The Causes, Costs, and Compensations of Inflation: An Investigation of Three Problems in Monetary Theory

The Causes, Costs and Compensations of Inflation In memory of Jenny Milne (1963–2005) The Causes, Costs and Compens...

Author: William Oliver Coleman

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The Causes, Costs and Compensations of Inflation

In memory of Jenny Milne (1963–2005)

The Causes, Costs and Compensations of Inflation An Investigation of Three Problems in Monetary Theory

William Oliver Coleman Australian National University, Australia

Edward Elgar Cheltenham, UK • Northampton, MA, USA

© William Oliver Coleman 2007 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical or photocopying, recording, or otherwise without the prior permission of the publisher. Published by Edward Elgar Publishing Limited Glensanda House Montpellier Parade Cheltenham Glos GL50 1UA UK Edward Elgar Publishing, Inc. William Pratt House 9 Dewey Court Northampton Massachusetts 01060 USA A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data Coleman, William Oliver, 1959– The causes, costs, and compensations of inflation : an investigation of three problems in monetary theory / by William Oliver Coleman. p. cm. Includes bibliographical references and index. 1. Money. 2. Inflation (Finance) 3. Quantity theory of money. 4. Risk I. Title. HG220.A2C65 2006 332.4´1—dc22 2006009125

ISBN 978 1 84542 484 8 (cased) Printed and bound in Great Britain by MPG Books Ltd, Bodmin, Cornwall

Contents vi vii

Acknowledgements Notation PART 1. PRELIMINARIES

1

1. A statement of the problem

3

PART 2. INFLATION IN A RISK FREE WORLD 2. 3. 4. 5.

9

The theory of the demand for money A theory of the supply of money The Quantity Theory of Money Inﬂation without a quantity of money: the Wicksellian approach

11 35 54

PART 3. INFLATION IN A RISKY WORLD

111

6. 7. 8. 9.

Technological risk and the social function of real debt Monetary risk and the social function of money debt The Quantity Theory in a risky world Wicksellianism in a risky world

86

113 138 160 185

PART 4. THE COST OF INFLATION

201

10. The cost of inﬂation as the cost of moneylessness 11. The cost of inﬂation as the cost of creditlessness 12. A summarization of results

203 225 251

References Index

254 257

v

Acknowledgements My thanks are due to the seminar of the School of Economics of the Australian National University, and especially to the comments of Ben Smith, Farshid Vahid and Timo Henckel. I also wish to thank Fiona Khoo and Akiho Sahara for carefully reading the manuscript.

vi

Notation B C D E H I K L M N P Q R S T U W X Y Z SL b c d eh h i_ i m n p r sh

real bonds consumption nominal bonds expectation operator, or endowment point nominal money demand investment capital labour outside money inside money money price level number of liquidations revision operator seigniorage time horizon utility wealth expenditure national product the total cost of inside money total shoe leather costs beneﬁt of money marginal social cost of public funds supplement to money endowment, in real terms elasticity of money demand to the nominal rate of return on capital real money demand the nominal interest rate benchmark interest rate quantity of real outside money quantity of real inside money probability real interest rate semi-elasticity of money demand to the nominal rate of return on capital vii

viii

v w x z h K L

Notation

rate of within period redemption of inside money real wage real residual demand for outside money cost of inside money per unit of money co-eﬃcient of money demand to the nominal rate of return on capital rate of time preference co-eﬃcient of reaction of the interest rate to the price level co-eﬃcient of reaction of the interest rate to the proﬁt rate implicit yield on money nominal rate of return on capital rate of growth of outside money semi-elasticity of the supply of inside money to the nominal rate of return on capital rate of inﬂation rate of proﬁt elasticity of technical substitution semi-elasticity of residual demand for outside money to the nominal rate of return on capital normalised deviation of marginal utility from its expectation proﬁt risk premium inﬂation risk premium marginal cost of inside money supplement to money endowment, in nominal terms costs of liquidations proﬁt bill wages bill

A starred variable indicates a ‘permanent’ measure of the variable.

PART 1

Preliminaries

1.

A statement of the problem

This study is concerned with three questions: ● ● ●

Why is there inﬂation? Why is inﬂation costly? Why is inﬂation beneﬁcial?

These questions have been exhaustively theorized and ‘answered’ many times. Yet all three questions remain open today. The ﬁrst question will appear closer than the others to a solution. There exists a theory of inﬂation that is built from mainstay economic principles, and has received repeated seeming empirical conﬁrmation: the Quantity Theory of Money. This Theory – that explains inﬂation as the reduction in the value of money that must take place if an increase in the money supply is to be demanded – had been successively and successfully sophisticated over several intellectual generations, and appeared to receive a clinching veriﬁcation during the Great Inﬂation of 1969–1983. It was at the height of that episode, during the 1970s, that the quantity of money took on the appearance of a golden magnitude of intellectual illumination. It was an unmatched resource of explanation. It explained nominal variables. But it also explained real variables. It explained goods market variables; labour market variables; and balance of payments variables. To know it, was to know (almost) all. This frame of mind soon passed. Almost at the moment of its apparent triumph over inﬂation, monetarism’s footing faltered. The chief solvent of the conﬁdence in monetarism was the increasingly manifest endogeneity of the money supply. The very attempts of monetarist policy to control the money supply had brought out that the market component of money (bank deposits) was a slippery, intractable, and contrary magnitude. Still more importantly, the experience of targeting money increased the awareness that central banks supplied money on demand, and could imagine no other way of conducting themselves. The upshot was that the money supply was now seen to be endogenous, not exogenous. And as an endogenous variable the money supply was something to be explained by inﬂation, rather than to explain inﬂation. 3

4

Preliminaries

That the Quantity Theory is inadequate as an explanation of inﬂation – or at least seen to be inadequate – has been received recent acknowledgement across a wide spectrum of economic opinion. It has been correctly observed that the LM curve – the relation that captures the equality of the demand and supply of money – has evaporated from thinking about macroeconomics. The LM curve no longer plays any role in serious analysis, having been supplanted by the assumption that the central bank controls the short-term nominal interest rate. (Alan S. Blinder 1997)1

Lucas and co-authors draw the implications of the Blinder type position for the Quantity Theory very plainly. A consensus has emerged among practitioners that the instrument of monetary policy ought to be the short-term interest rate, that policy should be focused on the control of inﬂation, and inﬂation can be reduced by increasing short-term interest rates. At the center of this consensus is a rejection of the Quantity Theory. (Lucas et al. 2001)

The Quantity Theory is dismissed, and, in most received recent academic treatments of inﬂation the money supply is (once again) almost invisible (Woodford 2002). But perhaps the most eloquent testimony to the demise of the Quantity Theory lies not in an 800 page book but in a stark, two line statement of the Federal Reserve of the United States of America. ‘On March 23, 2006, the Board of Governors of the Federal Reserve System will cease publication of the M3 monetary aggregate.’ (US Federal Reserve, 10 November 2005)

To any spectator of the battle over inﬂation in the 1970s, to traverse the ﬁeld again after 30 years is to walk in a landscape of giant, fallen statues. But what might be raised to do service in the place of the Quantity Theory? One approach to answering this need has been to take the stem of the Quantity Theory, and from it breed a brave, new progeny. Thus from the notion of monetary equilibrium – the root the Quantity of the Theory – has sprung the Fiscal Theory of the Price Level, in which the quantity of money is essentially irrelevant. An opposite response has been to completely discard any suggestion that inﬂation is a consequence of monetary equilibrium. In this spirit, there was an attempt at reviving the Phillips Curve (for example Akerlof et al. (2000)). This attempt was not, however, happily timed, and had the task of explaining away why ‘the long-lasting boom . . . during the nineties . . . did not lead to signiﬁcant inﬂation rates despite repeated record lows of unemployment’.2

A statement of the problem

5

A third response is to seek to determine inﬂation by reference to central bank policy over nominal interest rates. Wicksellian ideas have been revived here. In these Wicksellian models the price level is determined by the interaction of the return on capital with some rule that governs the nominal rate of interest at which the Central Bank supplies money. The present study considers neo-Wicksellianism the most meritorious approach of the three. But the neo-Wicksellian approach remains underdeveloped. It is yet to be assimilated outside of readers of academic journals. It is uninvestigated empirically. It has been little used in historical investigations of inﬂation. The impression is that the neo-Wicksellian approach remains recherché, and as if conﬁned behind glass. If the Quantity Theory was massive, granite landmark in the economist’s intellectual landscape, the Wicksellian approach is yet to be more than a shimmering hologram. The second question – why is inﬂation costly? – is swathed in a denser mystery. Core principles of economics suggest that in equilibrium inﬂation is neutral. In other words, inﬂation has no cost, despite the near-universal impression to the contrary. Monetary theorists are ready to evade this puzzling appearance of costlessness by arguing that inﬂation does have two costs even in equilibrium: Shoe-Leather Costs and Menu Costs. But these are surely weak grounds for expecting inﬂation to be costly. It can be argued the ﬁrst cost is trivial in magnitude for low to moderate inﬂation. And the second is, surely, ultra-trivial. Ingenuity can contrive many models of inﬂation with a social cost even in equilibrium, but ingenuity is rarely beholden to one side of any debate. These dexterous exercises would be more convincing if one could be satisﬁed that the same outlay of ingenuity could not produce models which demonstrate beneﬁts from inﬂation in equilibrium. The answer to the second question, is made no easier by the need to ask a third question – why is inﬂation (sometimes) beneﬁcial? For inﬂation is sometimes (apparently) beneﬁcial. The amount of inﬂation the world has experienced suggests an unwillingness to eliminate it. The speciﬁc prospect of a disinﬂation programme is treated coldly by the public, apprehensive about the impact of disinﬂation on unemployment. This bespeaks of some nominal rigidity, that in turn betokens some money illusion; phenomena are normally neglected or discounted in every other economic investigation. To summarize: ● ● ●

There is inﬂation, but no one is quite sure how. Inﬂation is usually bad, but no one really knows why. Inﬂation is sometimes good, but no one has a confident answer when.

6

Preliminaries

What this study does is to raise some possible answers to these questions; answers that are suggested by some general neoclassical principles of equilibrium and optimization; answers, in other words, drawn from a well-deﬁned stream of thought. Our contribution to the ﬁrst question – why is there inﬂation? – is twofold: First, we refurbish the Quantity Theory of Money. A theory of market money is advanced which is squarely based on optimization, and which does not compromise the Theory. Overall, the posture is defensive. The refurbishment is intended to repel ‘horizontalists’, Liquidity Traps, and the Fiscal Theory of the Price Level. The second contribution to the theory of inﬂation is to advance a simple and tractable neo-Wicksellian theory of inﬂation, in which the value of money adjusts to make the demand price of credit equal the supply price of credit, rather than the demand for money equal to the supply of money. Although this theory is strong where the Quantity Theory falters – the endogeneity of money – the posture with which the neo-Wicksellian theory is advanced is not aggressive to ‘monetarism’. For although neoWicksellianism totally discards the equilibrium of money demand and supply as an explanatory force, inﬂation in the Wicksellian theory remains very much a ‘monetary phenomenon’. Indeed, the present study shows there to exist a distinct parallelism between Wicksellian theory and the Quantity Theory; with a clear-cut correspondence between the categories of the two approaches. Thus the relation of the neo-Wicksellian and Quantity Theory is better seen as a matter ‘translation’ rather than contradiction. The contribution to the second question – why is inﬂation costly? – begins with an act of destruction: we contend that Shoe Leather costs is an insuﬃcient account of the costliness of inﬂation. We argue the costs that arise from inﬂation’s disturbance of the demand for money are supplemented by the costs that arise from inﬂation’s disturbance to the supply of money. But we conclude over all that the money market does not warrant any reliable conclusion about the cost of inﬂation. We then advance a new theory of the costliness of inﬂation. It is argued that unpredictable inﬂation damages the social function of debt markets – the eﬃcient sharing of risk. Risky inﬂation reduces welfare as there is less transfer of risk from locations where it is most painful, to locations where it is less painful. With respect to the third question – the beneﬁts of inﬂation – the present study does not turn to nominal rigidities, or any form of disequilibrium. Our answer springs from our analysis of the costs of risky inﬂation. The analysis of inﬂation and debt markets that was used to show unpredictable inﬂation is costly, is now extended to show that inﬂation sets up a process

A statement of the problem

7

by which wealth is redistributed in a manner that leaves it more equally divided. This redistribution also removes the social cost of inﬂation. Thus unpredictable inﬂation ultimately yields more equality, and at no social cost once the equality has been achieved. This may be construed as a beneﬁt.

NOTES 1. Blinder adds ‘It is high time we changed our teaching in this way too’. 2. ‘The Phillips Curve Revisited: Conference Announcement’, CEPR Conference, June 2003.

PART 2

Inﬂation in a risk free world Part 2 advances two apparently rival theories of the value of money: a ‘quantity’ of money theory, (the Quantity Theory, or monetarism), and a ‘price of money’ theory (that can be called Wicksellian). Despite their diﬀerences, a signiﬁcant homology between the two theories is demonstrated. Throughout the Part perfect foresight and general equilibrium are assumed.

2. The theory of the demand for money The Quantity Theory of Money seeks to explain the value of money by means of the most fundamental and successful of all economic models: the model of supply and demand. As the economist explains the value of oil by reference to the demand and supply of oil, so the Quantity Theory proposes to explain the value of money by reference to money’s supply and demand. The Quantity Theory’s undertaking may appear unpromising on account of the lack of any palpable monetary parallel of, say, the oil market. Two leading tasks for the Quantity Theorist have been, therefore, to explicate the notion of ‘the demand for money’, and ‘the supply of money’. In accordance with this imperative, this chapter develops and advances a theory of money demand. This chapter’s theory is built from one of three paradigms that may be used as a foundation for a theory of money demand. A theory might be founded upon consumer theory, and treat money as a consumer durable of the utility-maximizing household. Alternatively, a theory might be founded on production theory, so that money is interpreted as an input of ﬁrms. Third, a theory of money demand might be founded on ﬁnance theory, so that money is regarded as a ﬁnancial asset of wealth managers. Money, however, does not yield a tangible income, as ﬁnancial assets do. And the complete arbitrariness of its physical aspect would appear to preclude its interpretation as a factor of production. This chapter’s theory, then, interprets money as a consumer durable. The interpretation of money as a consumer durable requires some articulation and rationalization of the ‘utility’ that money ostensibly supplies, and we identify that utility with the reduction in liquidation costs that higher money balances achieve. The theory therefore amounts to an attempt to span the gap between ‘money utility in function’ approaches to money demand (Samuelson 1947, pp. 117–22; Patinkin 1965), and ‘inventory theoretic’ approaches to money demand (Baumol 1952). The resulting theory is well within the tradition of well known transactions theories of the demand for money, and its grosser contours are indistinguishable from those of longstanding and familiar investigations of 11

12

Inﬂation in a risk free world

money demand. But, however lacking in conspicuous innovations, some of the more precise variations in outline developed here are less well known, and will prove crucial in later chapters.

THE MARGINAL BENEFIT OF MONEY Our point of departure is the proposition that the demand for money rests on money being necessary for exchange. Whatever exchange may be done eﬃciently without money (by means of bilateral and multilateral trade, or credit granted by the seller to the buyer), a signiﬁcant fraction of spending requires money. This fraction we take to be given, and we will assume it is one. It is easily seen, however, that the necessity to hold money in order to spend cannot explain why more than a vanishingly small amount of money is ever held. For while a given ﬂow of spending may be executed by running down over a long period of time a large sum of money previously acquired, and thereby holding (on average) a large amount of money, the same ﬂow may also be executed by repeatedly acquiring small amounts of money moments before spending, and so holding (on average) a small amount. Or, to pursue the thought, it may also be executed by continually acquiring amounts of money only an instant before spending, and so hold (on average) a vanishingly small amount. The reason why more than a vanishingly small amount of money is ever held lies in the costliness of ‘acquiring’ money: the costliness of transforming (or ‘liquidating’) non-money assets into money. The costliness of liquidation means that the fewer the liquidations, the less the cost from liquidations. And since the more money held, the less call to liquidate, holding more money brings a beneﬁt in the form of a reduction in liquidation costs. The notion that money brings a beneﬁt in terms of lower liquidation costs can be sharpened by supposing that the total cost, in real terms, of liquidating non-money assets varies with number of liquidations, per period, Q. Total Liquidation Cost (Q)

(2.1)

Q the number of liquidations The plainest way of understanding these costs is to take them to be objective and technological, as costs usually are. (For example, fuel, phone calls, fees). However, the analysis lends itself best to taking them to be subjective (the ‘toil and trouble’ of arranging the liquidation). The cost under this interpretation remains objectively dimensioned – it would measured in terms of real income – but subjectively founded.

The theory of the demand for money

13

We will also assume that liquidation costs per period increase with the amount of total expenditure per period, X. This supposition can be rationalized in terms of an increase in the expenditure equivalent of the ‘toil and trouble’ of arranging a liquidation as expenditure rises. Total Liquidation Cost (Q, X);

Q 0, X 0

(2.2)

Xmagnitude of real expenditure Q must be reduced by higher money holdings. This negative dependency can formulated precisely in one situation that is highly special but which will be our exemplar. Let a person’s income be entirely derived from an asset – capital – which they must liquidate in order to maintain a constant stream of expenditure. In this situation the number of liquidations is half the ratio of real expenditure to average real money holdings, h. Q X/2h

(2.3)

h average amount of money held, in real terms Consequently, Q X 0 Q 2h2 h Q h

(2.4)

As the ‘beneﬁt’ is the negative of the cost: db d

(2.5)

bbeneﬁt of money one may write:

b X , X X Q 2h h 2h2

(2.6)

If we are willing to assume constant or increasing marginal costs of liquidation, then this equality implies several dependencies of b/h on h: b 0, 2b 0, lim h → 0 b , h h h2

lim h → b 0 h

(2.7)

14

Inﬂation in a risk free world

The ﬁrst inequality states the beneﬁt of extra money is positive. This is because extra money always reduces the number of liquidations, as Q X/2h. Therefore, as long liquidations are costly at the margin, Q 0, then b h 0. The second inequality states the marginal beneﬁt of money is diminishing, 2b h2 0. This is because as h becomes larger, any absolute increment in h will be smaller in proportionate terms, and so the reduction in the number of liquidations will be smaller. This reduction is suggestive of a ‘diminishing marginal beneﬁt of money’, and the inference is conclusive if QQ 0. The third result states that the marginal beneﬁt of money becomes indeﬁnitely large as money holdings approach zero. This is easily understood. As h becomes smaller, any increment in h will be larger in proportionate terms. And since the number of liquidations approaches inﬁnity as h approaches zero, the number of liquidations economized by holding more one more unit of money must approach inﬁnity as h approaches zero. This is strongly suggestive of the marginal beneﬁt of money becoming indeﬁnitely large as h gets indeﬁnitely small, and the inference is conclusive as long as the marginal costs of liquidation is constant or increasing, QQ 0. The fourth conclusion is that as h becomes indeﬁnitely large the marginal beneﬁt of money approaches zero but does not become negative. The logic is simply that extra money always reduces the number of liquidations, and therefore alway reduces liquidation costs, be the reduction ever so small. It is this last implication – that marginal beneﬁt of money is always positive – that has signiﬁcant implications, as we shall see in later chapters. It is also open to doubt from (at least) two considerations; one of which is extraneous to the ‘liquidation costs’ framework. First, the number of liquidations cannot be reduced below 1 over a lifetime. Once the money holding is suﬃciently large that all lifetime spending can be ﬁnanced from it, there can be no saving from having any more money. Second, and more important, it is plausible that the marginal beneﬁt of money may shrink to below zero owing to the direct costs of ownership: theft, loss and destruction. (The very attributes of money that make it a desirable means of exchange also make it an attractive object of theft, and something liable to be mislaid, or destroyed.1) The larger the real holdings of money, the greater theft, loss and destruction, or the larger the cost of preventing them. Evidently, at some point the marginal beneﬁt of money, net of these direct costs of owner, is zero. At a still greater quantity of money, it is negative (Figure 2.1).

15

The theory of the demand for money

b h Neglecting direct costs of ownership

0 h Including direct costs of ownership Figure 2.1 The marginal benefit of money allowing for direct costs of ownership b h aa Including direct costs of ownership 0

bb ff cc

dd

ee

h

Figure 2.2 The marginal benefit of money with constant direct costs of ownership of idle balances Direct costs of ownership will be recognized in this analysis, and it will be additionally – and plausibly – supposed that direct costs of ownership are smaller, at the margin, for balances which are not spent (‘idle’), than those which are spent. Brieﬂy, money in a sock has fewer direct costs than money in a wallet. The implications of this assumption for the marginal beneﬁt of money are developed in Figure 2.2. Let direct costs for idle balances be ff.

16

Inﬂation in a risk free world

Then all money in excess of bb will be idle, because letting money be idle has a lower net marginal cost than letting it be active. Consequently, the marginal beneﬁt schedule will not be aa cc dd, but be aa cc ee. For extra simplicity it will be supposed that the direct marginal cost of owning idle money is zero (ff0). This puts a ﬂoor of zero on the marginal beneﬁt of money. The marginal beneﬁt falls until the quantity of money is so high the marginal beneﬁt is zero. All extra money is idle and, under our present assumption, has a zero marginal beneﬁt. It turns out that this possibility of a zero magnitude of the marginal beneﬁt of money is signiﬁcant. The expression for b/h also implies (under broad conditions) some dependency on X, b h 0, X

b h b h 0 X1 h1

The ﬁrst result states that the marginal beneﬁt of money is increased by X. The marginal cost of liquidations being non-decreasing in X, X 0, is a suﬃcient condition. The second result is that the beneﬁt of extra money depends only on contemporaneous money and expenditure, (b h) h1 (b h) X1 0. Future money does not reduce liquidation costs today. Neither does past money. Future expenditure does not aﬀect liquidation costs today. Neither does past expenditure. The analysis is complete, except for a consideration of the scope of its claims. The analysis has supposed capital is the only source of income, and must be liquidated in order to ﬁnance some constant stream of money outlays. It is supposed, in other words that the money holder receives no money receipts, but undertakes a continuous money outlay. Modelling the money holder this way might appear to be highly restrictive, but by widening the scope of costs the same theory of money demand is generated in diﬀerent contexts. Consider the converse situation; a person who receives a constant ﬂow of money receipts, but wants only to transform these receipts into capital, and consume none of it. If we also suppose that there is a cost to transforming money into capital – an ‘illiquidation cost’ – then is easy to see that the ‘illiquidation cost evading’ property of money balances will create a beneﬁt from holding money, in the same way that the ‘liquidation costs evading’ property of money did. For rather than continually incurring ‘illiquidation costs’ as a result of holding minimal money, one can reduce costs by accumulating money, and episodically ‘illiquidating’ the accumulation. In terms of the modelling, X of the previous analysis may simply be re-interpreted as ‘income’ rather than ‘expenditure’.

The theory of the demand for money

17

In the presence of both liquidation and ‘illiquidation’ costs ‘the same theory of money demand is generated in the context of a simultaneous stream of money receipts, and money outlay. If the stream of money outlay exceeds stream of money receipts then the situation amounts to a ‘liquidation costs minimising problem’, where X is the excess of expenditure over revenue. Alternatively, if the stream of money receipts exceeds the stream of outlay the situation amounts to an ‘illiquidation costs minimizing problem’, and X is the excess of revenue over expenditure. The simultaneous stream of money receipts and money outlay may seem to present one problem for the theory: it may appear that money demand would be vanishingly small if the ﬂow of money receipts was strictly equal to the ﬂow of money spending. But an allowance for a third type of cost also generates a money demand in this situation. Suppose there is a cost associated with making an outlay, that is independent of the size of the outlay. (It might be the stamp on the envelope that sends a payment. Or the ‘shopping time’ to make the purchase.) The existence of such an outlay cost implies that to spend every instant is to incur high costs. To not spend every instant – but to accumulate money receipts over intervals as money holding, and spend that in bursts – has a cost reducing property. And a beneﬁt of holding a non-vanishingly small amount of money emerges once more.2 In summary in a variety of situations extra money brings a beneﬁt, in terms of lower costs, and this marginal beneﬁt of money falls with h, and rises with X, in accordance with the expression (2.7).

THE MARGINAL UTILITY OF MONEY It would prove useful to the analysis of the demand for money if it can be assimilated into utility framework. Fortunately, the beneﬁt of money can be represented by marginal utility concepts. As the marginal beneﬁt of money is in ‘real terms’ – it is a magnitude measured in consumption units – it is admissible to consider the beneﬁt as delivered in terms of so many consumption goods. For this reason the ‘marginal utility of money’ may be considered as equal to the marginal beneﬁt of money multiplied by the marginal utility of consumption. U b U h h C This mapping between: (a) the marginal utility of money; (b) the marginal utility of consumption; and (c) the marginal beneﬁt of money means that any utility function that is augmented to include money holdings

18

Inﬂation in a risk free world

U u(C, C1, C2 ... CT; h, h1, h2 ... hT ) implies a magnitude of b/h, conditional on C and h. Further, any utility function with positive but diminishing marginal utility of money U 0 h

2U 0 h2

implies a b/h that behaves in a way our earlier theory has predicted; b h 0, 2b h2 0.3 In point of fact, a positive but diminishing marginal utility of money is only a suﬃcient condition for the utility function mimicking the behaviour of b/h that is implied by the liquidations costs theory. It is not actually necessary. More generally, any utility function that exhibits ‘Pareto’s false principle of diminishing marginal substitution’ implies that a Uh/UC that shall ‘predict’ the behaviour of b/h that is implied by the liquidations costs theory.4 The mimicry can be pursued further. If we were to assume XC, then we can specify the particular utility function that predicts the particular behaviour of b/h implied by some particular liquidation cost function. Table 2.1 gives three examples of the functional forms of the utility function that mimic the behaviour of b/h implied by three particular functional forms of the liquidations cost function.5 Plausible properties of the liquidation cost function can put restrictions on the parameters of the ‘corresponding’ utility functions. So we see in the case of the Cobb-Douglas cost function that if g0 (the marginal cost of liquidation is positive) then 0 and 1 in the ‘corresponding’ utility function. And if g1 (the marginal cost of liquidation is increasing) then both and are 0. Some qualiﬁcations: First, the ‘correspondences’ of Table 2.1 are predicated upon the equality of X and C. If X does not equal C the mimicry by the utility function is at best approximate. Second, the cost functions, and their ‘corresponding’ utility functions, have no claim to generality or realism. But they are useful equipment in articulating the Quantity Theory, and will be used in the remainder of the analysis. Third, no more than ‘mimicry proposition’ has been advanced; a money in the utility approach can mimic the behaviour of the beneﬁt of money that is implied from a consideration of liquidation costs. There is no claim that the money in the utility function have been derived from liquidations cost.6

19

The theory of the demand for money

Table 2.1 Three liquidation cost functions and their corresponding utility functions Cobb-Douglas

Log Inverse

X fQ g

b h

f g gX 2gh1 g

[1 ln 2 ln h]X f1

[1 2gh]X1 f

U

C … h

C …h [1 ln h]

C …h h2

Uh/UC

C1 h1

[1 2 h]C1

Parametric correspondences

g

1fg

[ln Q ln X] f X Q

ln h 1 C

2f

Polynomial

gX 1 X 2f 4Q2 2Q

f

g

THE DEMAND FOR MONEY The previous section concluded that introducing money into the utility function captures the existence and behaviour of the marginal beneﬁt of money. This allows the demand for money to be derived in a utility maximization framework. A person ‘demands’ (that is holds) the quantity of money that solves the maximization problem: Choose

C, C1,... CT; h, h1,... hT Max

U u(C, C1, ... CT; h, h1, ... hT )

Subject to budget constraints: period 0 period 1 and so on.

PC H K1P Pw P 1K PK M P1C1 H1 K2P1 P1w1 P1 K1 P1K1 H 1

(2.8)

20

Inﬂation in a risk free world

Y

K1 1

w real wage Mendowment of nominal K capital H nominal demand for money supplement to the endowment of money The T 1 period budget constraints can be consolidated into a single lifetime budget constraint: C C [ ] C 1 1 [1 ][12 ] ... h[1 ][1 ] 1 h [ 1 11] ... [1 1 ] [11 ][1 ] 1

1

w w w 1 1 [1 ][12 ] ... K[1 1] 1 d2 d ... m 1 1 [1 ][1 1] h real demand for money m real endowment of money d supplementation of the real money endowment The ﬁrst-order conditions for the choice of C and h imply, Uh

UC 1 1 [1 ] [1 ]

(2.9)

This equality characterizes the optimal choice of money holding. This equality can also be inferred from the principle that the net beneﬁt of any feasible perturbation of h and C proﬁles from optimum must be zero. As it is always feasible to sacriﬁce one unit of consumption so as to acquire one unit of real balances in this period, and spend that unit of real balances in the next period, the net beneﬁt of that perturbation must be zero.7 U Uh UC 1 C1 0

(2.10)

The theory of the demand for money

21

The ﬁrst item is the utility of the money acquired; the second is the utility cost of the consumption sacriﬁced to acquire that money, and the third item represents the increment in consumption next period that is secured by spending next period the money accumulated this period. This utility impact of this spending equals the marginal utility of consumption next period, adjusted by the decline of the purchasing power of money between this period and next period. As intertemporal optimization implies UC1 UC /[1 ], a zero magnitude of the net beneﬁt of the perturbation implies: Uh

UC 1 1 1 1

(2.11)

which is no other than the ﬁrst order condition.

THE RATE OF RETURN OF MONEY AND THE RATE OF RETURN ON CAPITAL The optimization condition (2.11) can be interpreted in terms of the familiar precept that two assets must have the same rate of return. The two assets in the analysis are money and capital, so, in this context, the ‘Law of one rate of return’ says: The nominal rate of return on money the nominal rate of return on capital The Nominal Rate of Return on Money Any rate of return equals the sum of a yield term – the asset’s income (explicit or implicit) relative to its value – plus a capital gain term; the rate of increase in the asset value measured in the chosen numeraire (money). Since there can be no capital gain on money when money is the numeraire, the nominal rate of return on money is simply its yield. But what is the yield? Recall that Uh/UC b/hthe real beneﬁt of an extra unit of real balances. This is no more than the implicit income of money relative to its value. Thus Uh/UC, which we denote , equals the implicit yield on money, and the rate of return on money.8 U UC h

22

Inﬂation in a risk free world

The Nominal Rate of Return on Capital The rate of return on capital is the sum of a yield term and a capital gain term. This sum can be identiﬁed with the rate of increase in the nominal value of one unit of capital, inclusive of any income. This is usually understood as the increase in the nominal value of one unit of capital relative to the value in the commencing period. [1 ]P1 P P However, ‘the increase in the nominal value of one unit of capital’ could with equal validity be deﬁned as the increase in the nominal value of one unit of capital relative to the value in the concluding period. Thus, [1 ]P1 P [1 ]P1 It turns out to be more convenient to deﬁne the nominal rate of return on capital, , in this way:

[1 ]P1 P [1 ]P1

(2.12)

It is easy to check that this may be rewritten:

1 1 1 1 Therefore, the optimization condition: Uh

UC 1 1 1 1 states

(2.13)

The implicit yield on money equals the nominal rate of return on capital, (measured so that the concluding period, rather than the commencing period, is used as the ‘base period’).9

The theory of the demand for money

23

The implicit yield on money also approximately equals the conventionally measured nominal rate of return on capital, , that uses the commencing period as the ‘base period’. Table 2.2 reveals for low to moderate rates of inﬂation the divergence between the two measures of the nominal rate of return on capital is small.10 Table 2.2 Alternative measures of the nominal return on capital, ( 8.25 per period) per cent per period 0 1 10 100

1 1 1 1

7.62 8.54 16.02 53.82

8.25 9.33 19.08 116.5

8.25 9.25 18.25 108.25

Thus for moderate inﬂations, the nominal rate of return on capital conventionally measured serves as the approximate magnitude of the required nominal yield on money.

(2.14)

THE ‘MONEY DEMAND FUNCTION’ The equality of the implicit rate of return on money with the nominal rate of return on capital implies a negative relationship between h and the nominal rate of return on capital. This is because the implicit yield is a negative function of money held, (h)

(h) 0

(2.15)

and as we may say (h)

(2.16)

h 1 0

(2.17)

And so,

24

Inﬂation in a risk free world

h Figure 2.3 Money holdings are negatively related to the rate of return on capital In terms of a ﬁgure, the negative plotting of against h is identical to the plotting of against h. The relation represented in Figure 2.3 is not a Marshallian demand function. It is simply a reworking of a ﬁrst order optimization condition. Being simply a re-working, the full implications of the budget constraint have not been allowed for. Thus in this ‘demand function’ no endowment term ﬁgures; there is no ‘wealth eﬀect’, or ‘real balance eﬀect’. At the same time, consumption ﬁgures (implicitly) as an exogenous variable h in this ‘demand function’. Consequently, it excludes what a true Marshallian demand curve would not exclude (endowments), and includes what a true Marshallian demand curve would not include (demands for other goods). Nevertheless, conditional on prices, and consumption, it does give the demand for money. Table 2.3 gives the three ‘money demand functions’ that correspond to the utility functions dealt with earlier. The money demand function has one unusual feature, and one that deserves emphasis as it is important in demonstrating the existence of a unique monetary equilibrium: there exists a minimum level of money demand. With every other good or service it is presumed that the minimum level of demand is zero: there will be some price so high that demand is annihilated. This is not true of money demand, and this reﬂects the fact that the nominal rate of return on capital, as it is presently measured, has a maximum magnitude, of 1. , by contrast, has no maximum magnitude; as h

The theory of the demand for money

Table 2.3

25

Three money demand functions

Exponential

Log Linear

Linear

U

C … h …

C … h[1 ln h] …

C … h h2 …

h

exp C

C1

1 1

1

1 C1 2

approaches 0, approaches inﬁnity. There must be, therefore, a range of ‘small’ values of h such that exceeds 1, and so necessarily exceeds the nominal rate of return on capital. Such ‘small’ magnitudes of h could never be optimally held. Such ‘small’ magnitudes could be held, if the holder chose to do so, but the holder would never wish to hold such a small magnitude, no matter how high the nominal rate of return on capital. This means h has a minimum optimal magnitude, hmin. The logic of the minimum optimal magnitude of h is that if h was any lower than hmin then Uh would be so large that it exceeds UC. But if Uh exceeds UC then the net beneﬁt of acquiring an extra unit of money must be positive, rather than 0 (see equimarginal condition). Consequently money holders would not be in equilibrium since they would beneﬁt from holding more money than they do. More intuitively, the minimum applies where holding one extra dollar in money saves more than one extra dollar in liquidation costs over ‘the period’.11 There will always be a level of money holding so low that the reduction of costs from one extra dollar is so high that the reduction of costs over the interval of the period exceeds 1. Once such a low level of money holding is reached, the holder wants to hold more money than they do hold. And no amount of discounting, by way of the nominal rate on capital, of the reduction of costs outside that interval can discourage the holder from wanting to hold more money. There is analogy with an elementary piece of investment analysis: once the revenues of an investment exceed the investment outlay within the current period, no amount of discounting of future revenues can dissuade a wealth maximizer from undertaking the investment. A number of remarks about the minimum demand are in order. First, the magnitude of hmin shrinks as the length of the period shrinks. Consider someone who has resolved, for some reason, to hold exactly hmin. As the length of the period shrinks – as interval of time within which there is no time preference (or return on capital) shrinks – so shrinks the total of cost reductions that an extra dollar secures over the interval. If the reduction in costs over the interval secured by an extra dollar is to remain equal 1 – if the holding is to remain at hmin – then the eﬀectiveness of an extra dollar

26

Inﬂation in a risk free world

held in reducing costs must rise. But that is achieved only by a fall in the quantity of money held. The critical magnitude of h, hmin, falls. As an upshot, as the period becomes inﬁnitely small, this minimum becomes inﬁnitely small. As continuous time is approached the minimum vanishes. Second, whether the minimum be large or small is not a matter on which theory gives any guidance. Third, the phenomenon of the minimum demand for money should not be mistaken for zero elasticity in the demand for money above a certain rate of return on capital. The demand for money, in the theory, never has zero elasticity; the curve never ‘becomes vertical’. The situation is, rather, one where no matter how fabulously high the rate of return on capital the demand is still positive. Finally, and most importantly, the minimum is more than just a curiosity; it assumes a signiﬁcance in the demonstration of unique equilibrium that is dealt with in Chapter 4, and is required if the Quantity Theory is to have any power. The existence of a non-zero minimum to money demand prompts the question, is there a non-inﬁnite maximum? Is there a maximum quantity that someone might hold? No. There may be an amount of money that maximizes total implicit return; there may be a quantity of money such that the implicit return is zero. But that does not constitute a maximum demand. If the implicit rate remains at zero with a larger quantity of money, then money and capital are perfect substitutes, a money demand is no longer

1

hmin Figure 2.4

The minimum quantity of money demand

h

27

The theory of the demand for money

uniquely determined, and may be described as indeﬁnite, indeterminate, non-unique, or ‘inﬁnitely elastic’.12 The Sensitivity of Money Demand to the Rate of Return on Capital To conclude that money demand is reduced by a higher nominal rate of return is to beg the question about how much it is reduced. Table 2.4 provides measures of the sensitivity of money demand to inﬂation. The magnitude of the elasticities in the table can range between zero and inﬁnity. But restricting ourselves to the ‘exponential demand curve’ allows more speciﬁc conclusions. As then must be negative (see Table 2.1), the elasticity for the exponential demand function must be less than one. Indeed, if the marginal costs of the liquidation are increasing in Q, the elasticity for the exponential demand function must be less than 0.5.13 Table 2.4 Measures of the sensitivity of money demand (units of measurement are chosen so that C 1) Elasticity of demand

Semi-elasticity of demand

Co-eﬃcient of demand

Symbol

eh

sh

h

Deﬁnition

h h

1 h h

h

Exponential money demand

1 1

1 [1 ]

1 1 1

Log linear money demand

exp

1

1 l

1 2

Linear money demand

1 1 1

The magnitude of these measures can either rise or fall with the nominal rate of return. It will be seen that some of these sensitivity measures change substantially in size as changes.14 But others are completely invariant to . Taking advantage of this invariance, the exponential, log linear and linear demands will henceforth be referred to as the ‘constant elasticity’, the ‘constant semi-elasticity’, and the ‘constant coeﬃcient’ demand function, respectively.

28

Table 2.5

Exponential Log linear Linear

Inﬂation in a risk free world

Responses of measures of the sensitivity of money demand to Response of elasticity to

Response of semi-elasticity to

Response of coeﬃcient to

Unchanged Rises Rises

Falls Unchanged Rises

Falls Falls Unchanged

MONEY DEMAND’S DEPENDENCE ON INFIATION AND CONSUMPTION While money demand is negatively related to the return on capital, it proves more helpful in theory to draw an implication of this negative relation; money demand is negatively related to inﬂation;

(h) 1

(2.18)

h 1 1 [1 ][1 ] 2 0

(2.19)

So

Demand asymptotically approaches a minimum as inﬂation approaches inﬁnity. And – in accordance with our assumptions regarding the marginal beneﬁt of money net of direct ownership costs – demand becomes ‘perfectly elastic’ when inﬂation reaches negative , and the rate of return on capital reaches zero. Money Demand and Consumption We know U Uh

C

Thus if we are willing to apply Pareto’s False Principle, we can infer that a higher C increases . ( h , C ) 0

29

The theory of the demand for money

0

hmin

h

Figure 2.5

Money demand and the inflation rate

Thus, for a given , money demand is increased by consumption. Pure theory has little purchase on the magnitude of this sensitivity: the elasticity of money to C may be either 1, greater than 1, or less than 1.15

THE AGGREGATE DEMAND FOR MONEY As the supply and demand approach to the value of anything relies on the construction of an aggregate or market demand schedule for the good in question, so any ‘supply and demand for money approach’ to the value of money requires an aggregate, or ‘market’, demand curve for money. This is easily obtained by summing over the individual demand curves. h

1(,Cj)

(2.20)

But to be eﬀective, any ‘supply and demand approach’ to the value of anything requires the satisfaction of two more conditions. First, the good’s market demand function must be ‘independent’ of the good’s supply function, in the sense that the determinants of demand should (with exception of its value) be independent of the determinants of

30

Inﬂation in a risk free world

supply. If the demand and supply share signiﬁcant common determinants the analysis loses eﬃcacy. This ﬁrst requirement is problematical with respect to a ‘supply and demand’ approach to the value of money, as anyone’s money demand depends in part on their consumption, that in turn depends on their endowment of real balances, that depends in turn on their nominal balance; which is none other than one part of the supply of money. On the face of it, then, there is a dependence between the demand and supply of money. The second condition for the success of a supply and demand approach is that the market demand of the good in question is not signiﬁcantly dependent on the price of any other good whose demand depends signiﬁcantly on the price of the good in question. So, in terms of the example of oil, a supply and demand approach to the value of oil loses strength if the demand for oil depends signiﬁcantly on the price of coal, and the demand for coal depends signiﬁcantly on the price of oil. In that case, it is no longer the value of are determined by the supply and demand for oil; but the values of oil and coal are determined by the supplies and demands for oil and coal. A general equilibrium approach beckons. The second requirement is also problematical for the Quantity Theory, since the demand for money depends on the nominal rate of return on goods, which depends in turn on demand in the goods market, which (in some conceptions) depends in part on the value of money. The prospect of a supply and demand for money approach to the value of money sputter.16 What truly serves the Quantity Theory’s vision of the value of money determined by its supply and demand is a dichotomized model; where real activity is determined without any reference to the money supply. The well known Ramsey-Solow model can serve this purpose. A version is brieﬂy sketched here, that will be used repeatedly. Time is divided into discrete periods. Within each period there is no time preference, and no rate of return. The two factors of production, capital, K, and labour, L produce two outputs; consumption and capital; under constant returns to scale, and with a time-invariant technology. Capital equals capital in the preceding period plus investment, I. Investment equals output, Y, less consumption. The economy can be conceived as a vast pastoral estate, where capital is the mass of livestock, output is the oﬀspring of the livestock, consumption is that part of the oﬀspring that are slaughtered, and investment that part of the oﬀspring that are not slaughtered, but added to capital (or livestock).

YC I

K1 K I

Yy K L L

(2.21)

The theory of the demand for money

31

Each person maximizes an identical utility function, that is separable and homothetic in the proﬁle of consumption from the current period until some common terminal period, T. j C1j ... Uj C [1 ]

(2.22)

Utility maximization implies that the utility cost of acquiring an extra unit of capital (by sacriﬁcing current consumption) is just matched by the utility beneﬁt in the following period of consuming the income of this extra capital, , and the capital itself. j UCj [1 ]UC1

for all j

(2.23)

But Y

K1

(2.24)

1

Thus

UCj K 1 y L1 j UC1 1

all j

(2.25)

or

j [1 ] Cj C1

1

K 1 y L1 1

all j

(2.26)

Given the identicality of preferences this implies,

C [1 ] C 1

1

K 1 y L1 1

1

K 1 y L1 1

So a complete system is,

C [1 ] C 1

YC I

K1 K I

(2.27)

32

Inﬂation in a risk free world

C [1 ] C1 2

1 1 y

Y1 C1 I1

K2 L2

K2 K1 I1

and so on. This is a determinate system in 3T 2 equations and 3T 2 unknowns.17 This system explains all real variables: C, K, Y, , and so on, without any reference to M or P. Since all consumption is independent we can treat the real structure as a given, and solved, independently of the monetary structure. The real rate of return on capital, and the quantity of consumption, are independently determined of P, or its rate of change. They may therefore be suppressed from explicit consideration in the demand for money function, leaving for explicit consideration only that nominal component of the nominal rate of return on capital: the rate of inﬂation. Thus, h

1(,C j) h()

(2.28)

CONCLUSION The chapter concludes that real demand for money is a function of its ‘price’ or opportunity cost, the nominal rate of return on capital. Unsurprisingly, demand is a negative function of this ‘price’. But there are, critically, two anomalous features about this demand. There is a minimum quantity held, no matter how large its price; and, under the assumptions maintained, there is a minimum price paid no matter how large the quantity held.

NOTES 1. 2.

‘40 percent of Germany’s 48.5 billion coins now in circulation are regarded as irretrievably lost – either down the backs of sofas, used as decorations or thrown into fountains’. ‘Costs of the Change’, http://edition.cnn.com/SPECIALS/2001/euro/stories/euro.costs What if there is both liquidation/illiquidation costs and outlay costs? If capital is the only source of income, spending would now come in bursts that would coincide with liquidations. So, Total Liquidation Cost (Q, X) ‘outlay cost’ Q Money demand can be derived in the same way as before.

The theory of the demand for money lim h → 0U , h

3.

4.

5. 6. 7. 8.

9.

and

33

lim h → U 0 h

also implies the earlier ‘predicted’ limits on b/h. ‘Pareto’s False Principle of Diminishing Marginal Substitution’ states, in this context, that the marginal rate of substitution between C and h diminishes as h increases for some given C. It is a ‘false principle’ in that it rules out the inferior good possibility (Hicks 1939, 29). In the present context, it rules out current consumption being an inferior good; a restriction easy to tolerate. The second order conditions for a maximum of utility may be violated for certain values of Q. If theory implies prediction , theory may also imply , without implying . It is as if part of the stream of money receipts that was being spent is diverted to add to money balances so as to allow the liquidation of capital – necessitated by the excess of outlays over income – to be less frequent. An alternative demonstration that the Uh/Uc equals the yield on money: Uh/Uc is the beneﬁt measured in goods of one unit of real balances. Thus P Uh/Uc is the beneﬁt, measured in money, of one unit of real balances. But ‘one unit of real balances’ amounts to P of nominal money. So P Uh/Uc is the beneﬁt, measured in money, of P of money. Thus Uh/Uc is the beneﬁt measured in money of $1 of money. This nominal rate of return on capital can be expressed more compactly if we measure, as we are entitled to, the rate of inﬂation and the real rate of return on capital using the ﬁnal period as the base, rather than the initial period. Let, 1

and

1

then

1 1 1 [1 ] 10. 11.

12. 13.

14. 15.

16.

How low is ‘low’? If we took ‘the period’ to be one year in length, then inﬂation would have to be well over 10 per cent per annum before the divergence amounted to one percentage point. There is an ambiguity in ‘saving more than one dollar in liquidation costs’. What period of time is being referred to? The period of time is the length of ‘the period’. What is the length of the period? A month? A decade? Theory provides no answer. But a critical feature of the period is that within ‘the period’ there is no time preference, or return on capital. If the implicit yield on money actually falls below zero with a still larger quantity of money, then the larger quantity will be demanded as long as the nominal rate of return on capital falls correspondingly below zero. As the elasticity of liquidation costs to Q rises, the elasticity of money demand falls. Very inelastic demand means very elastic costs. If inﬂation rises one tries to reduce forgone income by holding less money; but the attempt is defeated by the concomitant rapid increase in transactions costs. With a constant elasticity money demand, a halving in the nominal rate of return would double the semi-elasticity of demand, no matter how small the nominal rate was. As Q C/2h, if the elasticity of h to C is unit then the number of liquidations does not change with C. If the elasticity of h to C1 then the number of liquidations falls with C (one of the rewards of being rich is fewer trips to the bank). If the elasticity of h to C 1 then the number of liquidations rises with C (that is one of the diseases of aﬄuence is endless trips to the ATM, and one of the consolations of poverty being less waiting in the bank queue). However problematic the non-satisfaction of these requirements are for the Quantity Theory, their non-satisfaction is not necessarily problematic for money neutrality, and the

34

17.

Inﬂation in a risk free world ‘quantity’ type propositions linked to money-neutrality (for example the proportionality of M to P), as Patinkin (1965) stressed. But the non-satisfaction of these requirements are problematic to any supply and demand approach to the value of money. 3T 2 equations: T 1 national income identities T 1 investment identities T equimarginal conditions. 3T 2 unknowns: T Ks, T 1 Cs and T 1 Is.

3.

A theory of the supply of money

From the demand for money we turn to its supply. The Quantity Theory makes a very simple assumption about the supply of money: it is a given, determined by the monetary authority. It is in this assumption that the Quantity Theory critic believed that that a key weakness of the Quantity Theory lies. The Quantity Theory critic contends that the money supply is not given, but is endogenous. For by far the greatest part of the money supply – the quantity of ‘inside money’ consisting of promises to pay outside money – is a decision of the market. The neglect the endogeneity of inside money does necessarily subvert the Quantity Theory. But it does recommend the theorist to construct an adequate theory of inside money, in order to ascertain whether the theory is, or is not, adequate in the face of endogeneous inside money. This chapter advances a theory of the supply of inside money. It is one squarely based on optimization; one impelled by Hicksian themes of competition between inside and outside money; and one that sets out from the question, ‘As outside money has an opportunity cost that a mere promise to pay outside money does not, why is outside money used at all?’.1 The theory of inside money advanced here identiﬁes the nominal rate of return on capital, , as the key determinant of the supply of inside money. As the nominal rate of return is the cost of demanding money, so the nominal rate of return is identiﬁed as the reward for supplying (inside) money. And as the demand for money is negatively related to the nominal rate of return on capital, so the supply of inside money is positively related to the nominal rate of return on capital. And just as the sensitivity of the demand for money to its cost plays a signiﬁcant role in the Quantity Theory, we will discover that the sensitivity of the supply of inside money also plays a signiﬁcant role in the Quantity Theory.

THE BENEFITS OF THE SUPPLY OF INSIDE MONEY We suppose, in accordance with the analysis of Chapter 2, that money provides a beneﬁt by reducing the frequency of costly liquidations of capital; a beneﬁt that is represented by the appearance of real money holdings in the utility function. 35

36

Inﬂation in a risk free world

We now make the assumption that money holdings, h, can consist of either outside money or inside money. Outside money is whatever is universally accepted, without cost, as tender. This is typically state money today; ‘ﬁduciary’ notes and coin. Inside money is a (credible) promise to pay outside money. More precisely, it is a credible promise to pay outside money to the bearer of the promise, on the demand of the bearer, and at no cost to the bearer.2 These promises will circulate within that network of people who have been persuaded of their credibility. We will usually think in terms of ‘individuals’ issuing these promises, but we can think – Kaldor style (Kaldor 1970) – of ‘ﬁrms’, who pay their workers and suppliers in ‘chits’, which circulate within that network of businesses who have been persuaded of their credibility. As inside money is a credible promise to pay the bearer outside money (at no cost), the beneﬁt of an extra unit of inside money is the same as the beneﬁt of an extra unit of outside money, Uh. From beneﬁts of inside money, we turn to its cost.

THE COSTS OF THE SUPPLY OF INSIDE MONEY The principal cost of the supply of inside money will be assumed to arise from making any promise to pay a credible promise to pay. Anyone can promise; but not everyone’s promises are credible. There are costs in making a promise believable, namely the costs of providing evidence of the solvency and honesty of the issuer of the promise. Evidence of solvency includes audited accounts, and perhaps investment in ‘conspicuous capital’ (for example ostentatious buildings). Evidence of honesty might include demonstrations of the willingness of persons of known honesty to associate with, and speak for, the issuer of the promise. These evidences are costly, and we will call these costs ‘credibility costs’. There is a second cost of the supply of inside money that we will sometimes consider. This turns on our assumption that the promise to pay money is a promise to pay money at no cost to the bearer, where ‘cost’ includes inconvenience and time loss to the bearer in being paid. The provision of honouring a promise in a way that is both convenient and timely to the bearer presumably also involves cost to the issuer. We might call these ‘convenience costs’. There is, third, the matter of ‘operational costs’. It may cost money to produce the physical embodiment of promises, and to produce them in a way that is not worth the while of a forger successfully forging them. We suppose that all three costs increase as the issue of promises requires. This is fairly obvious with respect to convenience and operational costs.

A theory of the supply of money

37

With respect to ‘credibility costs’ there are two reasons one has to spend more on credibility to increase issue: 1.

‘Credibility deepening’. As the magnitude of these liabilities rise there must be more scrutiny to establish whether the issuer can and will meet these expanded liabilities (‘John can pay $1000, but can he pay $10 000? More evidence is needed’).3 ‘Credibility widening’. If the issue is to expand, the network amongst which these promises are accepted must expand. More persons must be persuaded that the issuer is solvent and faithful to his promises.

2.

The increasing costs of issuing premises can be represented by letting Z be the total costs of issuing n of inside money. Z Z(n)

Z 0

(3.1)

nissue of inside money in real terms.4 The marginal cost of issue will prove to be signiﬁcant, and we symbolise it z. Z n Z(n) 0

(3.2)

(n)Z(n) is the cost of establishing the credibility of the nth dollar promised. Several points about z should be noted. ●

●

●

is dimensioned like the interest rate. It’s a cents per period per dollar type variable. But instead of being a rate of return, it is a rate of outlay. will be positive at n0. There is a minimum marginal cost of inside money. n 0. The marginal cost of establishing credit worthiness is increasing in n (‘increasing marginal costs’), at least for ‘low’ and ‘high’ magnitudes of n. This assumption is required for the existence of a maximum in the households’ inside money issue problem.

Two more issues merit airing at this point. Credibility Decay There is the question of how long credibility lasts, once it has been acquired. Once a promise has been made credible, how long will it be credible for? Forever? This period? A ﬁnite number of periods? We will begin by making

38

Inﬂation in a risk free world

(n)

n Figure 3.1

The marginal cost of inside money rises with inside money

the fairly extreme assumption that credibility lasts only ‘one period’. Promises made this period are credible for redemption at the opening of the following period, but otherwise have zero credibility in the following periods. This assumption will be relaxed later. Heterogeneity These cost functions may diﬀer from person to person. The cost is presumably lower for persons both wealthy and trusted, than for persons that are both poor and distrusted. The cost may be so high that it is prohibitive to issue any.

THE OPTIMAL SUPPLY OF INSIDE MONEY The Condition of Optimization The individual’s maximization problem is, Choose C, C1, … CT; h, h1, … hT

A theory of the supply of money

Max

39

U u(C, C1, …CT; h, h1, …hT )

subject to PC H K1P PZ(n) Pw P 1K PK M N

Period 0: Period 1:

P1C1 H1 P1K2 P1Z(n1 ) P1w1 P1 K1 P1K1 H N1 N (3.3) and so on. The period budget constraints can be consolidated into a single budget constraint, [ 1 11 ] C h [ ] 1 … C 1 1 … h [1 ][1 ] [1 1 ] [1 ][1 1 ] 1 [ 1 11 ] n

1 m n [1 ][1 ] [1 1 ] [1 ][1 1 ] 1 Z(n ) w Z(n) 1 1 … w 1 1 … K [1 1 ]

(3.4)

Optimization with respect to n implies,

Z(n) 1 1 1 1

(3.5)

The equality says that inside money is issued until the marginal credibility cost, , equals the nominal rate of return on capital, . This result can be explained by three equally valid arguments. The argument from increasing capital holdings Suppose that promises to pay are issued, and are immediately used to purchase capital. This capital is then sold next period to meet the redemption of the promise to pay. The addition to costs is . There is no beneﬁt from any reduced frequency of capital liquidations, since the quantity of money held is no higher. But there is a beneﬁt from the capital acquired. This is the income per dollar of capital that has been acquired, which is . So matching cost with beneﬁt, (n)

(3.6)

40

Inﬂation in a risk free world

Notice that the income per dollar of capital is nominal income per dollar. This reﬂects the fact that, if inﬂation is positive, not all of the proceeds of the sale of capital need go to meet redemption. Part of the sale – the nominal capital gain – is left over for the issuer.5 The above argument has an implication of considerable social signiﬁcance: there is an aspect of ‘printing money’ in issuing inside money. There is a net income to be derived from issuing it. The issuer gains a capital asset, and receives its income, less the costs of establishing credibility. Plainly, allowing people to literally build their own printing presses, and spend the outside money they print, is socially wasteful. And similarly there is waste in persons issuing inside money: resources are devoted to establishing credibility for the purpose of avoiding the opportunity cost of outside money. But the opportunity cost of outside money is purely a private cost, and involves no social cost. Thus the reduction in the holding of outside money, that the issue of inside money permits, produces no reduction in social costs, but it does involve costs. It is socially wasteful. The argument from increasing money holdings Consider a person who issues an extra quantity of promises to pay, holding consumption and capital holdings constant. The addition to costs is . The additional beneﬁt lies in the reduced frequency of capital liquidations, allowed by the higher quantity of money held. This beneﬁt is the marginal utility of money, Uh.6 Thus Net Beneﬁt of Marginal Issue of Inside Money Uh (n)UC (3.7) Inside money is issued until the net beneﬁt of an extra issue is zero: Uh (n)UC 0 or, U (n) U h C

(3.8)

But utility maximization also implies: Uh

UC 1 1 1 1

(3.9)

Substituting the optimization condition for money demand into the optimization condition for inside money supply yields the result:

A theory of the supply of money

41

An argument from consumption Suppose the issue of money was used to purchase one unit of consumption in the current period. This would have a credibility cost in consumption, . It would also entail a reduction in consumption in the next period to meet redemption. The net beneﬁt is: U UC UC 1 C1 Ensuring this is zero, and recalling UC [1 ]UC1, yields (3.5). The Expression for Supply The optimization condition can be inverted to relate n to ; that is, to derive a supply equation for inside money. (n) n 1 ()

(3.10)

n 1 n 1 () 0

(3.11)

(see Figure 3.2). The positive relation between inside money and the nominal rate of return can be rationalized in several ways. It can be understood in terms of the depreciation, in real terms, of the issuer of inside money’s liability that occurs under higher inﬂation, and so higher . The more one’s liability declines in real terms, the more incentive one has to issue the liability in order to acquire some capital. The positive relation between inside money and the nominal rate of return can also be understood in terms of the argument from increased money holdings. We may reason: An increase in the rate of return on capital, , reduces the amount of money held, and increases the implicit rate of return on extra money, , including extra inside money. That rate of return now exceeds the marginal cost of circulation of a dollar, . Therefore the supply of inside money will be expanded until it rises so high that reaches ( ).

To put it another way, as the rate of return on capital rises, the implicit rate of return on money rises. This amounts to the beneﬁt of persuading people of your credit rises; so you do persuade more people.

42

Inﬂation in a risk free world

(n),

n Figure 3.2

The supply of inside money is a positive function of

BANKNOTES, BANK DEPOSITS AND INSIDE MONEY How does the theory of inside money developed here relate, if at all, to the categories of inside money typically observed? The inside money we have dealt with is not bank money – banks have not been mentioned. But, historically, inside money has been commonly bank money: either commercial banknotes (in the nineteenth century) or bank deposits (in the twentieth century). So there is a distance between the categories deployed in the theory, and the categories of commercial life. Can the theoretical categories – promises to pay the bearer – be related to bank money? Can the analysis of promises to pay be construed to be an implicit analysis of bank money? It is easy to ‘tell stories’ in which the category used here – a promise to pay the bearer on demand – can be related to banking categories, and in a way such that the bank is merely a superstructure. We could suppose, for example, that every individual owns their own bank, BankofMyself. Instead of issuing to the public promises to pay, each individual borrows from their respective BankofMyself, which issues BankofMyself banknotes to eﬀect the loan, that each individual spends. Banknotes are now in

A theory of the supply of money

43

the form of inside money, but absolutely nothing of any substance has changed. No BankOfMyself exists, but only because of economies of scale, just as backyard steel foundries do not exist on account of economies of scale. We can imagine a group of persons reducing the costs of obtaining credibility for their promises to pay by grouping together, and forming a ‘bank’. Instead of an individual issuing $100 of promise to the bearer, they issue $100 of promise to intermediary – a ‘bank’ – that in turn issues them $100 of promises of ‘the bank’ to pay to the bearer. These promises to pay the bearer are ‘bank notes’, of the kind that circulated in the nineteenth century. It is the bank that does the promising, but the foundation of that promise lies in the promises made by the set of persons who have issued promises to the bank. The bank note, then, is just a way of tying together the promises of that set of persons. The credibility of the note still turns on the credibility of the persons. Nothing essential has changed. Can deposit money also be rationalized in terms of the concepts of this chapter? To think about deposit money, imagine that bank notes do not circulate, but are left in the custody of the bank. To be quite concrete about it, we might imagine a series of labelled bins, each bin pertaining to the certain person nominated on its label. In any given bin are placed all the notes, issued by whatever bank, that are the possession of the person named on the bin’s label. The amount in a given bin might be thought of the ‘deposit’ pertaining to the person. When this person wishes it, they may instruct the bank to transfer part of their note holdings to a diﬀerent bin. This captures the situation of chequable deposits. The key point is that the acceptability of this payment depends on the credibility of the persons who are ultimately the bank’s debtors.

MODEL EXTENSIONS Scale Effects It is likely that the cost of an issue of any given size is reduced by the scale of the wider economy. To issue $10 000 is presumably more costly when money demand is $10m than when it is $10B.7 One convenient way to capture this phenomenon is to suppose an x per cent increase in issue implies an only x per cent increase in the cost of issue, if total money demand also increases by x per cent. Equivalently, the average cost of issue is purely a function of the ratio of issue to total money demand.

44

Inﬂation in a risk free world

j Zj z n h h

z n h 1 z

(3.12)8

Then

(3.13)

(3.14)

j z n h

so j z n h

This implies: n j z 1 ()h

(3.15)9

This extension involves three signiﬁcant revisions of the theory. 1.

2.

3.

The supply of inside money is now unit elastic to total demand for money. This unit elasticity also implies a unit elasticity between hn and h. That is, a unit elasticity between m and h. To illustrate, if Z n1 h then n h [ [1 ]] 1 , and m h 1 [ [1 ]] 1 . A ‘money multiplier’ like relativity between total money and outside money emerges, although with a completely diﬀerent rationale from the standard rationale of the money multiplier. The elasticity of the supply of inside money to the nominal rate of return on capital is no longer necessarily positive. A large enough (negative) sensitivity of total money demand to the rate of return can outweigh any (positive) sensitivity of inside money to the rate of return. More formally, the semi-elasticity of the supply of inside money will be negative if the positivized semi-elasticity of total money demand exceeds the partial semi-elasticity of inside money supply.10 n as a proportion of h may be subject to a maximum. If Z n1 h then n h [[1 ]] 1 . The maximum magnitude of the RHS is n h 1 [1 ] 1 .

Credibility Decay We have supposed that outlays on credibility only secure credibility for a single period. But credit does not decay 100 per cent period. The proofs of credibility (that amount to proofs of solvency and honesty) must have some endurance over time. Increasing credibility today may favourably impact on credibility tomorrow. Therefore, the higher the

A theory of the supply of money

45

issue yesterday, the less the cost of accrediting a certain magnitude of n today. One way of modelling the endurance of credibility is to suppose: Zj z

j n j fn1 h h

(3.16)

f is a measure of the rate of survival in the ‘quantity of credibility’ in inside money. If f0, there is no survival in credibility beyond one period (as we have assumed so far). But if f 1, there is complete survival, and any outlays on credibility impact equally on the issue of inside in all future periods. If we choose units so that h 1 then, j ) Z j z(n j fn1

(3.17)

The key implication is that, for any proﬁle of credibility outlays, to spend more on credibility in period 0, increases the supply of issue in all future periods, but at a diminishing rate: net beneﬁt of an extra unit of real spending on credibility n U [Uh fUh,1 f 2Uh,2 … ] Z C (see note 11). At the point of optimum issue the marginal net beneﬁt is zero: Uh fUh,1 f 2Uh,2 … UC 0

(3.18)

where j ) z(nj fn1

This implies: 2 j n j z 1 ( f1 1 f 2 [1 ][1 1 ] …) fn1

(3.19)12

The supply of money is now a function of the sum of discounted future nominal rates, and a lag of the supply of money. The full impact of any increase of the nominal rate of return on the supply of inside money now comes with a lag.

46

Inﬂation in a risk free world

Honouring within Period Redemptions We have assumed that the whole of the issue circulates within the period it is issued, and none of it is presented within that period for honouring in terms of outside money, (or presented as payment to the issuer for a debt owed to the issuer). But presumably a proportion of these notes, v, will be presented within the period. How will this aﬀect the analysis? Using the Argument from Money Holdings, we can say that to issue $1 is not to increase money holdings by $1, but by only $[1v]. Therefore the optimization condition is, (n)[1v]

(3.20)

(n)[1v]

(3.21)

or

n is a function of nominal rate of return on capital factored down by the redemption rate. Constant Marginal Costs We have assumed that marginal costs are rising. But optimization is consistent with marginal costs being constant over some ﬁnite range, as long as it is rising outside that range (Figure 3.3). At the critical rate, the supply of inside money can be any magnitude, within the range of constant marginal costs. The existence of a ﬂat portion of the supply curve of inside money captures the kind of situation speculated about by Kaldor (1970) in his critique of monetarism. The supply of money has no unique magnitude at the critical rate. There is no such thing as ‘the money supply’. Fixed Costs We have implicitly assumed there are no ﬁxed cost in establishing credit. But it is plausible that there are some ﬁxed costs in establishing credit. This means that there will be a nominal rate of return on capital that is so low, such that if the rate falls below that then there is a discontinuous jump from positive issue to zero issue. In other words, there is ‘shut down’ rate of return on capital, such that if the rate falls below that rate no inside money is issued. Non-neutralities in Supply We have assumed that the real supply of inside money is completely independent of the nominal price level. But there will be such a dependence if

47

A theory of the supply of money

Money supply any amount between these limits critical

n Figure 3.3 Supply of inside money with constant marginal costs of credibility we plausibly allow that the costs of securing the credibility of any given size of issue falls within the wealth of the issuer. Suppose Z j Z(n j,W j )

Z j 0, Z j 0 n j W j

(3.22)

This reformulation does not aﬀect the ﬁrst order conditions as the wealth of person j is not j’s choice variable. Therefore: (n,W)

(3.23)

But if W 0 then higher wealth increases supply of n for a given . As wealth is partly composed of outside money, we can conclude W/P0. Thus the supply of inside money shrinks with P, for a given M. Interest Earning Inside Money We have assumed that the only way to induce other persons to accept one’s promises to pay is to go to the expense of demonstrating one’s credibility to them. But there is another way of inducing others to accept one’s promises to pay: pay others to accept one’s promises to pay. To illustrate: if

48

Inﬂation in a risk free world

the public believes, in the absence of outlays on credibility costs, that there is a 10 per cent risk that one’s promise will not be honoured, then paying $10 on every $100 promise to pay will compensate them for the risk. This payment takes the appearance of ‘interest’ on inside money, in. Does this possibility of securing circulation by ‘risk compensation’ interest undermine the theory of this chapter? To explore this, suppose that the population consists of two categories. Category One makes up a fraction 1 p of the population, and pays all of their promises. Category Two makes up a fraction p of the population, and pays none of their promises. In the absence of credibility outlays each person’s category membership is private information of the person. The net beneﬁt of issuing money in these circumstances is: Beneﬁt Uh inUC where in p

(3.24)

Suppose that for the ﬁrst category p. Then the Category One people will establish their credibility rather than pay risk compensating interest. The remaining persons will then be identiﬁed as persons who do not honour promises, so none of their promises are accepted. We are back to the theory of this chapter. In ignoring inside money issued on the basis of risk compensating interest payments we are making some sort of implicit assumption about the cheapness of establishing probity. Systemic Risk and ‘Bank Runs’ We have ignored that the possibility (and likelihood) that the total quantity of inside money exceeds the total amount of outside money. In such a situation it is impossible for all promises for outside money to be simultaneously honoured. In other words, it is possible that no promises have credibility. This is the vicious equilibrium of bank runs. Distinct Demands for Inside and Outside Money In this analysis there is no ‘demand for outside money’ distinct from some ‘demand for inside money’. Inside and outside money are perfect substitutes from the point of view of the holder, as the beneﬁt of money to its holder regardless of whether it is ‘inside’ or ‘outside’. There is only a ‘demand for money’.

49

A theory of the supply of money

INSIDE MONEY AND THE ‘RESIDUAL DEMAND’ FOR OUTSIDE MONEY This chapter has been concerned with the supply of inside money. But the analysis of inside money has implications for the ‘residual demand for outside money’. Although we have seen there is no ‘demand for outside money’ distinct from some ‘demand for inside money’, there will typically be a demand for money that is greater than the supply of inside money, and persons will seek to hold a certain quantity of outside money to make up their total demand. This quantity we will call the ‘residual demand for outside money’, and symbolize it as x. xhn

(3.25)

where n satisﬁes: n

n j

We can also identify x diagrammatically, It is easy to see that the residual demand for money declines as the rate of return on capital rises. An increase in reduces the demand for money, h

n

i

Inside money

Outside money h

n Figure 3.4

m

The real quantity of inside money and outside money

h, n

50

Inﬂation in a risk free world

and increases the supply of inside money; the diﬀerence between the two ( x) must fall. The sensitivity of the residual demand to turns out to be of signiﬁcance, and is worth examining. The sensitivity can be measured by a semi-elasticity of demand for outside money to the nominal rate of return on capital, that is symbolized . x x

(3.26)

is a linear combination of the positivized semi-elasticity of money demand, s, and the semi-elasticity of supply of inside money, . h s n hn hn s h h

(3.27)

n n

Plainly the magnitude of the semi-elasticity of the residual demand for outside money is greater the semi-elasticity of the demand for money as a whole, and may be ‘large’ in absolute terms. Table 3.1 illustrates by means of some moderately conservative assumptions about the size of s and . Table 3.1 Magnitudes of the semi-elasticity of demand for outside money, h/x 5

0 0.5 1.0

s 1.0

s 2.5

s 5.0

5 7 9

12.5 14.5 16.5

25 27 29

One reason is ‘large’ is because x is only a small proportion of h (0.25 in Table 3.1). So even if an increase in produces only small decrease in h, a large decrease in x may eventuate. So, for example, if h/m was 0.25 then a 1 per cent reduction in h makes necessary a 4 per cent reduction in m. But may be over-estimated by Table 3.1. The table assumes that Z j Z(n j ) , rather than Z j z(n j h)h. If we allow Z j z(n j h)h then, s n hn

(3.28)

51

A theory of the supply of money

where

1 zz 1 ()

Table 3.2 reports the smaller semi-elasticities under this assumption: Table 3.2 Magnitudes of the semi-elasticity of demand for outside money: n unit elastic to h (h/x5)

0 0.5 1.0

s 1.0

s 2.5

s 5.0

1 3 5

2.5 4.5 6.5

5 7 9

Note: 1. is time dimensioned. As the length of the period shortens the magnitude of increases. This is because inﬂation is time dimensioned: so any given rise in inﬂation will be measured as 1/12 as large if is measured monthly rather than annually. But since the reduction in x is the same for any increase in inﬂation, regardless of the period used to measure it, the reduction in x relative to the measured increase in will be 12 times as large if is measured monthly rather than annually. 2. may rise or fall with inﬂation. One of its positive determinants, h/x, unambiguously rises with inﬂation.13 But another positive determinant, s, may rise or fall. Only by a freak of fortune will be parametrical to inﬂation.

CONCLUSION The chapter has advanced a theory of the supply of inside money that turns on the thesis that inside money is supplied until the cost of making an extra dollar suﬃciently credible to be acceptable for circulation equals the income that an extra dollar of wealth can earn. The upshot of this assumption is that the supply of inside money is a positive function of the nominal rate of return on capital, and thereby, inﬂation. n n()

n() 0

(3.29)

52

Inﬂation in a risk free world

NOTES 1. 2. 3. 4. 5.

See Hicks on the competition between inside and outside money (Hicks 1939, 1989). Economic historians have extensively documented cases where the population at large have used as a medium of exchange such promises to pay. See Shann (1938: 52–3), and O’Connell and Reid (2005). The magnitude of n is assumed to be known or knowable. N is the nominal issue. Z is related to the real circulation of notes, N/Pn, not N. The same nominal issue may be considered either very extensive or very slight, depending on whether P is low or high. Proof: P P Uc UC1 1P UC1 0 1 But UC1 [1 ] UC , so 1 1 PP 1 0 1 1 1 1 Thus, 1 1 PP 1 1 PP 1 1 1 1 1 1 1

6.

7. 8.

9. 10.

The beneﬁt of the additional issue of inside money equals the marginal utility of money in the current period only, as on present assumptions, credit decays at a rate of 100 per cent per period. It is the higher holding of money allowed by the issue of inside money, that makes it possible to honour the issue at the opening of the next period, without cost to next period consumption. The more commercial traﬃc there is, the easier it is to circulate promises to pay. The quantity of one’s promises that any circle of creditors will bear will rise the greater their demand for money. For the Zj z(nj h)h speciﬁcation to capture the greater costliness of, for example, n $10 000 when h$10 m than when h $10B, it must be that (zn h) z 1. The elasticity of cost Z to n (normalized by h) must be greater than 1. If Zj z(nj h)h then m [1 z 1 () ]h() . As z(n h) , dn h n dh nd z hd dn nd semi-elasticity of inside supply less the positivized semi-elasticity of total demand. n z 1 (Z) fn 1

11. Thus,

n1 z 1 (Z1 ) fn and so on. Consequently, n1 n1 n n Z n Z fZ

A theory of the supply of money

53

and n2 n2 n1 n1 2 n Z n1 Z f Z f Z 12.

and so on. The optimization condition can be written: U U Uh2 ... UC1 Uh1 Uh 2 C1 C2 Uc f UC UC1 f UC UC1 UC2 z(n fn 1 ) 0 Substitution of the ﬁrst order condition for consumption (UC [1 ]UC1), and money demand ( Uh UC ) yields: z(n fn1 ) f

13.

1 2 f 2 ... 1 [1 ] [1 1 ]

(h [h n]) nh hn 0 (h n) 2

4.

The Quantity Theory of Money

With the beneﬁt of earlier chapters’ articulation of the supply and demand for money, this chapter undertakes the realization of the ‘supply and demand’ approach to the value of money: the Quantity Theory. In doing so, the chapter takes up the role of both critic and expositor of the Theory, with some innovations in both performances. The chapter begins by exploring two distinct possibilities that would critically damage the Theory. The ﬁrst is that there may be no value of money that will make the demand for money equal to its supply. The second is that there may be many values of money that will make the money’s demand equal its supply. Either of these possibilities spell the total failure of a Theory that would have the value of money explained by the supply and demand for money. The chapter argues that there do exist certain ‘pathologies’ of supply such that the Theory is undermined by these paths. But it will also be argued these conditions of supply are unlikely, and warrant only a minor purchase of our attention. Having defended the very possibility of the Theory, the chapter then turns to the content of the Theory. The Theory’s content remains a ﬁxed and well-lit landmark in every economist’s mental landscape. Thanks to ritual exposition, ‘everyone knows’ what the Theory, rightly or wrongly, claims: that the value of money is inversely proportional to the quantity of money. This claim, however, is both ambiguous and misrepresenting. It is ambiguous in that it begs questions about what is ‘the’ quantity of money. And it is misrepresenting in that the ‘inverse proportion’ summary of the Quantity Theory is true of the Theory only in a case which is polar even if analytically helpful. The chapter advances an answer to what is ‘the’ quantity of money, and explores the relation between the value of money and its quantity in non-polar settings. Overall, the chapter subjects the Theory to some fundamental attacks on the Theory’s adequacy, but reveals their eﬀect to fall short. It then extricates the content of the Theory from the oversummarizations of pedagogy and advocacy.

54

55

The Quantity Theory of Money

THE PROBLEMS OF EXISTENCE AND UNIQUENESS A Statement of Monetary Equilibrium The key contention of the Theory is that the demand for money equals its supply. equilibrium

hm n

(4.1)

demand for money

hh()

(4.2)

m M/P

(4.3)

nn()

(4.4)

real supply of outside money real supply of inside money

inﬂation

P1 P P1

(4.5)

The equality of demand and supply of money can be more concisely represented by means of the concept of the ‘residual demand for outside money’, x: xh – n

(4.6)

This allows monetary equilibrium to be represented as an equality of the real residual demand for outside money to the real supply of outside money: x()m

mM/P;

P1 P P

(4.7)

Equivalent conditions hold in future periods, which we will assume are inﬁnitely extended:

and so on.

x1(1)m1;

m1M1/P1;

1

P2 P1 P1

(4.8)

x2(2)m2

m2 M2/P2;

2

P3 P2 P2

(4.9)

56

Inﬂation in a risk free world

The Possibility of Multiple-Equilibria A critical issue for the cogency of the Theory is the unicity of equilibrium. Can the equalities above be satisﬁed by a single sequence P, P1, P2 . . .? If an equilibrium exists, is it unique? The problem of unicity can be made more patent with the help of a diagrammatic representation of current period equilibrium (Figure 4.1). The vertical axis measures 1/P, the value of money. The horizontal axis measures the real supply of outside money, m, and the real residual demand for outside money, x. The supply curve of real outside money slopes upwards, in the manner expected of supply curves. This is simply because an increase in the value of money, 1/P, increases the real value of the supply of outside money, M/P. And the demand for money slopes downwards, in the manner expected of demand curves, because an increase in value of money in the current period implies, as a matter of arithmetic, a smaller rate of increase in the value of money between the current period and the following one. The position of the supply curve is determined by the magnitude of outside money, M. And the position of the demand curve is determined by the value of money in the next period (1/ P1), since a higher value of money in the next period implies a higher rate of growth in the value of money between this and the next period. m 1/P

x(P1/P1)

The supply of outside money

The residual demand for outside money

x, m Figure 4.1

Monetary equilibrium

57

The Quantity Theory of Money

The value of money is determined by the intersection of the supply and demand schedules for money. But as the position of the current period’s money demand schedule depends, in part, on the value of money in the next period, any value of money this period could be validated by an appropriate value of money in the next period. Indeterminacy seems to loom. The question of determinacy can be scrutinized more closely with the help of a simple case that allows the analysis to abstract from the problem of the existence of equilibrium. Suppose the supply of outside, M, is growing forever at a constant rate (that exceeds ); that the rate of proﬁt is constant; and that the residual demand function for outside money, x( ), is unchanging. t

xt( )x( )

for all t

(4.10)

Inspection of the equilibrium conditions easily conﬁrms that under these conditions the following sequence of prices will satisfy the equalities: P M x()

P1

M[1 ] x()

P2

M[1 ] 2 x()

(4.11)

and so on, implying 1 2…

(4.12)

But is this equilibrium unique? Is there are any other sequence of Ps that will satisfy equilibrium conditions? We argue that the above equilibrium is unique on the grounds that any sequence of P that commences with magnitude of diﬀerent from will yield either an accelerating inﬂation, or an accelerating deﬂation, and both will ultimately end in disequilibria. Figure 4.2 provides the argument that any sequence of inﬂation that commences with magnitude diﬀerent from will yield either accelerating inﬂation, or accelerating deﬂation. The real supply of outside money is measured on the horizontal axis, along with the residual demand for real outside money, x. The magnitude of the residual demand for real outside money is plotted as a negative function of inﬂation. It is supposed for the moment that the supply of inside money is exogenously zero, n0, so the residual demand for real outside money equals the real demand for money. Suppose at period 0, . At that rate of inﬂation the money demand function implies a certain quantity of real balances x m. But as , then, purely as a matter of arithmetic, the supply of real outside money in

58

Inﬂation in a risk free world

1

0

0

Figure 4.2

x1 = m1

x=m x, m

The accelerating inflation ‘equilibrium’

the following period, m1, must be less than m. In order to reduce the demand for outside money, x1, to the reduced supply, m1, inﬂation, 1, must be increased, beyond the inﬂation in period 0. Inﬂation accelerates. But that rise in P between period 1 and period 2 necessitates a further reduction in the supply of real outside money in period 2, requiring a still larger inﬂation in period 2 to reduce money demand further to the reduced real supply of outside money and so on. So we see any positive value of inﬂation leads to an indeﬁnitely accelerating inﬂation. This is the accelerating inﬂation solution. It is easy to see that if, conversely, then an accelerating deﬂation ensures. This is an accelerating deﬂation solution. Further, there exist arguments to the eﬀect that neither any accelerating inﬂation path, nor any accelerating deﬂation path, is a true equilibria. These arguments conclude that these paths end up with an amount of money that is so small (or so large), so that money is either ‘priceless’ (or worthless), and monetary equilibrium is impossible. The argument from priceless money This is an argument that any accelerating inﬂation path constitutes a disequilibria.

The Quantity Theory of Money

59

The supply of inside money is assumed zero, and so the real supply of money is the real value of outside money, M/P. An accelerating inﬂation will, eventually, reduce the real supply of money to any arbitrarily small magnitude. But our theory of the demand for money implies there is a certain minimum demand for money, below which demand cannot fall, hmin. Therefore, equilibrium must ultimately become impossible. In economic terms, inﬂation eventually reduces m to an amount so small that Uh is raised so large that it exceeds UC. But if Uh exceeds UC then the net beneﬁt of acquiring an extra unit of money must be positive. That means the money holder is not in equilibrium, since they would beneﬁt from holding more money than they do. To put it another way, Uh exceeds UC describes the situation where holding an extra dollar in money saves more than a dollar in liquidation costs. The money holder is not in equilibrium. Therefore, the explosion of prices to an indeﬁnitely high level is not an equilibrium. The argument from worthless money This is an argument that any accelerating deﬂation path constitutes a disequilibrium. Suppose, in the manner explained in Chapter 2, that the implicit return on money falls to zero at some suﬃciently large quantity of money, and remains at zero for larger quantities. This means the demand for money function becomes coincident with the horizontal axis at some ‘suﬃciently large’ magnitude of money. If this is the case then the accelerating deﬂation path implies that the magnitude of deﬂation is ultimately so large that the nominal rate of return on capital falls to zero. The rate of deﬂation (approximately) equals the rate of proﬁt, forever. Prices fall forever at a rate of per period, and real balances grow forever at a rate of per period. But this scenario, it may be argued, violates general equilibrium. Recall that the household has a lifetime budget constraint. If we ignore, for the moment, inside money and supplements to the nominal money endowment, then summing over households yields an aggregative lifetime budget constraint of this form: C2 C … C 1 1 [1 ][1 1 ] h [ 1 11 ] … [ ] h [1 ][1 ] 1 1 [11 ][1 1 ] 1 w w2 … m w 1 1 [1 ][1 1 ] K[1 1 ]

(4.13)

60

Inﬂation in a risk free world

However, certain ‘national income identities’ will also be observed in the aggregate: Period 0

C K1 w 1K K

(4.14)

Period 1

C1 K2 w1 K1 K1

(4.15)

and so on. Substitution of these ‘national income identity’ type constraints into one another implies the following equality in aggregate, C2 C … C 1 1 [1 ][1 1 ] w w2 … w 1 1 [1 ][1 1 ] K[1 1 ] Comparing this ‘national income identity’ constraint with the aggregate life time budget constraint allows us to infer: h [ 1 1 11 ] [ ] … h [1 ][1 ] 1 1 [1 ][1 1 ] m 1

(4.16)

This is a kind of aggregate budget constraint for money demand. The LHS is the present value of real money demands. The RHS is the real endowment of money. It says, in essence, that the money endowment, m, ﬁnances only money demand. And how could it be otherwise? Money endowments cannot ﬁnance consumption spending in the economy as a whole. To ﬁnance consumption out of money is to run down money balances. But if everyone is running them down, who is running them up? The equality of demand and supplies of money (since hm, h1 m1, and so on) allows (4.16) to be written: 1 2 … 1 [1 ][1 1 ] [1 ][1 1 ][1 2 ] 1

(4.17)

where If the time horizon is inﬁnite then this equality will always be satisﬁed for any sequence of positive ’s.1 But it obviously cannot be satisﬁed for any

The Quantity Theory of Money

61

inﬁnitely long sequence of zero magnitudes of . And a zero magnitude of implies a zero magnitude of Thus an inﬁnite sequence of zero magnitudes of the nominal rate of return on capital implies a violation of the budget constraint. But the accelerating deﬂation path implies a zero magnitude of stretching out into inﬁnity. Therefore the accelerating deﬂation path implies a violation of the budget constraint. (The argument can be reworked with a growing money supply2 and inside money.)3 Intuitively, a zero nominal rate of return means the endowment of money cannot, as a matter of budget arithmetic, be ‘spent’ on money holdings: money holdings are free; they have no opportunity cost. Consequently, individuals attempt to spend part of money endowments on goods. But this cannot be done in the aggregate. Another intuition on why a permanently zero rate of return violates equilibrium is supplied by examining what a zero rate of return spells for the magnitude of real balances. It implies that real balances grow larger, at the rate of deﬂation, forever. They become indefinitely large. Money holders would want to spend part of their gargantuan real balances on consumption. But this cannot be done in the aggregate. Disequilibrium Taking these two arguments at face value, both the accelerating inﬂation path and the accelerating deﬂation path are not equilibria (as long as the implicit rate of return on money can drop to zero) and the only equilibrium is the steady state inﬂation path. How cogent are these two arguments for the disequilibrium character of accelerating and deaccelerating inﬂation proﬁles? Both the Argument From Priceless Money, and the Argument From Worthless Money, are somewhat vitiated by the restrictiveness of the assumptions they use. The Argument From Priceless Money has assumed the money supply is composed entirely of outside money, and this assumption demands to be relaxed. The argument may be reworked in the case where there is a supply of inside money, but it is no longer conclusive. The argument previously operated on the existence of a minimum (positive) magnitude of money demand (which we have argued in Chapter 2 is feature of the demand for money). If it is wished to introduce inside money, the argument would now proceed on there being some minimum, positive residual demand for outside money, xhn. If this assumption is admitted, then the argument can then run as before; an accelerating inﬂation will, eventually, reduce the real supply of outside money to any arbitrarily small magnitude. But, by present assumption, the residual demand for outside money, h n cannot fall below certain minimum, so ultimately there is disequilibrium. But the weakness in this reworking of the argument is that the theory, in its generality, does not require the existence of any minimum and

62

Inﬂation in a risk free world

positive residual demand for outside money. Our theory, in its generality, allows for the possibility that, at some suﬃciently high magnitude of inﬂation, inside money supply satisﬁes all the demand for money, and x h n is zero. In that case, under ‘the accelerating path’, inﬂation does not indeﬁnitely accelerate, but will asymptotically approach a maximum rate of inﬂation. That maximum being the rate of inﬂation at which inside money supplies all money demand. Inﬂation will not rise above that rate since at that rate the real supply of money no longer falls with the rise in P over time, since all money demand is satisﬁed by inside money, and the real quantity of inside money is neutral to P. And, assuming the associated demand for money at that rate of inﬂation is above hmin, a perpetual rate of inﬂation at this maximum will be an equilibrium. There is no violation of optimization. The Argument from Worthless Money is similarly vitiated by its assumption that if real money holdings are suﬃciently ‘large’ then the marginal utility of money is zero. It was observed in Chapter 2 that if the motive for money demand is purely a matter of saving in liquidation costs, then the implicit rate of return on money is always positive, no matter how much money is held. Under this assumption, the nominal rate of return on capital cannot be zero. An accelerating deﬂation means only that nominal rate of return on capital asymptotically approaches zero, without ever reaching it. And this is an equilibrium. There is no violation of optimization. How forceful are the two Arguments in the face of these criticisms? We argue both retain signiﬁcant force. The Argument From Priceless Money retains force because its force only requires that the residual demand for outside money be (a) negatively related to inﬂation, and (b) has some minimum, no matter how high the rate of inﬂation. If we allow for Cobb-Douglas scale eﬀects on the supply 1 of inside money (see 3.12) then x h()[1 [ (1 )] ] . This has a minimum value. Under scale eﬀects there is no limit on the acceleration of inﬂation. Every period the real supply of outside money falls. Ultimately, the real supply of outside money will fall below xmin, and optimization is violated.4 The Argument from Worthless Money also retains force since force only requires that there be some direct cost of ownership of money, since such a cost must ultimately outweigh the beneﬁt of money arising from the reduction in liquidation costs, and produce zero marginal utility. And that is plausible. We take these defences to be robust. Henceforth our presumption is that any monetary equilibrium that exists is unique.

The Quantity Theory of Money

63

The Impossibility of Equilibrium Thus far we have assumed away the issue of equilibrium. We now turn to a second potential vulnerability in the Theory, the very existence of equilibrium, and ask ‘Does an equilibrium exist?’. The demand and supply structure of the Quantity Theory invites dividing the possibilities of the non-existence of equilibrium into ‘something wrong with demand’ possibilities, and ‘something wrong with supply’ possibilities. The Theory does not allow for possibilities of ‘something wrong’ on the demand side. But there are menaces to the existence of equilibrium from the supply side. These menaces include one from the supply of outside money, and one from the supply of inside money. Chronic contraction in outside money Equilibrium is impossible if:

t 1 for all t

(4.18)

It is easy to see that in these circumstances the ‘steady state rate of inﬂation’, , is /[1 ], and consequently the nominal rate of return on capital is negative in all periods.

0

(4.19)

A negative rate of return on capital in all periods is inconsistent with general equilibrium since the ‘aggregate budget constraint for money demand’ cannot be satisﬁed. h [ 1 11 ] … [ ] h [1 ][1 ] 1 1 [11 ][1 1 ] m 1

(4.20)

Thus if money growth is more negative than for an inﬁnite number of periods then equilibrium is impossible. Notice that this inﬁnite interval need not begin in the current period. If: t t

for all tT

(4.21)

then disequilibrium is necessitated. For under this assumption there cannot be equilibrium from period T onwards (by the previously rehearsed argument). But if there is not equilibrium in T or later, there cannot be equilibrium in the earlier period. Thus disequilibrium does not actually

64

Inﬂation in a risk free world

require that be less than now; only that it become permanently less than at some date in the future. But a world in which the nominal money supply is contracting permanently seems not to be our world (and raises questions about negative seigniorage). Might equilibrium also be impossible if money growth is less than for only a ﬁnite period of time, but much less than ? Could equilibrium be made impossible by an ‘acute’ contraction in outside money? To explore the possibility, suppose there will be some large fall in the money supply between the current and the next period, and a constant money supply thereafter. Could a fall be so large as to make equilibrium impossible? If we assume the implicit return on money is always positive, a fall in M can only make equilibrium impossible if a positive nominal rate of return on capital was made impossible. And that would only be impossible if real money demand could not contract at least as fast as the nominal money supply is contracted.5 But if the implicit return on money is always positive, there is no limit on how fast real money demand might contract between the current and next period. And this is because there is no limit on how large real money demand can be in the current period. A similar argument applies if we assume the implicit return on money ultimately falls to zero, and remains there with any larger quantities of money. An acute contraction in M could only make equilibrium impossible if a non-negative nominal rate of return on capital was made impossible. But that would only be impossible if real money demand could not contract as fast as the nominal money supply is contracted. But if the implicit return on money ultimately becomes zero, there is no limit on how fast real money demand might contract between the current and next period. And this is because there is no limit on how large real money demand can be in the current period: once the nominal rate has hit zero, money and capital become perfect substitutes. Thus equilibrium remains in the face of an acute contraction of the money supply but, as is explained later, the comparative-statics change fundamentally. Excess supply of inside money A second possibility of disequilibrium, comes from an abundance rather than a shortage of money. And from inside money rather than outside money. For convenience of exposition, suppose that the nominal supply of outside money is constant. Given this, equilibrium is impossible if nh at 0. The possibility is depicted in Figure 4.3. Under these circumstances zero (or positive) inﬂation is inconsistent with equilibrium: there would be

65

The Quantity Theory of Money

h The demand for money

The supply of inside money n 0 h, n

Figure 4.3

The impossibility of equilibrium with ‘too much’ inside money

an excess supply of outside money. A positive demand for outside money to match the positive supply requires, in these circumstances, deﬂation. But, given a constant nominal supply of outside money, deﬂation leads to accelerating disinﬂation (on the grounds we have argued), and disequilibrium. Thus under these circumstances there cannot be an equilibrium. How real is this menace to equilibrium? Historical experience has given no suggestion there may be an excess supply of inside money at zero inﬂation. We have also explored credibility cost functions such that there is never an excess supply of inside money. Nevertheless, there is nothing in the theory in its generality that forbids the possibility of an excess supply of inside money at a rate of inﬂation equal to the rate of growth of outside money. It is a possibility that is made more real the cheaper the cost of establishing the credibility of inside money; and it is made more real the higher the proﬁt rate, increasing the reward for issuing inside money. If the menace did become real, a policy to remedy it would be to steadily contract the supply of outside money, so that the steady state rate of inﬂation is negative. The steady state rate of inﬂation can be made so negative so that the nominal rate of return is so close to zero that the supply of inside money becomes extinct. So whereas a falling money supply can destroy equilibrium, it can also restore it.

66

Inﬂation in a risk free world

THE CONTENT OF THE QUANTITY THEORY We henceforth assume that equilibrium exists, and is unique. Our attention, therefore, now turns to the content of the Theory: what it teaches as to how the value of money adjusts to reconcile the demand and the supply of money. The comparative-statics of the Theory turn on a key property of the system of equilibrium conditions: future endogeneous variables are exogenous with respect to current endogeneous variables. This can be seen by noting that P1 is absent from the equilibrium system ((4.7), (4.8), (4.9) . . .). Thus, on the assumption that the system is determinate, we infer that P is completely independent of P1. That is, P is exogenous with respect of P1 . But if P is exogenous with respect of P1, then P1 is exogenous with respect of P. Thus the equality, x(P1/P1)M/P

(4.22)

is a determinate system consisting of a single equation in a single variable P, or 1/P. The equilibrium implies several distinct inﬂuences on the value of money. The Impact of Shocks to the Current Supply of Outside Money on P Consider the impact on P of an increase in M in the current period, and no other period (that is a temporary increase in the supply of money). The equality (4.22) implies, PM 1 PM 1 [1 ]

(4.23)6

Figure 4.4 illustrates the comparative-static. Four observations follow. The elasticity of P to current outside money is positive PM PM 0

(4.24)

P increases with M. This is the classic Quantity conclusion. It is worthwhile to rehearse the logic. If the value of money is unchanged in the face of a greater quantity of outside money, an excess supply of money will result. A fall in the value of

67

The Quantity Theory of Money

1/P

m

x, m Figure 4.4 The impact of an increase in the current supply of money on the value of money money will eliminate the excess supply of money, by both reducing the supply of real money, and increasing the real demand for money. A fall in the value of money will increase the real demand for money, because an increased P must – as a matter of arithmetic – reduce the rate of inﬂation between the current and following period. A fall in the value of money will also reduce the real supply of money, and will do so by two paths: the real supply of money of outside money is reduced as a matter of arithmetic; and the real supply of inside money is reduced, as a higher P reduces the rate of inﬂation (as an arithmetic necessity), and so makes the issue of inside money less rewarding. The elasticity of P to current outside money is less than 1 PM PM 1

(4.25)

P is inelastic to changes in current M (that is ‘temporary’ changes in M). Thus the Quantity Theory does not warrant this sort of statement: ‘Policy alpha has made M 15 per cent higher than otherwise. Therefore policy alpha has made P 15 per cent higher than it would have been otherwise.’ This is not true of the Theory if the 15 per cent increase is temporary.

68

Inﬂation in a risk free world

P is inelastic to current M, rather than unit elastic, on account of money demand and supply being responsive to inﬂation. A unit elasticity would mean the real supply of outside money is unchanged by an increase in M. But any positive elasticity of P to M (be it unit or not) will imply a reduction in the magnitude of expected inﬂation, as P rises relative to a given P1; and that means an increase in the demand for money, h, as well as a decrease in the real supply of inside money, n. Thus if there was a unit elasticity the residual demand for outside money would have increased, rather than be unchanged, and there would be an excess demand for money. So a unit elasticity would mean money has lost too much value. To obtain equilibrium, P rises less than one to one with M. In summary, the prospect of deﬂation that accompanies any temporary increase in M mutes the impact of an increase in M. The elasticity of P to current outside money is reduced by the semi-elasticity of money demand Clearly, the more responsive are money demand, h, and supply, n, to inﬂation the smaller the elasticity of P to M. The elasticity of P to current outside money may be much less than 1 Table 4.1 gives the elasticities for a variety of assumptions. Table 4.1 The magnitude of the elasticity of P to M ( 8.25, h/m5)

0 0.5 1.0

s0

s 1.0

s 2.5

s 5.0

1 0.35 0.21

0.18 0.13 0.11

0.08 0.07 0.06

0.04 0.04 0.04

Notice that even if s0 (and h 0), inside money provides a stabilizing force with respect to shocks to outside money. Table 4.2 reports the elasticities if n is unit elastic to h. (In the table the partial semi-elasticity of n to ). Clearly the elasticity of P to M may be very small without making ‘extreme assumptions’. The elasticity of P to M falls as the length of the period gets briefer, and (correspondingly) gets larger. Thus a ﬂuctuation in money supply today, and today only, will have near zero impact on prices. This is because the briefer the period the larger the prospective deﬂation, measured at an annual rate, created by an increase in M, and so the larger the counteracting forces generated. To illustrate: if M rises 1 per cent for a period

69

The Quantity Theory of Money

Table 4.2 The magnitude of the elasticity of P to M: n unit elastic to h ( 8.25, h/m 5)

0.0 0.5 1.0

s0

s 1.0

s 2.5

s 5.0

1.00 0.42 0.27

0.52 0.27 0.18

0.30 0.19 0.19

0.17 0.13 0.11

lasting 12 months, before falling back, the money holder faces an annual rate deﬂation of about 1 per cent. But if M rises 1 per cent for a period lasting only a single month, before falling back, then the money holder faces an annual rate deﬂation of about 12 per cent. This steep decline winds money demand back. The Impact of Shocks to the Future Supply of Outside Money on Current P Not only the current money supply impacts on current prices; the future money supply also does so. The chain of causation of this impact begins with the impact future money has on future prices: P1 responds to M1, in the same manner as P responds to M: P1M1 1 P1M1 1 1 [1 1 ]

(4.26)

But the impact of a higher M1 is not restricted to P1. A higher P1 implies greater inﬂation between periods 0 and 1. This reduces the demand for money in the period 0, and increases the supply of inside money in period 0, and that reduces the value of money in the current period.7 Thus if the supply of money rises in period 1 the value of money in the current period falls. Algebraically, PM1 [1 ] 1 PM1 1 [1 ] 1 1 [1 1 ]

(4.27)

The upshot is that the impact of monetary expansion in period 1 is felt in both periods 1 and 0. The inside supply of money increases as well, as a result of the anticipated inﬂation. Thus the total money supply rises in period 0 ‘in anticipation’ of the increase in the supply of outside money in period 1.

70

Inﬂation in a risk free world

This conclusion that a monetary expansion in period 1 is felt in both periods 1 and 0 generalizes; so that an increase in money in period T will ripple ‘backwards’ in time until the present period, as each increase in price produces in the preceding period an expectation of inﬂation that reduces money demand in the preceding period, and so increases in price in the preceding period, and so on. [ 1 ]...T1 [ 1 T1 ] PMT 1 (4.28) MTP [1 [1 ]] [1 T1 [1 T1 ]] 1 T1 [1 T1 ] This implies impact on current prices of a future shock to money diminishes as the distance of the current period from the shock period increases: PM P1MT P2MT … PTMT P M 0 PMT P M P M T 1 T 2 T T T

(4.29)

The rate of diminution is governed by the magnitude of . If is close to inﬁnity there is barely any diminution: the impact of an increase in M is almost the same in the current period as in some distant period. The current money supply is barely more relevant to current price than the money supply years away. If is close to zero there is drastic diminution, and increase in distant M has almost no eﬀect on any earlier period. The Impact of Permanent Shocks to Money on P Having examined the impact of current money shocks, and future money shocks, we are now prepared to ask, what will be the impact an equiproportionate increase in the supply of outside money in all periods? That is, what will be the impact of a permanent proportionate increase of the supply of outside money? dM dM1 dM2 …dˆ M M1 M2 The earlier comparative-statics allow us to conclude: [1 ] dP 1 1 …1 Pd 1 [1 ] 1 [1 ] 1 1 [1 1 ] This is the ‘classical’ Quantity Theory result. The reference of the classical claims of the Quantity Theory, then, is to permanent increases in the money supply.

The Quantity Theory of Money

71

We see then that the inverse proportionality result pertains to a polar case. We might ask how much claim this last property of the theory has on our attention. Analytically, it serves as a benchmark as polar cases do. Pedagogically, it can be made a simpler case. In policy terms it has use too. But in terms of the usefulness of the Theory as illuminating experience it’s doubtful. Are all changes permanent changes? Have all shocks to the money supply in all wars endured forever? Or have some been partly reversed? Have all contractions in the money supply during recessions lasted forever or have many been cancelled by compensating policy movements? The Impact of Shocks to the Supply of Inside Money on Current P It is not only outside money that is inﬂationary. The exogenous element of the supply of inside money is also inﬂationary. It is just as inﬂationary as the supply of outside money. To explore this, suppose that the supply of inside money includes an exogenous ‘autonomous’ term: _

n n n()

(4.30)

n¯ autonomous supply of inside money; n() induced supply of inside money Then if we deﬁne, x h() n()

(4.31)

we may write: _

x() m n or

P ˆ _ x P1 1 m n

(4.32)

(4.32) is a single equation in a single variable, and we can infer from it the impact of an increase in the current period (and the current period only) of the autonomous supply of inside money. P 1 _ 0 nPP 1 [1 ] M

72

Inﬂation in a risk free world

The response of P to the autonomous supply of inside money is the same as the response of P to outside money. P P _ Mt ntPt P Mt P Mt

all t

An increase in autonomous inside money will have exactly the same impact on P as an increase in the supply of outside money, of the same nominal value. To summarize: in the model there is just ‘money’; and autonomous inside money is just as inﬂationary as outside money. What matters is not whether money is outside or inside, but whether it is autonomous or not. For this reason the monetary aggregate that is the most linked to prices is _

M nP The moral is that in seeking the source of inﬂation we should be looking at _ M nP. We should not be looking at M. Neither should we be looking at M N. The analysis therefore advances one answer to the question that has plagued Quantity Theorists: what counts as money. Is it solely currency? Or also bank deposits money? What counts for the value of money is exogenous money, regardless of whether it is inside or outside. Shocks to Autonomous Money Demand and P Just as shock increases to the money supply reduce the value of money, so shock increases to the demand for money increases them. To _ explore this we model shocks to money demand in an additive fashion, h. _

h h() h hˆ ()

(4.33)

xˆˆ hˆ () nˆ ()

(4.34)

If we deﬁne,

then; _

_

xˆˆ () m n h

The Quantity Theory of Money

73

Evidently, autonomous money demand operates like a negative supply of outside money. Assuming shocks to autonomous money demand are temporary, we can infer, _P P Mt htPt P Mt P Mt

all t

(4.35)8

An increase in autonomous demand for money in the current period will have exactly the same impact on P as a (temporary) increase in outside money, of the same nominal value. Temporary autonomous shifts to money demand will also induce changes to money supply. We have seen previously how a temporary increase in M reduces the inside supply of money, as outside money ‘squeezes out’ inside money. It is also true that a temporary reduction in money demand will reduce the inside supply of money, as money demand ‘drags down’ inside money. The logic is that a temporary reduction in money demand raises P, and so creates the expectation of a deﬂation, which in turn reduces the supply of inside money.9 The overall conclusion is that the supply of inside money is shaped by both the demand for money, and the supply of outside money. Shocks to the Profit Rate To obtain the impact of the proﬁt rate, the residual demand for outside money function can be augmented with a proﬁt rate term, x(, ) m where x [1 ] x [1 ]

(4.36)10

We may infer, 11

P P 1 [1 ]

(4.37)

The semi-elasticity of prices to the rate of proﬁt is positive, and somewhere between zero and 1. As approaches zero, the semi-elasticity approaches zero. As approaches inﬁnity, the semi-elasticity approaches 1.11

74

Inﬂation in a risk free world

SOME PARAMETERIZATIONS Paramaterizations are useful: they condense a model into a single expression, and they provide a tool that makes possible more particular analysis. Table 4.3 reports the parameterizations of the residual demand for outside money implied by the money the parameterizations of money demand advanced in Chapter 2, and on the assumption that inside money functions have the same parametric form as money demand functions. Table 4.3 Some parameterizations of the residual demand for outside money Constant elasticity of 1 Speciﬁcation of x

k

[1 ]P1 P [1 ]P

1

Constant semi-elasticity

Constant co-eﬃcient

k exp [ ln P1ln P]

P1 P P

None of these parameterizations have any particular claim on our credence. Not only is each form special, the assumption inside money functions have the same parametric form as money demand functions is totally arbitrary.12 The rate of return on capital is also approximated in the constant co-eﬃcient case.13 Nevertheless, paramaterized solutions are hard to resist, and Table 4.4 reports the expression of the value of money in terms of money supply measures (the proﬁt rate and other separated parameters are omitted).14 It will be seen that each parameterization of demand, the value of money is measured in a particular way. For constant elasticity the requisite measure is 1/P; for log demand the log of P, and for linear demand, P.

PERMANENT CONCEPTS The lengthy expressions for P can be written more compactly using ‘permanent concepts’. Consider the expression for lnP derived from the constant semi-elasticity demand function.

2 3 ln M ln P 1ln M 1 1 1 …1 1 1 1 2 ln k …1ln k 1 1 1 …

(4.38)

75

The Quantity Theory of Money

Table 4.4

Parametrical expressions of the value of money

Demand type

Value of money variable

Expression

1 P

k k k … M M1 [1 ] M2 [1 ][1 1 ]

Constant semielasticity of demand

lnP

ln M ln M1 ln M2 1 1 1 1 1

Constant coeﬃcient of demand

P

2M2 M1 M … 2 b[1 ] [ [1 ]] [ [1 ]] 3

Constant elasticity of demand of 1

2

…

This can be expressed as ln P ln M* * ln k*

(4.39)

where the starred variables are permanent variables: ln M*

[1 q] 1 2 1 q ln M 1 q ln M1 1 q ln M2 …

*

ln k*

[1 q] 1 1 2 q 1 q 1 1 q 2 …

[1 q] 1 1 2 q ln k 1 q ln k1 1 q ln k2 …

(4.40)

(4.41)

(4.42)

and q 1

(4.43)

The RHS of each of these identities is the ‘present value’ of the variable in question, using q as a ‘rate of discount’. The LHS of each identity deﬁnes a

76

Inﬂation in a risk free world

‘permanent’ magnitude of each variable, where the ‘permanent’ magnitude of each variable is that hypothetical constant magnitude that would have the same present value as the actual proﬁle of the variable in question. The ‘discount rate’ q – is the percentage change in the nominal rate of return on capital required to induce a one percentage point decrease in money demand. If the demand for money is inelastic then q is large, and discounting is steep. The permanent values can also be understood as a kind of average value: as the coeﬃcients on the RHS sum to unity, each starred variable is a weighted average of the current and all future magnitudes of the variable.15 The upshot is that the price level is a function of the permanent money supply, the permanent proﬁt rate, and the permanent level of money demand. The ‘permanent value’ representation of the price level also allows compact expressions of inﬂation.We can write, * [ *1 * ] [ln k*1 ln k* ] where

*

[1 q] 1 1 2 q 1 q 1 1 q 2 …

But since for any ‘permanent magnitude’, !* deﬁned by the ‘discount factor’ q, it is necessarily true that !*1 !* q[!* !]

(4.44)

* * [ln k*1 ln k* ]

(4.45)

we may write:

Inﬂation equals the growth rate in the permanent supply of outside money plus the excess of the permanent rate of proﬁt over the current rate of proﬁt less the growth rate in the permanent demand for money.

STABILITY RULES The expression for price (4.38) seems to oﬀer considerable control over the price level and inﬂation by means of appropriate manipulation of the money supply.

The Quantity Theory of Money

77

In particular, there are algorithms for the perfect stabilization of either P or . If we suppose that is the only disturbance, then this rule of money growth _

[ 1 ] _

1 [ 1 2 ]

(4.46)

and so on in all t, implies _

t all t

(4.47)

(4.46) produces a perfectly stable path of prices. Nominal Interest Rate Stability If we suppose that is the only disturbance, then this rule of money growth _

(4.48)

_

1 1 and so on, implies: _

t in all t (4.48) produces a perfectly stable nominal rate on capital.

ABNORMAL PRICE DETERMINATION Thus far the analysis of comparative-statics has been implicitly restricted: we have restricted our attention to cases where the elasticity of the residual demand for outside money is ﬁnite; neither zero nor inﬁnite. We now turn to ‘abnormal’ cases. One is where the demand for money is inﬁnitely elastic. And the other is where the supply of inside money is inﬁnitely elastic. In both these cases the comparative-statics are radically altered. ‘Liquidity Trap’ Price Level Determination The analysis of the liquidity trap begins with the observation that there exists a suﬃciently low magnitude of the value of money such

78

Inﬂation in a risk free world

that the rate of appreciation in the value of money equals the real rate of proﬁt. 1 1 P1 [1 ] P

(4.49)

At this magnitude of the value of money, capital and real bonds oﬀer the same real rate of return, and consequently become perfect substitutes. At this magnitude of the value of money the demand schedule for money becomes horizontal: a ‘liquidity trap’.16 ‘Normally’ the equilibrium occurs at the downward sloping section of the demand schedule. Normally the liquidity trap is irrelevant. But in situations where the current money supply is considerably larger than the future money supply, then the equilibrium may occur at the liquidity trap. To illustrate how a creation of large diﬀerence between current and future M will engender a trap, consider an increase in the current money supply, holding the future money supply constant. That will shift the supply of money schedule downwards. Evidently, if the current money supply is suﬃciently large relative to the future money supply (equivalently, if the future money supply is suﬃciently low relative to the current money supply) then the intersection may occur at the horizontal portion of the demand curve for outside money. Essentially, the prospective deﬂation

1/P

x m

‘Liquidity trap’

1 [1 ] P1 x, m

Figure 4.5

The monetary equilibrium allowing for a liquidity trap

79

The Quantity Theory of Money

between 0 and 1 has become so large that that the nominal rate of return on capital has dropped to zero, and money and capital have become perfect substitutes. Alternatively, consider a fall in the future money supply, holding the present money supply constant. That will increase the value of money in the future, and by the algebra of both the ‘trap’ and the normal demand curve, x, will shift up the demand curve for outside money equiproportionately with the increase in 1/P1. Evidently, if the increase in 1/P1 is suﬃciently large, then the intersection of demand and supply may occur at the horizontal portion of the demand curve for outside money. It is easy to see that in a liquidity trap equilibrium any increase in current money supply will have zero eﬀect on the value of money. Any increases are simply held. Conversely, any increase in autonomous money demand will have no eﬀect on the value of money; it is simply satisﬁed from idle balances already held. The Quantity Theory is completely thrown over. What theory of prices is to take its place? Under a liquidity trap: P [1 ] P1

(4.50)

This is the expression for the price level under a liquidity trap. P equals P1, factored up by 1 plus the rate of proﬁt. But this answer begs the question: if under a liquidity trap P is determined by P1, what determines P1? There 1/P

x

m

x, m Figure 4.6

The liquidity trap equilibrium

80

Inﬂation in a risk free world

are two possibilities. First, equilibrium in period 1 might be ‘normal’: it might be that the money supply in period 1 is not so diﬀerent than the money supply in period 2, so that 1 0, and a normal equilibrium ensures. The second possibility is that a liquidity trap prevails also in period 1. But if this is the case then P1 is determined by P2, according to P1 (1 ) P2, and the determination will rely on a normal equilibrium in period 2. Only if equals 0 in all future periods will this strategy – of ultimately anchoring the price level on a ‘normal’ monetary equilibrium in some normal future period – not work. But if equals 0 in all future periods then equilibrium is impossible anyway, by the arguments rehearsed in the ﬁrst section of the chapter. Some remarks: ●

●

The liquidity trap here is entirely diﬀerent from the Keynesian liquidity trap in one respect. There is no unemployment in the present liquidity trap.17 How ‘large’ is a decline in M that will engender a liquidity trap depends sensitively on the semi-elasticity of money demand, and the rate of proﬁt. If, for the sake of argument, the rate of growth in M from period one onwards is zero, then the liquidity trap equilibrium will materialize if: [1 ]

●

If is, say, 5, and the rate of proﬁt is, say, 5 per cent, then the decline in M between the current and next period must be greater than 25 per cent if there is to be a liquidity trap equilibrium. Notice, however, if the proﬁt rate was zero, then – regardless of the size of – there would be a trap merely if the money supply was simply constant over time. The money supply need not decline at all between periods 0 and 1 for there to be a trap in period zero. A trap in period 0 might be wholly engendered by a fall in M between periods 1 and 2. For a contraction in M between 1 and 2 might have raised the value of money in period 1 by so much, that the value of money in period 0 must rise by almost as much to prevent real balances ‘beating’ capital, and that rise in the value of money creates an abundance of real balances that drives the implicit return on money down to zero in period 0. The upshot is that the impact of a liquidity trap may extend beyond periods that neighbour any drastic fall in money. A suﬃciently drastic fall in the money supply between t and t 1 will annihilate the usual Quantity Theory positions for periods before t. A drastic fall in M in the future casts a long backwards shadow.

81

The Quantity Theory of Money ●

The liquidity trap only spells the irrelevance of purely temporary movements in the current money supply. If the money supply in all periods – current and future – rises then the current price level will rise in tandem.

Infinitely Elastic Supply of Inside Money Price Level Determination We now turn to the comparative-statics with an inﬁnite elasticity of supply of inside money. Suppose also that in all periods apart from the current period the elasticity of the supply of money is less than inﬁnite, but that in the current period the supply of inside money is inﬁnite at some critical rate, c. That means there is an abnormal equilibrium in period 0, and a normal equilibrium in all later periods. We already have a theory of P in ‘normal equilibrium’, so P1, P2 and so on, are already understood. The problem is P. To solve for P we use the equimarginal condition for inside money, and the nominal rate of return on capital. c

P1 [1 ] P P1

(4.51)

m 1/P x

x, m Figure 4.7 The irrelevance of the supply of outside money to the value of money when the elasticity of the supply of inside money is infinite

82

Inﬂation in a risk free world

Therefore: P [1 c]P1

(4.52)

P equals whatever P1 equals, factored up by 1 plus the rate of proﬁt less the critical rate c. The magnitude of M is completely irrelevant.18 So is the magnitude of autonomous money demand. Again the Quantity Theory is thrown over. This situation is one in which the supply of money may be said to be undeﬁned. This resembles the situation imagined by Kaldor (1970) where persons are paid in ‘transferable IOUs’, and the supply of outside money is an irrelevance. The likelihood of the situation is doubtful. Yet, like most extreme cases, its force does not rest on its strict truth, but on its potential for approximate truth.

CONCLUSION The chapter has argued that it is safe to presume there exists a value of money that equates the demand with supply of money, and that this equilibrium value is unique. ‘Generally’, an increase in the supply of money will reduce its value. But ‘generally’ the inverse proportion rule will not apply, as it pertains only to ‘permanent’ changes. And there is the possibility, that is not fanciful, that an increase in the supply of money will not reduce its value, on account of an ‘inﬁnite elasticity’ of the demand for outside money.

NOTES 1.

The sum

1 [1 ][11 1] [1 ] [1 21] [1 2] ... will equal 1, by familiar geometric progression, if 1 2 . . . and so on. If, instead, has any value, but 1 2 . . . and so on, then the sum becomes:

1 [1 1 ] [1 11] [1 11]2 ...

But as (1 [1 1 ]) (1 [1 1 ] 2 ) ... 1 (from familiar geometric progression) and as [1 ] 1 [1 ] 1 we can conclude that:

1 [1 1 ] [1 11] [1 11]2 ... 1

83

The Quantity Theory of Money

If, instead, has any positive value, and 1 has any positive value, but 2 3 . . ., then the sum becomes:

1 [1 1 ] "

2.

1 2 2 1 ... [1 1 ] [1 1 ] [1 2 ] [1 2 ] 2

But 1 [1 1 ] (1 [1 1 ]) [2 [1 2 ] 2 [1 2 ] 2...] 1 (see above) so this sum equals 1 as well. And so on . . . The argument has assumed zero growth in the nominal money supply. The aggregate Lifetime Budget Constrain that allows for growth in the nominal supply is: C

C1 h1 [ 1 1 11 ] ... [ ] ... h 1 [1 ] [1 ] [1 ] [1 1 ] [1 1 ]

w

d d2 w1 ...K[1 1 ] m 1 ... 1 1 [1 ] [1 1 ] d real supplement in money endowment

By the previously aired logic one may infer: h

3.

h1 [ 1 1 11 ] ... d d2 [ ] m 1 ... [1 ][1 ] [1 ] [1 1 ][1 1 ] 1 [1 ] [1 1 ]

As long as the ds are positive, then a permanent zero magnitude of the nominal rate of return on capital must violate this equality. The same conclusion is reached as when the ds were ignored. To allow for inside money we write the constraint C

C1 Z h1 [ 1 1 11 ] ... [ ] ...Z 1 ... h 1 1 [1 ][1 ] [1 ] [1 1 ] [1 1 ]

m n

n1 [ 1 1 11 ] … w

w 1 … K[1 1 ] [1 ][1 ] [1 ] [1 1 ][1 1 ] 1

If the ‘national income identity’ can be written, C Z K1 w 1K K then a parallel argument implies: h

h1 [ 1 1 11 ] ... [ ]

m n [1 ][1 ] [1 ] [1 1 ][1 1 ] [1 ] [1 ]

4.

n1 [ 1 1 11 ] ... [1 ] [1 1 ][1 1 ]

Monetary equilibrium; h n m, h1 n1 m1, h2 n2 m2,. . .; yields the same series as before that cannot be satisﬁed if 0 for an inﬁnite interval. If Z n1 h then n h [ (1 ) ] 1 . The RHS is always less than 1, so inside money never satisﬁes all of money demand.

84 5.

Inﬂation in a risk free world To be more precise, equilibrium requires:

0 ,

or

1

.

But h. Therefore, equilibrium requires: 6.

x((P1 P) 1) M P implies dP P[M P x[P1 P] ] dM P, that may be written [dP dM][M P][1 [x x]P1 P] 1. But x x i x[1 ] [1 ] 2. Eliminating x yields, (dP dM) (M P) [1 [1 ][1 ] ] 1. The deﬁnitional truth 1 1 [ ] secures the result. PP1 [1 ] PP1 1 [1 ]

7. 8.

.

h . 1

Equivalently, P P _ M ht t P Mt P ht

9.

An application of the theoretical dependence of n on h is the seasonal ﬂuctuations in money demand. Suppose autonomous, money demand is higher in the December quarter than the March quarter. P will be lower in the December quarter than the March quarter, and therefore in December inﬂation is expected for March. This expectation of inﬂation induces inside money growth in the December quarter. This expectation of inﬂation also induces a contraction in money demand in the December quarter. The upshot is that the fall in P in the December quarter, that is necessitated by higher autonomous money demand, is moderated by adjustments on both the supply side and demand side. x x

10. and

x x Thus 1 x x x [1 ] [1 ] 2 1 [1 ] 2 [1 ] 11. 12.

13.

Notice that if [1 ] [1 ] then P P P M [M P] . A one percentage point increase in the proﬁt rate would have more impact on P than a 1 per cent increase in the money supply. It is almost certainly true that [1 ] [1 ] . A constant semi-elasticity speciﬁcation of the residual demand for outside money requires the implicit supposition that both h and n have log functional forms, and, by some freak of fortune, that the semi-elasticity of the supply of inside money is identical to the semi-elasticity of the demand for money. There is an element of approximation in the constant coeﬃcient of demand model. In the parameterization of the constant coeﬃcient of demand model in Tables 4.3 and 4.4, the opportunity cost of money is measured at [P1 P] P rather than what Chapter 2 argued is the appropriate measure, [P1 P] P1.

The Quantity Theory of Money 14.

85

Each of the expressions of column 3 of Table 4.4 may be obtained by ‘leading and substitution’. So, to illustrate using the constant coeﬃcient parameterization, x m in each period requires, M [ P P 1] 1 P M1 P1 [ P2 P1 1] M2 P2 [ P3 P2 1] and so on, or: P P P M P1 P1 P1 P1 M1 P2 P2 P2 P2 M2 P3

15. 16. 17.

18.

and so on. Successive substitution out of P1, P2, P3 etc. will provide the expression for P. The weight on the current magnitude is q/[1 q] q. See Boianovsky (2004) for a survey of the concept of the liquidity trap. Notice that the constraint a liquidity trap puts on the power of M to aﬀect P does place a limitation on the power of the monetary authority to ‘validate’ an arbitrary monetary wage, and thereby secure full employment in the face of an arbitrary monetary wage. In a liquidity trap the monetary authority cannot increase P above (1 )P1. So if the arbitrary money wage, divided (1 )P1, exceeds the full employment marginal product of labour then there will be unemployment that the monetary authority cannot eliminate by printing money. Thus a drastic monetary contraction in the future combined with money wage rigidity may spell unemployment that current monetary policy cannot undo. In this light money wage rigidity becomes a greater menace to full employment than is usually thought. Nevertheless, there remains an apparently simple monetary solution to such unemployment: increasing the future money supply, and so eliminating the prospective deﬂation that caused the trap. The irrelevance of M to P when the elasticity of the supply of inside money can also be inferred from the comparative-static, P M[M P] 1 [1 [1 ] ] . As approaches inﬁnity, the impact of M on P approaches zero.

5. Inﬂation without a quantity of money: the Wicksellian approach This chapter turns from a theory that supposes that money has the value that makes its demand equal to its supply, and towards a theory that supposes that money has the value that makes the demand price of credit equal the supply price of credit. This is the Wicksellian theory of the value of money. The Wicksellian model is motivated by the menace posed to the Quantity Theory by the conditions of the supply of money in the modern economy. The Quantity Theory commonly takes the quantity of outside money as something given by the central bank. In truth, the Quantity Critic contends, the modern central bank stands ready to lend, at a certain interest rate, whatever amount of money the public wishes to borrow. Rather than being ‘given’, M is, ultimately, a matter of the public’s wishes. Rather than the elasticity of the supply of M to P being zero, it is inﬁnite. This consideration eviscerates the logic of the Quantity Theory. It can no longer be the task of P to equalize the supply and demand for money, since no such task need be performed. The central bank always supplies whatever amount is demanded. If we admit the critique as cogent, we are left with a problem: if value of money is no longer determined by the demand and supply for money, how is the price level determined? This chapter presents the Wicksellian answer to this problem.1 This Wicksellian solution locates price level determination in the credit market, rather than the money market. It supposes that the value of money aﬀects both the ‘supply price’ of credit ( the minimum nominal interest rate lenders will accept), and the demand price of credit ( the maximum nominal interest rate that borrowers will tolerate), and that the value of money adjusts so that the supply price of creditthe demand price of credit; and the rate of interest is no lower than lenders will accept, and no higher than borrowers will tolerate. The Wicksellian theory discards the equality of money demand and money supply as having any causal role to play in the determination of P. The Wickellian theory, therefore, constitutes a radical alternative to the Quantity Theory, in which the money supply disappears from relevance and is replaced by measures of central bank willingness to lend. 86

87

The Wicksellian approach

But although the Wicksellian model entirely discards any explanatory role of money market equilibrium, there remains a distinct parallelism between the Wicksellian and the Quantity Theory. This chapter shows that the categories in the Quantity Theory have counterparts in the Wicksellian theory, so that all Quantity Theory comparative-statics are mimicked by their Wicksellian theory counterparts. Indeed, we show in the concluding section that the Quantity Theory will reproduce the Wicksellian model under certain assumptions about the supply of inside money. Thus the Wicksellian model is simply one further-most point on an interval that includes the Quantity Theory. Thus endogenous money does not destroy the Theory. Rather it creates another theory that replicates the Theory, and even meets it.

THE STRUCTURE OF THE WICKSELLIAN MODEL The Common Core The Wicksellian model retains the key assumptions of the Quantity Theory. It retains the existence of general equilibrium. It retains the dichotomization into real and monetary sectors. Thus both Wicksellian and Quantity Theory’s accept as equilibrium conditions, capital

UC UC1[1 ]

(5.1)

money

U UC Uh 1 C1

(5.2)

bonds

i UC UC111

(5.3)

The Supply Price of Credit: the Interest Rate Reaction Function In only one assumption does the Wicksellian model diﬀer from the Quantity Theory: in the supply of money. The Quantity Theory supposes M is given. The Wicksellian model, by contrast, supposes the central bank is willing to lend, on certain terms, the public any amount of money they wish to borrow. The present modelling of the Wicksellian approach will assume, and this is critical, that the terms at which the central bank lends (or borrows) are not exogenous, but depend on the relativity of the actual price level, P, to a ‘reference price level’, PR. The higher the excess of P over PR, the higher the interest rate the central bank requires from its borrowers. All that is

88

Inflation in a risk free world

required is of the relation is that it be continuous and (positively) monotonic. But, purely for reasons of convenience, it is assumed here that the relation is in logs. _

i i [ln P ln PR]

0

(5.4)

This _ is the ‘interest rate reaction function’(IRRF). i is the ‘benchmark rate of interest’; it is the rate of interest at which the central bank supplies money when the price level equals the reference price level ( the rate of interest the central bank settles upon when it has the price level it wants). is the response in the interest rate to a proportionate deviation of P from PR. Its magnitude is an indicator of the sensitivity of monetary policy. The smaller the less sensitive the reaction in interest rates. Four points about the IRRF can be usefully made: ●

●

●

●

No optimizing rationalization for the IRRF is here advanced. The IRRF is a broadly plausible description of actual central bank behaviour.2 But the function is otherwise taken as a given, just as the nominal money supply is taken as a given in the Quantity Theory. The Rule does not make the interest rate a policy instrument, properly speaking. The central bank chooses the benchmark interest rate, and PR. But it does not choose i. Nevertheless, as gets close to zero, the IRRF approaches an interest rate peg, as very large variations in P imply on very small changes in i. The IRRF is an interest rate rule. There is nothing left to the central bank’s discretion, or judgement. There is nothing to judge. The IRRF could be executed automatically. It is assumed that the IRRF is known by all. It is part of market participants’ information set.

The deployment of an IRRF in place of a given money supply makes for a diﬀerence in the causal signiﬁcance of the optimization conditions that both theories share. The Quantity Theory uses the optimization conditions for capital and money to derive a demand for money, which it then equates to a given supply of money. The other optimization condition (for bonds) is disregarded as redundant to the determination of the value of money. The Wicksellian model, by contrast, disregards the optimization condition for money as redundant for the determination of the value of money. It instead takes the supply price of credit from the IRRF and equates it with

The Wicksellian approach

89

a demand price for credit obtained from the optimization conditions for capital and bonds. The Demand Price of Credit: the Fisher Condition We turn now from the minimum rate lenders will tolerate, to the maximum rate borrowers will accept. The equimarginal conditions for capital and bonds imply, [1 i] [1 ][1 ]. Taking logs: i lnP1 lnP

(5.5)3

This is the ‘Fisher Condition’ (FC), and gives the maximum rate borrowers will bear. The Fisher Condition is equally shared by the Quantity and Wicksellian models. But they diﬀer on how to interpret the direction of causality in the Condition. Quantity Theorists suppose causality in the Fisher Condition runs from inﬂation to the rate of interest. Inﬂation is the independent variable, and the interest rate is the dependent. The Wicksellian approach allows that there may also be a causal arrow running from i to . In other words, a higher rate of i generates a higher rate of inﬂation. How could this causal arrow be rationalized? By means of an elasticity of P to i. A higher i must be accompanied by a higher rate of inﬂation, by the Fisher Condition. But if P1 is exogenous, then a higher inﬂation rate will be secured by a fall in P, as a matter of arithmetic. To illustrate, consider a commodity that has a price of $100 in both the current period and the next. Its return is 6 per cent, and the interest rate is 6 per cent: there is equilibrium. But if i rises to 9 per cent, then the commodity will be sold oﬀ until its price has fallen to $97, thereby creating a prospective 3 per cent increase in price (and so return), that compensates for the 3 per cent increase in the rate of interest. By a parallel logic, the Wicksellian approach also allows a higher to aﬀect . For a given i, a higher must be accompanied by a lower rate of inﬂation. But if P1 is exogenous, the required lower inﬂation rate will be secured by a rise in P, as a matter of arithmetic. In contemplating this reverse causation we glimpse the shape of the Wicksellian theory; higher i means lower P, and higher means higher P. However, such a characterization of the Wicksellian approach is somewhat misleading, to the extent that the interest rate is not exogenous the Wicksellian approach. It is, quite critically, a function of the price level. So equilibrium is a matter of simultaneous causation.

90

Inflation in a risk free world

The Equality of the Demand Price of Credit with the Supply Price of Credit Equilibrium requires that the interest rate satisfy both the Fisher Condition and the Interest Rate Reaction Function. i

_

i i [ln P ln PR]

ln P1 ln P

(5.6)

Corresponding equalities hold for all other periods: _

i1 1 1

i1 i1 [ln P1 ln PR,1]

i2 2 2

i2 i2 [ln P2 ln PR,2]

1 ln P2 ln P1 (5.7)

_

2 ln P3 ln P2 (5.8)

and so on. The fundamental question to be asked of this system are: Can all equalities be satisﬁed by some sequence P, P1, P2 . . .? And, if they can, is this sequence unique? We will contend that if there is any such sequence P, P1, P2 . . . exists, then no other exists. The equilibrium, if it exists, is unique. We demonstrate this for a simple, special case where the rate of growth of the reference price level is constant over time, R R,1 R,2 ... R ln PR,1 ln PR and where the rate of proﬁt and the benchmark interest are constant over time:

1 2 ...

_

_

_

i i1 i2 ...

(5.9)

Inspection conﬁrms that the equalities will be satisﬁed by this sequence of prices: _

R i ln P ln PR

(5.10) _

R i ln P1 ln PR,1

(5.11)

_

R i ln P2 ln PR,2

(5.12)

91

The Wicksellian approach

and so on. Is there are any other sequence, that will satisfy equilibrium conditions? No. Any sequence of _inﬂation that commences with magnitude diﬀerent from lnPR [ R i ]/ will ultimately yield disequilibrium. Figure 5.1 provides the argument. The vertical axis plots i. The horizontal axis plots the deviation of the log of price from the log of target price. ~

ln P ln P ln PR

(5.13)

The IRRF is captured by the upwards sloping relation. The Fisher Condition is captured by the arrows: if i exceeds R then (by i ) ~ must exceed R, and that implies ln P is growing larger. ~ If ln P exceeds the steady state value then there is an accelerating ~ inﬂation solution. If ln P is exceeded by the steady state value then there is an accelerating disinﬂation solution. But neither the accelerating inﬂation or the accelerating disinﬂation are true equilibria, on the same grounds that were deployed in the treatment of the Quantity Theory. In the accelerating inﬂation case the equimarginal condition for optimal money holdings is eventually violated. In the accelerating deﬂation case the aggregate budget constraint is eventually violated when the interest rate hits zero, and money holdings become ‘free’. Thus the steady state is the only equilibria. i

R Steady state price ~ ln P Figure 5.1

The unicity of the Wicksellian equilibrium

92

Inflation in a risk free world

The analysis of price determinacy has assumed an IRRF that is continuous and monotonically rising.4 Continuity means that every interest rate is achievable by a price level; including, critically, the rate that equals the proﬁt rate. If the IRRF is discontinuous, then not every interest rate can be secured by an appropriate price level; and therefore, perhaps, the rate that matches the proﬁt rate cannot be secured. And therefore, inﬂation or deﬂation, and a journey to disequilibrium. With a monotonic and continuous IRRF the unicity and existence of equilibrium is assured. But this is not true of ‘trip-wire’ and ‘stepped’ reaction functions. Trip-wire reaction function It is plausible that the central bank does not adjust i in response to very small changes in P, but ‘tolerates’ ﬂuctuations in P within a certain margin of tolerance or ‘comfort zone’. Within the zone the reaction the IRRF is ‘ﬂat’. It only adjusts i once lnP has passed beyond a ‘comfort zone’. Figure 5.2 illustrates. We see in Figure 5.2 that, barring one special case, equilibrium is impossible. For every initial price level, either inﬂation will (eventually) accelerate, or deﬂation will eventually accelerate. The problem is that, due to the speciﬁcation of the reaction function, movements in the price level cannot

i

_ i

Margin of tolerance

ln P Figure 5.2

Trip-Wire interest rate reaction function

The Wicksellian approach

93

i

lnP Figure 5.3

Stepped reaction function

shift i to exactly equal to . The only possibility of equilibrium is where, by some ﬂuke, the i initially equals . Stepped reaction function (discontinuous adjustment) It is also plausible to suppose that the central bank restricts its choice of the interest rate to certain magnitudes, say multiples of one quarter of a percentage point. In this situation any adjustment in the interest rate is made in jumps, and this creates a ‘margin of tolerance’ around admissible magnitude of the interest rate. The IRRF takes on a stepped form. And those steps also spell disequilibrium. Again, a ﬂuke apart, there is no equilibrium. If by a ﬂuke the natural rate exactly equals one of the magnitudes that the interest rate can assume, then equilibrium exists but the price level is indeterminate with the ‘margin of toleration’. We will assume that the IRRF is continuous. Until the very last section of the chapter, we will assume the equilibrium exists and is unique.

COMPARATIVE-STATICS What makes P change? The comparative-statics of the Wicksellian model turn on a key property of the system of equilibrium conditions: future endogeneous variables are exogenous with respect to current endogeneous

94

Inflation in a risk free world

variables. This can be seen by noting that P1 is absent from the equilibrium system. Thus, on the assumption that the system is determinate, we infer that P is completely independent of P1. That is, P is exogenous with respect of P1. But if P is exogenous with respect of P1, then P1 is exogenous with respect of P. Thus the equalities, i ln P1 ln P

_

i i [ln P ln PR]

(5.14)

constitutes a system of two equations in two unknowns: P and i. The equality on the left gives the ‘demand price for credit’. The equality on the right gives the ‘supply price of credit’. The equality of these two implies, 1 ln P 1 [ _i] ln P 1 ln PR 1 1 1

(5.15)

Figure 5.4 represents this solution. It will be observed that in the expression for the price level (and in the Figure) the money supply is completely absent. The supply of money may still be supposed to equal the demand for money, but the requirements of the equality of money demand to money supply have nothing to do with the determination of P. Thus a shock to money demand have zero impact

i ‘Supply price of credit’

Equilibrium i

Interest rate reaction function

Fisher condition

_ i

‘Demand price of credit’

Equilibrium P lnPR

Figure 5.4

lnP1

The determination of P in the Wicksellian model

lnP

The Wicksellian approach

95

on P; it will be met by more lending from the central bank. And a shock to the money supply will also have no impact on P. Any helicopter drop of money will be unwanted, and used to repay debt to the central bank, or buy assets from them. Any helicopter drop of money is the equivalent of cancelling a debt the public owes to government. An increase in the autonomous supply of inside money also has no eﬀect on P. The public will simply have an excess supply of money, and it will use its holdings of outside money to pay oﬀ its debts to government. The Wicksellian model, then, is a model of the price level without a quantity of money. The Wicksellian model, also note, is not a ‘banking’ view of inﬂation: bank loans – any kind of loans – do not exist (except those of the central bank to money holders). It is not a ‘Banking and the Price Level’ theory; it is a ‘central banking and the Price Level’ theory. The Wicksellian model explains P by means of the reference price level, the benchmark interest rate and the future price level. We will go through these in turn. A higher reference price level increases P The expression for ln P implies, ln P 0 ln PR 1

(5.16)

To give an intuition for this result, imagine an adjustment process in which the nominal interest rate always equals the ‘supply price’ of credit, but, following a disturbance, only equals the ‘demand price’ of credit, , with a lag. Equivalently, i always is ‘on’ the IRRF, but, following a disturbance, only gets back ‘on’ the FC with a lag. P, at any period, is pre-determined, but adjusts upwards over time to any excess of the return on capital over i. The increase in P following an increase in PR may be read thus. An increase in PR for a given P, reduces P relative to PR. i is reduced, in accordance with the IRRF. But that fall in i means the rate of interest is now less than the rate of return on capital. Money is borrowed from the central bank in order to buy capital, and the money price of capital (and so output) is bid up. But that rise in P induces the central bank to raise rates, in accordance with the IRRF. That same rise in P reduces prospective inﬂation, as a matter of arithmetic, and so reduces the nominal rate of return on capital. These two responses combined restore the equality of the rate of interest to the nominal rate of return on capital. In terms of Figure 5.5 the IRRF shifts down. The immediate response is a fall in i, and no change in P. This leaves i below the FC. i and P move north-east.

96

Inflation in a risk free world

i

‘Supply price of credit’

Interest rate reaction function

Ultimate fall in i Initial fall in i

‘Demand price of credit’

Ultimate rise in P

lnPR

Figure 5.5

lnP

A higher reference price level increases P

By how much will P rise? P will not rise equiproportionately with PR. P is inelastic to PR. If P was unit elastic with PR, then i would be unchanged in accordance with the IRRF; yet expected inﬂation would (as a matter of arithmetic) still be reduced by the higher price level, leaving the nominal return on capital smaller than i. To preserve equilibrium, therefore, P rises by less than PR, and thereby producing a fall in i.5 A higher proﬁt rate increases P ln P 1 1 1

(5.17)

An increase in means capital earns a greater nominal rate of return than bonds. So proﬁt is made by borrowing money and buying capital. So money is borrowed, and capital is bid for. That raises prices. That rise reduces expected inﬂation as a matter of arithmetic, and so helps restore equilibrium. It also induces the central bank to raise i, and that also helps restore equilibrium. And so equilibrium is restored. How much P rises depends on the magnitude of . If is large – a sensitive IRRF – the interest rate will do ‘most of the work’ to restore the equality to the rate of return on capital, and P need barely rise. But if is near zero – an insensitive IRRF – then P ‘does most of the work’, and it

The Wicksellian approach

97

adjusts upwards so as to engender an expectation of deﬂation almost equal to the increase in the rate of proﬁt. These results are captured in the expression for the semi-elasticity of P to , which depends on the magnitude of . If is near zero the semi-elasticity is near one. A higher benchmark interest rate reduces P ln_P 1 1 i

(5.18)

Intutively, an increase in the benchmark rate increases i in accordance with the revised IRRF. Bonds now oﬀer a higher rate than capital, and capital is sold with the intention of lending the proceeds to the central bank. That process reduces P. That reduction in P increases prospective inﬂation, and causes part of the initial increase in i to be reversed, and thereby secures an equality between i and . As varies between zero and inﬁnity, the semi-elasticity of P to # varies between one and zero. An increase in future prices increases current P ln P 1 1 ln P1 1

(5.19)

Intuitively, an increase in P1 means capital initially outperforms bonds. Capital is bought, that raises prices, and that both raises i and reduces prospective inﬂation, and so restores equilibrium. Future Shocks The preceding analysis is incomplete: it begs a question about the determination of P1. That question can be answered by the equality of the demand price and supply price of credit in period 1. That equality implies: 1 ln P 1 [ _i ] ln P1 1 ln PR,1 1 2 1 1 1

(5.20)

and similarly for following periods: 1 ln P 1 [ _i ] ln P2 1 ln PR,2 1 3 1 2 2 and so on. Repeated substitution yields,

(5.21)

98

Inflation in a risk free world

ln PR,1 1 ln PR,2 1 ln P ln P 1 R 1 1 1 1

_ 1 [ _i ] 1 ... [ i]1 1 1 1

2

2

...

(5.22)6

The price level is, evidently, a matter of all reference prices, proﬁt rates and benchmark rates, both current and future. Thus any increase in the proﬁt rate – be it in the future as well as the present – will increase P. An increase in the benchmark rate of interest – be it in the future as well as the present – will reduce P. An increase in the reference price level – be it in the future as well as the present – will increase P. Table 5.1 gives details. Table 5.1 Sensitivities of the price level in the Wicksellian model: elasticity/semi-elasticity

t

PR,t

t 1 1 1 1

it

t

1 1 1

t

1 1 1

Elasticity/semi-elasticity Table 5.1 has implications regarding the relative importance of ‘the present’ and ‘the future’. A nearer shock is always more signiﬁcant than a further shock. But as approaches indeﬁnitely small (that is, as the interest rate approaches a peg), future magnitudes assume an almost equal impact with current ones. The larger (the more response i) the more present biased is price determination. As approaches inﬁnity, all future variables are irrelevant. The complete expression for P may be expressed more compactly by using ‘permanent’ magnitudes of the reference price level, and so on. _

ln P ln P*R

* i *

(5.23)

where ln PR,1 ln PR,2 1 ln P*R ln PR 1 ... [1 ]2

(5.24)

2

1 ...

* 1 1 [1 ]2

(5.25)

The Wicksellian approach _

99

_

i i2 1 _ i* i 1 1 ... [1 ]2

_

(5.26)

Evidently, it is the permanent magnitudes that count. Inﬂation A compact expression for inﬂation can be derived using the permanent concepts, including a ‘permanent rate of reference inﬂation’ _

_

*R * i i*

(5.27)

where*R, the permanent rate of reference inﬂation, is deﬁned as the hypothetical constant rate of reference inﬂation that has the same ‘present value’ as the ‘present value’ of the actual proﬁle of reference inﬂation rates: R,1 R,2 1 ... *R R 1 [1 ]2

(5.28)

The moral of (5.27)’s expression for inﬂation is that it is the permanent magnitude of R, *R, that is signiﬁcant for inﬂation. Not this period’s R. Or next period’s. Or any other single period. So, for example, R may be negative this period, but if the permanent rate is positive there will be inﬂation this period. The moral is that current inﬂation is insensitive to the ambitions of authorities for inﬂation in the current period. It is their ambitions over the long period that counts. Inspection also reveals that it is the excess of the permanent rate of proﬁt over the current rate, * , that creates inﬂationary pressure. If the tendency of future proﬁt rates is to exceed the current proﬁt rate, then prices will rise faster than they would otherwise. This makes sense. We have learnt a higher proﬁt rate produces a higher price level. So if future proﬁt rates are higher than current ones, then future prices will be higher than the current one: inﬂation. Thus it is the trend, or escalation in the proﬁt rate, yielding

* , that produces inﬂation, not the ‘height’ of the proﬁt rate. This implies that any change in the proﬁt rate which is permanent has no impact on inﬂation, except in the period in which it occurs. Conversely, inspection reveals that it is the excess of the permanent benchmark rate over the current benchmark rate that that creates deﬂationary pressure. Thus if the tendency of future benchmark rates is to exceed the current benchmark, rate then prices will be falling. We have learnt that a higher benchmark rate produces a lower price level. So if future benchmark rates are higher than current ones, then future prices are

100

Inflation in a risk free world

lower than the current one: deﬂation. Thus it is the trend, or escalation in _ _ the benchmark rate, yielding i* i, that produces deﬂation, not the ‘height’ of the benchmark rate. Thus any change in the benchmark rate which is permanent has no impact on inﬂation, except in the period in which it occurs. Notice that a permanent reduction in the benchmark rate does not create inﬂation. This underlines the property that there is no ‘inﬂation neutral’ benchmark rate.7 To try to summarize, if one was to ask ‘Why is there inﬂation?’, one acceptable answer would be to point to three circumstances which are suﬃcient for inﬂation: (a) A positive trend in the Reference Price Level (other factors unchanging); or (b) a positive trend in the rate of proﬁt (other factors unchanging); or (c) a negative trend in the benchmark rate of interest (other factors unchanging). If one was to ask ‘Why is there deﬂation?’ one answer would be to point to three circumstances which are suﬃcient: (a) A reducing Reference Price Level (other factors unchanging); or (b) a continuous reducing the rate of proﬁt (other factors unchanging); or (c) a continuous increase in the benchmark rate of interest (other factors unchanging). A continuous increase in the proﬁt rate, or decrease in the benchmark rate, is unlikely to be sustained for long, or to be very large. Thus if we are concerned with ‘secular’ rates of inﬂation it is the reference inﬂation rate that is crucial. Price trends are largely to be explained in terms of the change in the Reference Price level. If there is 15 per cent inﬂation it is on account of an unwillingness of the central bank to raise the interest rate except when prices rise more than 15 per cent – and a willingness to reduce it if prices rise by less than 15 per cent. It is such policy that produces an inﬂation rate of approximately 15 per cent. If there is to be 1 per cent inﬂation the central bank must be willing to raise the interest rate as soon as inﬂation exceeds 1 per cent – and lower it as soon as it is less than 1 per cent. If there is any inﬂation, it is on account of the central bank’s unwillingness to raise interest rates whenever prices are rising, be it ever so small.

INTEREST AND PRICES We have explored the Wicksellian model’s determination of prices. But the Wicksellian model also determines the rate of interest. This determination is worth explicating because of the presumption – which is borne out here – that in any Wicksellian model the nominal interest rate and the price level are tied up together. The expression for the interest rate is:

101

The Wicksellian approach _

ii

ln PR,1 1 ln PR,2 1 ln PR 1 1 1 1 1

_ 1 [ _i ] 1 ... [ i]1 1 1 1

2

2

... PR

(5.29)

This can be more compactly expressed: _

_

i *R * i i*

(5.30)8

The expression suggests that, with one exception, it is permanent magnitudes that determine the rate of interest. It is the permanent rate of reference inﬂation that features in the expression for i, not reference inﬂation in the current period. It is permanent magnitude of that appears, and not the current magnitude. The moral is purely temporary ﬂuctuations in the reference inﬂation rate or the proﬁt rate are pretty much irrelevant to the nominal interest rate; they are swamped by the action of future rates. However, the nominal interest rate does respond to, and on a one-to-one basis, temporary changes in the benchmark rate. Thus the interest rate is smooth (‘sticky’ in appearance) save for central bank playing about with benchmark rate. Co-movements and Paradoxes i and P are so tied up with one another that there is considerable comovement in the interest rate and the price-level, both positive and negative (Table 5.2). The ﬁrst row indicates that shocks to the proﬁt rate will push i and P in the same direction: upwards, and by the same amount. This one-for-one comovement means that if proﬁt rate shocks were the only shocks then the interest rate and the log of the price level would be perfectly ‘correlated’. This is reminiscent of Gibson’s Paradox.9 Table 5.2

PR _

i

Relative impacts on P and i of shocks in the Wicksellian model Derivative of lnP

Derivative of i

1 1

1 1

1 1

1

1

102

Inflation in a risk free world

The presence of other shocks, however, will spoil any perfect comovement in prices and nominal_ interest levels. The second row indicates that current shocks to ln PR and i will shift i and P in opposite directions, by equal amounts. _ However, future shocks to i will shift i and P in same directions, and same amount. And future shocks to lnPR will shift i and P in the same direction, and by the same amount.

i ln P 1 ln PR,t ln PR,t 1 1

i_ ln_ P 1 1 it it

t

t1

t

t1

(5.31)

(5.32)

What then is the impact of permanent changes on i, as compared with the impact of permanent changes on P? A permanent change in will impact on i. The contrary future and present impacts of the benchmark rate balance each other out in that a permanent change in the benchmark rate will have no impact on i. Similarly for lnPR. Interest and Price Stability The discussion prepares the way for the analysis of the requirements of price stability. Wicksell contended that price stability requires that the interest rate assume some ‘right’ level; the nominal rate of interest to equal the rate of proﬁt. For Wicksell’s contention to be a prescription for price stability10 the interest rate must be taken to be a policy instrument, and exogenous to prices. As the interest rate is endogenous within this model, Wicksell’s prescription cannot be strictly rationalized within this model. Nevertheless, the benchmark interest rate is exogenous, and a closely kindred prescription is true. Recall: _

_

*R * i i*

(5.33)

It can easily be inferred, that stable and zero inﬂation (and lnPlnPR) can be secured by a double pronged policy: R,t 0

_

and

it t all t

(5.34)11

Price stability is secured by the benchmark rate moving up and down with the rate of proﬁt.

The Wicksellian approach

103

Two remarks: ●

The price stabilization rule implies it t

for all t

(5.35)

but this implication should not be considered a prescription. A rule which amounted to an injunction to peg the nominal rate at the rate of proﬁt, it t

●

for all t

(5.36)

would not secure price stability. It would imply price-indeterminacy. It is the willingness to make i diﬀerent from , in the face of P diverging from PR, that provides price stability. The price stability in no way rests upon the central bank being willing to raise the interest rate by so much if P rises so much above the target. All price stability requires is that i be raised by some amount if there is any inﬂation in the current period, and increase it again by some amount more next period if there is any more inﬂation the following period (and increase it yet again by some amount in the period after if there is any more inﬂation the period after . . . and so on). It is the willingness to keep on raising i the further P departs from its reference value that is critical.12

THE PARALLELISM WITH THE QUANTITY THEORY The Wicksell model ﬁts in easily with central bank practice. It does not have the adversarial relationship that monetarism often had with central banks. Nevertheless it also has a kinship with the Quantity of Money. Although the money supply is not a determinant of P, nevertheless ‘monetary conditions’ remain relevant to inﬂation: the interest _ rate at which the central bank is willing _to lend money, as measured by i , is critical for prices. Tight _ money, a high i , will reduce P. Loose money, a low i , will increase P. We have turned from a ‘quantity of money’ theory of prices, to a ‘price of money’ theory of prices. If we assume semi-elasticities of money demand to are parametric, then the price level under the Quantity Theory is: ln M

ln P 1ln M 1 1 1 ... 1 1 1 1 ...

104

Inflation in a risk free world

The price level the Wicksellian model obeys:

ln PR,1 1 ln PR,2 1 ln P ln P 1 R 1 1 1 1

_ _ ... [ i] 1 [ i ] 1 1 1 1 1

2

2

...

A parallelism between the two theories is immediately evident. M in the Quantity Theory plays the same role as lnPR in the Wicksellian model. in the Quantity Theory plays the same role as in the Wicksellian model. And plays the same role as 1/. To press the point, if 1/ and if lnPR was to track the same path as M, then there will be the same path in P. To put it another way: instituting some growth ‘rule’ for M will yield exactly the same price level under the Quantity Theory as if the very same rule was applied to lnPR under the Wickselllian theory. (The assumption 1/ is, of course, totally arbitrary.) The Quantity Theory with Inﬁnitely Elastic Inside Money as Mimicking the Wicksellian Model The Quantity Theory approach – with the central bank setting the supply of outside money exogenously – can actually reproduce a key property of the Wicksellian model; the irrelevance of the supply and demand for money. This possibility occurs through the endogenous supply of inside money. Suppose that the elasticity of supply to inﬂation become inﬁnite at some rate (Figure 5.6). This critical rate of inﬂation corresponds to some critical nominal rate of return on capital. At this critical rate, the suppliers of inside money are essentially acting, like a central bank under an IRRF; supplying any amount of money at some critical rate. We have already observed that under such an inﬁnite elasticity, the price level is indeterminate under the Quantity approach if this inﬁnite elasticity is permanent rather than temporary. But suppose that critical rate at which supply becomes inﬁnite is itself a function of P. Suppose the relation in Figure 5.6 shifts upwards with a higher P. Under this assumption, the suppliers of inside money now eﬀectively have an ‘Interest Rate Reaction Function’; the critical nominal rate at which inside money is supplied with an inﬁnite elasticity rises with P. The market is now unwittingly replicating the role of a Wicksellian central bank, and the conclusions of the Wicksellian model apply.

The Wicksellian approach

105

n Figure 5.6

The supply of inside money as ‘inﬁnitely elastic’ at some rate

But why would the critical rate rise as P rises? Recall that in Chapter 3 it was suggested that cost of credibility might be a negative function of the wealth of the issuer. Z j Z(n j,W j )

Z j 0 W j

(5.37)

Suppose also that the marginal cost of issue is invariant to the quantity of issue but a negative function of the wealth of the issuer. n WW

(5.38)

But as we infer n WW As wealth in the aggregate, W, is the sum of endogenous real balances and exogenous capital, we are entitled to suppress capital from explicit consideration and simply write: n WM P

(5.39)

106

Inflation in a risk free world

and if this is combined with a Fisher Condition P P1 1 we arrive at a ‘Quantity’ system that is homologous with a Wicksellian system. Consider for example this Wicksellian system: _ P i i PR

(5.40)

P i P1 1 The Quantity Theory with a perfectly elastic supply of inside money is analogous to this Wicksellian theory. We could work out a theory of the price level, just along the lines of the Wicksellian model. PR corresponds to M. In the ‘Quantity Theory as Wicksellian model’ money demand is completely irrelevant. The model is, indeed, Wicksellian in this respect. Notice, however, that the money supply is not irrelevant. On the contrary, the price level is proportional to permanent changes in M. The endogeneity of money is no longer the feet of clay of the Quantity Theory. It is quite consistent with P being proportional to M. Of course, the mechanism is diﬀerent. M is now powerful by reducing the costs of supply of inside money. If M goes up, (the marginal cost of credit) is reduced below the nominal rate of capital, unless P goes up to restore that equality. (If it did not would be less than .)

ABNORMAL WICKSELLIAN PRICE DETERMINATION Just as in the Quantity Theory, there is the possibility that price determination is abnormal. The Zero Interest Bound The interest rate cannot be negative, or zero. In diagrammatic terms the IRRF has, to the left of some critical price level, a ‘ﬂat’ portion, coinciding with the horizontal axis. The central bank is willing to lend at zero, but no further reduction in P can make it willing to lend at any less. Normally this zero bound on the interest rate does not impinge on equilibrium: the Fisher

The Wicksellian approach

107

i

Interest rate reaction function

Fisher condition

0 lnP Equilibrium P Figure 5.7

Equilibrium at the zero interest bound

Curve intersects with the IRRF at a positive interest rate. But a suﬃcient increase in the current reference price level, without a matching increase in the future reference price level, may shift the IRRF so far to the right that the Fisher Curve only intersects with the IRRF at its horizontal portion, where the interest rate is zero (Figure 5.7). The zero interest solution rate does not menace full employment, but it does place a constraint on the power of the central bank to aﬀect P. Consider a situation where future policy is given. We know current P can be raised by reducing the reference price level and shifting the IRRF right. This also reduces the nominal rate of interest since the increase in P has reduced expected inﬂation. But there is a limit to which current P can be raised by shifting the IRRF right, because eventually current P will be raised so high that deﬂation is so high that the interest rate hits zero. Essentially, the increase in the current reference price level has engineered an expectation of a deﬂation so large that the interest rate is reduced to zero. At such an equilibrium changes in the reference price level would have no eﬀect. Diagrammatically, the upward sloping portion of the IRRF can shift right, but the horizontal portion of the IRRF is still intersected by the Fisher Condition at the same P. Economically, P cannot rise further: if it

108

Inﬂation in a risk free world

did the expectation of deﬂation would be so large the interest would be nonpositive. The price level is determined by P1[1 ] P 0 P1 or P[1 ]P1

(5.41)

The future price level, when the zero bound is no longer constraining, is determined in the normal way, and the current P is related to it by 1 . The powerlessness of the reference price level under a liquidity trap parallels the powerlessness of the money supply under the Quantity Theory. The expression for lnP is exactly the same expression for lnP as prevails under the Quantity Theory with a liquidity trap.

CONCLUSION The Wicksellian approach supplies a perfectly coherent theory of the value of money, one based on equilibrium and optimization no less than the Quantity Theory. But, despite the Wicksellian’s approach discarding the ‘supply and demand’ approach to the value of money, it has a strong homology with the Quantity Theory.

NOTES 1.

2.

3. 4. 5.

The model of this chapter borrows its name from Knut Wicksell (1851–1926), whose Interest and Prices (1898) supplied the one serious rival to the Quantity Theory. Although there are hints of Wicksellianism in the General Theory, it remained largely in the mausoleum of ideas until the 1980s. A reference price level may exist implicitly, even it does not exist explicitly. We might imagine, for example, a central bank adjusting the rate of interest in accordance with the deviation of the actual price of foreign exchange from some ‘reference’ exchange rate. If the actual exchange rate conforms with purchasing power parity then the central bank is, without necessarily realizing it, adjusting the rate of interest in accordance with the deviation of the actual price level from some reference price level. ln [1 i] ln [1 ] ln P1 ln P. We are measuring the interest and proﬁt rate as if it was continuously compounded within the period. , the coeﬃcient of reaction in the IRRF, is assumed to be positive. If it was negative, there would be an inﬁnity of equilibria. The size of the increase in P varies positively with the magnitude of . If is large then P rises by almost the same as PR: only a small deviation of P below PR is suﬃcient to secure a fall in i that is big enough to match the fall in .

The Wicksellian approach 6. 7.

8.

109

must be positive in order for the series to converge; another illustration of the necessity for to be positive. Contrary to the view of some central bankers, there is no ‘inﬂation neutral’ benchmark rate: there is no unique benchmark rate that will ensure zero inﬂation, continually over time. There will always be some magnitude of the benchmark rate that will secure zero inﬂation in a given period. But that magnitude will, ﬂukes aside, change over time. Nevertheless, the benchmark rate has a role in securing zero inﬂation. As the later part of the section argues, the disturbances to P from are eliminated if the benchmark rate moves in tandem with the proﬁt rate. We can ﬁnd the expression for i when all magnitudes are unchanging over time: i R

9. 10. 11.

Whether this is a plausible explanation is another question. Few economies had central banks in the eighteenth and nineteenth centuries. Wicksell’s prescription is presumably more than just a restatement of the Fisher Hypothesis, that trivially implies an equality of the interest rate and the proﬁt rate if there is price stability. The critical thing is for the benchmark rate to move one-for-one with the proﬁt rate. It is not necessary for it to equal the proﬁt rate. A stable and zero inﬂation can also be reached by, R,t 0

12.

_

and

i t t

The insigniﬁcance of the size of the coeﬃcient of reaction, , for the success of the price stabilization rule, might seem to ascribe great power to microscopic increases in the benchmark rate of interest. The model implies that a willingness to raise i by, say, one hundredth of a percentage point for every 10 per cent that P exceeds the stable counterfactual is suﬃcient to ensure price_ stability. Is this pausible? It may be that in the real world a merely minute increase in i may throw doubts on the central bank’s willingness to persist, and that persistance is critical.

PART 3

Inﬂation in a risky world The theories of the value of money explored in Part 2 assumed that the future is perfectly known. Part 3 relaxes this assumption, and reworks the Quantity and Wicksellian theories to allow for risk. This execution of this task requires some preparation; it requires the investigation of the distinct, but highly germane, topic of the social function of debt in risky economies. Chapters 6 and 7, therefore, are devoted to this problem, and establish the functionality of real debt and money debt in risky economies. With the groundwork of Chapters 6 and 7 completed, Chapters 8 and 9 reconstruct the Quantity and Wicksellian theories in an environment of risk.

6. Technological risk and the social function of real debt This chapter advances a theory of social function of real debt1 that will be repeatedly and crucially deployed throughout the remainder of this study of the causes, costs and compensations of inﬂation under an environment of risk. The theory of debt and interest that is advanced is ‘classical’, in that the quantity of debt and its price are determined by real phenomena. The theory is also ‘classic’, in that the model’s composing parts are entirely familiar; general equilibrium, competition, two factor production functions, identical and homothetic preferences. Yet the analysis denies the standard view as to what social function debt performs. In the standard interpretation debt relocates consumption across time. In a phrase: ‘consumption smoothing’. The present analysis contends that the social function of debt is not to smooth consumption; it is not to relocate consumption over time. The function of debt is to deal with risk. It transfers risk from those more risk vulnerable, to those less risk vulnerable, so that the exposure to risk is equalized across all members of society. This process might be called ‘risk evening’. The structure of the argument is to draw the contrast between the redundancy of debt under a perfect foresight model, with its usefulness under risk.

DEBT WITHOUT RISK? This section demonstrates that in a standard work-horse of economic analysis – a Ramsey-Solow growth model – debt is functionless the absence of risk. It will be recalled from Chapter 2 that, under the model’s assumptions: period 0:

C [1 ] C 1

1

K 1 y L1 1

113

YC I

K1 K I

114

period 1:

Inﬂation in a risky world

C [1 ] C2 1

1

K 1 y L2 2

Y1 C1 I1

K2 K1 I1 (6.1)

and so on. This is a determinate system in 3T 2 equations and 3T 2 unknowns. With the Ks determined the proﬁt rate is determined. The explanation of the rate of proﬁt is the familiar Fisherian one of the reconciliation of time preference and the productivity of capital. The rate of proﬁt equals both the rate at which it is possible to transform C today into C later, and the rate at which the consumer is willing to transform consumption today into consumption later. A theory of the rate of interest follows easily. Suppose there is a one period bond that pays a contractual real rate of interest, r. The equimarginal optimization condition would be: j UjC [1 r]UC1

(6.2)

which under homothetic preferences may be written:

j [1 ] Cj C1

1

1 r

(6.3)

which under identical preferences may be written in aggregates:

C [1 ] C 1

1

1 r

(6.4)

A comparison of equations (6.1) and (6.4) allows us to infer, r

(6.5)

We have, then, a theory of the rate of interest. The rate of interest equals the rate of proﬁt, which is, in turn, a matter of time preference and productivity of capital. But we have no theory of debt. The model provides no reason as to why anyone would wish to own, or owe, debt. To press the point: what is the quantity of debt in the model? The quantity is completely indeterminate. It is indeterminate since bonds have no function, either socially or privately. There is no beneﬁt to individuals in buying or selling them; anything a bond can do, capital can do just as well. Do you wish to transfer consumption

Technological risk and the social function of real debt

115

into the future? Buy capital today. Do you wish to transfer consumption from the future to the present? Sell capital today. (And if you do not have any: short sell it).2 Debt is not required for intertemporal trade in consumption. Bonds are a ﬁfth wheel. Neither does debt confer any beneﬁt to ‘society as whole’; the system is welfare eﬃcient without it. The welfare eﬃciency is established from the equality of all persons’ marginal rate of intertemporal substitution with the marginal rate of intertemporal transformation, [1 ][Cj Cj1]1 1 y(K1 L1 ) , which were derived without any reference to debt markets. The upshot is that most extreme anti-usury laws would not have the smallest signiﬁcance for the welfare eﬃciency of the model economy. If such laws completely wiped out any debt that might exist, the economy remains Pareto-eﬃcient. The argument above rests on a certain modelling of economy – a Ramsey-Solow modelling – which cannot claim to be perfectly general. The model assumes, critically, that the economy is capitalized; and it is the existence of capital that makes debt redundant for the intertemporal relocation of consumption. Nevertheless, the modelling is a common one.

RISK AS THE BASIS OF DEBT This section argues that debt acquires a usefulness in the presence of riskiness in technology.3 We show that it, in certain circumstances, provides perfect insurance against proﬁt risk caused by technology shocks. To introduce the argument we note that technological change can be represented by labour augmenting and capital augmenting parameters:

Y y $K %L %L

(6.6)

K may now be more precisely interpreted as the number of livestock, and $ as the weight of each member of the livestock, so that $K is then the mass of livestock, that accounts for the mass of livestock output, Y. To allow for riskiness in technology we suppose now that the state of technology in period 1 is not known but is a random variable; it changes unpredictably. So for any state of the world, s sY 1 s% L y 1 1

s$s K 1 1 s% L 1 1

(6.7)

116

Inﬂation in a risky world

There is, correspondingly, an ex post rate of proﬁt, s , for each state s. s

Y1 s . $s1K1 s. $ y K1 %s1L1 $ . s$

s$ 1

$ 1

(see note 4.) This riskiness in technology does not alter the national income identities, and capital growth equations. But the equimarginal conditions are changed. Assuming that decision-makers know the probabilities of the states of the world, and maximize expected utility. The equimarginal condition that matches the cost of acquiring capital (in terms of current consumption sacriﬁce) to the beneﬁt (in terms of future consumption gain) is now: UjC E[[1 ]UjC1]

Capital

(6.8)

Critically, there is now an equimarginal condition for bonds that does not merely ‘repeat’ that for capital. UjC [1 r]EUjC1

Bonds

(6.9)

The Two Group: Two State: Two Period Model We ﬁrst present a solution to the model with risk in terms of a further simpliﬁed model. We suppose there are just two periods: Period 0 and Period 1. We also suppose there are just two possible states of the world in Period 1: State 0 and State 1. State 1 is the high output state of the world. State 0 is the low output state of the world. We also assume, for the time being, that the high output state of the world is also the high proﬁt rate state of the world, and the low output state of the world is also the lower proﬁt rate state of the world. (This assumption can be relaxed.) The probability of State 0 is p. We additionally suppose that there are two groups or ‘classes’ that make up the population, F and G. The two groups are completely homogeneous in composition, but diﬀer from each other in their relative factor endowments. Group F has a relatively high amount of capital, and group G has a low amount of capital. The two groups have identical preferences.

117

Technological risk and the social function of real debt 0C G 1

1C G 1

E

0C F 1 0C F 1

Figure 6.1

Social and private endowments in a two state model of risk

Finally, we suppose, for the moment, that saving of the two groups in period 0 is exogenous. Consequently, the magnitude of the stock of capital in period 1 is exogenous. Further total saving in period 1 simply equals the negative of the capital stock, as there is no wish to have capital in period 2. Thus the consumption endowment of each group in period 1 equals their income from the factors they own, plus their endowment of capital.5 The consumption endowment can be represented in an Edgeworth Box Diagram (Figure 6.1). 0CF1 consumption of F in state of the world 1 in period 1, and so on. As F and G are expected utility maximizers, it helps to plot on the Box the ‘expected utility indiﬀerence curves’. The expected utility curves

118

Inﬂation in a risky world

Iso-expected utility

Iso-expected consumption

1C F 1

45 degrees 0C F 1

Figure 6.2 Preferences and expected consumption in a two state model of risk of F have a slope of p/[1p] along the 45 degree ray from the south-west origin.6 It also proves helpful to plot on the Box ‘iso expected consumption’ loci, each locus indicating the combinations of State 1 consumption and State 0 consumption which yield a certain level of expected consumption. These ‘iso expected consumption’ curves have a slope of p/[1p].7 Thus the expected utility indiﬀerence curves and ‘iso expected consumption curves’ are tangential at the 45 degree line (Figure 6.2).

Technological risk and the social function of real debt

119

0C G 1

45 degrees

1C G 1

1C F 1

45 degrees 0C F 1

Figure 6.3

Most consumption points are Pareto ineﬃcient

G’s expected utility indiﬀerence curves can also be plotted on the Edgeworth box. The expected utility curves of G have a slope of p/[1p] at the 45 degree ray from the north-east origin. Plainly, most distributions of consumption between the F and G are Pareto ineﬃcient (Figure 6.3). But it is also plain that there exist eﬃcient distributions, that are located at the tangency of the indiﬀerence curves of F and G. This equality of the slopes of the indiﬀerence curves may be written: 0&F 0&G 1&F 1&G

(6.10)

120

Inﬂation in a risky world

where s& j

sU j C1

EUC j

(6.11)

1

(6.11) provides a description of eﬃcient consumption: an eﬃcient distribution of consumption requires that the F’s marginal utility in the ‘up’ state relative to F’s marginal utility in the ‘down’ state (each normalized by the expectation of F’s marginal utility) must equal G’s marginal utility in the ‘up’ state relative to G’s marginal utility in the ‘down’ state (each normalized by the expectation of G’s marginal utility). Since the marginal utility of consumption depends on the level of consumption, this requirement of welfare eﬃciency must imply some sort of relation between F’s consumption in the up state, relative to the down state, with G’s consumption in the up state, relative to the down state. If we assume homogeneous and identical preferences the implied relation is, 0CF 0CG 0C 1 1 1 1CF 1CG 1C 1 1 1

(6.12)

Under identical and homothetic preferences, eﬃcient management of social risk means that if society’s consumption is x per cent higher in the favourable state compared to the unfavourable state, then every individual’s consumption is x per cent higher in the favourable state compared to the unfavourable state. All boats must rise and fall with the tide, equally. That is the welfare eﬃcient way of dealing with risk under identical and homothetic preferences. A diagrammatic corollary of the eﬃcient management of risk is that eﬃcient points lie upon the diagonal of the Box (Figure 6.4). We are now led to the question: What is the relation between the eﬃcient allocation of consumption and the market allocation of consumption? The analysis implies that the market for real bonds will, with some exceptions noted below, secure this welfare eﬃcient allocation. The critical observation to sustain this contention is that F can shift their consumption point from their endowment point by selling bonds in period 0 in order to buy capital in period 0. If bonds are not to dominate capital, or vice versa, it must be that 1 r 0 . This implies that F’s purchase of bonds, ﬁnanced by the sale of an equal value of capital, will reduce F’s consumption in period 1 by 1 r in state of the world 1, and increase F’s by r0 in state of the world 0. In other words, F’s consumption point is sent ‘south east’ by F’s purchase of bonds. Greater purchases of bonds by F will send F’s consumption further ‘south-east’ along a line with a slope in absolute terms of [ 1 r] [r 0 ] . Conversely, F’s purchase of capital, ﬁnanced by the sale of bonds, will shift the F’s consumption point ‘north west’ (Figure 6.5).

Technological risk and the social function of real debt

121

0C G 1 1C G 1

Diagonal = locus of efficient consumption

1C F 1

45 degrees 0C F 1

Figure 6.4

The locus of eﬃcient distributions in consumption

In the same way, G can shift themselves along the same frontier by either buying capital and selling bonds, or, selling capital and buying bonds. The Box Diagram shows the general equilibrium, where F wants to buy the same number of bonds that G wants to sell (Figure 6.6). At the maximizing point for, both G and F, the slope of the frontier ( 1 r) (r 0 ) equals the slope of the iso-utility frontier, p0UC1 [(1 p) 1UC1].8 Evidently, F and G at the equilibrium point are on a higher level of expected utility than at the autarkic point E. There are social gains from trading bonds for capital. How can we understand these gains? Essentially, those burdened with risk have paid others to relieve them of part of their risk. By moving south-east of their endowment, F has reduced their expected consumption; but, in approaching the 45 degree line, F has increased the security of their

122

Inﬂation in a risky world

F Buying K, F Selling B

E

F Selling K, F Buying B

Efficiency locus

1C F 1

45 degrees 0C F 1

Figure 6.5 The consumption point is shifted by the exchange of bonds and capital consumption. F in this situation is a ‘hedger’ (Figure 6.7). At the same time, G, by shifting their consumption ‘south-east’ of their endowment increases their expected consumption, but, by withdrawing further from their 45 degree line, reduces the security of their consumption. Group G are speculators. In the general equilibrium, one group will be Hedgers and the other Speculators: one group will be increasing their expected income at the cost to its security, and the other will be reducing their expected income, but to the beneﬁt of their security. Hedgers pay speculators to assume part of their risk. The social function of debt is established: real debt allows the relatively risk insulated to supply ‘insurance’ to the relatively risk exposed over unpredictable ﬂuctuations in factor prices.

123

Technological risk and the social function of real debt 0C G 1

1C G 1

Efficiency locus

E

C

1C F 1

45 degrees 0C F 1

Figure 6.6

Equilibrium

Debt provides an exchange of risk and expected consumption that is mutually improving. It gives consumption security to those who are consumption vulnerable, and consumption expansion to the (relatively) consumption invulnerable. It drives the variance of consumption of the two groups to equality. It shares out the risk between the two groups. Like many things in economics, this conclusion has an obvious aspect.9 It is suﬃciently ‘obvious’ that, historically, the critics of interest income have mocked it.10 It is not only eﬃciency that is served by the market for debt: equity might also be said to be helped. As the consumption point lies on the diagonal, we may conclude that consumption of the two groups goes up (or down) in the two states relative to period 0 by the same proportion. We have already pointed out that this is socially eﬃcient, but does it not also have an equitable

124

Inﬂation in a risky world 0C G 1

1C G 1

Efficiency locus

E C

1C F 1

Budget constraint 0C F 1

Figure 6.7

Hedging and speculating

ring about it? Both groups make the same proportional sacriﬁce in lean years; both reap the same proportionate gain in the fat years. Debt means that both groups will tighten their belts by the same proportion during the lean years, and loosen them in the same proportion in the fat years. Of course, the two groups may have a very diﬀerent share and it may also be suggested that equity would require those with a higher mean consumption to tighten their belts, not by the same proportion as those with less, but by more.11 Hedgers and Speculators/Workers and Capitalists We have analysed debt as a trade between those whose consumption endowments are more risk vulnerable, and those less risk vulnerable. The

Technological risk and the social function of real debt

125

more risk vulnerable experience disproportionately high incomes during the up, and disproportionately low incomes during the down. They become hedgers. Those less risk vulnerable experience income increases less than proportionate during the up, and decreases less than proportionate during the down. They become ‘speculators’.12 Can we relate the categories of hedger and speculator to other categories in the analysis? Can we link them to the factors which explain why some experience increases in income that are disproportionate? To the distribution of endowments of labour and capital? To the relative importance of labour augmenting and capital augmenting change? To the elasticity of technical substitution? With the help of these, can we identify ‘who’, and ‘under what conditions’ will be the hedgers, and ‘who’, and ‘under what conditions’, will be the speculators? No. No sweeping simplicities govern the identiﬁcation of hedger and speculator with economic interest. The point can be made by analysing the identiﬁcation of the hedger and speculator in the case of an extreme segregation in factor ownership; where the two groups can be identiﬁed as ‘capitalists’ (who own only capital and no labour) and ‘workers’ (who own only labour and no capital). The relativity of the consumption endowment of workers and capitalists is: ' $K

(6.13)

' wages bill profits bill If this ratio increases in the up state of the world, then workers are hedgers, and capitalists are speculators, as workers do disproportionately well in the up. If it decreases in the up state of the world, then workers are speculators, and capitalists are hedgers. If it is unchanged in the up state of the world, then there are neither hedgers nor speculators. An equal augmentations of % and $ in the up state will leave the ratio unchanged in the up state. Thus Hicks Neutral technical shocks will not cause the ratio to diﬀer in the two diﬀerent states of the world. The endowment point, E, is ‘on the diagonal’; consumption is eﬃciently distributed, without the assistance of debt. But the ratio will change if $ and % are augmented at diﬀerent rates. Suppose that in the up state $ is augmented at a greater rate than %. And suppose the elasticity of technical substitution, (K L , exceeds 1. Then the workers’ consumption endowment falls relative to capitalists’ in the up

126

Inﬂation in a risky world

state; workers miss out on the up state; capitalists do disproportionately well; and so capitalists are hedgers and workers are speculators. Proof. Assuming $ is augmented at a greater rate than %, the marginal product of capital grows at a lesser rate than $’s rate of augmentation, and so grows slower than $K. And because (K L , exceeds 1, the assumption that $ is augmented at a greater rate than %, means that ' grows still more slowly than .13 Therefore workers’ endowment falls relative to capitalists’ in the up state. Suppose now that in the up state $ is augmented at a lesser rate than %. And suppose the elasticity of technical substitution, (K L , exceeds 1. Then workers’ consumption endowment rises relative to capitalists’ in the up state; workers do especially well in the up state; capitalists do not do so well; and so capitalists are speculators and workers are hedgers. Proof. As $ is augmented at a lesser rate than %, the marginal product of capital grows at a greater rate than $’s rate of augmentation, and so grows at faster than $K. And because (K L , exceeds 1, the fact that $ is augmented at a lesser rate than %, means that ' grows still faster than . Therefore workers’ endowment rises relative to capitalists’. To summarize: if (K L exceeds 1 then a bias towards capital augmentation makes capitalists hedgers and workers speculators, while a bias towards labour augmentation makes capitalists speculators and workers hedgers. This constitutes ambiguity enough: both workers or capitalists may be hedgers, and both workers or capitalists may be speculators, depending on the bias of technical shocks towards labour augmentation or capital augmentation. But if (K L is less than 1 then even the conditional identiﬁcations of the previous paragraph are unwarranted. If $ is augmented at a greater rate than %, the marginal product of capital still grows at a lesser rate than $’s rate of augmentation, and so $K still grows faster than . But because (K L is less than 1, the fact that $ is augmented at a greater rate than %, means that ' grows faster than . We can no longer be certain the ratio falls. If, by contrast, $ is augmented at a lesser rate than %, the marginal product of capital still grows at a greater rate than $’s rate of augmentation, and so $K grows at a lower rate than . But if (K L is less than 1, the fact that $ is augmented at a lesser rate than %, means that ' grows slower than . Therefore, we can no longer be certain the ratio rises. The identiﬁcation of hedgers and speculators with the economy’s various economic interests cannot be secured with any conﬁdence theoretically, but only empirically. Experience teaches that the proﬁt bill is more unstable over the business cycle than the wage bill: we will therefore henceforth assume that the interest with the higher capital/labour ratio, F, is the hedger.

Technological risk and the social function of real debt

127

A SUPPLY AND DEMAND EXPOSITION The Edgeworth Box is a useful tool for introducing the topic of risk evening, but being two dimensioned it is too special. A supply and demand treatment has a desirable generality. The demand schedule and the supply schedule for bonds may be derived from the equimarginal conditions: UCj E[UCj 1[1 ]]

Capital: Real Bonds:

UCj [1 r]EUCj 1

Substituting: E[[1 ]UCj 1] [1 r]EUCj 1

(6.14)

E r cov( , & j )

(6.15)

Or:

where & j

UCj 1 EUCj 1

(6.16)14

This equality is the familiar risk premium type expression in ﬁnance: the excess of the expected return on capital over bonds is the negative of the covariance of the proﬁt rate with the normalized marginal utility of consumption. It will prove convenient to deﬁne: )j cov( , & j )

(6.17)

and rewrite the condition as, E r )j

(6.18)

The LHS of this equality may be considered the ‘price’ of bonds. The RHS is a magnitude that will change with the amount of bonds owned, or owed. We again think in terms of two groups, F and G; one with a higher endowment of capital relative to labour, F; and one with a lower endowment of capital to labour, G. E r )F

(6.19)

128

Inﬂation in a risky world

Er

F at B0

Demand for bonds

B Figure 6.8

The demand curve for bonds E r )G

(6.20)

At B 0 the magnitudes of )F and)G

are determined by the co-movement of each group’s consumption endowment with the proﬁt rate. The group with the more positive capital/labour ratio, F, is assumed to have the more negative co-movement in marginal utility with proﬁt, and so a more positive ) j . The group with the smaller capital/labour ratio, G, will have a less negative co-movement in marginal utility with proﬁt, and so a smaller positive ) j (and possibly even a negative one). We can plot )F and )G

as the vertical axis intercepts of a ﬁgure that plots E r on the vertical axis, and bond issues on the horizontal axis. As F’s ownership of bonds becomes positive, the magnitude of )F falls, by the logic explained earlier by means of the Box: the bond holder, as ‘hedger’, is reducing the comovement of their consumption opportunity with the proﬁt rate. So E r falls. We are tracing out a ‘downward’ sloping demand curve for bonds with respect to the risk premium (Figure 6.8). As G owes positive rather than zero, the magnitude of )G

rises, also by the logic explained earlier: the bond seller, as ‘speculator’, is increasing the comovement of consumption with the proﬁt rate. So E r rises. We are tracing out a ‘upward’ sloping supply curve for bonds with respect to the risk premium (Figure 6.9).

Technological risk and the social function of real debt

129

Er

Supply of bonds

G at B0 B Figure 6.9

The supply curve of bonds

It is easy to persuade oneself that at the point where the demand curve for bonds intersects the supply curve for bonds, the negative of the covariance of the proﬁt rate with the marginal utility of the two groups are the same. )F )G

(6.21)

But this equality is a signal of the eﬃcient distribution of consumption. For this equality is satisﬁed if: &F &G

(6.22)

in all states of the world, which, in the two state case implies: 0&F 0&G 1&F 1&G

(6.23)

But we have already concluded, from the Box analysis, that this is condition welfare eﬃciency in the two state case, and a property of the market equilibrium. The welfare eﬃciency of the market equilibrium is underlined by the fact that the intersection of the two schedules is interpretable as the quantity of debt that maximizes the sum of ‘creditors’ surplus’ and ‘debtors’ surplus’.

130

Inﬂation in a risky world

Consider: j’s marginal expected utility of a unit of capital E[[1 ]UCj1] E[1 ]EUCj1 cov( ,UCj1 ) j’s marginal expected utility of a unit of bonds [1 r]EUCj 1 Then marginal expected utility of a purchase of bonds by way of a sale of capital [1 r]EjUC1 E[[1 ]EUCj 1] cov( ,UCj 1 ) [r E cov( ,&j )]EUCj 1

(6.24)

Thus the current consumption equivalent of a purchase of bonds by way of a sale of capital is [r E cov( ,& j )]

EUCj 1 UCj

1 [ [E r] cov( ,&j )]1 r 1 [)j [E r]]1 r

(6.25)

For any exchange of bonds for capital ‘on the margin’, the magnitude of this expression is obviously zero, since in equilibrium E r ) j . But for any infra marginal exchange the magnitude is positive, and equals the excess of ) j over E r, that equals the distance between the demand curve, and the ‘price of bonds, E r. Another way of looking at it: the creditor in buying bonds loses an amount equal to B[E r], but unburdens, on the debtor, a ‘value’ of risk equal to aa bb cc dd. The creditor’s surplus in Figure 6.10 is the triangle aa bb ee. By a parallel logic the surplus of debtors is represented in Figure 6.11. For borrowers, they receive a net income equal to B[ r], but they pay a risk expense equal to the sum of inframarginal magnitudes of )G

.

131

Technological risk and the social function of real debt

Er Net benefit of creditors

aa

bb ee

B[r] dd

cc B

Figure 6.10

The ‘surplus’ of creditors

Er

Net benefit of debtors

aa bb

cc

B

Figure 6.11

The ‘surplus’ of debtors

Multiple States of the World The ‘supply and demand’ presentation, that crucially implied the equality of the covariances, in no way assumed just two states of the world. Debt performs its insurance function if there are many states of the world, rather than just two.

132

Inﬂation in a risky world

Multiple Groups Debt performs its insurance function if there are several groups rather than just two. The supply and demand presentation can be adjusted to allow for many distinct groups with a demand curve for bonds, yielding an aggregate demand curve for bonds. And it can be adjusted to allow for many distinct supply curves, yielding an aggregate supply curve of bonds. In equilibrium each group will have the same ) j .

THE DETERMINANTS OF THE QUANTITY OF DEBT AND INTEREST The Determinants of the Quantity of Debt Apart from its risk evening function, the quantity of debt will prove signiﬁcant in the later analysis of the cost of unpredictable inﬂation. The quantity of debt depends on how much risk is unequally shared; the disparity in relativities across states of consumption of the two groups, in the absence of debt. If there is a large diﬀerence – if the two groups experience very diﬀerent proportionate change between the two states – there will be much debt. If there is only a small diﬀerence, there is little debt. And if there is no diﬀerence there is no debt at all. If the two groups’ consumption rises and falls by the same proportion, in the absence of debt, – if the endowment lies on the diagonal – then debt will be zero. Risk is already eﬃciently shared, without the assistance of debt. (In terms of Figure 6.10, the demand and supply schedules intersect above the point of origin.) There are a number of circumstances under which risk be eﬃciently (equally) shared even without debt. One is if both groups have the same labour/capital ratio. In this case both groups’ consumption ﬂuctuates by the same amount; a proportion equal to the ﬂuctuation in consumption of society as a whole. In this case risk is already eﬃciently shared in the absence of debt. Debt has no function. But even if two groups have diﬀerent capital labour ratios, the technological shock may be such that there is no diﬀerence in the proportionate diﬀerence in the consumption of the two groups across the two states. This will occur if the technological shock is equally capital and labour augmenting. .

.

%$

(6.26)

Technological risk and the social function of real debt

133

The Determination of the Real Rate of Interest The preceding section has advanced a theory of debt. Might it also contain a theory of the rate of interest? The preceding analysis of equilibrium might suggest that it does, as the analysis of equilibrium has implied a rate of interest as part of that equilibrium. There is the temptation to think that there is being advanced some sort of risk evening theory of interest, whereby the interest rate is that which achieves eﬃcient risk evening. In fact the analysis cannot by itself constitute ‘theory’ of the rate of interest at all, since the analysis, and its implications, are conditional on some arbitrary assumption about the magnitude of aggregate saving. And aggregate saving will be determined by preferences and technology in the familiar Fisherian manner. Thus the equilibrium magnitude of the interest rate will reﬂect preferences and technology just as it does in the model without risk.

THE SOCIAL FUNCTION OF DEBT: SOME GENERALIZATIONS AND QUALIFICATIONS Wage Fluctuations Debt will not provide insurance over ﬂuctuations in wages per se. Debt cannot provide insurance against ﬂuctuations in wages that occur by themselves, and without any concomitant shock to proﬁt rates.15 Only wage movements that share a common cause with movements in the proﬁt rate can be insured. This can be easily seen with the two state case. If there are shocks to wages only then total consumption still varies with the shock to wages, but, by assumption, the proﬁt rate, , is the same magnitude in both states of the world. Borrowing to buy capital either increases income in both states of the world (if r ), or reduces income in both states of the world (if r ). But it will not increase income in one, and reduce it in another, and so yield the NW–SE consumption opportunity locus that will take society to the diagonal.16 Asymmetric Information The analysis does (and must) assume that the probabilities of the states of the world are known by all persons. But it does not require everyone know what the current state of the world is. Consider an example when technology shocks hit only proﬁts, and not wages. Workers need not know what state it is for the market’s sales of bonds and capital to secure synchronization of consumption.

134

Inﬂation in a risky world

Government Debt The preceding analysis of debt is an analysis of private debt, and does not pretend to analyse government debt. But does the presence of government debt qualify, modify or invalidate the analysis of private debt? Any government debt amounts to the existence of state of the world invariant incomes of a certain magnitude, equalling the liability to the debt holder less their tax liability on account of the debt. This means that even in a Hicks Neutral world there would be debts and credits: for those with positive state of the world invariant income (and so less risk vulnerable because of their ‘safety net’ of positive income ﬂows from the government regardless of the state of the world) would play speculator, and borrow from those with negative state of the world invariant incomes (who would be playing hedger on account of their greater risk vulnerable because of their ‘danger net’ of negative income ﬂow from the government regardless of the state of the world). Essentially, the government debt induces intra private debt. But public sector debt does no harm, and eﬃciency is still secured. State Contingent Government Transfers We have completely ignored the ‘real’ safety nets of welfare states. These could displace part of the function of debt. Suppose we have a Peter-to-pay-Paul transfer system, that by taxation and subsidies changes the share each individual gets. If these work, by making the bad times not so bad for the worst hit, and the good times not so good for the least hit, then the function of debt is displaced. Non-Identical Preferences We have assumed that preferences are identical, and homothetic. That has allowed us to move from the general eﬃciency condition: 0&F 0&G 1&F 1&G

to the more speciﬁc conclusion 0CF 0CG 1 1 1CF 1CG 1 1

135

Technological risk and the social function of real debt 0C G 1

Budget constraint

C

Efficiency locus

1C F 1

E

Iso-expected consumption locus

45 degrees 0C F 1

Figure 6.12

Equilibrium with non-identical preferences

Eﬃciency consequently required all boats to rise and fall with the tide. And that meant that lenders were those persons more exposed to ﬂuctuation, and borrowers less exposed. But if preferences are homothetic but not identical – if there are diﬀerent coeﬃcients of relative risk aversion – the debt still achieves its purpose of eﬃciently sharing out risk – but the sharing out may change; each person’s consumption will not rise and fall by the same proportion as total consumption. Consider the case where F is less risk averse than G. There is less curvature in F’s indiﬀerence curves, and the contract curve is no longer the

136

Inﬂation in a risky world

diagonal, but a curve to the northwest of the diagonal. It is no longer true that eﬃcient risk sharing requires all person’s consumption to go up and down by the same proportion. And it is no longer necessarily true that lenders are those more exposed to ﬂuctuation, and borrowers are those less exposed. It is no longer true that debt is functionless when all go up and down by the same proportion; if the endowment is such that all do go up and down by the same proportion, there shall still be a trade in risk as the relatively risk averse seek stability in their incomes at the cost of expected income (Figure 6.12). But it remains true that debt exists and serves a function when there is social risk, save in the ﬂuke that E is on the contract curve. It remains the case that in the presence of social risk debt will operate so that both F and G are better oﬀ, social eﬃciency is obtained, and )F )G

.

CONCLUSION Any economy faced by technological shocks is confronted with a task if welfare is not to be wasted: achieving in the face of these shocks a synchronization of consumption across interests. Debt performs that task.

NOTES 1. 2. 3. 4.

Real debt‘inﬂation indexed’ debt. Is there any diﬀerence between short-selling capital and selling a bond? There seems no operational diﬀerence in a riskless world. We take this to be saying that in a riskless world short-selling capital can do the job of bonds. All risk is technological risk. We assume bonds have no credit-risk. Proof: The rate of return dC1 dC 1 But C $[K1 K] Y and C1 $1[K2 K1] Y1 Keeping K and K2 constant, Y1 dC1 K $1 $K $ 1 y %1 $1 1 1 1$ dC 1

5. 6.

The expression indicates that with technological change the proﬁt rate may be negative; in the case of if is technical regress, or negative shocks. That is, natural disasters and so on may produce negative proﬁt rates. Or, more properly, the ‘mass’ of capital: $K. Given EUF pU( 0CF1 ) [1 p]U( 1CF1 ) then dEUF p0Ud0CF1 [1 p]1UdCF1 . As 0U 1U along the 45 degree ray, if dEUF 0 along the 45 degree ray then d 0CF 1 ([1 p] p)dCF1 .

Technological risk and the social function of real debt 7. 8.

137

Given ECF1 p0CF1 [1 p]1CF1 then dECF1 pd0CF1 [1 p]d1CF1 . If dECF1 0 then d0CF1 ([1 p] p)d1CF1 . This equality is simply the diagrammatic manifestation of the implication of two ﬁrst order conditions; [1 r]EUc1 E[[1 ] Uc1]. 1 r

p0Uc1 0 r [1 p]1Uc

9. 10.

1

implies [1 p]1 1Uc1 p0 0Uc1 r[[1 p]1Uc1 p0Uc1], or E UC1 rEUC1. And is well recognized in theory. Students of CAPM are familiar with the equalization of the painfulness of risk by means of some investors holding positive money, and some ‘negative money’ (debt). Keynes did not only disparaged the view that interest was a consequence of the productivity capital. He not only dismissed the idea that interest had something to do with thrift. Keynes completely disregarded the insurance aspect of debt. The whole ethic of prudence, sobriety and cosy nest-eggs and was antipathetic to the spirit of Bloomsbury. She learnt, to her horror, that Margaret, now of age, was taking her money out of the old safe investments and putting it into Foreign Things, which always smash. Silence would have been criminal. Her own fortune was invested in Home Rails, and most ardently did she beg her niece to imitate her. ‘Then we should be together, dear.’ Margaret, out of politeness, invested a few hundreds in the Nottingham and Derby Railway, and though the Foreign Things did admirably and the Nottingham and Derby declined with the steady dignity of which only Home Rails are capable, Mrs. Munt never ceased to rejoice, and to say, ‘I did manage that, at all events. When the smash comes poor Margaret will have a nest-egg to fall back upon.’ (E.M. Forster, Howards End, Chapter 3)

11. 12.

13. 14. 15.

The ‘sharing the pain’ aspect of debt is not neutral between groups F and G in that (assuming F is the hedger), the pain is ‘shared’ out in a way that reduces the expected income of F. It may be that the income of both groups increase in the ‘up’ state, but one disproportionately. Alternatively, it is possible that the income *f hedgers goes up so much in the ‘up’ that speculator’s income may be lower in the up than the down. (In the terms of the Figure, E is above G’s 45 degree ray.) Reﬂecting the fact that '/ is a negative function of the eﬀective capital/labour ratio if (K L 1. &j Ucj1. Since the marginal utility is arbitrary up to a multiplying factor we can always choose a multiplying factor so that EUc1j 1. Consider for example the technology,

Y g L L K K K

16.

where is stochastic. The proﬁt rate is non-stochastic g(L/K) –g(L/K)L/K. Conceivably, insurance over wage ﬂuctuations per se could be achieved by workers selling their labour for capital – a form of slavery where one enslaves part of oneself (say each Monday) in order to buy capital.

7. Monetary risk and the social function of money debt Chapter 6 advanced a theory of the social function of real debt. It was argued real debt minimized the welfare cost posed by unpredictable productivity shocks. This chapter turns to the social function of money debt. It demonstrates that money debt also has a social beneﬁt, and a private one. The chapter therefore provides an answer to a puzzle: Why is any lending or borrowing done in terms of money, when such money debt exposes the lenders’ wealth to inﬂation risk? The ‘received’ answer to this question might go as follows: All money bonds are ‘really’ real bonds; they just take the form of money bonds. In fact, all money incomes and outlays are ‘really’ real, but are merely expressed in the form of money. An oﬀer of a wage income, for example, is an oﬀer of a real wage, but will be expressed in the form of an oﬀer of a money wage. Similarly, a loan of a certain amount of real resources is expressed in the form of a loan of money.1 Wages or loans are in the form of money on account of the convenience of using the medium of exchange – money – as the unit of account, rather than (say) the consumer price index. The pursuit of this convenience does create the chance of some unanticipated inﬂation changing the real value of a debt, but that chance is neglected, or negligible.

The conclusion of this logic is that money bonds are just proxies for real bonds, proxies born of insuﬃcient appreciation, or a benign neglect, of inﬂation risk. The implication is that money bonds are redundant. Money bonds are without social function. Everything could be done and done better with inﬂation indexed bonds. The thesis of this chapter is that money bonds are not redundant. Money bonds do have a social function. If money bonds did not exist, it would be necessary to invent them. Or, it would, at least, be advantageous to invent them. The advantage of money bonds lies in the reduction they secure in the unpredictability of consumption that arises from the operation of real balance eﬀects in an environment of unpredictable money shocks. It is the very vulnerability of money bonds to inﬂation that makes them useful in immunizing the economy against unpredictable redistributions of purchasing power caused by real balance eﬀects. For it is the very dependence of the real value of money debt to the price level that allows money bonds 138

Monetary risk and the social function of money debt

139

to operate as a counter-weight to the dependence of the real value of money balances to price level. It is a case of taking a non-neutrality to beat a nonneutrality. The upshot of the analysis is that, rather than money bonds being born of a negligence of inﬂation risk, it is the very consciousness of inﬂation risk that creates a demand and a supply of money bonds. The chapter proceeds by ﬁrst demonstrating that money debt is functionless in economies without monetary risk, and then showing the functionality of money debt in economies with money risk.

MONEY DEBT AS REDUNDANT DEBT In a monetary economy there may be said to be two types of risk: the risk of a shock to technology (that will impact on the rate of proﬁt), and the risk of a shock to money (that will impact on the value of money). The presence or absence of these two types of risk creates a matrix of four possibilities: ● ● ● ●

both technological and monetary risk absent; technological risk present, but monetary risk absent; monetary risk present, but technological risk absent; both technological and monetary risk present.

This section demonstrates that money bonds are redundant in the ﬁrst two types of economy- economies without monetary risk. Both Technological and Monetary Risk Absent If both technological or monetary risk are absent then the optimization conditions are, capital

UC [1 ]UC1

(7.1)

real bonds

UC [1 r]UC1

(7.2)

money

U UC Uh 1 C1

(7.3)

i U UC 11 C1

(7.4)

money bonds

Inspection indicates that the equimarginal conditions for both real bonds and money bonds are redundant. They simply ‘repeat’ the relation between

140

Inﬂation in a risky world

current and future marginal utility that capital secures. The markets for real and money bonds do not do anything that the capital market does not do. Technological Risk Present, but Monetary Risk Absent Suppose now that the future path of the price level is known, but the proﬁt rate is risky. The optimisation conditions are: capital

UC E[UC1[1 ]]

(7.5)

real bonds

UC [1 r]EUC1

(7.6)

money

EU UC Uh 1 C1

(7.7)

money bonds

i EU UC 11 C1

(7.8)

Inspection reveals that real bonds now do not merely repeat the relation between future and current marginal utility secured by capital. As was argued in Chapter 6, the capital and real bonds conditions secure – in fairly wide circumstances – the socially eﬃcient sharing of risk imparted by technology shocks. The conjunction of markets for capital and real bonds secures the welfare eﬃcient ‘synchronization’ of the consumption of all interests in the face of these shocks; the consumption of all interests rise and fall in tandem with the social supply of consumption. This synchronization of consumption synchronization could be done by money bonds; as they ‘repeat’ the relation between current and future marginal utility that real bonds secure. But money bonds are not required for synchronisation. Money bonds remain a ﬁfth wheel.

MONETARY DEBT AS AN ANTIDOTE TO MONETARY RISK Monetary Risk but no Technology Risk We turn now to the third possibility: where the future path of the proﬁt rate is known but, the price level is risky. (This is a rather peculiar scenario, but it is analytically useful.) The four optimization conditions now are: UC [1 ]EUC1

capital

(7.9)

Monetary risk and the social function of money debt

real bonds money

money bonds

UC [1 r]EUC1

U UC Uh E 1 C1

i UC E UC111

141

(7.10) (7.11)

(7.12)

The real bonds condition is now redundant. It simply replicates the capital market, and does not do anything that the capital market does not do. By contrast, the money bonds’ equimarginal condition does not merely ‘repeat’ the equimarginal condition for capital. In fact, we will argue that money bonds secure the socially eﬃcient sharing of consumption in the face of risk in the money supply. To introduce this contention, it will be useful to ﬁrst consider how unpredictability in the money supply may impart unpredictability to the consumption of individuals. This possibility can be traced to the ‘real balance eﬀects’ which redistribute consumption from one individual to another whenever there is a diﬀerential in the growth rates of the nominal endowments of diﬀerent persons.2 j k 0

(7.13)

Any person who experiences a growth rate in their nominal endowments in excess of the growth rate of some other person will experience an increase in purchasing power relative to that other person. There will be a redistribution of consumption to the person with the larger rate of growth, and from the person with a smaller rate of growth. But redistribution by way of real balance eﬀects does not itself cause unpredictability in the consumption. If the diﬀerential does not vary with the ‘state of the world’: sj sk

for all states of the world s

(7.14)

that is, if it is perfectly predictable; then the redistribution is the same in all ‘states of the world’, and no unpredictability has been imparted to the consumption of any individual. But if the diﬀerential varies from state to state: sj sk

for all s

(7.15)

142

Inﬂation in a risky world

then the size of the redistribution varies, according to the state of the world, an unpredictability has been imparted to consumption.3 The signiﬁcance of the creation of unpredictability in consumption by unpredictability in money can be better appreciated by registering once more our conclusion that welfare eﬃciency requires the consumption of all individuals to be ‘synchronized’ with aggregate consumption. That is, under identical and homothetic preferences, the consumption of all individuals should rise and fall in proportion to aggregate consumption. In other words each individual’s consumption should have exactly the same riskiness as aggregate consumption. But when technology is perfectly predictable (as we are presently assuming) then there are no shocks to aggregate consumption, and so no unpredictibility (or riskiness) in aggregate consumption. Therefore, eﬃciency requires there be no unpredictibility (or riskiness) in the consumption of any individual. The variance of each individual’s consumption should be zero. But a less than perfectly predictable diﬀerential in money endowment growth rates, will (it would seem) impart some unpredictability to consumption, and thereby violate welfare eﬃciency. An Edgeworth Box Illustration The notion that an imperfectly predictable diﬀerential in money endowment growth rates poses a threat to welfare eﬃciency can be illustrated by a simple case that can be represented on an Edgeworth Box. The same Box can demonstrate that money bonds will remedy this problem. Let the economy consist of two persons – F and G. Let there be two possible money supply states, 0 and 1. In state zero there is no change in the money supply. In state 1 there is a shock increase in the money supply. Thus 1 0. Technology, however, is completely invariant, with the consequence that the total consumption endowments are the same in states 0 and 1, and the Box is a ‘square’. The action takes place in the change in the distribution of endowments within the square. Suppose, as the baseline case, that F and G’s nominal endowment grow at the same rate in state 1 (Table 7.1). Table 7.1 State of the world 0 1

F and G’s nominal endowments growing at same rate Aggregate money growth

F

F

FG

0

0

0

0 0

143

Monetary risk and the social function of money debt

In this case the diﬀerential is zero in both states. There is no redistribution of purchasing power in state 1. In terms of the Edgeworth Box the purchasing power of F and G lies on the diagonal, which under our present assumptions is identical to the 45 degree line. But suppose now that all of the increase in money in state of the world 1 is received by F (Table 7.2). In state 1 there is a redistribution of purchasing power from F to G. Thus, in terms of the Box, the endowment point E is pushed oﬀ the 45 degree line, and to E. Table 7.2

F’s nominal endowment growing faster than G’s in state 1

State

Aggregate money growth

F

G

FG

0

0

0

0

0

MF M

0

MF M

1

0C G

1C G

Efficiency locus E

E 1C F

45 degrees 0C F

Figure 7.1 Unpredictibility in the diﬀerential in growth in money endowments creates unpredictability in consumption

144

Inﬂation in a risky world

At E the expected utility indiﬀerence curves intersect, and this would seem to spell a welfare ineﬃciency. However, the existence of money bonds will secure eﬃciency. The critical observation to sustain this contention is that F can shift their consumption point from their endowment point, E, by having previously sold capital in order to buy money bonds. Assuming 1 i 0 (so that neither bonds nor capital dominate each other), a purchase of money bonds by F, ﬁnanced by the sale of F’s capital, will reduce F’s consumption by in state of the world 1 (when real interest rates are lower than the proﬁt rate), and increase F’s consumption in state zero (when real interest rates are higher than the proﬁt rate). More precisely, F’s consumption falls by 1 i in state of the world 1, and rises by i[ 0] in state zero. F’s purchase of bonds (from G), ﬁnanced by the sale of capital (to G) sends F’s consumption point south-east. Greater purchases will send F’s consumption further ‘south-east’ along a line with a slope in absolute terms of [ 1 i]/[i[ 0]]. Conversely, F’s purchase of capital, ﬁnanced by the sale of money bonds, will shift the consumption point north west. This frontier for F also constitutes a frontier for G, who will have their own optimizing sale and purchases of bonds and capital. The market equilibrium occurs where G and F’s iso-utility curves are tangential. This occurs when the consumption point is on the 45 degree line.4 In this equilibrium F sells capital to G, and buys bonds from G. G borrows money from F to buy some of F’s capital. The critical point is that at the equilibrium consumption point, C in Figure 7.2, both F and G have zero variance in their consumption. The consumption of each in both states of the world is the same. F’s consumption is no higher in state 1 than state 0. The very increase in the money supply that F receives in state 1 creates an inﬂation that reduces the real value of the bonds F has purchased. Thus a non-neutrality favourable to F (receiving the money shock) has been cancelled by a non-neutrality unfavourable to F (the erosion in the real value of the money principal F is owed by G). Similarly, G’s consumption is no lower in state 1 than state 0. The very increase in the money supply that F receives in state 1 creates in state 1 an inﬂation that reduces the real debt G owes to F. An unfavourable non-neutrality (the ‘inﬂation tax’ of the money shock) has been cancelled by an favourable non-neutrality (the erosion in the real value of the money principal they owed). Notice that the equilibrium does not eliminate F’s ‘good fortune’ in receiving a disproportionate amount of the monetary shock of state 1: F’s expected utility is still greater than it would be in the absence of the monetary shock in state 1. Indeed F’s consumption outcome in both states of the

Monetary risk and the social function of money debt

145

0C G

1C G

E Efficiency locus

C

1C F

45 degrees 0C F

Figure 7.2

Trading bonds and capital to an eﬃcient point

world is higher than it would be in the absence of the monetary shock. F’s beneﬁt is spread out across states. Correspondingly, G’s consumption outcome is lower in both states of the world than it would be in the absence of the monetary shock. G’s loss is spread out across states. So relative to no shock, F beneﬁts in both states, and G loses. (But relative to the shock, F’s is down in state 1 and G is up, while F is up in state 0 and G is down.) A Supply and Demand Illustration The capability of money bond markets to immunize the economy against monetary shocks can also be underlined by an analysis of the supply and demand of money bonds that, unlike Box analysis, is not restricted to two states. Supply and demand schedules can be derived from two optimization conditions. UjC [1 r]EUjC1

real bonds

(7.16)

146

Inﬂation in a risky world

1 j UjC1 [1 i]EUjC1 E 1 cov( , & )

money bonds (7.17)

where & j

UjC1 EUjC1

(7.18)

These imply:

(7.19)

1 1 r E 1 1 i i E r

(7.20)

i E r cov(& j, )

(7.21)

1 1 r j E 1 1 i cov(& , ) But as

we may write

The RHS, cov(& j, ) , is the negative of the covariance of marginal utility with the growth rate of purchasing power. The LHS, i E r, is the expected excess rate of return a money bond over a real bond. Thus if we take this excess to be positive – if we assume the real rate of return on a money bond will on average be greater than the rate of return on an inﬂation indexed bond – then negative cov(& j, ) can be identiﬁed as ‘the inﬂation risk premium’. This premium is the average excess rate of return that a money bond is (commonly expected) to secure over a real bond if real bonds are not to be preferred to money bonds, on account of the riskiness inﬂation imparts to the real return on money bonds. j i E r cov(& j, ) )

(7.22)

j r [i E] )

(7.23)

Equivalently:

In terms of F and G:

Monetary risk and the social function of money debt

147

r [i E] )F

(7.24)

r [i E] )G

(7.25)

The LHS may be considered as the ‘price’ of money bonds. The RHS is a magnitude that will change with the amount of bonds owned, or owed. Thus for F and G we can trace out a relation between the ‘price of bonds’ and the quantity of bonds demanded, or supplied. At D 0 (that is zero money bonds) the magnitudes of )F and )G are determined by the co-movement of each group’s consumption endowment with the negative inﬂation rate; the rate of growth of purchasing power. At D 0, the party with the greater growth of nominal endowment during high inﬂation, F, will have ● ●

● ●

a positive co-movement in consumption and inﬂation, a positive co-movement in marginal utility and the growth of purchasing power; a negative ) ; and so a positive ).

We can treat )F as the vertical axis plotting of F’s demand for bonds in a ﬁgure that plots r [i E] on the vertical axis, and money bond issues on the horizontal axis. As F’s ownership of bonds becomes positive (D 0), the magnitude of )F falls, by the logic explained earlier: the bond holder, as ‘hedger’, is reducing the comovement of their consumption opportunity with the deﬂation rate. r [i E] falls. A ‘downward’ sloping demand curve is traced out for bonds. We can also trace out a supply curve for bonds. At D 0, the party with the lesser growth of nominal endowment during high inﬂation, G, will have: ● ●

● ●

a negative co-movement in consumption and inﬂation; a negative co-movement in marginal utility and the growth of purchasing power; a positive ) ; and so a negative ).

We can treat )G as the vertical axis plotting of G’s supply of bonds in a ﬁgure that plots r [i E] on the vertical axis, and money bond issues on the horizontal axis. As G’s money debt (liability in money bonds) becomes positive, the magnitude of )G rises, by the logic explained earlier Box analysis. The money debt owed by G makes a high inﬂation state less of a bad state for G; the correlation between growth in purchasing

148

Inﬂation in a risky world

r[iE]

F

0

D G

Figure 7.3

Money bond market equilibrium

power gets smaller in absolute terms. )G gets less negative. An ‘upward’ sloping supply curve for bonds is traced out. At the intersection, the )F )G 0

(7.26)

So the premium for inﬂation risk is zero in equilibrium. It is zero because a positive premium depends on a negative covariance between marginal utility and the growth in purchasing power; so that when creditors get a good payoﬀ it is weighted weakly, and when they get a bad payoﬀ it is weighted strongly. But under the present assumptions the covariance is zero in equilibrium. A good payoﬀ is weighted no more or less than a bad payoﬀ.

GENERALIZATION TO S STATES AND J PERSONS? Is the capability of money bonds to ‘immunize’ the welfare from money shocks just a freak of a 2 person: 2 ‘states of the world economy’? Or does generalize it to an economy of J person and S states world?

Monetary risk and the social function of money debt

149

An apparent argument in favour of generalization may be derived from the equimarginal conditions of this economy. j r [i E] )

(7.27)

j As r, i and E are the same for all persons, ) is the same for all persons. All persons have the same tendency of co-movement of their normalized marginal utility with the growth of purchasing power. As the normalized marginal utility sUjC1 EUjC1 maps into a normalized consumption, sCj ECj we may say that every person has the same tendency of co1 1 movement of their consumption with the growth purchasing power. Indeed, we can go further: this common tendency of this co-movement is zero. For, under our assumptions, the states of the world are only distinguished by monetary shocks. Therefore aggregate consumption is the same in all ‘states of the world’. Given this, it is not possible for everyone to experience (say) an above average consumption in an above average purchasing power growth state, for total consumption is the same regardless of the growth in purchasing power. To put the same thought another way: as there is no co-movement of total consumption with monetary states, how can there be any common co-movement of individual consumption with monetary states? The only possible common co-movement is no co-movement at all. That is ) j 0 for all j. And this seems to suggest that consumption of each individual is state invariant, the economy is immunized against monetary instability. However, any such inference is too hasty. The following considerations establish that ) j 0 for all j does not ensure that consumption of each individual is state invariant. Recall that we have previously assumed a two states of the world situation. In Table 7.2, F’s indebtedness to G solves the potential ineﬃciency, by transferring to state of the world 0 part of F’s good fortune in state of the world 1, and transferring part of Gs ill fortune from state of the world 1 to state of the world 0. The ups and downs in F and G are straightened out. The potential ineﬃciency of the converse situation, where G receives all the money shock, can be solved in a parallel manner (Table 7.3). It is easy to see that G being in debt to F solves the ineﬃciency, by transferring part of Gs good fortune in state 1 to state 0, and transferring part of Fs ill fortune from state 1 to state 0. But now suppose that there are three states of the world. In two of these there is a money shock. In one of these two F gets the whole of the money shock, but in the second G gets the whole of it (Table 7.4). Is there a level of money debt that can ‘straighten out’ the consumption proﬁles, so that consumption is invariant to the state? No. There is no

150

Table 7.3

Inﬂation in a risky world

G’s nominal endowment grows faster than F’s in one state

State

Aggregate money growth

F

G

F G

0

0

0

0

0

0

MG M

MG M

1

Table 7.4 F’s nominal endowment grows faster than G’s in one state, and G’s grows faster than F’s in the other State

Aggregate money growth

F

G

F G

0

0

0

0

0

1

MF M

0

MF M

2

0

MG M

MG M

magnitude of money debt between F and G that can do that. If F was to lend to G, then F’s ill fortune in state 2 is made even worse by the fact that their loan has been eroded in real value, and Gs good fortune is made even better. Thus we do not get zero variance in both person’s consumption. Conversely, if G was to lend to F then G’s ill fortune in state 1 is made even worse by the fact of their loan, and Fs good fortune is made even better. We do not get zero variance. Do we therefore conclude that money debt cannot immunize an economy against money risk in an S state economy? No. There exists a monetary risk environment which is ‘amenable’ to money debt functioning to secure eﬃciency. Under this plausible modelling monetary risk environment money debt can function in S states as 2.

S STATES AND J PERSONS: A SUCCESSFUL EXTENSION We can introduce the ‘amenable risk environment’ by considering that money loans worked in the two state world because there was a perfectly

Monetary risk and the social function of money debt

151

coincidence between F’s ‘fortunate state’ and the high inﬂation state (and, correspondingly, a perfect coincidence between G’s ‘unfortunate state’ and the high inﬂation state). However, in the three state counter-example, there was no longer this coincidence: the high inﬂation state was sometimes a fortunate state for F, and sometimes an unfortunate state for F. It would seem that money lending works if the direction and magnitude of redistribution caused by the money shock can be mapped uniquely into the inﬂation state. One modelling of money shocks that conforms to this requirement supposes that each person’s money growth is a linear function of growth in aggregate money, sj j s

for all j and all s

(7.28)

This modelling can be interpreted as supposing that the share any person receives of the aggregate money shock is invariant to the size of the aggregate money shock.5 To double the aggregate shock is to double the size of each individual’s shock. The assumption implies the diﬀerential between F’s and G’s money growth is uniquely relatable to money growth. sF sG [F G]s

It can be shown that under this assumed modelling, and in spite of any variation across states of the world of the diﬀerential in the growth of money endowments, there exists a degree of monetary indebtedness between F and G such that the diﬀerence between the rate of nominal gains of F and G, after allowing for debt ﬂows, is the same in all states. It is proved in an Appendix that, assuming there is only F and G, this degree of indebtedness is: G DF 1 F 1 MF MG

(7.29)

DF F’s ownership of money bonds This quantity of debt will make for a uniform excess diﬀerential between state of the world 1 and state of the world s. The actual diﬀerential of F over G can be calculated to be [F G][i ]. Assuming that there is some expectation of inﬂation, then F will have a positive diﬀerential in each state if F’s growth in money endowment varies more strongly than G’s with the total money growth.

152

Inﬂation in a risky world

Notice the implication of (7.29) that the person who has the disproportionate share of the money shock will lend (that is positive D). They wish to spread out their gain, and they do so by lending. When inﬂation is high (when they receive the money shock) they are burnt by inﬂation; but when inﬂation is low (when they don’t receive the money shock) they beneﬁt. Those who are likely to get a disproportionate (that is above average) share of ‘inﬂation’ even out the beneﬁt by lending. On the other side, those who will get a below average share of monetary expansion, even out the pain by borrowing. They protect themselves against the possibility of expansion by getting into debt. When expansion comes and they get burnt, they are compensatingly rewarded by being debtors. The uniformity in the diﬀerential across states can be generalized_ to n persons. If we assume all persons have the same initial endowment M, the debt for F is H I ... D_F n 1 n F G F average of others n M Interpretations of the Amenable Risk Environment This section has argued that there is a certain environment of monetary risk that can be immunized against by money bonds. This environment of monetary risk is that where aggregate money growth is unpredictable but each person’s money growth is a linear function of aggregate money growth. Equivalently, the environment is where the diﬀerential in the rate of nominal gains of diﬀerent persons is perfectly correlated with aggregate money growth. Is this kind of risk environment a freak occurrence? Are diﬀerentials completely uncorrelated with the aggregate growth in the money supply? Or can this kind of risk environment be given are plausible rationalisation? There are several scenarios for rationalising a correlation between the growth in aggregate money and diﬀerentials in the rate of nominal gains of persons. Bretton-Woods style fixed exchange rates Suppose that we have two nations in the world, A and B, and two corresponding national monies, dollars and pounds. Bs are obliged to sell pounds for dollars at a ﬁxed rate whenever As tender dollars for pounds. But As are not obliged to sell dollars for pounds at any ﬁxed rate if Bs tender pounds for dollars. (This asymmetry has some similarities to the Bretton-Woods arrangements.) The peg creates in eﬀect a single world money, and the asymmetry means increases in the world money

Monetary risk and the social function of money debt

153

amount to A printing dollars, which accrue obviously to As. This is the kind of situation illustrated in Table 7.2. Thus, in the context of unpredictable monetary expansion by A, a welfare ineﬃcient unpredictability in A and B consumption threatens. But on the logic of this section, we infer that A will lend, in nominal terms, to B, and that will eliminate that ineﬃciency.6 Bond holders and tax payers Increases in the money supply commonly involve purchases of government debt by central banks from private holders of government debt. It will be argued here that these purchases amount to a gift of money to one group: tax payers, (and not bond owners). One may argue that Central Bank purchases of government debt are equivalent to money grants to tax payers from a scrutiny of simply granting money to tax payers. Suppose individuals are divided into two groups: owners of (inﬂation indexed) government debt and tax payers. Suppose also, for the sake of argument, that government debt owners pay no tax, and tax payers own no government debt. And suppose in this situation that the government unpredictably prints money, and simply gives it to tax payers. In this story, high money growth states redistribute consumption away from bond holders and towards tax payers. The price level, and the nominal demand for money, will rise in accordance with the growth rate of aggregate money endowment, but as the money endowment of tax payers has grown at a greater rate than this, tax payers have excess money balances that they partly spend on bonds. Bond holders, by contrast, have received no increase in their money endowment, and in the face of rising prices and their rising money demand, sell some of their bonds to tax payers. In this story tax payers will hedge their upside by lending money to government bond holders. The key question is: how can the purchase of government debt by the central bank be any diﬀerent from a grant of money to tax payers? In the case of the grant, money is given to tax payers who buy bonds, and thereby obtain relief against taxes. In the case of the central bank purchase, the money is granted to the central bank who buys government bonds, and thereby also obtain relief for the tax payer against taxes.7 It makes no diﬀerence whether the new money is given to tax payers to purchase bonds, or given to central banks to purchase bonds on tax payers’ behalf. In either case the increase in the money supply goes wholly to one group, and in the context of unpredictable inﬂation, a welfare-ineﬃcient unpredictability consumption threatens. And that may be removed by tax payers lending in nominal terms to holders of government debt.

154

Inﬂation in a risky world

Partial Risk Immunization Outside the ‘Amenable Risk Environment’ We have demonstrated that perfect risk immunization requires a certain risk environment. But does that mean there is zero risk immunization outside of that amenable environment? Consider Table 7.4. We have argued that in this context perfect immunization is not possible. Is therefore, none, possible? No. Suppose the probability of states 0, 1 and 2 were 0.5, 0.49 and 0.01 respectively. Would not F lending to G cause some improving diminution in the variance of consumption?

TECHNOLOGICAL AND MONETARY RISK CONCURRENTLY Thus far we have concentrated on the case where there is monetary risk, but no technological risk. What happens when there is simultaneously monetary risk and technological risk? Suppose the money shocks are perfectly correlated with technology shocks, and there is a two person, two state world. So there is a high output and high money state, and a low output and low money state. This situation can be represented in a Box diagram, and its inspection reveals there is no diﬃculty in reaching eﬃciency.8 But what if we don’t make those assumptions? Both Technology Shocks and Monetary Shocks Present capital

UCj E[[1 ]UCj 1]

real bonds

UCj [1 r]EUCj 1

money

Uj UCj Uhj E 1 C 1

money bonds

Uj U jC [1 i]E 1 C 1

These conditions suggest the achievement of a welfare eﬃcient distribution of consumption in the face of technological and monetary shocks, at least in the face of an appropriate risk environment. For the equimarginal for capital and real bonds conditions yield: E r ) j

Monetary risk and the social function of money debt

155

and this implies an identical magnitude of ) j for all j, a necessary condition for eﬃcient distribution of consumption in the face of technological shocks. And the equimarginal conditions for real bonds and money bonds yield:

1 rE 1 j 1 i 1 ) j which implies an identical magnitude of ) for all j. Because money is neutral in total consumption, the socially eﬃcient response of every person’s consumption to a money shock is no response at all. This suggests the necessity of a zero covariance between marginal utility and inﬂation for j all persons, ) 0. However, it may be that money shocks are positively correlated with total consumption, even in the absence of causal ﬂow from consumption to money. In that case, marginal utility will be observed to vary negatively with money shocks, even though marginal utility is not actually responding to the money shock. Thus it would be too strong to say the socially eﬃcient response to money shock requires for all persons no comovement between marginal utility and inﬂation. What it does require is no diﬀerence in the co-movement across persons, because diﬀerent comovements could not be rationalized in terms of a correlation between total M and total C. This only requires uniform co-movement between marginal utility and inﬂation across persons. And the conditions of equilibrium seem j to obtain a common co-movement; ) is the same (and positive) for all j. However, the previous section has stressed that a common co-movement is not suﬃcient. Suppose that whenever the money supply is increased the increase is randomly distributed across persons. This will impart a socially ineﬃcient riskiness to consumption that money bonds can do nothing to eliminate. And because money is randomly distributed there will be a common (zero) covariance between inﬂation and consumption. So, just as in the previous section, the usefulness of money bonds requires there be a certain monetary risk environment: (7.28).9

CONCLUSION Money shocks threaten a welfare loss by creating an ‘artiﬁcial’ riskiness in individual consumption; artiﬁcial in that it does not reﬂect technology and could be totally eliminated. In certain circumstances money bonds can eliminate that risk. Thus the chapter has advanced a social functionality to money bonds. In doing so it has also imputed a social functionality to monetary equilibrium,

156

Inﬂation in a risky world

and the price level instability that it implies. In models without monetary risk, it is (strangely) unclear what social function is actually served by monetary equilibrium. On the contrary, for a given value of money in the future it would seem advantageous to make the current real money supply as large as possible, by forcing P way below the equilibrium P. In the present chapter, by contrast, it is a good thing if P increases in response to an increase in current M. Because it is the increase in P, combined with money bonds, that will stop the money increase from causing riskiness in consumption.

NOTES 1.

2.

‘If you lent me so much labour and so many commodities; by receiving ﬁve per cent you always receive proportionate labour and commodities, however represented, whether by yellow or white coin, whether by pound or by ounce’. David Hume, Of Interest, 1754. The real balance eﬀect (or the windfall in purchasing power) dMj P (Mj P) (dP P) dhj. Assuming no change in real demand for money, the real balance eﬀect of a permanent increase in money is

dMj M dP Mj dMj dP Mj dMj dM Mj[j ] P P P P Mj P P Mj M P Conclusion: it is a diﬀerential between the growth in j’s money endowment and the aggregate money growth that creates a real balance eﬀect for j. But as sjj skk

3.

4.

5. 6. 7.

or [1 sk]j skk

or j sk[j k]

we could equally say that in a two person economy, the real balance eﬀect of a permanent increase in money is (Mj P)sk[j k]. It is a diﬀerential between the growth in j’s money endowment and the growth in k’s money endowment that creates a real balance eﬀect for j. Evidently, unpredictability in the money supply does not by itself impart risk to consumption. It is possible that the growth rate in total M is unpredictable, but in each outcome all persons experience the same diﬀerential in the growth rate in their nominal endowment. In that case the same redistribution occurs in all states, and so consumption remains money state invariant. Equilibrium requires the slope of the frontier [ 1 i]/[ i[ 0]] equal that common slope of the ‘expected utility indiﬀerence curves’. At the 45 degree line the slope of the ‘expected utility indiﬀerence curves’ slope p/[1 p]. Thus [1p][ 1] p [ 0]i the expectation of the nominal rate on capital. This makes sense; there is zero risk premium on capital due to riskless technology. The modelling implies Mj M jMj M . Country B can increase world currency to the extent that it chooses to run down its reserves of A currency. But there is, obviously, a limit to the extent that this can be done. This case for the equivalence of an open market purchase of bonds with a gift of money to tax payers does assume indexed government debt. But introducing nominal money debt simply introduces an element of debt cancellation. The central bank’s purchase of nominal debt is equivalent to a transfer of money to tax payers, combined with a can-

157

Monetary risk and the social function of money debt

8. 9.

cellation of a certain amount of the debt. This does create complications: debt holders will want a higher rate of interest to compensate for the probability of cancellation by way of inﬂation. The cov( , & j ) are the same for F and G, but now non-zero, since the & now varies across states. In order to eliminate any welfare ineﬃciency caused by money shocks any redistribution from monetary eﬀects must be state-invariant, that is perfectly predictable. The real balance eﬀect for F:

dMF MF dP MF dMF dP dhF P P P dhF P MF P hF

Assuming the monetary shock is permanent this can be rewritten:

. . M dM dh M PF M F dM dh F PF[F h hF] M h h F F .

.

But the immunization against real shocks by real bonds side will ensure h hF 0, as everybody’s real consumption grows by the same rate in all states. So, the real balance eﬀect for F: M PF[F ] So we want the cross-person diﬀerentials in the growth in nominal endowments to be the same in all states. Let s index monetary shocks, and let prime index technology shocks, then a state-invariant diﬀerential in the growth in nominal gains between F and G requires the satisfaction in all states of: [1 s] s 1 M M [DF DG][ ]

[F G]

[1 s] s 1 M M [DF DG][ ]

[F G]

[1 s] s 1 M M [DF DG][ ]

[F G]

and so on. But we know (see Appendix) that DF and DG can be chosen to satisfy one of these. But to satisfy one is to satisfy all since [s 1] [s 1] [s 1] and so on.

158

Inﬂation in a risky world

APPENDIX This appendix demonstrates that if sj Mj js M then the equality of the diﬀerentials corresponds with certain distribution of money debt. The equality of the diﬀerential in the nominal gains of F and G, in state s and state 1, may be stated as: 1F DF[i [1 ]]

MF

[1G DG[i [1 ]] MG

sF DF[i [s ]]

MF

[sG DG[i [s ]]] MG

where DF the amount of bonds owned by F. Thus: 1F DF 1

MF

[1G DG1] sF DFs [sG DGs] MF MG MG

or 1F

1G

MF MG

sF

sG

MF MG

DF D G 1 [ s] MF MG

But assuming s sj jM Mj

then equality across states requires

1 s F G M [F G] M D F D G [1 s] M M

or F G [F G][1 s] D F D G [ 1 s] M M

This by the Quantity Theory reduces to DF DG F G MF MG

Monetary risk and the social function of money debt

159

This pattern of debt will make the diﬀerential in the growth in nominal endowments invariant across states (see note 1). This expression of the indebtedness between a pair that secures an invariant diﬀerential in the growth in nominal endowments across states allows us to obtain the corresponding expression when there are three persons: DF DG DH 0

Market Equilibrium

DF DG MF MG F G

F and G synchronized (see above)

DF DH MF MH F H

F and H synchronized

This is an expression in three equations and three unknowns: DF, DG, DH. Magnitudes of DF, DG and DH that secure an invariant diﬀerential exist.

NOTE 1.

We have used an approximation which ignores cross products. The exact formulation would be, DF D G 1 i MF MG 1 [F G]

8. The Quantity Theory in a risky world The classic theoretical terrain of the Quantity Theory is one of perfect knowledge. The preceding chapters have accordingly assumed that the key determinant of money’s supply and demand – the nominal rate of return on capital – was a matter of knowledge. The nominal rate of return is, however, a matter of surmise, rather than a matter of knowledge. The Quantity Theorist therefore seems obliged to rework the Theory in an environment of risk. This chapter does so by introducing risk into technology and the money supply. The tendency of the analysis is generally conserving of the content of the perfect-foresight model, rather than subverting. The Theory’s substance is not so much changed, as made more intelligible. The allowance for risk makes possible the analysis of the impact of things that the perfect foresight model was incapable of addressing: the impact of mistakes and revisions, the degree of conﬁdence, diﬀerences in information. The analysis of the theory under risk also provides a theory of lags, that the theory under perfect foresight was largely incapable of. The model with risk, then, emerges as a superior vehicle for presenting the Quantity Theory’s ideas. Throughout the chapter it is assumed, critically,1 that there exist markets for both real bonds and money bonds, and that the environment of monetary risk is ‘amenable’ to the achievement of welfare eﬃciency (in the manner described in Chapter 7). The chapter begins by reworking the demand and supply of money.

THE DEMAND FOR MONEY UNDER RISK The demand for money under risk derives from the conditions of optimization under risk, which may be written, capital

UC [1 E ) ]EUC1

(8.1)

real bonds

UC [1 r]EUC1

(8.2)

160

The Quantity Theory in a risky world

money

1 ) UC Uh EUC1 E1

money bonds

1 ) UC [1 i]EUC1 E1

161

(8.3)

(8.4)

The conditions for money and money bonds imply that the implicit yield on money (approximately) equals the rate of interest. U i i Uh 1 i

(8.5)

(h)

(8.6)

C

As

we can infer

i h 1 1 i

(8.7)

The demand for money is negatively related to the rate of interest. A useful indicator of the negativity is the semi-elasticity of money demand, that we will measure as, ih

h

(8.8)

i h 1 i

which approximately equals the more conventional semi-elasticity of h to i, h ih. The inference that the demand for money is negatively related to the rate of interest is, however, a question begging one, as the interest rate is itself a dependent variable. The possibility of substituting money bonds and capital requires the interest rate satisfy: 1 i

1 E ) 1 ) E1

(8.9)

Ignoring cross product terms, and supposing ) to be ‘small’ relative to 1, we may infer the interest rate satisﬁes: i E E ) )

(8.10)

162

Inﬂation in a risky world

) is the ‘proﬁt risk premium’. It indicates the superiority of the average return on capital to indexed bonds, that must be enjoyed by capital if indexed bonds are not to be preferred to capital, on account of capital’s riskiness. In the present context, however, it is more useful to register that ) may be equally interpreted as the ‘proﬁt security discount’, that indicates the inferiority (expressed as an absolute number) of the return on inﬂation-indexed bonds, relative to capital, that must be endured by such bonds if they are not to be preferred to capital, on account of their immunity to proﬁt risk. As money bonds share this immunity to proﬁt risk, the nominal interest rate must be depressed by the extent of this. We have also learnt that ) may be considered an ‘inﬂation risk premium’, assuming that ) is positive. It indicates the superiority of the return on nominal bonds, relative to real bonds, that must be enjoyed by nominal bonds, on account of their inﬂation risk, if real bonds are not to be preferred to nominal bonds. As money shares this vulnerability to inﬂation risk, the yield on money must be raised by the extent of ). To summarize, money’s yield must match the forecast nominal rate of return on capital less a ‘security discount’ reﬂecting money’s insulation from proﬁt risk plus a ‘risk premium’ reﬂecting money’s vulnerability to inﬂation risk. The equality i E E ) ) permits the inference: h h(E , E, ) , ) )

(8.11)

An alternative route to the same conclusion is by the substitution of E E ) ) and (h) . Three comparative-statics follow. 1.

The Expected Rate of Inflation and the Expected Rate of Profit

Increases in either of these diminish the demand for money. 2.

The Profit Risk Premium and the Demand for Money

The proﬁt risk premium – understood as a negative correlation between the proﬁt rate and the marginal utility of consumption – increases the demand for money. It does so by creating, on account of money’s invulnerability to proﬁt risk, a ‘security discount’ for money: the yield on money must be

The Quantity Theory in a risky world

163

reduced if it is not to be preferred to capital. The yield is reduced by holding a larger quantity of money. 3.

The Impact of Inflation Risk on Money Demand

The impact of inﬂation riskiness on money demand turns entirely on the covariance of purchasing power with the marginal utility of consumption. A positive ‘inﬂation risk premium’ – understood as a negative correlation between the growth rate of purchasing power and the marginal utility of consumption – will increase the demand for money. It does so by necessitating – if capital is not to be preferred to money – a rise in the yield of money, which is in turn secured by a smaller holding of money. This conclusion may be rationalized more thoroughly: a negative correlation between the growth rate of purchasing power and the marginal utility of consumption reduces the expected utility of consuming a previously accumulated unit of real balances – as a negative covariance means that whenever the rate of growth of purchasing power is above average, marginal utility consumption is below average. So when the payoﬀ aﬀorded by investing in real balances is high, the utility weighting on that payoﬀ is low. This obviously constitutes a deterrent to holding money.2 A zero ‘inﬂation risk premium’ will have no impact on the demand for money. If the ‘premium’ is zero – if the correlation between the growth rate of purchasing power and the marginal utility of consumption is zero – then both above and below average rates of growth in purchasing power receive the same utility weighting. As a consequence, the riskiness of inﬂation is not a deterrent to holding money. A negative ‘inﬂation risk premium’ will have a positive impact on the demand for money. If the ‘premium’ is actually negative – if the correlation between the growth rate of purchasing power and the marginal utility of consumption is positive – then whenever the rate of growth of purchasing power is above average (and so when that quantity of consumption aﬀorded by investing in real balances is high), then marginal utility is high. In other words, when the quantity of consumption aﬀorded by investing in real balances is high, the utility weighting is high. This constitutes an inducement to holding money. To summarize: inﬂation variance need not reduce the demand for money. It might increase the demand for money. It might leave it unchanged. It is the sign of the covariance of purchasing power with marginal utility that is critical.3 It is a negative covariance that reduces money demand.

164

Inﬂation in a risky world

The semi-elasticity of money demand to these determinants are approximately equal: h h h h 0 hE hE h) h)

(8.12)4

A Condensed Formulation Some arguments in the money demand function may be suppressed from explicit consideration on account of their being exogenous with respect to other arguments. Thus and ) are exogenous with respect to E on account of the economy’s dichotomization, and may be suppressed. It is more suppositious if ) can be suppressed on account of it being exogenous with respect to monetary changes. But one may say that ) is exogenous with respect to monetary shocks that are uncorrelated with the state of consumption; say, an increase in the money supply that occurred in all consumption states. On the assumption that we restrict ourselves to monetary shocks that are uncorrelated with the state of consumption we can treat ) as exogenous to monetary phenomenon. Thus we are left with the money demand function, h h(E)

(8.13)

THE SUPPLY OF INSIDE MONEY UNDER RISK Just as the implicit return on money is equalized to the rate of interest, so the marginal cost of inside money is equalized to i.5 i 1 i

(8.14)

But as: (n) we may write:

i n 1 1 i

(8.15)

The Quantity Theory in a risky world

165

or n n(E , E, ) , ) )

(8.16)

n ) 0. The proﬁt risk premium ) reduces the supply of inside money. This can be understood as a consequence of the disincentive that proﬁt risk poses to issuing inside money for the purpose of investing in capital. n ) 0. The inﬂation risk premium ) increases the supply of inside money. A risk premium on nominal loans means there is a tendency for above average growth in purchasing power to occur with a below average marginal utility. But this implies there is a tendency for the real liability of the issuer of inside money to be larger when the marginal utility of money is smaller. In other words the burden of the issuer is largest when their need is least. And that creates an incentive to supply inside money. So we meet another symmetry. Expected inﬂation reduces money demand but increases money supply. Inﬂation risk reduces money demand but increases money supply.6

MONETARY EQUILIBRIUM UNDER RISK The conditions of monetary equilibrium under risk are: Demand for money:

hh(E)

Supply of outside money:

mM/P

Supply of inside money:

nn(E)

Demand supply:

h m n

Or, xm where xhn Including deﬁnitions, the equilibrium in period zero is described by, x(E) m;

mM/P;

EEP1/P1

(8.17)

166

Inﬂation in a risky world

For period one the same equalities hold for each state of the world, s: x(sE11) sm1;

sM 1 sm sP ; 1 1

sE P 1 2 11 sP 1 1

sE

all s

and so on. Similarly for period 2, and all later periods. An equilibrium exists if there is a set of P in all periods and all states of the world such that all equalities are satisﬁed. The counting of equations and unknowns creates a presumption of the existence of equilibrium as it reveals there are as many unknowns as equations.7 However, an equilibrium need not exist (just as in the perfect foresight model). One example will suﬃce. Suppose that there is one state of the world in the next period such that, if that state of the world comes to pass, all subsequent periods will experience the same rate of monetary growth. In other words, there is one state of the world in the next period such that, if that state of the world comes to pass, then risk completely evaporates for the remainder of time. And suppose that that rate of growth of the money supply in all subsequent periods is more negative than . By the arguments pursued in Chapter 1, there is necessarily no equilibrium next period if this state of the world comes to pass next period. But that possibility of disequilibrium in the next period makes for an impossibility of equilibrium this period. For how is the mathematical expectation today of the price next period, EP1, to be computed if for one state of the world there is no equilibrium price? The non-existence of equilibrium, notice, does not rest on the probability of permanent monetary contraction being of a certain size. As long as the probability is larger than zero, it does not matter how small the probability is: equilibrium will not exist. The conclusion is that risk magniﬁes the problem of non-equilibrium. It is not enough that money growth not contract too rapidly; it must be impossible that money growth contract too rapidly. We will, however, assume henceforth that equilibrium does exist, and turn to the response of equilibrium to shocks.8

COMPARATIVE-STATICS UNDER RISK As in the perfect foresight model, the comparative-statics of the Quantity Theory turn on a key property of the system of equilibrium conditions: future endogeneous variables are exogenous with respect to current endogenous variables. This can be seen by noting that P1 is absent from the equilibrium system. Thus, on the assumption that the system is determinate, we

167

The Quantity Theory in a risky world

infer that P is completely independent of P1, that is exogenous to P1. But if P is exogenous with respect of P1, then sP1, (and so EP1) is exogenous with respect of P. Thus the equality, x

EP1 M P 1 P

constitutes a single equation in a single variable P. The solution is also presented in Figure 8.1. Equilibrium is at the intersection of the schedules. Notice that the intersection could occur at the horizontal portion of the demand for money schedule. This is the liquidity trap solution. In an environment of risk the liquidity trap solution occurs where the expected nominal rate of return on capital, adjusted down for proﬁt risk and up for inﬂation risk, is zero. (Equivalently, where the risk adjusted expected growth rate in purchasing power equals the risk adjusted expected rate of proﬁt.) The innovation in the present analysis of the liquidity trap, relative to its analysis under perfect foresight, is that a liquidity trap does not require a drastic fall in money to actually occur in periods subsequent to a trap. It is now merely a matter of it being possible, rather than certain. A drastic fall need not even be likely to occur for a liquidity trap to arise: given the arithmetic of

m

1/P x(EP1/P1)

The supply of outside money

The residual demand for outside money

1 ∼E /[1 E ) )] P1 x, m Figure 8.1

Monetary equilibrium under risk

168

Inﬂation in a risky world

mathematical expectations it may be that the deﬂationary states of the world have only a small probability: but if they are severe enough in magnitude the expected nominal rate of return on capital is zero. We have a kind of ‘peso problem’ where a small possibility of a considerable contraction creates a liquidity trap. Henceforth, however, we will assume that equilibrium occurs in the downward sloping portion of the demand for money schedule in Figure 8.1. The equilibrium condition implies exactly the same elasticity of P to M as in perfect foresight model, PM 1 PM 1 1 i where

x i x1 i

What will be elasticity of P to an increase in the future money supply? The question is ill put. There is no such thing as ‘the’ future money supply. There are a range of possible future money supplies. It is sensible to ask: what will be the impact of a change in the money supply in a given state? The answer turns on the fact that a change in the money supply in a state s of the next period is felt this period through its impact on the expectation of inﬂation. P EP1 P sM1 EP1 sM1 As EP1 s sP1 sM1 p sM1 the impact may be written in elasticity form, s sp sP P M1 1 1 s M1 P 1 1 i 1 1 Esi EP1 1

(see note 9.)

The Quantity Theory in a risky world

169

This comparative-static parallels the elasticity of P to M1 in the riskless model, the central innovation simply being that the impact of an increase in the money supply in some state is weighted by the probability of that state. (This comparative-static becomes identical to that of the riskless world for an increase in the future money supply when sp 1.) To summarize: P depends not upon ‘the’ future money supply, but on each and every possible future outcome of the money supply. We face a glut of inﬂuences. One way of coping with this glut is to conﬁne our attention to shocks that are common to all states. Consider, for example, a shock that involves a proportionate increase in the money supply in all states in the next period. Let the size of the proportionate increase be ˆ . dsM1 M

all s

Then, dP P 1 d 1 1 i 1 1 sEi

1

This expression says that any increase in all money supply outcomes of the proportion, , will increase the price level by a proportion approximately equal to times the semi-elasticity of money demand. The RHS of the expression is perfectly analogous to the elasticity of P to M1 in a riskless environment. So by sharply restricting the kind of disturbance these inﬂuences are subject to – by assuming a proportionate shock to all possible money supplies – one may cope with the glut of inﬂuences, and mimic the comparative-statics of perfect foresight. An alternative and more ambitious response to the glut of inﬂuences of future possibilities is to seek some suﬃcient statistic that captures the impact of all the money supply possibilities. The obvious summary measure is the mathematical expectation of the money supply EM1. Alas, the mathematical expectation of the money supply will not, generally speaking, serve as a suﬃcient statistic; one cannot, in general, uniquely relate P to EM1. For if the mathematical expectation of M is to be a suﬃcient statistic, then the dependency between P and the realizations of the M must be linear. But one does not expect linear relations in economics, and there is no reason to expect them here. In this context linearity requires:

170

Inﬂation in a risky world

P s sM1 p And since: s P P1 sp P 1 s s M1 M1 EP1 1 1 1 i 1 sEi1

linearity requires: sP 1 sM 1

1 P for all s EP1 1 1 1 i 1 sEi1

There is no general reason why this might hold.

MATHEMATICAL EXPECTATION MODELS Nevertheless, there are several special cases of the money demand function that can be solved in terms of mathematical expectations, at least with some approximation error. Table 8.1 lists various money demand functions that allow the (at least approximate) expression of some measure of the value of money in terms of the mathematical expectation of some measure of the money supply. The constant semi-elasticity function provides the solution that is commonly used, and will be used here. Taking logs of the monetary equilibrium condition x k exp [E E )( ) )()] Table 8.1 Three mathematical expectations models of the Quantity Theory under risk Demand type

Value of money variable

Money supply variable

Constant elasticity of demand of 1

1 P

1 M

Constant semi-elasticity of demand Constant co-eﬃcient of demand

lnP

lnM

P

M

The Quantity Theory in a risky world

171

and re-arranging yields 1 ln M E ln P [E ) ) ] 1 ln k ln P 1 1

1 1 1 Taking ) and ) to be parametrical, and leading forward and substituting yields,

E ln M 2 ln P 1ln M 1 1 1 ... 1 E 1 E 1 ... This expression has exactly the same form as the perfect foresight expression for lnP, but with expectations of future variables substituted in for actual future variables.10 It can be more compactly expressed in permanent variables form: ln P Eln M* E * This expression can be manipulated to provide an expression for inﬂation:11 E* R1ln M* E * R1 * where revision terms are deﬁned: Rtln Mt Etln Mt Et1ln Mt Rt t Et t Et1 t and permanent variables are deﬁned E1 E2 [1 q] q E 1 q [1 q]2 ...

E*

[1 q] 1 1 2 q E 1 q E 1 1 q E 2 ...

E *

[1 q] 1 1 2 q R1ln M1 1 q R1ln M2 1 q R1ln M3 ...

R1ln M*

172

Inﬂation in a risky world

[1 q] 1 1 2 q R1 1 q R1 2 1 q R1 3 ...

R1 *

q 1 The new element in this expression for inﬂation in comparison with the perfect foresight expression are the revision terms. But how important are they? Revisions in a given period have little impact. But if a revision aﬀects both present and future money supplies the impact may be much greater. This can be illustrated with a parameterization. Suppose the magnitude of the expectations of current and all future money supplies are increased by the same percentage, a. In other words, there is a ‘permanent’ shock to the level of the money supply. for all t

R1ln M1 R1ln Mt ... a

then E* a The coeﬃcient is one on the revision term: permanent shocks have the oneto-one eﬀect of the simple Quantity Theory. Now suppose the magnitude of the expectations of current and all future money supplies growth rates are increased by the same percentage, g. In other words, there is a ‘permanent’ shock to the growth rate of the money supply. R1ln M1 g; R1ln M2 2g; R1ln M3 3g;... This implies, E* g[1 ] (See note 12.) Given the possible size of , it is evident, that current inﬂation will be strongly aﬀected by permanent shocks to money supply growth.

MODELLING SURPRISES: AN ILLUSTRATION The previous section concluded that inﬂation is sensitive to revisions of expectations. But it is less than satisfactory to try to explain inﬂation in terms of a variable (revisions) that is both unobservable, and (in any case)

173

The Quantity Theory in a risky world

only a proximate cause; expectations not being ‘exogenous’ but determined by exogenous forces. It would be desirable to advance modellings of money growth that will explain expectations of money growth, and thereby resolve inﬂation into observables. Consider a simple and convenient formulation: ln M1 ln M +1 ,+

(8.18)

+unpredictable white noise shock term with mean zero ,fraction of shock that is temporary Each period the money supply receives a shock, , of which is removed the following period, and 1 , of which is permanent. We may infer, E ,+;

E1 E2 E3 ... 0;

E* 1 ,+

(8.19)

, + R1ln M* 1 , 1 1

The implications for dimensions of inﬂation are reported in Table 8.2. The fundamental explicative forces of inﬂation are the surprise element in money growth, +1, combined, with a negative impact, the preceding period’s surprise. These surprises may have a large or small impact. Table 8.2 Inﬂation under various money supply disturbances in the Quantity Theory under risk 1 ,0

, 1

, 0

Autoregressive moving average

White noise

Random walk

E

,+ 1

+ 1

0

, + , +1 1 , 1 1

+1 + 1

+1

E[E]2

, var+ 1 , 1

var+ [1 ]2

var +

Process

2

174

Inﬂation in a risky world

White Noise At the extreme end of temporariness, surprises have a ‘small’ impact. This is due to the stabilizing forces brought into play by the fact that all changes are expected to be temporary (regardless how long they last). The fact that every increase in M is expected to be temporary means that every increase in M brings with it an expectation of deﬂation. That increases money demand, and reduces inside money supply, as n responds to expected deﬂation. So in the white noise M case every positive shock to outside money induces a contraction in inside money supply. This is stabilizing. This conclusion is just a re-working of the notion of Chapter 4 that temporary changes have little impact. Under white noise all changes are expected to be temporary, even those that (by chance) endure some time. Random Walk At the other extreme end of permanence, surprises have a ‘large’ impact. All changes are expected to be permanent, even those that are quickly reversed. The simple Quantity Theory is restored.

LAGS Allowing for risks, and modelling disturbances in the manner of an autoregressive moving average, allows a rationalization of the apparent signiﬁcance of lags. With an ARMA formulation of the money supply, inﬂation (putting aside polar cases) can be mapped into observable present and past money growth rates. Table 8.3 indicates as long as , is not at one pole ( 0) or the other ( 1), inﬂation can be expressed as a function of the lags of monetary growth. Here is a puzzle. Table 8.2 indicated that ‘true’ lags are very limited: the

Table 8. 3 Current inﬂation and past money growth in the Quantity Theory under risk

1 , 0

, 1

, 0

[1 ,] 2 [1 ,] [1 ,] 1 1 1 1 ,1 1 , 2 ...

1

The Quantity Theory in a risky world

175

money growth in any period prior to period 0 is completely irrelevant to price growth between 0 and 1. But Table 8.3 suggests that inﬂation depends upon past growth of money. The appearance of ‘lags’ in Table 8.3 can be explained by a thought experiment. Suppose the money supply has been constant over time, but experiences a single, one-oﬀ, and sustained, increase in the money supply of size + in period 0. This requires a series of epsilons, ...0, 0, 0, +, ,+, ,2+, ,3+,...

(8.20)

or +t 0, t0

+t ,t+,

t0

This series of epsilons will cause a permanent, one-oﬀ step up in the money supply in period zero equal to +. (Ignore the minute probability of this sequence of epsilons: the sequence is possible, and that is enough for this purpose.) In accordance with Table 8.3, P will rise in the period of the shock, but by less than , and will continue to rise in subsequent periods, as if still responding each period to the (ever receding) original increase in M (Figure 8.2). Why? The answer lies in the fact that P depends on the expectation of the ‘permanent money supply’. P does not rise proportionately with the lnP, lnM

lnM lnP

Time

Figure 8.2

The profiles of M and P under a sustained increase in M

176

Inﬂation in a risky world

current money supply because that expectation of the money supply in future periods is little aﬀected. (In economic terms, the prospect of deﬂation impedes the ﬂow-through of the increase in M into P.) But why has the expectation of future M not risen by as much? Because, rationally, part of the increase in M is not expected to last; part of it is expected to be reversed in the following period. But as time passes, and as it is not reversed, the expectation of future M rises. This is because there have been (and must have been) more favourable shocks in subsequent periods, so as to compensate for the cessation of the temporary component of the original shock. It is this accumulation of more favourable shocks that has made the original increment enduring, and increases the expectation of the future money supply. So the reason why an enduring shock takes time to exert its full eﬀect is because it takes time to discover that an enduring shock will endure. The economy only responds to enduring M; but it takes time to see that it will endure. Notice that there is no lag between the change in the ‘permanent money supply’ and the change in P: that link is immediate. But there is a lag between any change in the money supply, and an associated change in the permanent money supply. This is the theory of lags in model: an actual change can become a permanent change only with a lag. The theory also is informative about the ‘long and variable’ nature of lags. The lags may be ‘long’ because it may take a long time for any increase to establish a permanent increase. Suppose , is large. Most of the shock is expected to be shortly reversed, and so the estimate of the ‘permanent money supply’ is much lower than the actual money supply. As time passes, contrary to (rational) expectation, the increase in the money supply is not reversed owing to a succession of positive shocks. But even these shocks will only have a small impact on the estimate of the ‘permanent money supply’ as they themselves are expected to be mostly reversed, owing to the large ,. To summarize: in the face of a large , it will take a long time for an enduring increase to be registered as one.13 The lags in adjustment extend to nominal and real interest rates. In the thought experiment the long-run eﬀect of a sustained increase in the nominal money supply has no long-run impact on nominal and real interest rates. But the nominal interest rate will initially fall. And then rise back only over time.14 Additionally, the ex post real interest rate will initially fall, given the high and unexpected inﬂationary shock. The succession of unexpected inﬂationary shocks will mean that the ex post real interest rate will not return immediately to normal, but will slowly rise back over time. So the temporary reductions in nominal and ex post real interest rates that are associated with increases in the nominal money supply are predicted here.

The Quantity Theory in a risky world

177

Overall, the theory under risk, combined with an assumption about the money supply process, provides a rationale of the stylized facts of the dynamics of the adjustment of prices and interest rates to a sustained increase in the nominal money supply. It is a rationalization entirely diﬀerent from those that rely on rigidity in nominal wages or prices (and the non-neutrality of output to money). All prices are ﬂexible, and output is money neutral. Neither does the rationalization rest on any illusion, or rationality. Rational expectations prevail throughout; ex ante optimization prevails throughout. What the existence of dynamics rest upon is the deﬁance by the money supply of its rational expectation. The same analysis will provide to the appearance of lagged adjustment to other shocks. If

1 + ,+ 1 then the price response to an enduring proﬁt shock will be lagged. Similarly for autonomous money demand.

THE PROBLEM OF HETEROGENEOUS INFORMATION AND HOMOGENEOUS EXPECTATIONS The introduction of risk introduces a problem for the existence of equilibrium that we have not yet faced. The analysis has assumed that the probabilities of the possible magnitudes of the probabilistic variables are known, by all. But here arises an ambiguity. Probabilities are conditional on information, with the consequence that diﬀerent information leads to diﬀerent probabilities, and diﬀerent expectations. As diﬀerent persons will presumably have diﬀerent information sets, they will have diﬀerent expectations. So whereas we have spoken about ‘the’ expectation of the money supply, if information is heterogeneous, there can be no such a thing as ‘the’ expectation of the money supply, but only a suite of diﬀerent expectations. But if expectations are heterogeneous how can the ﬁrst order conditions be satisﬁed? Consider that the conditions for nominal and real bonds imply, i r E ) If expectations diﬀer across persons then it would seem inevitable that the RHS will diﬀer across persons, and the equality cannot be satisﬁed for all persons.

178

Inﬂation in a risky world

The heterogeneity of information, therefore, seems to pose a menace to equilibrium. Since there is heterogeneity of information, equilibrium becomes impossible to understand. The line of thinking is, however, erroneous. Contrary to appearances, the ‘problem’ of information heterogeneity presents no genuine diﬃculty for the existence of an equilibrium. The existence of an equilibrium – a set of prices such that every actor is doing the best they can subject to their information set – does not require the homogeneity of information. To be more speciﬁc: in the spirit of the Eﬃcient Markets Hypothesis, it is contended that a situation whereby P, i and r reﬂect the totality of information in the economy will constitute an equilibrium, regardless of any dispersion across persons of elements of that totality of information.15 By ‘reﬂect’ is meant that the magnitudes E ) and E ) that determine i, r and P are those rationally implied by the totality of available information in the economy. By ‘will constitute an equilibrium’ is meant that everyone is doing the best they can subject to their information set. One may go further: a situation whereby P, i and r do not reﬂect the totality of information in the economy cannot constitute an equilibrium. An argument for the ﬁrst contention runs: let P, i and r reﬂect the totality of information in the economy. Anyone in the possession of that totality of information will be in equilibrium. Anyone with less than that totality will also be in equilibrium because, although they do not have that totality, they know the nominal interest rate is such that even the best informed cannot borrow money to buy capital with advantage. But how do they realize this? How will holders of a contracted information set appreciate that they hold a contracted information set, and that the interest rate is based on the best information? Brieﬂy, how will they know that they don’t know? The answer lies in the pattern of errors of forecasts based on the more contracted information set. This pattern will indicate to the possessors of a contracted information set that it is lacking some available information. In equilibrium the covariance between their information set and the errors of the forecasts implied by their information set will be non-zero. Consequently their use of forecasts implied by that information set would be irrational. Thus those who don’t know best will know that they don’t know best. We will give two corroborative and illustrative examples of the thesis that the ‘reﬂection’ of i and r of the totality of information in the economy will constitute an equilibrium, regardless of any dispersion across persons of elements of that totality of information.16

179

The Quantity Theory in a risky world

Example 1: ‘Perfects and Zeros’ Suppose that variable parameters, except for M which is a random walk. lnM1 lnM +1

(8.21)

Suppose that the realization of + in every period is perfectly known from the beginning to the best informed, ‘the Perfects’. But the less informed – ‘the Zeros’ – know only that + is white noise, and so predict + will be zero. Zeros and Perfects are evidently diﬀerently informed about central bank behaviour. The equilibrium price level will reﬂect the information set amounting to the total of information set of all participants; in this case the information set of the Perfects.

+ + + 1 1 1 2 1 1 3 1

2

...

(8.22)

(8.22) coincides with universal perfect foresight. The Zeros and Perfects have the same rate of return, as capital and loans oﬀer the same rate, and so all portfolios have the same rate of return. The Zeros could make forecasts upon the basis of their own limited information endowment, and if they did so they would mistakenly infer that the rate of return on capital and bonds was unequal. But the Zeros know they don’t know. And they know they don’t know on account of a non-zero autocorrelation between the error in their inﬂation forecast and their inﬂation forecast error in the previous period. E Zeros’ information 0, Therefore:

+ + + E Zeros’ information 1 1 1 2 1 1 3 1

2

... (8.23)

But also:

+2 2 ... + +1 1E11 Zeros’ information 1 1 1 1 1 (8.24)

180

Inﬂation in a risky world

Therefore, cov( E Zeros’ information, 1 E1 1 Zeros’ information) 0 (8.25) To summarize; the equilibrium is based on the information of the best informed. Example 2: ‘Epsilons and Xis’ Suppose, lnM1 lnM +1 -1 1 The Epsilons know the epsilon shocks, and the Xis the xi shocks, but neither knows the third type of shock, chi. In accordance with our ‘Eﬃcient Markets Hypothesis’, the equilibrium rate of inﬂation is, + - [+ - ] 11 1 12 1 1 ... 1

(8.26)

In this situation the excess rate of return on borrowing money to buy capital is equally unknowable to Epsilons and Xis; 1. They are both passive. They could each forecast the nominal rate on capital, but they know that they don’t know. For both Epsilons and Xis the errors in the forecasts that could be drawn from their limited information set will be autocorrelated. E Epsilons’ information 1 1 1 1 2 1 ... + + E Xis’ information 1 1 1 1 2 1 ... cov( E Epsilons’ information, 1 E 1 Epsilons’ information) 0 cov( E Xis’ information, 1 E1 Xis’ information) 0 Thus if both were surveyed, both would test out as inconsistent with rational expectations. Therefore, they will not forecast on the basis of their limited information. They will instead forecast inﬂation (if they need to know it) on the basis of the nominal interest rate. Note also:

The Quantity Theory in a risky world ●

●

●

●

181

No one is disadvantaged with respect to the exchange of money bonds and capital by having a restricted information set. The ‘implicit expectation’, that is implied by the total information set, may not correspond to any person’s explicit expectation, drawn from an individual information set. So the answer to ‘whose expectations?’, may be ‘No one’s expectations’. The implicit expectation, corresponding to the total information set, is most closely related not to the most common information set but the information set that yields the greatest conﬁdence amongst its possessors. If var + var - then (a) the Xis are more conﬁdent than the Epsilons in that the var(error) of the Xis is less than the Epsilons, (b) The implicit expectation, corresponding to the total information set is better correlated with the explicit expectation of the Xis rather than the Epsilons. So: what matters or signiﬁes is not the number of those who hold a viewpoint but the (rational) conﬁdence with which the viewpoint is held. The important expectation is not the common one; it is the conﬁdent one. Weight those who are conﬁdent. The acquisition of information (‘learning’) may not be relevant to the equilibrium outcome. It will only be relevant if no one else has acquired it.

The equilibrium of (8.26) may seem peculiar. How is this equilibrium obtained? We will oﬀer one remark: There can be no equilibrium if prices do not reﬂect the totality of information. So, to illustrate, in Example 1 the equilibrium cannot be that reﬂecting only the information set of the Zeros. That would imply, +1 E 0 i Therefore,

i +1 As the +1 is perfectly known to the Perfects (but not to the Zeros) the excess rate of return on capital is perfectly known to the Perfects. There are inﬁnite riskless proﬁt opportunities to the Perfects. And corresponding inﬁnite losses to the Zeros. Zeros would either buy information from Perfects, or just abandon playing the game. Any such withdrawal means prices are shaped by the Perfect; and the interest rate becomes perfectly correlated with .

182

Inﬂation in a risky world

In Example 2 if equilibrium was conditioned only on the information of the Epsilons then + + 1 1 1 2 1 ... -1 1 and the return on borrowing to invest in capital is:17

i -1 1 This rate of return does not have a conditional expectation of zero for Xis. Their knowledge of - will allow them to get on the right side of the bonds versus capital choice, on average. Their decision would not always be right ex post. But it’s a bet that is unfair in the favour of the Xis, and a bet unfair in one’s favour cannot be so risky that its value is zero. (Arrow 1971, 100). So the Xis would start ‘betting’. And the Epsilons could only improve themselves by exiting.18 The Eﬃcient Markets equilibrium is the only equilibrium; there is no ‘ineﬃcient markets’ equilibrium. So the question of how the Eﬃcient Markets equilibrium is obtained has no more significance than the question of how any market obtains equilibrium. The Eﬃcient Markets equilibrium is a supply and demand equilibrium, and is no more or less diﬃcult to obtain than any other equilibrium. These remarks serve our limited purpose: that the heterogeneity of information sets is not problematic for the existence of equilibrium.19

NOTES 1.

2.

3. 4.

The existence of markets for real bonds and money bonds, combined with an ‘amenable’ environment of monetary risk (see Chapter 7), is required for the identicality across persons of the proﬁt risk premium and inﬂation risk premium. This identicality is in turn required for dichotimization, and the use of the representative agent. Put more informally, high inﬂation (or below average growth in purchasing power) tends to occur when one’s needs are most pressing; so when one’s needs are most pressing one’s money balances are worth the least. This is an undesirable attribute, and this will prompt a switch away from money. The variance of inﬂation has no impact in itself. An increase in variance unaccompanied by a change in covariance will be completely irrelevant. These equalities provide a handle on the size of the impact of the unpredictability of the proﬁt rate, something which empirical studies have shed little light on. If &

+, (+ white noise) then cov( , &) var( ) (2 , or ) (2 . Thus: ) d) 2 ( d( 2( d(

183

The Quantity Theory in a risky world dh 2) si ( h hd(

So if the semi-elasticity of h to expected inﬂation, sih, was 5, and the (percentage) proﬁt risk premium was 1, and the standard deviation of the (percentage) proﬁt rate was 1, then an increase in the standard deviation of (percentage) rate of proﬁt of 1 would increase money demand by 2.5 percentage points ( 5*0.01/0.02). 5. Spending on credibility costs, allows the spender to lend $1. That loan provides $i of interest next period, providing $i/[1 i] of income in present value terms. The principal on the loan is repaid next period, and used to meet the money issue returning for redemption. 6. So to illustrate: during the Great Depression the increased riskiness of capital would have increased the demand for money, but any unpredictability in the price level that amounted to an inﬂation risk premium would have reduced the total demand for money. Perhaps the same conjunction took place in the 1970s. 7. Let there be S money supply states in each period, producing in each period S price levels. In each period, therefore, there will be 3S equations ( S equations for x m, S equations for m, and S equations for E), and 3S variables (S Ps, S ms and S Es). 8. But might the destructive consequences for equilibrium of this ‘possibility’ make for its impossibility? s s s P M1 P s P1 M1 9. sM1 P EP1 p sM1 P But P1 EP1 sP1 sM1 1 and EP1 P 1 sM1 sP 1 1 i 1 Esi1 10.

Substitution of these elasticities secures the result. The corresponding solution for the constant unit elasticity of demand is

P1 M x kE P P1[1 ] P 1 1 P x M kE

1 1 PP1

1

To the extent that E

1 1 PP1

1

1 1 E PEP1

1

then

1 E 1 k (1 E ) P P1 M and by leading and substitution an expression can be derived for 1/P in terms of future E[1/M].

11.

ln P1

ln M1 E1ln M1 ... E1 1 E1 2 1 1 1 1 1

2

...

184

Inﬂation in a risky world Using the deﬁnition of revision terms: ln P1

E ln M1 R1ln M1 E ln M2 R1ln M2 ... [E 1 R1 1] ... 1 1 1 1

Taking the diﬀerence from lnP and deploying the necessary truth E *1 E * q[E * ] R1 * 12. 13. 14. 15. 16.

17.

secures the result. Given arbitrary magnitudes of b and B (1), the progression bB, 2bB2, 3bB3, . . . has the inﬁnite sum bB [1 B]2. E ln M* ln M , + (1 ) and E ln M*t ln M ,,t+ (1 ) . As E ,+ (1 ) , we may say that i for t 0, and i (,t 1+ 1 ) for t 0. This is, of course, Eﬃcient Markets Hypothesis. Is the information of the central bank part of ‘available’ information? If the central bank does not play the market its information is irrelevant. But if it does play the market, it plays with price-making power. Given its competitive assumptions, the model implicitly assumes a passive central bank. If the equilibrium reﬂected just the knowledge of the Epsilons then, + +1 2 ... -1 1 1 1 1 + + E Epsilon 1 2 ... 1 1 1 + + i 1 2 ... 1 1 1

18. 19.

The less informed Epsilons could also improve themselves by purchasing information. There are other issues that these observations ignore. Error over parameters has been ignored, as has disagreement over parameters. The issue of the production of information is also completely ignored.

9.

Wicksellianism in a risky world

A key assumption of the Wicksellian model as developed so far is that the proﬁt rate is a known. This chapter reworks the Wicksellian approach under the assumption that future magnitudes of proﬁt rate, the reference price level, and the benchmark interest rate are a matter of surmise rather than knowledge. The resulting analysis tends to be conserving, rather than undermining, of the conclusions of the perfect foresight treatment. Indeed, Wicksellian ideas in some ways lend themselves better to a model with risk than a model with perfect foresight. With the incorporation of risk, the Wicksellian notion that higher interest reduces inﬂation becomes rationalizable as ‘unexpectedly high interest rates causes unexpectedly low inﬂation’. The lagged eﬀects of impacts of dependent on independent variables, too, become rationalizable. The analysis of this chapter assumes the existence of markets for money bonds and real bonds.

THE DEMAND AND SUPPLY PRICE OF CREDIT UNDER RISK Under risk, the core of the Wicksellian model remains the adjustment of the price level to equate the demand price of credit to the supply price of credit, just as it was under perfect foresight. The demand price for credit under risk can be obtained from the optimization conditions: capital

UC E[UC1[1 ]]

(9.1)

real bonds

UC [1 r]EUC1

(9.2)

money

U UC Uh E 1 C1

(9.3)

money bonds

i UC E UC111

(9.4)

185

186

Inﬂation in a risky world

The capital and money bonds condition imply (ignoring cross product terms): i E E ) )

(9.5)1

This expression can be interpreted as the maximum rate of interest that will be tolerated by someone borrowing money in order to buy capital: the demand price of funds under risk. The demand price is adjusted down by the risk premium on capital; the maximum rate at which persons will borrow to buy capital is reduced. On the other hand, the demand price is adjusted up by the ‘inﬂation risk premium’, ) – the negative of the covariance of purchasing power with marginal utility means. This is because such a negative covariance means that debt is most costly to the debtor (biggest in real terms) when income is least needed (marginal utility lowest), and that makes borrowing for the purpose of buying capital more attractive. And that increases the demand price for funds.2 How is the supply price of credit aﬀected by risk? We have previously assumed the supply price of credit is determined by the Central Bank according to an IRRF: _

i i [ln P ln PR]

(9.6)

It is not easy to judge how this should be revised in an environment of risk; as an ad hoc formulation it is hard to convincingly argue any revision of (9.6). But there are some potential points for reconsideration. It might be argued that the riskiness of the future magnitudes of and PR should make their current magnitudes, too, a matter of surmise to the public, rather than knowledge. Similarly, the riskiness of future magnitudes of P might suggest that current P is not a matter of knowledge to the Central Bank. We will, however, suppose that, whatever probabilistic process that governs them, current magnitudes of PR and P are known to all. The interest rate reaction function is assumed to be unchanged by risk. Collecting the conditions for supply and demand price: Demand Price of Credit

i E E ) )

Supply Price of Credit

i i [ln P ln PR]

_

E E ln P1 ln P and solving:

187

Wicksellianism in a risky world

i Supply price of credit

Fisher condition

_ i

Demand price of credit

E

lnPR Figure 9.1

Interest rate reaction function

Equilibrium lnP

ElnP1

lnP

Equilibrium in the Wicksellian model under risk _

ln P E ln P E ) ) i ln P 1 R 1 1 1

(9.7)

Figure 9.1 above depicts the solution. The equilibrium is analogous to perfect foresight, and the comparativestatics are analogous. The only innovations in the comparative-statics are the impact of the proﬁt risk premium and inﬂation risk premium. A greater proﬁt risk premium reduces the demand price of credit.3 That necessitates a reduction in its supply price, that can only be achieved by a fall in P. ln P 0 )

(9.8)

A greater inﬂation risk premium raises the demand price for credit,4 and that necessitates a rise in the supply price, and so P rises. ln P ) 0

(9.9)

188

Inﬂation in a risky world

The algebraic expression for the price level also has a positive response to the expected future price level. This begs questions about the determination of the future price level. To complete the model we take each equivalent expression for price equilibrium in each future period: _

E ln P E1 1 ) ) i ln P1 1 ln PR,1 11 2 1

_

E ln P E2 2 ) ) i ln P2 1 ln PR,2 12 3 1

(9.10)

and so on. Take expectations and substituting: ln P ln PR1 E ln PR,1

_ 1 ... [E ) ) i] 1 [1 ]2

_ 1 [E 1 ) ) Ei1] 1

2

...

(9.11)

Dropping the risk premium parameters, and using permanent variables based on (9.11) may be written more compactly: _

ln P ln EP*R

E * Ei*

(9.12)

From this an expression for inﬂation can be derived using the identity that equates the actual outcome to its expectation in the previous period, plus the current revision. _

_

_

E*R E * [i Ei*] R1ln P*R

R1 * R1i*

(9.13)5

WICKSELLIAN INFLATION: AN ILLUSTRATION The expression for inﬂation (9.13) is general but not very informative. The inﬂation equation can be given content and precision by making a speciﬁc _ assumption about the disturbances to PR, , and i . Suppose there are no disturbances, save for the proﬁt rate. Suppose each period experiences an unpredictable disturbance, of which a fraction, ,, is temporary, and 1, endures into the next period.

189

Wicksellianism in a risky world

Table 9.1

Wicksellian inﬂation under proﬁt disturbances

Type of Process

E E[ E]2

0, 1

, 1

, 0

ARMA

White noise

Random walk

1 ,+ 0 [1 ,]2 var+ 2

0 0 0

1 + ,+ 1

+ 0 var+ 2

(9.14)

+ is a white noise variable, of which the current period magnitude is not known. Table 9.1 reports the relation of P and inﬂation to proﬁt shocks. The topmost row indicates that under the present assumptions a very simple model of inﬂation holds: inﬂation equals the permanent shock to proﬁt, divided by the interest rate response parameter. As the degree of permanence of a proﬁt shock falls, the impact of a proﬁt shock falls; ultimately to zero, if the shock is entirely temporary. As the size of the interest rate response falls, the impact rises. As the interest rate response parameter may be any positive magnitude, the impact on inﬂation of a proﬁt rate shock is extremely variable, varying between a minimum of zero to maximum of indeﬁnitely large. The middle row indicates that, despite the possibility of very large inﬂation movements, the expected rate of inﬂation is always zero.6 This makes the expectation of the squared errors of the inﬂation forecast capable of being very large (the bottom row). But it will be lowered by a larger coeﬃcient of reaction in the interest rate, . The more sensitive interest rate policy is to price deviations, the more predictable inﬂation. Table 9.1 was derived under the assumption that the only disturbance is to the proﬁt rate, and that the Central Bank makes no contribution to inﬂation; the reference price level and benchmark rate are assumed constant. Suppose now that the reference level is revised each period in an unpredictable manner,7 and only part of that revision is itself revised away the following: ln PR,1 ln PR +1 ,+

(9.15)

In this expression a high , means that any innovations in policy tend to be eliminated. Policy innovations do not endure. A low , means any innovations in policy tend to endure. Policy is always changing direction.

190

Inﬂation in a risky world

The expectational variables under this modelling are, ER ,+;

ER,1 ER,2 ER,3 ... 0

E*R 1 ,+

R1ln PR*

1, 1 +1

(9.16)

With the help of the preceding section’s expression for inﬂation, expressions for prices and inﬂation corresponding to certain magnitudes of , can be obtained, and are reported in Table 9.2 below. The topmost row indicates that, with the exception of an extreme case, current inﬂation is a matter of the current shock to the price level target, and – with a negative impact – the preceding shock to the price level target. This can be understood in terms of the previous decomposition of inﬂation into the expected rate of growth in the permanent reference price level plus the revision. E*R R1ln P*R. The term involving the preceding period’s shock is the E*R term; a big shock in the preceding period means an expectation of a negative E*R as most of the shock is expected to be reversed. The term involving the current period’s shock is the R1ln P*R term; a big shock in the current period raises prices in the current period. The bottom row indicates that the more enduring are policy innovations (the higher 1,), the higher the typical error of expected inﬂation. In other words, the subsequent ‘correction’ of disturbances of policy makes for more predictable inﬂation than just going along with them, and treating them as so much water under the bridge. The bottom row also indicates the typical error of expected inﬂation is made larger the larger the interest rate reaction parameter. So whereas sensitivity in interest rate policy mutes the unpredictability imparted by proﬁt volatility, central bank interest sensitivity exacerbates the unpredictability imparted by the reference price level volatilTable 9.2

Inﬂation under disturbances to the reference price level

Type of process

1, 0

,1

, 0

ARMA

White noise

Random walk

1 [+1 +]

+1

1 +

0

[ 1 ,] 1 +,1

+1

E

1 ,+

E(E)2

var+

1, 1

2

var+ 1

2

var+

191

Wicksellianism in a risky world

ity. It is as if unpredictability in one element of monetary policy (the price level target) will be compensated for by rigidity in another (the interest rate).

‘LAGS’ Allowing for risks, and modelling disturbances in the manner of an autoregressive moving average manner, allows a rationalization of the apparent signiﬁcance of lags. With an ARMA formulation of the reference price level, inﬂation can (putting aside polar cases) be mapped into lagged values of past rates of reference inﬂation, as long as 1 , 0 (Table 9.3). Table 9.3

Inﬂation and the lags of reference inﬂation

1, 0

,1

,0

[1 ,] [1 ,] [1 ,] R 1 1 ,R,1 1 ,2R,2...

R1

R

This mapping suggests that actual inﬂation is a lagged function of target inﬂation. This suggestion, however, is misleading. Table 9.2 indicates that the only shocks to the target price level in 0 and 1 have any relevance for inﬂation between 0 and 1. So how do we explain the apparent dependence of actual inﬂation on past rates of target price inﬂation? A thought experiment is useful. Consider a one-oﬀ increase in the target price level that is sustained, exactly. Equivalently, consider a solitary positive realization of R, followed by a succession of zero realizations of R. The dynamics in Table 9.3 indicate that P will rise in the period of the shock, but by less than R. The table also indicates that in subsequent periods P will continue to rise, as if still responding each period to the (ever receding) past increase (Figure 9.2). Why does P seem to adjust with a lag? The answer lies in the fact that P depends on the permanent magnitude of the reference price level. P does not rise proportionately with the current money reference price level because the expectation of the reference price level in future periods is little aﬀected. And why has the expectation of the future reference price level not risen by as much? Because, rationally, part of the increase in lnPR is not expected to last; part of it is expected to be reversed in the following period. But as time passes, and as it is not reversed, the expectation of future lnPR rises. This is because there have been (and must have been) more favourable shocks in subsequent periods, so as to compensate for the cessation of the

192

Inﬂation in a risky world

lnP lnPR lnP

Time

Figure 9.2 The Wicksellian response in price to a one-off and sustained increase in the reference price level temporary component of the original shocks. It is this accumulation of more favourable shocks that has made the original increment enduring, and increases the expectation of the future lnPR. So it is not as if the original increase is ﬁnally being ‘fully felt’. Rather, in this thought experiment, the original increase is being transformed by the succession of positive shocks into a permanent increase. It is that permanent-making eﬀect of the implicit assumed succession of positive shocks that has imparted the upward climb in the price level. The conclusion of the preceding paragraph is underlined by the observations: if all of the shock is permanent (,0) then there is no ‘lag’; if all of the shock is temporary (,1) then there is no ‘lag’. In the ﬁrst case there is no lag because it takes no time for any shock to become enduring; it is instantly enduring. In the second case there is no lag because no shock ever becomes enduring.

INTEREST AND PRICES AGAIN Incorporating risk not only allows a rationalization of lags, it makes possible a rationalization of a negative relation between inﬂation and nominal interest rates; a relation that seems at odds with the Fisher hypothesis of a positive relation, but which is part of the Wicksellian thesis, as well as the common market place appreciation of the relation between inﬂation and

193

Wicksellianism in a risky world

Table 9.4 Surprise inﬂation and surprise interest rate movements under temporary shocks _

i

PR

lnP E1lnP

R# 1

i E1i

R# 1

RlnPR 1 RlnPR 1

R 1 R 1

lnP E 1lnP i E1i

1

1

1

nominal interest rates. For, once risk is introduced, there emerges, in the context of certain temporary shocks, the negative comovement between the inﬂation surprise in a period, ln P E 1ln P, and the interest rate surprise for that period, i E 1i. Table 9.48 enumerates the types of shocks we are concerned with, and column 1 the types of responses we are concerned with. The bottom row reports the ratio of the unexpected change in prices (in the face of a temporary shock) to the unexpected change in the interest rate (in the face of a temporary shock). The row indicates that if the temporary shock involves the benchmark rate (see the leftmost column) then there is a negative onefor-one comovement between the unexpected change in prices and the unexpected change in the interest rate. A 1 percentage point surprise in the interest rate goes with a 1 per cent surprise fall in the price level. If the temporary shock involves the reference price level (see the centre column) then there is also a negative one-for-one comovement between the unexpected change in prices and the unexpected change in the interest rate. The co-movement of i and P is not negative in the face of all temporary shocks. If the temporary shock concerns the proﬁt rate (see the rightmost column) then there is a positive co-movement between the unexpected change in prices and the unexpected change in the interest rate. The co-movement of i and P is also not negative in the face of permanent shocks. Any upward revision to a future benchmark rates will cause a reduction in the future price level, which will ‘ripple backwards’ to reduce the present price level, but still leave a downward trend in prices over time that will, consistent with Fisher, reduce the current interest rate. In other words, lower prices and lower interest rates: the co-movement is positive. In a parallel manner, any downwards revision to a future reference price level will reduce future prices, which will cut present prices, still leaving a downward trend in prices, which will, in accord with Fisher, reduce the

194

Inﬂation in a risky world

current interest rate. Again, lower prices and lower interest rates: the comovement is now positive. To summarize: temporary shocks to central bank policy create a negative relation between price surprises and interest rate surprises.

STABILIZATION RULES The issue of risk has signiﬁcance for the rules for stability advanced in earlier chapters. Stability in Prices Perfect price stability ( zero mean and variance of inﬂation, and zero mean and variance of inﬂation error) was secured in the perfect foresight model by holding the target price level constant, R 0, and moving the benchmark interest rate in tandem with the real rate of return on capital, _ i . It would appear that the return on capital being unknown presents a problem for this price stability rule. For if the return on capital is unknown, how can the central bank know what to make of benchmark rate? In truth, inspection of the expression for P reveals that perfect price stabilization can be secured by moving the benchmark interest rate in tandem with the expected rate of return on capital. R,t 0

_

for all t

and

it E t ) )

(9.17)9

Stabilization merely requires knowledge of the expected rate of return, not the actual. This implies that any ‘wrongness’ in the expectation E – its divergence from – does not harm the eﬃcacy of price stabilization. That is good news for price stabilization. Price stabilization seems to survive imperfect information. However, one diﬃculty still lurks. While stabilization does not require that Central Bank know , it does require that the Central Bank know E , the market’s expectation of . And how does the central bank know ‘the market expectation’ of the rate of proﬁt? This market expectation is not conventionally measurable. And the problem is still thornier than the measurement of what is contained inside a mind. For the expectation may be in no mind: recall it is entirely possible that no single member of the market need expect the rate of proﬁt that ‘the market’ expects. ‘The market’ expectation is the sum of all information. ‘The market’ expectation is something disembodied from any member, and greater than any single member. Does this present a diﬃculty for stabilization? It is helpful to distinguish three situations: (a) the market knows everything the

Wicksellianism in a risky world

195

Central Bank knows, and more; (b) the Central Bank knows everything the market knows, and more; (c) and the market and the Central Bank both know some things the other does not. With respect to (a) the application of the stabilization rule will still leave _ some deviation between E and i , reﬂecting the diﬀerence in the information sets of the market and the central bank. Perfect stabilization is now lost. For that part of the market’s information set that the central bank does not possess will not be cancelled by the benchmark rate, and so will impart instability. But this conclusion – that there is some instability, and not perfect stabilization – is a weak one: it would be more signiﬁcant to know if the stabilization rule stabilized relative to a policy of keeping the benchmark rate constant. And it will. If the market expectation is a sum of independent factors, those independent factors that are known to the central bank will still have their impact erased the stabilization rule. With respect to (b) it may seem that application of the stabilization rule _ will also leave a deviation between E and i , owing to the deviation of the information sets. But if the market knows that it knows less than the central bank it will adjust its expectation to the benchmark rate. The market expectation is adjusted to be coincident with the central bank’s expectation (that equals the bench mark rate), and stability is thereby achieved. (The Bank could, of course, simply make available whatever information it has to the public.) With respect to (c) again it may seem that application of the stabilization _ rule will also leave a deviation between E and i , owing to the deviation of the information sets. But again if the central bank sets its benchmark rate to its expectation of the proﬁt rate, the market will incorporate the benchmark rate into its information set. But the market’s expectation does not become coincident; it knows more than the bank. So perfect stabilization is lost. But the stabilization rule is still stabilizing compared to a policy of benchmark rigidity. For the central bank’s movement of the benchmark will still cancel the disturbing impact of any information that both the market and the central bank had in their information sets. Only if there is no overlap at all in the information sets – only if all the central bank’s information is ‘new’ to the market – will the central bank’s adjustment in the bench mark rate do no more than cancel the impact of information that the market never had in the absence of the adjustment.10 Stability in the Interest Rate The stabilization of the nominal interest rate may be achieved by this policy: ln PR,1 ln PR

(9.18)

196

Inﬂation in a risky world

In each period the target price is set equal to the target price the previous period less the realized rate of proﬁt.11 Equivalently, the target price is set so that the target rate of inﬂation plus the real rate of return is a constant. This is effectively targeting the nominal rate of return, as it says ln PR,1 ln PR .

FACT AND THEORY The Wicksellian analysis has been motivated in these pages by the more intelligible modelling of the money supply process it provides. But the real test of its worth lies elsewhere. Does it throw any light on historical episodes of inﬂation and deﬂation? Does it do this any better than the Quantity Theory? In evaluating the Wicksellian model’s predictive performance it is helpful to distinguish between trend and cycle components of inﬂation. The Wicksellian model nominates proﬁt rates and benchmark rates as the leading candidates for explaining cyclical movements in inﬂation. The Wicksellian model therefore seems to disregard almost every indicator that has been found useful, or thought to have been found useful, as a proximate cause of short run movements in prices: scarce inventories, booming commodity prices, falling exchange rates, oil shortages, bottlenecks, labour market pressures. The relevance of several of these things to inﬂation is, however, consistent with the Wicksellian emphasis on the proﬁt rate. Clearly inventories have a rate of return, and that will be high whenever inventories are scarce. Commodities, too, are assets, whose prospective rate return will often be high when they have been appreciating in the recent past. A reduction in oil inputs may increase the rate of return on capital. Bottlenecks may also increase the rate of return on capital as a whole, and thereby be inﬂationary on account of Wicksellian theory. Technical change may cause a simultaneous increase in the demand for labour and, critically, capital. Even more deﬁnitely in favour of Wicksellianism is the tendency for inﬂation to drop in a recession, as rates of return drop. Trend movements in inﬂation are much less easily explained by proﬁt rates and benchmark rates, as they themselves must have a trend in order to explain a trend. The Wicksellian resource for explaining the trend of inﬂation is in the trend in the reference price level. So, to illustrate, the Wicksellian explanation of the origin of the Great Inﬂation of the 1970s is that the reference price level increased more quickly than it did in the 1960s; the actual price level, in other words, was allowed to rise more quickly than it did before without the Central Banks raising the interest rate.12 And the Wicksellian explanation of the end of the Great Inﬂation was a decline in the rate of reference price inﬂation: the actual price level,

Wicksellianism in a risky world

197

in other words, could no longer rise so quickly as before without the Central Banks raising the interest rate. In Wicksellian conception, the unexpectedly high interest rates of Paul Volker’s chairmanship of the Federal Reserve signalled that. But is there any evidence that the reference price level increased more quickly in the 1970s? A great diﬃculty in answering that question is that in being a behavioural parameter, and not a ‘thing’ (as the money supply is), the reference price level is diﬃcult to observe – and has not been observed. One response would be to brave the diﬃculties and seek to gather observations, even if the ‘observations’ can be no more than ‘soft data’. For example, a monetary authority’s prediction of inﬂation, that is unaccompanied by predictions of higher interest rates, might be taken to be a report of the rate of increase in the reference price level. An inﬂation target may be even more readily identiﬁed with the rate of increase in the reference price level, as long as no change in the interest rate is foreshadowed. Another way of assessing whether the reference price level increased more rapidly in the 1970s would be to see if the central bank’s behaviour changed in this decade: if the interest rate chosen in response to the price level changed. Was there, in other words, a structural shift in the interest rate reaction function? Such an inquiry would be easier if the reference price level was thought to have undergone a one-oﬀ shift upwards in level in the 1970s. Such a one-oﬀ shift in level would be manifested as a change in the constant term in the interest rate reaction function, and a change in a constant term can be assessed by standard methods. But the Wicksellian hypothesis of the Great Inﬂation is, surely, that the rate of increase in reference price level increased. Under this hypothesis there is no constant term in the interest rate reaction function, but instead an (unobserved) exogenous variable (the reference price level) increasing according to a trend. Nevertheless, the Wicksellian hypothesis might still be investigated by treating the interest rate as a function of the ‘trend adjusted price level’; and taking the price level deﬂated by the trend of the price level as a proxy for the reference price level. One might investigate how well such a reaction function that was estimated for the 1950s and 1960s, performs out of sample in the 1970s. If it under predicts the interest rate in the 1970s, then one might infer there was a ‘loosening in interest rate policy’ in the 1970s; in other words the interest rate in the 1970s was not raised in circumstances where in previous decades it would have been. It is not obvious that there was a ‘loosening of interest rate policy’ in the 1970s. Interest rates were much higher than the 1960s. But this surely reﬂects the operation of the Fisher Hypothesis, which is also an integral part of the Wicksellian model. We confront, then, the problem that in the Wicksellian model both the interest rate and the price level are endogenous

198

Inﬂation in a risky world

variables, and trying to estimate a reaction function in the context of this endogeneity is beset with diﬃculty. We are left with estimating a reduced form equation for the price level, but still beleagured by the unobservability of the reference price level. Nevertheless proxies might be advanced to allow the investigation of the implication of the Wicksellian model that the price level is negatively related to those proxies.

NOTES

E[UC1[1 ]] E UC1 1 i 1

1.

EUC1[1 E ] cov(UC1, ) EUC1E 1 i cov UC1, 1 i 1 1 EUC1[[1 E ] cov(&, ) EUC1 E 1 i cov &, 1 i 1 1

1 E cov(&, ) 1 i E cov(&, ) i E cov(&, ) E cov(&, ) 2.

3.

At the same time, a more negative covariance between growth rate in purchasing power and the marginal utility of consumption necessitates a larger compensation for creditors if they are to lend in money terms; as their above average payoﬀs coincide with low marginal utility. This is a compensation that debtors are willing to pay, for reasons explained. How large is this eﬀect? If ( the marginal utility of consumption)

then ) (2 , and so ln P ) 2 ( 1 (

4.

Debtors are more willing to pay, as a greater inflation risk premium means whenever the rate of growth of purchasing power tends to be above average, (and the quantity of consumption lost by repaying debt is above average) the marginal utility consumption is below average. When debt is most costly it is least needed. _

5.

E * Ei* ... ln P ln PR E ln PR,1 1 [1 ]2 _

E * E i* ... 1 1 E1ln PR,2 1 [1 ]2

ln P1 ln PR,1

[ER,2 R1ln PR,2] 1 [1 ]2

[ER,1 R1ln PR,1] ...

_

_

E1 * E * E i* 1 Ei*

199

Wicksellianism in a risky world

Using the_ necessary truth, E1 * E * [E * E ] R1 *, and the corresponding truth for i yields the result. 6.

E E ln P1 ln P E

7.

8.

,+1

,+ 1 E ,E+

,+ 1 1

1 ,+ 1

,+ 1 0 1

That is, unpredictable to the public. Its past and present magnitudes are known by the public, but its future magnitudes are not, and – as epsilon is white noise – are rationally forecast by the public at zero. ln P E 1ln P R ln PR ... R ln PR,1 1 [1 ]2

_ _ [R Ri] 1 [R 1 Ri1] 1 1 1

2

...

and _

i E 1i Ri [R ln P R ln PR] Thus, if we restrict to current benchmark shocks, _ ln P E 1ln P Ri 1 1 _ _ _ _ i E 1i Ri R ln P Ri Ri 1 Ri 1 1 1

9.

A second method of price stabilization is, ln PR E

10.

11.

_

and

it

The ﬁrst method keeps the reference price level; constant, and jiggles the benchmark rate up and down. The second keeps the target rate constant but jiggles the reference price level up and down. Under both these policies the nominal interest rate hovers up and down with the real rate of return on capital. They are equivalent. One case of this is where the market knows nothing apart from the parametric component of the proﬁt rate. If it knows nothing, everything the central bank knows is ‘new’ to the market. But if the market knows nothing then there would be no instability anyway, as the market’s expectation is just parametric. This is most easily demonstrated using the central bank reaction function, _

i i [ln P ln PR] _

i i ln PR

... E ln PR,1 1 [1 ]2

_ _ [E i] 1 [E 1 i] 1 1 1

i ln PR

2

... ln PR

... E 1 E 1 1 E ln PR,1 1 1 1 [1 ]2

2

... ln PR

200

Inﬂation in a risky world

i ln PR [ln PR E E 1 2] ... [ln PR E ] 1 [1 ]2 [1 ]3

E 1 E 1 1 1 1

2

... ln PR

2 ... [1 ]2 [1 ]3

i 12.

It is not necessary for the success of the hypothesis to suppose Central Banks thought in terms of a relation between the interest rate and the price level. The institutional arrangements might have led them to act according to such a relation. A commitment, for example, to a ﬁxed exchange rate and unchanged interest rate in the face of imported inﬂation is, in eﬀect, a decision to let the reference price level inﬂate more rapidly.

PART 4

The cost of inﬂation This part turns from the causes of inﬂation to its costs. Two theories of this cost are advanced: one largely familiar, and one largely an innovation; one based on inﬂation’s disturbance of the money market, the other on inﬂation’s disturbance of the bond market; one turning on the costliness of having less outside money, and the other costliness of having fewer bonds; one turning on the increased liquidation costs from inﬂation, and the second on the disruption to the trade in risk of inﬂation risk. It is argued that costliness of using less outside money makes only a weak case for the desirability of price stability. We contend that the theory based on the disruption to the trade in risk gives a stronger rationale to the goal of price level stability.

10. The cost of inﬂation as the cost of moneylessness This chapter outlines the costs of inﬂation that arise from inﬂation’s disturbance of the money market. The chapter begins by rehearsing the cost that can be traced to inﬂation’s disturbance to money demand: the ‘shoe leather cost of inﬂation’. It is then argued that, despite managing to account for inﬂation’s costliness by means of well founded empirical and theoretical relations, the theory provides only an awkward case for the desirability of price stability. For not only are these shoe leather costs small, their reduction, it will be argued, may actually recommend price instability. The chapter then turns to identifying the costliness of inﬂation that may be traced to inﬂation’s disturbance to money supply. It is argued, on the basis of Chapter 3’s model of the supply of inside money, that inﬂation induces an increase in the supply of inside money that is socially costly. This is because inﬂation creates private incentives for inside money to supplant outside money, and that this involves a social cost without any social beneﬁt. The costs of inﬂation that result from a reduction in money demand are, therefore, augmented by the costs of inﬂation that result from an increase in money supply. But the introduction of costs from the disturbance to inside money not only augments the total costs of inﬂation; it critically changes the character of the distribution of the costs of inﬂation. For while all money holders bear part of the shoe leather cost of inﬂation, some suppliers of inside money may experience a net gain from the increased supply of inside money under inﬂation. Thus the theory of the supply of inside money establishes the possibility of a situation where inﬂation is costly to the population as whole, but may be beneﬁcial to some part of that population. Inﬂation is no longer a common enemy, but may be welcome by part of the community. The costly consequences of inﬂation’s disturbance of the money market will be analysed under the assumptions maintained in Parts 2 and 3; including the existence of real bonds as well as nominal bonds, and the dichotomization into real and monetary sectors. The theory developed applies equally to Quantity and Wicksellian models. 203

204

The cost of inﬂation

THE SHOE LEATHER COST OF INFLATION1 The shoe leather cost of inﬂation is bottomed on a basic premise of the theory of money demand: that holding more money reduces liquidation costs. Since higher inﬂation reduces real money holdings, higher inﬂation increases liquidation costs. This is the shoe leather cost of inﬂation. The idea of shoe leather cost can be ﬁxed more clearly by means of Figure 10.1.2 The gain from holding some amount of money, cc, rather than having none at all, is the area under the plotting of the implicit return on money: aa dd cc bb. The loss from holding cc of money, rather than the maximally useful amount of money (ee), is dd ee cc. The amount of money actually held is that for which the implicit yield on money, , equals the nominal rate of return on capital, : Thus if the nominal rate of return on capital is ﬀ (see Figure 10.2), then cc is the amount actually held, and the loss from not having the maximum amount of money is dd ee cc This is the shoe leather cost implied by a nominal rate of return of ﬀ, and it is denoted SL. aa

dd

bb

Figure 10.1

cc The benefit of money

ee

h

205

The cost of inﬂation as the cost of moneylessness

, aa

ff

bb Figure 10.2

dd

ee

cc

h

The shoe leather cost of inflation

It is easy to see from Figure 10.2 that an increase in inﬂation, by increasing the nominal rate of return on capital, will increase shoe leather cost. In other words, ‘the marginal shoe leather cost of inﬂation’ is positive. dU dU dh dSL d dh d h 0 UC UC d

(10.1)

Whether the marginal shoe leather cost of inﬂation increases or decreases with inﬂation is uncertain. d 2SL h h d2

(10.2)

If h0 then d 2SL d2 is of ambiguous sign. The ambiguity can be resolved only by resort to speciﬁc functional forms of money demand. Table 10.1 reports the response of marginal shoe leather cost to inﬂation for constant elasticity, constant semi-elasticity and constant coeﬃcient demand functions. Figure 10.3 brings out the possibly positive or possibly negative magnitudes of d 2SL d2 by plotting dSL d against for the three functions.

206

Table 10.1

d2SL d2 h

The cost of inﬂation

The marginal shoe leather cost of inﬂation Constant elasticity

Constant semielasticity

Constant coeﬃcient

e21e 0

expss[1 s]

0

e

exp s

-

dSL d

Constant semi-elasticity

Constant coefficient, 2SL 0 2

Constant elasticity, 2SL 0 2

0 Figure 10.3

The marginal shoe leather cost of inflation

A signiﬁcant revision in the theory of shoe leather cost is required once riskiness of inﬂation is allowed for. Money is no longer held until its implicit yield equals the actual nominal return on capital. Instead money is held until its implicit yield equals the expected nominal return on capital, adjusted down for proﬁt risk and up for inﬂation risk. E E ) )

(10.3)

The upshot is that actual inﬂation is no longer determinant of money demand; and so no longer a determinant of the size of shoe leather cost. But expected inﬂation, and the inﬂation risk premium do impact on the yield, and therefore on the amount of money held, and amount of shoe leather cost. The upshot is that under risky inﬂation there are two inﬂation type variables that impact negatively on money holdings, and so positively on

The cost of inﬂation as the cost of moneylessness

207

shoe leather costs: a positive expectation of inﬂation, E, and a negative covariance of the growth of purchasing power with the marginal utility of consumption, ) 0. The expected rate of inﬂation and the inﬂation risk premium take the place of the actual rate of inﬂation as ‘shoe leather costly’. dSL i h 0 1 i dE

dSL i h 0 1 i d)

(10.4)

SHOE LEATHER COSTS AS A RATIONALE FOR PRICE STABILITY? The theory of shoe leather cost teaches that higher inﬂation – or higher expected inﬂation – is costly. Nevertheless, shoe-leather costs of inﬂation give little warrant for making price stability a leading goal of public policy. This will be argued on four grounds. The ﬁrst three are well aired in the literature, and the fourth is not. The Shoe Leather Costs of Inflation are Small The marginal shoe leather cost of inﬂation, as a proportion of national product, can be expressed, dSL e h 1 e h hy 1 hy dy

(10.5)3

This magnitude can be quantiﬁed. Estimates of the elasticity of money demand to the interest rate are various, but a recent survey of several hundred money demand studies reveals that the median estimate is 0.25 (Knell and Stix 2004). The ratio of h to y is easily measured; in the western world annual national income is in the region of ﬁve times the value of transactions money, M1. Consequently, ehhy 0.05 Thus an inﬂation of 1 per cent per annum costs the equivalent 0.05 of 1 per cent of national income per annum. Equivalently, 5 per cent inﬂation amounts to a loss of 0.25 of 1 per cent of national income; not much more than the loss of 1/2 of a day’s work in a year. The shoe leather cost of an extra inﬂation is, seemingly, ‘small’. Can price stability be made an imperative on such a small cost? The insigniﬁcance of shoe leather costs is underlined by the fact that, although shoe leather is a cost that affects everyone, no one has ever protested it – except economists. It is a cost that might be noticed and

208

The cost of inﬂation

resented by all, but is, in fact, unnoticed by anyone, except relatively advanced students of economics. Nevertheless, the cost is not zero, and even a small cost should be minimized. But where does the minimization of shoe leather cost take inﬂation? The Minimization of Shoe Leather Costs does not Warrant Zero Inflation; It Warrants Negative Inflation Shoe leather costs are minimized when money holdings are maximized, and the marginal utility of yield has been reduced to zero: 0 But that implies:

0 or Thus if shoe leather costs are to be minimized, policy should not be content with zero inﬂation, but seek to reduce inﬂation to the negative of the rate of proﬁt (Friedman 1969). To put it another way, if shoe leather costs constitute a case to reduce inﬂation from 5 per cent to 0 per cent, then they also constitute a case to reduce it from 0 to per cent. The upshot is that the policy maker should run a surplus in order to drain the economy of its money so as to induce a deﬂation. The Shoe Leather Costs of Inflation are Smaller than the Tax Relief Benefits of Inflation Insofar as inﬂation is accompanied by increased outside money, inﬂation is accompanied by a contribution to revenue; ‘seigniorage’. S dM dtP

(10.6)

Insofar as seigniorage is a concomitant of inﬂation, inﬂation will constitute an alternative to other sources of revenue. The possibility therefore arises that inﬂation, with its associated seigniorage, is a less welfare costly source of revenue than alternative sources of revenue. Inﬂation becomes a desirable part of ﬁscal policy (Phelps 1973).

The cost of inﬂation as the cost of moneylessness

209

To make such a conclusion, or to deny it, seigniorage needs to be compared with taxes, in terms of: ● ● ●

equity – horizontal and vertical; administrative and compliance costs; allocational costs.

With respect to horizontal equity, seigniorage is surely superior to alternative revenue sources. The ‘inﬂation tax’ is very diﬃcult to either evade or avoid. Under seigniorage one category of persons – tax evaders and avoiders – are not, in eﬀect, taxed at a lower rate than more tax dutiful persons in the same ﬁnancial circumstances. With respect to vertical equity, the superiority of seigniorage over conventional revenue sources is debatable. Person j’s seigniorage burden is proportional to their money holding; S j h j. Thus seigniorage is ‘regressive’ if money demand is inelastic to income, and progressive if it is elastic to income. The theory of money demand deployed in this study allows money demand to be either elastic or inelastic. And time series surveys seem almost equally accommodating of both possibilities. A survey of 201 timeseries studies (Knell and Stix 2004) had a median income elasticity of 0.87, but with a standard deviation of 0.44, leaving open the possibilities of the elasticity often being equal to or greater than 1. Further, some crosssectional studies of major economies (the United States, Japan) have concluded that money demand is elastic to income.4 With respect to administrative costs, the superiority of seigniorage is plain.5 With respect to compliance costs the superiority of seigniorage is also obvious.6 Thus on equity, administration and compliance considerations the balance is in favour of seigniorage over taxation. So the question that now presents itself is: can the shoe leather costs of the inﬂation tax tip the balance against inﬂation, and in favour of revenue sources which do not have shoe leather costs? To explore this question, it will be supposed that the rate of growth of money supply is permanent and foreseen, and that rate of growth of real money demand is zero. Therefore:

(10.7)

S h()

(10.8)

dS d[h h]

(10.9)

and

210

The cost of inﬂation

SL S

eih 0

S

Figure 10.4 The marginal shoe leather cost seigniorage: the ‘well-behaved’ case Combined with expression (10.1) for shoe leather costs, we can obtain the amount of extra shoe leather costs brought by an extra amount of seigniorage: ‘the marginal shoe leather cost of seigniorage’. dSL eh 0 1 sh dS

(10.10)7

If follows: If then dSL 0 dS (see note 8.) If 0

then

dSL e h dS

If → s1

then

dSL → dS

h

(10.11)

These conclusions point to a rising marginal social cost of seigniorage, illustrated in Figure 10.4.

211

The cost of inﬂation as the cost of moneylessness

SL S

eih 0

S

Figure 10.5 The marginal shoe leather cost seigniorage: the not-so-well behaved case There is an important qualiﬁcation to the generality of Figure 10.4 that concerns the indeﬁnitely high magnitude of the marginal cost of seigniorage that is depicted at a certain size of S. The approach of an inﬁnitely high marginal cost turns on the fact that when approaches the inverse of the semi-elasticity of money demand, the quantity of seigniorage approaches its maximum, and, consequently, the marginal cost of seigniorage approaches inﬁnity as the increment in seigniorage from any increment inﬂation becomes inﬁnitely small. But must a maximum amount of seigniorage exist? Equivalently, can the inﬂation rate approach 1/s? If the elasticity of demand is constant, and less than 1, the the quantity of seigniorage has no maximum (see, for example, Marty 1999). This implies that if the elasticity of demand is constant, and less than 1, then there is an upper bound on the marginal cost of seigniorage.9 In either case, if we abstract from equity issues, optimal seigniorage would equate the marginal shoe leather cost of seigniorage with the marginal cost of public funds (assuming they can be equalized). This is illustrated, where c is the marginal social cost of public funds by non-seigniorage sources (Figure 10.6). Evidently, optimal seigniorage may be negative, zero, or positive. It is easy to see that whether seigniorage should be zero, positive or negative is determined by a comparison of the cost of public funds with the elasticity of money demand at zero seigniorage. If the cost of public funds

212

The cost of inﬂation

SL S

c

Optimal S

eh 0

Figure 10.6

S

Optimal seigniorage

exceeds the elasticity of money demand (evaluated at zero seigniorage) then it is optimal to have positive seigniorage. If the cost of public funds equals the elasticity of money demand (evaluated at zero seigniorage) then it is optimal to have zero seigniorage. If the cost of public funds is less than the elasticity of money demand (evaluated at zero seigniorage) then it is optimal to have negative seigniorage. What does this rule indicate? In recent decades Western economies have not been far from price stability, so one may say that that the elasticity at zero seigniorage is not far from 0.25. Thus the deadweight loss of a dollar of seigniorage is $0.25, at zero seigniorage. But c is commonly estimated to exceed 0.25. Therefore, assuming an unbounded rise in the marginal cost of seignoirage, we would expect optimal seigniorage to be positive. How positive? To answer this question we draw on the condition for optimality, e c 1 hs

(10.12)

h

and the three functional forms for money demand that we have used: constant elasticity, constant semi-elasticity, and constant co-eﬃcient. For each of these functional forms we can calibrate the rate of inﬂation that satisﬁes the equality above, as long as we are content to make assumptions about

The cost of inﬂation as the cost of moneylessness

213

Table 10.2 The optimal rate of inﬂation, per cent, eh 0.25, sh 5, 5

c0.2 c 0.3 c 0.4 Expression for the optimal inﬂation rate

Constant elasticity of money demand

Constant semielasticity of money demand

Constant coeﬃcient of money demand

2.5 10.0 n.a.

[c e] e c ce

0.8 0.8 2.1 c s s[1 c]

0.7 0.6 1.7 c- [1 c] 1 2c

the numerical magnitudes of the elasticity and semi-elasticity of money demand, and the rate of profit.10 Table 10.2 shows the results. We see that for the constant semi-elasticity and constant coeﬃcient cases the optimal rate of inﬂation may be positive or negative, but clustered near to zero. It is the constant elasticity of demand case that is subversive of shoe leather costs as a rationale for inﬂation minimization. Assuming a constant elasticity of demand, with ‘low’ costs of taxation (c0.2) the optimal rate of inﬂation is negative, and with moderate costs of taxation (c0.3) the optimal rate of inﬂation becomes 10 per cent. But it is with ‘high’ costs of taxation that massive seigniorage becomes triumphant. For with a constant elasticity of demand of 0.25, we are in the case of the ‘not so well-behaved’ marginal cost of seigniorage depicted in Figure 10.4. For ‘low’ and ‘moderate’ values of c welfare maximization occurs at the intersection of the horizontal marginal cost of public funds relation, and the upward sloping marginal cost of seigniorage relation. But for ‘high’ values of c there is no intersection. The horizontal marginal cost of public funds relation lies above marginal cost of seigniorage. In that situation is optimal to fund all government spending from seigniorage. And that is the result when c0.4 under the assumptions of Table 10.2. With a constant elasticity of demand of 0.25 the maximum marginal cost of seigniorage is 1/3, and as 1/3 is less than 0.4, seigniorage is less costly than public funds at all levels of seigniorage if c 0.4.11 In these circumstances, all revenue should be raised by seigniorage. And the optimal inﬂation rate is whatever secures that rate of inﬂation. Since the elasticity of demand is commonly estimated to be smaller than the marginal cost of public funds, a constant elasticity of demand assumption will have powerfully inﬂationary implications. But the analysis of Table 10.2 is vitiated by its implicit assumption that all money is outside money. The allowance for inside money implies a

214

The cost of inﬂation

substantial revision since while the ‘inﬂation tax’ only operates on outside money, the shoe leather cost of inﬂation operates on all money held. If h/x is assumed to be exogenous then: h dSL ehx dS 1 sh

(10.13)

Evidently, seigniorage is now more welfare costly, by the ratio of h/x, as the expression for dSL dS with inside money equals the expression without inside money multiplied by h/x. For example, if h/x is 5 the marginal shoe leather cost of an extra unit of seigniorage at zero inﬂation is, not 0.25, but 1.25. Seignoirage now appears prohibitively expensive in welfare terms.12 Indeed, given that the marginal cost of public funds is likely to be less than 1.25, deﬂation would be the welfare maximizing course of action. Table 10.3 brings this out by recalculating the optimal rates of inﬂation once inside money has been allowed for in the manner described in this paragraph. Deﬂation is now the rule. What is the logic? Inside money means there is so little ‘seigniorage per unit of inﬂation’ that there is very little tax relief in the seigniorage from inﬂation – and very little tax burden arising from the negative seigniorage that is concomitant with deﬂation. The Phelps critique of the Friedman rule is undermined: the policy maker should run a surplus in order to drain the economy of its money so as to induce a deﬂation, and reap shoe leather gains.

Table 10.3 Optimal rate of inﬂation with exogenous inside money eh 0.25, sh 5, 5, h/x5

c 0.2 c0.3 c0.4

Constant elasticity

Constant semi-elasticity

Constant coeﬃcient

4.8 4.6 4.5

4.0 3.6 3.1

3.9 3.3 2.9

h

c ex

Expression for optimal inﬂation rate

h c ce ex

h c sx

h c s x

h c c- x h x 2c

The cost of inﬂation as the cost of moneylessness

215

The Minimization of Shoe Leather Costs does not Warrant Price Stability; It may Warrant Unpredictability in Inflation The common notion of ‘price stability’ involves two requirements. E 0 Var 0 The ﬁrst requirement is that inﬂation is on average zero, the second that it be constant. The two together imply inﬂation is constantly zero.13 The previous section has argued that shoe leather cost minimization does not serve the ﬁrst requirement of price stability. This section argues it does not serve the second. This argument consists of the observation that shoe leather costs depend only upon the expectation of inﬂation, and not on the variance of inﬂation. Equivalently, shoe leather costs depend only upon the expectation of inﬂation, and not on the standard error in the expectation. The seriousness or frequency of errors – the seriousness of unpredictability – is irrelevant to money demand, and so shoe leather costs. The standard error might be 100 per cent or 1/10 of 1 per cent; it will make no diﬀerence to money demand. The welfare loss would be exactly the same.14 The suggestion that variance in inﬂation is irrelevant to shoe leather costs invites a riposte. For the theory of money demand does aﬃrm that money demand is reduced by an ‘inﬂation risk premium’; a negative of the covariance between the growth in the value of money and the marginal utility of consumption. The upshot of this is that there exist circumstances where a greater variance of inﬂation will spell a greater shoe leather cost. Consider a situation where the growth in purchasing power, , is negatively related to marginal utility. &

white noise

(10.14)

(This might happen if monetary policy is tightened during an upturn, so that purchasing is growing the most when marginal utility is lowest.) In this situation an increase in shall both increase the variance of inﬂation, and make more negative covariance of the growth of purchasing power with marginal utility (that is, the inﬂation risk premium).15 In these circumstances an increase in the variance of inﬂation will be concomitant with a fall in money demand, and so an increase in shoe leather costs. But this attempt to implicate inﬂation volatility in the size of Shoe Leather Costs is not diﬃcult to rebut. The argument has assumed is positive; an above average growth rate of purchasing power goes with a below average

216

The cost of inﬂation

marginal utility. But may be negative; an above average growth rate of purchasing power goes with an above average marginal utility. In this circumstance an increase in the absolute value of will both increase the variance, and increase the inducement to hold money (by reducing the inﬂation risk premium). Higher variance, but lower shoe leather costs. Further, the argument above has turned on inﬂation variance being changed on account of changes in the responsiveness of the growth of purchasing power to marginal utility; changes in in (10.14). But the variance of inﬂation may change without any change responsive of the growth of purchasing power to marginal utility. The variance in inﬂation may change on account of an increase in the volatility of other independent factors, such as white noise in the money growth, represented by . An increase in the variance of inﬂation on account of an increase in the variance of will have no eﬀect on the risk premium. Indeed, it can be argued that an increase in inﬂation variance, on account of increased volatility of independent inﬂuences, may reduce expected shoe-leather costs. Thus the minimization of expected shoe-leather costs may actually recommend increasing the variance of inﬂation. The case for this thesis can be introduced by recalling our earlier observation that marginal shoe leather cost of inﬂation may either rise or fall with expected inﬂation.16 The possibility of diminishing marginal shoe leather cost would be without signiﬁcance if in each period E was a choice variable of the policy maker. The policy maker could then obtain, with certainty, whatever feasible level of shoe leather costs they thought best, by choosing the magnitude of E appropriately. But what if E is not a choice variable of the policy maker? What if the policy maker’s choice variables have no more than a probabilistic relation with E, with the consequence that E is probabilistic for any setting of monetary instruments? Then E has a variance. As a consequence the magnitude of shoe leather costs is no longer certain for any setting of monetary instruments; it, too, is probabilistic and has a variance.17 In this context, the policy maker would presumably wish to minimize the mathematical expectation of shoe leather cost. It is easy to see that any mean-preserving increase in the variance of E will reduce the expectation of shoe leather costs under ‘diminishing marginal shoe leather costs of inﬂation’. The intuition is that the addition to shoe leather costs of E assuming a high realization is smaller than the reduction in shoe leather costs of E assuming a low realization. The ups in expected inﬂation are not as bad as the downs are good. So: have ups and downs in expected inﬂation. The upshot is that the shoe leather cost minimizing social planner would prefer E to have some volatility.

217

The cost of inﬂation as the cost of moneylessness

Further, under plausible circumstances, this volatility in E may be stim¯ + and supposulated by unpredictability in inﬂation. Using lnM lnM ing no disturbances to money demand, we may infer, var 2var E

(10.15)18

A higher variance in inﬂation means a higher variance in the expectation of inﬂation, and so lower expected shoe leather costs. Notice, further, that a policy maker concerned to minimize expected shoe leather costs would seek to maximize unpredictability in inﬂation. The conclusion that an increase in inﬂation unpredictability reduces shoeleather costs is contingent on an assumption of declining marginal shoe leather costs. But marginal shoe leather costs need not decline with inﬂation. It will be increasing under a constant co-eﬃcient of demand. It will be increasing for ‘small’ inﬂation rates with constant semi-elasticity money demand (see Table 10.1). In these cases, a mean-preserving increase in the variance of expected inﬂation would increase expected shoe leather costs (assuming a white noise money supply process). And a wish to minimize expected shoe leather cost would recommend a minimization of inﬂation unpredictability. But the assumption of declining marginal shoe leather costs is plausible, and no less plausible than that of increasing marginal shoe leather costs. The fact that minimization of shoe leather costs may recommend the unpredictability of inﬂation is a difficulty for any attempt to found the desirability of the stabilization of prices in shoe leather costs.

INFLATION AND THE COST OF THE SUPPLY OF INSIDE MONEY We have argued that the costs of inﬂation rising from its disturbance to money demand give little warrant to a goal of price stability. But the costs of inﬂation are not limited to those arising from its disturbance to money demand. Chapter 3’s theory of the supply of inside money implies inﬂation is also costly on account of its disturbance to money supply. Recall that in its simplest modelling the supply of inside money was a positive function of the nominal rate of return on capital. For a nominal rate of return of aa, the supply is ee (Figure 10.7). The total variable costs of supplying inside money of ee is cc bb ee dd, as the supply curve is interpretable as the marginal cost curve. The gross beneﬁt to suppliers is aa bb ee dd. The net beneﬁt to suppliersaa bb cc.

218

The cost of inﬂation

aa

bb

cc

dd

n

ee

Figure 10.7 The ‘private benefit’ and social loss of the supply of inside money But the critical point here is that there is no social beneﬁt in the supply of ee. For if the suppliers, contrary to their private interest, had supplied zero inside money, where would be social loss? Or – to vary the thought – if inside money was banned, where would be the social loss? The same amount of real money would be held, so the same amount of liquidation costs would be economized upon. The diﬀerence is that the whole of real money would be held in the form of outside money – and from that the cost of the supply of inside money is therefore saved. Thus inﬂation induces ‘unnecessary’ and costly establishment of credit. Inﬂation induces the replacement of socially cheap outside money with socially costly inside money. Figure 10.8 illustrates. Let bb be the rate of return when inﬂation is zero. Then the ‘credibility cost’ of inﬂation increasing from zero to aa bb equals cc dd ee ﬀ . (There are also ‘shoe leather costs’ of gg hh ii jj.) How large is this cost? Let . be the cost of the supply of inside money: d. dU dn 1 d dn d UC

(10.16)

Suppose also the very simplest modelling – inside money only lasts one period. Then dU zUC dn, and z so:

The cost of inﬂation as the cost of moneylessness

219

Credibility costs dd

aa

gg hh

bb

Shoe leather costs

cc

ff Figure 10.8

ee

jj

ii

h,n

The ‘credibility cost’ of inflation d. dn d d

Thus: d. e n n d where en dn n d

INFLATION COSTS AGAIN Might the cost of the supply of inside money retrieve the warrant of a goal of price stability? ‘The Cost of Inflation is Small’ We can now calculate the marginal ‘money market cost of inﬂation’. d(SL .) eh hy en ny yd

220

The cost of inﬂation

We have argued that eh h y is small. We now add an extra term; enn y. So the marginal money market cost of inﬂation is larger. But how much larger is hard to say, as theory allows en n y to vary from the indeﬁnitely small to the indeﬁnitely large. But strong ﬂuctuations in inﬂation have not led to large changes in n. So the marginal money market cost of inﬂation is still looking ‘small’. The Minimization of Money Market Costs does not Warrant Zero Inflation; It Warrants Negative Inflation The force of this criticism remains undiminished. ‘The Shoe Leather Costs of Inflation are Smaller than the Tax Relief Benefits of Inflation’ This contention is signiﬁcantly weakened by the allowance for an endogenous supply of inside money. Once inside money is allowed to respond to inﬂation, the ‘base’ on which the inﬂation tax operates, m, is reduced more by any given amount of inﬂation, as inside money expands to take the place of outside money. Any given amount of seigniorage therefore requires more inﬂation. Thus any given amount of seigniorage has greater costs.19

d[SL .] dS

h e h 1 eh x n x 1

Let h/x be about 5. Then if we assumed (arbitrarily) eih ein the seigniorage costs of an extra percentage point of seigniorage is multiplied by 4. The tax relief beneﬁts of seigniorage – already compressed by the consideration of an exogenous supply of inside money (see Table 10.3) are now looking still smaller. The Minimization of Money Market Costs does not Warrant Predictably ‘Low’ Inflation; It may Warrant Unpredictability in Inflation The ‘may’ is still valid. The analysis of the impact of unpredictability on expected shoe leather costs is unrevised. The only question is if a complementary analysis can be extended to the impact of unpredictability on expected ‘credibility costs’. The answer is yes. If the marginal cost of establishing credibility of inside money is diminishing then a mean-preserving increase in the variance of expected inﬂation shall reduce the expectation of the credibility costs.20

The cost of inﬂation as the cost of moneylessness

221

PERCEIVED GAINS, TRUE LOSSES, AND THE WINNERS AND LOSERS FROM INFLATION The previous section argued that inﬂation causes social losses both from aﬀecting the demand for money (shoe leather costs), and by aﬀecting the supply of money (‘credibility costs’). There is however a sharp diﬀerence between these two losses in the judgement by the public. Every money holder loses from the reduction in the demand for money induced by higher inﬂation. Diagrammatically, every money holder perceives a loss of the area gg hh bb aa. By contrast, every money issuer gains – or appears to gain – from the increase in the money supply induced by higher inﬂation. Diagrammatically, every money supplier has a gain of the area aa dd cc bb (Figure 10.8). Here is a paradox. How can each person measure a beneﬁt from the impact of inﬂation on the supply of inside money if there is a social loss from the impact of inﬂation on the supply of inside money? Recall that, in the judgement of the issuer, issuing inside money has allowed them to purchase an income earning asset (capital) with something that has no cost apart from the credibility costs. In other words, each issuer believes that by issuing inside money they acquire more capital. But each cannot acquire more capital; the capital stock is given. To be more precise, if everyone has the same supply function of inside money, no one will own any more capital than they would if no one issued inside money. The only thing the issue of money does is in this circumstance is to raise the price level; and the issued money goes, not into buying capital, but in restoring real balances in the face of this higher price level. If, for some reason, no one was issuing money, but then one person did, that one person would beneﬁt, and they would do so by purchasing other people’s capital. But if everyone issues money, no one owns more capital, and all have wasted resources in establishing credibility. We have a ‘paradox of action’; a ‘fallacy of composition’, or a ‘prisoner’s dilemma’. Everyone believes it advantageous to stand on the box rather than sit, but if everyone stands on the box, no one sees any better. And resources are wasted on boxes. Issuing inside money is like supplying everyone a box. Origins of Inflation The perceived net losses from inﬂation is the excess of gg hh bb aa over aa dd cc bb (Figure 10.8). It is not diﬃcult to see that this ‘perceived net loss’ from inﬂation must be negative. That is, there cannot be a perceived net gain: the perceived gross gain cannot exceed the losses. The population perceives a net cost of inﬂation. The perception is exaggerated: it ignores the

222

The cost of inﬂation

growth of outside money endowments that the expansion of inflation has brought. But presumably the decline in real outside money’s value that is a consequence of inﬂation will be perceived with suﬃcient intelligence. Thus there might be hopes that intelligence will reinforce the perception, and so prevent the socially costly inﬂation that harms everyone. Unfortunately, intelligence will not, alone, be able to prevent all possible socially costly inﬂation. That is because socially costly inﬂation need not harm everyone. Suppose that a few have low credibility costs, while the rest have prohibitively high ones. Then under inﬂation only those with low costs issue money: those with high costs do not. Those with low costs of issue might made better oﬀ by inﬂation. They do increase their capital by inﬂation. The increment in prices due the induced issue of inside money will induce those who cannot issue money to sell their capital (to those who do) in order to restore their real balances. Thus we may have private interests who do have an interest in inﬂation. There may be a rational constituency for inﬂation.

CONCLUSION Changing the rate of inﬂation will change welfare through the impacts of inﬂation on the demand and supply of money. But theory does not say whether these welfare impacts will be minimized by a rate of inﬂation that is positive, negative or zero. Neither does it suggest that the optimal rate of inﬂation will be approximately zero. The conclusion is that the impacts of inﬂation on the demand and supply of money provide only a weak case for price stability.

NOTES 1. 2.

The classical exposition, and seemingly entirely seminal one, is Bailey (1956). For the sake of simplicity, Figure 10.1 assumes, contrary to the theory of Chapter 2, that the implicit return on money has an upper limit. This infringement of assumptions does not aﬀect the validity of the conclusions. h dSL h h h d h h

3. But

1

implies 1 [1 ][1 ]2

The cost of inﬂation as the cost of moneylessness and so

4.

5. 6.

7.

223

h [1 ] 1 h eh h h h 1 h h [1 ][1 ]2

A cross-sectional study of the income elasticity of the demand for narrow money that used the 50 US states and data over 1929–90 concluded that the income elasticity lies ‘between 1.3 and 1.5, signiﬁcantly above one’ (Mulligan and Sala-i-Martin, 1992). They add ‘money demand is a stable function over an impressive sample period, 1929–1990’. Estimates in the 1.2 to 1.4 range are found for Japan by Fujiki and Mulligan (1996). The Australian Taxation Oﬃce had 19 318 employees on 30 June 2002. Note Printing Australia has no more than 210. The ATO estimates administration costs of the tax system is 1.1% of revenue: 1.8 billion dollars in 2002/3 Compliance costs have been estimated to amount to 7.9 per cent of total tax revenue for the Australian tax system (Pope 1993), or about $13B (1.5 per cent of GDP) in 2002/ 3.They are, surely, zero for the inﬂation tax. They are commonly assumed to be about 10 per cent for the USA. dSL dh and dS d[h h] implies dSL dS [h h] [1 h h] . Using h (h ) " ( ) and the necessary truth i (1 ) (1 ) one may infer eh eh dSL 1 sh dS 1 s h 1

8. 9. 10.

If then the nominal rate of return is zero, and so the elasticity of money demand to must be zero, and so the marginal cost of seigniorage is zero. A suﬃcient but not necessary condition for d 2SL dS 2 0 is h / 0. The expression implies e c 1 e if the elasticity of money demand is a parameter, e; s (1 s) c, if the semi-elasticity of money demand is a parameter, s; and - c 1 -

11.

12.

if the coeﬃcient of money demand is parameter, , and where - h . The approximate identity , will yield the expressions in Table 10.2. As sei, and s 1 (- i) , assuming numerical magnitudes for e and s ties down a numerical magnitude of -. It was stressed in Chapter 2 that there is no maximum yield on money. But - can be interpreted as the maximum yield on money that is implied by a linear approximation of the actual demand function. eh e 1 sh 1 e

if the elasticity of demand is parametrical. The upper limit on this expression is e (1 e) . But we are left with the query: How can inﬂation be a welfare-expensive form of revenue, if an extra bit of inﬂation seems to have only small shoe leather costs? Simply because inﬂation provides very little revenue. Suppose as we have that h is 1/5 of y, and x is 1/5 of h. Then x is 4 per cent of GDP. And an inﬂation of 1 per cent amounts to a 1 per cent expansion in x, and that is 4 per cent of a 1 per cent of GDP (that is 1/2500 of GDP). It was earlier argued that a 1 per cent inﬂation will produce a reduction in welfare equal to 5 per cent of 1 per cent of GDP: this is ‘very small’, but so is the amount of revenue it raises!

224 13.

The cost of inﬂation The second requirement of price stability, var 0, can be equivalently interpreted as inﬂation being perfectly predictable: the squared error in the rational forecast of inﬂation is approximately zero, Var E(E)2 E(e)2 0,

14.

e E

A corollary is that white-noise errors in the money supply growth rule are costless in terms of shoe leather. If t t randomt in all t then it would remain the case that Et E t in all t, and so still E

R R1ln M1 R1ln M2 ... 1 R 2 1 1 1 1 1

2

...

and so still E E . 15. 16.

17.

var( ) 2var(&) var(+)

and

covar( , &) var(&)

2SL h [ h][ E ) ) ] , 0

E2 A necessary condition for this to be negative – and marginal costs to be ‘diminishing’ – is h0. This would not seem to be a diﬃcult condition to pass, but is excluded by the constant coeﬃcient money demand function. To illustrate, consider a white noise process for the supply of money, _

ln M ln M + _

18. 19.

where the policy instrument is ln M, and + is an uncontrollable money supply shock. E + [1 ] and E has a variance of E+2 [1 ]2. If lnM lnM + and there are no disturbances to money demand, then E + [1 ] and [+1 +] [1 ] (see Table 8.2). Thus varE var+ [1 ]2 and var 2var + [1 ]2. dSL [ ih in]d and dS [x x]d. Thus ih in dSL ih in h h x x x x dS h h

20.

h ih in x h h 1 x x

d . dn dE dE d 2. d 2n dn , 0 dE2 dE2 dE A necessary condition for the negativity of this is d 2n dE2 0.

11. The cost of inﬂation as the cost of creditlessness Chapter 10 advanced a theory of the cost of inﬂation that turned on the reduction in money demand, and the increase in the money supply, that is caused by a higher expected level of inﬂation. This chapter presents an entirely diﬀerent theory of the cost of inﬂation; one that turns on the contraction of the bond market caused by the risk of inﬂation. In this theory the cost of inﬂation lies in its reduction of the socially beneﬁcial exchange of risks that a bond market secures. Thus, in turning to this theory, our attention is shifted from the money market to the bond market; and from a theory where expectation of inﬂation is critical (and the unpredictability of inﬂation is irrelevant), and towards a theory where the expectation itself is irrelevant (and the degree of unpredictability is critical). The theory of the present chapter turns on a certain property, explored in Chapter 6, of a standard Ramsey-Solow style model: bond markets transfer technological risk from those interests where risk is most concentrated (and so most painful) to interests where it is less concentrated (and so less painful), and thereby produce a social gain. To illustrate: if we suppose that capital owners are more vulnerable to technological risk than workers, then debt allows workers to borrow from capitalists and acquire some capital, and capitalists to disburden themselves of capital and acquire bonds. This trade in risky capital allows capitalists to hedge, and workers to speculate, to the mutual beneﬁt of both. Capitalists reduce the risk on capital: workers get the premium of capital. This successful transfer of technological risk by bond markets relied on the assumption of bonds being ‘real’ (that is inﬂation indexed). Real bonds, however, are a marginal and atypical feature of the real world. How would the model behave if real bonds did not exist? What if all borrowing and lending was conducted through money loans?1 The chapter argues that if all borrowing and lending was conducted through money loans then the useful trade in risks is impeded by unpredictable inﬂation. Lending to workers to ﬁnance their purchase of capital does not now reduce the capitalists’ ‘total risk’ as much as it did before, because the capitalist lender is now exposed to being burnt by episodes of 225

226

The cost of inﬂation

unexpectedly high inﬂation. At the same time, borrowing now increases the workers’ total risk more than it did before, because worker debtors are now exposed to being burnt by episodes of unexpectedly low inﬂation.2 This upshot is that less risk is traded. Less risk is transferred from where it is most painful to where it is least painful. The chapter’s analysis further suggests that inﬂation is not only reducing of the welfare of the whole, it is harmful to the welfare of all economic interests: it is a ‘lose-lose situation’. However, the analysis also suggests inﬂation may not be entirely a sad story. It is argued the very interference in the trade in risks imparts an egalitarian dynamic to the distribution of wealth, that over time that erodes the cost of inﬂation to the point of extinction. Thus in the long run unpredictable inﬂation has no cost and re-distributes wealth in an egalitarian direction. We are thereby led to conclude the chapter with a consideration of the compensations of inﬂation It assumed throughout this chapter (save for one part), that inﬂation is white noise, and uncorrelated with the proﬁt rate: cov( , ) 0

(11.1)

We also suppose that for each state s the money endowment of each person j grows at the rate of the aggregate monetary endowment: sj s

all j, s

(11.2)

and are thereby justiﬁed in neglecting the potential disturbance of real balance eﬀects that was examined in Chapter 7. We otherwise assume the same theoretical structure as previously, except that real bonds are absent.

THE IMPLICATIONS OF A MISSING REAL BOND MARKET FOR THE EVENING OF RISK The implications of a missing real bond market for the transfer of risk can be brought out by recapitulating the optimization conditions: Capital

UC E[UC1[1 ]]

(11.3)

Real Bonds

UC [1 r]EUC1

(11.4)

Money

U UC Uh E 1 C1

(11.5)

227

The cost of inﬂation as the cost of creditlessness

Money Bonds

i UC E UC111

(11.6)

If real bonds are present then, we may infer from the capital and real bonds conditions: E r ) j and so the magnitude of ) j is the same for all j. As explained in Chapter 6 this condition (with some exceptions) secures eﬃciency in consumption with respect to technical shocks. So, if real bonds are present the only role of money bonds is to immunize consumption against money shocks that redistribute purchasing power (and even that role has been assumed away in this chapter by supposing s j s for all j, s). But if real bonds are absent then money bonds will assume the function of real bonds of transferring technological risk. But in place of the risk premium equality above, we infer from the conditions for capital and money,

i )j )j 1 E E 11

Given the uniformity of the LHS across j one may infer that the magnitude j of ) j ) is the same for all j. Clearly this equality does not insure ) j is the same for all j. Indeed, as will be seen ) j will generally be diﬀerent for diﬀerent j in the presence of risky inﬂation. This implies consumption is not shared out in the eﬃcient manner.

THE IMPACT OF INFLATION RISK ON THE BOND MARKET The impact inﬂation risk has on the demand and supply of bonds can be used to explicate the contention that inﬂation risk harms the eﬃcient sharing of consumption. Demand and supply schedules for bonds can be derived from the equality:

i )j )j 1 E E 11

(11.7)

228

The cost of inﬂation

The LHS is the excess return capital is expected to earn over money bonds, and can be considered the price of money bonds. As both the terms on the RHS are a function of the quantity of money bonds, D, we can trace relations between the price of bonds and the quantity bonds. To facilitate demand and supply analysis we assume (as was done before), that the population falls into two groups; one with a higher endowment of capital relative to labour, F; and one with a lower endowment of capital to labour, G. The ﬁrst group, with a high exposure to risk, are ‘hedger-capitalistcreditors’. The second are ‘speculator-worker-debtors’. For each group there is an associated condition for the excess return on capital:

(11.8)

(11.9)

i )F )F 1 E E 11

i )G )G 1 E E 11

At D0 the magnitude of both )F and )G is zero, given our assumption that inﬂation is white noise. At D0 )F and )G

are determined by the comovement of each group’s consumption endowment with the proﬁt rate. As F has the more higher capital/labour ratio, F will have the more negative comovement in marginal utility with proﬁt, and so a more positive ) j than G. At D0, )F )G

and

)F )G 0

(11.10)

We can represent )F and )G

at D0 as the vertical axis intercepts of a ﬁgure that plots E r on the vertical axis, and bond issues on the horizontal axis. These two intercepts may be interpreted as the demand price of money bonds when D0, and the supply price of bonds at D0, respectively. To trace out the demand curve, we note that as D becomes positive then )F diminishes, by the process explained in Chapter 6: risky capital is exchanged for (more) reliable bonds. But as D becomes positive there is also now a new, additional element in tracing out the demand curve; )F assumes a non-zero magnitude as D becomes positive. Once D is positive then for F (a creditor), an above average rate of growth in purchasing power must coincide with an above average consumption of F – simply because creditors are richest when the growth in purchasing power is highest. So an above average rate of growth in purchasing power must coincide with a below average marginal utility. Thus )F (cov( , &F )) becomes positive. And so the demand price for bonds is reduced. This reduction in the demand price for bonds can be understood as a manifestation of an

The cost of inﬂation as the cost of creditlessness

229

E[iE]

F兿

D Figure 11.1 Inflation risk contracts the demand for bonds by capitalist hedgers undesirable property of money bonds in the circumstances we are analysing: for creditors money bonds give their best return when return is least ‘needed’ (that is, marginal utility is lowest). On account of this undesirable property the demand price for bonds falls. As D becomes larger )F becomes still more positive, as the largeness of D makes for a still stronger comovement of the growth of purchasing power and creditor consumption, and so demand price shifts down more. The construction of the supply curve for bonds follows a parallel logic. As D becomes positive then )G

rises, by the process explained in Chapter 6: risky capital is bought by the sale of relatively reliable bonds. But as D becomes positive there is also now a new, additional element in tracing out the supply curves. )G assumes a negative magnitude as D becomes positive. This is because for G (debtors), an above average growth rate in purchasing power must coincide with below average consumption – simply because debtors are poorest when the growth rate in purchasing power is highest. So an above average rate of growth in purchasing power must coincide with an above average marginal utility of debtors. Thus )G (cov( , &G ) ) becomes negative. The ‘supply price’ of bonds is increased. This rise in the supply price for bonds can be understood as a manifestation of an undesirable property of money bonds in these circumstances: they are most costly to debtors when debtors are most in ‘need’ of

230

The cost of inﬂation

E[iE]

D Figure 11.2 Inflation risk contracts the supply of bonds by worker speculators income (when the marginal utility of consumption is the highest). And they are least costly debtors when debtors are the least ‘need’ of income (when the marginal utility of consumption is the lowest). On account of this undesirable property the supply price for bonds rises.3 Figure 11.3 illustrates the combined impact of the shift inwards in the demand for bonds, and the shift upwards in the supply of bonds. Three observations can be made about the impact of inﬂation risk: ● ●

●

First, the quantity of bonds unambiguously contracts. Second, the expected real interest rate on money bonds is unchanged. The lack of response is a result of the mutual cancellation of two contrary forces. Inﬂation risk has deterred capitalist-creditors; an eﬀect which would, operating alone, drive up the rate. But inﬂation risk has also deterred worker-debtors, an eﬀect which would, alone, drive down the rate. The two eﬀects cancel each other out, and the impact of inﬂation risk on the expected the real interest rate, and the premium on capital is zero.4 Third, the magnitudes of )F and )G

are driven apart. Consumption of the two interests is no longer synchronized in the face of technical shocks. There is a loss of welfare.

Figure 11.3 can be adapted to bring out the social ineﬃciency of noisy inﬂation. Under our assumptions, welfare in the absence of risky inﬂation

231

The cost of inﬂation as the cost of creditlessness

E[iE]

F

Supply of bonds

F兿 G兿

G

Demand for bonds D

Figure 11.3

Bond market equilibrium under inflation risk

was the area that lay that between the demand curve that prevailed in the absence of risky inﬂation, and the supply curve that prevailed in the absence of risky inﬂation; aa ﬀ ee in Figure 11.4. The contraction in these demand and supply curves signals a loss in welfare.5 Welfare in the presence of noisy inﬂation is the contracted area underneath the demand curve, and above the supply curve, that prevail in the presence of noisy inﬂation; aa cc ee. A boomerang shaped area – aa ﬀ ee cc – is the loss in consumer welfare.6 In algebraic terms it can be shown (Appendices 2 and 3) that: ~

~

F G D the loss of welfare [)F )G] D 2 [) ) ] 2

(11.11)

To summarize: risky inﬂation reduces welfare by impeding the synchronization of consumption across interests. In the absence of unpredictable inﬂation a money bonds market would be capable of perfectly synchronizing consumption, but under white noise inﬂation it cannot.7 That synchronization of consumption that would take place at the old volume of lending is now disturbed by inﬂation surprises, which impact on the wealth of creditors and debtors in an unpredictable fashion. Discouraged from lending and borrowing, the amount of lending and borrowing shrinks. Capitalist-creditors cannot obtain the security they want, and worker-debtors cannot obtain the return they want.

232

The cost of inﬂation

E[iE] aa

bb

ff

cc

dd

ee D

Figure 11.4

~

D

D

The impact of white noise inflation on welfare

INFLATION RISK WITHOUT COST The contention that risky inﬂation causes a welfare loss by discouraging the trade of risk must be qualiﬁed in two ways. First, the contention assumes that there is some beneﬁcial trade in risk to be done. This will not be the case where technological shocks are such that workers and capitalists are equally exposed to risk; Hicks neutral technical progress. There will also be no beneﬁcial trade in risk to be done if there is an equal distribution of capital so that there are no ‘workers’ and ‘capitalists’ but simply worker-capitalists. Second, the contention does rest on the assumption that inﬂation is white noise. If we suppose that inﬂation is not white noise but varies with complete predictability with the rate of proﬁt, then inﬂation risk will have no impact upon aggregate social welfare. Recall:

i )j )j E[1 ] E 11

all j

(11.12)

The cost of inﬂation as the cost of creditlessness

233

If there is perfect linear relation with the rate of proﬁt #

(11.13)

j #) j )

(11.14)8

then

Therefore,

i ) j [1 #] all j E [1 ] E 11

(11.15)

The equality of the LHS for all j ensures ) j is equal for all j.9 This is a signal that consumption is eﬃciently shared. For the two person two-period, two output state case, it is easy to use the Edgeworth Box to bring out the costlessness of inﬂation risk when the inﬂation is perfectly conditioned on the rate of proﬁt. Suppose, as we have before, that there are two possible states; a high output/high proﬁt state, and a low output/low proﬁt state. Given # , there is for each of the two proﬁt states a uniquely associated rate of inﬂation. So it may be said there is a high output/high proﬁt/high inﬂation state, and a low output/low proﬁt/low inﬂation state. The consumption endowment of F and G – how much each can consume in the absence of debt – can be plotted on the Box as before. As long as 1 1 i 0 0 F can shift their consumption point from their endowment point by lending money to G to ﬁnance G’s purchase of F’s capital. In the low output/low proﬁt/low inﬂation state F increases their consumption from a dollar of lending by i 0 0. In the high output/high proﬁt/high inﬂation state F reduces their consumption from a dollar of lending by 1 1 i. F shifts their consumption point south-east. As debt rises the consumption outcome will shift away from the endowment point. The slope is dependent on the two possible payoﬀs from lending a dollar. Thus we can draw a consumption frontier for F, and in the same manner for G. Evidently, there is some nominal interest rate that will secure equilibrium. Evidently, the consumption outcome is eﬃcient, and the same as in the presence of real bonds. As in the presence of real bonds F has reduced risk at the cost of expected consumption, and G has increased expected consumption at the cost of risk. Inﬂation has no social cost. Although, inﬂation has no social cost if it is predictable conditioned on the proﬁt rate, inﬂation risk is not completely without impact in these

234

The cost of inﬂation 0C G 1

1C G 1

Efficiency locus

E

C 1C F 1

45 degrees 0C F 1

Figure 11.5 The irrelevance of inflation risk when inflation is perfectly correlated with profit circumstances: it will impact on the rate of interest. Proof: the equimarginal condition can be rewritten:

i E [1 ] ) j [1 #] E 11

(11.16)

This implies that the ex ante real interest rate is a negative function of #, the index of procyclicality of inﬂation. As the degree of procyclicality rises, the expected real rate of interest falls. How is this so? A positive correlation between inﬂation and proﬁt entails a negative correlation between the growth in purchasing power and proﬁt, and so a positive correlation

The cost of inﬂation as the cost of creditlessness

235

between the growth in purchasing power and marginal utility (given that higher proﬁt goes with higher consumption). That positive correlation amounts to a reward for lending, and a penalty for borrowing. Debtors seek a lower interest rate, and creditors are willing to grant it.10 If #0 then the interest rate rises. A negative correlation between inﬂation and proﬁt entails positive correlation between the growth in purchasing power and proﬁt, and so a negative correlation between the growth in purchasing power and marginal utility. That amounts to a penalty for lending, and a reward for borrowing. Creditors seek a higher rate, and debtors are willing to grant it. (See Obstfeld and Rogoﬀ 1998 for an exploration of like ideas). A procyclicality in inﬂation has one more impact; on the magnitude of debt. If #0 then quantity of real debt falls. This conclusion can be understood in terms of the impact of # on interest. A positive # reduces the interest rate. But the expected proﬁt rate will be unaﬀected by #. So any given amount of lending will reduce the expected income of creditors more than it did before, and increase the income of borrowers more than it did before. Thus the reduction in the expected income of creditors, that is requisite of optimality, will be secured with less debt. If you like, debt is more ‘powerful’. A dollar of debt shifts the consumption outcome towards the eﬃcient point more than before. If #0 then the quantity of real debt rises by a parallel logic. To summarize: whereas white noise inﬂation will reduce welfare, reduce debt, and have no eﬀect on the ex ante real interest rate, inﬂation that is perfectly correlated with the rate of proﬁt may increase debt, will aﬀect the ex ante real interest rate, and will not reduce welfare. But this qualiﬁcation of the original proposition is unlikely to be strong, for inﬂation is unlikely to be perfectly correlated with the proﬁt rate. Rather inﬂation is likely to have one component that is related to the rate of proﬁt, and a second that is unrelated to white noise. # +

(11.17)

That part of inﬂation that is systematically related to proﬁt has no social cost, but that which is white noise will be costly through discouragement of the trade in risk.

THE COSTS OF INFLATION RECONSIDERED Does the preceding theory of the cost of inﬂation change at all the appearance of the common account of the costs of inﬂation?

236

The cost of inﬂation

‘This Arbitrary Rearrangement of Riches’ The theory throws some light on the sentiment that it is the unpredictability in inﬂation, rather than its average level, that is harmful, as ‘choices’ are transformed into ‘gambles’.11 A more speciﬁc variant of this ‘unpredictability thesis’ – and one that is an old saw in the discussion of the costs of inﬂation – has it that that inﬂation is harmful on account of the unpredictable redistributions it secures between creditors and debtors. But how these redistributions are harmful to eﬃciency – as apart from equity – is barely touched.12 The present analysis provides an answer. These unpredictable redistributions between creditors and debtors damage welfare eﬃciency by reducing the synchronization of consumption across interests. Unpredictable redistributions reduce synchronization directly and indirectly. Directly (through surprise inﬂations’ impact on the real value of credits and debts), and also indirectly by discouraging the transfer of risky capital from those with much to those with a little. Unpleasant Surprises The cost of inﬂation risk that has been advanced here is not only a cost in its own right, it is a cost that taints some of supposed beneﬁts of inﬂation. It puts a taint on the beneﬁts of inﬂation unpredictability that have sometimes been argued to exist. It has been argued that the merit of preserving smooth tax rates validates unpredictable seigniorage, and a concomitant unpredictable inﬂation, as seigniorage should ﬂuctuate in accordance with unpredictable temporary shocks to government spending so as to allow tax rates to be invariant in the face of such shocks. In this logic a surprise increase in G should be ﬁnanced by an (equally surprising) increase in S, relieving any need for higher taxes, and producing a concomitant inﬂation surprise. Whatever beneﬁt that unpredictable seigniorage may have, it will, according to the theory of this chapter, be mingled with a cost; the cost of the disruption of capital markets. The argument for the cost of inﬂation risk is also an antidote to the recommendation of inﬂation risk that (we have seen) the theory of shoe leather costs implicitly makes in the presence of declining marginal shoe leather costs of inﬂation. Minimizing Shoe Leather Cost The theory of the costs of unanticipated inﬂation also taints the beneﬁts of deﬂation that is implied by the theory of shoe leather costs. Recall that shoe leather costs are minimized if:

The cost of inﬂation as the cost of creditlessness

237

The present theory is well prepared to analyse the impact of such a policy on capital markets. The shoe leather cost minimizing rule is a special case of: # We have previously concluded that # 0 was no bar to eﬃcient trade of risk. But this conclusion implicitly assumed that # 1. # 1, or , is totally destructive of the risk shifting function of bond markets. For if then money bonds have the same return as capital in all states. F cannot shift their consumption point by lending because in both states the payoﬀ is the same (zero). The risk trading function of money bonds is destroyed. There is, therefore, a strict conﬂict between minimizing shoe leather costs and maintaining the risk managing properties of the bond market. To strictly minimize shoe leather costs is to destroy the trade in risk. The conﬂict may not seem very serious as the conﬂict only occurs in the extreme case of #1. If # was close to 1, the nominal rate of return on capital would be close to zero without actually being zero, and so shoe leather costs would be close to minimized, without the trade in risk being destroyed. But if we allow that the systematic variation of with will be combined with some white noise disturbances, then even the mere reduction (not the actual abolition) of shoe leather by making # close to 1 will be still associated with the increased desynchronization of consumption. Consider, # +

(11.18)

Letting # approach 1 does not, directly, harm the trade in risk, as we have noted. But letting # approach 1 does aﬀect the size of D, as was argued in the previous section. As # gets more negative, D gets larger (for a given synchronization in C). But as D gets larger, the greater the volatility in consumption imparted by the white noise inﬂation disturbances, and so the greater welfare cost. So a more negative # is costly in welfare once we allow for operation of white noise disturbances to inﬂation. Thus serving the minimization of shoe leather costs, by making # more negative, does have a welfare cost. There is a conﬂict between minimizing the costs of technology shocks and minimizing the cost of liquidations.

238

The cost of inﬂation

QUANTIFYING THE WELFARE LOSS OF UNPREDICTABLE INFLATION How large is the welfare loss from the reduction in the trade of risk caused by unpredictable inﬂation? Theory gives very little indication. It could be as low as zero: it will be zero if, for example, F and G had the same endowment of capital and labour. And if it can be zero, it can also be minute. On the other hand, the welfare loss could be as large as aa ﬀ cc (Figure 11.4). This is where inﬂation risk approaches inﬁnity, and debt is driven down to zero. Some help in quantifying its size can be derived from: ~

~

F GD welfare cost [)F )G]D 2 [) ) ] 2

(11.19)

One~ could hazard numbers on the diﬀerence between the risk premia, and D, and draw the inference. So if we were ~ willing to suppose the diﬀerence in risk premia is 0.01 (in decimals), and D is twice C, then the cost of inﬂation risk is 0.01 of C, (or 1 per cent). A more reliable procedure would be to try to derive expressions for the risk premia, and then calibrate. Appendix 11.3 shows: D )F [1 ](2r sC

)G [1 ](2r [1 Ds]C

(11.20)

s ECF /EC Thus: total cost of inﬂation variance as a proportion of C ~

D (2r D C C [1 ]

1 2s(1 s)

(11.21)

(11.21) still has the considerable disadvantage of involving a counterfactual ~ variable whose size can only be speculated: D, which is the magnitude of ~ debt if inﬂation was zero. However, since D D, we can say, total cost of inﬂation variance as a proportion of C

2 1 (2r D C [1 ] 2s(1 s)

This inequality provides a ﬂoor to the cost.

(11.22)

The cost of inﬂation as the cost of creditlessness

239

Some illustrative ﬁgures suggest this ﬂoor could be very small. If we use the year as the period, and assume a ( equal to its US average over 1900–2004 (4.8 per cent), a real interest rate of 2 per cent, an elasticity of intertemporal substitution of 0.25, s 1/3, and D/C of 2, then the cost ﬂoor is 0.0016, or barely one sixth of one per cent. In considering this estimate some things should be kept in mind. ~ The ﬂoor is only a ﬂoor. Algebraically, it assumes D D. In diagrammatic terms, the ﬂoor measure omits the triangle bb dd ﬀ in Figure 11.4. Economically, it is neglecting the desynchronization caused by the reduction in the transfer of capital, and accounts only for the desynchronization caused by shocks to the real value of debts. This neglect does, however, not ~ account for the smallness of the estimate of the cost. If we assumed D was twice D (an expansive assumption), then total cost would be twice the ﬂoor, but still small (1/3 of 1 per cent rather than 1/6). The source of the smallness is in the very smallness of the implied estimates of the risk premia. In terms of the calibrating assumptions above; )F 0.0011, or 0.11 of 1 per cent, or 11 basis points; and )G 0.0005, or 0.05 of 1 per cent, or ﬁve basis points. Are these estimates arguable? They assume the propensity to consume out of wealth is the interest rate, or 0.02. A more plausible estimate would be 0.06, and you would get a cost of inﬂation of 0.5 of 1 per cent. But the smallness of the cost is rooted more in the logic of expected utility. Expected utility does not give much space for a large risk premium. Consider speciﬁcally the size of the inﬂation risk premium; the excess a nominal bond will have over a real bond, under hypothetical circumstances. Suppose inﬂation can assume, with equal probability, a magnitude of 1 or 1 per cent, with a corresponding 1 per cent deviation in marginal utility. The covariance between marginal utility and inﬂation is then 0.01 of 0.01. That amounts to a risk premium of a single basis point. A single basis point will compensate a holder of a money bond for the inﬂation riskiness of its return.

THE COMPENSATIONS OF INFLATION The Egalitarian Dynamic of Inflation Risk The welfare cost of a given degree of riskiness in inﬂation is not constant over time. It changes over time, as the welfare cost creates its own dynamic. This dynamic turns on a critical non-neutrality of the riskiness of inﬂation. Risky inﬂation makes the propensity to consume of the debtor smaller than the propensity to consume of the creditor, as risky inﬂation makes ‘the risk

240

The cost of inﬂation

corrected rate of return’ on capital lower for the capitalist than for the worker. Recall that, if a secure real bond existed, then r E ) Thus E ) j may be treated as risk corrected rate of return for j. In the presence of real bonds the correction to the rate of return on capital is the same for all, and so the risk corrected rate of return on capital is the same for all. But we have seen that unpredictable inﬂation makes ) j more positive for capitalist-creditors, and less positive for worker-debtors. Having been discouraged from selling their risk to workers, it is truer than it was before that the high reward from capital comes when the capitalist needs it least, and the low return comes when he needs it most. The risk corrected return to capital for the capitalist, E )F , has fallen. The reward to a capitalist-creditor for saving is less. Current marginal utility falls relative to forecast future marginal utility, as 1 E )F implies: Uc F EUc1 1 E )

(11.23)

Consequently, current consumption becomes higher. The capitalistcreditors consumption proﬁle is more present concentrated. By contrast, capital has become less risky for worker-debtors. It is now less true than before that the high reward from capital comes when the worker needs it least, and the low return comes when he needs it most. The risk corrected return to capital E )G

has risen for workers. Current marginal utility falls relative to forecast future marginal utility, as 1 E )G

implies: Uc G EUc1 1 E )

(11.24)

Worker current consumption becomes lower. Worker-debtor’s consumption is less present concentrated. This means the increment to consumption from some favourable wealth disturbance will be less present–concentrated for debtors than for creditors. The marginal propensity to consume for worker-debtors is lower. So a policy of unsystematic inﬂation stimulates the saving of workers, and diminishes that of capitalists. And that will feed back over time on the costliness of any risk of inﬂation. For in consequence of the reduction in capitalist thrift, and the increase in worker thrift, the amount of capital owned by workers rises over time relative to capitalists. The gap between the two risk adjusted rates of return will fall. But as long as the capital stock

The cost of inﬂation as the cost of creditlessness

241

is not equally distributed the workers will still have a greater risk adjusted rates of return, they will still have a larger inducement to save, and they will still gain on capitalists. Workers will gain on capitalists until their incentive for extra thrift disappears: )F )G

. But that is the situation where workers and capitalists have the same capital/labour endowment, and there are no workers distinct from capitalists any more. And under that situation the bond market ceases to have a function anyway. And so the costs of inﬂation risk fall to zero. So we see that the costliness of inﬂation is self-eradicating. As long as inﬂation is costly, that cost is redistributing wealth so as to make it less costly. Ultimately, wealth is redistributed so that inﬂation risk has no cost. At this ultimate state, workers have higher wealth and consumption than they would have had in the absence of inﬂation risk. Capitalists have lower wealth and consumption than they would have had in the absence of inﬂation risk. Total wealth will be the same as it would have been in the absence of inﬂation risk. It is worth re-emphasizing that inﬂation risk harms the welfare of workers, as well as capitalists. Workers would prefer the absence of inﬂation risk, even after allowing for the long run gain we have identiﬁed. Their ultimate increase in wealth does not compensate them for the risk cost of acquiring that extra wealth. Nevertheless, they are in the end better oﬀ on account of inﬂation risk. The egalitarian impacts of discouragement of the trade in risk has an enlightening parallel in the egalitarian impacts of the discouragement of the trade in capital in a global economy. Consider a world composed of two Ramsey-Solow economies sharing identical homothetic preferences, and an identical two factor production function, but one owning more capital than the other relative to its labour force – Plutovia – and one owning less, Pauperia. Under Perfect Capital Mobility (PKM) part of the capital owned by Plutovians shall operate in Pauperia. The freedom to shift capital between Plutovia and Pauperia will beneﬁt both economies, and preventing those shifts would harm both. To be more speciﬁc, a transition from Perfect Capital Mobility to Zero Capital Mobility (ZKM), will immediately reduce the total factor incomes of the inhabitants of both Plutovia and Pauperia.13 Any Pauperian interest that is beneﬁted by that transition (for example Pauperian capital owners) cannot compensate Pauperian losers and still be better oﬀ themselves. But such a transition to Zero Capital Mobility will not, however, reduce the long run output per head of the labour force in Pauperia (or Plutovia) relative to their long-run outcomes under Perfect Capital Mobility. For, regardless of the degree of capital mobility, long run output per head will be governed wholly by technology and preferences; long run output per head will be determined by

242

The cost of inﬂation

the capital/labour ratio that yields a marginal product of capital equal to the rate of time preference. Even more, ZKM will actually increase consumption per head in Pauperia in the long-run steady state. This is because under PKM part of the capital that operates in Pauperia is owned by Plutovia. Under ZKM Plutovia cannot own any capital of Pauperia, so that in the long run consumption in Pauperia must equal the whole of its output per head. The economic logic is that the transition to ZKM increases the proﬁt rate in Pauperia, and so provides an incentive for Pauperia to accumulate the whole of the quantity of capital that yields a marginal product equal to the rate of time preference. By making the proﬁt rate higher in capital scarce Pauperia than capital rich Plutovia, ZKM entails that Pauperia has more incentive to save than Plutovia, and that induces Pauperia ultimately to catch up with Plutovia in terms of capital owned per head. PKM, by contrast, provides the same incentive to save to both countries, and so while there is saving in both populations, there is no catch up, no convergence, and a preservation in the proportionate diﬀerential in the consumption of both countries. Discouragement of trade in capital ﬂows remains a cruel policy for Pauperia. ZKM will make at least some of the inhabitants of Pauperia worse oﬀ, perhaps all. Any winners in Pauperia cannot compensate the losers and still be better oﬀ. The potential Pareto superiority of PKM suggests ZKM could only be forced on Pauperia. Neverthless, on account of its long run impact ZKM has a ‘kind’ ultimate outcome for Pauperia. Discouragement of the trade in risk has the same ‘cruel to be kind’ logic as the discouragement of capital ﬂows. For in the absence of inﬂation risk, there is no convergence by economic class. But under inﬂation risk there is.14 Welcome Non-neutralities of Inflation We have argued that risky inﬂation raises the propensity to save of the capital poor, and reduces propensity to save of the capital rich. This has an implication for the distribution of wealth, as we have noted. But it also has implication for the creation of wealth; it makes real saving responsive to changes in the nominal money supply. A positive realization of inﬂation will produce higher total saving relative to a negative realization of inﬂation, since in a money debt economy a positive realization reallocates real wealth from creditors to debtors, and thereby reallocates wealth from the high propensity to consume group (capitalist-creditors) to a low propensity to consume group (capitalist-debtors). Saving and investment are therefore, ceteris paribus, positively correlated with the price level (and the money supply).

The cost of inﬂation as the cost of creditlessness

243

It can be retorted that as positive realizations of inﬂation are balanced by negative realizations the net eﬀect of inﬂation riskiness on saving must be neutral. And the retort is correct. Riskiness in inﬂation will average out as neutral to cumulative saving. But a certain kind of inﬂation is not subject to that retort. Consider a ‘moment shift’ in the distribution of price levels – say, all money supply outcomes in all periods and all states being factored up by some common proportion. This will increase saving, and growth, by the causal chain just alluded to: namely, higher prices reallocating wealth from the high propensity to consume group (capitalist-creditors) to a low propensity to consume group (capitalist-debtors). And because it is a moment shift it cannot be retorted that an up shift will be predictably neutralized by a down shift. Moment shifts, being moment shifts, cannot be predicted. We conclude that a positive moment shift in prices, in an environment of inﬂation risk, produces a stimulus to growth, while a negative moment shift in prices, produces a contraction in growth. We have Keynesian outcomes on neoclassical logic. An upward shift in prices stimulates growth by increasing the propensity to save. A downward shift in prices (the Great Depression?) harms growth by reducing the propensity to save. A fall in prices enriches capitalist creditors who go on a consumption binge, that reduces investment and therefore national product. A rise in prices enriches worker debtors who stint consumption, and thereby increase investment and so, later, national product.15 It might be asked sceptically, is there necessarily any social beneﬁt in increasing saving? Is there any reason for believing inﬂation will increase welfare by increasing saving? There is. The economy, recall, is not operating in a welfare eﬃcient manner under inflation risk; risk is not properly exchanged, and the risk premia are not uniform. A ‘moment-shift’ inﬂation will hasten it to operate in a welfare eﬃcient manner by evening up the premia. We are left with the paradox that the ‘radical’ unpredictability of a moment-shift increase in prices will mend some of the injury of the ‘normal’ unpredictability associated with a given probabalistic distribution of prices. But this is not to say there is any policy prescription of inﬂation here. Obviously, if in the light of the beneﬁts mentioned in the previous paragraph, a policy was adopted to increase the price level, that would no longer be a moment shift. It would be predicted. The beneﬁt of a moment shift only occurs in the context of risky inﬂation, which is itself costly. Thus the moment shift is only (partly) remedying the costs created by risky inﬂation.16 A policy of perfect price predictability would remain dominating.

244

The cost of inﬂation

NOTES 1.

This question, ‘what happens if real bonds are absent’, begs the question; why should there not be real bonds in the model? The present model does not supply an answer: the absence of real bond markets is ad hoc given the assumptions of the model. But arguments for the inferiority of a ‘CPI indexed’ bond can be advanced by reference to considerations our model has ignored; such as the existence of preference heterogeneity. Suppose there are n goods, but each person consumes only one of the n. The relevant real rate for each person varies from person to person, and that rate might be more predictable by lending in terms of money than a CPI indexed return. For the person who consumes only good k, the relevant ‘real’ return on an CPI indexed bond is: r

n

jj k j1

j weight on good k in the CPI while the relevant real return on a money bond is:

2.

3. 4.

5.

i k Evidently, the ‘real’ return on a money bond might be less unpredictable than the ‘real’ return on a real bond. Debtors also face the welcome ‘risk’ of unexpectedly high inﬂation. But the upsides and downsides are asymmetrical in impact, owing to the diﬀerent marginal utilities associated with high and low price outcomes. For debtors, the nasty thing about inﬂation risk is that debt is most costly when income is the most needed; for when the ex post real interest is highest, the debtor is (as a consequence) the poorest. Similarly for creditors, inﬂation risk means credit is the most remunerative when income is the least needed; for when the ex post real interest is highest, and the creditor is (as a consequence) richest. Inﬂation risk is bad for both capitalist creditors and worker debtors. Capitalist creditors want a smaller opportunity cost in lending; that puts an upward pressure on i. Workerdebtors want a greater gain in borrowing; that puts a downward pressure on i. Imagine an individual, j, with a capital/labour endowment representative of the economy as whole. As their endowment is the same as the rest – they are ‘sitting’ on the diagonal of the Box – they cannot do any mutually improving borrowing or lending. So, because their j holding of bonds is zero, ) must0, regardless of any change in the riskiness of inﬂation. Further, because their holding of bonds is zero, ) j must be unchanged in the face j 1 E E [(1 i) (1 )]. of any change in the riskiness of inﬂation. But ) j ) So 1 E E [(1 i) (1 )] must be unchanged in the face of any change in the riskiness of inﬂation. Marginal Expected Utility of a Unit of Capital E[UCj 1[1 ]]

Marginal Expected Utility of a Unit of Bonds E UCj 1 1 i 1

So, Marginal Expected Utility of a Purchase of Bonds by way of a Sale of Capital

E 1 i cov( , & j ) E [1 ] cov( , & j ) E jUC1 1

cov( , & j ) cov( , & j ) E [1 ] E 1 i 1

UC1 1 E cov( , & j )

The cost of inﬂation as the cost of creditlessness

6.

7.

245

So the downward shift in the demand curve is the loss of utility for bond buyers. Similarly for bond sellers. On an analogy with the welfare loss of tax on the demand and the supply of a good, it might be thought that the loss in welfare is the triangle bb dd ﬀ. But the analogy does not hold. A tax on the demand and the supply of a good, yields revenue to the state (aa bb cc and ee cc dd) that the standard analysis assumes is used to provide tax relief. But in analysing the cost of unpredictable inﬂation there is no revenue. (It has been implicitly assumed here that any seigniorage is not used to annex resources and so permit a reduction in taxes). ~ If lending stayed at D then )F and )G

would remain equal to each other. Despite this, consumption would not be synchronized. The equality of )F and )G

is a only necessary condition for synchronization. It is not suﬃcient. The volatility imparted to real debts and credits by inﬂation shocks prevents synchronization.

8.

1E [ E]& 1cov( , &) 1 ) ) cov( , &) E [ E ]& # # #

9.

)j must be the same for all j, too. And this is puzzling. Should not )j diﬀer across interests? Would not workers (as debtors) lose from a high growth in purchasing power, and capitalists (as creditors) win? And would that not create a negative correlation between marginal utility and for capitalists (as creditors), and a positive correlation for workers (as debtors)? No. This is because (by assumption) high and low inﬂation states are linked to proﬁt states. So while the capitalist does gain from a high growth in purchasing power augmenting the real value of loans, that does not make for a negative correlation between marginal utility and , as a high growth in purchasing power state is also a low proﬁt state. The conjunction of low proﬁt and high purchasing power growth leave the marginal utility of the capitalist uncorrelated with purchasing power growth. A parallel logic applies to the worker debtor. Creditors receive relief when they ‘need it’ (that is marginal utility is high), and are burdened when they do not so much need it. Debtors, by contrast, are burdened when they ‘need’ it. Notice the contrast with white noise inﬂation. Under white noise inﬂation, when creditors did not receive relief when they ‘needed it’. At the beginning of the modern era of inﬂation (that is, the French Revolution) Edmund Burke complained of ‘the policy of systematically making a nation of gamesters’.

10.

11.

12.

With you [ inﬂationary France] a man can neither earn nor buy his dinner without speculation. What he receives in the morning will not have the same value at night . . . Industry must wither away. Oeconomy must be driven from your country. Careful provision will have no existence. Who will labour without knowing the amount of his pay? Who will study to increase what none can estimate? Who will accumulate, when he does not know the value of what he saves? (Reflections on the Revolution in France, Penguin 1969, p. 310). The sight of this arbitrary rearrangement of riches strikes not only at security, but at the conﬁdence in the equity of the existing distribution of wealth. Those to whom the system brings windfalls, beyond their deserts and even beyond expectations and desires, become ‘proﬁteers’ . . . As the inﬂation proceeds and the real value of the currency ﬂuctuates wildly from month to month, all permanent relations between debtors and creditors, who form the ultimate foundation of capitalism, become so utterly disordered as to be utterly meaningless, and the process of wealth-getting degenerates into a gamble and lottery. (John Maynard Keynes, The Economic Consequences of the Peace, 1919) It may be worth noting that the present chapter’s notion of inﬂation risk getting in the way of the eﬀective intermediation between borrower and lender is aired in the General Theory. In section iv of Chapter 11 Keynes discusses the impact that an ‘adverse change in the monetary standard’ has on what he calls ‘lender’s risk’ and ‘borrower’s

246

13. 14.

15. 16.

The cost of inﬂation risk’. The upshot, he says, is a ‘duplication’ of risk margins, ‘which has not hitherto been emphasised’. The proﬁt rate will rise in Pauperia but wages will fall even more, reducing total factor incomes there. Inﬂation risk may compare favourably with other policies designed to achieve convergence: such as a progressive income tax, that funds a Guaranteed Minimum Income. This can be a powerful tool for convergence. Yet convergence will occur at lower levels of total capital than under inﬂation risk. Nevertheless, there remains a comparison unfavourable to risky inﬂation. All interests would prefer riskless inﬂation to risky inﬂation. But some interests may prefer a progressive income tax with a GMI, to no income tax and no GMI. If we were to endogenize labour supply eﬀects there may be immediate eﬀects on output. Higher prices reduces aggregate consumption, and thereby leisure, and so national product increases. Neither has any argument been made to the quantitative signiﬁcance of these eﬀects. They might be minute.

247

The cost of inﬂation as the cost of creditlessness

APPENDIX 11.1 This appendix provides an argument for the invariance of the interest rate to white noise inﬂation. It is necessarily the case:

F G C cov , C F [1 %] cov , C G % cov ,EC EC EC

and

F G C cov , C F [1 %] cov , C G % cov ,EC EC EC

where EC % ECG If cov( , C EC) and cov( ,C EC) are unchanged by white noise inﬂation one may infer:

F G cov , C F [1 %] cov , C G % 0 EC EC

F G cov , C F [1 %] cov , C G % EC EC

or

F F cov , C F [1 %] cov , C F [1 %] EC EC

G G cov , C G % cov , C G % EC EC

But with homothetic and identical preferences

U C1j & j 1 EUC1 1 [ 1] 1 EC1j C1 or cov( ,C j )

cov( , & j ) 1

248

The cost of inﬂation

Similarly for cov(,C j ). Thus [1 %][cov( ,&F ) cov( ,&F )] %[ cov( ,&G ) cov( ,&G )] j , this can be written as: Using 1 E E [[1 i] [1 ]] ) j )

i [1 %] 1 E E 11

i % 1 E E 11

The only way to satisfy this is [1 E E[[1 i] [1 ]]] 0

APPENDIX 11.2 This appendix relates the algebraic and diagrammatic measures of the cost of inﬂation. The welfare loss is the sum of three triangles: aa bb cc; ee cc dd; and bb ﬀ dd.

E[iE] aa

ii

jj

kk

bb

kk

gg

ff

cc

dd

hh

D

~ D

ee

Figure 11.6 The welfare triangles analysis of the cost of unpredictable inflation

D

The cost of inﬂation as the cost of creditlessness

249

bb ﬀ dd bb ﬀ cc cc ﬀ dd ~

bb gg ff cc cc ff hh dd [D D] [)F )G] 2 2 2

aa bb cc aa bb kk kk bb cc ii kk cc kk bb cc ii bb cc

ii bb cc jj )FD 2 2

ee cc dd )GD 2 (by parallel logic) Welfare loss aa bb cc ee cc dd bb dd ﬀ ~

~ [D D] F D )G D [)F )G ]D ) 2 2 2 2

[)F )G]

APPENDIX 11.3 This appendix derives measures of )F and so on. )F cov(&F, ) E&F[ E]

F F )F [ 1] C EC E( E) ECF

The proportionate deviation in C from expectation that is caused by a surprise inﬂation equals the propensity to consume out of wealth times times the loss of wealth. Assuming the propensity to consume is the interest rate: CF ECF rE( E) D ECF ECF Therefore: )F [1 ]var r DF C and by the same logic, )G [1 ]var r DG C But

~

bb ﬀ dd 0.5[)F )G][D D]

250

The cost of inﬂation

~

0.5 var r DF [1 ] var r DG [1 ] [D D] C C

~ 1 [D D] s(1 s)

0.5 var rD C[1 ]

the area under the demand curve for bonds: 2

intercept0D slope0D2

Therefore aa bb cc the area between the old and new demand curves 2

[slope0 slope0 ]D2 But

)F rvar slope0 slope0 D [1 ] ECF Therefore 21 aa bb cc rvarF[1 ]D2 rvar C [1 ]D s EC the area between the old and new supply curve 2

[slope0 slope0 ]D2 But

)G rvar [1 ] slope0 slope0 D ECG Therefore D 1 ee cc dd rvarG[1 ]D2 rvar C [1 ] 2 1 s EC 2

2

Therefore

~

1 D s(1 s)

Total cost0.5 var rD C[1 ]

12.

A summarization of results

The opening pages of this study undertook to investigate the causes, costs and consequences of inﬂation. The closing pages will summarize the salient theses of the analysis that followed.

THE CAUSES OF INFLATION AND PRICE INSTABILITY ●

●

●

●

●

●

Under certain plausible assumptions, there is only one value of money that equates the demand to the supply of money. Monetary equilibrium, in other words, exists and is unique (Chapter 4). The unicity of equilibrium turns on two anomalous properties of demand for money: that there is minimum quantity of money no matter how high its ‘price’, and that there is a minimum ‘price’ of money no matter how high its quantity (Chapter 2). Wealth maximization by issuers of inside money induces the supply of inside money to increase in response to an increase in the nominal rate of return on capital (Chapter 3). The supply of money that is relevant to the value of money is the supply of autonomous money. So whether money ‘counts’ or not, depends not on whether it is inside or outside, but whether it is autonomous or endogenous. (Chapter 4). There exists the possibility of a liquidity trap that will annihilate the impact on prices of temporary movements in the money supply. The possibility is created by a decline in the money supply, and magniﬁed by a low rate of proﬁt (Chapter 4). The possibility need not be ‘exotic’; if the proﬁt is near zero then the liquidity trap will apply with a near constant money supply. There exists also the possibility of an ‘inside money liquidity trap’, whereby any increase in outside money is exactly matched by an equal decrease in inside money. This possibility is realized when the marginal cost of the supply of inside money is constant (Chapter 4). The existence of dispersed information does not menace the existence of monetary equilibrium. An equilibrium – a set of prices such that everyone is doing the best they can subject to their information 251

252

●

●

●

The cost of inﬂation

set – does not require the homogeneity of information. The equilibrium will be that which reﬂects all available information. The existence of imperfect information provides a rationalization of the appearance of lags between changes in the money supply and a change in prices. Only enduring increases in the money supply will impact on prices, and time must pass before one can (rationally) tell whether an increase in M is going to endure. The Wicksellian approach to prices may be interpreted as the theory that has the value of money adjusting so as to secure the equality of the demand price of credit and the supply price of credit. The Wicksellian approach and the Quantity Theory are kindred theories in that they both conceive of inﬂation as a ‘monetary phenomenon’. And there is a striking homology between the two apparently discordant theories. In the Wicksellian model the reference price level plays the role that the quantity of money does in the Quantity Theory.

THE COSTS OF INFLATION AND PRICE INSTABILITY ●

●

●

●

●

●

Real bonds secure a socially eﬀicient sharing of consumption in the face of technological shocks (Chapter 6). Under plausible conditions, money bonds secure a socially eﬃcient sharing of consumption in the face of monetary shocks (Chapter 7). The shoe leather cost of inflation provide no reliable warrant for the desirability of either a zero mean of inﬂation or a zero variance of inﬂation (Chapter 10). In the absence of real bonds, inﬂation unpredictability will harm the socially eﬃcient sharing of consumption in the face of technological shocks (Chapter 11). In the absence of real bonds, inﬂation predictability spells the disappearance of a risk premium that is common across the population. There will be, in other words, no such thing as the proﬁt risk premium, or the inﬂation risk premium: the premia will vary according to the circumstances of the individual (Chapters 6 and 7). In the absence of real bonds and inﬂation unpredictability, the minimization of shoe leather costs compromises the socially eﬃcient sharing of consumption in the face of technological shocks (Chapter 11).

A summarization of results

253

THE COMPENSATIONS OF INFLATION AND PRICE INSTABILITY ●

●

●

●

Movement in the price level following a monetary shock allow, in the presence of money bonds under plausible circumstances, the elimination of the socially costly riskiness in consumption arising from disturbances to consumption by real balance eﬀects (Chapter 7). Low cost issuers of inside money will beneﬁt from inﬂation, as their expansion in issue under inﬂation is not matched by the expansion of high cost issuers, allowing low cost issuers to purchase more capital than in the absence of inﬂation (Chapter 10). In the absence of real bonds, unpredictability in inﬂation imparts an egalitarian dynamic to accumulation, that ultimately produces a convergence in the wealth of economic interests, that will not occur in the absence of inﬂation unpredictability (Chapter 11). In the absence of real bonds, a ‘moment shift’ in the money supply, that yields a completely unpredictable increase in money prices, shall increase saving, and reduce the social waste of inﬂation risk (Chapter 11).

References Akerlof, George A., William T. Dickens, George L. Perry, Truman F. Bewley and Alan S. Blinder (2000), ‘Near-rational wage and price setting and the long-run Phillips Curve’, Brookings Papers on Economic Activity 1:2000, 1–60. Arrow, Kenneth J. (1971), Essays in the Theory of Risk Bearing, London: North-Holland. Bailey, Martin (1956), ‘The welfare cost of inﬂationary ﬁnance’, Journal of Political Economy, 64 (2), 93–110. Baumol, William J. (1952), ‘The transactions demand for cash: an inventory theoretic approach’, Quarterly Journal of Economics, 66 (4), 545–56. Blinder, Alan S. (1997), ‘Is there a core of practical macroeconomics that we should all believe?’, American Economic Review, 87 (2), 240–43. Boianovsky, Mauro (2004), ‘The IS-LM model and the liquidity trap concept: from Hicks to Krugman’, History of Political Economy, 36, 92–126. Evans, C., J. Pope and J. Hasseldine (eds) (2001), Taxation Compliance Costs: A Festschrift for Cedric Sandford, Sydney: Prospect. Friedman, Milton (1969), The Optimal Quantity of Money and Other Essays, Chicago: Aldine Publishing. Fujiki, Hiroshi and Casey B. Mulligan (1996), ‘A structural analysis of money demand: cross-sectional evidence from Japan’, Bank of Japan Monetary and Economic Studies, 14 (2), 53–78. Hicks, J.R. (1939), Value and Capital: An Inquiry into Some Fundamental Principles of Economic Theory, Oxford: Clarendon Press. Hicks, J.R. (1989), A Market Theory of Money, Oxford: Clarendon Press. Hirshleifer, J. and John G. Riley (1979), ‘The analytics of uncertainty and information – an expository survey’, Journal of Economic Literature, 17 (4), 1375–421 Kaldor, Nicholas (1970), ‘The new monetarism’, Lloyd’s Bank Review, 95, 1–18. Knell, Markus and Helmut Stix (2004), ‘Three decades of money demand studies. Some diﬀerences and some remarkable similarities’, Oesterreichische Nationalbank Working Paper, 88. Lucas, Robert E. Jr, Fernando Alvarez and Warren E. Weber (2001), ‘Interest rates and inﬂation’, American Economic Review, 91 (2), 219–25. 254

References

255

Marty, Alvin L. (1999), ‘The welfare cost of inﬂation: a critique of Bailey and Lucas’, Federal Reserve Bank of St Louis Review, 81 (1), 41–6. Mulligan, Casey B. and Xavier X. Sala-i-Martin (1992), ‘US money demand: surprising cross-sectional estimates’, Brookings Papers on Economic Activity, 2:1992, 285–343. Obstfeld, Maurice and Kenneth Rogoﬀ (1998), ‘Risk and exchange rates’, National Bureau of Economic Research, Working Paper No. 6694. O’Connell, Sean and Chris Reid (2005), ‘Working-class consumer credit in the UK, 1925–60: the role of the check trader’, Economic History Review, 58 (2), 378–405. Patinkin, Don (1965), Money, Interest, and Prices. An Integration of Monetary and Value Theory, (second edition), New York: Harper & Row. Phelps, Edmund S. (1973), ‘Inﬂation in the theory of public ﬁnance’, Scandinavian Journal of Economics, 75, 67–82. Pope, Jeﬀ (1993), ‘The compliance costs of taxation in Australia and tax simpliﬁcation: the issues’, Australian Journal of Management, 18 (1), 69–89. Samuelson, Paul Anthony (1947), Foundation’s of Economic Analysis, Cambridge, MA: Harvard University Press. Shann, E.O.G. (1938), An Economic History of Australia, Cambridge: Cambridge University Press. Wicksell, Knut (1936), Interest and Prices, New York: Augustus M. Kelly. Woodford, Michael (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton, NJ: Princeton University Press.

Index Akerlof, George T. 2 Arrow, Kenneth J. 182 asymmetric information 133 autonomous money demand 72–3, 79, 82, 84, 177 impact on price level 72 Bailey, Martin 222 bank money 42–3, 72 Baumol, William J. 11 benchmark interest rate 88, 95, 97, 102, 194 Blinder, A. 4, 7 Boianovsky, Mauras 85 bonds demand for 128, 147, 229–31 ‘price’ of 127 supply of 129, 197, 227, 230–31 Bretton-Woods Agreement 152 Burke, Edmund 245 central banks 3, 103, 109, 153, 196–7, 200 coeﬃcient of reaction 88, 108–9, 189 coinage 32 convenience costs of supply of inside money 36 compensations of inﬂation 239–42 cost of inﬂation risk 231 quantiﬁcations of 238–9 credibility costs of supply of inside money 36, 44–5, 48, 183, 219, 221 debt determinants of quantity of 132 social function of 113–15, 123, 133–6 demand price of credit 84, 89, 89–90, 94–6, 185–217, 252 under risk 186

eﬃcient consumption, 119–20, 133, 136, 140, 142, 155, 231, 236–7, 239, 245 eﬃcient markets hypothesis 178, 180, 184 elasticity of money demand to consumption 33 to nominal rate of return on capital 27, 28 and optimal rate of inﬂation 211–13, 223 endogenous money supply 3, 35, 42–3, 47, 64–5, 81–2, 86, 87, 104–5, 220 exchange rates falling 196 pegged 108, 152, 200 expectations and heterogeneous information 178–82 Federal Reserve of the United States 4 Fiscal Theory of the Price Level 4, 6 Fisher Condition 89–91, 94, 106–7, 187 Forster, E.M. 137 Friedman, M. 208, 214 Fujiki, Hiroshi 223 Gibson’s Paradox 101 government debt 134, 152–3, 156 Great Depression 183, 243 Great Inﬂation 3, 196–7 hedgers 122, 124–6, 137, 229 Hicks, J.R. 33, 35, 52 Hicks neutral technical change 125, 132, 134, 232 Hume, David 156 implicit yield on money 1–3, 33, 161, 204

257

258

Index

inﬂation and banking 95 private beneﬁts of 218–19, 221–2 social beneﬁts of 239–42 inﬂation rate and money demand 27–9, 32 optimal rate of 213–14, 217 Quantity Theory determinants of 76–7, 171–4, 188 seasonal pattern 84 and semi-elasticity of money demand 50–51 Wicksellian determinants of 99–100, 188–91, 196–8 inﬂation risk social cost of 231, 235–41 inﬂation risk premium 146, 155, 162, 252 and the bond market 227–32 and the demand for money 163 and the demand price of credit 186 and the liquidity trap 167 and price level determination 87 and shoe leather costs 207, 215–16 and the supply of inside money 165 inﬂation target see reference inﬂation rate inside money autonomous supply of 71–2, 95 elasticity of supply of to total money demand 44 elasticity of supply to return on capital 44 impact of supply of on prices 71–2 inﬁnite elasticity of supply of 81 interest upon 47 semi-elasticity of supply of 50 social cost of supply 40, 217–19, 221–2 supply of 41–2, 164–5 interest rate 89 determination of 101 and inﬂation 247 and relation to price level 100–102, 192–5 stability 77, 195–6 zero bound 91, 106 interest rate reaction function 87–8, 92–3, 186, 200 international capital mobility 241–2

Kaldor, N. 36, 46, 82 Keynes, J.M. 137, 245 Knell, Markus 207, 209 lags in the impact of the money supply 174–7 in impact of reference price level 191–2 ‘long and variable’ 176 liquidation costs 12–18, 19, 21, 25, 32–3, 59, 62, 204, 218 liquidity trap 77–81, 85, 108, 167–8, 251 LM curve 4 Lucas, R. 4 M3 4 marginal beneﬁt of money 13–14, 16, 33 and direct costs of money holding 15 marginal utility of money 17–18, 52, 62, 165 Marty, Alvin L. 211 Monetarism 3, 46, 103 see also Quantity Theory of Money monetary equilibrium existence 63–5, 166 uniqueness 57–62 monetary policy 76–7, 102–3 see also Price Stabilisation; Central Banks money debt determinants of size of 152, 230 money bonds demand for in the absence of real bonds 22–9 ‘price’ of 147, 148 social function of 138, 140–48 supply of in the absence of real bonds 227–9 money demand aggregate 30 coeﬃcient of 27, 75, 84–5, 205, 213–14, 223–4 elasticity 27–8, 33, 75, 211–13 empirical studies 223 functional forms 25 minimum level of 24–6

Index semi-elasticity of 27–8, 75, 68, 80, 161, 164, 169, 211, 223 sensitivity of 27 theory of under risk 160–64 money neutrality 33 see also non-neutrality of money money supply permanent shocks to 70–71, 106, 157, 172–3, 175 permanent measures of 74–6, 171 money wage rates 85 Mulligan, Casey B. 223 natural rate of interest see proﬁt rate nominal rate of return on capital deﬁned 22, 33 see also proﬁt rate non-neutrality of money 176, 242–4 non-neutrality of price level 46 Obstﬁeld, Maurice 235 O’Connell, Sean 52 operational costs of supply of inside money 36 ‘Optimal Quantity of Money’ and costs of inﬂation risk 236–7 outside money residual demand for 49–50 semi-elasticity of demand for 50 Pareto’s False Principle of Diminishing Marginal Substitution 18, 28, 33 Patinkin, Don 11, 34 permanent magnitudes 75–6, 99, 101, 171 permanent rate of growth of outside money 75–6 permanent rate of proﬁt 75–6, 99 permanent reference price level 98–9, 190 Phelps, E.S. 214 Phillips Curve 4 Pope, Jeﬀ 223 price level determination of 86–7, 95 elasticity to inside money supply 72 elasticity to outside money supply 67–9, 81, 85 elasticity to reference price level 95 semi-elasticity to benchmark rate of interest 97

259

semi-elasticity to proﬁt rate 73, 96 price stability 102–3, 194–5, 215–17 proﬁt rate determination of 36, 114 and inﬂation 99–100, 189 and price stability 102 proﬁt risk premium 127, 162, 165, 252 and price level determination 187 Quantity Theory and the classical dichotomy 30, 33 comparative statics 66–73, 166–70 deﬁned 3 foundations of under risk 160–66 and the inﬁnitely elastic supply of inside money 82 and the liquidity trap 79–80 mathematical expectations versions of 170–72 price level determinacy 57–61 and Wicksellianism 103–6 Ramsey-Solow model 30–32 real balance eﬀects 24, 138, 141, 156, 157 real interest rate 133 and inﬂation risk 230, 234–5 and money shocks 176 reference inﬂation rate 99–101 reference price level 87, 95–6 and inﬂation 190 and the interest rate 193 and the price level 193 Residual demand for outside money see outside money risk and debt 113–15, 122–3 trade of 241 risk evening 113, 133 risk premium see inﬂation risk premium; proﬁt risk premium Samuelson, P.A. 11 seigniorage 208 optimal quantity of 212–13 social cost of 209–11, 220, 223, 226 semi-elasticity of money demand to nominal rate of return on capital 27, 28, 84 to rate of interest 161

260 Shann, E.O.G. 52 shoe leather cost of inﬂation 204–5 and inside money 214 as a rationale for price stability 207 and sensitivity of money demand 206–7 speculators 122, 124–6 supply price of credit 86, 88, 90, 94–7, 185–7 taxation compliance costs 223 technological change 115–16, 125–6, 140, 154, 225, 252

Index see also Hicks neutral technical change unemployment 4, 80, 85 Wicksell, K. 102, 108 Wicksellian theory of prices comparative-statics 95–9 equilibrium 94 existence of equilibrium 92–3 inﬂation 99–100 parallelism with Quantity Theory 103–6 and zero interest bound 106 Woodford, Michael 4