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
Several approaches can be found in the literature for introducing the role of money into the intertemporal optimizing framework.
1
To specify the distinctive roles of currency and deposits in an infinite horizon model, we modify Sidrauski’s 1967 general equilibrium
single-monetary-asset model by separating M1 into its currency and demand deposit components and apply a required reserve ratio to the deposit component. In this currency–
deposit model, we think about the money stock as “making life easier” because it allows people to get consumption goods without having to go to the bank and transform bonds
into consumption goods all the time.
We will assume that the economy is populated by infinitely lived consumers or dynasties who derive utility from the only consumption good c
t
and from the real money stock, which is composed of non–interest-bearing real currency m
t
and interest-bearing real demand deposits d
t
. Each household is assumed to have access to a production function that is homogenous of degree one in its two inputs, capital and labor. We will also
assume that labor is supplied inelastically so that the production function with diminishing marginal product of capital can be written as
y
r
5 f~k
t
1 where y
t
is real output, k
t
is real capital and dy
t
dk
t
5 f9k
t
. 0, d
2
y
t
dk
t 2
5 f0k
t
, 0. Here capital is treated as output that is not consumed, so its price is the same as that of
consumption good. The households in the economy begin period t with certain amount of money and pay a lump sum tax H
t
or, if H
t
, 0, receive a lump sum transfer from the government. They can save by accumulating nominal currency M
t11
and nominal demand deposits D
t11
, by investing in real capital k
t11
, and by buying government bonds B
t11
. All four variables denote quantities held at the beginning of period t 1 1. The bonds that agents buy in period t, B
t11
, are sold at the nominal price z
t
and yield one unit of money in period t 1 1, so that the nominal rate of return on bonds between t and t 1 1
is i
t11
5 1 2 z
t
z
t
. The gross real rate of return on bonds is, therefore, defined as 1 1 r
t11
5 1 1 i
t11
1 1 p
t11
, where p
t11
5 P
t11
2 P
t
P
t
is the inflation rate between
1
Within the infinite horizon model, four approaches have been adopted to incorporate the role of money. The first is to incorporate its role as a medium of exchange through the so-called cash-in-advance constraint,
originally proposed by Clower 1967 and then generalized by Lucas and Stokey 1983 to include “cash” and “credit” goods. The second is the “transactions-time” approach developed by McCallum 1983 and McCallum
and Goodfriend 1987. The third, originally due to Sidrauski 1967, is to introduce money directly into the utility function. The fourth approach introduced by Driffill et al. 1990 views money as having value because
other financial assets require real transaction costs associated with real purchases.
236 T. G. Bali
t and t 1 1. The number of bonds B
t
divided by the price of the commodity P
t
in that period is denoted by b
t
. We will assume that competitive banks pay interest on deposits but are subject to a fractional reserve requirement on demand deposits. Assuming that
banks face no operating costs, the zero-profit condition under perfect competition yields i
d
t 5 1 2 m
t
i
t
implying that the opportunity cost of holding demand deposit is m
t
i
t
: the difference between i
t
and the nominal interest on demand deposits i
d
t. In each period, households have the following budget constraint
M
t11
1 D
t11
1 P
t
k
t11
1 z
t
B
t11
5 M
t
1 D
t
~1 1 i
d
~t 1 P
t
k
t
~1 2 d 1 B
t
1 P
t
y
t
2 P
t
c
t
2 H
t
2 in nominal terms and thus
~1 1 p
t11
m
t11
1 ~1 1 p
t11
d
t11
1 k
t11
1 ~1 1 r
t11 21
b
t11
5 m
t
1 d
t
~1 1 i
d
~t 1 k
t
~1 2 d 1 b
t
1 f~k
t
2 c
t
2 h
t
3 in real terms, where h
t
5 H
t
P
t
, and d is the depreciation rate. We will further assume that the household preferences are time separable, that is
O
t50 `
~1 1 r
21
u~c
t
,m
t
,d
t
, 4
where r is the subjective rate of time preference. Households maximize equation 4 subject to equation 3, and the first-order Euler conditions for the maximum problem can
be written as equalities holding for each period:
u
c
~c
t
,m
t
,d
t
2 V
t
5 0, 5a
u
m
~c
t
,m
t
,d
t
1 V
t
2 ~1 1 r~1 1 p
t
V
t21
5 0, 5b
u
d
~c
t
,m
t
,d
t
1 ~1 1 i
d
~tV
t
2 ~1 1 r~1 1 p
t
V
t21
5 0, 5c
V
t
2 ~1 1 r~1 1 r
t 21
V
t21
5 0, 5d
V
t
~1 1 f9~k
t
2 d 2 ~1 1 rV
t21
5 0, 5e
where V
t
and V
t21
are the Lagrangian multipliers attached to the budget constraints at time t and t 2 1, respectively. From equations 5d and 5e, we have
V
t21
V
t
5 ~1 1 r
t
~1 1 r 5
~1 1 f9~k
t
2 d ~1 1 r
, which implies that the real rate of return on capital equals the real rate of interest on bonds: f9k
t
2 d 5 r
t
. Dividing equations 5b and 5c by equation 5a, and using the fact that
V
t21
V
t
5 ~1 1 r
t
~1 1 r where 1 1 r
t
5 1 1 i
t
1 1 p
t
, we get the following conditions:
u
m
~c
t
,m
t
,d
t
u
c
~c
t
,m
t
,d
t
5 21 1 ~1 1 r~1 1 p
t
V
t21
V
t
5 i
t
, 6a
Welfare Cost of Inflation 237
u
d
~c
t
,m
t
,d
t
u
c
~c
t
,m
t
,d
t
5 21~1 1 i
d
1 ~1 1 r~1 1 p
t
V
t21
V
t
5 i
t
2 i
d
~t, 6b
which imply u
m
~c
t
,m
t
,d
t
5 i
t
u
c
~c
t
,m
t
,d
t
, 7a
u
d
~c
t
,m
t
,d
t
5 ~i
t
2 i
d
~tu
c
~c
t
,m
t
,d
t
. 7b
In order to find a closed from solution for real currency and real deposit demand functions, suppose that the period utility function takes the following CES-isoelastic form:
u~c
t
,m
t
,d
t
5 g
1 1u
c
t ~u21u
1 g
2 1u
m
t ~u21u
1 g
3 1u
d
t ~u21u
u~u21 12
1 s
1 2 1
s .
8 where u . 0 is the intertemporal substitution elasticity. Imposing CES-isoelastic prefer-
ences on first-order conditions, equations 5a, 5b, and 5c, yields a generalization of asset demand functions that take the following double-log form with constant elasticity of
u:
m
t
5
S
g
2
g
1
D
i
t 2u
c
t
9a d
t
5
S
g
3
g
1
D
~i
t
2 i
d
~t
2u
c
t
9b where the consumption elasticities of demand for real currency and real deposits equal
one.
2
2
Imposing the same interest rate elasticity on currency and deposit demand may seem too restrictive to the reader. However, in this paper, a currency–deposit model is developed to estimate the “steady-state” welfare cost
of inflation and the welfare cost analysis is compatible with a steady state only if the velocity of money remains constant. In turn, constant velocity requires a unitary scale elasticity of demand for money. Therefore, it is crucial
that the scale elasticity is unity. If this were not the case, we could not use steady-state analysis. The use of unitary scale elasticity in our theoretical model leads currency and deposit demand functions to have the same
elasticity or semi-elasticity with respect to their own opportunity costs. To illustrate this, instead of using the CES-isoelastic utility function given in equation 8, we can employ its more flexible version,
u~c
t
,m
t
,d
t
5
S
1 1 2 v
DFS
1 1 2 g
D
~c
t 12g
2 1 1
S
w 1 2 a
D
~m
t 12a
2 1 1
S
f 1 2 b
D
~d
t 12b
2 1
G
12 1
v
, in our intertemporal optimizing framework, which yields the following double-log currency and deposit demand
functions: m
t
5 w
1a
i
t 21a
c
t ga
, 9a9
d
t
5 f
1b
~i
t
2 i
d
~t
21b
c
t gb
, 9b9
where ga and gb are, respectively, the scale elasticities of demand for real currency and real deposits. 1a and 1b are, respectively, the elasticities of demand for real currency and real deposits with respect to their own
opportunity costs. It is clear from equations 99 that, in order for the scale elasticity to be unity which is required for the steady-state assumption, a 5 b 5 l must hold, which implies that the asset demand functions must have
the same opportunity– cost elasticities in this currency– deposit model.
238 T. G. Bali
The monetary model developed in this section is quite new in the sense that it yields a specific estimable form for asset demand functions as in equations 9a and 9b, which
are used to derive the welfare cost function in a currency– deposit framework for both the consumer surplus and the compensating variation approaches. In addition, the currency–
deposit model is more general than the earlier monetary models because, as discussed in the previous section, it nests the currency-only and deposits-only models.
As presented above, standard utility functions yield double-log demands for money although the semi-log form has been widely used in money demand studies, more a matter
of convenience than theoretical or empirical attractiveness. The convenience comes from its simplicity, the fact that steady-state seigniorage is easily defined at a constant nominal
interest rate and the fact that the seigniorage-maximizing interest rate as well as the maximum seigniorage are easily derived. However, the semi-logarithmic form has its
drawbacks. The micro-foundation of the semi-log specification in either “the shopping time” or “the money-in-the-utility function” approach is somewhat ad hoc. We have to
impose a specific form on the utility function or on the shopping time technology in order to arrive at the constant semi-elasticity Cagan-type demand for money. The utility
function and the similarly shaped shopping time technology take the following unusual forms, respectively:
u~c
t
,m
t
,d
t
5 m
t
~1 1 w 2 ln m
t
a 1
d
t
~1 1 v 2 ln d
t
b 1 v~c
t
, u
m
. 0,u
d
. 0,u
c
. 0 c~c
t
,m
t
,d
t
5 m
t
~ln m
t
2 w 2 1
t
a 1
d
t
~ln d
t
2 v 2 1
t
b 1 v~c
t
, c
m
, 0,c
d
, 0,c
c
. 0 which imply the following Cagan-type demand for currency and deposits:
ln m
t
5 w 2 av
c
i
t
, ln d
t
5 v 2 bv
c
~i
t
2 i
d
~t, where the marginal utility of consumption is assumed to be a positive constant, v
c
. 0.
III. The Specification of the Money Demand Function
