A note on the integrability of partial-equilibrium measures of the welfare costs of inflation

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A note on the integrability of

partial-equilibrium measures of the welfare

costs of inﬂation

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Rubens Penha Cysne

1

Fundacß~aao Getulio Vargas, Rio de Janeiro, Brazil Received 19 January 2001; accepted 6 June 2001

Abstract

Multidimensional measures of the welfare costs of inﬂation have been employed in the literature without an explicit concern of how the demand for the respective mone-tary assets are generated and without an investigation of the respective integrability con-ditions. This note establishes conditions under which such welfare measures are well deﬁned.

JEL classiﬁcation: E40; E60

Keywords: Inﬂation; Welfare; Integrability

1. Introduction

This work investigates the integrability of multidimensional partial-equilibrium measures of the welfare costs of inﬂation. Such measures, which in a certain way

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q

A previous version of this paper circulated with the title ‘‘Integrability and the demand for monetary assets: an alternative approach to an old problem’’.

E-mail address:rpcysne@midway.uchicago.edu(R.P. Cysne).

1_{Rubens Penha Cysne is a professor of economics at the Fundacß~}_a_{ao Getulio Vargas (FGV) and a}

researcher at the IBRE/FGV.

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extend Bailey’s (1956) seminal contribution, are particularly aimed at economies where more than one asset perform monetary functions. They have been used in the literature (Marty and Chaloupka, 1988; Marty, 1994, 1999; Baltensperger and Jordan, 1997), though, without an explicit concern about the issue of path indepen-dence, as well as the issue of how the demands for the respective monetary assets are derived. By showing how this can be done and under what conditions these measures are well deﬁned, their applicability is strengthened.

This is even more important due to the fact that, elsewhere, Lucas (2000), for the one-dimensional case, as well as Simonsen and Cysne (2001) and Cysne (2001), for the n-dimensional case, have shown that such partial-equilibrium measures can be regarded as very good approximations, especially for moderate rates of inﬂation, of their general-equilibrium counterparts.

Our procedure can be understood as paralleling the search, in classical static con-sumer theory, of particular specifications of the utility function which assured the in-tegrability of Marshallian demands (the two classic textbook cases being those of homothetic preferences and of parallel preferences with respect to the numeraire). Likewise, we look for conditions that a certain transacting technology must fulfill in order to make a specific welfare measure well defined.

Our basic conclusion is that, in a shopping-time framework, the partial-equilib-rium measure of the welfare costs of inflation is well defined (path independent) if and only if the transacting technology is blockwise-weakly separable with respect to the monetary variables and shopping time. Expressed differently, the marginal rates of substitution between the two monetary assets must be independent of the total shopping time.

Our derivation of the demands for the monetary assets is based on Simonsen and Cysne (2001) model, which, in turn, draws on Lucas (1993, 2000) previous analysis of welfare costs of inﬂation in shopping-time economies. To simplify, we present a model using currency and one kind of alternative monetary asset, which we call de-posits. Our basic results, though, can be easily extended to the case when more than two assets perform monetary functions.

This note presents two more sections. Section 2develops the basic model. Section 3 presents the partial-equilibrium measure of the welfare costs of inﬂation, establishes conditions for its path independence and displays two contrasting exam-ples.

2. The model

This section draws heavily on the exposition of the model in Simonsen and Cysne (2001). Households are assumed to maximize discounted utility subject to their bud-get constraint and to the constraint that the total time spent producing the consump-tion good and shopping must sum to one. A transacting technology speciﬁes how currency and deposits permit agents to economize on the amount of time spent on transactions in the goods market.

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Household preferences are determined by Z 1

0

egtUðcÞ dt ð1Þ

where U : X0! R, X0 Rþ, is a strictly concave function of the consumption

at instant t and g > 0. The household is endowed with one unit of time that can be used to transact or to produce the consumption good with constant returns to scale:

yþ s ¼ 1: ð2Þ

Here, y stands for the production of the consumption good and s for the fraction of the initial endowment spent as transacting time. Households can accumulate three assets: currency ðMÞ, bonds ðBÞ and deposits ðX Þ.

In their maximization, households take as given the nominal interest rate of bonds, i, and the opportunity cost of holding deposits, j¼ i ix, where ixis the

in-terest rate paid by X and 0 < j < i. Letting P ¼ P ðtÞ be the price of the consumption good, the household faces the budget constraint

_ M

Mþ _BBþ _XX ¼ iB þ ixX þ P yð cÞ þ H :

H indicates the ﬂow of currency transferred to the household by the government and the dot over the variable its time derivative. Making a ¼ ðM þ B þ X Þ=P , p ¼ _PP =P (inﬂation rate), m¼ M=P , x ¼ X =P and h ¼ H =P , and taking Eq. (2) into account, the budget constraint reads

_a

a¼ ið pÞa þ 1 s im jx c þ h: ð3Þ Compared to currency, or to deposits, bonds are obviously preferable as a reserve of value. However, currency, as well as deposits, are useful because they save trans-action time, as the transacting function describes:

c¼ F ðm; x; sÞ: ð4Þ

F is supposed to be diﬀerentiable and strictly increasing in each of its variables, with decreasing marginal returns.

The household maximizes Eq. (1) subject to the budget constraint (3) and subject to the time-transacting technology (4).

The steady state is characterized by _mm¼ _bb¼ _xx ¼ 0. Also, in steady state, the rate of inﬂation is determined so that the seigniorage matches the real value of the net transfers made by the government (h¼ pm, where p ¼ _MM =M).

In steady state, Euler’s equations lead to the equilibrium relations: i¼ p þ g;

Fm¼ iFs; ð5Þ

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In equilibrium, since the consumption good is non-storable and since all house-holds are equal, y¼ c. Using Eqs. (2) and (4), we get the fourth equation that com-pletes the model:

1 s ¼ F ðm; x; sÞ: ð7Þ

We use Eqs. (5)–(7) to determine, locally, i, j and s, as functions of m and x. In this model, the shopping-time variableðsÞ can be considered as a general-equi-librium measure of the welfare costs of inﬂation (Simonsen and Cysne, 2001; Lucas, 2000). Indeed, since ﬁat money can be produced with a zero social cost, the totality of the transacting services demanded in the economy could be produced at no cost. Time dedicated to shopping is therefore a waste.

In what follows, though, we will be solely interested in the partial-equilibrium measure of the welfare costs of inﬂation PE, to be deﬁned in the next section. Simon-sen and Cysne (2001) show, in the particular case when j is constant, that PEðBðiBÞ

in their paperÞ is an upper bound to s. Cysne (2001) extends this result, under certain conditions on the path followed by the parameters of the model, by relaxing the as-sumption of a constant value of j.

3. Integrability

We will be interested in the evaluation of iðm; xÞ and jðm; xÞ along paths CðtÞ ¼ ðmðtÞ, xðtÞÞ, CðtÞ V , V an open set of R2

þþ and a 6 t 6 b. With currency and

interest-bearing deposits, the partial-equilibrium measure of the welfare costs of in-ﬂation is given by the line integral

PE¼ Z

C

Kdr ð8Þ

where K¼ ðiðm; xÞ; jðm; xÞÞ and dr ¼ ðdm; dxÞ. PE can be interpreted as a general-ization of the area under a demand curve, although it is a diﬀerent object from the mathematical point of view.

For PE to be well defined as a welfare measure, it must take a unique value for different paths of m and x, when the initial and final points are the same (path inde-pendence). It is a well-known result from Calculus, sometimes referred to as the ‘‘po-tential function theorem’’ (see e.g. Apostol, 1957; Lang, 1987), that such path independence happens iff K is the gradient of some function b in all points where K is defined. Locally, this is equivalent to having:

oi

oxðm; xÞ ¼ oj

omðm; xÞ ð9Þ

in all points of V.

The formal deﬁnition and an encompassing analysis of separability can be found in Leontief (1947). For our purposes, the function Fðm; x; sÞ is said to be blockwise-weakly separable when there are functions G and H such that Fðm; x; sÞ ¼

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HðGðm; xÞ; sÞ. It also follows from the analysis made by Leontief that this condition is equivalent to having the marginal rate of substitution between m and x indepen-dent of s:

o

osðFm=FxÞ ¼ 0:

The following proposition uses this deﬁnition to establish our main result:

Proposition 1. The welfare measure PE is well defined (path independent) if and only if the transacting technology Fðm; x; sÞ is blockwise-weakly separable with respect to the monetary aggregator and the shopping-time variable, s.

Proof. Considering Eq. (4) and the equilibrium conditions, taking the partial de-rivatives of Eqs. (5)–(7) and making

D¼ det Fs 0 iFss Fms 0 Fs jFss Fxs 0 0 1þ Fs 2 4 3 5 ¼ F2 sð1 þ FsÞ > 0: We have Doi ox ¼ Fs½ðiFsx FmxÞð1 þ FsÞ iFð ss FmsÞFx; Doj om ¼ Fs½ðjFsm FxmÞð1 þ FsÞ ðjFss FxsÞFm: Therefore, Fsð1 þ FsÞ oi ox oj om ¼ ðiFsxþ jFsmÞð1 þ FsÞ þ FssðiFx jFmÞ þ ðFmFxs FxFmsÞ:

Substituting Fm=Fsand Fx=Fs, respectively, for i and j, in the above expression, one

concludes that oi=ox¼ oj=om for any values of m and x if and only if FmsFx¼

FmFxs() oðFm=FxÞ=os ¼ 0.

Example 1. Consider an economy with transacting technology: Fðm; x; sÞ ¼ Aðm1=2_{þ sÞx}1=2_; _{A >}

0:

This technology is not weakly separable, since Fm=Fx6¼ Fms=Fxs. Given the ﬁrst order

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iðm; xÞ ¼ 1 2m1=2;

jðx; mÞ ¼ 1þ m

1=2

2xð1 þ Ax1=2_Þ;

and the partial-equilibrium welfare measure PE¼ Z C 1 2m1=2dmþ 1þ m1=2 2xð1 þ Ax1=2_Þdx: ð10Þ

Since in this example we are interested only in comparing the values of PE along different paths, we make A¼ 1 to simplify the calculations. Let us then suppose that this economy presents an initial value ofðm; xÞ given by ð0:04; 0:04Þ, and a final value ofð0:01; 0:01Þ. We consider two different paths. In the first, C1, the economy moves

fromð0:04; 0:04Þ to ð0:01; 0:01Þ along the straight line x ¼ m ¼ t. The second path is a union of two intermediate paths. In the ﬁrst, C21, in which m is always kept

constant, the economy moves fromð0:04; 0:04Þ to ð0:04; 0:01Þ. In the second, C22, in

which x is always kept constant in its new level, 0:01, the economy moves from ð0:04; 0:01Þ to ð0:01; 0:01Þ. By ﬁrst calculating the value of PE for C1, making

x¼ m ¼ t and dx ¼ dm ¼ dt in Eq. (10), we get PE1¼ 0:1 þ log 2. On the other

hand, along C22, PE22¼ 0:1, whereas along C21we have

PE21¼ 0:6

Z 0:04 0:01

1

xð1 þ x1=2_Þdx:

Since PE2¼ PE21þ PE22, PE1¼ PE2() PE21¼ ln 2. However, this does not occur,

as the inequality below shows: PE21¼ 0:6 Z 0:04 0:01 1 xð1 þ x1=2_Þdx > 0:6 Z 0:04 0:01 1 1:2xdx¼ 0:5 Z 0:04 0:01 1 xdx ¼ log 2:

Any evaluation of PE for this economy would therefore have to consider the path followed by the variables.

Example 2. Here we consider the technology Fðm; x; sÞ ¼ Gðm; xÞs ¼ Bma_x1a_s,

B >0. This technology is separable, since Fm=Fx ¼ Fms=Fxs¼ ða=ð1 aÞÞðx=mÞ.

Along a path C, this technology leads to the following measure of PE: PE¼ Z C a m 1 Bma_x1a_{þ 1}dmþ 1 a x 1 Bma_x1a_{þ 1}dx: ð11Þ

We prove path independence by showing that this problem admits a potential function. By making Gðm; xÞ ¼ Bma_x1a_{and solving the partial diﬀerential equations}

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oCðm; xÞ om ¼ a m 1 1þ G; oCðm; xÞ ox ¼ 1 a x 1 1þ G:

One easily ﬁnds the potential function to be Cðm; xÞ ¼ logðG=ð1 þ GÞÞ. Hence, for initial and ﬁnal values of m and x given by, respectively, mð 1; x1Þ and ðm2; x2Þ:

PE¼ ðCðm2; x2Þ C mð 1; x1ÞÞ

whatever the path taken byðm; xÞ between these two points.

Acknowledgements

This work beneﬁtted from conversations with Robert Lucas Jr. Humberto More-ira, Paulo Klinger Monteiro, Samuel Pessoa and two anonymous referees made valuable comments and suggestions. Remaining errors are my responsibility. The work was carried out while I was visiting the Department of Economics of the Uni-versity of Chicago. I wish to thank the Department for its hospitality and the ﬁnan-cial support from Capes.

References

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Bailey, M.J., 1956. Welfare cost of inﬂationary ﬁnance. Journal of Political Economy 64, 93–110. Baltensperger, E., Jordan, T.J., 1997. Seigniorage, banking, and the optimal quantity of money. Journal of

Banking and Finance 21, 781–796.

Cysne, R., 2001. Divisia indexes, inﬂation and welfare. Working paper 397, FGV, and the University of Chicago. Annals of the 2001, Summer Meeting of the North American Econometric Society. Lang, S., 1987. Calculus of Several Variables, third ed. Springer, Berlin.

Leontief, W., 1947. Introduction to a theory of the internal structure of functional relationships. Econometrica 15 (4), 361–373.

Lucas Jr., R.E., 1993. The welfare costs of inﬂation. Working paper. University of Chicago, Chicago. Lucas Jr., R.E., 2000. Inﬂation and welfare. Econometrica 68, 247–274.

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