U.S. Money Demand and the Welfare Cost of Inflation in a Currency–Deposit Model
Turan G. Bali
This paper emphasizes that it is crucial to identify the proper specification of money demand as well as the appropriate monetary aggregate to find the exact welfare cost of
inflation. The econometric test results obtained from the nonlinear form of money demand with Box–Cox restriction indicate that not the semi-logarithmic form but the double-log
form with constant elasticity of less than one is a more accurate characterization of the actual data. The welfare cost estimates for the U.S. economy from Bailey’s 1956
consumer’s surplus and Lucas’s 1994 compensating variation approaches imply that for each monetary model currency–deposit, single-asset and each scale variable income,
consumption the double-log function, compared to the constant semi-elasticity Cagan- type demand for money, yields substantial welfare gains in moving from zero inflation to
the Friedman optimal deflation rate needed to bring nominal interest rates to zero. This paper also shows that the estimated welfare cost of inflation is proportional to the money
stock and because M1 is about three times the monetary base in the United States, identifying M1 without modeling the distinctive roles of currency and deposits as the
relevant definition of money overestimates the true welfare cost of inflation.
© 2000 Elsevier Science Inc.
Keywords: Currency–Deposit Model; U.S. Money Demand; Welfare Cost of Inflation JEL Classification: E31, E41
I. Introduction
The estimated welfare cost of inflation is proportional to the money stock, so the choice of an observed monetary aggregate to serve as “money” in the formula is crucial. Lucas
1994 chooses to use M1 as the relevant money and runs his welfare integral from a zero nominal rate to a positive rate—a procedure that lumps currency and interest-bearing
Department of Economics, Queens College, City University of New York, New York Address correspondence to: Dr. T. G. Bali, Department of Economics, Queens College, CUNY, 65-30
Kissena Boulevard, Flushing, New York 11367-1597.
Journal of Economics and Business 2000; 52:233–258 0148-6195 00 –see front matter
© 2000 Elsevier Science Inc., New York, New York PII S0148-61959900035-1
deposits together and treats this composite as non-interest bearing. In this regard, it is relevant to return to Bailey’s 1956 classic article and review his treatment of currency
and deposits. Bailey used Cagan’s 1956 hyperinflation data to estimate the welfare cost of inflation. He set the real rate of interest at zero r 5 0 so that the cost of holding
currency was the anticipated inflation p a proxy for the nominal interest rate i. He assumed that competitive banks pay interest on deposits but are subject to a sterile reserve
requirement. A zero-profit condition was imposed: the revenue from a bank’s interest bearing assets was completely dispersed in interest payments on deposits: i
d
5 1 2 mi where m is the required reserve ratio and i 5 p in his setup. The zero-profit condition
under perfect competition implies that the opportunity cost of holding deposits is i 2 i
d
5mi or mp. Because deposits are partially indexed, Bailey assumed that at very high rates of inflation, the public uses only deposits and all currency would be held as bank
reserves. In this case, Bailey’s welfare integrals run from zero to mp to account for the partial indexing of the deposit rate to inflation. In contrast to Lucas, there is a substantial
reduction in welfare cost of inflation when interest on deposits is partially indexed against inflation.
What is at issue is the polar choice of two very incomplete models. On the one hand, we can lump currency and deposits together and assume that both pay no interest. We can
then run our integrals from zero to a positive interest rate. This is Lucas’ choice. On the other hand, we could assume with Bailey that all high-powered money is held as reserves
and run the welfare integrals from a nominal rate of zero to mi, the opportunity cost of holding interest bearing deposits. Lucas’ choice overstates the welfare loss of deviating
from the Friedman 1969 rule. The assumption in Bailey that all deposits pay interest underestimates the welfare loss. Without a theory of banking in which the distinctive roles
of currency and deposits are modeled, it remains uncertain how large is the degree of overestimation or underestimation.
As an alternative to these two polar cases, we try to model the distinctive roles of currency and deposits in Ramsey–Sidrauski infinite-horizon model. We measure the
welfare cost of inflation using the traditional approach, developed by Bailey 1956, by computing the appropriate areas under currency and deposit demand curves. In our
currency– deposit model, it is possible to run separate integrals: that for currency running from a zero nominal rate to a positive rate, and the integral for deposits running from zero
to mi. As an alternative to Bailey’s consumer’s surplus argument, we also provide the quadratic approximation and the square root formula, [originally developed by Lucas
1994 in a single-monetary-asset model], for evaluating the welfare loss in the currency– deposit model.
In the current literature, the welfare cost of inflation is developed for models involving a money stock made up of currency only, or competitively priced bank deposits only. The
quantitative analysis of the welfare loss is not extended to a framework guided by a model with both currency and demand deposits. For example, articles by Bailey 1956, Calvo
and Fernandez 1983, and Marty 1994 each discuss what amounts to a special case of the model we develop here, where currency and deposits are perfect substitutes, so that
when i
d
. 0 no currency is held, which we refer to as a deposits-only model. Other earlier studies, such as Marty 1976 and Marty and Chaloupka 1988, assume that the required
reserve ratio is either zero or one, which reduces the currency– deposit model to a currency-only model. If the reserve ratio equals one, so that demand deposits pay no
interest, then currency and deposits are of the same stuff. On the other hand, if the reserve ratio is zero, then demand deposits produce neither seigniorage nor a welfare loss; we are,
234 T. G. Bali
in effect, in a currency-only model because the monetary authority receives no revenue from deposits. In addition to the above-mentioned studies that deal with money demand
and welfare cost analysis either in a deposits-only world or in a currency-only world, Eckstein and Leiderman 1992, Lucas 1994, and Dotsey and Ireland 1996 quantita-
tively assess the welfare losses in a general equilibrium single-monetary-asset model without separating money into its currency and deposit components.
This paper extends the current literature by developing a currency– deposit model in an intertemporal optimizing framework and deriving the welfare costs of inflation in a world
of both currency and demand deposits. Because currency and deposit demand functions reach a specific estimable form in the theoretical model introduced in the paper, the
welfare cost function can be derived for both the consumer’s surplus approach of Bailey 1956 and the compensating variation approach of Lucas 1994. In estimating the
welfare costs of deviating from a zero inflation policy and the costs of deviating from the Friedman optimal deflation rate in the currency– deposit model, the paper makes a
contribution to the literature regarding the welfare costs of low inflation rates.
This study also emphasizes that the estimated welfare cost of inflation is sensitive to the specification of the money demand function and to the definition of money. Thus, it
is crucial to identify the proper specification of money demand as well as the appropriate monetary aggregate to serve as “money” in the formula. To determine what form the U.S.
money demand takes, the Box–Cox transformation is applied to the nominal interest rate and the proper specification of the money demand function is identified with the aid of
two different econometric tests. The empirical results indicate that, for each monetary aggregate and each scale variable, the double-log function with constant elasticity is a
more accurate characterization of the actual data and yields substantial welfare cost estimates for the U.S. economy compared to the constant semi-elasticity Cagan-type
demand for money.
Lucas 1994, Dotsey and Ireland 1996, and Lucas 1981 use M1 to calculate the welfare cost of inflation for the American economy. As will be discussed in the paper,
identifying M1 instead of the monetary base as the relevant definition of money leads to an overestimate of the true welfare loss because the estimated welfare cost of inflation is
proportional to the money stock, and M1 is about three times the monetary base in the United States. Within his framework, and with his preferred constant elasticity double-log
function scaled with income, Lucas 1994 finds the welfare cost of a 10 percent nominal interest rate is about 1.3 percent of GDP. In a general equilibrium monetary model, Dotsey
and Ireland 1996 find the welfare cost of a sustained 4 percent inflation rate is over 1 percent of GDP when M1 is defined as money. Fischer 1981 and Lucas 1981 find the
cost of inflation to be surprisingly low. Fischer computes the deadweight loss from an increase in inflation from zero to 10 percent as just 0.3 percent of GNP, identifying money
with the monetary base. Lucas 1981 places the cost of a 10 percent inflation at about 0.45 percent of GNP, using M1 as the measure of money.
According to our welfare cost estimates with Bailey’s 1956 consumer’s surplus and Lucas’s 1994 compensating variation approaches, the double-log function specified with
M1 and scaled with income yields the welfare cost of 1.08 percent of GDP when evaluated at the 10 percent nominal interest rate. In contrast, when the monetary base is used as the
appropriate monetary aggregate, the same functional form with the same scale variable yields the welfare cost of 0.37 percent of GDP, which is about one-third of the welfare
cost estimated with M1.
Welfare Cost of Inflation 235
This paper is organized as follows. Section II describes our currency– deposit model in an intertemporal optimizing framework and outlines the microfoundations for the double-
log form of money demand. Section III sets out the estimation techniques employed in money demand regression equations and provides the econometric test results for the
specification of the U.S. money demand function. Section IV not only measures the costs of deviating from both a zero inflation and an optimal deflation policy but studies the
sensitivity of the estimated welfare cost of inflation to the specification of money demand in both the single-monetary-asset M1 or monetary base model and the currency– deposit
model as well. Section V concludes the paper.
II. The Currency–deposit Model
