©International Monetary Fund. Not for Redistribution€¦ · A reduction in the costs of inflation may induce an increase in the rate of inflation. This increase in inflation in

IMF WORKING PAPER

© 1991 International Monetary Fund

This is u working paper and the author would welcome anycomments on the present text. Citations should refer to anunpublished manuscript, mentioning the author and thedate of issuance by the International Monetary Fund. Theviews expressed are those of the author and do not neces-sarily represent those of the Fund.

WP/91/1 INTERNATIONAL MONETARY FUND

Research Department

Welfare Costs of Inflation. Seigniorage, and Financial Innovation

Prepared by Jose De Gregorio I/

Authorized for Distribution by Peter Wickham

January 1991

•Abstract

This paper examines the welfare effects of mitigating the costs ofinflation. In a simple model where money reduces transaction costs, a fallin the costs of inflation is equivalent to financial innovation. This canbe caused by paying interest on deposits, indexing money, or "dollarizing."Results indicate that financial innovation raises welfare in low inflationeconomies while reducing it in high inflation economies, due to theoffsetting indirect effect of higher inflation to finance the budget.

JEL Classification Number:310, 320

I/ This is a revised version of the first essay of my Ph.D. dissertationat MIT. I am grateful to Rudi Dornbusch, Pablo Guidotti, FedericoSturzenegger, and Peter Wickham for valuable comments. I also thankChristophe Chamley, Larry Kotlikoff, Jeff Miron, Carlos Vegh, andparticipants at seminars at Boston University, MIT, The World Bank, UCLA,and UC-San Diego for helpful discussions. Any remaining errors are my own.

©International Monetary Fund. Not for Redistribution

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Contents Page

I. Introduction 1

II. A Simple Monetary Model 2

III. Welfare Effects of Financial Innovation 5

IV. Financial Innovation with a Banking System 12

V. Endogenous Government Spending and Optimal Taxation 16

VI. Concluding Remarks 21

References 24

Appendix: The Optimal Tax Problem 27


I. Introduction

A reduction in the costs of inflation may induce an increase in the rateof inflation. This increase in inflation in turn may be large enough toreduce welfare. The issue of reducing the social costs of inflation wasrecently addressed by Fischer and Summers (1989). Their model is based on amonetary policy game, following Barro and Gordon (1983) model. In thesemodels, inflation arises because government cannot commit to the optimal rateof inflation (assumed to be zero) and tries to exploit a Phillips curverelationship. In its simplest form, reducing the costs of inflation willreduce welfare because of the more than offsetting increase in theequilibrium rate of inflation.

Fischer and Summers (1989) have extended the model to consider imperfectcontrol of inflation. They show that mitigating the costs of inflation maybe desirable only in high inflation economies, since in low inflationeconomies reducing the costs of inflation may lower welfare.

In this paper, the problem of reducing the costs of inflation is studiedin the context of a simple monetary model. In the model, inflation arisesbecause of the need to finance government spending rather than as a result oftrying to exploit any short-run trade off between output and unexpectedinflation. This approach is particularly relevant to the analysis of theexperience of high inflation, where the evidence shows that the underlyingcause is the existence of a large fiscal deficit. I/ Thus, seigniorageconsiderations play a crucial role in determining the welfare effects ofinflation mitigation.

A monetary model is developed in which money is introduced through atransactions technology, and an exogenous constant flow of governmentspending is assumed. Only one case of reducing the costs of inflation isconsidered, namely, decreasing the necessity of people to hold money. Thereare several potential ways of achieving this: financial deepening, payinginterest on deposits, lowering the reserve-deposit ratio, indexing money orallowing foreign exchange denominated deposits, among others.

In all of the above cases, the shift in money demand will require anincrease in inflation to finance the deficit. The rise in inflation willproduce a further reduction in real balances and results opposite to those ofFischer and Summers (1989) will be shown to hold, i.e., at low levels ofinflation a reduction in the social costs of inflation is more likely toincrease welfare. In the extreme case where seigniorage equals zero,reducing the costs of inflation will allow people to economize on real moneybalances without affecting the rate of inflation (zero in this case). At theother extreme, when inflation is close to the revenue maximizing rate ofinflation, the resulting shift in money demand will reduce welfare. Thelarger the interest rate elasticity (in absolute value) of money demand, thelarger will be the required increase in inflation to offset a fall in realbalances. Provided that the rate of inflation is not on the "wrong side of

I/ For example, Sargent (1983), Helpman and Leiderman (1988), Liviatan andKiguel (1988) and Dornbusch and Edwards (1989).


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the Laffer curve", the elasticity is largest at the maximum level ofseigniorage.

The transactions technology assumed in this paper allows to interpretreductions in the welfare costs of inflation as financial innovation. Thus,the purpose of this paper can be reinterpreted as an exploration of thewelfare effects of financial innovation in economies that are subject toseigniorage constraints. Therefore, the main result can be reformulated asfollows: financial innovation will increase welfare in low inflationeconomies, and will reduce welfare in high inflation economies, because ofthe offsetting indirect effect of financing the budget.

The paper is organized in six sections. Section II presents a simplemonetary model where the only asset and medium of exchange is money. Peopledemand money because it facilitates transactions. So, money reducestransaction costs. In Section III the question of a reduction in requiredmoney holdings, which is related 1:1 to the welfare costs of inflation, isanalyzed. Section III also contains the main results of the paper: given therate of inflation, a reduction in required money holdings is welfareimproving. Once the indirect effect on inflation is added the result isambiguous. The direction of the welfare change depends, however, on thelevel of inflation. For low inflation countries the reduction in the socialcosts of inflation will increase welfare while at high inflation rates, andconsequently high seigniorage, welfare will decrease.

Section IV extends the model to the case where deposits are alsoavailable for making transactions. Deposits are assumed to pay a fixedinterest rate and are imperfect substitutes of money. Analogous to SectionIII, it is shown that increasing the interest paid on deposits at high ratesof inflation reduces welfare, and increases welfare at low rates ofinflation. If the interest rate were determined in a competitive bankingsystem and a reduction in reserve requirements would occur, the same resultswould hold.

The main part of the paper treats government spending as exogenous andinflation as being the only source of revenue. In Section V, however, bothassumptions are relaxed. First, it is assumed that government spendingconsists of providing a public good, which is optimally chosen. It is shownthat the results obtained for a the case of an exogenous government spendingare still valid. Next, Section V extends the model to an optimal taxationscheme, in which two taxable goods are assumed: money and consumption goods.It is shown that, with some qualifications, the main results from theprevious sections still hold. Finally, Section VI provides the conclusionsand discusses other possible extensions of the results.

II. A Simple Monetary Model

The economy is populated by a constant number of identical infinitelylived individuals. The representative consumer maximizes the presentdiscounted value of a concave instantaneous utility of consumption u(c ) :


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where w is inflation, m is real money balances and g are government lump-sumtransfers. \J

Per capita government transfers are denoted by g and are financedexclusively through the inflation tax. Alternatively, it could be assumedthat seigniorage is not returned to consumers. The model would be basicallythe same, because the main result concerns with a constant level of g.

0F(m) represents the transactions technology. Modeling money as anintermediate input in transactions has been used recently by Fischer (1983),McCallum (1983), Kimbrough (1986a,b), Benhabib and Bull (1987), Faig (1988)and Vegh (1989), among others. As shown by Feenstra (1986) there is a closerelationship between this approach and the introduction of money into theutility function.

In contrast to the traditional formulation, it is assumed in this paperthat transaction costs are independent of the level of consumption. This isa convenient shortcut to take which does not alter the results. 2J

The transactions (or shopping) technology represents real resources thatare foregone in transactions. 0F(m) will depend on institutionalconsiderations, for example the degree of development of credit markets,where 8 is a parameter reflecting those aspects of the transactionstechnology.

The technology assumed has the advantage that 6 can be also interpretedas a parameter of the welfare costs of inflation. In the present modelinflation is a distortionary tax, whose welfare costs depends positively on

I/ Money is the only asset in this economy, although in generalequilibrium an interest bearing bond can be introduced at the margin, payinga real interest rate equal to 6. The case of debt financing is notconsidered. Since g is constant, the constant rate of inflation can beinterpreted as the result of optimal tax smoothing (Mankiw (1987) and Barro(1988) for recent applications).

2/ This is equivalent to assuming that money provides "shopping services"as suggested in Dornbusch and Frenkel (1973). The individual budgetconstraint would be: c+m+7rm=y(l+v(m))+g, so -yv(m)=0F(m). Hence, thetransactions technology would depend on y rather than c. But, since y isconstant, it is omitted as an argument of F.

There is no production and the individual has a constant flow endowmentof y. The only asset is money which is used because it facilitatestransactions. The budget constraint is (time subscripts are omitted):


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The reason for (A3) will be clear later. It is a natural requirementsince it will guarantee that, other things being equal, a reduction of 9 willreduce the amount of resources spent in transactions, thus it will increasewelfare.

The necessary conditions for optimality of consumer plans are:

where A is the current marginal utility value of an extra unit of money att

time t.

Now we can proceed to characterize the general equilibrium. As is wellknown from Brock (1974) and Calvo (1979), this economy may have multipleequilibrium paths. However, there is only one bubbleless equilibrium inwhich m - 0. In this equilibrium, inflation is equal to the rate of moneygrowth. This unique saddle path stable equilibrium will be examined. \J

Because there is no capital and prices are fully flexible the model hasno inherent dynamics. Hence, the economy is always at the steady state.This equilibrium is characterized by the following equations: 2/

I/ Bubbles can be ruled out assuming that the transactions technologysatisfies lim -F' (m)m > 0, Blanchard and Fischer (1989), Chapter 4 and

references therein.

2/ To save in notation and given that the economy is always in steadystate, superscripts or subscripts to denote equilibrium values are omitted.

The following assumptions with respect to the transactions technologyare made:


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A caveat: The optimal quantity of money and maximum seigniorage

Two recurrent issues in the literature on inflation and inflationarytaxation can be reproduced with this model. First, the optimal rate of moneygrowth, which equals the rate of inflation, is the Friedman (1969) rule: TT =-6, which is a zero nominal interest rate. This is checked after maximizingconsumption in the steady state. This rate of inflation is optimal since itequates the marginal productivity of money with the marginal cost ofproduction. Second, the maximum level of seigniorage is found by solving:

which yields the following expression for the revenue maximizing rate ofinflation:

* *

which together with (4) determines (K , m ). !_/ The revenue maximizing rateof inflation is given by the standard rule that the money-demand inflationelasticity has to be equal to one. Also, it can be checked using theenvelope theorem that dg /d# > 0.

Following the tenets of public finance it is known that, up to a firstapproximation, the deadweight loss of distortionary taxation is proportionalto the square of the tax rate. In this model the deadweight loss ofinflation will be proportional to the square of n+8, which by (4) impliesthat it will be proportional to the square of 6. Therefore, 8 has a directinterpretation in terms of welfare.

III. Welfare Effects of Financial Innovation

This section looks at the effects of a reduction of 8 on welfare in twocases. Because the economy is always in steady state and consumption is theonly argument in the utility function, it is enough to analyze the effects of6 on consumption.

The fall in 8 is interpreted as financial innovation, which allowspeople to require lower money balances to carry the same amount oftransactions. In Section IV a more concrete example will be discussed, where8 is related to deposits and interest rates paid on deposits. The casedeveloped in this section, however, shows the basis of the model and its main

I/ Assumed is the second order condition holds: 2F''(m )+F'''(m )m > 0.A sufficient condition for a unique interior solution is that F'(m) goes to-co when m goes to zero.


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implications in the simplest setting. Two issues are addressed. First, thewelfare effect of a change in 8 given the rate of money growth is examined.The second is the welfare effect of a change in 9 with the constraint thatthe rate of money growth and the equilibrium real balances must satisfy thegovernment flow budget constraint.

The reason for having inflation in the present model is to finance thebudget deficit. Therefore, the only purpose of the first exercise is to showthat, other things being equal, a fall in 6 will increase welfare. I/ Thisformally shows that the welfare costs of inflation are positively related to

Proposition 1: Given the rate of inflation, a decrease in 8 increaseswelfare.

Proof: Differentiating (3) and (4) we have:

and

When 8 falls, equilibrium money holdings will fall. This effect doesnot, however, offset the direct effect on welfare. When the government isconstrained to raise a given amount of revenue through an inflation tax, theprevious result changes. Denoting as g (6), the maximum seigniorage for agiven level of 6 we have:

Proposition 2: For a given g, the welfare effect of a change in 8 isambiguous. For g close and below g a decrease in 8 willdecrease welfare. For g equal to zero a decrease in 8 willincrease welfare.

Corollary: If in addition we assume that F'''>0, there exists g(0) suchthat a decrease in 6 increases welfare for all g e [0, g) anddecreases welfare for all g e (g, g ).

\J This exercise assumes that the government budget constraint, equation(5), does not hold.

therefore:


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Proof: Differentiating (3), (4) and (5):

and

therefore:

which is negative for g-ir-0, and goes to +« as rr approaches from below to therevenue maximizing rate of inflation.

To have monotonicity in dc/d0, d c/d0 < 0 is required, so that Ac/Adwill start from +» for the lowest feasible 8, denoted as 8_: this is the value

that satisfies g - g (8) . As 6 increases, g will become small relative to

g (0) and Ac/Ad will become negative. Therefore, for any g, there will be aunique 6 , 6 , such that Ac/A6 - 0. Define this function as 6 - 6(g). Theinverse is g - g(0). Hence the rest of the proof consists of showing thatthe function c - c(0) is concave.

using again the fact that D < m^F'', it can be seen that the two positiveterms in the brackets, -F'0 and D, are less than the absolute value ofF'm0 F''/D and -2mF''0, respectively. Therefore, for F'''> 0, K is strictlynegative, and hence d c/d0 is negative •

where


In contrast to Proposition 1, a decrease in 6 may be welfare reducing,specially for economies with large fiscal deficits which are financed throughinflation tax. The reason is that as 6 falls, real money balances also fall.When inflation remains constant the benefit of the reduction in 6 is largerthan the cost of holding lower real balances, as was shown in Proposition 1.For a given g, however, the government has to increase inflation because ofthe drop in the tax base. This increase in inflation reduces even more realmoney balances. In the end, the tax revenue requirement may reduce realmoney balances up to a point where the increase in F outweighs the fall in &.The above corollary provides a sufficient condition to generalize the resultfor all relevant g.

The results of Propositions 1 and 2 are shown in Figure 1. For 8 = 9 ,

(8 >d ) the demand for money, given g, is at A. In the space (c,m) point A

corresponds to A' . A reduction in 8 from 6 to 8 can be decomposed into two

parts. The first part is a reduction in m, given the rate of inflation (A toPI in the upper panel and A' to PI' in the lower panel). Proposition 1 showsthat consumption at PI' is larger than consumption at A'. However, since gis fixed, inflation will have to rise. Therefore there is a furtherreduction in real money balances (from PI to P2) . This effect will offsetthe increase in welfare, since consumption falls from PI' to P2 ' .

Proposition 2 shows that the total welfare effect of a reduction in 8 (Ato P2) is ambiguous, and its sign will depend on the size of g. For g closeto g (the value of g that produces tangency between the hyperbola irm = g andmoney demand for 8 ) consumption decreases. On the other hand, for g — 0,

there is no need to increase the rate of inflation from PI to P2 .

That welfare increases in the case of g=0 is a particular result ofProposition 1. The interesting outcome is the reduction of welfare for gclose to g . Looking at equation (7), what drives the result is that dm/ddgoes to infinity as g goes to g . The intuition for this result can beprovided through analysis of a standard multiplier effect.

Consider a small reduction in 6, which reduces m by A percent. Since gis constant, inflation has to rise A percent to offset the fall in m. Thisincrease in inflation induces an additional fall of eA percent in m, where eis the money- inflation elasticity (in absolute value). Then, a new increasein inflation of eA is required with a consequent e A percent reduction in m.A new increase in inflation will then be required, and so on. Therefore, thetotal fall in m in response to a small fall in 6 is equal to A(l+c+e +...).Hence ,

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The money-demand inflation elasticity varies between 0 and 1 in the"right side" of the inflation-tax-Laffer-curve. For g-0, the elasticity ofmoney demand is zero, so that there is no indirect effect from inflation tomoney demand. As g goes to g the indirect effect also increases because eis increasing. In the limit, a second order change in 6 causes a first orderchange in m because of the adjustment in the inflation rate needed to financethe budget. From the welfare (consumption) point of view, the increase inF(m) caused by the reduction in real balances is larger than the directeffect of the fall in 0.

Finally, note that the result is only concerned with inflation rates atthe "right side" of the inflation-tax-Laffer-curve. That is, in theincreasing portion of the seigniorage-inflation schedule. \/ Since theeconomy is always in steady state, there is no reason to assume that thegovernment is collecting seigniorage with excessive inflation.

Numerical example

Before explicitly introducing a banking system, a numerical example mayhelp to clarify the nature of the results. The example is illustrative andis not intended to replicate an actual economy. The figures are, however,consistent with empirical evidence on seigniorage and inflation (Fischer,1982; Dornbusch and Reynoso, 1989; and Giavazzi and Giovannini, 1989).

Table 1 provides the structure of the example. A quadratic transactionstechnology defined for m > £) /ft is assumed . Because there is a one-to-one

mapping from 9 to g , the simulations can be presented in two ways. Thefirst approach is depicted in Figure 2, where there are two curves. Theyrepresent the relationship between consumption and 9 under the assumptions ofPropositions 1 and 2. The hump-shaped curve considers a constant value of gequal to 0.25; this is seigniorage equal to 2.5 percent of potential GNP (yis normalized to be 10). This value for seigniorage is the maximum revenuefor 0 = 1.1,

0.25 = g*(0-l.l)

Therefore, 6 has to be restricted to be larger than 1.1. Otherwise itwould not be possible to collect g = 0.25 through seigniorage. As 9 changes,the rate of money growth, and hence inflation, will adjust to keep g = 0.25.In this simulation money over GNP is between 5 percent and 10 percent. Thehump shape reflects Proposition 2: the curve increases vertically when gcorresponds to the maximum revenue, g (0-1.1). As 0 increases therelationship between consumption and 0 is negative. In terms of Proposition2, as g becomes low with respect to g welfare decreases monotonically in 9.

I/ Eckstein and Leiderman (1989) estimate the relationship of seigniorageand inflation for Israel. In their model they find that the Laffer curve isalways increasing, becoming almost flat for rates of inflation above5 percent per quarter.

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For 0-1.1 the rate of inflation that maximizes revenue is 52 percent.Any inflation rate above that value will be at the "wrong side of the Laffercurve". For the range of 0 depicted in figure 2, the rate of inflationchanges from 52 percent for 9 — 1.1 to 30 percent for 6 - 2.1.

The monotonically decreasing curve corresponds to consumption, keepingthe rate of money growth, and hence inflation constant. Given this rate ofinflation (assumed to be 0) we can find m from the money demand for any valueof 0. Therefore this curve illustrates the result of Proposition 1: that therelationship between consumption and 9 is negative everywhere when inflationis constant--at zero in this example. Note that the optimal rate ofinflation should be -5 percent, therefore in both exercises there is awelfare loss due to inflation tax. This loss varies between 1 percent and2.5 percent of potential GNP. I/

Propositions 1 and 2 have been illustrated in terms of the slope of theconsumption-6 schedule. This is done in Figure 3. The curves are drawn fora fixed value of 6 (1.1), and g is allowed to vary. Proposition 1 says thatfor any constant rate of inflation the slope is negative. This is the curveat the bottom of Figure 3. For any initial g, a change in 6 withoutadjusting inflation will cause consumption to move in the opposite direction.Once we incorporate the restriction that g has to be constant, and inflationis adjusted accordingly, the welfare effect of a change in 0 is ambiguous.This is reflected in the other two curves, drawn for different values of /3 .

2

As g approaches g (6-1.1)-0.25, the marginal change in welfare goes toinfinity. That is c(0) becomes vertical as in Figure 2.

The two curves are drawn for different assumptions about /9 (1 for the

curve above and 0.85 for the other) to show that the value of g, at whichdc/d0 changes sign (g in prop. 2), is very sensitive to the parameters. Forft "I welfare falls when 8 falls for any g larger than 2 percent of potential

GNP. Instead, for ̂ -0.85, welfare falls when 8 falls for any g larger than

0.8 percent of potential GNP.

The gap between the curve for the inflation constant and the one for gconstant can be interpreted as the welfare cost arising from the adjustmentof inflation to raise the required revenue. In terms of Figure 15, this isthe change in c from PI' to P2'.

i/ Assuming an inflation rate of 7 percent and 0=1.1, the welfare loss is1.2 percent of potential GNP. Fischer (1981) computes a welfare loss("inflation triangle") of 0.3 percent of GNP due to a 12 percent deviation ofinflation from its social optimum.

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Figure 2

Figure 3(8 = 1.1)

Seigniorage (% of potential GNP)


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Table 1: Numerical Example

Equations:

Transaction technology:

Consumption:

Money demand

m -

Maximum seigniorage, and money balances at the maximum:

Given g, required money balances (inflation tax curve as function of m)


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IV. Financial Innovation with a Banking System

In this section the monetary model is extended to incorporate deposits.Deposits are imperfect substitutes of money, but they also providetransaction services. \J In this case what is called money is more preciselya monetary base which under the absence of reserves corresponds to currency.

The transactions technology is now F(m,e), where e denotes deposits.Assumptions (Al) to (A3) become:

m

This last assumption simply says that money as well as deposits arenormal goods. 2/

Financial intermediation is made through banks. Deposits are the bank'sliabilities. On the assets side, because there is no capital in this model ,it is assumed that banks have exclusive access to a storage technology with areturn equal to 6. I/ The marginal cost of providing deposits is zero andbanks pay an interest rate equal to i. Further, it is assumed that theinterest rate paid on deposits is fixed by the government and there are noreserves. The interest rate is kept below its competitive level,representing a financially repressed economy.

Now the consumer's problem is to maximize (1) subject to the followingbudget constraint:

c + F(m,e) + m + e + ir(m +e) - y + g + f + i e (11)

f is banks profits. The nominal return of banks is ic+6. Therefore the

I/ Fischer (1983), and recently Brock (1989), introduce deposits in thetransactions technology. Romer (1985) put deposits in the utility functionand Walsh (1984) in a cash in advance constraint. In all cases deposits areassumed to be imperfect substitutes of money.

2/ This assumption is equivalent to (A3) noting that an increase in 9 isequivalent to a fall in e.

/̂ This is equivalent to assume that banks are the only ones allowed tohold foreign assets. The world interest rate is 6. In addition, it shouldbe assumed that PPP holds, and the nominal exchange rate grows at the samerate as money and domestic prices.


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competitive interest rate is equal to *+8. I/ If i is less than ir+8 , f isequal to (?r+5-i)e.

At the competitive interest rate, and for any i < 8+x, people do no wantto save, although they still maintain deposits since they, as well as money,facilitate transactions.

The steady state is now characterized by:

and the government budget constraint (5).

Note that again the optimal rate of inflation (first best) is equal toFriedman's rule. At that optimum, the interest rate is i-0. In this case,the marginal productivity of both,-money and deposits, are equated to theirmarginal cost of production. This optimum is attained setting the rate ofdeflation equal to S and allowing banks to pay the competitive interest rate.

Note also that by (Al')( the problem can have interior solution only fori < 5+7T. In the rest of this section the focus will be on the interest rateon deposits between 0 and 8+ic. The question is what are the welfare effectsof an increase in the interest rate towards its competitive level. By makingthe financial system more competitive financial repression is alleviated.

Using equations (13) and (14) it is possible to show that the rate ofinflation that maximizes government revenue is given by: 2/

Now, the results from Propositions 1 and 2 can be extend to thefollowing proposition:

Proposition 3: (i) For a given inflation rate an increase in the interestrate will increase welfare.

(ii) For a given g, an increase in the interest rate willincrease welfare for g around zero and will decrease welfarefor g around, but below, g .

\/ In the first version of this paper it was assumed zero return on thisstorage technology, therefore banks were only providers of an alternativeasset to make transactions. The competitive interest rate in this case is ir,so they provide "indexed money".

2/ It is also assumed that the second order condition holds and thesolution is interior.


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Proof:

(i) Differentiating (12):

(ii) Using, from (5),d7r =-TT dm/m in (17), the system can be written as:

Lets call H' the square matrix on the LHS. Note that for 7r=0, H'-H, hencethe first part of (ii) is true from (18).After solving the system, we have that:

For TT < TT : det(H') > 0, and for TT = TT : det(H')=0. So provided that theexpression in square brackets is non-zero, dc/di will diverge to ± »depending on whether K is larger or less than zero, respectively.Replacing in K the value of TT from (15), we have that:

Therefore close to g , dc/di goes to -<*>, and welfare falls B

The intuition for part (i) is simply that an increase in the interestrate will increase the productivity of holding deposits. The negativeeffects of economizing in money holdings is smaller given the assumption thatmoney and deposits are normal goods.

Part (ii) follows from Proposition 2 in the previous section, which saysthat an increase in the interest rate paid on deposits will decrease welfarewhen there is a high budget deficit, and hence a high rate of inflation. In

Now, differentiating (13) and (14):

Denoting as H the hessian of F(•,•) and considering dn=0, we have that:


contrast, increasing interest on deposits will increase welfare at low ratesof inflation.

As in the previous section, the increase in the interest rate paid ondeposits requires a further increase in inflation to collect the requiredrevenue. This effect reduces money balances even more, leading to anunambiguous welfare loss at high rates of inflation. As was already shown,the welfare effect depends on the elasticity of the money demand, whichdetermines the change in inflation and its implied further reduction in realbalances required to finance the budget.

Reserve-deposit ratio in a competitive banking system

The critical element for the results of this paper is the unitaryelasticity of the money demand at the revenue maximizing rate of inflation.The results may be not robust under some specifications that do not satisfythe unitary elasticity rule, which may be the case with positivereserve-deposit ratio.

Brock (1989) shows that for a positive reserve-deposit ratio theelasticity of the money demand (currency) that maximizes revenue is not one.The reason for this is that the tax base is high powered money. If p is theactual reserve-deposit ratio, seigniorage, which by assumption is equal to g,will be given by:

g = (m + pe)?r = hit

Therefore, the rate of inflation that maximizes revenue is that at whichthe elasticity of h (in absolute value) with respect to TT (€ ) is equal to

h

one. Hence, the elasticity of currency (m) will, in general, be differentthan one. In what follows it will be shown that the result of Proposition 3can be extended to this case, and reinterpreted in the context of acompetitive banking system where the reserve-deposit ratio falls.

The increase in the interest rate paid on deposits can be envisioned asa reduction in reserve requirements in a competitive banking system (Calvoand Fernandez, 1983). If banks were allowed to compete and required to holda fraction p of their deposits as non-interest bearing reserves, thecompetitive interest rate would be (5+jr)(l-p).

In this case (14) would become:

The effect of a change in p on consumption is:

and the effect on high powered money is:

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Substituting (24), (22) and (13) in (23), the following expression forthe welfare effects of a change in p is obtained:

The same intuitive argument explaining Proposition 2 can be used to showthat at high levels of seigniorage welfare falls when p falls. It is enoughto note that dh/dp goes to infinity as the elasticity of high powered money(e ) goes to one:

h

A change in p that causes a A percent change in h, will require anincrease of A percent in inflation to offset it. The increase in inflationwill reduce high powered money by an additional £ A. Inflation will then be

h

required to increase further by e A percent, where h will consequently fall2 h 2 3by e A. and so on. Therefore the total fall in h will be A(l+e +e +e + . . .),J h h h h

which goes to » when e goes to one. Then, according to (25) welfare willh

unambiguously fall "close" to the maximum inflation-tax revenue because ofthe excessive increase in the inflation rate and the fall in real balances.

V. Endogenous Government Spending and Optimal Taxation

In previous sections the government has been assumed to have a passiverole by imposing an exogenous tax structure to finance a given budget. Thissection examines government behavior by analyzing two separate cases. First,government spending is assumed to be a public good which is optimallyprovided, and hence one that will be modified when financial innovationoccurs. Second, an optimal taxation approach is followed by assuming that,in addition to the inflation tax, the government can resort to a consumptiontax in order to finance a given level of government spending.

Endogenous Government Spending

This section considers g to be a public good which enters into theindividual utility function. For simplicity the-utility function is assumedto be separable in consumption good and public good. In this case there willalso be an optimal response of g to a reduction in 8 (financial innovation).The response will be a reduction in the provision of the public good as thewelfare cost of providing it increases.

It is shown below that the main results of previous sections hold: inhigh inflation economies financial innovation will reduce welfare while inlow inflation economies welfare increases.


- 17 -

The first term at the RHS represents the standard direct effect of areduction in transactions costs, which increases welfare when 6 falls. Thesecond term is the welfare-reducing effect of financial innovation, by whichthe provision of the public good will be reduced because the increasing costsof raising revenue. Although, in this case dW/dtf will not necessarilydiverge to -H» as inflation approaches the level that maximizes governmentrevenue, thus the same results from Propositions 1 and 2 hold. To see this,we can replace (27) in (28) to obtain:

It is clear from (29) and (30) that the direct effect (first term withinsquare brackets) is of a second order when compared to the indirect effectof rates of inflation close to the revenue maximizing rate of inflation(equation (6)). When inflation is low, because g is low, the direct effectwill dominate. Therefore, the result from Propositions 1 and 2 apply to thecase of endogenous government spending.

The effects of financial innovation can be determined using the envelopetheorem (arguments of the functions are omitted):

Thus, the government implicitly chooses m, by setting accordingly therate of inflation. The first order condition to this problem is:

where both, u(•) and v(•) are increasing and strictly concave functions. Thegovernment will maximize this utility function subject to consumer behaviorand its budget constraint. That is, the government problem is to maximizeU(c,g) subject to (3), (4) and (5). This problem can be conveniently writtenas:

Consider the same model as that of Section II, but now the instantaneousutility function is:


-Optimal Taxation

In addition to the inflation tax, governments use several instruments toraise revenue. Throughout this paper g has been financed via inflation only.Therefore, g can be interpreted as the fraction of government spending thatcannot be financed through nondistortionary taxes. In high inflationcountries, the heavy reliance on seigniorage may be due to the large size ofthe underground economy or the inefficiency of the tax system. It may alsobe caused by political factors as has been discussed recently in Cukierman,Edwards and Tabellini (1989). They argue that political instability and thedegree of polarization are key determinants of seigniorage.

Under a non-single tax system, when the base of one tax falls, thegovernment will, in general, adjust several other tax rates. Hence, apermanent shift in money demand will be accommodated not only by an increasein inflation, but also through an increase in other taxes.

Since Phelps (1973) , a large body of literature has focused on inflationas part of an optimal tax system. In this case the optimal structureconsists of equating the social marginal costs of different distortions. \J

It is assumed for the model in Section II, that money is the only assetand is used in transactions. Government optimally sets taxes on moneyholdings and consumption to finance g of government spending, which is thenreturned to consumers as a lump sum transfer.

The consumer problem is the same as before, but now consumption is taxedat a rate T:

subject to:

Note that despite having taxes on the only two goods, consumption andreal balances, lump sum taxation cannot be reproduced. The reason is that0F(m) plays the role of a third good which is not taxed.

Individual behavior is characterized by the following equations:

I/ Poterba and Rotemberg (1990) and Grilli (1989) cast serious doubts onthe empirical validity of the optimal seigniorage theory, when tested inindustrialized countries.

- 18 -


- 19 -

Since '̂ e (0,1), /i is greater than 1. In the case where ^>'=0, i.e.there are no collection costs, n=l. Solving (35) for the case of nocollection costs, the following equation is obtained for the optimal taxproblem:

I/ The specification of the problem in this section is consistent with theresult of Kimbrough (1986b), who extended the inflation tax problem inDiamond and Mirrlees' (1971) principle that intermediate inputs should not betaxed. As pointed out in Guidotti and Vegh (1990), however, these resultshold true only under particular characteristics of the transactionstechnology, which in the above case applies.

2/ For a full derivation of the remaining equations, see appendix.

Although taxes and g are related through the government budgetconstraint, the individual takes g as given. These equations describeconsumption and real balances as functions of the two tax rates, which haveto be considered in solving the optimal tax problem. Note that m dependsonly on the inflation tax.

As will be shown later, in the present model, an optimal tax structureinvolves no tax on money, where all the tax burden should fall on theconsumption tax. I/ Therefore, to make an inflation tax positive, the modelhas to be amended. For this purpose real costs in collecting consumption taxare, realistically, assumed, whereas the cost of collecting an inflation taxis zero (Aizenman, 1983 and Vegh, 1989). Collection costs are increasing inthe amount of revenue collected. They are described by a function <J>(TC) ,where 0 < <£'(•) < 1 and <£''(•) > 0. The case <£'(-)=Q is equivalent to the nocollection cost case. Since utility depends only on consumption, which isconstant in equilibrium, the optimal tax problem consists of:

It can be shown that the first order conditions from the governmentproblem are: 2/

subject to:

The lagrangian of this problem is:


-- 20 -

The direct effect (dc/dd) will be the same as the total effect computedin Proposition 1, which by the assumptions made on F(-) guarantee this to benegative (financial innovation increases welfare). I/ The other effects willhave the opposite sign since taxes will have to be risen to offset the fallin the inflation-tax base. After some algebra it can be shown that:

where a is the share of inflation tax revenue on total spending (G=g+<£) , andx denotes the percentage increase in the tax rate x (x = TT, r) on account ofthe fall in 6 .

When the tax rates are not adjusted, allowing a fall in revenue, a fallin 9 will increase welfare. Therefore, only the direct effect matters. Thisresult is merely a generalization of Proposition 1; thus, given the taxrates, a fall in 6 is welfare improving.

Nevertheless, once we consider the government budget constraint, thereis a negative effect because of the increase in the tax rates. A reduction

I/ One could suspect by the envelope theorem that only the direct effectof 8 matters. Equations (28) and (33) show why this intuition is wrong. Thegovernment does not set 3c/3(tax rate) equal to zero, but rather d£/d(taxrate) equal to zero. In fact, both taxes are distortionary, so a decrease in6 will cause both tax rates to increase such that at the margin thedistortions are equated, but the indirect effects through the tax ratescannot be eliminated.

In this case, both money and consumption goods have to be taxed.

The question now is what is the welfare effect of a change in 8 ,considering that all taxes will be optimally adjusted. The total effect canbe written as:

/I

Then, optimal taxation calls for 6+jr=0, which recovers Friedman'soptimal money rule as in Kimbrough (1986b). In the presence of collectioncosts, however, the optimal inflation tax departs from the zero nominalinterest rate rule. In this case combining (35) and (36) and substitutingdm/dx by -1/0F''(m), the optimal tax problem is reduced to:


in the inflation tax base requires an adjustment in tax rates. Then, a fallin 8 implies that the term within square brackets is positive, offsetting thebeneficial direct effect.

It is not possible to generalize Proposition 2 without making additionalassumptions, although it can be argued that under general conditions theresults still hold. Looking at equation (39)) two remarks that support thispresumption can be made. First, the negative effect of a fall in 8 ispositively related to g, although the change in tax rates and their share inrevenue will also depend on g. Second, the increase in the rate of inflationrequired to raise a given amount of seigniorage is increasing in theelasticity of money demand. Therefore, the larger the elasticity of moneythe larger the increase in the rate of inflation and the more distortionarycommodity taxation will become to offset the distortion in money holdings.

It is possible, however, that a fall in 8 will always increase welfare.The fact that the indirect effect of tax rate changes is of a first orderwhen compared to the direct effect of financial innovation--which resolvedthe ambiguity-- is not a general proposition under the assumptions of thissection. The reason is that when the elasticity of money demand is close toone, the required revenue will be raised mainly through commodity taxation.Thus, the welfare effect will also depend on the elasticity of the demand forconsumption goods. Nevertheless, the direction of the result is the same:the larger the seigniorage, the larger will be the distortion introduced inconsumption because of the increase in the tax rate.

VI. Concluding Remarks

Only two interpretations have been fully developed for the change in themarginal productivity of money, and consequently for the reduction in thewelfare costs of inflation. The results of Section IV can, however, beeasily extended to include other sources of reduction in the welfare costs ofinflation.

Deposits can be interpreted as indexed money which is an imperfectsubstitute for non-indexed money. More importantly, it represents analternative asset to money. This asset can be used in transactions and ityields higher interest than money. Policies that make the use of this assetmore attractive will increase the velocity of money. Consequently theinflation tax base will fall. This is the case of "dollarization", wherethere is a shift from domestic to foreign money. Dollarizations are usually

- 21 -


observed when a country's inflation rate increases (Fischer, 1982). l_/2./

In the model presented here, for a constant rate of money growth anincrease in velocity is welfare improving. When a given revenue has to beraised through inflation tax, however, the result can be reversed. The mainconclusion of this paper is that negative effects on welfare will occur inlarge seigniorage economies. In contrast, the lower the inflation tax, themore beneficial is the reduction in inflation costs.

Faced with high inflation, people seek institutional changes that willprotect them from inflation. For example, wage indexation and short-termfinancial instruments become very important. Usually there are demands forgovernment to introduce changes that reduce the costs associated withtransactions, such as a reduction in the reserve-deposit ratios. Theperverse effect that some of these changes have, when the deficit is financedmainly by seigniorage, may, however, explain why governments are reluctant toaccept changes that may look, other things being equal, positive.

Note that the results of this paper could easily be extended to any formof taxation and its relationship to technical progress. The parameter 8 isequivalent to a Hicks neutral parameter of technical progress and m isequivalent to an input. Therefore technical progress can be immiserizingwhen it produces a fall in the demand for taxed inputs. The requiredincrease in the tax rate, and hence in the degree of distortion, may end upreducing welfare. For example, imagine an economy where the only availabletax is a tax on gasoline. Technical progress that saves on inputs willreduce the use o.f gasoline. Therefore, the required increase in the gasolinetax rate may end up reducing welfare.

Inflation and capital flight are frequently observed in inflationaryeconomies. In this paper, Propositions 1 and 2 assume that B is exogenous.Making this variable endogenous may explain inflation and capital flight as acoordination failure. If 8 is interpreted as a parameter of the structure ofcredit markets, it can be related to the extent of capital flight.Proposition 1 shows the private incentive to reduce 8 because it takesinflation as given: individual decisions have no effect on inflation. Underthe assumptions of the model, the private incentive to reduce 8 is alwayspositive. Instead, Proposition 2 shows that the total effect is uncertainand depends on the current level of inflation (through seigniorage).Therefore, a welfare reducing increase in inflation may be the result ofspillover effects from capital flight to real balances, and hence toinflation.

Finally, this paper connects the degree of inflationary finance with

I/ For the case of Mexico, see Ortiz (1983). In November 1989 in themiddle of an unsuccessful stabilization program in Argentina the governmentallowed deposits in foreign currency, which were to be guaranteed.

2./ Arrau and De Gregorio (1990), in a framework similar to the one in thispaper, estimated empirically the role of financial innovation in money demandequations for Chile and Mexico. The results showed that an importantcomponent of money demand fluctuations corresponds to financial innovation.

- 22 -


developments in financial markets, as was recently addressed by Dornbusch andReynoso (1989). Financial deepening may lead to a deterioration in welfarethrough an increase in the rate of inflation. Therefore, a requisite for theremoval of financial repression is fiscal discipline. Relying heavily onseigniorage to finance the budget may outweigh the advantage of amore developed financial market.

In summarizing the results of the original question, what are thewelfare effects of a reduction in the costs of inflation, this paperconcludes that they are positive (negative) in low (high) seigniorageeconomies. In a broader interpretation, concerning the welfare effects offinancial innovation, it can be concluded that the benefits of improvedfinancial intermediation may be offset by the negative effects of a higherrate of inflation.

- 23 -


References

Arrau, P. and J. De Gregorio, (1990), "Money demand and financial innovation:Theory and empirical implementation", mimeo.

Aizenman, J. (1983), "Government size, optimal inflation tax, and taxcollection costs", Eastern Economic Journal, 9:103-105.

Barro, R. (1988), "Interest-rate smoothing", NBER Working Paper No. 2581.

Barro, R. and D. Gordon, (1983), "A positive theory of monetary policy in anatural rate model", Journal of Political Economy, 91: 101-121.

Benhabib, J. and C. Bull, (1983), "The optimal quantity of money: A formaltreatment", International Economic Review, 24: 101-111.

Blanchard, 0. and S. Fischer, (1989), Lectures on Macroeconomics, MIT Press.

Brock, P. (1989), "Reserve requirement and the inflation tax", Journal ofMoney, Credit and Banking, 21: 106-121.

Brock, W. (1974), "Money and growth: The case of long run perfect foresight",International Economic Review, 15: 750-777.

Calvo, G. (1979), "On models of money and perfect foresight", InternationalEconomic Review, 20: 83-103.

Cukierman, A., S. Edwards and G. Tabellini, (1989), "Seigniorage andpolitical instability", mimeo.

Diamond, P. and J. Mirrlees, (1971), "Optimal taxation and publicproduction", American Economic Review, 61: 8-27 and 261-278.

Dornbusch, R. and S. Edwards, (1989), "Macroeconomic of populism in LatinAmerica", NBER Working Paper No. 2986.

Dornbusch, R. and J. Frenkel, (1973), "Inflation and growth: Alternativeapproaches", Journal of Money, Credit and Banking, 5: 141-156.

Dornbusch, R. and A. Reynoso, (1989), "Financial factors in economicdevelopment", NBER Working Paper No. 2889.

Faig, M. (1988), "Characterization of the optimal tax on money when itfunctions as a medium of exchange", Journal of Monetary Economics, 22:137-148.

Feenstra, R. (1986), "Functional equivalence between liquidity costs and theutility of money", Journal of Monetary Economics, 17: 271-291.

Fischer, S. (1981), "Toward an understanding the costs of inflation: II", inK. Brunner and A. Meltzer (eds), The Costs and Consequences ofInflation, Carnegie Rochester Conference Series on Public Policy, vol.15, pp. 5-42.

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(1982), "Seigniorage and the case for a national money", Journalof Political Economy, 90: 250-313.

(1983), "A framework for monetary and banking analysis", EconomicJournal, Conference papers, pp. 1-16.

Fischer, S. and L. Summers, (1989), "Should governments learn to live withinflation?", American Economic Review, Papers and Proceedings, May, pp.382-387.

Friedman, M. (1969), "The optimal quantity of money", in The Optimum Quantityof Money and Other Essays, Chicago: Aldine.

Giavazzi, F. and A. Giovannini, (1989), Limiting Exchange Rate Flexibility:The European Monetary System, MIT Press.

Guidotti, P. and C. Vegh, (1990), "The optimal inflation tax when moneyreduces transactions costs: A reconsideration", mimeo, IMF.

Grilli, V. (1989), "Seigniorage in.Europe", in M. De Cecco and A. Giovannini(eds), A European Central Bank?, Cambridge (U.K.): Cambridge UniversityPress.

Helpman, E. and L. Leiderman, (1988), "Stabilization in high inflationcountries: Analytical foundations and recent experiences", CarnegieRochester Conference Series, vol. 28., pp. 9-84.

Kiguel, M. and N. Liviatan, (1988), "Inflationary rigidities and orthodoxstabilization policies: Lessons from Latin America", World Bank EconomicReview, 2: 273-298.

Kimbrough, K. (1986a), "Inflation, employment, and welfare in the presence oftransaction costs", Journal of Money, Credit and Banking, 18: 127-140.

(1986b) , "The optimum quantity of money rule in the theory ofpublic finance", Journal of Monetary Economics, 18: 277-284.

Mankiw, G. (1987), "The optimal collection of seigniorage: Theory andevidence", Journal of Monetary Economics, 20: 327-341.

McCallum, B. (1983), "The role of overlapping-generations models in monetaryeconomics", Carnegie Rochester Conference Series on Public Policy, 18:9-44.

Ortiz, G. (1983), "Dollarization in Mexico Causes and consequences", in P.Aspe, R. Dornbusch and M. Obstfeld (eds.), Financial Policies and theWorld Capital Market: The Problem of Latin American Countries, Chicago:Chicago University Press, pp. 70-95.

Poterba, J. and J. Rotemberg, (1990), "Inflation and taxation with optimizinggovernments", Journal of Money, Credit and Banking 22: 1-18.

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Romer, D. (1985), "Financial intermediaries, reserve requirements and insidemoney", Journal of Monetary Economics, 16: 175-194.

Sargent, T. (1983), "Stopping moderate inflation" in R. Dornbusch and M.Simonsen, (eds.)i Inflation, Debt and Indexation, MIT Press.

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Walsh, C. (1984), "Optimal taxation by the monetary authority", NBER WorkingPaper No. 1375.

- 26 -


Substituting (A5) in (A8), this first order condition becomes

where m is a lagrange multiplier. The first order conditions of this problemare (subscripts denote partial derivatives):

The lagrangian of this problem is:

subject to:

The government optimal tax problem consists of

The direct effect, is the one total derivative from Proposition 1, that isassuming all tax rates as constants:

The effect on consumption, then welfare, of a change in 8 is given by:

Note that at the optimum c will be a function of both taxes, r and it,while m is only a function of JT. Although in equilibrium g equals ?rm + CTthe individual takes g as a given transfer. The following partialderivatives can be obtained from (Al) and (A2):

Consumer behavior is given by the following equations:

The Optimal Tax Problem

- 27 - APPENDIX


- 28 - APPENDIX

Since <j>' e (0,1), n is greater than 1. In the case that <f>'=0, i.e. there isno collection cost, /i=l.Using (A4) and (A5) in (AlO), it can be written as:

From this equation it can be seen that when '̂=0 and hence /i=l, (A12)collapses to:

So optimal taxation calls for 5+?r=0 which recovers Friedman's optimal moneyrule as in Kimbrough (1986). But in the presence of collection costs theoptimal inflation tax departs from the zero nominal interest rate rule.Replacing (All) and (A3) in (A12) we obtain the following expressioncharacterizing the optimal tax scheme, ]L/

Substituting (A3), (A4) and (A5) in (A6) (consumer behavior in equation forthe total effect of 6 in consumption) we obtain

which after using the optimal tax rule (A14) becomes equation (39) in thetext:

Note that this equation can be further reduced to find expression forthe percentage change of the tax rate by totally differentiating thegovernment budget constraint.

I/ The final solution could be obtained replacing m as a function of TTfrom (A2) in (A13) to have a single equation for jr. Then, substituting cfrom (Al), as a function of TT and T, in the government budget constraint thesolution for T would be obtained.


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