwise Independence: Let x,y -
-uh (x) > uh - {y), 1
I
I
•
J = 1, .. , k and
e '""
-
= u'(y) for all i. Then 1
(uh Cuh
(x) k
(x)
> uh > uh
Does not allow interpersonal comparisons: Let
(y) or k
-
n)
or k
-
n) *
k
k
1
1
cui,
(y) k
-u'(x) 1
= fj
l
-u (x)
for all i
+
0:
or k
-
n)
(y) or k
-
n)
1
1
and x, with 13 > 0 for all i. Then for all x,y e X l
x .
•
F(u , ... , u ) y • · 1 I - (y) for -uh (x) -> uh J
•
• Thus,
•
J
-
1,
...
k and
J
- (y) for J - 1, .. , -uh (x) -> uh J J y. x F(u', ... , u') I 1 F( u , ..·., u ) = F(u' ... , u'). I 1' I 1 •
k and
(uh (uh
(x) k
(x)
> uh (y) > uh
k
*
k
k
•
(b) When n=l, F can be generated from a social welfare function W(u
, ... , 1
u
1
)
= uh . When n>l, F cannot be generated from a social welfare 1
22-15
function. since it induces lexicographic social preferences on utility profiles (uh , uh ) (for simplicity we keep utilities of all other 1
2
agent constants), and as we know from Example 3.C.l, such preferences cannot be represented by a utility function. (c) Let k be the dictator. Suppose in negation that b -
·-
I
-
J
>F
0 for some j. Then •
-
take any x,y e X and (u , ••. , u ) e U such that u (x) = u
-u
k
-u
(x) =
1
k
(y) + b /2,
J
and
-u (x)
I
=
J
I
-u {y)
for i::t:j,k,
Then agent k prefers x to y,
- b .
J
I
(y)
•
k
while W(~ (x), ••• , ~ (x)) - W(~ (y}, ••• , ~ (y)) • b b /2 - b b < 0, i.e. 1
I
1
I
Jk
Jk
socially y is preferred to x. This contradicts agent k being a dictator.
We will prove Arrow's impossibility theorem (Proposition 21.C.l) only
22.D.S.
for the case where preference domain allows no indifference: ,4. = 1J 1 'R ,
To extend the proof to the domain ,4. =
1
•
one needs to bring the definition of
of a Paretian social welfare functional defined on preferences, Definition · •
2l.C.2, in correspondence with the corresponding definition for a functional
defined on utility profiles, Definition 22.0.2.)
•
For every social welfare functional F() defined on preference profiles from 1J we can construct a social welfare functional letting
F (U
-u
1
1
1
.. ,
F'()
on utility functions by
) "" F(>- , .. , >- ), where >- is the preference relation
1 .
-r
-1
induced by the utility function
-1
-u ( • ). I
F'O inherits the Pareto property from FO: (i) If
-u (x) 1
?:
-u (y)
for all i, then either x >- y 'di or x = y. In the first
1
I
case, we must have x F (>- , p -1
··1
>- ) y since F() is Paretian (Definition
-1
2l.C.2). In both cases, therefore, we have x FC>-, •• , >-) y, and -a -I
correspondingly x (ii)
If
-u (x) 1
?:
F'(u
-u (y)
1
I
"I
-u ) y. I
for all i, then x >- y Vi.. Since F() is Paretian I
1
•
22-16
1
(Definition 2l.C.2), we must have x F (>- , •• , >- ) y, and correspondingly p -1
•
F'(u ,
x
1
p
•. ,
-u
I
-I
) y.
Thus, both parts of definition 22.D.2 are satisfied. F'() the Pairwise Independence property from F():
"""' Suppose that u (x) = u'(x) and u (y) = u'(y) for all I I l 1 x >- y .,. u (x) ~ u (y) .,. u'(x) :s u'(y) ..,. x >-' y, -1 l 1 1 l -1 . y >- x .,. u (y) u (x) .,. u'(y) :s u'(x) ... y >- x. -1
~
i!:
I
l
t. This .implies that and that
1
I
I
-1
Since FO satisfies PI (Definition 2l.C.3), we therefore have
F'(u ,
x
•• , u ) y 1 I F'(U U
.,. x
1
... x F{>- , •• , >- ) y .,. x Fb·', .. , >-') y ... -1
1
•• ,
1
1
I
)
-I
-1
-I
y.
Therefore, F'() satisfies PI (Definition 22.0.3).
•
F'() does not allow interpersonal comparisons of utility:
Suppose that for all i. and x we have
-u (x)
= f3
1
-U (x) 1
1
1
a. • with f3 > 0. Then
+
I
1
for all x, y we have
x >- y ...,. u (x) ~ u (y) ...,. f3 u'(x) -1 1 1 I I ._. U (x) ~ u'(y) ..,. x >-' y.
•
+ a.
I
~
f3
I
-u'(y) I
+ a.
1
1
I
I
-1
-
Therefore, >and >represent the same preferences, and -1 -1 1
F'Cu'1
x
.. ,
1
•
...,. x
-u')
F'(u
I
I
.. ,
I
y ..,. x FC>-', •• , >-') y .,.. x F(>- 1 .. , >-)
-u ) I
-1
-I
-1
-I
y.
Therefore, F'O does not allow interpersonal comparisons of utility (Definition 2.0.5). A dictator for F 0 is a dictator for F(): 1
Suppose that h is a dictator for F'(), i.e ..• that for every pair x,y e X
-u
h
(x)
>u
•
h
(y)
x >- y h
implies that x F o+
-u
h
(x)
>u
h
1
p
(U
(y) -. x
I
"I
I
-u )
F'(u ~· p
1
I
"I
y. Then for every x,y e X we have
-u )
Therefore, h is a dictator for F() . • • •
22-17
I
y ,. x F C>-
p -1
I
.. ,
>-) y.
-I
Therefore, for every Paretian, PI social welfare functional F() defined on •
preference profiles we can construct a Paretian, PI social welfare functional F'(} on utility functions which does not allow interpersonal comparisons of •
utility, so that a dictator for F'() is a dictator for F(). Proposition 22.0.3 establishes that F'() admits a dictator, and therefore, so does FO.
This
establishes Arrow's impossibility theorem.
•
•
22.D.6.
(a) Since SWF is defined up to a monotonic •
transformation, we can always normalize the nonsymmetric utilitarian SWF W(u)
= I: 1
W(u) =
I: I
I:
b u so that 1 1
b u
1 1
=
-u
b = 1. Then we can write I
1
+
I:
b (u I
I
1
-u).
-
'
•
(b) [First printing errata: we need to assume concavity of negation that g(u - u, ... u 1
every i we have
1
1 - u) < 0, where u =
I:
1 J
I
u .
we· ). J Suppose
in
Observe that for
j
u ( ) = u, where the sutn is taken over all permutations of
,f. n. n 1 n
agents. Since g( · ) is symmetric, we must have g(u
nm
-
-u, ...u
nm
-u)
-
for every permutation n of agents.
=
g(u
Let u
1
-
-u, ... u
1
-
-u)
<0
= (u ( )' ... , u ) for every n n 1 nm
1 permutation n. On one hand, we have we ,I: u > = n. n n
weu, ..., u)
=
-u
- geOl =
-u.
On the other hand, we have
-u, ... u
= 1 ~ eli - g(u n! L 1 n
= u
- g(u
1
-
-u, ... u
I
I
-
-u))
- u) > u =
which contradicts the concavity of
=
1
we n., t
n
we· ).
u n >.
Intuitively, concavity of W · () means
that the SWF favors equality. In order for inequality to be penalized, we need •
22-18
to have g(u
1
-
-u, •••u - u) 1
gco>
~
= 0.
•
(c) The symmetric Rawlsian welfare function can be written as
-u
W(u) = min{u , .• , u } = 1
l
+ min{u
1
-
-u,
-u}.
.. , u I
As for the asymmetric
Rawlsian social welfare function, it is invariant to identical changes of •
origins and therefore cannot be expressed in the form (22.0.1). For example, take W(u ,u ) = min{b u , b u } with b I
2
ll
22
> b > 0. Take two utility vectors
2
1
u' • (1,0), u" = (O,l). Then W(u') = W(u") = 0, and at the same time •
(d) Take, for example, W{u) =
-u - L k
-2
I
with k
- u)
(u
i'!::
0. This SWF
l
punishes for a variance of utilities; it is more sensitive to inequality than the utilitarian SWF, but less sensitive than the Rawlsian SWF.
•
(e) If g( ·) is homogeneous of degree one, g(«z) = « g{z) for all o:
0.
i'!::
Differentiation with respect to o: yields z·Vg(az) = g(z). Substituting o: = 0,
-
we obtain g(z) = z · Vg(O), i.e. g( • ) is linear . •
22.0.7.
(a) W (i\.u)
p
= li:
(i\.u ) 1-p]1/(l-p> I
I
= i\.
(I:
u•-P]l/11-P> = i\. W (u)
p
1
I
'
therefore, the social preferences are invariant to common changes in units. •
•
•
(b) Let W'(u) =
p
W (u+a). When does W'(u) induce the same social preferences
p
p
•
as W (u+a)? To answer the question, we can look at marginal rates of p
substitution: aw (u)/au p. J MRS = l,J BW (u)/Bu
p
1
uJ -p
; MRS'
u
l,J
-
8W'(u)/Bu p J-
u
8W'(u)/8u
u
-
p
I
1
J I
+
a
-p •
+
a
•
For social preferences to be invariant to a, we need to have u + a -p u -p J J This holds for p MRS =MRS' , which implies • l,j I,J u + a u I
l
•
22-19
•
-
0
(both sides are equal to one) and p =
(both sides are equal
CD
0 when u > u , one when u = u ). For 0 < p < 1
J
·
1
J
·
u
l =
substitution can only be equalized if
u •
unless u
I
= uJ
or a
u
1
when uJ < u
-
CD,
marginal rates of
+a
J u
CD
, which cannot hold
a
+
I
= 0. •
(a) If f( · ) is an increasing function, we have
22.0.8.
W(f(u ), •..• f(u )) 1
I
= min{f(u 1),
= f(min{u 1••.•• Therefore, W(f(u
1
), ••••
u }) I
...• f(u )} • I
= f(W(u 1•••••
u
)). 1
f(ui)) induces the same social ordering as •
W(u , ... , u ). 1
I
(b) If f( • ) is an increasing function, we have W(f(u ), ...• f(u )) = max{f(u l ..... f{u )} = I
= f(max{u,1
I
... , u }) I
1
= f(W(u,1
I
... , u )). I
•
Therefore, W(f(u ), ... , f(ui)) induces the same social ordering as 1 W(u , ... , u ). 1
·
I
(c) If k the dictator, we can use W(u) = u. If f( · ) is an increasing k
function, we have W(f{u ), ... , f(u )) = f(u ) = f(W(u .... , u )). 1
I
k
1
1
•
Therefore, W(f(u
1
), ... ,
f(u
1
))
induces the same social ordering as
W(u , ... , u ). I
I
•
•
(d) We can find a monotone increasing transformation of the utility ax1s which increases u•. u•, while keeping u , u 1
a
E
2
1
2
unchanged. For example, pick
(0,1) and b > 1 and consider the following ordinal transformation:
•
22-20
•
u, u
-
f{u}
u u
I
2
u
1
:!i
u
•
2
+ a(u-u ), u < u
•
I
1
+
b u-u
2 '
u > u
•
2
Suppose that W(u', u') = W(u, u ). Since W is Invariant 1
2
1
2
to [, we must also have W(f(u'), f(u')) = W(f(u ), f(u )). And from the way f 1
was constructed, W(u , u ) I
2
2
= W(f(u 1),
I
2
f(u )). All this together implies 2
• •
W(u', u') = W(f(u'), f(u')). But this contradicts W being increasing, since I
2
1
2
> f(u'), u' > f(u'). 1 12 2
Therefore, we cannot have W(u', u') = W(Li , u ).
u'
12
12
Geometrically this implies that social indifference curves above the 45
0
line
•
•
cannot have a slope between zero and infinity. The
argument, of course,
works below the 45° line (by switching u and u ). Thus, we are left with four 1
2
possibilities regarding slopes of indifference curves above and below the 45° line:
- 0 above the line, 0 below the line: 2 is a dictator; - 0 above the line,
Ill
below the line: anti-Rawlsian SWF;
-
Ill
above the line, 0 below the line: Rawlsian SWF;
-
Ill
above the line,
m
below the line: 1 is a dictator.
•
22.E.l. (a) Egalitarian solution f {U)(Example 22.E.l) solves max mm u . I e U j - IUO is· satisfied by construction (see footnote 26 in text) . •
- Paretian: Suppose
• u
= f (U). Suppose in negation that u' e
E
U and
•
u' >> u. We then have max min u' >max mm u , which contradicts
u
I
I
u
I
I
•
egalitarian solution . .
- Symmetry: follows from Exercise 22.C.l.
• - Individual rationality: Suppose u •
max mm u and 0
U
I
I
E
= f (U). Since e
U, we must have min
• u
I
22-21
I
~
0.
• u
solves
• u
being the
(b) Utilitarian solution f (U) (Example 22.E.2) solves u
rI
u .
E
U and
I
- IUO is satisfied by construction (see footnote 26 in text) . - Paretian: Suppose u'
»
• u
u. We then have
.
= f {U). Suppose in negation that u' u
L u' I
>}:u, which contradicts
1
1
• u
being the
I
utilitarian solution. •
•
•
- Symmetry: follows from Exercise 22.C.l. - Individual rationality: f (U) z: 0 by definition, since the maximization u
domain is restricted to the positive orthant.
(c)
Nash solution (Example 2.E.3) solves
LIn u 1
I
1
- IUO is satisfied by construction (see footnote 26 in text) . - Paretian: Suppose u'
»
• u
= f (U). Suppose in negation that u' n
r
u. We then have
ln u' > I
I
L In
u
1
1
,
which contradicts
U and
E
• u
being the
Nash solution.
•
- Symmetry: follows from Exercise 22.C.l. - Individual rationality: f (U) z: 0 by definition, since the maximization n
domain is restricted to the positive orthant. (d) Kalai-Smorodinsky solution (Example 2.E.4) is f (U) = k
where
• t
= max {t I t(u {U),u (U)) 1
2
E
• t (u
(U),u (U)),
1
2
•
U}.
- IUO is satisfied by construction (see footnote 26 in text) . - Paretian: Suppose
• u
= f {U). Suppose in negation that k
u' and u' » u. Let t = min (u' /u (U)) > I
•
I
1
• t . We then have t(u (U),u {U)) :s I
and therefore t(u (U),u (U)), which contradicts 1
2
• u
being the
Kalai-Smorodinsky solution. - Symmetry: If U is symmetric, then u
(U) =
1
•
22-22
u (U), and therefore 2
2
1
f
kl
(U)
= tu 1(U) .= tu 2 (U)
= f k2 (U).
- Individual rationality: since t = 0 is feasible, we must have Therefore, f (U)
2:
k
(a)
~
0.
0.
Take a vector of weights (i\ , ••• , i\ )
22.£.2.
• t
1
I
»
0. Then define
max min Non-symmetric egalitarian solution: solves •
u
1
(b) Non-symmetric utilitarian solution: solves max U
{c) Non-symmetric Nash solution: solves max U
l: I
~
rI
~
u.
1 1
u. I 1
i\ In u . 1
1 •
(d) Non-s:ynuuetric Kalai-Smorodinsky solution: f (U) where
• t =
k
•
•
max {t I t(i\ u (U), i\ u (U)) 1 1
E
2 2
• = t (i\
u (U), i\ u (U)},
1 1
2 2
U}.
These solutions satisfy all the axioms which the "standard" solutions satisfy, except symmetry. Weights (i\ ) can be interpreted as the relative importance of l
individuals in bargaining. •
(a) The Nash solution f = (f, f )
22.£.3.
•
f
= argmax
W(u)
U
= E In
1
2
E
U is given by
u . Take the line which separates U and
I
I .
V = (u I W(u)
>
W(f)}. (The existence of such a line follows from the
separating hyperplane theorem, Theorem M.G.2 in the Mathematical Appendix.) If the boundary of U is smooth, then this line is tangent to it. We know that the •
boundary of V is smooth, therefore this line is tangent to its boundary. From the equation of the boundary of V, a normal to the line can be written as (aW /au , aw /au ) = (1/f, 1/f ). Since this line passes through N = ([ 1
2
1
2
its equation can be written as u / f + u / f 11
22
= f /f
11
+
f /f = 2. 22
•
• •
• I
22-23
1
,
f
2
),
From this we can see that this line crosses the vertical axis at A = (0, 2f2 ) • •
and the horizontal axis at B = (2f , 0). Using similarity of triangles (see 1
Figure 22.E.3(a)), one ·can deduce that N is the midpoint of AB.
v
•
•
(b) The idea here is that the rescaling should transfoam the tangent line AB
into a line with slope -1. Take the rescaling u' = (3 u. Then A is mapped into 1
I I
mapped into B' = (2(3 f , 0). For A'B' to have slope -
-1, we can take, for example, (3
= 1/f, 1 1
1 1
for this gives us A'
= (1, 0). The Nash point in the new utilities will then be N'
= ((3 1f 1,
(3 f
2 2
)
= (1,
1). (See FiiUI"e 22.£.3(b).)
•
•
U'/ /
1 •
•
1
• 22-24
•
= (0,
1) and B'
Now it is clear that in the new utilities the Nash point N' is both the utilitarian solution (since U is supported at N' by a line
AB with slope -1)
and the egalitarian solution (since N' is at the intersection of the 45° line and on the boundary of the convex set U). •
22.E.4.
(a) The Kalai-Smorodinsky solution (Example 2.E.4) can be written as
f
• t (u
k
(U) =
• t
•
(U),u (U)), where
1
2
= max {tl t(u (U),u (U)) e U}. Take a linear 1
2
rescaling u'= f3 u , and let this rescaling transform U into U'. We need to I
l 1
• t' (u
figure out how f (U') = k
(U'),u (U'.)) relates to
1
2
f (U). First, it is k
clear that u (U') = f3 u (U). Then 1
• t'
I I
= max (t' I t'(u (U'), u (U')) e U'} = 1
2
= max {t' l t'(f3 u (U), ·1 I
Thus, fk(U') =
f
kl
(U')
• t {u
= /31f kl (U),
f3
u {U)) e U'} = max {t' I t'(u (U), u (U)) e U} =
2 2
1
=
(U'), u (U'))
• t ({3
2
•
u (U), [3 u (U)). This means that
1 1
1
2
i.e. f
is independent of utility units.
(b) Take U = {u I u
1
k
+ u
2
:=s
• t •
2}, U' = {u I u
2 2
1
+ u
2
:=s
2, u
:=s
2
l}. We have U' c: U,
and by symmetry f (U) = (l,ll. At the same time, u (U') = 2 and u (U') = 1, 1
k
which implies that f
k2
2
(U')/f (U') = 1/2. Therefore, f (U') kl
k
*
= f (U).
(l, 1)
k
Thus, the Kalai-Smorodinsky solution does not satisfy IIA .
•
22.E.S.
e (i)
(a) The egalitarian solution can be written as f (U) =
= (1, ••• ,1),
e
• and t
= max
{t I
• t e,
where
te e U}.
First we show that the egalitarian solution satisfies the four properties.
The properties of IUO, Pareto, and symmetry are shown in Exercise 22.E.l. Monotonicity: let U c: U'. Then and therefore f (U') = e
• t' e
~
• t' =
max {t I te e U'}
f (U) = e
• t e.
22-25
~
• t = max
{t I
te e U},
(ii) Now we show that a solution that satisfies the four properties is
egalitarian. We will show this only for utility possibility sets for which the •
•
weak and strong Pareto properties coincide. (The general proof is more difficult.)
•
Independence of utility origins allows us to normalize the threat point to be the origin, and to write our solution as f(U). Now, consider an arbitrary UPS U and let its egalitarian solution be f (U) = (a, ..•,a). e
Define
u· = {u
IR
E
r
1
I +
u
1
1
:s I a}. The properties of Pareto and symmetry of f(.) •
imply that f(U') = (a, ••• ,a). Since U u U' c: U, by monotonicity of f( ·) we
= (a, ... ,a).
must have f(U u U') :s f(U')
•
But we know that (a, ..• ,a) e U u U'.
By (strict) Pareto optimality of f(U'), we must therefore have Now, since U u U' c: U, by monotonicity of f( ·) we must ·
f(U u U'} = (a, ... ,a).
have f(U)
~ f(U
= (a, ...,a) = f e (U).
u U')
But we know from (l) that f (U) is e
a (weak) Pareto optimum in U. Therefore, it isalso a strict Pareto optimum, and we must have f(U) = f (U). (b) Independence of utility origins allows us to e
normalize the threat point to be the origin, and to write our solution as f(U). Let us start with looking at the family of symmetric UPS with linear
boundaries: U(a)
= {u.e
•
I
Lu
IR + I
:s a}, and let us define v(a) = f(U(a)). First we
1
I •
•
establish that v(a) is continuous in a e [O,+Cil), By Pareto optimality
I: I
v
1
(a)
= a for all a.
v(a+£) e {u e
I v(a+£)
- v(a)
Using monotonicity,
1
IR I [ u +
I :s
1
£.
I
=
r
v (a) + £, u
1
~ v(a)}.
This implies that
l
Therefore, v( ·) describes a continuous curve. By
definition, v(O) = 0. Also, since
L v 1(a)
= a, we must have
1
= al
-1/2
, therefore the curve
v( ·)
•
•
22-26
•
is unbounded.
•
Now, take an arbitrary UPS U. On one hand, we have v(O) = 0 e U. On the othe •
•
hand, since v( ·) is unbounded and U n IR for some a
0
an a' e (O,a
1 +
is bounded, we must have v(a
0 )
~ U
e
(O,+Cil). Since v( ·) is continuous and U is closed, there exists
)
such that v(a') lies on the boundary of U. Denote U' = U(a'),
0
and apply the argument of pa•·t (a){ii) of this exercise to show that
= f(U') = v(a').
f(U)
Thus, for every UPS U, f(U) is the intersection of the
boundary of U with the curve v( • ).
22.E.6.
The Kalai-Smorodinsky (K-S) solution {Example 2.E.4) can be written
as f (U)
• = t (u
k
(i)
1
(U),u (U)), 2
where
• t
= max {t I t(u (U),u (U)) e U}. 1
2
First we show that the K-S solution satisfies the five properties. The
properties of IUO, Pareto, and symmetry are
shown in Exercise 22. E.l. The
property of IUU is shown in Exercise 22.E.4. Partial monotonicity: let U' expand U in the direction of agent j
~
i., i.e.
U c: U' and u (U) = u {U' ). Observe that U c: U' implies u (U) :s u (U'). Let j 1 J J
'f = f J(U)/u J{U')
:s f (U)/u (U) J J
= t.
Then (see Flpre 22.E.6(a)).
•
'
•
22-27
•
A
Therefore, t(u (U'), uJ(U')) e U c U'. This implies that 1
Therefore, f
kJ
(U')
= t'
•
A
u (U')
u (U')
?: t
J
J
A t u {U)= f (U). J J
i!:
(ii) Now we show that a solution that satisfies the five properties has to •
coincide with the K-S solution. Independence of utility origins allows us to •
normalize the threat point to. be the origin, and to write our solution as · •
f(U).
Independence of utility units allows us to rescale utilities to
nonualize u (U) = u (U) = 1. Let f (U) be the K-S solution for U. Once we show 1
2
k
that f(U) = f k (U), independence of utility units and origins of both f(U) and
f k (U) will allow us to extend this equality to all bargaining problems. To show that f(U) • f (U), define k
•
u· •
k1
(U)) u
1
+
I I I I
1
u
+
1
(1-f
kl
f
kl
(U)
(U)) u
1
u s f 2
s
f
kZ
kl
(U),
(U)}.
•
•
•
/ /
' 1 •
1 Geometrically, U' Is bounded by two line.: one connecting f,.(U) and (0,1), and •
the other connecting f (U) to (1,0). (See Figure 22.E.6(b).) The properties of k
symmetry and Pareto imply that f(U') = f (U). By convexity of U we have U' c k
U. We also have u (U') = u (U) = 1, u (U') = u (U) = 1. Now we can use partial 1
1
2
2
•
22-28
monotonicity ln. both directions: U expands U' in the direction of agent 1 .. f (U) l
f
2
f (U') = f
it
1
f
(U) it
2
(U')
kl
= f
(U). Also, U expands U' in the direction of agent 2 ..
k2
To summarize, f(U)
(U).
2'!
f (U). As the Kalai-Smorodinsky k
solution for I = 2 is always strictly Pareto optimal (check), this implies f(U) =
f
(U).
k
Let S c: I. For any u
22.E.7.
Us(u
-s
= {u
) 5
In words, U (u
-s
)
-s
-s
e IRs l (u , u
s
IR
E
-s
)
l\S
•
, define
E U}.
is the bargaining set left for coalition
·s
when utilities of
the other players have been fixed at u . Consistency then means that for any
-s
S c I, t
1
s
•
= fs(~C/ (U))).
(U)
-s
Our solution for any· I is given by /CUl = argmax U
r
g(u ), and uniqueness of
I
I
argmax is guaranteed by strict convexity of g( ·) and convexity of U. Suppose In negation that /CU) is not consistent, i.e. that for some U and 5
some S we have u' = f (U s u'
= (u', s
i -s (U))
E U
and
C/-s(U))) ~ icul. s that r g(u, ) =
5
S
This means that
L:
gCu
s
Sl
>
>
l
I: gtiwn. s
But then we have •
L g(u')I = L g(u'Sl ) I
S
+
L I \S
I g(f (U)) I
>
L S
•
I g(f CU)) + 1
L I \S
I g(f (U)) I
=L I
I g(f (U)), I
I
which contradicts the definition of f (U).
3
22.£.8. Take I = 3, and let U = {u e IR 1 u 1
1
:S
1, u /2 + u 1
2
+ u
3
:S
3
2
lt is easy to- see that u (U) = u (U) = u (U) = 1. To find the Kalai-Smorodinsky solution, therefore, we solve Max {t l (t,t,t) e U}. This gives t = 0.4, and therefore f (U) = (0.4, 0.4, 0.4). k
Now, define • 0
• •
•
•
22-29
1}.
l
U'
= {(u1, 2 u }I
(u , u , 0.4} 12
E
U}
= {(u1 , u )I u 21
1
~
1, u /2 + u •
l
2
::s 0.6}.
2
It is easy to see that u (U') = 1, u (U') = 0.6. To find the Kalai-Smorodinsky solution for this two-dimensional UPS, we solve Max {t I (t, 0.6t) e U'} . •
This gives t = 6/11. Therefore, 2
/u' >(U') = (t, 0.6t) = (6jll, 3.6jlll -. (0.4, 0.4}. k
.
Therefore, the Kalai-Smorodinsky solution is not consistent . •
22.£.9.
(i) The utilitarian solution satisfies IUO, Pareto, and syuunetry
(Exercise 22.E.l). Since it is given by. a maximization problem it satisfies
•
IIA. The utilitarian solution requires comparing utilities across agents, so it clearly does not satisfy IUU. (ii) The solution given by f(U) = 0 TIU clearly satisfies all the properties
except Pareto . •
(iii) The solution /(U) = (u (U), 0) satisfies all properties except symmetry. 1
•
(iv) The Kalai-Smorodinsky solution satisfies IUO, Pareto, symmetry (Exercise 22.E.l), and IUU (Exercise 22.E.4), but does not satisfy IIA (Exercise 22.E.4).
(v) Take f(U,
• u ) =
f (U) - the Nash solution where the threat point is ta~en n
to be the origin regardless of
• u .
Just as the Nash solution, this solution
•
satisfies Pareto, symmetry, IUU and IIA, but it does not satisfy IUO, since now utility origins matter.
22.£.10.
Take U = {u
Observe that U'
E
E
2
IR I u
1
+
u ::s 2}, U' = {u 2
E
2
IR 1 u ::s 1, u 1
2
::s 1}.
U, but and that we can take, for example, /(U) = (0,2),
while we must have f(U') = (1,1) . •
22-30
22.E.U.
(a) Take a SPE in stationary strategies, and let v be player i's 1
continuation payoff in this equilibrium at a stage when he makes an offer. 1
Denote the players' inverse utility functions by ~ ( ·) = u~ ( • ), i = 1,2.
1
•
(Observe that t/J ( ·) is increasing and convex as the inverse of the mcreasmg •
1
•
concave function u ( · ). ) In a stationary SPE, player i will demand 1
~
(v )
1
1
every time, and his offer wi'll be accepted. Indeed, if he demanded something other than
~
(v ) and his offer were accepted, then his continuation utility
1
1
would be different from v . On the other hand, if his offer were rejected, 1
then in equilibrium it would always be rejected, and we would have v = 0, 1
which can be easily ruled out as a SPE outcome (show this). This argument implies, in particular, that player j's strategy (j•il should involve •
~ (v ). 1 I
rejecting any offer giving him less than m -
Now, in order for player j to accept player i's offer in equilibrium, he must be at least as well off doing this as he would be by rejecting i's offer ~
and demanding
J
(v) at the next stage, which would be accepted. On the other j
hand, if he was strictly better off accepting player L's offer , player i •
•
could raise his demand a little bit and player j would still accepted it. Therefore, player j should be as well off accepting player i's offer as he •
would be rejecting it and making an offer at the next stage. This can be written as u (m - t/J (v )) J
11
= c'5vJ,
or equivalently m -
~ (v ) 11
=
~
(c'5v ). J J
Substituting in turn i=l and i=2, we obtain a system of equations:
•
t/> Cv ) 1
1
+
t/> (c'5v ) = m 2
2
•
t/> (c'5v ) 1
I
+ ~ (v ) 2 2
= m
To show uniqueness of the solution to (•), we need the following lemma: Lemma: 4> (x) 1
~ (c'5x) 1
is increasing in x for x > 0.
22-31
•
•
Proof: Take x' > x > 0. We can have the following two cases: • Case (i): cS x' ?i x. Then by the Mean Value Theorem,
•
where cS x (/>'(y') I
x'
~ cS
~
~
y'
x'. From convexity of 4> ( • ) we have 1
l
(x') -
1
(1-cS)
?:
x
~
y
(/>'(y ). Therefore,
?:
4>
~
4>
I
= (1-cS) x' f/>'(y') > (1-cS) x f/>'(y')
(cSx')
x
I
¢'(y ) = 4> (x) I
I
4>
I
I
(cSx}.
Case (ii): cS x' < x. Then by the Mean Value Theorem,
4>
l
¢ (x)
(x') -
=
1
•
(x' - x) (/>~(y'),
where cS x :s y ::s cSx' < x ::s y' :s x'. From convexity of 4> ( ·} we have l
f/>'(y') I
¢'(y ). Therefore,
?:
I
•
4> ?:
(x') -
1
·
4>
I
(x)
cS (x' - x)
= (x' - x) ¢'(y') > cS (x' - x) ¢'(y') I
I
?:
(/>'(y ) = 4> (cSx') - 4> (cSx). I
1
I
This can be rewritten as 4> (x') - 4> (cSx') > 4> (x) - 4> {cSx). c I
1
1
I
Now, suppose in negation that (v', v') and (v", v") are two different 1
2
1
2
solutions to (•), and without loss of generality suppose that v" > v'. Then 1
•
I
•
each of the two equations implies that v" < v'. Subtracting the second 2
2
•
equation in (•) from the first, we obtain ¢ (v ) - ¢ (cSv ) = 4> (v ) - 4> (ov ), 11
I
1
22
2
which should hold for every solution, in particular (v', v'). But then the l
2
•
Lemma implies ¢ (v") 11
4>
(ov")
11
> 4>
(v') - ¢ (ov') = ¢ (v') - ¢ (cSv')
11
11
22
2
2
> 4>
(v") -
22
4>
~ov"), 2
z
which means that (v", v") cannot be a solution to (•) -contradiction. 1
2
(b) The utility possibility set can be written as •
•
•
22-32
•
2
U = {(v ,v ) E IR I
•
2
I tfJ
2
I
t/J 2 (v
(v ) + I·
:s m}.
)
2
The Nash bargaining solution solves max lln u
u
+
In u
1
2
).
The first-order
condition for this maximization problem can be written as u ¢'(v ) = u I
I
1
¢'(v ). Since the Nash bargaining solution has to lie on the
2
2
2
boundary of U, it can be described by the following system of equations: u t/J'(v ) = u I
'
1
1
tfJ (v )
+
1
1
t/J'(v )
2
2
• •
2
tfJ (v ) = m. 2
2
Now, we will show that the solution to (•) goes to the solution to (.. ) as 1. First, both equations in (•) converge to the second equation in (.. ).
-+)
Second, let subtracting the second equation in (•) from the first results in
tfJ (v ) - ¢ Cav ) = t/J 11
1
I
(v ) -
22
of both sides around
a=
t/J cav ). Taking the first-order Taylor expansion 2
2
•
1 yields:
means that asymptotically as satisfied. Therefore, as
(c) Let
-v
l
and v
-1
a
a )
u-a) v1 t/J'{v ) :::: (1-a) v t/J'(v ), which 1 1 2 2 2 1 the first equation in (•) becomes
> 1 the solution to (•) converges to the solution
be respectively the maximum and the minimum
continuation payoffs achieved by player C when he makes an offer in a SPE, i. = 1,2. Observe that • •
(i) Player j
~
•
i will always reject an offer leaving him less than av , smce -J
in any SPE starting next period he will get at least v . Thus, player i. will -J
not demand more than m - t/J cav ), and his continuation payoff in any SPE J -J
cannot exceed equivalently ¢
. I
(ii)
Player J
u (m - ¢ (av )). This implies that l j -J
(v
~ i
-v
I
:s u (m - ¢ (av )), or I
j
-J
) + ¢ (av ) :s m . I J -J
will always accept an offer giving him more than
in any SPE starting next period he will get at most
22-33
-v . J
avJ,
•
smce
Thus, player i can
always demand almost as much as m - t/J c~v J
any SPE cannot fall short of u (m - ""(~v 1
t/J C~v )), J
or equivalently t/J (v )
J
+
l -l
'f'
t/J (~v J
J
J
>. J
)}. J
and his continuation payoff in
This implles that v z: u (m -1
I
) z: m.
Substituting in turn i = 1 and i = 2, we obtain a system of four inequalities: •
t/J2(~v2}
t/J1(~1) +
-
z: m, ·
•
•
t~J (v
2 2
t/J
} +
{~v) ~ m, 1
t/J
+
-·
(~v
1
1
) z:
m.
We will show that these inequalities. imply
-v
1
-v
= v, -1
2
= v . Suppose in -2
negation that it is not true, and without lose of generality
-v
1
> -1 v .
· Then subtracting the first inequality from the second, we obtain
t/J
2
{~v
2
) - t/J
2
(~v
-2
) z:
t/J Cv
1 1
-v
•
) - t/J (v ) > 0, which implies 1 -1
.
2
> -2 v . Then adding up
the first and the third inequalities and subtracting the second and the fourth inequalities results in [(t/J +
(v
1
1
t/J (~v
) -
1
[(t/J2(v2)-
1
t/J
)) - (t/J (v ) 1 -1
t/J2(~v2))-
(~v ))] 1 -1
+
(¢2(~2)- t/J1(~~2))) :s 0 ·
-v
But the Lenuna proven in (a) implies that whenever
1
> -1 v and
-v
•
2
> v-2 • the
expressions in both square brackets are positive. Thus, we obtain a contradiction. Now, when we substitute
-v = v = v -v = v = v 1 -1 1' 2 -z 2
in the system (•••), the
system is reduced to {•). Therefore, the payoffs achieved in a SPE are unique.
22.F .1.
IUO allows us to normalize the threat point to be the 1
origin. The UPS in the TU case can be described as U = {u e IR I
r 1
Pareto property implies that
Lf 1
f
(U) 1
= ... = f
I
(U).
(U) =
0.
u
1\
I
Symmetry implies that
I
Therefore, we must have f (U) I
22-34
= ...
1\
= f (U) = u/I. I
~ u}.
S1
(
•
n
lnvariance to changes in utility origins: Take u' = u 1
v'(S)
= v(S)
+ a: for every i. Then in new utilities
1
I
I:
+
a: for every S
J
JES Sl
•
c I. Substituting in (•), we obtain + •
n
~
£.
a:
J
je :Snun
r
-
- v((h I n(h) < n(i)})
JE< 1
= I!
I:
-
a:J J
[v((h I n(hl :S n(i)}) - v((h I n(h) < n(i)})
(l))
+ a:
n
J=
1
1
= f Si{v) + I! reel = fSI(v) + eel.
•
n Invariance to common changes in utility units: •
u' = 13 u
Take
I
for every i. Then in new utilities v'(S) = 13 v(S)
1
for every S c I. Substituting in (•), we obtain
f
Sl
(v')
1
=
II
E l/3
vC
:S n( i)})
n
- (3 v({hl n(h)
<
13 f
n(i)})) =
Sl
(v).
•
Pareto: •
1
Sl
•
n(h)
n
<
n(i)})] =
1
1
= I! [ [v(S) - v(0)] = v(S). n Symmetry: Suppose v(S) =
v'(~-t(S) l
for all S c I, where
~-t:
I
-+
I is a permutation of
agents. Then from (•) we can write (replace h with
1
-
I! [
-
1
v'((~-t(h)
I
<
n(h)
n~-t(i)}))
=
~-t(h))
[v'({~-t(h)
I
n~-t(h) :S n~-t(i)}) -
v'({~-t(h)
I
n~-t(h)
<
n~-t(i)}))
=
n~-t(i)}))J
=
n II
I:
[v'C~-tC{h
I
n~-t(h) :S n~-t(il})l
n •
• •
22-35
-
v'C~-tC
I
n~-t(h)
<
•.
(let n' = ffll and use v'(ll(S)) = v(S)) 1 I! I:,lv({hl n'(h)
=
~ n'(i)}) -
v({hl n'(h) < n'(i)})] =
n = f
Sl
(v).
The dummy axiom: If v(Sv{l}} = v(S) for all S c I, then from (•) we obtain , Sl
=
22.F.3.
f
Sl
•
1
n r o = o. n
•
(a)
•
~
(v+v') =
n(i)}) + v'({h
I
n(h) :!> n(i)}) -
n - v({hl n(h)
<
n(i)})- v'({hl n(h)
<
n(i)})}
n(h) :£ n(i)}) - v'({h I n(h)
<
n(i}})) =
n = f
Sl
(v') +
f
Sl
(v').
(b) Suppose that the agents do not know which of the two bargaining situations
v and v' will occur, and they expect each situation to occur with probability 1/2. Linearity of a bargaining solution then implies that the agents are indifferent as to whether the solution is applied before or after uncertainty is resolved: E- f V
I
(v)
= 1/2 f (v) + 1/2 f I
(vI) 1
•
=
(by IUU) = f {1/2v) + f {1/2 v') = = f (1/2v + 1/2 v') = f I
22.F.4.
I
1
(a) Let v be a T-unanimity game, where Tci.
and each Sci we have •
22-36
I
(Ev).
Then for each t
E
I\T
v(I), TcS
=
v(Sv{i})
•
= v(S). 0 otherwise
Therefore, by the dummy axiom, f (v) = 0 for each i e I\T. By the Pareto l
property, •
r
•
fl(v)
= r
leT
= v(l) .
f."ll)
iei\T
Sincy by summetry we must have f
• f (v) for each t,j e T, we must have
l ( v)
J
f (v) = 1/ITI v(I) for each i e T. 1
•
•
(b) Let v, v' be any two TU characteristic form games on the set I of agents, and ex. be a real number. Then we can define two new TU characteristic function games on I, v+v' and o:v, so that for any Sci, (v+v')(S) (a.v )(S)
= v(S) + v'(S).
•
= cx.v(S).
It is easy to check that with addition and multiplication by a real number defined as above, the set of TV characteristic function games is a linear
•
space, which we will denote as V. Since a TU characteristic function game
v
v is described
For any Tel, define the normal T-u.nanimity game v •
T
by
•
1, TcS
0 otherwise.
We are going to show that normal unanimity games constitute a basis in V. 1
First, there are 21 I different normal unanimity games. Second, we will •
show that normal unanimity games are linearly independent, i.e. that for any 0:
.IR2 E
I
such that
L o:T Tci
v (S) = T
t
o:
1
=0
'f/S, we must have o:
1
=0
VTci.
TcS
We will show this by induction on the number of elements in T. Setting 5=0,
22-37
•
E a:T· = a:~ = 0.
we find
Further, suppose that a:
T
=0
for all Tel such that
Tc0 •
•
IT I s k < I I 1. Then, for any Sci such that IS I = k
+
1, we must have
•
a:
s
L
+
a:T
=a:
s
+0=
a: = 0,
s
TcS, IT l ::sk+l i.e. the statement is true for all subsets of I containing k+l elements. By induction, a:T = 0 'dTci, which proves linear independence of normal •
•
unanimity games. Now it is easy to prove that weak linearity of a cooperative solution f( ·) implies its linearity. Weak linearity of f( ·) means that for any
unanimity game v and any v' e V, f(v) + f(v') = f(v+v' ). Now, take instead any two characterestic function games v, v' v' e V. Let a: e IR
21
be the
coordinates of v in the basis of normal unanimity games, i.e. let
v =
[ a:
T
v . Since a: T
T
v
T
is a unanimity game for all Tci, we can use
Tci
weak linearity iteratively to obtain
Tel
Tci
v ) + f(v') = f(v) + f(v' ), T
i.e. f(•) is linear.
(c) It is straightforward to check that the Shapley value satisfies all the mentioned properties. Here we will show that any cooperative solution f( · ) which satisfies all the mentioned properties coincides with the Shapley
i IR
value. Take any characteristic function game v, and let a: e
coordinates of v in the basis of normal unanimity games, i.e .
•
•
22-38
be the
v =
r '\
VT. Using linearity,- invariance to common changes of utility units,
Tel
and the result of part (a), we can write
L fl(a.T
fl(v) =
VT) =
Tel
L a.T
L
fl(vT) =
L
a.T fl(vT) =
I
I
Tel leT
Tel
cx.,JITI.
Tel leT
Now we can show that f( ·) satisfies Definition 18.AA. 7 of the Shapley value in •
•
Appendix A to Chapter 18 of the textbook (p.681). For this purpose. we define •
IT I. Clearly. f
(v) =
1
f (l,v), and it thus remains to show I
Tesj ieT •
that f (S,v) satisfies both conditions of U8.AA.2). To see that the first l
condition (preservation of utility differences) is satisfied, take any •
S e I and any i,h e S, and observe that /
(S,v) / (5\{h},v) 1 1
=
L
L
a.T/ITI -
=
a.T/ITI
Tc:S\fh) I ieT
Tc:SI ieT
L
a.T/ITI.
TeSj f i ,h)cT
Since the last expression is symmetric in i and h, it is clear that
f 1(S,v) - f
(<=\. {h},v) =
.... ,
f h (S,v) - f h (S\{i},v), i.e. utility differences are
preserved. To see that the second (adding up) condition is satisfied, take any S e I and observe that
I: ieS
ieS
a.r/ITI =
TcSI LeT
I:
ITI a.
ITI =
r
a.T
TcS
Tc.S
= r
c..T VT(S)
= v(S).
Tel
•
•
(We have reduced the double summation to a single sum, noting that each set TcS is encountered IT I times in the double summation. The last equality follows from the defintion of a. as the vector of coordinates of v in the basis of normal unanimity games.) Therefore, f (S,v) satisfies both parts I
of (18.AA.2), and f( ·) coincides with the Shapley value as defined in Definition 18.AA.7. For the relation between this and other definitions of Shapley value, see e.g. A. Mas-Colell and S. Hart, "Potential, Value, and Consistency", Econometrica 1989, 57(3), p.589(26).
22-39
•
22.F .5.
Suppose in negation that the two lowest excesses are not equal.
(a)
Without loss of generality, suppose that e(u,{l.2}) < e(u,{l,3}) = m (u)
e(u,{2,3}) • m (u).
:!:
2
(1)
I
(The individuals could always be renumbered to obtain this.) First, we show that u + u 1
> 0.
2
Indeed, suppose in negation that u + u 1
•
-
feasibility of the utility profile we must have u definition of excesses,
3
2
= 0, then by exact
= v(l). Then using the
by (1) we must have
e(u,{l,2}) = v0,2} < e(u,{l,3}}
= v(l,3)
This implies that v(l) < v(l,3) - v(1,2)
~
- v(l).
v(1,3), which contradicts the
assumption that v(S) :s v(l) for any coalition S. Therefore, we must have either u > 0, or u I
> 0, or both.
2
When u > 0, take a new exactly feasible utility point I
u' = (u -
£,
e(u' ,{2,3})
= m I (u)
1
u , u 2
+ £),
3
-
£,
e(u' ,{1,2}) = e(u,{l,2})
m (u) -
£
:!:
1
where
1
= m 2 (u)
e(u' ,{1,3})
+ £
:s m (u), and that 1
•
< min{e(u' ,{2,3}), e(u' ,{1,3})}. If -
m (u), the above implies that 2
= e(u' ,(2,3}) = m 1(u)
m (u')
> 0 is small enough. Then we know that
£
-
£
< m (u), I
which contradicts the assumption that u' is in the nucleolus. If, instead,
m (u) -
£
1
< m (u), the above implies that 2
m (u') = e(u' ,{1,3}} = m (u) :s m (u), 1
2
m (u') 2
= e(u' ,{2,3}) = .m I (u)
I
-
£
<m
2
(u),
which again contradicts the assumption that u' is in the nucleolus. When u 2 > . 0, take a new exactly feasible utility point
u' = (u , u I
2
-
£,
u
3
+ £),
where
> 0 is small enough.
£
e(u' ,{2,3}) = m (u), e(u' ,{1,3}) = m (u) 1
e(u' ,{1,2})
2
= e(u,{l,2})
+
£
£
Then we know that
< m (u), and that 1
< min{e(u' ,(2,3}), e(u' ,{1,3})}. -
•
22-40
This
implies that m (u') = e(u' ,{2,3}) • m (u}, and 1'
1
m Cu') = e(u' ,(1,3}) = m (u) -
£
2
2
<m
2
This contradicts the assumption
(u).
that u' is in the nucleolus. Therefore, we have shown that in all cases (1) contradicts the assumption that u' is in the nucleolus. •
(b)
Suppose that the nucleolus contains t"YO utility profiles, u and u'. If u
is In the nucleolus, by definition of the nucleolus we must have m (u) 1
m (u') 1
m (u' ).
:!:
Symmetrically, if u' is in the nucleolus, we must have
1
~
m (u).
Therefore, If both utility profiles are in the nucleolus, we
I
must have •
= m 1(u' ).
m (u) 1
(2.1)
•
•
Using (2.1) fact and the fact that u is in the nucleolus, we must have m (u) :s m (u' ). Symmetrically, using (2.1) and the fact that u' is in the 2
2
nucelolus, we must have m (u') :s m (u). 2
2
Therefore, if both utility profiles
are in the nucleolus, we must also have m (u) 2
= m (u' ). 2
(2.2)
•
•
Now we are going to show that there exists a two-agent coalition S such that m (u) = e(u, 5) 1
true.
= m 1(u') = e(u',
Sl..
Suppose in negation that this is not
For definiteness, suppose that •
m (u)
= e(u,
m (u')
= e(u', {1,3}) = m (u)
1 1
{1,2}}
=
m (u') 1
1
>
e(u', {1,2})
= e{u',
>
e(u, {1,3})
= e(u, {2,3}),
{2,3}),
where the last equality in each line follows from the result of part (a) . •
Now, take u' '· = 1/2 u
+
1/2 u'.
Using linearity of the excesses in utility
profiles, and the above relations, we can write e(u' ', {1,2})
= 1/2 e(u,
{1,2}) + 1/2 e(u', {1,2}) < •
= m (u), 1
•
• • •
•
22-41
e(u'
1
= 1/2 e(u,
{1,3})
,
<
{1,3}) + 1/2 e(u', {1,3}) •
< 1/2 e(u'
m (u) + 1/2 m (u I
1
)
1
= m 1(u),
1 1/2 e{ ul {213}) + 1/2 e( u_ I {2,3}}
'1 {2,3}) =
< 1/2
<
m (u) + 1/2 m (u') = m (u). I
•
.
I
I
This implies that m (U 1
1 1
)
= max{e(u' 'I
{112}), e(u' ', {1,3}), e(u' 'I {213})} < m (u), 1
which contradicts the assumption that u is in the nucleolus.
Therefore, we
there must exist a two-agent coalition 5 such that m (u) I
=
e(u, Sl
=
m (u') I
=
•
e(u', S).
For definiteness, suppose that S = {1,2}, I.e. m (u) 1
= m 1(u') = e(ul
{1,2}) = e(u', {1,2}). •
Using the definition of excesses, this can be rewritten as u
1
+ u
2
=
U
1
1
+
1
U
{3.1)
•
2
Equation (2.2), together with the result of part (a), implies that •
m
2
= m
(u)
2
(u'}
=
e(u, {1,3})
=
e(u, {2,3}) =
•
= e(u', (1,3}) = e(u', {213}). Using the definition of excesses, these equalities imply that u
1
+
u
3
= u'
u'.
(3.2)
= ul + ul.
(3.3)
1
+
3
2
3
Equations (3.1), (3.2), and (3.3) together imply that u = u'. Therefore, the nucleolus cannot contain two different utility profiles. Suppose for definiteness that v(1,2) = v(l,3). We are going to show that
(c)
in that case u u
2
*
2
= u
3
at the nucleolus solution.
u . Define u' = (u , u , u ) 3
I
3
e(u, {1,2}) = v(1,2) - u
e(u, {1,3} l = v(l,3) - u
I I
2
- u - u
2 3
*
Suppose in negation that
u. We then have
= v(l,3) - u'
- u'
= e(u',{l,3}),
= v(l,2) - u'
- u'
= e(u' ,{1,2}),
I I
•
22-42
3
2
e(u, {2,3}}
= v(2,3)
- u
2
- u
= v(2,3)
3
= m1(u)
Therefore, we must have m (u') 1
u'
-
= e(u' ,{2,3}),
- u'
2
3
= m 2 (u).
and m (u') 2
This implies, by
definition, that u' is also in the nucleolus, which contradicts uniqueness of the nucleolus solution proven in part (b). •
(d)
= v(l,2) = v(l,3)
Agent 1 being a dummy means that v(I) - v(2,3)
= 0.
We have two cases to consider: (i) m (u) = e(u, {2,3}) at the nucleolus . 1 solution, and
m (u) -. e(u, {2,3}) at the nucleolus solution.
(ii)
1
Case (i}: m (u) 1
= e(u,
{2,3}) at the nucleolus solution. Using the result
of part (a), we must then have e(u, {1,2})
=-
=-
u - u = e(u, {1,3}) I
2
Using the fact that u + u I
2
+ u
3
1
3
= v(I), we can express u
u = u = [v(I) - u 1/2 2
u - u , which implies that u
3
1
0 as long as u
?:
1
and u
2 :!:
= u3 .
2
through u:
3
I
•
v(I).
Substituting these expressions into the formula for excesses, we obtain m (u) = e(u, {2,3}) = v(I) - u 1
- u
2
= u
3
I
?:
0,
•
e(u, {1,2}) = e(u, {1,3}) = - u - u I
= - [v(I)
2
where the inequalities hold as long as u
i?!:
1
u 1/2 < 0
+
1
~
m (u), 1
Now, if we had u > 0,
0.
1
•
we could define u' = (0, vCilj2, v(Ilj2). At this new point, we have
m (u') = 1
e(u', {2,3})) = 0
< u = m (u), I
1
•
which contradicts u being the nucleolus solution. case (ii): m (u) 1
'It
e(u, {2,3}) at the nucleolus solution. For
definiteness, suppose that m (u) = e(u, {1,3}) 3}). 1
(a), we must then have e(u, {2,3}) = v(Il - u At the same time, we know that u
1
equations, we can express u and u 2
u
2
= v(I) + u · u 1'
3
+
3
u
+ u
2
2
3
- u
Using the result of part
= e(u,
3
= v(I).
through u : 1
= -2u . 1
22-43
•
{1,2}} = - u - u .
Using these two
1
2
•
We see that we can have u
= (u 1•
u • u ) 2
3
~
0 only if u
1
= 0.
•
(e)
First, let us ·make sure that there exists an exactly feasible
utility point which euqalizes the three excesses. This point should satisfy •
the following conditions: v(l,2) - u
u
u
+
1
2
Let
• u
2
1
2
+ u
u , u • u 1
- u ·= v(l,3) - u - u
3
3
it
1
3
= v(2 , 3) - u 2 - u 3 ,
= v(I), 0.
denote the solution to the linear equations above.
Solving for
• u •
we
-
obtain: •
• u
= 1/3 [v(I) - 2v(2,3) + v(l,2) + v(l,3)],
• u
= 1/3 [v(I) - 2v(l,3)
+
v(l,2) + v(2,3)],
• u
= 1/3 {v(l) - 2v(l,2)
+
v(l,3) + v(2,3)].
1
2
3
Using the assumption that v(S) ~ v(l)/2 for any coalition S of two agents, it •
is easy to see that all components of
• ui
• u
are nonnegative. For example,
= 1/3 [v(I) - 2v(2,3) + v(l,2) + v(1,3)] ~ ~ 1/3 [v(I) - 2v(2,3) + vCil/2 + v(l)/21
=
= 2/3 [v(I) - v(2,3)) ~ 0 .
Substituting
• u
• u
into the formula for excesses, we obtain that each excess at
is equal to 1/3 [
L v(S)
- 2v(I)).
Therefore,
IS I =2
m
• (u )
1
= 1/3 [
L v(S)
- 2v(I)).
IS I =2 •
In further analysis, we will use the following formula, which is easily derived from the definition of the three excesses: m (u) + m (u) + m (u) 1
2
3
=
L v(S)
(4)
- 2v(l),
IS I =2
where m (u) stands for the smallest excess at the utility point u. 3
•
•
22-44
•
•
Now, suppose in negation• that the nucleolus solution u is different from •
• u.
•
1.e. it does not equalize the three excesses. Using the result of part
(a),
we must then have m (u} > m (u) = m (u). 1
m (u) 1
2
2
3
3
I:2 v(S)
> m (u) = m (u) = [
Using (4), this implies that
- 2v(I))/2 - m (u)j2. 1
IS I=
Using this inequality and the expression for
• m (u ) 1
obtained above, we see • •
•
that •
m (u) 1
>
1/3 [ L v{S)
- 2v(l))
IS1=2
= m
1
•
• (u ),
which contradicts the assumption that u is the nucleolus solution.
(f)
The characteristic function described satisfies v(S)
v(Il/2 = 3 for all
i'!::
two-agent coalitions S. By the result of part (e), this. implies that the •
equalizes the three excesses, and it is given
nucleolus solution of this
• u
by the point
• u
=
l
computed above.
• 4/3. u 2
=
• u 3
Substituting the numbers, we obtain
= 7/3.
To compute the Shapley value, observe that the three agents can enter a room in six different orderings.
For example, let us focus on agent 1. There are
two orderings in which he comes first, and in those orderings he contributes • •
v(l) - 0 = 0.
There are two orderings in which he comes last, in which case
•
he contributes v(I) - v(2,3) = 1. There are two orderings in which he comes second.
When he comes after agent 2, he contributes v(l,2) - v(2) = 4.
When he comes after agent 3, he contributes v(l,J) - v(3) = 4.
Weighting
each ordering with the probability one-sixth, we obtain the Shapley value of agent 1
as
his average contribution over orderings:
us = (2/6). 0 + {2/61 ·1 + 0/6). 4 + Cl/6). 4 = 5/3. 1
In the same manner, we can compute the Shapley values of the two other agents, which are us = us = 13/6. 2
3
22-45
Comparing to the nucleolus solution, we see that the Shapley value is more equitable: u
s 1
= 5/3 > u
•
= 4/3, which means that agent 1, whose
1
contributions are smaller than those of agents 2 and 3, is punished less in the Shapley value than he is in the nucleolus.
(g)
By definition of the core, an exactly feasible utility profile u is in
-
•
the core if and only if u + u
2
1
u u
1
2
u
+
3
+ u
3
•
i'!::
v(l,2),
i'!::
v(l,3),
i'!::
v(2,3).
(5)
If the core is non-empty, we can add up those inequalities to obtain 2v( I)
[ v(S).
i'!::
•
(6)
•
lSI =2
Suppose in negation that u, the nucleolus solution, is not in the core, i.e. at least one of the inequalities (5) is not satisfied.
For definiteness,
•
suppose that u + u < v(1,2). 1
2
m (u)
i'!::
1
•
This implies that
•
e(u,{l,2}) = v(l,2) - u + u 1
2
We are now going to construct an exactly feasible
> 0. utility profile u·' such
that m {u' ) :s 0 < m (u), which will contradict the assumption that u is the I
1
nucleolus solution .
•
First, let us see if the point
• u
which equalizes the
•
three excesses is fit for this role. (e).
This point has been computed in part
We only need to check under what conditions the point gives non-negative
utility levels to all agents. Using the expressions for
• u
from part (e) and
the inequality (6), we see that
• u 1
= 1{3 [v(I) +
[ v(S) - 3v(2,3))
i'!::
IS I :2
= 1/2
L v(S)
1/3 [3/2
[ v(S) - 3v(2,3)) = IS I : 2
- v(2,3).
IS I =2
Similarly, we obtain that
•
22-46
• u
i'!::
z
I:
1/2_
v(S) -
vU~3),
v(S) -
vU~2).
lSI =2
• u
i'!::
3
I:
1/2
IS I =2
I:
Now, as long as v(T) :s 1/2
v(S) for every two-agent coalition T, the
IS I =2
utility profile obtain m m
• u •
• (u ),
we can use (4):
1
• (u )
= 1/3
1
which equalizes the three excesses, is non-negative.
• (m (u ) 1
by the inequality (6).
+ m
To
•
2
• (u )
+ m
Therefore, m
1
3
• (u·))
I:
=1/3 [
v(S) - 2v(1)] :s 0
IS 1=2
• (u )
:s
0
< m 1(u), which contradicts
the assumption that u is the nucleolus solution.
I:
We are left to consider the case where v(T) > 1/2
v(S) for some
IS I =2 •
two-agent coalition T.
I:
Suppose for definiteness that v(1 1 2) > 1/2
v(S).
IS I =2 •
•
Let us now find the exactly feasible utility profile u' such that e(u' 1
u;
= 0 and
{1,3}) = e(u', {2,3}). Solving a system of two linear equations with two
unknowns (u'. u' ), we obtain 1
u~
u;
2
= lf2 [v(I) + v(1,3) - v(2,3)], •
= 1/2 [v(I) + v(213) - v(ll3)].
Now we can compute the three excesses at u'. We obtain: e(u' ,{1,3}) = e(u' 1{213})
= lj2· [
L v(S)
- v(l,2) - v(I)] <
IS I =2
L v(S)
< 1/2 [1/2
- v(I)] :s 0
IS I =2
(we have used the assumption that v(l,2) > 1/2
L v(S),
and the inequality
IS I =2
(6)). Also, e(u ,{1,2}) = v(l,2) - v(l) :s 0.
Putting the above inequalities
1
•
together, we see that m (u') = Max e(u' 1
1
S) :s 0 < m (u), which contradicts 1
ISI=2
the assumption that u is the nucleolus solution.
22.F.6.
(a) The first-best price equals to marginal cost: • •
•
•
22-47
• p
=
• c'(q ) .
But since c( ·) is concave, average cost
• c'(q ).
marginal cost:
• • c(q )/q
>
•
Therefore, the firm's profits at the first-best price are negative:
• • p q
-
• • c(q ) = [c'{q )
-
• • • c(q )lq J q
< 0.
•
(See Figure 22.F.6(a) for an illustration. The average cost (AC) at a point on •
the cost curve is the slope of the line connecting the point with the origin. The marginal cost (MC) at this point is the slope of the tangent to the curve at this point.
The average cost always exceeds the marginal cost for a convex
cost cn!'ve.) •
•
22.F. On the other hand, If costs are covered and p have S' (q
0
)
=p
0
> c'(q
0
),
0
0
0
• c(q )/q , then we
which implies that production is socially
suboptimal. (See Figure 22.F.6(b)). •
S'(
I I
AC
MC
q
Fi 22-48
·
(b) The second-best welfare problem can be written as max S(q) - c(q)
s. t. S'(q)q
?:
c(q).
If the constraint were not binding, we would obtain the first-best solution, •
but from part (a) we know that then the costs would not be covered contradiction. Therefore, the • constraint Is binding, and the second-best quantity is given by S'(q)q = c(q), i.e. price equals average cost .
•
(c) We can interpret every unit of output as a "project", with c(q) describing •
how much it costs to run q projects at once. The Pareto property of the Shapley value in this context means that the costs have to be fully covered, while the symmetry property implies that the price for every "project" should • be the same. Thus, the Shapley value suggests that every unit of output should be priced at the average cost.
22.F.7.
(a) The second-best problem can be written as S (q ) + S Cq ) - c(q ) - c(q )
max
q 1 ,q2
1
2
1
s.t. S'(q ) q 11
2
1
+ S'(q ) q
I
22
2
•
2
- c(q ) - c(q ) 1
2
?:
0.
If we. denote the Lagrange rimltiplier with the constraint by >.,
•
the first-order conditions obtained by differentiation can be written as (1+>.) S'(q) - (l+A.) c'(q) + >.S'(q) q 1
I
1
1
I
I
= 0,
= 1,2.
i
S"( q ) q
If we denote the elasticities of demand by
£
I
(q) = I
-
1
I
1
the first-order conditions can be rewritten in the form •
(S:(q
) 1
c'(q ))/S;(q 1
) 1
=
a.fc 1(q1)
for some
Thie coincides with the Ramsey taxation formula (22.8.1 ). •
22-49
a. > 0.
•
•
(b) By symmetry of the problem, the Shapley value cost allocation should •
allocate the same value (c ) to all projects of type 1, and the same value • 1
•
(c ) 2
to all projects
of type 2. Take a unit project of type 1, and take a
random ordering of the large number of small unit projects. Suppose that the proportion t of all the projects precede the given project. Since by the Law of
I
.arge Numbers with a very high probability our project will be preceded by
an almost perfect sample of all projects, therefore the output of all preceding projects will be (tq
, 1
tq/ The contribution of our project to the
costs will therefore be c' (tq ). Since every t is equally likely, the Shapley 1
value for type 1 projects is obtained by averaging: 1
1
1
c' (tq ) d(tq ) = c(q ljq .
c' (tq ) dt = Cljq )
c =
•
1
0
I
1
0
1
l
I
Similarly, the Shapley value for type 2 projects is
•
c2 = c(q1)/ql.
(For a rigorous derivation of Shapley value with a continuum of agents, see
R. Aumann, "Value of Markets with a Continuum of Traders", Econometrica 1975, 43, pp .. 611-646)
(c) If outputs are priced according to their Shapley values obtained in part (b), this results in average cost pricing for each output. •
production levels
(q , q 1.
2
The resulting
) will be determined by the intersection of demand
curves and average cost curves:
Clearly, production is lower than the first-best levels, as long as costs are strictlty concave and average costs exceed marginal costs. In general, this is also different from the second-best allocation obtained in part (a).
Indeed,
under average cost pricing, production level for each output is entirely determined by demand and cost functions for this output, and there is no
•
22-50
•
cross-subsidization.
In
contr~st,
under Ramsey pricing in obtained in (a),
production level for each output is in general determined by demand elasticities for both outputs (which enter from the break-even condition defining «}, and in general we should expectcross-subsidization. For example, if demand for good 1 is perfectly inelastic, under the Ranasey pricing good 2 is priced at marginal cost and subsidized from the non-distortionary mark-up on good 1.
22-51
23
23.1.1.
In (a) and (d) agent 2 is indifferent between truth telling
or·lying, and therefore is willing to tell the truth. In (b) and (c) agent 2 strictly prefers truth telling. However, in (e) the agent.always 0
•
prefers to lie because
23.1.2.
z~ (8')y,
2
Agent 2's problem is:
which is equivalent to:
and
y~ (8')z.
2
·
max
...
2
' 2-
max A
•
Taking the FOC (note that the SOC is satisfied) we get
This result
•
is similar to the understatement in the first-price auction in example 23.b.4. (see footnote 6). •
23. B. 3.
Showing this is basically following the argument at the end of
example 23.B.4.
Agent i can choose bi(8i)-aili where aie[O,l]. Assume all •
Agent i is then indifferent between a e 1 .
, 1 ] 'i
(lower values of a.L will give him zero utility).
If there exists j s.t.
ajlj>8i then agent i is indifferent between a e[O,l]. 1
weakly dominant strategy .
•
23 - 1
Therefore, a.-1 is a L
23.B.4.
(a). Normalize the seller's (agent 1) and buyer's (agent 2)
no-trade utility levels to zero.
The parties'
payoffs as a function of •
•
' 2-
To find a Bayesian equilibrium with
•
first note that a seller of type 1 •
1
will bid b
to maximize•
1
his expected payoff:
max
bl+a2+J32 1 2
Eu1 -
2
-
1
· 11
I
a2+J32I2>bl
(bl-Q2)/J32
the FOC yields (the SOC is satisfied): •
(i) •
Similarly, the buyer maximizes her expected payoff:
(b2 -Ql) /,61 8 2
0
the FOC yields (the SOC is satisfied):
•
(ii)
The two equations (i) and (ii) above yield: a Trade occurs if and only if
1
1
2
--3
--4
.
b (1 )
strategies, trade occurs if and only if
1
+
1
4
<
s2
. The transfer in case
of trade is function is not Pareto optimal because
9 < 9 does not guarantee trade, and 2 1
efficient trade may not always occur .
• • •
23 - 2
(b) The social choice rule above has
and •
•
•
Ve will check if it is
truthfully
impleme~table
in a
direct revelation mechanism,
that truth-telling is a Bayesian Nash equil
i.e.,
in the direct •
•
mechanism.
The seller,
assuming that the buyer reveals herself •
truthfully, solves:
... ...
•
the FOC yields (the SOC is satisfied):
-
1 -4
1
-
- -6 - ' 1
- 0
... which implies that 1 -1 . 1
1
Similarly, the buyer solves: •
A
A
1
max
- -]dl 6 1
A
•
0
the FOC yields (the SOC is satisfied): 1
'2 -
-6 -
1 ... 1 -(1 - -) - 0 3 2 4
•
A
which implies t:hat 9 -8 . 2 2
23.C.l.
Therefore truth telling is an equilibrium.
Assume the preference reversal property is satisfied for all i,
'i·
9j:0 and 9 _ , and assume in negation that f( ·) is not truthfully implementable 1 in dominant strategies.
That is, there exists i,
'i· 'i·
and '-i' such that
but this contradicts the preference reversal property, therefore, f( ·) must ·be truthfully implementable in dominant strategies.
23 - 3
2
23.C.2.
for which x 1>x 2 , 'i' The majority voting social choice •
•
function is defined by {let x
be chosen as a tie-breaker):
1
f(l)•
•
It is easy to see that if agents are asked to vote one way or another (i.e., •
a "direct mechanism") then no agent has an incentive to lie no matter what his preferences, or the other agent's announcements are. In cases where an agent is pivotal then he will strictly prefer to tell the truth (telling the truth will cause his preferred outcome to be chosen while lying will •
cause the other outcome to be chosen). he is indifferent.
In all other cases (no influence) then
Therefore, £(·) is implementable in dominant strategies . •
23.C.3.
Let
~.-'P 1
for all i, £(·) be ex post efficient, and assume in
negation that there exists all i, then there exists and for all i.
iex
-1-(1-
1
such that for all le8, f(l)~i.
, ... ,1
1
)ee_
such that
Since ~ -'P for
1
for all xex,
This, however, contradicts£(·) being ex post efficient,
therefore f(8)-X . •
23.C.4.
Assume that f:8 -+ X is truthfully implementable in dominant
strategies when the set of possible types is {by definition) that for all i and for
al~
e1 for i-l, ... I.
This implies
s1eei,ui(f(li,l-i),li) >
u 1 (f(li·'-i),l 1 ) for all 9{E8i and all B_iee·i·
This implies that for all i,
A
all l_iee·i' which in turn implies the desired result.
23.C.S.
Define
F( ·)
to be the pairwise majority voting social welfare
23 - 4
functional (note that single-peaked domain ensures that this functional is rational).
Observe
that
_satisfies
F( ·)
non-negative
responsiveness.
0 0
Therefore, if we define f(·) to be the'social choice •
tion that selects a
Condorcet winner, i.e., the maximizer ofF(·), we can apply Exercise 23.C.7 below to see that f(•) is truthfully
le in dominant strategies. •
• 0
23.C.6. and
(a)
Let X-{x,y) and let there be two type profiles 1'-(li•····li)
IN-(Ii·····li)
where all agents are indifferent between x andy no
which type profile occurs.
Let f(l' )-x: and f(lw)-y.
example for f(·) satisfying IPM and not being monotonic.
tter
This is a trivial •
•
•
preferences are:
••
•
•
z
w
X
X
y
X
X
y
w
z
y w z
y w z
0
and
The
only
two
preference changes for which the "if" part of the monotonicity definition is satisfied •
are the change from (li,12) to (1i,l2)• and from (1i,12) to
(li,li>·
Indeed,
the "then" part of the definition is also satisfied for these two changes. For all other preference changes, the "if" part of the definition is not •
satisfied, therefore the definition is vacuously satisfied. f(·) is monotonic.
Now observe that
conclude that
f(91,92) E L (f(Bi,82),1i>· 1
IPM is not satisfied. (c)
~e
Therefore, •
From Proposition 23.C.2 we know that£(·) satisfies IPM if and only if •
it is truthfully implementable in dominant strategies.
Now consider two
• 0
profiles of types, 9 and 6', such that
Li(f(l),li) C Li{f(6),8i) for all i •
23 - 5
•
e~ist
(if no such two profiles
in
•
e
then f(·) is vacuously monotonic).
We
•
•
need to show that £(8')-f(l), and this can be done by following the proof of •
1 in the proof of Proposition 23.C.3.
Claim (d)
Suppose in negation that f(·) does not satisfy IPK, i.e., there exists
'i·'f.· and '-i such that
•
that
f(l)~f(li·'-i)
Since ~i-7' i this implies
.f(l)llLi(f(lf.·'-iLif.). •
and that
•
•
•
•
ui(f(l),lf.) > ui(f(lf.,l_i),lf.>· Consider the strict preference
'i
such that f(l) is at the top, f(lf.·'-i) is •
second, and all other alternatives are below these two in some arbitrary order.
This construction implies that Li(f(l),li) c Li(£(1),11), therefore •
monotonicity
implies
that
f(8i·'-i)-f(l).
Note,
however,
that
this
construction also implies that Li(f(lf.·'-i),lf.) c Li(f(lf.·'-i),li), therefore monotonicity implies that
f(81,1. 1)-f(8f.·'-i)' a contradiction .
•
•
23.C.7. is
single-valued.
responsiveness,
'We want to show that if
F( ·)
satisfies non-negative
then f( ·) is implementable in dominant strategies. By the
Revelation Principle, it suffices to check the direct revelation mechanism, i.e., to show that for all i: •
Suppose
in
negation
that
there
exists
an
agent
who
can
gain
by
A
misrepresenting, that is, there exists iei, 9-(8i,B-i)' and 8i, such that izes F( ·), and is single valued, so we have: f(B)•x
implies
x
Fp(~i(l ),~_
1
A
A
1
(8_
1
))
y
(i)
•
Now we apply non-negative responsiveness of F( ·) to (ii), by noticing that when we move from {li,8-i) to I, y can only rise relative to x for agent i's
23 - 6
preferences, and does not move for the other agents. y Fp (~i (6 i_) ·~·i (6 ·i))
which contradicts {i) above.
Thus we must have:
X
Therefore, no agent can gain by misrepresenting
his preferences.
23.C.8.
(a)
It is easy to check that we cannot improve upon f(·) for one •
•
•
agent without hurting the other, therefore f(·) is ex post efficient. It is again easy to check that given the preferences, E{·) satisfies the
(b)
property identified in Proposition 23.C.2. (c)
Truth·telling is not the unique (weakly) dominant strategy · another one •
is for each agent i to always announce 6 -61, which causes outcome a to be 1
implemented.
The agents are indifferent between this strategy and truth· •
telling because with preferences (6i,62) the agents are indifferent between a and b, and for all other preferences,
alterr~tive
a would have been
•
implemented anyway. •
\le begin by showing that if f( ·) is truthfully implementable in
23.C.9
dominant strategies then both conditions are satisfied.
Actually, the second
condition was shown to be satisfied in section 23. C,
and is exactly the
condition in equation (23.C.l3).
ak(6)
as i
~ ~
0·
Therefore, we only
Let 9-(8 , ... ,6 ) and take 6i_>6 . 1
1
1
optimal, we must have: •
(i) and (ii) imply: •
23 - 7
need to show that
Because truth-telling is always
(iii)
let 'i•1 +dli, which for 11 dli implies that (ignoring higher order 1 dk- Bk(l)·d9' and we can rewrite (iii) as ) 88
i'
i
av1 (k(l),lt) 8k(l) . . dl
av 1 (k(l),li) Bk(l)
it.
•
• dl i •
• •
which is equivalent to: Bk(l) ·--·dli a1i
-
~
(iv)
0
•
but since 11>1 , and we ass•wed that 1
•
•
brackets in (iv) is strictly positive. (recall
that dli>O) we must have
Therefore, for (iv) to be satisfied
8k(l) Bl i
~
0
co.
•
We •Fe lefc to show the converse, i.e., that both conditions imply that£(·) is truthfully implementable in dominant strategies.
As
in negation that •
there exists an agent i, such that given types 1-(li,l-i), agent i strictly prefers lying and announcing that he is of type. I i.
Define
agent i' s
A
•
utility, given types are (li,l-i) but he announces lias: A
•
A
•
A
Our negation ass A
'
'
•
•
au(81,r)
Bv(k(r,l_i),li) Bk(r,l_i)
-......;~-ddr-
----~__...;;;.._ •
ae(r,l_i) +
ak
ar
dr> 0
(v)
ar
'
' A
From the first condition, i.e., k(8) is increasing in •
' '
8i' we know that
8k(r,8 _ ) 1
~
0, and from the ass
23 - 8
tion
we know that: •
8v(k{r,6_i),r)
av(k(r,8_i),6i) for all
- - - - - - - l!::
ak
(vJ.)
Bk
•
using (v) and (vi) we get: A
6
•
Bv(k(r,l_i),r) 8k(r,6_ )
8e(r,6_ ) 1 + _ _ _..;;:..._ dr - 0 Br
1
-------·---·~__;;;;-
Br
Bk 6
•
because the bracketed tera equals zero for all
(see equation (23.C.l2)) •
...
This, however, contradicts our negation as
that u(li,Bi)-u(li,6i)>O
so£(·) must be truthfully implementable. A
~Suppose
(vi)
6i
•
We can proceed as before, however the inequality in
above will be reversed,
and we will have a
sign before the
integral, so we will get the same contradiction . •
23.C.l0
· At the end of the first paragraph insert:·
[
"Assume throughout that conditions are such that necessary
condition
for
{k*(·)
•
,el (·), ••• ,e(·)I)
implementable in dominant strategies."
holding is a
(23.C.8)
to
be
truthfully
Also, in the second line of part c)
•
insert the word "implementable" before
"ex post efficient social choice
function".] a) $uff1ciency: Suppose.that we can write V*(6) - LiVi(6_i).
Consider the
transfer functions of the form + h. (6 i) • 1.
-
where for all i, •
•
h {6 .) - -(I i ·L
for all 6 -i .
By proposition 23.C.4, {k*(·),e (·), ... ,e(·) ) is truthfully implementable in 1 1 23 - 9
dominant strategies.
Moreover, for all I we have,
•
•
j~i
i
i
- (I - l)V*(I) - (I - 1)L V (I i
i
-i
) - 0
• Suppose (k*(·),t 1 (·), ••. ,t(·)I) .is ex post efficient and is •
truthfully implementable in dominant strategies.
Since (23.C.8) is necessary •
(by as
tion)
for truthful
tion,
this means that there exist
I
functions (hi(8_ 1 )Ji-l such that (I- l)V*(I) +
r
hi(l-1)-
i
I .r
vj(k*(l),lj) j"'i
i
+
r i
hi(l 1) -
-I ei<s> - o i
•
But this implies that by defining •
L Vi(l_i)
we can then write V*(l) -
i
•
b)
·
If vi(k,6 1 ) - 1 1k-
1
2
k
2
for all i, then, k*(l) - Argmaxk(Lili)k-
r '
for all 6, and so the FOC implies
V*(l)
-
3
Ii'i
6 Ii i
-
ri
3
1 ri9i
. Hence, 2
6 1 l1 i
'i - 2
~3~
- (61+ 62+ 63)[61+ 62+ 83 1
~
2
2
2
- 2C.t.i 9 i)
2
- (81 + '2 + 83 + 29182 + 28183 + 28293) . •
We now define, Vl(l2,l3)-
,2 + ,2 2 3 2
6 21 + 2 3 '
,2 + ,2 3 1 9 28 V2(61,93) 2 + 1 3 ' 23 - 10
•
•
•
•
•
0
•
and the result then follows from part a) above since V*(l) - Vl(l2,l3) + V2(ll,l3) + V3(ll,l2) . 0
1
c)
then clearly
a V*CI> al ···al 1
I
- 0. •
BV* all 2 a v* as 1a1 2
avl
-
Bk + ak
2
ak avl a1 + Bl ' 1
1
2 a v2
2
a v2 + 2 2 ak ak
a vl
-
av2
2 _a vl
ak ak ak ak • • + a1 1 as 2 + BkBI 2 a1 1 akal as · 1 2
Since, •
we have, •
- -
,
•• •
which in turu implies that
- -
"' 0 •
thus proving the statement . •
23.C.ll and as
Let agent 1' s Bernoulli utility function be •
ui (vi (k,li) +
in negation that Proposition 23.C.4 no longer holds.
-m
+ t 1 1 That is,
A
there exists i, 1 , 1 1 , and l_i such that: 1 A
A
Substituting from (23.C.8) we get: A
A
• 0
Given ui(·) > 0 this implies that: I
,..
l
I
vj(k*(li,l-i)'lj) >
j-1
l
vj(k*(B),Ij)
j-1
a contradiction to k*(·) satisfying (23.C.7).
Thus, £(·) must be truthfully •
implementable in dominant strategies . •
23.D.l
(a) The set of alternatives X are all pairs (k,y) where
k-1,2, ... is the period in which trade occurs, and yeR is the transfer from the buyer to the seller (the price). (b)
•
s-o
It is clear that when 1-9, the seller will not want to announce that
since then he will receive a net payment of (-4)
ately.
The relevant
6k(9.5 - 0) ~ 5, i.e., that a
incentive constraint is therefore
will not misrepresent given that trade will occur after k periods. this yields the minimum k required: k
23.D.2 . (a)
l!::
s-o
type
Solving
[ln(l0/19)]/ln(cS).
Letting the seller be agent l, it is optimal to trade if and
only if 1 >8 , so we must have: 2 1
and of course, v (k)•8 1
(and zero otherwise).
0
if
8 s8 1 2
1
if
91>82
if k-0 (and zero otherwise) whereas v (k)•9 if k-1 1 2 2 From equations (23.0.8) and (23.D.9)we can derive the
transfer functions for each agent. - E
For the seller:
82
[
s2
23 - 12
- -E •
'1
[ '1
I s 1~s 2 1
2
1
-
'2 - 1
-I s1ds 1 -
2
'2
and since the optimal transfer for agent 1 is the sum of these two
we
have:
•
•
• •
, it is clear that:
By
•
(b)
We will show that for each agent 1,
truth-telling is optimal in
expectations given the belief that agent j"'i is always truth-telling. The problem for the seller is: •
Max
"
•
which reduces to: •
Max
(i)
.
which yields the FOC (the SOC is satisfied): 1
"
-e -o 1 1
• •
For the buyer, the problem is: •
Max ,.. A
which symmetrically gives us the FOC: 1 -1 2
23.D.3.
-o. 2
It is easily seen that the conditions for the Revenue Equivalence
Theorem are satisfied for the two auction settings.
We will now verify that
•
both examples 23. B. 5 and 23. B. 6 yield the same expected revenue to the •
seller.
In the sealed-bid first-price auction we have: •
•
1 1 11 -·Prob( 1 ~1 ) 2 1 2
0 0
1
1
-
1
't
ldl2dll 0
2
--
2
'
0
6
•
0
• •
•
for player 2 we get that E[-e (1)] - 1/6 as 2 Now consider the sealed-bid second-price auction:
Since the problem is well.
1 1
0 0
•
1 1
-
-6
0
•
0
tric for player 2 we get that E[-e (1)] - 1/6 as 2
Since the problem is s well.
0
Therefore, both the first and second price auctions yield an expected
revenue of 1/3 for the seller {for a general fotmula with any I and with valuations in [0,1] see exercise 23.D.6).
23.D.4.
(a)
'i
Let
<
'1·
let b' and b• be agenc i's bids associated in negation that
with his two types, and as
b' >b-.
Denote by
K'
bids
and
and
•
the
probabilities
respectively. that
(I'
i
i
'
• •
and similarly,
~
winning
the
auction
given
lity (using a revealed-preference ar
By op
- b' )'If'
of
(I'
i
- b"')K"'
i
which implies:
K' I' - b"' i i (i) I' - b' 'If"' i i i!:. (I"' - b' ) 'If' which implies: (I"'i - b"')'lf"' i i i •
23 - 14
b' i
b"'
t) we have
i
«'
I" - b"' i i
(11)
I"' - b' i
i
•
(i) and (ii) imply (after some simple algebra) that •
li>Cbi - bi> ~
o
that I"' - I' > 0 and b" - b' < 0 i i i . i
which is a contradiction to our in negation that there is a s
(b)
<11 -
tric equilibri
,1~).
with (I' •
such that b*(l)-b for all le(l' ,1"').
Let «>0 denote the (strictly positive) •
probability that all agents' types are in this interval (it this
case
all
agents
get
the
good with equal
t be that in
probability because
the
• •
equil
is
).
Let pi(l ) be the probability that agent i gets 1
the good if" he bids b*(l i). deviates and bids
-b +
t
> b.
Now as For
that for all eie(l' ,1"') agent i , the loss is pi(li)·t, while the
t •
gain is that he will get the good with probability 1 in the event that all types are in (I' ,1"'), and will increase the probability of getting the good in all other events. Ignoring the second gain, the total change is then at least: •
•
So for small enough
t
this deviation is beneficial.
Therefore, we must have
b*(li) strictly increasing. (c)
'We need to verify that the conditions for the Revenue Equivalence
Theorem are satisfied. positive density.
First, all valuations are drawn from an interval with
Second, for the lowest valuation each agent gets the good
with probability zero so the expected utility is zero. Finally, since b*(l ) 1 •
is strictly .increasing,
then the highest valuation agent gets the good with
probability one and all others get nothing.
These conditions are equivalent
to those that would be satisfied for a sealed-bid second-price auction, and therefore the Revenue Equivalence Theorem holds. •
23 - 15
•
23.D.5.
'1
Let
<
'i•
let bi and bi be agent i's bids associated •
with his two types, and assume in negation that the
tr"'
probabilities
respectively.
'
of
winning
b' > b"'
Note that
the
bl > bt.
auction
implies that
a revealed-preference argument) we have that
•
•'
given ~
Denote by
1r'
bids
and
b' i
and b"' i
•"'· By optimality (using ~
6'1f' - b' i i
I
i 1r"'
-
bt which
•
implies:
b"' i
Similarly,
6"'1r" -
i
b"' ~ i
6'(1r"'- •')
-·b'~
i
(i)
i
'iff' - bi
which implies:
'i(1f"'- tr') ~- bi- bi
(i) and (ii) imply that
('Ill -
', )( ,.....
i
i
have a contradiction to our this holds with
-
(ii)
ff' )
ion chat 9"' i
ity this implies that
If this is strict, we
0.
'i
> 0 and
1r" -
~
1r'
0.
If
bi - bt - 0, a contradiction to •
our as
bt - bi > 0.
ion that
Now to show that the bid must be strictly increasing in the type's valuation, an almost identical argument to that in part (b) of exercise 23.D.4 can be •
used (see above).
Therefore, the argument of part (c) of exercise 23.D.4
applies here as well and this concludes the answer . •
23.D.6.
Going to stand in line t hours before 9:00am is equivalent to •
•
biding a monetary amount of Pt dollars (ignoring the cost of the ticket which is assumed to be captured in 6).
Since this is equivalent to a sealed-
bid first-price auction, we know from the Revenue Equivalence Theorem that the expected revenue from the auction (i.e., the expected highest bid) is equal to that of a sealed-bid second-price auction.
Therefore, we need to
•
calculate
the
expected
straightforward to see valuation
is
second-highest valuation of that
I· (I-1) ·F(t)
the
agents.
It
is
the density function of the second highest
I-2
second highest-valuation is:
·f(t) • (1-F(t) J, •
•
23 - 16
and
therefore
the
expected
1 •
t·t
E[b] - I(I·l) •
1-2
·l·(l·t) _dt- I(I·l)
t
•
I+l
I+l
•
0
indivi~sa1
E[t] •
Therefore,
if fJ doubles,
I-1
-
I+l
0 t-b/~,
Given a monetary bid •of b, the respective waiting time is expected waiting time of the first
1
so that the
is:
I·l JJ(I+1)
•
the waiting time goes down by 50',
and if I
doubles, the waiting time goes up but this increase diminishes to zero as I goes to infinity. •
From the solution to Exercise 23.0.2, we know that the seller will
23 .E.l.
•
tell the truth,
and his expected utility will be given by the function
(this is (i) from the solution to 23.0.2, and
taking 1 -1 ): 1 1
1 1
•
•
+-
1 +-
2
6
0
•
,2 Therefore, the seller will gain from not participating if is, if
, 1" > 1 -
2/3
•
1
-1+, that '>1 2 6
Note that since the buyer's reservation utility is
0, he will always be better off participating under the assumption that the seller is always participating.
23.E.2
(a)
The Myerson-Satterthwaite Theorem
tells us that there is no
•
Bayesian incentive compatible (BIC)
and interim individually rational (IIR)
social choice function (SCF). We know that if a SCF is incentive compatible (DSIC), then it is BIC.
dominant strategy
Therefore, a SCF which is not
BIC, is not OSIC. This concludes that there is no SCF that is OSIC and IIR. •
•
23 - 17
•
We know that if a SCF is ex post individually rational, then it is IIR . •
(b)
•
•
Therefore, a SCF which is not IIR is not ex post individually rational. This, •
with the Myerson-Satterthwaite Theorem concludes that there is no SCF which is BIC and ex post individually rational . •
23.E.3 •
•
implies that k(l,l )-k(3,4)•1 and k(3,2)-o (where k-1 denotes trade and k-0
2
no trade).
we
efficiency.
Now set
Now for individual rationality we must have the following:
t (1,2)
1
tl(l,62)
1
t2(11,2)
tl(3,62)
3
t2(81,4)
-
t (1,4)
1
-
2,
t
1
(3,2)
trically set
-
0,
-2 ~
-4
and
This example gives ns
the
required result, which concludes that without a strictly positive density function (i.e., with discrete values) the Myerson-Satterthwaite Theorem may •
not hold.
23.E.4 .
Since both agents are risk neutral, we can interpret this
•
game in the following way: If trade occurs at date t, this is equivalent to t-1 trade· occurring with probability 6 due to the discounting.
Then, trade
occurring in the first period is equivalent, in the interpretation model, to trade occurring with probability one.
But the Myerson-Satterthwaite theorem
exactly tells us that in this setting we cannot have trade ocuuring with •
probability one.
23.E.S.
(a)
•
The expost efficient trading rule will have all the highest
valuation agents owning a unit of the.good.
i.e., we may have only buyers,
only sellers or a mixture of both ending up with the good, as long as there
23 - 18
~ I*
is soae I* such that all agents with I others do not.
The total
have one unit of the good and all
of good is given by the cont
(b)
Define the following "competitive" social choice function as follows:
Let q
and q denote market supply and demand, and be defined as: 0 8
qD -
.
- - !2 '2 -,2 - p
for p ~ !2 for
for p
0
The market equilibr the efficient
!2
0
for p ~ !1 for !1 s p s 0
~ p
2:::
qs -
s j2
-'2
p -
!1 -' !1 1
for p ~
-'1
;
1
.
price p* will cause efficient trade, and will lead to
outcome
described in
(a)
above.
There will
be no
need
for announcements, so that is is incentive compatible (vacuously) and it is individually rational since a b':lyer will buy if and only if p:SI , and a 2 seller will sell if and only if
~e
1
.
This example concludes that for a
continuum of buyers and sellers the Myerson- Satterthwaite Theorem no longer holds.
23.!.6.
• •
(a)
The efficient trading rule is such that if
sell the good to i for a price (b)
1
and b
then j should
pe[9i,8j).
Let bi denote agent i's bid.
the relative values of b
li~lj
2
The utilities of the agents, conditional on are: bl2::b2
Restricting the bids to linear bids, biexpected utility:
•
23 - 19
Q
+ /Ji I i, agent 1 maximizes his 1
•
1
0
Using Leibnicz's rule we obtain the FOC: •
• •
1
Odl2 - 0 0
which yields (after some simple algebra):
(i)
•
(ii)
3 •
(i) and (ii) imply that a
1
a
-o 2
and therefore bi-
s1
for i-1,2 .
•
(c)
The social choice function that is implemented is: 8i>Bj implies that i
buys the good from j at price
e1 .
Bayesian incentive compatible, individually rational.
The analysis in (b) above shows that it is and it is clearly ex post
The result differs
efficient and
from the Myerson-Satterthwaite
Theorem because of the symmetry of the agents - each agent is a buyer and a seller so there is always an efficient trade . •
23.E.7
(a) •
•
•
Let y(B ,e ) be the probability of trade and t(8 ,e ) be the 1 2 1 2 •
payment from the buyer to the seller given that they announced (B 23 - 20
,e ). 1 2
ex
pose efficiency entails that y(B 1 ,B )-1 if 1 <1 , and y(B ,s )-o otherwise . 2 1 2 1 2 •
Denote the expected payment that the announces
'1
~eller
anticipates to get given that he
by:
For incentive compatability on behalf of the seller we must have that truth •
telling solves :
s ·Prob(li>l ) 2 ,, 1
Max
•
+
-t (B' ) 1 1
1
Given that y(B ,1 ) is ex pose efficient, the objective function is just
1
2
11-9 1
to be a solution we must have the following FOC:
dt 1 (Bl)
for all
e1
•
Integrating this we get: 1
1
•
Note that from interim IR, we must have
t 1 (l)t!:O, otherwise the seller can •
the good, getting a utility of 1. Using this, the integral
leave and -
above yields: e <1 )
1
1
~
1
1 2
2- 2' 1 .
Therefore, the interim expected utility of
the seller, given that truth telling is optimal (IC), is given by: 2
Similarly, denote the expected payment that the buyer anticipates to pay given that he announces
92
by:
-e2(B2> •
For incentive compatability on behalf of the seller we must have that truth telling solves :
•
•
Max 9' 2
•
Given that y(l , 8 ) is ex post efficient, the objective function is just 1 2
23 - 21
Note that from interim IR, we must have
-t
2
(0):SO, otherwise the buyer can
leave and get a utility of 0. Integrating the FOC from 0 to 1 , and using this 2 1 2 interim IR condition yields: t 2 <1 2 ) ~ 2' 2 · Therefore, the interim expected •
utility of the buyer, given that truth telling is opt
1 (IC), is given
by:
Adding the interim expected utilities, we get that the ex ante sum of expected utilities is: 5
--6
:!!:!+ 2
•
0
Remark:
The result above can be
derived using inequality
(23.E.4).
The
minimum total expected utility for the two agents should be the expected utility of the seller wit:hout: trade, plus the minimum •
expected
gains
from
trade.
The
latter
can
be
calculated
by
rearranging (23.E.4) as follows: 1 1
•
•
•
0 0
0 0 •
The right hand side of this inequality is the expected gains from trade, and solving the left hand side with our parameters gives
1
3· 1
Adding the seller's expected utility without trade, which is 2' •
gJ.ves us
5
6
as calculated above . •
23 - 22
(b)
If the ·true observations are (1 ,1 ), the total utility that can be 1 2 • •
produced is at· most overcome.
Kax(l
, 1 ), even if all info 1 2
ional
ies are
Thus, the ex anee total expected surplus can be at most: •
1 2
--3 0 •
Therefore, no SCF can have the
23.F.l
of expected utilities exceeding
2
. 3
We prove the statement in three steps: ~
•
If E( ·) -
(k( ·), t ( ·), .•. , ti( •)) is ex post classically efficient 1
then it is ex-ante classically efficient. j'roof: Suppose not, A
i.e. , there exists a feasible social choice function
A
A
(SCF) E(·) - (k(·),t (·), ..• ,ti(•)) such that , 1 A
(i) •
with strict inequality for at least one i.
This implies that
•
A
A
E1 [L 1 t 1 (1)) - E1 [Liti(l)] ~ E1 [vi(k(l),li)] - E1 [vi(k{l),8i)].
(ii)
From ex post efficiency of £( ·) we must have that the right hand side of (ii) A
is non-negative, and that E [Liti(l)] - 0. 1
However,
feasibility of £( ·)
A
implies that E [Liti(9)] ~ 0, a contradiction. 1 Claim 2: If£(·) - (k(·),t (·). ... ,ti(·)) is ex ante classically efficient 1 and £( ·)
·E
F* then it is ex-ante incentive efficient.
Proof: By definition {see Definition 23.F.l and the preceding paragraph). 3; If £( ·) - {k( ·), t ( ·), ... , t ( ·)) is ex ante incentive efficient 1 1 then it is interim incentive efficient. Proof: his is just restating Proposition 23.F.l. •
•
23 - 23
2.3.P.2
think of the buyer as
th9 a&ent,
principal, in the seetlng of example 2l.F.l. u (1,•,t) - lv(x) • 1
and a higher con
t
and the aonopolist as tbe
The agent's utility is glven by
(so that a higher I results in a higher utillty level ucility
~a~ginal
-
latter
the
The sellers utility is given by
is
•etngle crossing•
the
u (B,x,t:)- t - e•x. 0
the revelation principal we can concentrate on a direct
mecbants~
Using
(x(f),t(l))
which solves the sellers proble•: Max x(l) .t(B)
c·~(l))
E[c(l) •
(X(f),t(l)) is Bayeaian ve eompatibh and indivf.dually rational.
s.t.
We now follow the analysis of linear utility 1n section 23.D [with k-x, and v(l)•v(x(l))).
u1 (1)-
Lcttin&
fv(x(l}) • t(l) denote the utility of type
from tluth-telling, proposition 23.0.2 implies that the principal's
I
problem h
equivalent to (recall that
V' ( • )>0
so that conatToi.nt (i) ean be
•
11fr1 tten a• •hown below):
Kex
•
E{lv(x(B)) - U(B) • cx(6)J
x(l) ,U(I) s.t.
(i) X(·) is non-dacrea£1ng
(11) U(l) • U{B)
j
•
(iii)U(S)
~
' + f ,•
•
for all Be(e• ,IJ
v(x(s}}ds
for all te(e,i)
0
-·
Following the same steps as in example 23.F.l the problem
beco~es:
-
' Max
rev(x(l))
·
cx(l)]~(8)dl
- U(l)
-
x(ll') ,U(e)
' •
s.t.
(i) x(·) is non-deereas1ng (ii) U(l)
-
~
0
2J • 24
'Clurly~ruu w111 blnd -.c a solution. and wt:ing integradon by pares the •
proble• becoaes: •
,.- .
·~':
... .
-I
l'i.;
••
.
.
••
.'
• • "
••••
Kax
v
.
v(x(B))
- c.x(f) #(l)dl
x(l)
'•
x(•) is non-decreasing
s.t.
Isnortng the eonatra1nt. and ass.,qtn.g an lnt:erlor solution (see in chapter 23)
w•
footno~:e
22
•
get that the function x(l) which solves the problem
~st
satisfy the FOC:
' .
1 · ~(I)
• c • 0
~(9)
for all fE(I,I] -
To see that the solution x(·) 1s indeed non-decreasing, the same argument as ln example 23.F.l applbs hen as well.
Note that the hi&hesc: •
ve get
val~tion
'Itle soc is sac1sf1ed since v•(·)
consumer h set at the
f1rst·b~st
level since
•
v' (x(B) )I - c - 0, but all other eonsUJHr types are distorted (this
•
v1ll
We
23.F.3.
follow
the
an~lysls
particular the result of equation (23.F.9).
!-o,
there edsts I such th;:,.c J i (
lgpleaent
~:he
''1 )<0
of
exa~plc
23.F.2,
and
in
FrODJ symmetry and the fact that
for all l.e{l,f). l. -
Therefore,
Ye
~an
r•sult of (2l.F.9) using a second-price sealed-bid suction
with a reserve price of I.
In the general ease,
f~r
sell•r can then offer
each bidder 1 there exists 1
such that J (6l)
auction where each agent l 1• restricted to submit
A
bids greater than
't·
and the •sent which gene~ates the highest Jl(li) wins
"
the anetian.
If agent 1 von the auction. ve calculate th• value
-t
1
such thot •
.
' ' " ' ..
•
13 • 2S
A
generates the second. highest J( ·) sene rated), and agent 1 pays NaxU , i L 1 1
on thls •ee Bullow and
(For
23.F.4.
Robe~t•
(1989]).
The analysis of example 23.F.2. applie• here es well, yet the --
objeetive funetion in (23.F.8) changes and
b•co~s: .
•
•
•
•
1
'1 .
•
+
'o
1 n - 1 <• 1 >Jds 1 ···dl 1
1
i-1
~1 and therefore, (2J.F.9) becomes:
if
yl(l) - 1 •
•
and
q3.F.S.
Letting p Cx ) 1 1
(a)
•
d~note
agent i's inverse des.nd function, the
monopolist' s· probleat b: •
Using Kuhn-Tucker, the FOes are:
•
Therefore, depending of the x
(b)
1
on
the deMand f1.1.n4:Uon.s we will have one, both, or none·
pos 1 tive, and there sum baing lass than one.
The results here resemble
~ose
of example 2J.F.2.
The monopolist sets • ••
priees
(quantities)
tl)
aa.ximlu hll payoff.
Interpreting q1.1antltbs as •
•
prohab111ties, and prices es valuations, tha results here are ai•ilar to the •
23 • 26
•
(1-). fle C1le •
ttw fta. tt - p•.-(p) +
~.
'!Mit.. •tC.,I) •
11"1 ~ t:H
•
•
1lil
Y
(k) + 1
't•
PrOf!INildoa 23 •.D.2 tel!. u
lf - ' only f)) ••
(l)
+ ·1. _.
v.1 Ca.l) •
Wtala
. . [p(l),!(l)).
[,(f)w%(1)} U
(Ul
11·-t
if·~
• ••• ' p(l) ...
u<•> .. • • , v1 c.,. •
t
••J•t
t:o u(l)
a
0 for 1111 ' ' . . •
(f(l),u(l)).
-
a'b.v.. tlnu a(l) - f(l) ·ia(f(l)) the objectl.,.
... -<•>• • (u(l) + lx(p(l))
J(f)
• p(l)a(p(l))l
... • ( f ) f(l)
1
-
u(l)~(l)dl
-
Substituting, we want to
u(l).(ll
' 1 -
-
-
'
1
-
u'(l).(l)dl
' - u(l) + x(p(l)).(l)dl . 1 choose [p(l),u(l)] to solve,
•
-
- (1-a)u(i) •
s.t.
u(i) ~ 0
(i)
(which implies u(l) ~ 0 for all I)
(ii) p(8) is non-decreasing. Start by setting
u(f)
- 0, and let's ignore constraint (ii).
Taking the FOC
w.r.t. p(l), we get, -x(p(f)) + x(p(l)) + (p(l) - l)x' (p(B)) rearranging and dividing by x' (p(l)) we get, p(l) - ' + So,
•(8)
~(I)
nondecreasing and
{1-Q)
•<•> (1-a)~(B)
.
(*)
> 0 implies that constraint
satisfied, and (*) is the regulator's optimal price rule. want p(8) -
9.
(ii)
If a - 1 then we
If, however, a > 1, then we have no solution:
for any
transfer function e{·), we can increase "welfare" by raising it to e(·) + for
E
is
E
> 0.
23.F.7.
This question is analyzed in Dana-Spier (1994).
Following the
analysis in section 2 of the paper, one can derive the following "vir •
welfare function" corresponding to a monopoly awarded to f paper for
this
1 (see the
definition) : (1-c) 8
2
+
{1-c) 8
2
- '
for i-1,2
and the virtual welfare function associated with a duopoly is:
23 - 28
1
2
F(Bl) - (l-1) f(l ) 1
1 2 ri(B)-Max(l¥ (1) ,r.r (6) ,l'ld(l)),
The goverument awards a monopoly to firm i if .
d
~_2
1
for i-1,2
d
and a duopoly right if W (6)-Kax(l'l (l),r.r-(1),1'1 (9)).
23.F.8.
For both IL and 18 , the buyer
does.
s the good more than the seller
If y(l)
utility than the seller looses, and we can therefore increase the transfer e( 6) to (more than) compensate the seller for his loss so both agents are better off.
23.F.9.
Therefore,
(a) Ye can rewrite (23.F.l5) and (23.F.l6) as follows: t t
H H
Both the above yield that (b)
any ex post classically efficient social choice
-
tL
40(yH - yL)
(23.F.l5)
-
tL s 20(yH - yL)
(23.F.l6)
~
20(yH - yL)
~
(tH - tL)
~
40(y8 - yL)' which implies
t~tL.
ya--
ex post efficiency implies that yL-y -1. 8
This, with (23.F.l3) imply
•
that tL
~
~
tH 40.
40.
This in turn, together with ytL (from (i) above) imply that
Using these conclusions, we could evaluate the expected utility of
the buyer: Ell·
so that
2
-
.2(50 -
(23.F~l4)
~)
+ .8(30 - tL)
is violated.
s
.2(50 - 40) + .8(30 • 40)
(23.F.l6) together with
-6
Therefore, no feasible social choice function
can be ex post efficient. (c)
s
y~O
implies that:
tL - 20yL ~ tH - 20yh ~ tH - 40yH . •
23 - 29
23.F.10.
The set of interim efficient social choice .functions are those
that solve (23.F.l7) for some u ~
o· and ulH (the change is that we no longer 2
require u
:W o > . 1
The logic leading to 23. F .19 and Figure 23. F .l in the
textbook is unchanged from the logic in the textbook.
-u
Now, however, because
H can take on negative values the set of (yH,tH) pairs in interim incentive 1
efficient social choice functions
is the standard set in Figure 23. F .10
below:
1 The
boundary
at y -1 H
coaes
froa
the
feaaibility
requirement
that
Following, again, the same sort of logic as in the textbook (and of Figure 23.F.3 there),
the set of (yH,tH)
in ex ante incentive efficient social
choice functions is the heavily traced boundary at yh-1 in figure 23.F.l0 above.
Thus,
incentive
when trade
efficient
is not voluntary for
social
choice
functions
the seller, have
trade
all ex ante occurring
with
probability one.
23.AA.l.
Consider as a mechanism the following game where player 1 chooses
rows, 2 chooses columns, and the mechanism chooses the resulting outcome in the matrix:
23 - 30
player 2 1 m r
u player 1
b
a
c
C a
a
d
c
d
e
B
'i then U is his u~ique type 'i then C is his unique player 2's type is 'i then 1 is when she is type 'i then m is her
It is easy to check that if player l's type is •
.
(weakly) dominant strategy, and when he is (weakly) dominant strategy.
Similarly, if
her unique (weakly) dominant strategy, and unique (weakly) dominant strategy.
23.AA.2.
The answer to the first part is no.
This follows from the fact
that every possible valuation function arises from 8i for all i, so that for •
all aSR, and every 8i' there exists a '1 such that vi(kn,l1) - v 1 (k0 ,1i) +a. Therefore,
these two valuation functions, vi(kn,l1) and vi(kn,9i) not only
rank the projects ordinally the same, but measure the differences in utility between projects identically.
This implies that k * (li,l -·i)
which in turn implies that ti(8i,9-i) - ti(Bi,B-i) so agent i is indifferent between announcing Bi or the shifted '1· This
is
the
only
type
of modification
that
would
keep
the
agent
indifferent, implying that the answer to the second art of the question is no.
If
the
agent
"anchored" to zero,
chooses
to misrepresent,
since
the
value
of
k0
is
then any misrepresentation must involve changing the
monetary values of the differences between some alternatives, which might change the decision of the optimal project, which in turn will change the utility of the deviant agent.
therefore, truth-telling will be the unique
(weakly) dominant strategy.
23 - 31
23.BB.l. described,
1 form game associated with the extensive
(a) In the no
agent 1 has three strategies,
(Ly,Lz,cy,cz}
(where "aP" means
•announce a, and if challenged choose p•) and agent 2 has two strategies: (A,C} for "Accept" or •Reject".
and extensive f
The
games can be
depicted as shown in the following figure: 2
A Ly 1
Lz cy cz
X X X X
L L
C X X
1
L L
X
2
L
A
c c
X
z
c
z
y
It is easy to check that (cz,A) and (La,C), ae(y,z} are both Nash equilibria game, and therefore, any strategy profile which
outcoaes in the normal
supports these outcoaes will be a Nash equilibri (b) Looking at the extensive form of the
•
and using backward induction
for each type profile, it is easy to check that the only strategy profile which is a sub game perfect equilibrium is:
23.81.2. equilibr
Clearly,
Ly
if 1 -L
c
cz
if
e1-c
A
1
0
any equilibrium
in dominant
strategies
, so the answer to the first part is "yes".
is
a
Nash
If the dominant
strategy equilibrium is strict (i.e., not weakly dominant) then clearly we would have strong implementation. d0111inant equilibrium is weak.
However, this
is not the case if the
Consider the following example:
Two agents, 1 and 2, with the following preferences:
23 - 32
b-e a
and the social choice function (SCF) is f(1i•'2)-a. f(6i,62)•c, and f(6i,62)f(6i,12)-b. dominant
It is easy to check that this SCF is implementable· in ·(weakly) However,
strategies.
to
see
that
it
announces herself to be a 6i type.
'i
not
strongly. Nash
of the direct mechanism:
implementable, the following is a Nash equilibri agent 1 always announces hiaself to be a
is
type,
and agent 2 always
This is a NE that always implements
outcome b, no matter the types, and therefore it does not implement£(·).
~
23.88.3.
Property (i) must be satisfied.
· By Proposition 23.BB.l. if f(·) is Nash implementable the f(·)
Let 6-(li,l-i) be such that k * (6)-1, and let l'•{li•'-i) satisfy
monotonic.
* Similarly, if k {6)-0, and
* 61 > 6i, so monotonicity implies that k {9')-1. ) satisfies 81 1. - i
6'-(6'
is
6
s 8i, then
*
k -(9'
)-0.
Now
in negation that
•
property (i) didn't hold, e.g., w.l.o.g. assume that ti(8f.'6-i) > t 1 (6i,6-i). Then when agent i's type is 61, and all others' are 6-i' then agent i would prefer to misrepresent and announce 6i instead, contradicting that f(·) is Nash implementable. ~
P~onf:
Therefore, we must have ti(81,6-i)- ti(6i,6-i).
Property (ii) need not be satisfied.
Consider a SCF that is implementable in Strong Nash equilib
proposition 23.BB.l such a SCF must satisfy monotonicity. if k * (6'.,6 i) 1
However,
-
this
1 and k * ( ei. , 6 _ 1 ) -
imposes
no
•
By
This implies that
0 then we must have that 61 > 6 i .
r~strictions
on the
change
in
the vector
of
Therefore, any change will be compatible with monotonicity, and we need only to satisfy incentive compatibility which can be satisfied in many ways.
23 - 33
u (1,x,t) - lv(x) - t (so that a higher I results in a higher utility leve 1
and a higher marginal condition).
utility
-
the
latter
The sellers utility is given by
is
the
•single
u (1,x, t) 0
t
-
crossing•
c·x •
Using
the revelation principal we can concentrate on a direct mechanism (x(l),t(l)) which solves the sellers problem:
Max
E[t(l) - c·x(l)]
x(l),t(l) s.t.
(x(l),t(l)) is Bayesian incentive compatible and individually rational.
Ye now follow the analysis of linear utility in section 23.D
-v(f)-v(x(l))].
Letting
u1 (1)
-
[with~.
and
lv(x(l)) - t(l) denote the utility of type
I from truth-telling, proposition 23.D.2 implies that the principal's
problem is equivalent to (recall that Y(·)>O so that constraint (i) can be written as shown below): Max x(l) ,U(I) s.t.
E[lv(x(l)) - U(l) - cx(l)] (i) x(·) is non-decreasing 9
(ii) U(O) - U(!) +
(iii)U(I) ~ 0
J v{x(s))ds '-
for all ee[9,i]
-
for all le[e,i]
Following the same steps as in example 23.F.l the problem becomes:
-9 Max x(9) ,U(O)
- ex ( e) ]; (·I ) de
[6v(x(9))
's.t.
(i) x(·) is non-decreasing (ii) U(l)
-
?;
0
-
u(e- )
23.F.6.
Let x- (p,T) where p is the price and T is the
revenue given
•
to the firm, T- p•x(p) + t. Then, ui(x,l)- 1[-x(p)] + T, and using the notation on page 887 of the textbook we obtain that u (x, I) - vi (k) + ti, 1
Which in tULb is equal tO Vi(k). a)
Consider a mechanism as [p(l) ,T(I)].
Then Proposition 23.D.2 tells us
that [p(B),T(I)] is implementable if and only if, (i)
-x(p(f)) is non-decreasing, i.e., p(l) is non-decreasing,
(ii) u(l)- u(!) +
' vi(s)ds 1 8
- u(l) -
-
-
b)
u(l) +
1
x(p(s))ds
' x(p(s))ds
1
The regulator wants to choose [p(B),T(I)], or equivalencly, [p(l),u(B)],
to maximize,
' 'subject to u(8) above.
~
0 for all 8,
and che two conditions given in part a)
Since u(9) - T(l) -9x(p(l)) the objective function can be rewritten
as,
' 9 ' - 9 •
x(s)ds - (u(8) + lx(p(l}) - p(9)x(p(8))} p(l) x(s)ds p(9)
+ au(l) ;(l)d9
- [p(l) -l]x(p(l)) - (1 - a)u(l) ;(l)dl .
Using integration by parts we have,
