ALVARO RODRIGUEZ Rutgers Uniuersity

New Brunswick, New ]ersey

The EMtence of an Optimal Inflation Rate in a Cash-In-Advance Economy

This paper examines fiscal and monetary policies for maximizing steady-state utility in a cash-in-advance model. We consider a case in which government revenues come from an inflation tax on monetary assets and from a linear tax on income. It is shown that there is no optimal combination of these alternative forms of taxation since long-run welfare only depends on the sum of the revenues generated by the two taxes.

1. Introduction Studies about the optimal rate of inflation have usually fol-

lowed two approaches. The first, starting with Friedman (1969), as- sumes that there is no commitment on part of the monetary au- thority to raise revenues and help finance government expenditures. Friedman proposes that in such a case the rate of deflation must be made equal to the intertemporal rate of discount. The validity of Friedman’s rule for a cash-in-advance economy has been inves- tigated by Grandmont and Younes (1973).

A second approach to the study of inflation follows Phelps (1973) in looking at this issue as an optimal taxation problem. The gov- ernment must choose the combination of different taxes that gen- erates the desired revenues and minimizes welfare losses. In such a case, decisions about inflation and income taxes are interrelated. There are two recent studies that look at this issue in the context of a cash-in-advance economy. Lucas and Stokey (1983) focus on the conditions for obtaining time consistent policy rules. Kimbrough (1986) considers a different cash-m-advance constraint and shows that Friedman’s rule is optimal when the government has a sufficiently rich menu of alternative taxes. Inflation, in such a case, has no role in raising government revenues.

This paper focuses on an aspect of the problem that has been ignored. We examine how the existence of a trade-off between in- flation and growth affects the optimal tax system. The studies men- tioned previously consider models without capital accumulation, so they cannot take into account the effects of inflation on growth. We

Journal of Macroeconomics, Spring 1989, Vol. 11, No. 2, pp. 309-314 309 Copyright 0 1989 by Louisiana State University Press Olf+I-0704/89/$1.50

Alvaro Rodriguez

examine the choice between inflation and a linear income tax in the context of the cash-in-advance model proposed by Stockman (1981). In such a model, the only distortion generated by the two taxes is a reduction of the rate of return on capital. We analyze the steady-state solution to the model and show that there is no optimal combination of these alternative forms of taxation. The reason for this indeterminacy is that, in the long run, welfare only depends on the sum of the revenues generated by the two taxes.

2. The Model We modify Stockman’s (1981) model by introducing a govern-

ment sector that provides a public good and raises taxes on income and nominal assets.

The economy uses capital and labor to produce a homogenous good. The capital stock at time t + 1 is equal to the capital stock at the beginning of period t plus the excess of output over con- sumption of private and public goods:

K t+l = K, + W,, L) - C, - G ; 0)

where K stands for capital, L for the labor force (which it is as- sumed to be constant), G for government expenditures, and C for private consumption. The production function is assumed to be con- cave and exhibits constant returns to scale. There is perfect com- petition, and both capital and labor are paid the value of their mar- ginal productivity.

The government sector operates in the following way: there is a flat tax on income, s, and the excess of expenditures over in- come taxes is financed by issuing money. If M, denotes the nominal stock of money at the beginning of period t and if P, is the price level, the government budget constraint can be stated as

M t+l - W = Pt [G - sF(K,, Ql . (2)

Government expenditures are a fraction of current output:

G = @Ok L) , O<g<l. (3)

Consumers have an infinite horizon and maximize the dis- counted sum of their utility in different periods. It is assumed that the utility function depends upon private consumption, C, and upon

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Existence of an Optimal Inflation Rate

government expenditures, G. (For simplicity we make the labor supply exogenous since we want to concentrate on the effect of taxes on savings decisions.) When the utility function is additively sep- arable, the consumer’s problem is to maximize

m

c. P’ NW3 + V(G)1 (4) t=o

subject to the following constraints. The first states that the rate of asset accumulation is equal to the excess of income over expendi- tures:

ptwt+1 - Kt) + Mt+1 - M, = (1 - s)P,F(K,, L) - C,P, . (5)

The second restriction is the cash-in-advance constraint which requires that all private purchases of capital and consumption goods must be made with the stock of money available at the beginning of the period:

Mt = PtK+1 - Kt + G) . (6)

The consumer’s maximization problem can be solved using Bell- man’s principle of optimality. First, we combine (5) and (6) to ob- tain

M t+1 = (1 - 4PtFWt, L) . (7)

On the other hand, from (6) it follows that C, is given by

Ct = UW’t) - Kt+, - Kt . (8)

Given (7) and (B), the value function satisfies the following recursive relation:

JK M,, PJ = ma+,+, {WWPt) - &+I + &I

+ PJ[Kt+l> (1 - s)PtFK U Pt+,II .

Differentiating the right-hand side of (9) with respect to K,,, one obtains

- u: + PJ:” = 0. (10)

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Alvaro Rodriguez

The partial derivatives of J can be computed applying the envelope theorem to Equation (9).

J”1 = u: + f3J:” (1 - S)P,Fk = 0. (11)

J; = u:p, . (12)

Updating (11) and (12) and substituting into (10) one obtains

- u; + pu:+’ + p”u:‘” (P,+r/P,+2)(l - s)Fx = 0. (13)

The above equation together with (1) and (2) jointly determine the trajectory of prices and consumption along a competitive equilib- rium path.

3. Optimal Taxation In this section we look at the monetary and fiscal policies that

maximize steady-state utility. First, we characterize the stationary equilibrium.

There are three equations that define a steady-state equilib- rium. The first one concerns the rate of return on capital. From (13) it follows that, when consumption is constant, the marginal pro- ductivity of capital is given by

F K = (1 - PN + m

p2 (1 - s) ; (14)

where Il stands for the rate of inflation, P,+JP, - 1. Equations (1) and (3) imply that the rate of consumption as-

sociated with a constant capital stock is equal to

c = (1 - g)F(K, L) . (15)

Finally, it is necessary to determine the long-run rate of inflation. From (2) one can obtain an expression for the change in real bal- ances:

Since at the steady state real money balances are constant,

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Existence of an Optimal Inflation Rate

M Mt t+1 -c-z,

P t+1 pt . (17)

The above equation, combined with (3) and (16), implies that

mII = (g - s)F . W

Recalling (6) and (15) one gets

m = (1 - g)F . 09)

Equations (18) and (19) yield the following expression for the long- run rate of inflation:

n = k - 4

(1 - d .

cm

The system of equations ([14], [15], and [20]) uniquely deter- mines the long-run equilibrium. According to (14), in such an equi- librium, inflation decreases the capital stock. An interpretation of this relation is provided in Stockman (1981). When an individual decides to consume less now and to buy capital, his higher future income can be converted into consumption only through holding additional money balances. When the rate of inflation is high, money is more costly to hold, so the profitability from investment is lower. On the other hand, when s is high, the after-tax return on capital is low and investment is discouraged. It will be shown that, in the long run, the distortion generated by the two taxes is the same.

The highest steady-state utility is obtained by maximizing U[(l - g)F] + V(gF) with respect to g and s subject to (14), (15), and (20). Given g, utility depends only upon K, so the planner would like to minimize F= To show that FK is independent from s, it suffices to substitute (20) into (14) and obtain

FK = (1 - d-’ LO - P)/P”l . (21)

Because the long-run capital stock is independent from s, so is steady- state utility and there is no optimal rate of inflation. There is, how- ever, a value of g that maximizes welfare.

The above analysis can be extended in several ways. One is to allow for taxes on consumption rather than on income. The in-

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troduction of this type of tax is likely to restore the validity of Friedman’s (1969) rule and corroborate the findings of Kimbrough (1986). However, until drastic changes in the present tax system are introduced, the case considered here is the most relevant. Moreover, since a consumption tax has no impact on the long-run capital stock, it has to be analyzed in a context that takes into ac- count the transitory effects. Such a study will have to face the prob- lems associated with time inconsistency and, therefore, is beyond the scope of this study.

Received: August 1987 Final version: June 1988

References Friedman, M. “The Optimal Supply of Money.” In The Optimum

Supply of Money and Other Essays, edited by M. Friedman. Chicago: Aldine, 1969.

Grandmont, J., and Y. Yuones. “On the Efficiency of a Monetary Equilibrium.” Review of Economic Studies 40 (1973): 149-65.

Kimbrough, K.P. “The Optimum Quantity of Money Rule in the Theory of Public Finance.” Journal of Monetary Economics 18 (1986): 277-84.

Lucas, R., and N. Stokey. “Optimal Fiscal and Monetary Policy in an Economy Without Capital.” Journal of Monetary Economics 12 (1983): 55-93.

Phelps, E. “Inflation in the Theory of Public Finance.” Swedish Journal of Economics 75 (1973): 67-82.

Stockman, A. “Anticipated Inflation and the Capital Stock in a Cash- in-Advance Economy.” Journal of Monetary Economics 8 (1981): 387-93.

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