A General-Equilibrium Closed-Form Solution to the Welfare Costs of Inflation

This work presents closed-form solutions to Lucas's (2000) general equilibrium expression for the welfare costs of inflation, as well as to the difference between the general-equilibrium measure and Bailey's (1956) partial-equilibrium measure. In Lucas's original work only numerical solutions are provided.


Introduction
In this work I derive a closed-form solution to Lucas's general-equilibrium expression for the welfare costs of in ‡ation when the money demand function is double-logarithmic 1 . Next, I use this closed-form solution to derive an expression which delivers, also in closed-form, the di¤erence between the general-equilibrium and Bailey's (1956) partial-equilibrium measure of the welfare costs of in ‡ation. This work bene…ted from conversations with Robert E. Lucas Jr. Remaining errors are my responsability.
y Key Words: In ‡ation, Welfare, Closed-Form. JEL: C0, E0. z EPGE/FGV and, in 2004, Visiting Scholar at the Department of Economics of the University of Chicago. Present Address: 5020 S. Lake Shore Drive # 1402 N., Chicago, Ill, 60615, USA. E-Mail: rpcysne@uchicago.edu. 1 Lucas (2000) argues that this is the functional speci…cation of the money demand that best …ts the United States historical time series.
1 In Lucas's (2000) original paper, both the solution of the underlying nonlinear di¤erential equation leading to the general-equilibrium welfare …gures, as well as the comparison with Bailey's estimates, are based only on numerical methods.
The work is divided as follows. Section 2 presents a continuous-time, no-growth version of Lucas's shopping time model. Given the correspondent interpretation of the variables in each case, both the discrete and the continuous approach, with or without growth, lead to the same non-linear di¤erential equation describing the welfare costs of in ‡ation (equation 5.8 in the original paper and equation (6) in Section 2 of this work). We therefore present the continuous-time no-growth model for the sake of simplicity in the exposition, with no loss in generality.

The Model
In Lucas's (2000, sec. 5) analysis of the welfare costs of in ‡ation the representative consumer is supposed to maximize utility from the consumption (c) : subject to the households budget constraint (2) and to the transactionstechnology constraint (3): In these equations, s stands for the fraction of the initial endowment spent as transacting time (the total endowment of time being equal to the unity), m for the real quantity of money, for the rate of in ‡ation, U (c) for a concave utility function; h for the (exogenous) real value of the ‡ow of money transferred to the household by the government, g > 0 for a continuous-time discount factor (Lucas uses 1=(1 + ) for the discrete case) and F (m; s) = m (s); 0 (s) > 0; for the transacting technology.
Intertemporal optimization leads to the …rst order condition: Equilibrium in the goods market reads: In the steady-state solution m converges to a constant …gure, the rate of interest r equals the rate of in ‡ation plus the discount factor (r = + g); the in ‡ation equals the rate of monetary expansion and the real transfers (h) equal the in ‡ation tax (h = m; standing for the rate of monetary expansion).
Solving the system given by (4) and (5) for s = s(r) and m = m(r) yields s 0 (r) > 0 and m 0 (r) < 0. The problem of deriving s(r) from m(r) without knowing (s) is solved by eliminating (s) and 0 (s) using (4) and (5) which determines the welfare cost s(r) as a function of the money-demand m(r).
Lucas (2000) argues that the double-logarithmic functional speci…cation …ts the United States data better than the alternative semi-log speci…cation. Making m (r) = Ar a , 0 < a < 1; A > 0; (6) leads to: s(r 0 ) = s 0 ; r 0 > 0 Proof. It is easy to see that, with r bounded away from zero, v(r; s) 2 C 1 , and, by the mean-value theorem, and for a certain constant L > 0; satis…es the Lipschtz condition j v(r; s 1 ) v(r; s 2 ) j L j s 1 s 2 j for each par (r; s 1 ); (r; s 2 ) in D: It follows from a standard result in ordinary di¤erential equations based on the contraction mapping theorem (see, e.g., Coddington and Levinson (1955)) that there exists an interval containing r such that a solution to (7) exists, and that this solution is unique. It is then immediate that such a solution can be continued to the right to a maximal interval of existence [r 0 ; +1) : Even though existence has been easily proved in Proposition 1, it is by no means clear that this non-separable, non-linear di¤erential equation presents a closed-from solution. For example, it is well known that a simple equation like ds dr = w(r; s) = s 2 r cannot be expressed as a …nite combination of elementary functions or algebraic functions and integrals of such functions. I shall show, next, that such a problem does not happen with (7) and and (8).

Proposition 2 The solution to (7) and (8) is given by
Proof. Start by considering r 0 > 0 and the initial condition Suppose s(r) is a solution to (7), given (10). Then, since s 0 (r 0 ) > 0; the inverse function r = r(s) is de…ned in a su¢ ciently small neighborhood of the point s 0 and: This type of equation is generally called a Bernoulli equation, which can be easily solved by an adequate change of coordinates. Consider the di¤eomorphism that associates with each r > 0; t = r 1 a : Then (11) is equivalent to the equation: Multiplying both sides of this equation by the integration factor exp( Integrating in s and using the fact that t(0) = 0: Solving for the integral of 1= (1   ) : Use the fact that t = r 1 a to get (9).

A Direct Comparison with Bailey' s Measure
Lucas provides numerical simulations in order to compare his general-equilibrium measure (6) and Bailey's partial-equilibrium measure (B) of the welfare costs of in ‡ation. Having obtained a closed-form solution for the former allows us to provide a closed-form expression for the di¤erence between these two measures.

Proposition 3
The di¤erence between the general-equilibrium (s) and Bailey's partial-equilibrium (B) measure of the welfare costs of in ‡ation is given by: Proof. Bailey's measure, in di¤erential form, is given by the area-underthe-inverse-demand-curve: