Welfare costs of inflation when interest-bearing deposits are disregarded: a calculation of the bias

Ensaios Econômicos

Welfare costs of ination when

interest-bearing deposits are disregarded: A

calcu-lation of the bias

Rubens Penha Cysne, David Turchick

Os artigos publicados são de inteira responsabilidade de seus autores. As

opiniões neles emitidas não exprimem, necessariamente, o ponto de vista da

Fundação Getulio Vargas.

ESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA Diretor Geral: Renato Fragelli Cardoso

Diretor de Ensino: Luis Henrique Bertolino Braido Diretor de Pesquisa: João Victor Issler

Diretor de Publicações Cientícas: Ricardo de Oliveira Cavalcanti

Penha Cysne, Rubens

Welfare costs of inflation when interest-bearing deposits are disregarded: A calculation of the bias/

Rubens Penha Cysne, David Turchick  Rio de Janeiro : FGV,EPGE, 2010

(Ensaios Econômicos; 700) Inclui bibliografia.

Welfare costs of in‡ation when interest-bearing

deposits are disregarded: A calculation of the bias

Rubens Penha Cysne,

David Turchick

January 2010

Abstract

Most estimates of the welfare costs of in‡ation are devised considering only noninterest-bearing assets, ignoring that since the 80’s technological innovations and new regu-lations have increased the liquidity of interest-bearing deposits. We investigate the resulting bias. Su¢ cient and necessary conditions on its sign are presented, along with closed-form expressions for its magnitude. Two examples dealing with bidimen-sional bilogarithmic money demands show that disregarding interest-bearing monies may lead to a non-negligible overestimation of the welfare costs of in‡ation. An intu-itive explanation is that such assets may partially make up for the decreased demand of noninterest-bearing assets due to higher in‡ation.

1

Introduction

Feldstein (1997), Lucas (2000), Mulligan and Sala-i-Martin (2000) and Attanasio et al. (2002) are examples of papers providing theoretical foundations and/or estimates of welfare

Keywords: Welfare, In‡ation, Money Demand, Divisia Index, Interest-Bearing Monies. JEL: E40, D60.

y_{Corresponding author. Rubens Penha Cysne is a Professor at the Graduate School of Economics of the}

Getulio Vargas Foundation (EPGE/FGV). Address: Praia de Botafogo 190, 11o andar –Rio de Janeiro, RJ – 22250-900 – Brazil. Telephone number: +55 (21) 3799-5871. Fax: +55 (21) 2553-8821. E-mail address: rubens.cysne@fgv.br.

costs of in‡ation (for unconstrained consumers) based on Bailey’s (1956) unidimensional area-under-the-inverse-money-demand-function formula.1 _{Attanasio et al.’s welfare}

calculations, following the work of Mulligan and Sala-i-Martin, also consider adoption decisions by households concerning the di¤erent …nancial technologies available in the economy. But the underlying welfare formula is still Bailey’s.

To simplify the notation throughout this paper, let M 1 denote the noninterest-bearing component of M 1, and M 2 n M1 a monetary aggregate composed of all interest-bearing deposits belonging to M 2 (according to some de…nition), but not to M 1 . Both Bailey’s (1956) and Lucas’s (2000, sections 3 and 5) welfare measures have been devised for monetary aggregates which do not pay interest. That is, irrespective of considering only the monetary base or M 1 as the relevant monetary aggregate, they ignore a fact that has been thoroughly documented since the 80’s: the presence of several interest-bearing

deposits (M 2 n M1 ) providing monetary services.

As Teles and Zhou (2005) put it, "technological innovation and changes in regulatory practices in the past two decades have made other monetary aggregates as liquid as M 1". Attanasio et al. (2002, p. 341), for instance, report that 58:7 percent of the households in their sample (originated from the Italian economy) hold, besides the two monetary assets currency and interest-bearing bank deposits, at least one other interest-bearing

non-monetary asset (e.g., bonds). Regarding the United States, Mulligan and Sala-i-Martin (2000, p. 962) report, following the 1989 Survey of Consumer Finances, that 35 percent of all households hold bank deposits and at least one additional interest-bearing asset.

In such settings, Bailey’s (1956) or Lucas’s (2000) unidimensional formulas may be misleading because they disregard, in the calculation of the welfare costs of in‡ation, the existence of a possible trade-o¤ between more-liquid noninterest-bearing and less liquid

1_{The dimension here corresponds to the number of monetary assets considered in the economy. See, e.g.,}

interest-bearing monies.2 _{This paper investigates the bias in the welfare calculations arising}

from this fact. Put in another way, we investigate the error arising from using

unidimensional, rather than bidimensional (or multidimensional, since the extension to n di¤erent interest-bearing monies is straightforward) measures of the welfare costs of in‡ation.3

Monetary aggregates are here classi…ed and aggregated solely according to their user costs. Let m and x stand for the real quantities of M 1 and M 2 n M1 , respectively.4 _The

aggregation of all interest-bearing deposits in one single asset x, as is done here, implicitly assumes that they all have approximately the same user cost (de…ned as their opportunity cost relative to holding bonds). Otherwise, additional dimensions may be used, an

extension that requires no signi…cant cost. The present analysis concerns using only m in the calculations of the welfare costs of in‡ation, vis-à-vis using an aggregation of m and x. We argue that the latter should become the standard procedure.

We assume that all monies are costlessly issued by the government. When it comes to the closed-form solutions for the bias arising from disregarding interest-bearing deposits, our calculations assume, as Jones et al. (2004) do in their theoretical model and Attanasio et al. (2002) verify in their empirical assessment with Italian data, that the user cost of the interest-bearing money is constant. The underlying approach to the problem, though, could be extended to the more general case in which this fact is not taken for granted, using for example the multidimensional formulas in Cysne and Turchick (2007).

To make our point clear right from the outset, let R stand for the interest rate on bank deposits, and RB for the interest rate on a non-monetary …nancial asset. Since bank

deposits provide monetary services, we must have RB > R 0. Bailey’s unidimensional 2_{Lucas (2000) acknowledges this point in the concluding section of his work.}

3_{Cysne and Turchick (2007) provide an upper bound to another type of error: the one arising from using}

one of the di¤erent types of multidimensional measures available in the literature, vis-à-vis the others.

welfare formula BU, which reads5

BU(RB) =

Z RB 0

m0( )d , (1)

has been widely employed in the literature in the past …fty years, a recent contribution being that of Ireland (2009).

We shall see that even when the opportunity cost of holding bank deposits (RB R) is

constant, welfare formulas for households holding both bank deposits and bonds (in addition to currency) should also take into account the e¤ect of variations of the interest rate paid by bonds on the demand for bank deposits (x0(RB)). The use of equation (1)

should therefore lead to a bias, the sign of which depends on the sign of x0; i.e., whether currency and deposits are substitutes or complements.

In the remainder of this text we will proceed as follows. Section 2 shortly presents multidimensional measures of the welfare costs of in‡ation based, respectively, on the McCallum-Goodfriend (1987) shopping-time dynamic framework and the Sidrauski (1967) money-in-the-utility-function (MUF) framework. They are developed for households holding currency, bank deposits and bonds. The main purpose here is to show why a second integral (besides the one in Bailey’s unidimensional formula) has to be considered in the calculations of the welfare costs of in‡ation.

Section 3 concentrates on the sign of the bias arising from the use of unidimensional measures instead of formulas which also take into consideration the existence of

interest-bearing deposits. The comparison could be based on the multidimensional formulas arising from the use of the McCallum-Goodfriend or of the Sidrauski model. To simplify, we de…ne the bias and proceed with the calculations using an approximation to both of these formulas. For analogy reasons, we shall call this approximation formula "Bailey’s multidimensional formula for the welfare costs of in‡ation", BM. Next, we

5_{The subscript "U " used here stands for unidimensional. The subscript "B" in R}

determine when Bailey’s unidimensional measure BU should be expected to overestimate or

underestimate BM.

Section 4 introduces particular functional forms of the utility and monetary-aggregator function in order to o¤er closed-form expressions of the bias. Two examples of a particular case leading to bilogarithmic money-demand functions are developed in order to illustrate the fact that the size of the relative bias can be far from negligible. Section 5 concludes.

2

Multidimensional approaches to the welfare costs of

in‡ation

The two next subsections reproduce (with minor adjustments) results obtained in Simonsen and Cysne (2001), Cysne (2003) and Cysne and Turchick (2007). They are included here for two reasons: …rst, to introduce the notation and assumptions necessary for the derivations of our main results in the forecoming section; and second, for

convenience of the reader.

2.1

The shopping-time approach

In this section we shall not consider growth (it makes no di¤erence concerning our …nal results). Let y stand for real output and normalize it to 1. The consumer has a (…xed) time endowment equal to unity and gains utility from real consumption (c = C=P , P standing for the price index) of a single non-storable consumption good, with preferences determined by

Z 1 0

e gtU (c)dt, (2)

where U is a strictly increasing and concave function and g > 0.

Consumers can accumulate three assets: currency (M ), interest-bearing deposits (X) and bonds (B). The interest rates paid by each one of these assets are, respectively, zero, R

and RB. Let b = B=P , m = M=P , x = X=P , h = H=P (H indicates the (exogenous) ‡ow

of money transferred by the government to such consumers). Let s stand for shopping time and = _P =P for the in‡ation rate (the dot over the variable represents its time

derivative). Then, the budget constraint reads:

_

m + _x + _b = 1 c s + h + (RB ) b + (R )x m. (3)

The transacting technology is given by

c = F (m; x; s) = G(m; x) (s), (4)

with Fm > 0, Fx > 0 and Fs > 0. The twice-di¤erentiable monetary-aggregator function G

is assumed to be …rst-degree homogeneous, concave and such that Gmm < 0, Gxx < 0. The

microfoundations for a transacting technology of type (4) are based on the inventory technology found in the works of Baumol (1952), Tobin (1956) and Miller and Orr (1966). Lucas (2000, p. 265 –see, in particular, ft. 13) discusses this point.

Households maximize (2) subject to the budget constraint (3) and to the

time-transacting technology (4). In the steady state, assuming interior solutions, Euler equations lead to the necessary conditions

8 > > > > < > > > > : RB = + g RB = F_Fm_s = Gm(m;x) 0_(s) RB R = F_Fx s = Gx(m;x) 0_(s) .

The equilibrium in the goods market is described by

1 s = F (m; x; s).

m(RB; RB R), x(RB; RB R) and s(RB; RB R).

As shown in Cysne (2003), in this model the welfare costs of in‡ation are given by the di¤erential form of a Divisia index:

ds = (1 s) [RBdm + (RB R)dx] 1 s + RBm + (RB R)x

, (5)

with s(0; 0) = 0. In this 2-dimensional context, a reasonable approximation to (5) for low values of RB can be given by the Bailey-like multidimensional di¤erential formula

dBM = [RBdm + (RB R)dx], (6)

with BM(0; 0) = 0.6

Expressions (6) and (5) extend the 1-dimensional ones o¤ered by Bailey (1956) and by Lucas (2000, section 5), respectively. Note that as RB ! 0, provided RBm and (RB R)x

also go to 0, we get ds dBM and, upon integration and usage of the initial conditions,

s(RB; RB R) BM(RB; RB R).7 As an example, if currency demand takes the

conventional log-log form with respect to RB, the assumption limRB!0RBm = 0amounts

to the condition that this demand be inelastic with respect to its opportunity cost (and similar reasoning applies to limRB R!0(RB R) x = 0).

2.2

The MUF approach

The purpose of this subsection is to establish the robustness of the results derived in the previous one concerning the general form of the expressions which should be used for

6_{As a counterpart to the subscript "U " used in (1), the subscript "M " used here stands for}

"multidimen-sional".

7_{Here "f g" means asymptotic equivalence, that is, f =g ! 1 in the limiting process considered. Lucas}

welfare calculations in economies with more than one type of monetary asset. We show that the use of an alternative setting (Sidrauski’s money-in-the-utility-function setting, rather than the shopping-time one) leads to welfare expressions similar to (5) and (6), but now extending the unidimensional measure given in Lucas (2000, section 3).

We incorporate growth in the model by normalizing y0 to 1 and making (as in Lucas

2000)

yt= e t, (7)

with > 0.

Also as in Lucas (2000), the utility function (here written in terms of the output shares c, m and x) is given by

U (c; m; x) =

c' G(m;x)_c

1

1 ,

where > 0_{, the twice-di¤erentiable function ' : [0; +1] ! [0; +1] is such that '}0 _{> 0}

and '00_{< 0 over 0; M (where M 2 (0; +1]) and constant thereafter, and}

G : [0; +_1]2 _{! [0; +1] is as in the previous subsection. Given (7), consumers maximize} Z +1 0 e (g (1 ) )tU (c; m; x)dt (8) subject to _ m + _x + _b = 1 c + h + (RB ( + )) b + (R ( + ))x ( + )m. (9)

Euler equations for this problem give 8 > < > : RB = U_Um_c RB R = U_Ux_c . (10)

De…ne the domains

D :=_{f(m; x) 2 [0; +1]}2 : G(m; x) M_g,

D :=_{f(m; x) 2 [0; +1]}2 : G(m; x) M_g

and

D := D _{\ D .}

The points d 2 D are those corresponding to the maximum available value for '. Call it ' := supm;x' (G (m; x)) = '(M ). Note that ' , as M , is allowed to equal in…nity.

Repeating the procedure used in Cysne (2003), we shall henceforth work with the welfare-cost function w de…ned as a function of the vector of monetary aggregates, rather than as a function of the interest rates. The initial condition reads w (d) = 0 for any d 2 D . We shall be speci…cally interested in paths : t_{7! (m; x) such that (0) = d 2 D} (note that each coordinate di of d can be equal to in…nity) and _ 0. The reason for

taking (0)_{2 D is that this is how one gets the lowest possible values for the opportunity} costs, making this the benchmark used for measuring the welfare cost of in‡ation.

Making c = 1, (10) becomes 8 > < > : RB = ' 0_(G(m;x)) '(G(m;x)) G(m;x)'0(G(m;x))Gm(m; x) =: m_{(m; x)} RB R = ' 0_(G(m;x)) '(G(m;x)) G(m;x)'0_(G(m;x))Gx(m; x) =: x(m; x) . (11)

Note that (11) gives the opportunity cost of holding each asset as a function of m and x_{. Call this function : D ! [0; +1]}2, with := ( m(m; x); x(m; x)). It is a

di¤erentiable function, given the twice-di¤erentiability of '.

Write down the equations de…ning Lucas’s measure of the welfare cost of in‡ation using the de…nition of ' :

for some d 2 D, or

(1 + w)' 1

1 + wG (m; x) = ' M = ' . Totally di¤erentiating this expression gives

' 1 1 + wG (m; x) 1 1 + wG (m; x) ' 0 1 1 + wG (m; x) dw+ '0 1 1 + wG (m; x) (Gm(m; x) dm + Gx(m; x) dx) = 0,

or, using (11) together with G’s homogeneity,

dw = m 1

1 + w(m; x)(m; x) dm +

x 1

1 + w(m; x)(m; x) dx . (12)

(12) is the di¤erential form of the Divisia index which gives the welfare costs of in‡ation for this economy. Obviously, as (m; x) approaches D, w(m; x) ! 0, and

m _{(1 + w(m; x))} 1

(m; x) m(m; x) and x (1 + w(m; x)) 1(m; x) x(m; x). Therefore, from (11) and (6), dw dBM and w BM, as shown for s in the previous

subsection. In words (and again, as shown concerning the unidimensional case in Lucas 2000), as w gets closer and closer to zero, the welfare measure which emerges from the multidimensional money-in-the-utility-function approach leads to (6), the generalization of Bailey’s formula to a multidimensional setting.

The main conclusion to be drawn from this section is that when either the

shopping-time or the Sidrauski setting is considered, (5) or (12) are the correct welfare measures to be used for households holding currency, bank deposits and bonds, rather than (1). And that, when reasonable interest rates are considered, (6) can be used as a good approximation to both.8

8_{It can even be shown that, irrespective of interest rates, one always gets s < B < w (see Cysne and}

3

The sign of the bias

From the previous section we conclude that, in economies with currency, interest-bearing deposits and bonds, at least three alternative formulas can be used to calculate the welfare costs of in‡ation: (5), (6) or (12). In such economies, the use of Bailey’s (1956) or Lucas’s (2000) unidimensional formulas may be misleading. Even when RB R is independent of

in‡ation, as considered to be the case in Attanasio et al. (2002), the use of unidimensional measures overlooks the fact that the e¤ect of changes in the nominal interest rate RB on

the demand for bank deposits also has to be taken into consideration in the welfare calculations (since such deposits also provide monetary services).

Indeed, by assuming RB R = K constant in the previous multidimensional formulas,

and taking into consideration that RB K (since R 0), one obtains the following

particular cases:9 s(RB) := s RB; K = Z RB K (1 s ( )) m0_{( ) + Kx}0_{( )} 1 s ( ) + m( ; K) + Kx( ; K)d (13) when the shopping-time model is taken as reference;

BM(RB) := BM RB; K =

Z RB K

m0( ) + Kx0( ) d (14)

when a multidimensional Bailey-like approximation is taken as reference; and

w (RB) := w m(RB; K); x(RB; K) = Z RB K 2 6 4 m 1 1+w( ) m( ; K); x( ; K) m0( )+ x 1 1+w( ) m( ; K); x( ; K) x0( ) 3 7 5 d (15)

when a money-in-the-utility-function approach is taken as reference.

9_{For notational ease only, we write m}0_{( ) and x}0_{( ) for (@m=@R}

B) ; K and (@x=@RB) ; K ,

There are three conceptual di¤erences between (1) and any one of formulas (13), (14) and (15).

First, since R 0, the dummy variable in the formulas above should run from K to RB, rather than from 0 to RB. Note, though, that K can be equal to zero.

Second, one has to take into consideration the e¤ects of changes in RB on the demand

for bank deposits (x).

Third, and in general as well, all functions should be written as depending both on RB

and RB R, rather than on one variable only.

Note that (13), (14) and (15) can be split into two Riemann integrals, the second one based on x0_{( ) (evaluated for a …xed value of R}

B R equal to K), whereas (1) cannot.

Since (13), (14) and (15) are asymptotically equivalent (for small interest rates or large amounts of monies demanded; and under the assumptions at the end of Section 2.1), a comparison between (1) and any one of these three measures also conveys relevant

information about the di¤erence between (1) and the other two. For simplicity, we shall in the remainder of this section compare (1) with its multidimensional extension (14)

(accordingly, the bias will be de…ned in the next section as the absolute value of the relative di¤erence between these two formulas).

(14) will lead to welfare …gures less than or greater than (1), depending if x0_{( ), the}

cross derivative of the demand for interest-bearing deposits with respect to the opportunity cost of holding currency, is, respectively, positive or negative. In the former case, currency and interest-bearing deposits are said to be substitutes; in the latter case, to be

complementary to each other. It is therefore important to study the sign of x0_{( ) which}

may emerge from the shopping-time or the MUF model.

From this point on, we work this out in the context of the MUF model. This choice bears on generality reasons, since it can be shown that money demand speci…cations derived from the shopping-time model can also be obtained through the Sidrauski model, or an adequate extension including interest-bearing deposits (Cysne and Turchick 2009).

Totally di¤erentiating the equilibrium equations (11) yields: 8 > < > : dRB = mmdm + mxdx d (RB R) = mxdm + xxdx , where ij = 1 ' G'0 ' 0_G ij + ''00 ' G'0GiGj

and the functions G, ', '0, '00 are calculated at the equilibrium. Inversion of this system gives:

8 > < > : dm = 1 mm xx 2mx[ xxdRB mxd (RB R)] dx = 1 mm xx 2mx [ mxdRB+ mmd (RB R)] ,

implying the equilibrium derivatives 8 > < > : @m @RB = xx mm xx 2mx @x @RB = mx mm xx 2mx . (16)

This can be rewritten as (see Appendix A for details)

m0(RB) = mx (' G'0₎ G2_G mx''0'00 G2_x''00+ Gxx'0(' G'0) < 0 (17) and x0(RB) = mx (' G'0₎ G2_G mx''0'00 (GmGx''00+ Gmx'0(' G'0)). (18)

From the subgradient inequality applied to the concave function ', we know that ' (G) G'0_{(G) > 0 if G > 0. Since G}

m is 0-degree homogeneous, Euler’s formula gives, for

any (m; x) 2 R2

++, 0 = Gmm(m; x) m + Gmx(m; x) x, so that Gmx > 0. This explains the

3.1

Results under general

G and ' functions

We now look for a su¢ cient condition regarding functions G and ' so that BU > BM,

irrespective of interest rates. By comparing (1) and (14), one easily concludes that a su¢ cient condition for BU > BM is that currency and interest-bearing deposits are

substitutes to each other (x0 _{> 0).}10 _{Indeed, for any R}

B, we have BU(RB) BM(RB) = K Z RB K x0( ) d . (19) Therefore,

BU > BM if, and only if, x (RB) > x K ;8RB > K.

This leads to Proposition 1.

Proposition 1 Consider an economy with currency, interest-bearing bank deposits and bonds, where the spread on bank deposits can be considered constant. Then a su¢ cient condition for Bailey’s unidimensional measure of the welfare costs of in‡ation to overestimate (underestimate) the true welfare costs of in‡ation is that the following expression is negative (positive):

GmGx''00+ Gmx'0(' G'0) , (20)

where G, Gm, Gx and Gmx are evaluated at each pair (m; x) 2 [0; +1]2, and ', '0 and '00

are evaluated at G (m; x).

Proof. This comes directly from (18) and (19).

The condition given in the above proposition, which makes x0_(R

B) > 0, can also be

written in terms of elasticities in the following way. Let "m

RB and " x

RB be the elasticities of, 10_{Following the explanation above, in this economy an adaptation in Bailey’s formula (1) must take place:}

respectively, cash and bank-deposits demands, with respect to the nominal interest rate paid by government bonds. Let represent the elasticity of substitution between cash and bank deposits, all three elasticities being evaluated at the same point. That is,

"m_RB = d log m d log RB , "x_R B = d log x d log RB , and = d log (x=m) d log (RB= (RB R)) = d log (x=m) d log (Gm=Gx) . Since RB R is constant, we have

= d log (x=m) d log RB log K = d (log x log m) d log RB = "x_R B " m RB. (21) Therefore, "x

RB > 0 if, and only if, + " m RB > 0, that is, " m RB < (since " m RB < 0, by

(17)). This leads to Proposition 2.

Proposition 2 Consider an economy with currency, interest-bearing bank deposits and bonds, where the spread on bank deposits can be considered constant. Then Bailey’s

unidimensional measure of the welfare costs of in‡ation overestimates the true welfare costs when the elasticity of demand for currency is (in absolute value) always lower than the elasticity of substitution between currency and bank deposits:

"m_R

B < . (22)

Conversely, an underestimation occurs when the elasticity of demand for currency is (in absolute value) always greater than that elasticity of substitution:

"m_R

Proof. Done.

Remark 1 (The Cobb-Douglas case, = 1.) Note that "m_RB 1 is unusual in theoretical monetary models. In models with constant real interest rates, like the two presented in Section 2, it would make the in‡ation tax negatively correlated with in‡ation, leading to potential instability. Lucas (2000), for instance, presents welfare formulas which are well de…ned only when "m

RB < 1.

11 _{This implies that in the Cobb-Douglas case one should}

expect, from Proposition 2, an overestimation of the welfare costs of in‡ation when Lucas’s or Bailey’s unidimensional formulas are used.

3.2

Results under a CES

G function and a general ' function

Note that condition (20) establishes a property which depends on both functions G and '. In order to get further results and insights, this subsection still assumes a general ', but particularizes to the case in which the aggregator function G has a constant elasticity of substitution : Assumption G. G (m; x) = 8 > < > : m 1 + (1 ) x 1 1 , if > 0 and _{6= 1} m x1 _{, if = 1} , where 2 (0; 1).

Proposition 3 gives a su¢ cient condition for overestimation related to ' only, provided the elasticity of substitution between m and x is high enough.

Proposition 3 Consider an economy with currency, interest-bearing bank deposits and bonds, where the spread on bank deposits can be considered constant, and Assumption G is valid. Then, a su¢ cient condition for Bailey’s unidimensional measure of the welfare costs of in‡ation to overestimate (underestimate) the true welfare costs is that, 8G 2 0; M , the

following expression is positive (negative):

+ '

0_{(' G'}0₎

G''00 , (24)

where ', '0 _{and '}00 _{are evaluated at G.}

Proof. Using the expressions obtained in Appendix B for the partial derivatives of G, one has sgn (GmGx''00+ Gmx'0(' G'0)) = sgn G m 1 (1 ) G x 1 ''00+ (1 ) G 1+2 m1x1 '0(' G'0) ! = sgn ''00+ ' 0_{(' G'}0₎ G = sgn + '0_{(' G'}0₎ G''00 .

Proposition 1 …nishes the job.

Remark 2 The sign of the expression in (24) coincides with that of

(1 ) G'00_{=' (G'}0_=')0_{. Thus, in the = 1 case (that is, if G is simply a}

Cobb-Douglas-type aggregator function), BU ? BM if (G'0=')0 7 0.

At this point it is natural to ask what condition would be necessary so that BU = BM.

Proposition 4 below takes care of that.

Proposition 4 Consider an economy with currency, interest-bearing bank deposits and bonds, where the spread on bank deposits can be considered constant, and Assumption G is valid. Then BU = BM if, and only if, the expression in (24) equals zero. Moreover, in this

case, if = 1 (the Cobb-Douglas G case), the demand for m and for x depend only on their respective user costs, as in12

8 > < > : m = _{1 R} B x = _{1 R}1 B R .

Proof. From (19), BU and BM are identical if and only if x0 is identically null, which,

from (18), happens if and only if the expression in (20) is identically null. Or still, from the math done in the proof of Proposition 3, + '0_{(' G'}0_{) = (G''}00_{) = 0.}

In particular, if = 1, Remark 2 gives (G'0_=')0 _{= 0, whence ' is such that}

' (G) = G , for some > 0 and _{2 (0; 1). Given (11), this leads to} 8 > < > : RB = _{1 m} RB R = ₁ 1_x .

3.3

Results under a CES

G function and a CES ' function

For de…niteness, we impose, besides Assumption G, a functional form on ' implying the general CES-type expression for the utility function:

U (c; m; x) = c 1 + (1 ) 1G (m; x) 1 1 1 1 . That is, Assumption '. ' (G) = 8 > < > : + (1 ) ( G) 1 1 , if > 0 and _{6= 1} ( G)1 , if = 1 , where 2 (0; 1) and > 0.

Here M = +1, and properties '0 _{> 0 and '}00 _{< 0 are straightforward. We then obtain}

the following very simple test:

Proposition 5 Consider an economy with currency, interest-bearing bank deposits and bonds, where the spread on bank deposits can be considered constant, and Assumptions G and ' are valid. Then BU R BM if (and only if ) R . In words, BU overestimates

a greater (lesser) degree of substitutability than that between the consumption good and the monetary aggregate.

Proof. As shown in Appendix C, (24) becomes . Thus Propositions 3 and 4 provide the result.

The relation between Propositions and is established by equation (29), to be derived in the next section.

Under the additional Assumption ', since (24) depends on parameters only, either BU

always overestimates BM, or always underestimates BM, or always equals it.

4

Gauging the bias

The last section presented a simple reason why unidimensional calculations of the welfare costs of in‡ation may be biased. We now shift our concern from the direction of the bias to its size. That is, we are interested in assessing the measure

(RB) := BU(RB) BM(RB) BM(RB) = K x (RB) x K BM (RB) , (25)

the error one incurs when taking BU instead of BM. The second equal sign is derived from

(19). Evidently, in the extreme K = 0 case, in which bonds and deposits are perfect substitutes, (1) and (14) would yield BM(RB) = BU(RB), so that = 0. From now on,

K > 0is assumed. Assumptions G and ' are also supposed to hold. For the reader’s sake, auxiliary expressions relative to the _{6= 1 case are left to the appendices.}

As shown in Appendix C, given Assumptions G and ', the demand system (11) in the = 1 case becomes, after inversion (regardless of whether is equal to unity or not),

8 > < > : m = 1 1 _R 1 B 1+( 1) (1 ) (RB R) 1 ( 1)(1 ) x = 1 1 _R 1 B ( 1) (1 ) (RB R) 1 1+( 1)(1 ) , (26)

the bidimensional log-log money-demand speci…cation.

Putting RB R = K, we get m and x as functions of RB only. Formula (1) (with the

lower limit of the integral adjusted to K as before) gives (see Appendix D)

BU(RB) = 8 > > > > > > > < > > > > > > > : 1 _{K R} B+ lnR_KB , if = 1 1 1 (1 ) 1 (1 ) 1 1 K(1 )(1 ) _R B(1 ) K(1 ) , if _{6= 1} . From (14), we have BM(RB) := BM RB; K = BU(RB) K RRB K x0( )d = BU(RB) K x (RB) x K , whence BM(RB) = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : BU(RB) 1 (1 ) (1 1) = 1 K RB+ lnR_KB , if = 1 BU(RB) 1 1 ( 1) (1 ) 1+( 1)(1 ) K(1 )(1 ) _R(1 ) B K (1 ) = 1 ₁ (1 )1 1 K(1 )(1 ) _R B(1 ) K(1 ) , if _{6= 1} . (27) Therefore, we get

Proposition 6 Consider an economy with currency, interest-bearing bank deposits and bonds, where the spread on bank deposits can be considered a positive constant, and

Assumptions G and ' are valid. Then (RB) = 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : 0, if = 1 and = 1 (1 )(1 ) , if = 1 and _{6= 1} (1 ) (1 ) K1 ( R 1 B +(1 ) K 1 ₎ 1 _(K1 _{( +(1 ) ))} 1 ln( R1_B +(1 ) K1 _{) ln(K}1 _{( +(1 ) ))} , if _{6= 1} and = 1 (1 )(1 ) K1 ( R_B1 +(1 ) K1 ₎1 _{K ( +(1 ) )}1 ( R1_B +(1 ) K1 ₎ 1 1 _K1 _{( +(1 ) )} 1 1 , if 6= 1 and _{6= 1} . (28) Proof. It’s just a matter of noting that the amount K x (RB) x K was

conveniently left in explicit form in the calculation of BM above (as well as in Appendix

D), and then applying the second expression for in (25).

It may be noted that, in any of the four cases presented in Proposition 6, neither nor (the parameters associated with the share of importance the individual gives to

consumption of the good vis-à-vis the money aggregate) have any e¤ect on this bias. Obviously, the absolute-value bars may be dispensed with when BU > BM. From

Proposition 2, we know this will be the case when "m

RB < . From (26) and RB R = K,

"m_RB = "m_RB = @ log m @ log RB

= 1 + ( 1) = + (1 )

when = 1. As calculated in Appendix C, also in the _{6= 1 case one obtains for "}m RB a

weighted average between and :

"m_RB = R

1

B + (1 ) K

1

R1_B + (1 ) K1 . (29)

Thus R if and only if R "m

Proposition 5.

The two examples ahead illustrate the two main possibilities, overestimation and underestimation, of the welfare cost of in‡ation, when interest-bearing deposits are not taken into account13_{. For simplicity, they use the Cobb-Douglas G case ( = 1), where}

neither RB nor K a¤ect the bias. For K, we use the value 0:0242.14

Example 1 = 1, = 0:5, = 0:2, K = 0:0242, and any ( ; )_{2 (0; 1) R}++ making

BU(0:1) = 0:016 (as in Lucas 2000, p. 258).15 Formula 26 gives, in this case,

(m; x) 0:062R_B0:6; 2:557R0:4_B . Propositions 5 and 6 imply that, in this case, BU would

overestimate the welfare cost of in‡ation in =_j(1 0:2) (1 0:5) =0:2_{j = 200%.}

That is, while for yearly interest rates of 10%, BU is at 1:6% of GDP, the measure BM

leads to just 0:53% of GDP, a threefold di¤erence.

13

Although, from a theoretical point of view, the partition we propose here is between M 1 and M 2nM1 ; from this point on, when using monetary data (all of it related to the U.S. economy), we will not distinguish between M and M : This is to say that we will associate m with the available de…nition of M 1 (disregarding the possible existence of interest-bearing deposits in it, such as the NOW accounts) and x with M 2 n M1, both as fractions of the U.S. GDP (M 1 was entirely noninterest-bearing only before 1980). We do that for practical reasons only.

14_{Considering the 6 1/2 year-period from 01/2003 to 06/2009, the yearly average interest rate on the}

ten-year constant-maturity Treasury Bill was 4:23%, whereas that on M 2 n M1 was approximately 1:53%. The relative share of M 2nM1 in M2 was around 0:81. The average spread and the unconditional standard deviation (both calculated using monthly data) were 2:42% and 0:70%, respectively (data available from the Federal Reserve Bank of St. Louis - FRED).

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03

nominal interest rate R

B w el far e cos t of inf lat ion B U B M

Although our purpose here is more focused at providing an illustration of the application of our theoretical results than empirical estimates of the bias, a word of comparison with previous investigations of this same issue may be valuable.

Bali (2000) concludes, akin to the numbers above, that assuming M 1 is noninterest bearing leads to an overestimate of the welfare cost by a factor of three. Bali’s result, though, is not comparable with ours, for he assumes a utility function separable in currency and deposits, equivalent to making _{! +1.}

The overestimation obtained by Bali is a consequence of assuming that the user cost of interest-bearing deposits is proportional to the nominal interest rate (determined by a competitive banking system operating under constant reserve requirements). Under such an assumption it is easy to show that the welfare costs are proportional to the real value of the monetary aggregate used in the calculations.16 _{Since the historical ratio of M 1 to the}

monetary base is equal to three, an overestimation by a factor of three emerges.

We present below an example where the use of Bailey’s welfare formula leads to an

underestimation, rather than an overestimation, of the true welfare costs of in‡ation (BU < BM). As noted in Remark 1, because it implies (when G is Cobb-Douglas) an

elasticity of demand for currency greater than one, this case can be considered to be less likely to hold in the real world.

Example 2 = 1, = 0:5, = 2, K = 0:0242, and any ( ; )_{2 (0; 1) R}++ making

BU(0:1) = 0:016 (as in Lucas 2000, p. 258).17 Formula 26 gives, in this case,

(m; x) 1:633 10 3_R 1:5

B ; 6:748 10 2R 0:5

B . Propositions 5 and 6 imply that, in this

case, BU would underestimate the welfare cost of in‡ation in =j(1 2) (1 0:5) =2j =

25%. That is, while for yearly interest rates of 10%, BU is at 1:6% of GDP, the measure

BM leads to 2:13% of GDP. 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03

nominal interest rate R

B w el far e cos t of inf lat ion B U B M

As previously mentioned, setting = 1 in Examples 1 and 2 was arbitrary, done for illustrative reasons only. Starting with Chetty (1969), empirical assessments of the U.S. elasticity of substitution between noninterest-bearing and interest-bearing monies found in

the literature have varied in a very wide range. First, because in the last 30 years …nancial systems all over the world have gone through considerable technological innovations. Second, di¤erent de…nitions of M 1 and M 2nM1 (which we are taking in this Section as proxies for M 1 and M 2nM1 ) have been used in this period18_{. Third, on account of}

di¤erent model and econometric speci…cations and methods.

Values reported for this parameter have been as low as 0:024 (Husted and Rush, 1984, p. 179) and as high as 30:864 (Chetty, 1969, p. 278). Edwards (1972, p. 566) reports 2:417; Boughton (1981, p. 383), 4:63; and Gauger (1992, p. 250), 0:1400. Estimates for this same elasticity using post-80’s U.S. data include 9:73 (Sims et al., 1987, p. 123); 0:0981 (Gauger, 1992, p. 251); 1:691 (Fisher, 1992, p. 150); and 1= (1 0:269) 1:368 (Poterba and Rotemberg, 1987, p. 229).

Figure 3 uses the same set of parameters used in Example 1 (except ) and depicts the behavior of the bias function for _{2 f0:02; 0:05; 0:1; 0:2; 0:5; 1; 2; 5; 10; 20; 50g.}

0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 bias 0.05 0.1 0.15 0.2 0 20 40 60 80 100 120 140 160 180

nominal interest rate R

B ξ=1 ξ=0.5 ξ=0.02 ξ=0.05 ξ=0.1 ξ=0.2 ξ=5 ξ=10 ξ=20 ξ=50 ξ=1 ξ=2

The 200% (or 2) bias corresponding to the Cobb-Douglas-G case, already calculated in

18_{Not to mention the fact that some of the measurements of the elasticity of substitution found in the}

Example 1, is repeated in Figure 3. As anticipated in Proposition 5, when = 0:2 (only because we have set = 0:2) there is no bias at all. The …gure suggests that in the

low- /left-side (high- /right-side) case a larger interest rate tends to reduce (increase) the relative di¤erence between these two measures.

Although it is not our purpose here to make a sensitivity analysis regarding , it is clear from Figure 3 that the bias is usually of a relevant magnitude. Consider, for instance, a 10% yearly interest rate. Then taking = 0:02 leads to a BU about 28% lower than BM,

whereas with = 2 BU turns out to be about 585% higher than BM.

5

Conclusions

Several papers in the literature on the welfare costs of in‡ation use the one-dimensional formulas provided by Bailey (1956) and by Lucas (2000), in which interest-bearing monies are not explicitly considered. This procedure (as acknowledged in Lucas 2000) is at odds with the huge process of …nancial innovations which has occurred in several economies since the 80’s, and which has led to the widespread use of di¤erent interest-bearing monies.

This paper has built upon the literature on the welfare costs of in‡ation with di¤erent monies to provide the sign and a measure of the error when interest-bearing monies are not explicitly introduced in the calculations.

Specializing to the case in which the user cost of the interest-bearing asset is constant, we have provided conditions under which the use of unidimensional formulas can lead to overestimation or underestimation of the true welfare costs. The general results assume only a homothetic utility function. The discussion was then further specialized to a CES utility, including the case in which the money demand has a bilogarithmic structure. We have used the bilogarithmic money demand to illustrate with reasonable parameter values that Bailey’s and Lucas’s measures of the welfare costs of in‡ation may easily overestimate the true welfare costs of in‡ation by a factor as high as 3.

Underestimation may also occur, but only if the interest-rate elasticity of M 1 is, in absolute value, su¢ ciently high (as stipulated by Proposition 2). For instance, in the Cobb-Douglas G case, leading to the bilogarithmic money demand (26), it would have to be greater than one, an assumption usually not supported by empirical evaluations. The intuition for the overestimation implied by the use of unidimensional formulas is clear: such measures capture the welfare costs caused by in‡ation due to a decrease in the equilibrium money demand. However, they ignore the fact that the existence of

interest-bearing monetary assets in the economy may partially o¤set the drop in the real equilibrium value of M 1, by these means leading to welfare costs that may be lower than those calculated by such formulas.

Finally, we have also depicted the behavior of the relative bias between Bailey’s uni- and multidimensional welfare measures as a function of the interest rate for several values of the elasticity of substitution between monetary aggregates found in the literature. Although highly dependent on the value of this elasticity, the bias is usually of a relevant magnitude.

Acknowledgement

Appendix A

Here we derive (17) and (18) from (16). Firstly, note that

mm xx 2mx = 1 ' G'0 2 2 6 4 '0_G mm+ '' 00 ' G'0G2m '0Gxx+ '' 00 ' G'0G2x '0_G mx+ '' 00 ' G'0GmGx 2 3 7 5 = 1 ' G'0 2 2 6 4 ' 02_(G mmGxx G2mx) + ''0'00 ' G'0 (GmmG2x+ GxxG2m 2GmxGmGx) 3 7 5 .

From Gm’s and Gx’s 0-homogeneity, Euler’s relation for homogeneous functions gives

8 > < > : 0 = mGmm+ xGmx 0 = mGmx+ xGxx ,

so that Gmx = mGmm=xand Gmx = xGxx=m, and GmmGxx G2mx = 0.

Also, GmmGx GmxGm = Gmx( xGx=m Gm) = (Gmx=m) (mGm+ xGx) =

GGmx=m, where the last equality comes again from Euler’s relation. Similarly,

GxxGm GmxGx = GGmx=x, so that GmmG2x+ GxxG2m 2GmxGmGx = GxGGmx=m

GmGGmx=x = G2Gmx= (mx), where once again Euler’s relation is used in the last step.

Thus mm xx 2mx = G2Gmx mx ''0'00 (' G'0₎3

which, together with (16), gives (17) and (18).

Appendix B

Case _{6= 1: G (m; x) =} m 1 + (1 ) x 1 1 _{) G}m = m 1 m 1 + (1 ) x 1 1 1 = (G=m)1 and, analogously, Gx = (1 ) (G=x) 1 . Case = 1: G (m; x) = m x1 _{) G}m = m 1x1 = G=m = (G=m) 1 and Gx = (1 ) (G=x) 1 .

So one can continue with the calculations from this point on as if it were a single case:

Gmm = G m 1 ₁ Gmm G m2 = G m 1 ₁ Gxx m2 = (1 ) G 1+2 m1+1x 1+1 ; Gxx = (1 ) G 1+2 m 1+1x1+1 ; Gmx = G m 1 ₁ Gx m = (1 ) G 1+2 m1x1 . Appendix C

Our objective here is to show how the money-demand speci…cation (26) arises from (11), Assumptions G and ', as well as evaluating expression (24) under these assumptions.

From ' (G) = 8 > < > : + (1 ) ( G) 1 1 , if > 0 and _{6= 1} ( G)1 , if = 1 , we get '0(G) = 8 > > > > > > > > < > > > > > > > > : + (1 ) ( G) 1 1 1 (1 ) 1G 1 = (1 ) 1G 1' (G)1 , if > 0 and _{6= 1} (1 ) 1 _{G = (1 ) G} 1_{' (G), if = 1} = (1 ) 1 ' (G) G 1

and ' (G) G'0(G) = 8 > > > > > > > < > > > > > > > : ' (G)1 ' (G) 1 G (1 ) 1G 1 = ' (G)1 , if > 0 and _{6= 1} ' (G) G (1 ) 1 G = ' (G), if = 1 = ' (G)1 . Then (11) gives 8 > > < > > : RB = (1 ) 1('(G)_G ) 1 '(G)1 Gm = 1 1 G1 1m 1 RB R = (1 ) 1('(G)_G ) 1 '(G)1 Gx = (1 ) 1 1 G1 1x 1 , (C1)

which can be inverted (…rst in the more general _{6= 1 case) by noting that}

RB1 + (1 ) (RB R)1 = 1 1 1 G(1 )(1 1) m 1 + (1 ) x 1 = 1 1 1 G 1, whence G = 1 1 RB1 + (1 ) (RB R)1 1 . Plugging this into (C1) yields the demand system

8 > < > : m = 1 1 _R 1 B R 1 B + (1 ) (RB R)1 1 x = 1 1 (1 ) (RB R) 1 R_B1 + (1 ) (RB R) 1 1 . (C2)

For the = 1 case, (C1) reads 8 > < > : RB = 1 1 m 1 1x(1 ) 1 RB R = (1 )1 1 m 1x(1 ) 1 1 , readily implying (26).

We also have, from the expression above for '0(G),

'00(G) = 1 (1 ) 1 ' (G) G 1 ₁ '0_{(G) G ' (G)} G2 = 1 1 ' (G) G 1 ₁ ' (G)1 G2 = (1 ) 1' (G) 2 ₁ G1+1 , so that '0_{(' G'}0₎ G''00 = (1 ) 1 '(G)_G 1 ' (G)1 G' (G) (1 ) 1'(G) 2 ₁ G1+ 1 = ,

and the expression in (24) equals . Finally, for _{6= 1, (C2) gives}

"m_RB = @ log m @ log RB = @ log 1 1 _{log R} B+ ₁ log RB1 + (1 ) K1 @ log RB = + 1 (1 ) R_B R1_B + (1 ) K1 dRB d log RB = + ( ) R 1 B R1_B + (1 ) K1 ,

Appendix D

Here we calculate the welfare cost measures BU and BM. We are still under

Assumptions G and '. Case = 1 and = 1: BU(RB) = Z RB K dm ( ) = m ( )_jRB K + Z RB K m ( ) d = 1 1_jRB K + Z RB K 1 d = 1 K RB+ ln RB K . BM is shown in (27). Case = 1 and _{6= 1:} BU(RB) = Z RB K dm ( ) = m ( )_jRB K + Z RB K m ( ) d = 1 1 1+( 1) (1 ) K 1 ( 1)(1 ) (1 ) RB K + Z RB K 1+(1 ) _d = 1 1 1+( 1) (1 ) K 1 ( 1)(1 ) " (1 ) ₊ (1 ) (1 ) RB K = 1 1 (1 ) 1 (1 ) 1 1 K(1 )(1 ) RB(1 ) K(1 ) . Again, BM is shown in (27).

Case _{6= 1 and} = 1: BU(RB) = Z RB K dm ( ) = m ( )_jRB K + Z RB K m ( ) d = 1 2 6 6 4 1 1 _{+ (1 ) K}1 1 R B K + RRB K 1 _{+ (1 ) K}1 1_d 3 7 7 5 = 1 2 6 4 1 1 + (1 ) K1 1₊ 1 (1 )ln 1 _{+(1 ) K}1 RB K = 1 2 6 4 1 +(1 ) R1_B RB1 +(1 ) K1 + 1 (1 )ln R_B1 +(1 ) K1 K1 _{( +(1 ) )} 3 7 5 . Using BM(RB) = BU(RB) K x (RB) x K : BM(RB) = BU(RB) 1 (1 ) K1 " 1 R1_B + (1 ) K1 K 1 + (1 ) # = 1 1 1 ln R1_B + (1 ) K1 K1 _{+ (1 )} .

Case _{6= 1 and 6= 1:} BU(RB) = Z RB K dm ( ) = m ( )_jRB K + Z RB K m ( ) d = 1 1 2 6 6 4 1 1 _{+ (1 ) K}1 1 RB K + RRB K 1 + (1 ) K1 1 d 3 7 7 5 = 1 1 2 6 6 4 1 1 + (1 ) K1 1 + 1 (1 ) 1 _{+ (1 ) K}1 1 1 RB K = 1 1 2 6 4 1 + 1 +(1₁ ) K1 1 _{+ (1 ) K}1 1 RB K = 1 1 1 1 + (1 ) K1 1 + (1 ) K1 1 RB K = 1 1 1 2 6 6 6 6 6 4 R1_B + (1 ) K1 R1_B + (1 ) K1 1 K1 + (1 ) + (1 ) 1 3 7 7 7 7 7 5 . Again using BM (RB) = BU(RB) K x (RB) x K : BM (RB) = BU(RB) 1 ₁ (1 ) K1 2 6 4 R_B1 + (1 ) K1 1 K + (1 ) 1 3 7 5 = 1 1 1 2 6 4 R1_B + (1 ) K1 1 1 K1 _{+ (1 )} 1 1 3 7 5 .

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Figure legends

Figure 1. Overestimation of the welfare costs of in‡ation due to disregarding other monetary assets.

Figure 2. Underestimation of the welfare costs of in‡ation due to disregarding other monetary assets.

Figure 3. Bias function for several values of the elasticity of substitution between m and x.