Cysne, Rubens Penha

English

Research Papers in Economics

 
 
 
 
 
 
 
 
Nº 562  ISSN 0104-8910
 
A general-equilibrium closed-form solution to the welfare costs 
of inflation(Forthcoming, Revista Brasileira de Economia) 
  
Rubens Penha Cysne 
 
 
 
 
 
Agosto de 2004 A General-Equilibrium Closed-Form Solution
To The Welfare Costs of In￿ ation
(Forthcoming, Revista Brasileira de
Economia)￿ y.
Rubens Penha Cysnez
August 30, 2004
Abstract
This work presents closed-form solutions to Lucas￿ s (2000) general-
equilibrium expression for the welfare costs of in￿ ation, as well as to
the di⁄erence between the general-equlibrium measure and Bailey￿ s
(1956) partial-equilibrium measure. In Lucas￿ s original work only nu-
merical solutions are provided.
1 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-logarithmic1. 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.
yKey Words: In￿ ation, Welfare, Closed-Form. JEL: C0, E0.
zEPGE/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.
1Lucas (2000) argues that this is the functional speci￿cation of the money demand
that best ￿ts the United States historical time series.
1In Lucas￿ s (2000) original paper, both the solution of the underlying non-
linear di⁄erential equation leading to the general-equilibrium welfare ￿gures,
as well as the comparison with Bailey￿ s estimates, are based only on numer-
ical methods.
The work is divided as follows. Section 2 presents a continuous-time,
no-growth version of Lucas￿ s shopping time model. Given the correspon-
dent 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.
2 The Model
In Lucas￿ s (2000, sec. 5) analysis of the welfare costs of in￿ ation the rep-
resentative consumer is supposed to maximize utility from the consumption
(c) : Z 1
0
e
￿gt U(c)dt (1)
subject to the households budget constraint (2) and to the transactions-
technology constraint (3):
_ m = 1 ￿ (c + s) + h ￿ ￿m (2)
￿c + m￿(s) ￿ 0 (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:
￿ (s) = rm￿
0 (s) (4)
Equilibrium in the goods market reads:
1 ￿ s = m￿ (s) (5)
2In 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
s0(r) > 0 and m0(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).
The result is the di⁄erential equation [Lucas (2000, equation 5.8)]:
s
0 = ￿
r(1 ￿ s)
1 ￿ s + r m
m
0 (6)
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:
ds
dr
= v(r;s) =
(1 ￿ s)(aAr￿a)
1 ￿ s + Ar1￿a (7)
s(r0) = s0; r0 > 0 (8)
Lucas does not provide a closed-form solution to this equation. His wel-
fare ￿gures, as well as his comparison with Bailey￿ s measure, are based on
numerical calculations.
3 A Closed-Form Solution for the Welfare Costs
of In￿ ation
I start the formal analysis by demonstrating existence and uniqueness.
Proposition 1 Consider s and r in a closed, bounded and convex region
D ￿ R2
++; with r bounded away from zero. Then there exists a unique solution
to (7) and (8).
Proof. It is easy to see that, with r bounded away from zero, v(r;s)
2 C1, and, by the mean-value theorem, and for a certain constant L > 0;
satis￿es the Lipschtz condition j v(r;s1)￿v(r;s2) j￿ L j s1￿s2 j for each par
(r;s1);(r;s2) in D: It follows from a standard result in ordinary di⁄erential
3equations 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 [r0;+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) = s2 ￿ 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
r =
￿
a ￿ 1
A
(1 ￿ s)
h
1 ￿ (1 ￿ s)
￿1=a
i￿ 1
1￿a
(9)
￿ Proof. Start by considering r0 > 0 and the initial condition
s(r0) = s0 (10)
Suppose s(r) is a solution to (7), given (10). Then, since s0(r0) > 0; the
inverse function r = r(s) is de￿ned in a su¢ ciently small neighborhood
of the point s0 and:
dr
ds
+
￿1
a(1 ￿ s)
r =
1
aA
r
a (11)
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 = r1￿a: Then
(11) is equivalent to the equation:
dt
ds
￿
(1 ￿ a)
a(1 ￿ s)
t =
1 ￿ a
aA
Multiplying both sides of this equation by the integration factor exp(￿
R s
0
1￿a
a(1￿￿)d￿) :
d
ds
￿
texp(￿
Z s
0
1 ￿ a
a(1 ￿ ￿)
d￿)
￿
=
1 ￿ a
aA
￿
exp(￿
Z s
0
1 ￿ a
a(1 ￿ ￿)
d￿)
￿
Integrating in s and using the fact that t(0) = 0:
texp(￿
Z s
0
1 ￿ a
a(1 ￿ ￿)
d￿) =
Z s
0
1 ￿ a
aA
exp(￿
Z g
0
1 ￿ a
a(1 ￿ ￿)
d￿)dg
4Solving for the integral of 1=(1 ￿ ￿) :
t =
a ￿ 1
A
(1 ￿ s) +
1 ￿ a
A
(1 ￿ s)
a￿1
a
Use the fact that t = r1￿a to get (9).
4 A Direct Comparison with Bailey￿ s Mea-
sure
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 mea-
sures.
Proposition 3 The di⁄erence between the general-equilibrium (s) and Bai-
ley￿ s partial-equilibrium (B) measure of the welfare costs of in￿ation is given
by:
B ￿ s = a(1 ￿ s)(￿1 + (1 ￿ s)
￿1
a ) ￿ s (12)
Proof. Bailey￿ s measure, in di⁄erential form, is given by the area-under-
the-inverse-demand-curve:
dB = ￿rm
0(r)dr; B(0) = 0
By substituting the double-logarithmic money demand function into the
above expression and integrating:
r =
￿
B (1 ￿ a)
aA
￿ 1
1￿a
(13)
Solve (13) for B and use the value of r given by (9) to obtain (12).
5 Comparing the General-Equilibrium and the
Partial-Equilibrium Measures
Both Lucas (2000), through numerical simulations, and Simonsen and Cysne
(2001), analytically, have shown that Bailey￿ s measure is an upper bound
to Lucas￿general-equilibrium measure, and that the di⁄erence between B
and s in an increasing function of s. A ￿nal Proposition shows that both
conclusions are consistent with equation (12).
5Proposition 4 B(s) ￿ s and the di⁄erence B(s) ￿ s is an increasing func-
tion of s:
Proof. Make B(s) ￿ s = g(s). Then, g(0) = 0 and
g
0(s) =
h
(1 ￿ s)
￿ 1
a ￿ 1
i
(1 ￿ a)
Hence, g0(s) > 0 for any s > 0: It follows that B > s for any strictly positive
values of s and that the di⁄erence B ￿ s increases with s:
References
[1] Bailey, Martin J., 1956, Welfare cost of in￿ ationary ￿nance, Journal of
Political Economy 64, 93-110.
[2] Coddington, E. A. and N. Levinson, Theory of Ordinary Di⁄erential
Equations, McGraw Hill, 1955.
[3] Lucas, R. E. Jr., 2000, In￿ ation and Welfare. Econometrica 68, No. 62
(March), 247-274.
[4] Simonsen, Mario H. and Cysne, Rubens P. , 2001, Welfare costs of in￿ a-
tion and Interest-bearing deposits￿ . Journal of Money Credit and Bank-
ing, February, 33-1, 90-101.
6 
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- Agosto de 2004 – 06 págs. 

A general-equilibrium closed-form solution to the welfare costs of inflation (Forthcoming, Revista Brasileira de Economia)

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A general-equilibrium closed-form solution to the welfare costs of inflation (Forthcoming, Revista Brasileira de Economia)

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