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clevelandfed.org/research/workpaper/1995/wp9504
Working Paver 9504
INTEREST RATE RULES VS. MONEY GROWTH RULES:
A WELFARE COMPARISON IN A CASH-IN-ADVANCEECONOMY
by Charles T. Carlstrom and Timothy S. Fuerst
Charles T. Carlstrom is an economist at the Federal
Reserve Bank of Cleveland, and Timothy S. Fuerst
is an assistant professor of economics at Bowling
Green State University and a research associate of
the Federal Reserve Bank of Cleveland.
Working papers of the Federal Reserve Bank of
Cleveland are preliminary materials circulated to
stimulate discussion and critical comment. The
views stated herein are those of the authors and
not necessarily those of the Federal Reserve Bank
of Cleveland or of the Board of Governors of the
Federal Reserve System.
June 1995
clevelandfed.org/research/workpaper/1995/wp9504
ABSTRACT
This paper considers the welfare consequences of two particularly simple rules for
monetary policy: an interest rate peg and a money growth peg. The model economy
consists of a real side that is the standard real business cycle model, and a monetary
side that amounts to imposing cash-in-advance constraints on certain market transactions.
The paper also considers the effect of assuming a rigidity in the typical household's
cash savings choice. The competitive equilibrium of the economy is not Pareto efficient,
partly because of two intertemporal distortions:
a distortion on the capital
accumulation decision, and a distortion on portfolio choice that arises from the assumed
rigidity. The principal result of the paper is that the interest rate rule (but not the
money growth rule) entirely eliminates these two intertemporal distortions and is thus
the benevolent central banker' s policy choice.
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clevelandfed.org/research/workpaper/1995/wp9504
1. Introduction
One of the oldest debates in monetary economics concerns the appropriate target for
monetary policy:
interest rates?
Should central banks target money supply growth rates or nominal
Friedman (1990) provides an introduction to this voluminous literature.
Much of the early work follows Poole (1970) and Sargent and Wallace (1975) and conducts
the analysis within an ISILM-type aggregative framework.
In contrast, the more recent
studies, led by Sargent and Wallace (1982), address the issue in the context of general
equilibrium models.
The present paper belongs to this latter tradition. In the monetary
economy analyzed below, the competitive equilibrium is not Pareto efficient, but is
instead distorted relative to the Pareto optimum by one intratemporal distortion and two
intertemporal distortions.
monetary policy rules:
The paper considers the welfare consequences of two simple
1) a constant money growth rate (in which case the nominal
interest rate is endogenous), and 2) a constant nominal interest rate (in which case the
money growth rate is endogenous).
The principal result is that an interest rate rule,
but not a money growth rule, entirely eliminates the two intertemporal distortions and is
thus the benevolent central banker's policy choice.
Our analysis is carried out in an economy in which the real side is the standard
real business cycle model.
Money is introduced by imposing cash-in-advance constraints
on the representative household's consumption purchases and the representative firm's
wage bill.
As is well known, real variables in this monetary economy generally behave
quite differently from their counterparts in the corresponding real economy run by a
Pareto planner.
For example, the cash constraint on labor demand imposes an inflation
tax on labor market activity and thus lowers equilibrium work effort (see, for example,
Cooley and Hansen [1989]). In contrast to this intratemporal distortion, we focus on two
potential intertemporal distortions arising in the monetary economy.
First, the cash
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constraint on consumption imposes a distortion on the capital accumulation decision.
In
particular, capital accumulation is affected by the time path of the nominal rate of
interest (see Fischer [I9791 and Fuerst [1994a]).
Second, a non-Fisherian component of
interest rate determination enters into the model under the assumption that the
household's cash versus bank deposit portfolio decision is made in the absence of full
contemporaneous information.
Models incorporating this type of portfolio rigidity are
something of a growth industry, partly because they are consistent with an increase in
the money growth rate (temporarily) driving down the nominal rate of interest (see Lucas
[1990], Christian0 and Eichenbaum [1992, 19941, Fuerst [1992, 1994b1, and Carlstrom
[1994]).
The objective of the present paper is to show how a simple interest rate rule
can eliminate both this portfolio rigidity and the capital accumulation distortion.
A common criticism of interest rate rules is their potential for giving rise to
price-level indeterminacy and sunspot behavior.
For example, Smith (1988) argues that
one possible justification for the money growth regime in the Sargent and Wallace (1982)
environment is that it precludes the possibility of sunspot equilibria.
Issues of this
type do arise below, but we sidestep some of them by limiting our analysis to stationary
rational expectations equilibria.
In particular, we ignore the possibility of self-
fulfilling hyperdeflations and hyperinflations. We make this choice because:
a) we have
nothing new to contribute in this regard, and b) as demonstrated by Woodford (1994) in a
comparable environment without capital, these equilibria are not unique to interest rate
regimes, and in fact are in some sense more likely under money growth regimes.
Even
within the class of stationary equilibria, price-level indeterminacy does arise below.
In two of the three model variants, this indeterminacy is purely nominal and would thus
have no effect on real welfare comparisons.
A novel result is that in the case of
portfolio rigidities, this indeterminacy becomes a real indeterminacy, so that some care
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must be taken in defining monetary policy.
The next section lays out the basic model.
Section 3 addresses the interest rate
versus money growth issue in a deterministic setting, while the two sections that follow
carry out the corresponding analysis in increasingly complicated economic environments.
Section 4 considers the case of stochastic shocks without portfolio rigidities, while
section 5 considers the case of stochastic shocks with portfolio rigidities.
Section 5
also presents some computational results of a numerical welfare comparison of the two
monetary policy regimes.
Section 6 discusses the real indeterminacy problem mentioned
above, and section 7 concludes.
2. The Model
The economy consists of numerous agents of three types:
intermediaries.
households, firms, and
Since all behave as atomistic competitors, we will restrict our
discussion to a representative agent of each type.
We will first describe the
optimization problem of each agent, then turn to an analysis of equilibrium behavior.
The typical household is infinitely lived, with preferences over consumption (ct)
and leisure (I-Lt) given by
00
where Eo is the expectation operator,
E
(0,l) is the personal discount rate, Lt denotes
household labor supply, and the household's leisure endowment is normalized to unity.
The household begins period t with Mt dollars and must decide how much of this cash to
keep on hand for contemporaneous consumption and how much to deposit in the intermediary,
where it will earn a gross nominal return of Rt.
Let Nt denote the amount of cash
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deposited in the intermediary, a choice that we assume is fixed until the next period.
An important issue below is the information the household has when making this portfolio
decision.
We consider two distinct possibilities:
In the case of a portfolio rigidity
(PR), the household selects Nt before knowing the current innovations in technology and
government spending, while in the case of no portfolio rigidity (NPR), the household
knows the current innovations when choosing Nt.
In either case, after making its
portfolio decision, the household makes its consumption and labor supply decisions with
full information on the current state of the world. Consumption purchases are subject to
a modified cash-in-advance constraint.
In particular, households can use cash not
deposited in the intermediary, as well as current labor income, to purchase consumption:
Ptct 5 Mt - Nt
+ WtLt
where Pt and Wt denote the price level and nominal wage, respectively. At the end of the
period, the household receives a cash dividend payment from both the firm and
intermediary, as well as principal plus interest on its deposits at the intermediary.
Hence,
f
M t + l = Mt
where
ntf
and
nfi
i
+ (Ril)Nt + WtLt + nt + I'It - Ptct - PtTt
denote the profits of the representative firm and intermediary,
respectively, and Tt denotes the real lump-sum taxes imposed by the fiscal authority.
The representative firm uses its accumulated capital stock (kt) and the labor it
hires from households (Ht) to produce current output via its stochastic production
technology:
etf(kt,Ht), where €It is the time t state of technology and f is a
neoclassical production function.
The firm keeps part of this output to augment its
capital stock (It) and sells the rest to households (on a cash basis) for consumption.
The firm also faces a cash constraint in that the current wage bill must be financed with
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cash loans from the intermediary. These loans are at the gross rate Rt, and are repaid
at the end of the period.
The firm chooses its production and investment levels to
maximize the discounted value of its dividend payments:
with
ntf and It given by
Note that in the terms of Lucas and Stokey (1987), labor is a cash good for the firm,
while investment is a credit good.
The technology variable is assumed to evolve
according to the following stochastic process:
where pg is the autocorrelation coefficient, ct is an i.i.d. shock, and the nonstochastic
steady state of Bt is 8.
Finally, the typical intermediary accepts deposits of Nt from households and
receives the current monetary injection of M;(Gil)
from the central bank, where Gt =
MS+I /MS,
t and M: is the money supply per household. All of this cash is then loaned out
to firms at the rate Rt. This implies that IIi = R~M:(G~-~).
To close our description of the model, we need to specify fiscal and monetary
policy. To begin with the former, real government expenditures are exogenous and follow
the stochastic process
gt = (l-pg)g + Pggt-l + ^It
where p is the autocorrelation coefficient, yt is an i.i.d. shock, and the nonstochastic
g
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steady state of gt is g.
Because the model is otherwise Ricardian, we abstract from
government debt by assuming that Tt = gt V t.
We consider two schemes for the conduct of monetary policy. Under a money growth
rule, Gt = Gss V t and Rt is endogenous. In contrast, under an interest rate regime, Rt
=
R V t and Gt is endogenous. For ease of comparison, we set R = Gss/P, so that the
nonstochastic steady state of the model is unaffected by the choice of monetary regime.
There are four markets in this economy: the goods market, the labor market, the
money market, and the credit market. The respective market-clearing conditions are given
by
The model's equilibrium is defined by the household's and firm's optimization conditions
evaluated at these equilibrium conditions.
To make the model stationary, we normalize
all nominal variables by Mt and define the following new variables: pt = Pt/Mt , wt =
Wt/Mt, nt = Nt/Mt.
normalize by Mt+l,
Given the timing of the model, a more natural choice might be to
since this represents the money stock available for time t
transactions. However, in the PR model, this choice would not be appropriate because Nt
must be chosen before Mt+ is known. Hence, to maintain symmetry between the two models,
we will use Mt as our normalization. An equilibrium is given by the Lt, kt+l, wt, pt,
nt, and Gt
% stochastic processes that satisfy the following Euler equations:
EsUc(')'pt
=
EsP%Uc0+ l)/pt + 1Gt
(1)
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E t ( ~ t / G t+~l)Uc(t+
t
l ) = EtP(~t
+ 1lGt+ l ~+21[et
t
+ lfk(t+ 1)+(1-6)1Uc(t+2)
(6)
where Es = Et in the NPR model and Es = Et-l in the PR model.
We will now turn to an analysis of the economy's behavior under the alternative
monetary regimes, beginning with a deterministic version of the model and then turning to
the NPR and PR cases.
I
3. The Deterministic Case
Suppose that
et
=
9 and gt = g V t, and that monetary policy is nonstochastic.
Then, solving (2), (3), and (5) for pt, wt, and nt, we have:
Pt = Gt/ct
wt = ULGt/CtUc
nt = ULGtLt/ctUc - (Gt-1).
Substituting these back into the remaining three equations, we are left with the
following three Euler equations in Lt, kt+ l , and Gt g_r Rt:
Uc(')ct = RtPUc(t + l)ct + 1lGt+ 1
(7)
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The two distortions in this economy are apparent.
First, there is an intratemporal
distortion on work effort in equation (8) that arises because of the transactions
Second, notice that by substituting (1) into (6),
constraint on the firm's wage bill.
the capital accumulation equation (9) collapses to something resembling the optimal
growth equation.
The difference is that the two respective marginal utilities are scaled
by the corresponding nominal rates of interest.
This intertemporal distortion arises
because of the cash constraint on consumption.
If the firm decides to increase its
capital stock by one unit, then there will be pt fewer dollars to distribute to the
household at the end of period t.
At the beginning of period t, the household could
borrow against this expected dividend flow and finance p{Rt
dollars of consumption.
Hence, the private utility cost of increasing capital by one unit is Uc(t)/Rt.
Next
period, this capital will produce a profit flow of pt+ l[Ofk(t+ 1)+(l-6)] dollars that will
be paid out to households at the end of the period.
At the beginning of t + l , the
household could borrow against this cash flow and finance pt
dollars of consumption.
+
+
[Ofk(t + 1) (1-6)]/Rt
+
Hence, the private utility gain of increasing capital by one
unit is P[Ofk(t 1)+(1-6)]Uc(t + l)/Rt l.
margins.
+
+
The optimizing firm equates these two private
Note that both of these private margins are distorted relative to the social
margins by the corresponding nominal rate of interest. This observation is formalized in
Proposition 1 below.
Consider the economy's behavior under two different monetary regimes: i) a money
growth regime in which Gt = Gss V t, and ii) an interest rate regime in which
V t.
Rf
=
GSs@
Note that the economy's unique steady-state capital stock (kss) is identical under
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either regime.
However, the economy's behavior along the accumulation path is quite
different under the two policies. Under a money growth regime, Gt = Gss V t, and (7)-(9)
determine the paths for Lt, kt+l, and Rt.
constant along the accumulation path.'
Note in particular that Rt is generally not
In contrast, under an interest rate rule, (8)-(9)
determine the paths for kt+ and Lt, while (7) then determines Gt+ l. We immediately have
the following:
Proposition 1:
In the deterministic model, if monetary policy operates under an interest
rate regime, equation (9) collapses to the accumulation equation from the optimal growth
problem, that is, the intertemporal distortion on capital accumulation is entirely
eliminated.
Proposition 2:
In the deterministic model, if labor supply is inelastic, the optimal
monetary policy is an interest rate rule.
Proposition 2 cannot, in general, be extended to the case of elastic labor because
then we have a second-best problem.
Under an interest rate rule, there is no distortion
on the capital accumulation equation and a constant distortion on the labor supply
decision.
In contrast, under a money growth rule, there is a varying distortion on both
margins. The preferred regime will, in general, depend on preferences. However, we can
state the weaker result that an interest rate policy of Rt = 1 dominates a money growth
policy of Gt+l =
P, since the latter does not guarantee a zero nominal interest rate
along the accumulation path.
(Woodford [1990] makes a similar point in a variety of
'The one exceptional' case is separable preferences with log preferences over
consumption, in which case the money growth and interest rate regimes are identical. See
Fuerst (1994a) for more discussion.
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models without capital.)
As an aside, note that under an interest rate regime we need an extra initial
condition, that is, (7)-(9) impose no conditions on Go and thus none on po, wo, and
%.
This is the standard result of nominal indeterminacy under an interest rate rule (see
Sargent [I9791 or Sargent and Wallace [1975]), which can be eliminated in the current
context by specifying the initial money stock (see McCallum [1981, 1986]).2 Given the
timing of the monetary injection in the model, the money stock available for use in time
0 is MOGO Because we have implicitly set Mo = 1 under our normalization above, we can
eliminate the nominal indeterminacy by specifying Go. Note that in any case, there is no
indeterminacy in the real variables.
,-
4. The Stochastic Case without Portfolio Rigidities
Now, suppose that Bt and gt are stochastic, but that Nt is chosen after the current
innovations are observed. Once again we can eliminate nt, wt, and pt. Using the law of
iterated expectations, we have:
Uc(t)ct = RtPEtUc(t+ 'kt +
RtUL("/UC(')
=
pt+
1
B,fL(t)
Uc(')/Rt = pEt[et+lfk(t+l)+(l-s)luc(t+l)/Rt+ll
Propositions 1 and 2 apply here as well:
(10)
(11)
(I2)
An interest rate rule eliminates the distortion
on capital accumulation and thus is clearly the optimal monetary policy if we abstract
2Woodford (1994) demonstrates how the homogeneity property that gives rise to this
nominal indeterminacy can also be eliminated by assuming that changes in the money supply
are brought about through open market operations rather than through lump-sum monetary
transfers.
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from elastic labor supply.
As in the previous section, this result does not immediately
generalize to the case of elastic labor supply, because then we have a second-best
problem. However, once again, a peg of Rt = 1 dominates a money growth policy of Gt+
=
P.
The nominal indeterminacy (under an interest rate peg) discussed in the previous
section takes on a slightly modified form here.
determine the behavior of Lt and kt+l.
Under a peg of R, (11)-(12) uniquely
This behavior is identical to that in the
corresponding real business cycle economy, where the marginal utility of leisure is
proportionally scaled upward by R.
Given this real behavior, (10) then imposes the
following restriction on the money growth process:
(13)
(PR)-' = (uc(t)ct)-l EtUc('+ INt+ lzt+
where zt+
= (1IGt + l ) The earlier nominal indeterminacy arises here in that there is no
restriction on the initial Go.
However, even with such a Go specified, there are an
infinite number of money growth processes satisfying (13).
For example, if U is
logarithmic, we have:
( P ~ 1 - l= Et(zt++
In this economy, only the conditional mean of zt+l matters; there is no restriction on
the variance of z ~ + nor
~ , on its covariance with the technology shocks.
This is an
economy in which only expected money growth matters. (Lucas and Stokey [I9871 make a
similar point in a similar context.)
This indeterminacy is something of a nuisance, but
has no consequence for real variables.
A natural restriction on Gt+l is to assume that
, ~ ~ , 0 t ) ,
it is a time-invariant function of (kt,gt,Ot), that is, Gt+l = ~ ~ ~ ~ ( k ~ with
Gnpr (kss,g,O) = Gss.3
A loose interpretation of this restriction is that the Fed does
3McCallum (1983, 1986) calls restrictions of this type the "minimal state vector
solution. "
11
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not "play dice" with the money growth rate. Under this assumption, we can solve (10) for
Gt+ 1'
"pr k
Gt+l = G ( ,,gt,03 = RPEt[Uc(t+ l)ct+ l/Uc(t)ct].
Returning to the example of log preferences, the no-dice restriction implies
Gt+l = Gss
"
t.
5. The Stochastic Case with Portfolio Rigidities
The previous two sections demonstrated that under an interest rate regime, the
intertemporal distortion on capital accumulation is entirely eliminated.
In this
section, we add another distortion to the economic environment, namely, that household
portfolio allocations respond sluggishly to innovations in technology and government
spending.
This rigidity is of particular interest because many recent models of the
monetary business cycle use it as a means of modeling monetary non-neutrality (see, for
example, Carlstrom [1994], Christian0 and Eichenbaum [1992, 19941, and Fuerst [1992,
1994bl).
The principal result of this section is that an interest rate rule also
eliminates this distortion.
In the PR case, nt is a predetermined variable, so we must alter our solution
procedure.4 In particular, we will solve (2), (3), and (5) for pt, wt, and Lt:
4Blanchard and Kahn (1980) call a time t variable predetermined if it is a function
only of variables known at the end of time t-1. In the present context, nt is a function
only of (kt,Ot-l), both of which are known at the end of time t-1.
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The equilibrium is now given by the kt+l, Lt, nt, and Gt
Rt that solve:
The effect of the portfolio rigidity is most easily seen in (15).
In contrast to
equation (10) in the NPR case, in the PR case the nominal interest rate is equal to
Fisherian fundamentals only "on average."
Innovations in technology alter the shadow
value of cash in the goods market (the left-hand side of [15]) and in the financial
intermediary (the right-hand
side of
differences cannot be arbitraged away.
interest rate determination.
the capital market.
responsive to shocks.5
[15]).
Since portfolios are rigid,
these
Hence, there is a non-Fisherian component to
This portfolio distortion affects both the labor market and
As for the labor market, the rigidity tends to make labor less
For example, if U is separable and logarithmic in consumption,
then (14) implies that under a money growth regime, labor is invariant to productivity
and government spending shocks.
The portfolio rigidity also alters the distortion on
capital accumulation, since the non-Fisherian component of interest rate determination
implies that (16) cannot be collapsed into (12).
This latter point suggests that if an
interest rate peg eliminates the portfolio distortion, then it will also eliminate the
capital accumulation distortion.
The goal of this section is to demonstrate this
explicitly. We will begin with an observation about the portfolio rigidity.
5Christiano and Eichenbaum (1994) also emphasize this point.
13
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Proposition 3: In a PR economy without capital, if monetary policy operates under an
interest rate regime, then a) the real behavior of the economy. is identical to the
corresponding NPR economy, b) there exists a unique time-invariant central-bank reaction
function, Gt = ~ ~ ~ ( n ~ , ~with
~ ,G
0 t~) ~, ( ~ ~=~ Gss,
, ~ , that
B ) supports the interest rate
peg, and c) nominal interest rates are purely Fisherian.
In summary, an interest rate
regime eliminates the portfolio rigidity.
Proof:
With no capital and a constant interest rate, (17) uniquely determines Lt as a
function of gt and Bt, a relationship that is common to both the NPR and PR models.
Substituting this Lt into (14), we can uniquely solve for the time-invariant central-bank
~ ~ ,supports
B t ) the interest rate regime, where
reaction function Gt = ~ ~ ~ ( n ~ , that
~ P ' ( n ~ ~ , ~=, 0Gss,
)
and nss denotes the value of n in the nonstochastic steady state.
This Gt choice implies that the share of the money stock in the intermediary,
(nt+Gt-l)/Gt, is ultimately the same in both the NPR and PR models. This implies that an
agent in the PR economy would have no desire to vary nt in response to gt and Bt. Hence,
nominal interest rates are purely Fisherian, and (15) is trivially satisfied.
Although Proposition 3 implies that an interest rate regime leads to identical real
behavior in the NPR and PR models, the behavior of the current money growth rate (Gt) is
quite different.6
From (14), the key variable is the share of the time t money stock
that is in the intermediary, st = (nt+Gt-l)/Gt.
In the case of NPR, the previous
6As an aside, since pt = Gt/ct, differences in the conditional variability of Gt in
the two models (NPR versus PR) imply stark differences in the variability of the price
level.
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section's no-dice
npr
G
restriction implies that
Gt
is
predetermined,
that
is,
Gt
-
I+
to ensure that st is at the level
needed to support the response of Lt to gt and Bt.
In contrast, in the PR model, nt is
Bt-1)
so that the household adjusts
predetermined, and the central bank adjusts Gt to ensure that st is at the level needed
to support the response of Lt to gt and Bt, that is, Gt = #r(nt,gt,~t).
It is in this
precise sense that an interest rate rule enhances the ability of the PR economy to
respond to real shocks.
Returning to the model with capital, note that the proof of Proposition 3
immediately generalizes to prove a weaker result:
Proposition 4:
In a PR economy with capital, if monetary policy operates under an
interest rate regime, there exists a time-invariant central-bank reaction function, Gt =
# r ( k t , ~ , g t , ~ t ) , with #r(kss,nss,g,B)
=
Gss, such that a) the real behavior of the
economy is identical to the corresponding NPR economy operating under an interest rate
regime, and b) nominal interest rates are purely Fisherian.
Hence, an interest rate
regime can eliminate both the portfolio and capital accumulation distortions.
Proof: Under an interest rate rule, the NPR economy uniquely determines the behavior of
Lt and kt+l in response to gt and Bt. Substituting these values into (14), we can solve
for the unique Gt = #r(kt,~,gt,Bt), with #r(kss,nss,g,B)
real behavior.
= Gss, that supports this
As in the proof of Proposition 3, st is ultimately the same in both the
NPR and PR economies, so that nominal interest rates are purely Fisherian.. This implies
that (15) is trivially satisfied, and (16) collapses to (12).
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To close this section, we will present a quantitative assessment of the welfare
advantage of an interest rate policy over a money growth policy. The numerical analysis
is carried out in three steps.
First, the equilibrium Euler equations are linearized
about the nonstochastic steady state, and the method of undetermined coefficients is used
to calculate the two sets of linear decision rules characterizing the economy under the
money growth regime and the interest rate regime.
Second, after taking a quadratic
approximation of the value function and utility function, the method of undetermined
coefficients is used to find the value function under the two monetary regimes.7 Third,
and finally, the constant level of capital subsidy needed to equate the unconditional
expectation of the two value functions is calculated.
To be precise, let VR and VG
denote the value functions under an interest rate and money growth regime, respectively.
Then-in table 1 we report the value of A that solves
EoVR(kl.el,gl) = E0VG(kl + Akss,el'gl)
where kl, e l , and gl are integrated over their steady-state joint distribution, and A is
expressed as the percentage increase in steady-state capital that must be given to
households in the money growth regime to make them as well off as households in the
interest rate regime.
Functional forms and parameter values were chosen to be consistent with the
literature. Preferences are given by U(c,l-L) = [ ( ~ ~ - ~ - l ) l ( l +
- o Aln(1-L)],
)
where the
constant A is chosen to imply a steady-state level of labor of .3. We experimented with
7To approximate the utility function, we need the equilibrium decision rule for
consumption. For these calculations, we used a linear approximation of the aggregate
resource constraint to determine consumption behavior. Of course, there are other
possibilities, including substituting the linear decision rules for capital and labor
into the actual resource constraint and backing out a nonlinear rule for consumption. In
current work, we are exploring the consequences of using these alternative methods (along
with the possibility of using log-linear decision rules).
For a discussion of these
alternatives, see Dotsey and Mao (1992).
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several values of o , all with broadly similar results. We report results for o = 1 and o
=
5.
We set
p
= .99 (implying a 4 percent steady-state annual real rate of interest).
Technology is Cobb-Douglas, with a capital share of .36 and a capital depreciation rate
of 6 = .0175 per quarter. We chose g to imply a steady-state gt/Yt ratio of .08. For
the stochastic shocks, we utilized the benchmark estimates in Burnside, Eichenbaum, and
Rebelo (1993): p8 = .986, o8 = .0089, p = .982, o = .015, and corr(y ,E ) = .308.
g
g
t t
Finally, for monetary policy, we set G = .0075 per quarter for the money growth rule, and
R = G/P (or about 7 percent annually) for the interest rate rule.
The numerical results are presented in table 1.
Note that the welfare gain is
relatively large (as welfare numbers go) for either technology shocks alone or for
technology and government spending shocks. In the latter case, a value of 2 percent of
the aggregate capital stock is a benchmark estimate.8 With U.S. aggregate net worth now
at approximately $24 trillion, the welfare gain amounts to $480 billion--a sizable free
lunch.
We have two comments on this result.
sensitive to the variance of the shocks.
First, these welfare numbers are quite
For example, as pg increases and the
unconditional variability of 8 rises, the welfare gain of the interest rate regime grows
exponentially.
quarter.
Second, by assumption, the portfolio rigidity disappears after one
This implies that the basic difference between the two regimes is that under a
money growth regime, the market economy responds to shocks with a one-period lag. To the
extent that the portfolio rigidity is more long-lived, possibly because of portfolio
adjustment costs as in Christian0 and Eichenbaum (1992), the advantage of an interest
8In comparison, Lucas (1987) estimates that the welfare gain of eliminating all
consumption variability is only about .008 percent of aggregate consumption into
perpetuity, or (in present value) about .048 percent of the aggregate capital stock (we
are using the model's steady-state real rate of interest of 4 percent, and
consumption/capital ratio of .24, to make this transformation). Note, however, that
Lucas' calculation is a partial equilibrium exercise and is thus not strictly comparable
to the number we report.
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rate rule will be even larger.
6. A Real Indeterminacy in the Case of Portfolio Rigidities
Proposition 4 demonstrates that in the PR model with capital, there exists a timeinvariant central-bank reaction function that supports the interest rate peg and produces
the same real dynamics as in the NPR model. However, this is not the only real behavior
consistent with an interest rate peg in the PR model.
construction.
state.
We will demonstrate this by
To begin, linearize the system (14)-(17) about the nonstochastic steady
For simplicity, we will set gt = g V t.
Suppose that the central bank supports
the interest rate peg with the following reaction function:
"pr k
Gt+l = G ( t,g9etI
+ alEt + a2Et+1 + a3vt + a4Vt+1
(18)
where G~~~ is the central-bank reaction function in the linearized NPR model, ct is the
time t innovation in the technology shock, vt denotes an extraneous or sunspot process
that is uncorrelated with the technology process, and Etml(vt) = 0.9
By construction,
(18) satisfies (15). Substituting (18) into (16) yields the following linear equation:
npr
EtQ (kt,kf+l'kt+2.Lt,Lt+1,et,et+l)
where
+ 4(a1ct + a3vt)
anPris the equation that results in the corresponding NPR
=
0
economy and q is a
constant. Combining this equation with (17) gives us the law of motion for capital and
labor:
9As an aside, note that in the previous two sections the choice of Go was entirely
arbitrary, since it was an initial condition that only scaled all future nominal
variables. However, in the PR case, Go is not an initial condition, since the choice of
% occurs prior to the revelation of Go.
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where KnPr and LnPr denote the corresponding relationships in the NPR model and the a ' s
are constants. Note that if al = a3 = 0, the real behavior of this PR economy will be
identical to the corresponding NPR economy.
Substituting the law of motion for capital
into (14) yields another linear expression for labor:
These two expressions for Lt must, of course, agree. If a 1 = a3 = 0, then since nt is
predetermined, a4 = 0 and 9is uniquely determined. Therefore, if we restrict the money
growth rule to depend only on a minimal state vector, then the real behavior of the
economy is unique and identical to the NPR model, and there exists a unique reaction
function to support the interest rate peg.
(This is just Proposition 4.)
However, this
is clearly not the only reaction function that will support the interest rate peg.
In
particular, there is nothing to pin down either al or a3, since nt can respond freely to
past shocks. Given values for al and a3, there will exist unique values for
9and a4.10
Hence, an interest rate target can also be supported with a reaction function depending
on sunspots.
Since the past innovation in technology is not part of the minimal state
vector that is necessary to support an interest rate target, it is also in some sense a
sunspot.
These sunspots are reminiscent of our discussion of the NPR model. To uniquely
determine nominal variables, a no-dice restriction had to be imposed. In general, money
1ONote that there is nothing special here about the technology shock and the
indeterminacy of al and a2. A similar situation would arise for the case of government
spending shocks.
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growth in the NPR economy could depend in an arbitrary way on a sunspot term. Similarly,
in the PR economy, money growth could depend on sunspots, but unlike the NPR case, these
sunspots will have real consequences. Because of these real consequences, money growth
will need to depend on past sunspots (those that portfolios can react to), and on current
sunspots as well, in order to support an interest rate target."
An intuitive explanation may be helpful.
A positive technology innovation
increases the demand for labor and indirectly raises the demand for loanable funds. The
latter effect will tend to increase the nominal interest rate.
One natural way of
preventing this is for the central bank to increase Gt by exactly the amount needed to
support NPR behavior.
However, we have just argued that this is not the only method.
One alternative is to keep Gt the same but to reduce labor supply so that the implied
increase in real wages will eliminate the increased demand for loanable funds. To reduce
labor supply, the central bank needs to stimulate current consumption by lowering capital
accumulation.
The desired effect can be achieved by varying Gt+l and thus altering
expected inflation.12
At a more basic level, the real indeterminacy under an interest rate peg arises
here because the standard nominal indeterminacy conflicts with the model's assumption of
a nominal rigidity (that is,
nf
is predetermined).
In the previous two sections, the
standard nominal indeterminacy is easily eliminated by specifying the initial money stock
"Note that the real indeterminacy problem we are highlighting is quite different
from the indeterminacy problem discussed in Blanchard and Kahn (1980), who provide
restrictions on the eigenvalues of the matrix governing deterministic dynamics that
ensure the existence of a unique path to the nonstochastic steady state. In contrast,
under an interest rate rule, the deterministic dynamics of the present model are unique
(because the model is identical to the corresponding real business cycle economy, with
the marginal utility of leisure proportionally increased by the nominal rate of
interest). Instead, the indeterminacy problem that arises here concerns the impulse
response to a technology, fiscal, and/or sunspot innovation.
12This discussion highlights why real indeterminacy is not a problem in the model
without capital.
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and assuming that the Fed does not play dice. In the PR case, the issue is a bit more
complicated, since
activity.
nf
is chosen before Gt is observed, so that Gt potentially alters real
Our approach to resolving this problem is to restrict Gt to be a time-
invariant function of the state variables--what Proposition 4 calls the central bank's
reaction function, Gt = ~ ~ ~ ( k ~ , n ~ , ~(This
~ , f is
3 ~the
) . assumption we used in our
numerical calculations at the end of section 5.)
This assumption of a stationary
reaction function is analogous to the no-dice restriction in the NPR case.
Hence, to
fully articulate an interest rate policy in the PR model, one must specify both R and the
reaction function the central bank uses to support R. (McCallum [1986, p. 1481 analyzes
a nonoptimizing model and comes to a similar conclusion.)
7. Conclusion
Poole's (1970) classic analysis of the targeting debate concluded that, in an
environment with numerous money demand shocks, an interest rate rule is preferred because
it lowers the volatility of output.
This observation raises three issues:
a) What is
the nature of money demand in our model? b) Are there money demand shocks in our model?
and c) How do our conclusions relate to Poole's? We will address each of these issues in
turn.
The typical criticism of the cash-in-advance constraint (relative to a more general
transactions-cost technology) is that it does not allow for endogenous fluctuations in
velocity in response to movements in the nominal interest rate.
seems unwarranted in the current context.
However, this criticism
It obviously does not apply to the interest
rate regime where, by assumption, the nominal rate of interest is constant. It also does
not alter our negative conclusion on money growth rules unless one makes the heroic
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assumption that endogenous movements in velocity can replicate the welfare-improving role
of a constant nominal rate of interest.
Are there money demand shocks in this model?
Our cash-in-advance assumption
implies that there are no shocks to the payments technology--one dollar of cash is always
needed to conduct one dollar of transactions.
for transactions.
However, there are shocks to the demand
Positive technology innovations increase the firm's demand for
workers, and thus their demand for cash.
Similarly, positive government spending
innovations drive down the real wage by increasing labor supply, and thus once again
increase the firm's demand for cash. Although in a general equilibrium environment it is
difficult (if not impossible) to cleanly demarcate IS from LM shocks, it is clear that
the shocks in this model do have money demand consequences.
This leads us back to Poole (1970).
If we follow the previous discussion and
interpret the model's shocks as money demand shocks, our conclusion is similar to
Poole's.
We find this quite remarkable, since our modeling strategy and welfare criteria
could not be more different.
point of the paper.
The differences in welfare criteria illustrate a central
Poole advocates an interest rate rule (in the stochastic money
demand environment) because it reduces the variability of output.
This paper advocates
an interest rate rule because it increases the typical household's expected lifetime
utility by providing more flexibility in responding to real shocks.
For example, in the
NPR economy operating under a money growth rule, a technology shock will generally cause
the nominal rate to deviate from the steady state, and a time-varying path for the
nominal rate of interest distorts the capital accumulation decision (see equation [12]).
Similarly, in the PR economy operating under a money growth rule, the response of labor
input to a technology shock is greatly muted (see equation [14]). The remarkable fact is
that a simple interest rate rule entirely eliminates both of these distortions and allows
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the household to respond more efficiently to technology and government spending shocks.
In sharp contrast to Poole, this increased flexibility improves welfare by actually
increasing output variability.
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Table 1
Welfare Gain of an Interest Rate Rule
(expressed as percentage of steady-state capital stock)
ft3jT-t
8 shocks
g and 8 shocks
Source: Authors' calculations.
@FedFRASER