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Working Paper Series
The Welfare Cost of Inflation in General
Equilibrium
WP 94-04
Michael Dotsey
Federal Reserve Bank of Richmond
Peter Ireland
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Working Paper 94-4
THE WELFARE COST OF INFLATION
IN GENERAL EQUILIBRIUM
Michael Dotsey*
and
Peter Ireland*
Research Department
Federal Reserve Bank of Richmond
March 1994
*We would like to thank William English, Jeffrey Lacker, Robert Lucas, Kevin
Reffett, Thomas Sargent, seminar participants at Duke University and the
Federal Reserve Bank of Richmond, and participants at the Federal Reserve
System Conference on Business and financial Analysis. September 1993. for
their helpful comments and suggestions. The views expressed here do not
necessarily reflect those of the Federal Reserve Bank of Richmond or the
Federal Reserve System.
Abstract
This paper presents a general equilibrium monetary model in which
inflation distorts a variety of marginal decisions. Although individually none
of the distortions
is very large, they combine to yield substantial welfare
cost estimates. A sustained 4% inflation like that experienced
in the U.S.
since 1983 costs the economy the equivalent of 0.41% of output per year when
currency is identified as the relevant definition of money and over 1% of
output per year when Ml is defined as money. The results illustrate how the
traditional,
partial equilibrium
cost of inflation.
approach can seriously underestimate
the true
I.
Introduction
A sound judgment regarding the desirability of price stability as the
principal goal of monetary policy requires an accurate assessment of the
consequences of sustained price inflation. Thus, monetary economists have
devoted considerable effort to measuring the welfare cost of inflation.
The traditional approach, developed by Bailey (1956) and Friedman (19691,
treats real money balances as a consumption good and inflation as a tax on
real balances. This approach measures the welfare cost by computing the
appropriate area under the money demand curve.
Applications of the Bailey-Friedman analysis, most notably those of
Fischer (1981) and Lucas (19811, find the cost of inflation to be
surprisingly low. Fischer computes the deadweight loss generated by an
increase in inflation from zero to 10% as just 0.3% of GNP using the
monetary base as the definition of money. Lucas places the cost of a 10%
inflation at 0.45% of GNP using Ml as the measure of money.
Since these
estimates appear small relative to the potential cost of a disinflationary
recession, they provide little support for the idea that price stability is
an essential goal for monetary policy.
The inflation tax, however, may distort economic decisions along
margins that the partial equilibrium approach of Bailey and Friedman
ignores. This paper, therefore, takes a general equilibrium approach to
assessing the welfare cost of inflation. A unique feature of the model
developed here is an explicit transactions technology that gives rise to a
money demand function resembling those estimated with data from the US
economy. Thus, the analysis begins by accounting for Bailey-Friedman costs
1
of inflation of the magnitude estimated by Fischer and Lucas.
The Bailey-Friedman approach, however, turns out to capture only a
fraction of the total cost of inflation in this model.
By explicitly
focusing of the role of money in facilitating transactions, the model
identifies several other distortions associated with the inflation tax.
First, as in Cooley and Hansen (1989, 19911, inflation causes agents to
inefficiently substitute out of market activity and into leisure. Second,
as suggested by Karni (19741, inflation causes agents to devote productive
time to activities that enable them to economize on their cash balances.
When adapted to a general equilibrium setting, Karni's specification
implies that inflation draws a fraction of the labor force out of goods
production and into a distinct financial sector.' Finally, the model takes
its specification for goods-producing technologies from Romer (19861, so
that the allocative effects of inflation can potentially influence the
growth rate, as well as the level, of aggregate output. Although none of
the additional distortions is very large, they combine to yield estimates
of the welfare cost of inflation that are more than three times the size of
the Fischer-Lucas estimates.
Black et al. (19931, Coleman (19931, De Gregorio (19931, Gomme
(19931, Jones and Manuelli (19931, Marquis and Reffett (19931, and Wang and
Yip (1993) also examine the effects of inflation in endogenous growth
settings.
Thus, this paper extends previous work by adding a novel
transactions technology to a familiar monetary growth model.
Unlike more
conventional cash-in-advance specifications, the transactions technology
used here can be parameterized to generate a money demand function that is
as interest-elastic as those estimated with US data.2 Consequently, the
2
model is ideally suited for comparing the partial equilibrium BaileyFriedman cost to the full general equilibrium cost of inflationary policy.
Indeed, the results show how the traditional, partial equilibrium approach
can seriously underestimate the true cost of inflation and thereby
understate the case for price stability.3
II.
A.
A General Equilibrium Model of the Inflation Tax
The
Economic Environment
The economy consists of a continuum of markets, indexed by iE[O,l),
arranged on the boundary of a circle with unit circumference. In each
market, a distinct, perishable consumption good is produced and traded in
each period t=0,1,2,.... Hence, the economy’s consumption goods are also
indexed by is[O,l), where good i is sold in market i.
Large numbers of identical households, financial intermediaries, and
goods-producing firms inhabit each market i.
Enough symmetry is imposed on
these agents’s preferences, endowments, and technologies that the analysis
considers without loss of generality the behavior of a single
representative household, a single representative intermediary, and a
single representative firm.
The representative agents all live at market
0, so that the index i measures the distance of market i from their home.
The government, which otherwise plays no role in the economy,
provides households with noninterest-bearing fiat money.
It supplies each
household with rn:units of money at the beginning of period t=O and
augments this supply by making identical lump-sum transfers h t to all
households at the beginning of dates t=0,1,2,.... Hence, the per-household
money supply mF+I at the end of date t satisfies
ms
(1)
t+1
=
(l+gt)m:,
where the rate of money growth gt is given by
(2)
gt
=
ht/ml.
The government announces the complete sequence {gt)yxo of money growth
rates at the beginning of period t=O. There is no uncertainty, and all
agents have perfect foresight.
B.
Households
and Their
Trading
Opportunities
The representative household at market i=O has preferences over
leisure and the entire continuum of consumption goods as described by the
utility function
(3)
tfr[
d InIct(iIldi + BJt ] ,
/3e(O,lI,B>O.
Thus, et(i) denotes the household's consumption of good i and Jt its
leisure at time t.
Following Lucas and Stokey (19831, the representative household is
imagined to consist of two members: a worker and a shopper. During each
period t, the representative worker rents out his household's capital stock
kt at the real rate rt and supplies 1: units of labor at the real wage w
to goods-producing firms.
t
He also supplies If units of labor to financial
intermediaries. The worker makes his labor-supply decisions subject to the
time constraint
(41
1
I
Jt + 1; + 1:
at each date t.
The representative shopper, meanwhile, travels around the circle in
order to acquire goods for his household’s consumption. As in Prescott
(1987), Schreft (19921, and Gillman (19931, the shopper chooses between two
alternative means of making purchases in each market i.
alternative is to use government-issued money.
His first
Since competition equates
the nominal price p, of consumption goods across markets, the shopper may
acquire et(i) units of good i in exchange for p,c,(i) units of money at
time
t.
The shopper’s second alternative for purchasing good i is to enlist
the services of a financial intermediary. At a cost of y(i) units of
labor, an intermediary verifies the shopper’s identity and guarantees his
ability to pay, so that a firm in market i is willing to sell its output on
credit at time t.
The communications and record-keeping costs of
facilitating a credit transaction do not depend on the size of the purchase
but increase as the shopper travels farther from home.
Hence, 7 is a
strictly increasing function of i. Under the additional assumption that
lim ;y(i)~=m,
some goods will always be purchased with cash, and there is a
i+l
well-defined demand for money.
In exchange for its services at time t, the intermediary in market i
charges the representative household the real price qt(i). Since the
intermediary’s cost r(i) is independent of the size of the transaction but
depends nontrivially on i, competition ensures that the function qt(i)
satisfies these same properties. Thus, the representative shopper may
acquire et(i) units of good i on credit at time t at a total nominal cost
5
of pt[ct(i)+qt(iI]:ptct(i) to pay for the goods themselves and ptqt(iI to
compensate the intermediary.
Let the indicator function c,(i)=0 if the representative shopper
purchases good i with money at
intermediary instead. Let
the shopper into
time
t;
mt
time
t,
and let Et(i)=1 if he uses an
denote the nominal cash balances carried by
these are augmented at the beginning of the period
by the government transfer ht. Since the shopper must use money whenever
he chooses not to hire an intermediary, he faces the cash-in-advance
constraint
m +h
t
t
(5)
Pt
1
L
I
Cl
[l-<t(i)lct(i)di
in each period t.
After consuming its purchases at the end of time t, the household
participates in a centralized asset market, where it receives its rental
payments rtkt and wages wt(lr+l:I and pays for the goods that it bought on
credit earlier in
time
t.
The household uses any excess funds to
accumulate the cash balances mt+l that it will carry into period t+l and to
purchase unsold output from the representative firm, which it combines with
its depreciated capital stock (1-6)kt in order to carry kt+I units of
capital into period t+l.
The representative household is also permitted to borrow from and
lend to other households in the end-of-period asset market by issuing
purchasing one-period, nominally-denominated discount bonds.
Off bt+l
or
Bonds paying
units of money in the time t+l asset market sell for b /Rt units
t+1
of money in the time t asset market, where Rt is the gross nominal interest
rate between t and t+l.
Since these bonds are available in zero net
6
supplys bt+l=0 must hold as an equilibrium condition in each period t.
As sources of funds in period t, the representative household has its
initial money and bond holdings, its beginning-of-period government
transfer, its rental and wage receipts, and its capital stock after
depreciation. As uses of funds it has its purchases of consumption goods,
its payments to intermediaries, and the capital, money, and bonds that it
It therefore faces the budget constraint
will carry into period t+l.
m
(61
+b
’
+h
t
’
Pt
+ rtkt + w,(l~+lf) + (1-6)kt
1
2
f
1
ct(i)di +.
0
in each period t.
m
b
t+1
t+1
Et(i)qt(i)di + kt+l + + -
0
Pt
ptRt
The representative household chooses sequences for
et(i), E,(i), Jo, ly, lf, kt+l, mt+l, and bt+l to maximize the utility
function (31 subject to the time constraint (41, the cash-in-advance
constraint (51, and the budget constraint (61, taking the sequences for ht,
and Rt as given.
r,w
t
t’ Pt* qt(i),
It also takes its initial holdings of
capital ko>O, money mo=mi, and bonds bo=O as given.
c*
The.Representative
Intermediary’s
Problem
An intermediary in market i hires ;r(i)units of labor and charges
qt(i) if the representative shopper purchases good i on credit at time t.
Thus, the representative intermediary chooses total labor input ni to
maximize profits
1
(7)
7r;= I
Ct(i)qt(i)di - wtn:
0
subject to the technological constraint
7
1
ni 5~
(8)
s
0
Et(i)T(ildi
at each date t, taking wt, 5,(i), and qt(i) as given.
D.
The Representative
Goods-Producing
Firm's
Problem
The representative goods-producing firm in market i=O hires kt units
of capital and n: units of labor from households in each period t in order
to produce output of consumption good i=O.
(9)
lrz
=
Its profits in period t are
A(ktla(n~)‘-a(Kt)s- rtkt - wtng
t,
cE(O,l), Tpo.
The production function in equation (9) contains Kt, the aggregate
capital stock-per household at time t.
Following Romer (19861, capital is
interpreted broadly here to include stocks of human capital and disembodied
knowledge in addition to physical capital. While goods production features
constant returns to scale at the firm level, spillover effects associated
with the accumulation of human capital generate increasing
aggregate level.
returns at the
Increasing returns make the economy’s growth rate
endogenous and possibly dependent on the inflation rate.
The
representative firm takes the aggregate capital stock Kt as well as the
factor prices rt and wt as given when maximizing (9).
E.
Equilibrium
Conditions
A competitive equilibrium in this economy consists of sequences for
prices and quantities that are consistent with the solutions to the
optimization problems for households, intermediaries, and firms outlined
above.
Given the initial conditions ko=Ko>O, m,=mi, and bo=O, equilibrium
prices and quantities must also satisfy the zero profit conditions
(10)
the consistency condition
k
(11)
t+1 =
Kt+ls
and the market-clearing conditions for goods, labor, money, and bonds
(1.2)
A(ktIa+T)(n;I1-”
+ (l-6)kt =
kt+I + j ct(i)di,
0
(13)
nf
=
l:,
(14)
n:
=
1
t’
m
(15)
t+1
f
m9
t+1’
=
and
b
(16)
t+1
=0
in each period t .
III.
The General Equilibrium Effects of the Inflation Tax
Part A of the appendix demonstrates that in equilibrium, there exists
a borderline index st for each date t such that the representative
household purchases all goods with indices isst on credit and all goods
i>s with cash.
t
(17)
This borderline index is determined by the solution to
7(Stl =
Iln(ht+~,I-ln(ht)l/wh ,
t t
where ht is the nonnegative multiplier on the budget constraint (6) and p,
is the nonnegative multiplier on the cash-in-advance constraint (5) from
the household’s optimization problem. As in Schreft (1992) and Gillman
(19931, the shopper uses credit close to home and cash far from home, since
intermediation costs increase with distance.
9
The representative household’s optimal et(i) is a step function at
each date t:
+/At
for isst
et(i) =
(18)
cz=l/(ht+ptlfor i>st
Since /.L,ZO,
c:EcF.
AS
in Prescott (19871, the shopper makes larger
purchases on credit and smaller purchases with cash, since the
intermediation costs are independent of the size of the transaction.
Equation (18) and the cash-in-advance constraint (51 determine
equilibrium money demand as
(mt+ht)/pt =
(19)
(l-sp~.
The technological constraint (81 and the market-clearing condition (14)
determine employment in the financial sector
S
t
1; =
(20)
as
I
;r(i)di.
0
The inflation tax causes the household’s cash-in-advance constraint
to bind, so that higher rates of inflation tend to be associated with
larger values of the multiplier pt. Since iy is increasing as a function of
i, equation (17) suggests that higher inflation rates are also associated
with higher values of s .
t
That is, under higher rates of inflation the
representative household purchases a wider range of goods with the help of
intermediaries.
Equation
(18)
then indicates that the inflation tax distorts
consumption and production decisions in two ways.
First, since c:>cF, the
representative household purchases different consumption goods in different
quantities; its marginal rate of substitution between cash and credit goods
10
deviates from the corresponding marginal rate of transformation. Second,
since cy is decreasing as a function of cc,,the representative household
purchases cash goods in smaller quantities so that overall, market activity
is reduced. These are the marginal effects of the inflation tax studied by
Cooley and Hansen (1989, 1991). Here, however, the production technology
described in equation (9) allows these allocative effects of inflation to
change the growth rate, as well as the level, of aggregate output.
Equation (19) suggests that the representative household economizes
on its cash balances in the face of a positive inflation tax both by
purchasing a wider range of goods without money (i.e., by increasing stI
and by consuming less of those goods that it purchases with money (i.e., by
decreasing cil. Thus, the demand for money is interest-elastic and gives
rise to the Bailey-Friedman cost of the inflation tax.
Finally, equation (20) indicates that as the household increases s
t
in response to a higher inflation tax, the size of the labor force employed
in the financial sector increases. The diversion of labor resources out of
productive activity and into finance also contributes to the welfare cost
of inflation. Again, the goods-producing technology in (9) provides a
channel through which this allocative effect can influence the economy’s
long-run growth rate.
Thus, the model associates a number of distortions with the inflation
tax.
It is not possible, however, to assess the magnitude of any of these
distortions analytically. Hence, the following sections apply numerical
methods to determine the quantitative effects of inflation in general
equilibrium.
11
IV.
Model Parameterization
In order to apply numerical methods, specific values must be assigned
to the model’s parameters. The household’s discount rate is set at p=O.99
and the depreciation rate at.6=0.025 so that each period in the model
corresponds to one quarter year.
Sustained, balanced growth occurs when
the aggregate production function is linear in the capital stock, so ct=O.4
and -r)=O.6.With A=0.265, the economy grows at a constant annual rate of 2%
(the US average since 19591 under a constant annual inflation rate of 5%
(again, the US average since 1959). The representative worker devotes 20%
of his
time
to labor (the figure used by King and Rebel0 1993) under 5%
inflation when B=4.25.
The magnitude of the Bailey-Friedman cost of inflation hinges on two
numbers : the size of the tax base and the interest elasticity of money
demand.
(21)
When the intermediary’s cost function is specialized to
a(i)
=
~[i/(l-i)le,
;y>o,e>o,
the parameters 7 and 8 can be chosen so that the size of the tax base and
the interest elasticity of money demand in the model match corresponding
figures ‘in the US economy.4
The next section constructs equilibria for two
specifications, one in which money is defined as currency and the other in
which money is defined as Ml.
The alternative definitions of money require
different sets of values for 7 and 8.
Following Cooley and Hansen (19911, the size of the inflation tax
base in the US economy is measured by the fraction of all purchases that
are made using money.
Avery et al. (1987) report that in 1984, when
inflation was about 4%, US households made 30% of their transactions with
12
currency and 82% of their transactions with Ml.
to the value of
These fractions correspond
1-st under 4% inflation in the model.
Annual data from 1959-1991 yield estimates of the money demand
equations
ln(vc) =
2.88 + 2.73R
ln(v1) =
1.24 + 5.95R,
and
where vc is the income velocity of currency, vl is the income velocity of
Ml, and R is the 6-month commercial paper rate.5 The OLS coefficients on R
in these equations measure the long-run interest semi-elasticity of money
demand.
An analogous statistic in the model economy is
[ln(vIo)-ln(vo)
l/(Rlo-Ro),
where v
10
and v. are the constant annual velocities of money and Rio and R.
are the constant annual nominal interest rates that prevail under constant
annual inflation rates of 10% and zero.
To match the tax base and the elasticity figures in the data and
model, ~=0.00075 and 8=2.45 for the currency specification and r=O.O0933
and 8=0.333 for the Ml specification. With these combinations of iyand 8,
the annual velocity of money under 5% inflation in the model economy is
19.9 for currency and 7.6 for Ml.
Annualized velocity in the model varies
inversely with the assumed length of each period; a shorter period length
implies a higher annual velocity. Thus, the fact that these figures for
velocity are similar in magnitude to the averages of 21.6 for currency and
5.4 for Ml found in US data since 1959, they are also consistent with the
identification of one model period as one quarter year.
13
V.
The Quantitative Effects of Inflation in General Equilibrium
This section computes the welfare cost of monetary policies that call
for constant rates of money growth. These policies give rise to steadystate equilibria in which all variables grow at constant rates.
Part B of
the appendix outlines a method for constructing these steady-state
equilibria.
Table 1 describes steady-state equilibria under the benchmark policy
that yields a constant zero inflation rate.6
It compares these equilibria
to those obtaining under constant 4% (the US average since 19831, and 10%
(the alternative policy considered by Fischer 1981 and Lucas 1981) annual
rates of inflation.
It also reports results from adopting the Friedman
(19691 rule, under which the money supply is contracted at the rate of time
preference so as to make the nominal interest rate equal to zero.
With an
annual rate of time preference of about 4% and a 2% annual rate of output
growth, the steady state real interest rate in this economy is
approximately 6%.7
Thus, following the Friedman rule generates a 6% annual
rate of price deflation.
Under a constant rate of inflation, the representative shopper makes
a constant fraction of his purchases with cash.
The model is parameterized
so that with 4% annual inflation, this constant fraction is about 30% if
money is defined as currency and about 80% if money is defined as Ml.
Table 1 indicates that for either specification, the shopper uses money in
a smaller range of transactions when inflation is higher.
Thus, the
steady-state velocity of money rises with the inflation rate.
The model is parameterized so that the representative worker devotes
14
approximately 20% of his time endowment to labor. Table 1 shows that as
the inflation rate rises, the representative household tends to substitute
out of market activity, which requires either money or costly financial
services, and into leisure, which can be enjoyed without the use of a means
of exchange.
In addition to this substitution effect, however, there is a
negative wealth effect associated with an increase in the inflation tax.
While the substitution effect always dominates in Cooley and Hansen's
(1989, 1991) models, Cole and Stockman (1992) find that the wealth effect
can easily dominate in their version of the cash-in-advance model in which
the use of money can be circumvented at a cost in terms of real resources.
The wealth effect can dominate here as well, so that an increase from 4% to
10% inflation actually increases the household's labor supply under the Ml
specification.
Higher rates of inflation shift the allocation of the labor force in
addition to changing the total labor supply. Table 1 shows that while the
fraction of the labor force working for intermediaries is always less than
l.S%, this share rises with the inflation rate.
The substitution of labor
out of goods production and into leisure and finance tends to reduce the
growth rate of output via the spillover effects of aggregate productive
activity. But the relationship between inflation and growth is not
generally monotonic; the wealth effect that increases the household's labor
supply going from 4% to 10% inflation also increases the economy's growth
rate under the Ml specification. In general, the effects of inflation on
growth are small: 10% inflation reduces the annual growth rate from 2.12%
to 2.07% under the currency specification and from 2.03% to 1.97% under the
Ml specification.
15
The welfare cost of inflation is measured, following Cooley and
Hansen (1989, 19911, by the permanent percentage increase in the
consumption of all goods that makes that representative household as well
off under a positive rate of inflation as it is under zero inflation. This
figure is converted from a fraction of consumption to a fraction of output
by multiplying it by the constant ratio of consumption to output under
positive inflation. Table 1 shows that when money is defined as currency,
a sustained 4% inflation like that experienced in the US since 1983 has a
cost that is equivalent to a permanent 0.41% decrease in output.
inflation costs almost 0.92%
of
OUtpUt.
A 10%
When money is defined as Ml, a 4%
inflation costs 1.08% of output, and a 10% inflation costs 1.73% of output.
Table 1 also shows the welfare gain from adopting the Friedman (19691,
equivalent to a permanent 0.91% increase in output in the currency
specification and a permanent 2.22% increase in output in the Ml
specification.
Table 2 reports the Bailey-Friedman cost of inflation in the model
economy, computed as the area under the money demand curve that is lost as
the steady-state inflation rate increases. Since by construction the model
gives rise to a money demand curve that resembles those estimated with US
data, the Bailey-Friedman costs are quite similar to those reported by
Fischer (19811 and Lucas (1981). With money defined as currency, the
Bailey-Friedman analysis puts the cost of a 10% inflation at about 0.06% of
output.
Both the tax base and the elasticity of demand are larger when
money is defined as Ml.
Hence, the Bailey-Friedman cost of inflation is
higher as well: a 10% inflation costs about 0.42% of income.
The Bailey-Friedman approach also indicates that the welfare gain
16
from adopting the Friedman rule is substantial, equal to 0.59% of output,
under the currency specification. Recall that for currency, the parameters
of the transaction technology (211 are set so that under 4% inflation, the
representative household makes only 30% of its purchases with money.
The
representative household makes all of its purchases with cash under the
Friedman rule, since the zero nominal interest rate eliminates the
opportunity cost of holding real balances.
In order to reduce the fraction
of cash transactions from 100% under the Friedman rule to 30% under 4%
inflation, the model must give rise to a money demand function that is
extremely elastic at low nominal rates of interest. Here, as in Lucas
(19931, the high elasticity of money demand at interest rates close to zero
implies that there are large welfare gains from moving to the Friedman
rule.
Comparing the welfare cost estimates in tables 1 and 2 illustrates
that the partial equilibrium analysis of Bailey and Friedman generally
captures only a fraction of the total cost of the inflation tax.
In
addition to its effects on velocity, the inflation tax causes agents to
inefficiently allocate productive labor across its various uses.
labor-supply effects may seem small, but they contribute to
The
estimates
of
the total welfare cost of inflation that are much larger than those
obtained using the Bailey-Friedman approach.
In order to isolate the contribution of labor-supply effects to the
welfare
cost
estimates
reported in table 1, table 3 considers a version of
the model with exogenous growth. This version of the model replaces the
production function shown in equation (9) with the more conventional CobbDouglas specification
17
Instead of the aggregate spillovers emphasized by Romer, sustained growth
is now driven by exogenous technological change:
X
As before, a=O.4;
t =
PXt-,.
the economy grows at the annual rate of 2% in steady
state when p=(l.OZl”*.
Table 3 demonstrates that, for the most part, the effects of
inflation do not depend on whether growth is endogenous or exogenous. The
changes in velocity, total labor supply, and the share of the labor force
in finance shown in table 3 are almost identical to those in table 1.
The
welfare cost estimates, however, are much smaller under exogenous growth.
Consider, for example, that in the currency specification with endogenous
growth, the annual growth rate falls from 2.12% under zero inflation to
2.07% under 10% inflation. As emphasized by Lucas (19871, policies that
induce even
small
changes in an economy’s growth rate have substantial
welfare consequences.
In fact, comparing tables 1 and 3 reveals that
growth effects increase the welfare cost of 10% inflation from 0.20% to
0.91% of output under the currency specification and from 0.92% to 1.73% of
output under the Ml specification.
Since these results indicate that growth effects play a large role in
generating the welfare
cost
estimates
reported in table 1, it is worth
noting that empirically, Kormendi and Meguire (19851, Fischer (19911, and
De Gregorio (19931 find that differences in inflation do contribute
significantly to explaining cross-country differences in growth.
Levine
and Renelt (1992) argue that results from cross-country studies such as
these are not generally robust. However, the results in table 1 also
18
explain why the inflation-growth rate link may be difficult to detect in
the data: the changes in growth rates are small and not always monotonic.
Finally, note that although the economy's growth rate increases going from
4% to 10% inflation under the Ml specification, welfare still decreases.
Policies that promote growth do not always increase welfare.
VI.
Summary and Conclusions
In the general equilibrium model developed here, the inflation tax
distorts a variety of marginal decisions. Agents inefficiently economize
on their holdings of real cash balances. They substitute out of market
activity by taking more leisure. They divert productive resources out of
goods production and into finance.
The model shows that individually, none of these distortions is very
large. By construction, the model's money demand function matches those
estimated with US data.
Hence, the Bailey-Friedman cost of inflation in
the model is similar in magnitude to the figures obtained with US data by
Fischer (1981) and Lucas (19811, which are too small to justify the expense
of a disinflationary recession. Similarly, the effects of inflation on the
total labor supply and its sectoral allocation are small.
These labor-
supply effects are allowed to influence the economy's long-run growth rate,
but a 10% inflation turns out to reduce the growth rate by only 0.05%
compared to a regime of price stability.
The various small distortions, however, combine to yield substantial
estimates of the total cost of inflation. A 4% inflation like that
19
experienced in the US since 1983 costs the economy 0.41% of output per year
when currency is identified as the relevant definition of money and over 1%
of output per year when Ml is defined as money.
These higher estimates
strengthen the case for making price stability the principal objective for
monetary policy.
More generally, the findings demonstrate the usefulness of general
equilibrium models for policy evaluation.
In this case, a partial
equilibrium approach to measuring the welfare cost of suboptimal policy
grossly underestimates the true welfare effects.
Only when all of the
policy-induced distortions are allowed to interact can reliable estimates
be obtained.
20
Appendix
This appendix derives the equilibrium conditions presented as
equations (17)-(20) in the text.
It also outlines methods for numerically
constructing competitive equilibria under policies that call for constant
money growth rates.
A.
Derivation
of
Equilibrium Conditions
In the representative household’s optimization problem, let At be the
nonnegative multiplier on the budget constraint (6) and let ~1,be the
nonnegative multiplier on the cash-in-advance constraint (5). Let c:(i) be
the household’s consumption of good i at time t if it purchases this good
with money; let c:[i) be the household’s consumption of good i at time t if
it purchases this good on credit. The first order conditions from the
household’s problem are
(A.l)
c:(i) =
ci =
l/(At+pt)
(A.2)
c:(i) =
ct =
l/At
(A.3)
c(i)
t
[l-<t(i)lci+ <,(i)c’
t
(A.4)
et(i) =
=
1 if ln(c:)-Attc:+qt(i)lrln(c~)-(ht+Cl,)c~
i
(A.5)
(A.6)
(A.7)
htwt
ht
=
%‘pt
=
0 otherwise
B
@Atc;[rt+1+(1-6)1
=
P(At+1+cct+l)/P
t+1
21
(A. 8)
=
%'ptRt
pht+*'pt+l -
The first order condition from the representat.ive intermediary’s problem is
(A.91
n:
=
s Et(iIT(iIdi.
0
The first order conditions from the representative goods-producing firm’s
problem are
(A.101
r
(A.111
w
t
t
=
aA(kt)a-‘(n~l’-a(Kt)~
=
(l-a)A(kt)a(n~)-a(Kt)l).
Equation (A.91 and the zero-profit condition nf=O imply that
(A.12)
qt(i)
=
wtr(il.
Equations (A.l)-(A.41 and (A.121 imply the existence of the borderline
index st satisfying
(A.13)
~ln(ht+~tI-ln(htll/wh
t t
;Y(s,) =
such that
(A.14a)
ct(i)=l, ct(i)=c:
for iSst
(A.14b)
ct(i)=O, ct(iI=cF
for i>st.
Equations (A.131 and (A.141 correspond to (17) and (181 in the text.
Equation (A.141 and the cash-in-advance constraint (51 imply equation (19)
in the text.
In light of (A.141 and the market-clearing condition (141,
equation (A.91 can be rewritten as equation (20) in the text.
B.
Construction
of
Steady-State
Equilibria
Given the initial conditions ko=Ko>O, mo=mz, and bo=O, a competitive
equilibrium consists of the sequences {ht, gt, mI+I, nz, rt, wt, kt+l,
K
t+1’ et(i), E,(i), Jt,
l:,
I:,
mt+l, bt+l,
22
P,, qt(i), Rt, nf}to.
When
monetary policy calls for a constant rate of money growth, gt=g for all
t=0,1,2,... and equations (1) and (21 in the text determine {ht, m~+l>~=o.
In light of the consistency condition (111 and the market-clearing
conditions (13)-(161, the task of constructing a competitive equilibrium
reduces to finding sequences {rt, wt, kt+l, ct(il, 5,(i), Jt, lz, lf, p,,
qt(iL Rt}yzo that satisfy the remaining equilibrium conditions.
Equation (A.51 can be used to solve for wt=B/At. Substituting this
solution into (A.131 yields
(A.151
=
r(y)
[ln(ht+~t)-ln(ht)l/B.
Equations (A.91 and (A.141 determine lf=nf, E,(i), and et(i) in terms
of st, At, and cc,. Equation (19) in the text can be rewritten as
(A.161
p,
=
m~+l(ht+~tl/(l-st),
which determines p, in terms of st, Xt, pt, and the exogenously-given ms
t+1*
The household's first order condition (A.81 then determines Rt in terms of
s
s
and the constant g, while equation (A.121
%+I'
t' t+1' At, %+1. I-(,,
determines qt(il=Br(iI/ht in terms of ht.
When a+q=l, equations (A.101 and (A.111 determine lz=nT and rt in
terms of kt and Xt:
(A.171
1; =
[(l-a)Ahtkt/B]"U
(A.181
r
t
orA[(l-a)Ahtkt/B]'l-a"a.
=
The solutions for 1: and lf and the household's time constraint (41 then
determine Jt as a function of kt, st, and At.
Substituting these results into the goods market-clearing condition
(12) and the household's first order conditions (A.61 and (A.71 yields
23
(A.191
kt+l =
Ak [(l-a)Ahtkt/Bl"-a"a
t
+ (l-6)kt - st/ht - (1-st)/(At+Pt)
=
(A-20)
A
t
(A.21)
ap-St)
pAt+l{aA[
=
(l-a)A~tkt/Bl'l-a"a+(l-~)}
[P/(l+gH(ht+pt)(l-s 1.
t+1
Equations (A.151 and (A.19)-(A.21) represent a system of four equations in
{ktt1' st' At. P,};=,Define the variables k:=kt+,/kt,h:=Atkt, and p:=p,k,. In equilibria
where k:=k*,
* +a*,
St=S '
and p:=p*, equations (A.15) and (A.19)-(A.21)
become
[ln(A*+fl*)-ln(h*)
l/B
(A.22)
?+*I
=
(A.23)
k*
=
A[(l-a)AA*/BI(l-Q)'a+ (l-6) - s*/A* - (1-s*)/(h*+p*)
(A.24)
k*
=
B{aA[
(A.251
A*
=
[/3/(l+g)
I o*+l.l*1,
(1-a)AA*/BI('-a)'a+(l-6)}
which can be solved numerically for the constants k*, s*, A*, and p*.
The initial condition ko>O and the definition k*=kt+l/ktcan then be
used to ‘find kt+l for all t=0,1,2,.... The definitions s*=st, A*=htkt, and
*
p =ptkt determine s
and p, for all t=0,1,2,.... With the sequences
t' At,
{k
s
h
p )w in hand, all other equilibrium prices and quantities
tt1' t' t' t t=o
can be deduced.
All of these prices and quantities are either constant or
grow at a constant rate.
24
Notes
'This general equilibrium interpretation of Karni's insight is consistent
with Yoshino's (1993) finding that inflation and employment in banking have
been positively correlated over time in the US and other countries.
21n Cooley and Hansen's (1989) single-good cash-in-advance model, for
example, the velocity of money is practically constant. Thus, money demand
is highly interest-inelastic. Cooley and Hansen (1991) use a multiple-good
cash-in-advance model that, in principle at least, allows velocity to vary
with the inflation rate. Benabou (19911, however, demonstrates that this
alternative cash-in-advance formulation also generates a very inelastic
money demand function.
3See Imrohoroglu's (1992) work for a very different general equilibrium
model that also yields the result that the Bailey-Friedman analysis
captures only a fraction of the total cost of inflation.
4Cooley and Hansen (1991) choose one parameter to match the size of the
inflation tax base in their model with the analogous figure from the US
data. Similarly, Lacker and Schreft (1993) choose one parameter to match
the interest elasticity of money demand in their model and the US data.
Thus, the approach taken here combines the methods of these earlier studies
by setting two parameters in order to match both the size of the tax base
and the interest elasticity in the model and data.
5All data are taken from the Economic
Report
of the President
(1993).
The
series for velocity are constructed using GDP as the measure of income.
?ero inflation serves as a benchmark here since it is also used as a
benchmark by Fischer (1981) and Lucas (19811 and since, as noted by
Carlstrom and Gavin (19931, price stability is the most widely-cited
objective for monetary policy in the US economy.
7Although this 6% real interest rate may seem high, it is similar in
magnitude to those that typically arise in models of sustained growth.
King and Rebelo's (1993) model, for example, yields a 6.5% real rate in
steady state.
25
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28
Table 1.--The Welfare Cost of Inflation
Annual Inflation Rate
0
Friedman
Percent
Rule
4
Percent
10
Percent
Currency Specification
Annual Rate of Money Growth
Fraction of Transactions Using Cash
Annual Velocity of Money
Total Labor Supply
Fraction of Labor Force in Finance
Annual Rate of Output Growth
-0.0394
1.0000
5.7129
0.2016
0.0000
0.0218
0.0212
0.3452
16.7262
0.2008
0.0017
0.0212
0.0618
0.1228
0.3012
0.2646
19.3199 22.2562
0.2005
0.2002
0.0028
0.0041
0.0210
0.0207
Welfare Cost (Percentage of Output)
-0.9142
0.0000
0.4087
0.9155
Annual Rate of Money Growth
Fraction of Transactions Using Cash
Annual Velocity of Money
Total Labor Supply
Fraction of Labor Force in Finance
Annual Rate of Output Growth
-0.0394
1.0000
5.7129
0.2016
0.0000
0.0218
0.0203
0.9482
6.0247
0.1987
0.0007
0.0203
0.0605
0.8051
7.1356
0.1981
0.0042
0.0197
0.1216
0.5173
11.3350
0.2003
0.0148
0.0197
Welfare Cost (Percentage of Output)
-2.2159
0.0000
1.0767
1.7273
Ml Specification
Table 2.--The Bailey-Friedman Cost of Inflation
Annual Inflation Rate
Friedman
Rule
0
Percent
4
Percent
10
Percent
-0.5876
0.0000
0.0147
0.0605
-0.0137
0.0000
0.0573
0.4220
Currency Specification
Welfare Cost (Percentage of Output)
Ml Specification
Welfare Cost (Percentage of Output)
Table 3.--The Welfare Cost of Inflation With Exogenous Growth
Annual Inflation Rate
Friedman
0
Rule
Percent
4
Percent
Annual Rate of Money Growth
Fraction of Transactions Using Cash
Annual Velocity of Money
Total Labor Supply
Fraction of Labor Force in Finance
Annual Rate of Output Growth
-0.0394
1.0000
5.7043
0.2013
0.0000
0.0200
0.0200
0.3470
16.6176
0.2006
0.0017
0.0200
0.0608
0.1220
0.3021
0.2650
19.2472 22.2107
0.2004
0.2001
0.0027
0.0041
0.0200
0.0200
Welfare Cost (Percentage of Output1
-0.1569
0.0000
0.0875
0.2032
-0.0394
1.0000
5.7043
0.2013
0.0000
0.0200
0.0200
0.9488
6.0194
0.1986
0.0007
0.0200
0.0608
0.8035
7.1520
0.1982
0.0042
0.0200
0.1220
0.5158
11.3744
0.2004
0.0149
0.0200
0.2739
0.9231
10
Percent
Currency Specification
Ml Specification
Annual Rate of Money Growth
Fraction of Transactions Using Cash
Annual Velocity of Money
Total Labor Supply
Fraction of Labor Force in Finance
Annual Rate of Output Growth
Welfare Cost (Percentage of Output)
@FedFRASER