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9 8 1 9
Price-Level and Interest-Rate
Targeting in a Model
with Sticky Prices
by Charles T. Carlstrom and
Timothy S. Fuerst
FEDERAL RESERVE BANK
OF CLEVELAND
Working Paper 9819
Price-Level and Interest-Rate Targeting in a Model with Sticky Prices
by Charles T. Carlstrom and Timothy S. Fuerst
Charles T. Carlstrom is an Economist at the Federal Reserve Bank of Cleveland.
Timothy S. Fuerst is Associate Professor of Economics at Bowling Green State University,
Bowling Green, Ohio, and a visiting consultant with 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 on research in progress. They
may not have been subject to the formal editorial review accorded official Federal Reserve
Bank of Cleveland publications. The views stated herein are those of the authors and are
not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of
Governors of the Federal Reserve System
Working papers are now available electronically through the Cleveland Fed's home page
on the World Wide Web: http://www.clev.frb.org.
August 1998
Price-Level and Interest-Rate Targeting in a Model with Sticky Prices
by Charles T. Carlstrom and Timothy S. Fuerst
This paper examines a standard sticky price monetary model. The equilibrium conditions of the
model are perturbed relative to the canonical real business cycle model by two varying distortions:
marginal cost and the nominal rate of interest. The paper explores the implications of two monetary
policies that are frequently advocated: (1) an inflation target and (2) an interest rate target. Under
an inflation rate target, marginal cost is stabilized while the nominal rate is variable. In contrast,
under an interest rate target, the nominal rate is stabilized but marginal cost is (in general) variable.
Both policies are subject to sunspot fluctuations arising from the endogenous movement of the
money stock. These fluctuations can be avoided by eliminating the contemporaneous response of
the money stock to innovations in the environment.
Price-Level and Interest-Rate Targeting
in a Model with Sticky Prices
Charles T. Carlstroma
Timothy S. Fuerstb
a
Federal Reserve Bank of Cleveland, Cleveland, Ohio, 44101, USA.
b
Department of Economics, Bowling Green State University,
Bowling Green, Ohio, 43403; and Consultant, Federal
Reserve Bank of Cleveland.
August 1998
ABSTRACT: This paper examines a standard sticky price monetary model. The
equilibrium conditions of the model are perturbed relative to the canonical real business
cycle model by two varying distortions: marginal cost and the nominal rate of interest.
The paper explores the implications of two monetary policies that are frequently
advocated: (1) an inflation target and (2) an interest rate target. Under an inflation rate
target, marginal cost is stabilized while the nominal rate is variable. In contrast, under
an interest rate target, the nominal rate is stabilized but marginal cost is (in general)
variable. Both policies are subject to sunspot fluctuations arising from the endogenous
movement of the money stock. These fluctuations can be avoided by eliminating the
contemporaneous response of the money stock to innovations in the environment.
Key Words: General Equilibrium; Money and Interest Rates; Monetary Policy. JEL
Classification: D51; E42; E52.
The views stated herein are those of the authors and not necessarily those of the Federal
Reserve Bank of Cleveland or those of the Board of Governors of the Federal Reserve
System.
1. Introduction
This paper examines two traditional monetary policies in an economy in which
money is non-neutral because some nominal goods prices must be set in advance. The
assumed nominal rigidity leads to a real rigidity: in the absence of central bank action,
the market economy responds sluggishly to real shocks to the environment. A
benevolent central banker will attempt to eliminate this real rigidity by varying money
growth in response to the real shocks. However, informational restrictions make it
unlikely that the central bank can vary the money supply in response to fundamental
shocks. Instead, the central bank must rely on a more readily observable target. This
paper considers two frequently advocated targets: a price level target, and an interest
rate target.
The model’s equilibrium conditions are perturbed relative to the canonical real
business cycle model by two varying distortions: marginal cost and the nominal rate of
interest. The first distortion arises because of sticky prices, while the latter arises from
the cash-in-advance constraint. The two targets—price level and interest rates--have
opposing effects on these two distortions. Under a price level target, marginal cost is
stabilized but the nominal rate is variable. Under an interest rate target, the nominal rate
is stabilized but marginal cost is (in general) variable. Thus choosing the target is
isomorphic to choosing which distortion is variable and which is stabilized.
Both policies are subject to sunspot fluctuations arising from the endogenous
movement of the money stock. That is, there is not a unique money reaction function
that supports either a price level or an interest rate target, and each of these reaction
functions leads to different real outcomes. These sunspot fluctuations can be avoided
1
by specifying the money reaction function. One natural way of doing this is to limit the
contemporaneous response of the money stock to innovations in the environment. The
disadvantage of this approach is that it hinders the economy’s response to real shocks.
There is a voluminous literature on optimal monetary targeting. A selected
review of papers close in spirit to the current analysis include Carlstrom and Fuerst
(1995), Ireland (1996), and Goodfriend and King (1997). Carlstrom and Fuerst (1995)
examine interest-rate targeting in a limited participation environment and demonstrate
that one advantage of an interest rate peg is that it enhances the private economy's
ability to efficiently respond to shocks.1 Ireland (1996) demonstrates that in a sticky
price model the first-best monetary policy implies pegging the nominal rate to zero.2
Finally, Goodfriend and King (1997) use a sticky price framework to demonstrate the
advantages of price-level targeting. An interesting implication of all these analyses is
that by overcoming the source of monetary non-neutrality, a welfare-enhancing
monetary policy exacerbates output fluctuations arising from technology shocks.
The paper proceeds as follows. The next Section outlines the basic model.
Sections 3 and 4 examine the effects of a price-level target and interest-rate target,
respectively. Section 5 considers these targets under the additional constraint that the
1
See Lucas (1990), Fuerst (1992), and Christiano and Eichenbaum (1992), for analyses of limited
participation models.
2
The phrase “first-best” is not quite accurate. The distortion created by the assumption of imperfect
competition cannot be undone by any monetary policy. Ireland (1996) demonstrates that the optimal
monetary policy in this distorted world is still characterized by pegging the nominal rate to zero.
Carlstrom and Fuerst (1998) demonstrate that there are some concerns with this first-best approach to
policy. The foremost problem is that the combination of indeterminate real cash balances and demanddetermined output opens up the possibility of sunspot equilibria that would not arise if the nominal rate
were pegged above zero. The reason these equilibria do not arise in the case of a positive nominal rate is
that a binding cash constraint places an additional restriction on equilibrium behavior.
2
money supply is predetermined. Section 6 concludes.
2. A Model with Sticky Prices
We follow Chari, Kehoe, and McGrattan (1996), and utilize a model of imperfect
competition in the intermediate goods market.3 Since the model has become fairly
standard, we will be quite economical in our presentation.
Households are infinitely lived, with preferences given by
∞
E 0 ∑ β tU (ct ,1 − Lt )
t =0
where E0 denotes the expectation operator conditional on time-0 information, β
∈ (0,1) is the personal discount factor, ct is time-t consumption, Lt is time-t labor, and
the leisure endowment is normalized to unity. To purchase consumption goods,
households are subject to the following cash-in-advance constraint:
Pt ct ≤ M t + M ts (Gt − 1) − Bt
where Pt is the price level, Mt denotes beginning-of-period cash balances, Mst denotes
the per household money supply, and Gt denotes the (gross) money supply growth rate,
Gt ≡ Mst+1/ Mst. Note that monetary injections are carried out as lump sum transfers at
the beginning of each period. Bt denotes the household's choice of one-period nominal
bonds, each promising to pay Rt dollars at the end of time t. The existence of these
bonds has no effect on the equilibrium of this representative agent economy, but simply
allows us to determine the equilibrium nominal rate of interest. The household's
3
Chari, Kehoe, and McGrattan (1996), are in turn building on the imperfect competition model of
3
intertemporal budget constraint is given by:
M t +1 = M t + M ts (Gt − 1)+ Pt ( wt Lt +rt K t ) − Pt ct − Pt [ K t +1 − (1 − δ ) K t ]+Π t + Bt ( Rt − 1)
where wt and rt are the wage and rental rates, Kt is the capital stock, δ is the rate of
capital depreciation, and Πt denotes firm profits. Note that we are assuming that capital
accumulation occurs at the household level (and that investment is a credit good).
Final goods production in this economy is carried out in a perfectly competitive
industry that utilizes intermediate goods in production. The CES production function is
given by
1
Yt = { ∫ [ y t (i ) (η −1)/η ]di}η /(η −1)
0
where Yt denotes the final good, and yt(i) denotes the continuum of intermediate
goods, each indexed by i ∈ [0,1]. The implied demand for the intermediate good is thus
given by
y (i) = Yt[Pt(i)/ Pt]-η
t
where Pt(i) is the dollar price of good i, and Pt is the final goods price.
Intermediate goods firm i is a monopolist producer of intermediate good i.
Fraction ν of these firms set their prices flexibly within each period, while the
remainder (1-ν) must set their price one period in advance. The variable ν is thus a
measure of price flexibility. Below we will refer to three natural variants of the model:
(1) with ν = 1, this is a flexible price model, (2) with ν between zero and one, this is a
sticky price model, and (3) with ν = 0, this is a model with rigid prices. Other than the
difference in the timing of pricing, the firms are all symmetric, so we will henceforth
Blanchard and Kiyotaki (1987).
4
drop the firm-specific notation. Let Pt f denote the flexible price, while Pt s will denote
the pre-determined (or sticky) price. The final goods price (or aggregate price level) is
given by the appropriate average of these two prices:
{
Pt = (1 − ν ) Pt
s (1− η )
+ ν Pt
f (1− η )
1
1− η
}
.
(1)
The flexible price is given by a constant mark-up over the marginal cost (zt) of
production:
η
Pt f =
Pt z t .
η − 1
(2)
The term in brackets will appear frequently below, so we define z ≡ (η-1)/η < 1. In a
model with flexible prices, equation (2) and the assumption of symmetry implies that zt
= z. Combining (1)-(2), we have
1−ν
s
Pt = h( z t ) Pt , where h( zt ) ≡
1−η
zt
1 − ν
z
1/(1−η )
(3)
The function h is increasing so that innovations in marginal cost correspond with
changes in the price level. As for the sticky price, it is given by the solution to the
following maximization problem:
−η
Pt s Pt s
Pt =arg max Et −1 µ t +1 Pt Yt − z t
Pt Pt
s
where Et-1 is the expectation conditional on time t-1 information. The intermediate
goods firm is owned by the household, and pays its profits out to the household at the
end of each period. Because of the cash-in-advance constraint on household
5
consumption, the firm discounts its profits using µt+1 ≡ βUc(t+1)/Pt+1, the marginal
utility of $1 in time t+1. The firm's optimal preset price is given by:
{
η Et −1 µ t +1 Ptη +1 z t Yt
Pt s =
η
η − 1 Et −1 µ t +1 Pt Yt
{
}
} .
Using (3), this can be written as
[
]
[
]
Et −1 zµ t +1Yt [h( z t )]η = Et −1 z t µ t +1Yt [h( z t )]η+1 .
(4)
As for production, the intermediate firm rents capital and hires labor from households
and utilizes a CRS production function,
y t = θt f(K t ,H t ) ≡ θt K tα H t1−α
where θt is a measure of aggregate productivity, Kt denotes capital, and Ht denotes
labor. With zt as marginal cost, we then have rt = ztθtfK(Kt,Ht) and wt = ztθtfH(Kt,Ht).
As for the technology variable, we assume that it evolves in the standard fashion:
θt = exp(ζt), where ζt = ρθζt-1 + εθt, Et-1(εθt) = 0,
where ρθ is the autocorrelation coefficient, and εθt is the innovation in technology.4
Monetary policy is given by a process for Pt or Rt. In either case, Gt is
endogenous with the central bank varying the money supply growth rate passively to
achieve the targeted price level or nominal rate. The extreme versions of these two
policies are an inflation rate peg Pt+1 = πPt with π > β, and an interest rate peg, Rt = R >
1.
There are four markets in this economy, the labor market, the money market, the
bond market, and the goods market, and the respective market-clearing conditions are
4
Although we restrict the analysis to technology shocks, the principle theoretical conclusions of the paper
(eg., Propositions 1 and 2) are valid for a wider set of shocks including fiscal behavior.
6
given by Lt = Ht, Mt = Mst, Bt = 0, and
ct + Kt+1 = θt f(Kt, Lt ) + (1-δ) Kt.
(5)
A recursive competitive equilibrium is given by stationary decision that satisfy (3), (4),
(5), and the following:
U1(t)/Uc(t) = ztθtf L(Kt, Lt)/Rt
(6)
U (t + 1)[z t +1θ t +1 f K (K t +1 , Lt +1 ) + (1 − δ )]
U c (t )
= β Et c
Rt
Rt +1
(7)
U ( t + 1)
U c (t )
= β Rt E t c
Pt
Pt +1
(8 )
Ptct = MtGt
(9)
Before proceeding, a few observations about the sticky price equilibrium are in
order:
First, (6)-(7) are remarkably similar to the standard real business cycle (RBC)
Euler equations for labor choice and capital accumulation. They differ only in that both
are distorted from the Pareto optimum by the nominal rate of interest (Rt) and the
marginal production cost (zt). As neither distortion enters into the resource constraint
(5), Rt and zt can be interpreted as distortionary taxes that are rebated to the
representative household. Using this public finance interpretation, τc = (1-1/Rt) can be
7
viewed as a consumption tax, and τw = (1-zt) as the tax rate on wage and rental income.
Standard arguments from public finance suggest that constant tax rates would be
preferred to variable ones. Hence, one natural goal of monetary policy is to stabilize Rt
and zt.
Second, a log-linear approximation to the system implies that we can use (4) to
set zt+1 = z in the capital accumulation equation (7). That is, fluctuations in the implicit
tax on factor income are not persistent but arise only in the period of a shock. Thus
fluctuations in the marginal cost distortion have an effect on current labor input, but (to
a log-linear approximation) have no effect on capital accumulation. We will henceforth
refer to these fluctuations as movements in the wage tax. This behavior is in contrast to
the nominal interest rate distortion where fluctuations are likely to be persistent. For
example, under an inflation rate target a shock that leads to persistent movements in the
real rate will lead to persistent movements in the nominal rate.
Third, recall that if all prices are flexible then marginal cost is constant, zt = z.
Hence, the sticky price assumption affects real behavior via movements in the marginal
cost distortion on factor income. Conversely, for a given interest rate path, a monetary
policy that stabilizes zt is a policy that causes the sticky price model and the
corresponding flexible price model (with the same interest rate path) to be
observationally equivalent in real variables.
In the next two sections, the economy’s behavior under price level targeting
(Section 3) and interest rate targeting (Section 4) are discussed. There are two central
themes. First, either target is potentially consistent with the sticky price economy
responding to shocks as would a flexible price economy. Second, both targets are
8
susceptible to real indeterminacy and sunspot fluctuations. To avoid these bad
equilibria, Section 5 analyzes the two targets under a pre-determined money supply.
Such a policy avoids the sunspot equilibria, but also hinders the economy’s response to
real shocks.
3. Equilibrium with Price Level Targeting.
In a model in which some firms must pre-set prices, there are clear advantages to
the central bank eliminating uncertainty about the future price level. In particular, we
have the following:
Proposition 1: Consider a monetary policy given by a pre-determined price level policy
pt = p(kt-1,θt-1), where pt ≡ Pt/ Mt. Consider two economies, one with flexible prices and
one with sticky prices. Assume that monetary policy is conducted according to the same
pre-determined price level policy in the two economies. Then the sticky price economy
and the flexible price economy are observationally equivalent. This equivalence
includes the behavior of the nominal rate of interest and the money growth process.
Proof: With Pt pre-determined, (3) implies that zt is predetermined. From (4) we then
have that zt = z. Thus, real behavior in the sticky price and flexible price economies are
identical. The supporting money growth process is then given by the cash constraint
(9). The implied nominal rate is given by the Fisher equation (8). ♦
Remark: The equivalence in Proposition 1 does not necessarily hold in the case of
9
completely rigid prices (ν = 0). With rigid prices, the relationship between marginal
cost and the price level (3) is broken, so that a predetermined price level does not predetermine marginal cost. Thus there exist equilibria in which marginal cost responds to
innovations (real or sunspot). There is of course an equilibrium in which marginal cost
is constant and Proposition 1 is satisfied. The rigid-price case is an interesting
counterexample to the equivalence result, but is likely empirically irrelevant, so
henceforth we will ignore it by assuming ν > 0.
♦
The intuition for Proposition 1 is quite clear. In a flexible price economy a predetermined price level policy implies that firms would be willing to set their prices one
period in advance. Hence, such a monetary policy causes the flexible price economy to
act as if it were a sticky price economy. Or, in other words, in the sticky price economy
operating under such a policy the "sticky" firms would not want to alter their prices
even if they could do so costlessly. This monetary policy eliminates the sticky price
rigidity by making prices rigid.
Proposition 1 of course holds for a price level peg. This is the policy advocated
by Goodfriend and King (1998) and King and Wolman (1997). These papers
demonstrate that in a Calvo-style (1983) price-setting environment, the observational
equivalence of Proposition 1 occurs only with a price level peg.5 A price level peg is
required for observational equivalence in a Calvo setting because some firms wait an
arbitrarily long amount of time before having the option of adjusting their prices.
5
The counterexample in the Remark (where no firms are contemporaneously flexible) also holds in a
Calvo-style economy.
10
Proposition 1 is a natural extension of this result to an economy in which prices are
fixed for a finite number of periods.
Despite the simplicity of a price-level peg there are problems with such a policy.
Carlstrom and Fuerst (1998) demonstrate that in a flexible price economy an inflation
peg causes real indeterminacy. This implies an immediate corollary to Proposition 1: in
a sticky price economy, a policy of targeting inflation causes real indeterminacy. 6 This
arises because under such a policy the nominal rate moves too closely with the real rate.
Since the nominal rate acts as a distortionary consumption tax, this policy implies
raising consumption taxes when consumption is low. This policy creates self-fulfilling
behavior: by lowering current consumption, agents increase the real rate, which (under
an inflation rate peg) implies that the implied consumption tax rises, thus validating the
initial consumption reduction. Under an inflation rate target, marginal cost is constant,
but there is nothing to pin down the real and hence nominal rate of interest.
While this indeterminacy result under inflation targeting is robust to a wide range
of parameter values, it is particularly easy to see in a special case. Suppose that labor is
inelastic and the shocks are iid. Under a price level peg, we can use the Fisher equation
(8) to express the capital equation (7) as
U c (t + 1) = β Et {U c (t + 2)[ zθ t +1 f K ( K t +1 ,1) + (1 − δ )]}
Log-linearizing this equation yields
~
~
~ ~ ~
Et Q( K t +3 , K t + 2 , K t +1 ,θ t + 2 ,θ t +1 ) = 0
6
Goodfriend and King (1997) ignore these problems by ignoring the effect of the nominal rate of interest
on the work effort decision (6) and the capital accumulation decision (7).
11
where the ~'s denote percent deviations from the steady state. Note that neither current
capital nor current productivity enter directly into this equation. For determinacy, we
need two explosive roots. If we do have two explosive roots, then the only equilibrium
path is for capital to immediately jump to the steady state and stay there forever. This
equilibrium is not only peculiar, it is also unlikely. For all plausible parameter values
there will be only one explosive root. In this case, the equilibrium (scrolled back to
time-t) is given by
~
~
~
K t +1 = λK t + γθ t .
In this case, λ is given by the stable root of the following equation:
1
β f ′′U ′ 1
λ 2 = 1 + λ +
− .
U" β
β
The larger root exceeds one, so we are interested in the smaller root. For ease of
discussion, let us assume that this root is positive.7 The corresponding characteristic
equation for the RBC model is given by
1 β f ′′U ′
1
λ − .
λ 2 = 1 + +
U"
β
β
The stable root of the RBC equation will (of course) not solve the characteristic
equation of the inflation peg economy. Since this RBC root is positive and less than
one, casual inspection reveals that the root of the inflation peg is smaller than that of the
RBC model. That is, in comparison to the optimal path, convergence to the steady-state
is faster under an inflation peg. The intuition is as follows. Suppose that capital starts
7
For all plausible calibrations, this smaller root is positive. If it is negative, there are oscillatory
dynamics.
12
below the steady state. Along the path the real rate will be falling. Under an inflation
peg this implies that the nominal rate or implicit consumption tax is falling along the
path. The rational agent thus postpones consumption and increases capital
accumulation thereby speeding the movement to the steady state.
As for the response to the innovation, γ is free--the equilibrium is consistent with
any response of investment to the productivity shock! (And we can, of course, add an
arbitrary sunspot random variable to this solution.) The intuition is similar to the
previous paragraph. Suppose that we consider the case of γ = 0. In this case, investment
is constant and consumption moves sharply with the technology shock. Under an
inflation peg, an increase in current consumption implies that the real and nominal rates
fall. Referring back to the capital equation (7), this temporarily low nominal rate
stimulates consumption and deters investment to such an extent that capital does not
respond to the shock.
To eliminate this real indeterminacy under inflation rate targeting, we need
another real variable to be pre-determined. We take up this issue in Section 5.
4. Equilibrium with Interest Rate Targeting.
The previous discussion suggests that to avoid indeterminacy the central bank
should dampen the nominal rate’s response to the real rate. This suggests an interest
rate peg will eliminate this form of indeterminacy. However, Carlstrom and Fuerst
(1998) demonstrate that at a zero nominal rate there is real indeterminacy in a sticky
price model. The next proposition extends this result to the case where nominal rates
are positive. In an exact reversal of the indeterminacy under inflation rate targeting, this
13
indeterminacy arises because marginal cost is not pinned down.
Proposition 2: Suppose that in an economy with sticky prices the central bank pegs the
nominal rate Rt = R. Then the economy's real behavior is indeterminate. This
indeterminacy arises because the money supply is endogenous, and there is no
restriction on the response of marginal cost to innovations (including sunspots).
Proof: With a fixed nominal rate, real behavior is given by (4)-(7). The only restriction
on marginal cost is given by (4). In particular, there is essentially no restriction on the
response of zt to innovations (either real or sunspot). This behavior is then supported by
the endogenous money supply process implied by (9).
♦
This real indeterminacy and potential sunspot behavior arises because the money
supply is endogenous and not uniquely determined, and fraction (1-ν) of firms are not
operating on their labor demand curves but agree to supply whatever is demanded at the
pre-set nominal price. These firms' hiring needs are thus determined by the monetary
authority--higher money growth implies a higher nominal and, since prices are sticky,
real demand for output. The firm absorbs the resulting fluctuations in costs in its markup: the mark-up (1/zt) varies negatively with the rate of money growth.
An immediate corollary to Proposition 2 is that there exists an equilibrium in
which marginal cost is constant. That is, there exists a supporting money growth
process that pegs the nominal rate and marginal cost. Since marginal cost is constant,
this supporting money growth process also predetermines the price level. Thus, a
14
combination policy of pegging the nominal rate with a predetermined price level: (1)
stabilizes the interest-rate distortion, (2) stabilizes the marginal cost distortion on factor
prices, and (3) is not subject to real indeterminacy.8 This “nirvana” policy is probably
of little practical interest since it assumes a great deal of information on the part of the
central bank.
5. Targeting with a Pre-Determined Money Supply.
To summarize the previous two sections, an inflation rate peg is advantageous as
it stabilizes the marginal cost distortion. But such a peg is clearly inconsistent with a
constant interest rate as the nominal rate moves one-for-one with the real rate. This
latter fact implies that there is real indeterminacy and sunspot equilibria under an
inflation rate peg. To avoid these sunspot equilibria, the variability of the nominal rate
must be dampened. An interest rate peg can be consistent with a constant marginal cost
and, of course, a constant nominal rate. The disadvantage of such a policy is that there
are sunspot equilibria in which marginal cost fluctuates.
To avoid the sunspot equilibria under the two possible targets, we need an extra
constraint on monetary policy. This Section explores one natural possibility: the central
bank pre-determining the money stock, ie., eliminating the money stock’s response to
contemporaneous innovations. The disadvantage of this restriction is that real cash
8
As in Goodfriend and King (1997), this policy implies that the central bank varies the money supply to
achieve the desired level of real balances without movements in the price level. In particular, policy
responds to the innovations in consumption spending that would arise in the corresponding flexible price
economy. This is quite convenient as the flexible price economy is essentially a standard RBC economy,
and we have a great deal of knowledge about consumption spending in that model. Hence, a transient
(persistent) productivity increase should be matched by a relatively small (large) increase in the money
supply, while a transient (persistent) government spending increase should be matched by a relatively
small (large) decrease in the money supply.
15
balances are sticky, thus making the economy’s initial response to a shock suboptimal.
We begin the analysis by working out some simple examples, and then use the intuition
developed to discuss the impulse response functions in a realistically calibrated business
cycle model.
A Labor-Only Example:
Suppose that U(c,1-L) = ln(c) – L, f = (θL)1-α. The most interesting difference in
the economy’s behavior under the two monetary targets is in the response to an
innovation in the environment. Hence, for simplicity assume that the technology shocks
are iid. As point of comparison, recall that RBC behavior implies that labor is constant
in this well-known example.
Interest Rate Peg
Suppose first that the central bank pegs the nominal rate with a predetermined
money supply. Log preferences and predetermined money growth imply that money
growth is constant ((8)-(9)). Approximating the function h around the steady-state of zt
= z, we have dln(h)/dln(zt) = ν/(1-ν). Expressing the cash constraint (9) in log
deviations (hereafter denoted with a ~) yields:
ν − 1 ~
~
zt =
ct
ν
where we have used the fact that with iid shocks Pst/Mt is a constant. Substituting this
into (6) and the resource constraint yields
16
~ (ν − 1)(1 − α ) ~
Lt =
θ t .
(1 − α + αν )
Consider the response to a one-time shock. Contemporaneous labor input falls, with the
strength of the decline determined by the fraction of firms that are sticky (1-ν). Labor
input falls because marginal cost zt falls (the wage tax rises). Output rises as long as
prices are not completely rigid (ν = 0). In the next period, labor returns to its steady
state as marginal cost returns to normal. Hence, relative to RBC behavior, the unusual
effect of a shock arises only in the impact period.
Inflation Peg
In contrast, suppose that the central bank pegs the price level. In this case, the
cash constraint implies that consumption (and thus output) is predetermined, so that
labor responds sharply downward. In particular, (6), (8), and the resource constraint
yield:
α ~
~
~
~
Lt =
Lt −1 + θ t −1 − θ t
(α − 1)
Note that the contemporaneous decline in labor is equivalent to the behavior under the
interest rate peg only if prices are completely rigid (ν = 0). If some firms adjust prices
contemporaneously, labor responds more sharply downward under an inflation rate peg.
The irony is that while an inflation peg makes the sticky price economy behave as if all
prices were flexible, the economy behaves with little flexibility. The contemporaneous
decline in labor occurs because the nominal rate (consumption tax) rises. With inflation
pegged and current consumption fixed, the real and thus nominal rate rises because
17
future consumption (and future labor input) rises. Thus labor responds positively with a
lag to the shock, and thereafter the consumption dynamics are damped oscillations back
to the steady-state. These oscillations are supported by corresponding movements in the
nominal rate. Relative to the behavior under an interest rate rule, the perverse response
to a transient shock is remarkably persistent.
An Example with Capital and Labor:
Suppose that U(c,1-L) = ln(c)-L, f = Kα(θL)1-α, δ = 1, and the technology shocks
are iid. As point of comparison, recall that RBC behavior implies that consumption
and investment move one-for-one with output and that labor is constant.
Interest Rate Peg
In the case of an interest rate peg, we have:
(ν − 1)(1 − α ) ~
~
Lt = ~
zt =
θ t
(1 − α + αν )
~
~ ν (1 − α ) ~
c~t = K t +1 = αK t +
θ t .
(1 − α + αν )
The shock leads to a fall in marginal cost zt (an increase in the wage tax). The extent of
this decline depends upon the level of price rigidity (ν). At the extreme of full price
flexibility (ν=1) marginal cost and labor are constant as in the corresponding RBC
model. For less than perfect flexibility, the temporary tax change produces a negative
labor response and thus a dampened response of consumption and investment (relative
to RBC). As before, the perverse behavior of labor in response to a shock is over after
18
the period of the shock (once prices adjust).
Inflation Peg
In contrast, under an inflation peg, the perverse behavior is persistent.
Straightforward calculations imply:
~
(1 − α)(1 − αβz ) 2
~ (1 − αβ z )(1 − α) ~
Lt =
K
−
t 2
θt
(1 − αβz ) + (1 − α)
α βz (1 − αβz ) + (1 − α)
~
(1 − α 2 βz )
~
ct =
Kt
(1 − αβz ) + (1 − α)
(1 − α)[(1 − α) + (1 − αβz )] ~
~
~
K t +1 = αK t + 2
θt .
α βz (1 − αβz ) + (1 − α)
A few observations are in order.
First, the coefficient of capital-on-capital is exactly α, the RBC value. This is
surprising given Section 3’s demonstration that in an inelastic-labor model, convergence
to the steady-state is faster under an inflation peg, that is, the capital-on-capital
coefficient is less than α. The difference arises because of the effect of labor. When
capital is below the steady state the nominal rate is relatively high and is expected to
decline. This temporarily high nominal rate leads to two offsetting effects on capital
accumulation. Since investment is a credit good, there is a stimulus to investment (the
coefficient of consumption on capital is strictly greater than α). However, since leisure
is also a credit good, there is a negative effect on current labor input and thus output (the
coefficient of labor on capital is positive in contrast to zero for the RBC model). These
two effects exactly cancel leading to a coefficient of capital-on-capital that is identical
to RBC.
19
Second, the response of labor to an innovation is negative, but not as sharply
negative as in the inflation-peg case without capital. The investment channel is critical.
Although consumption is predetermined, since investment can respond output is not
predetermined. Hence, output responds positively to the shock. Since consumption is
predetermined movements in output must be entirely absorbed by movements in
investment. For example, notice that the coefficient of investment on the shock is
strictly greater than the RBC value of (1-α), and that output responds less strongly than
in the RBC economy.
Finally, in the previous model without capital, the labor response is always
larger (less negative) in the interest-rate-peg economy than in the inflation-peg
economy. However, in a model with capital, the initial labor responses in the two
economies are identical at the degree of price rigidity given by
(1 − ν ) =
(1 − αβ z ) 2
.
(1 − α 2 β z )
With β = .99, α = 1/3, z = 10/11, we have a bound of (1-ν) = .54. If prices are more
sticky than this, then the initial labor response is smaller (more negative) in the interestrate peg economy than in the inflation-peg economy.
Impulse Response Functions in a Calibrated Model
The previous examples illustrate the basic mechanisms at work in the numerical
simulations. A positive technology shock will cause the marginal cost distortion (wage
tax) to rise under an interest rate peg, and the nominal rate distortion (consumption tax)
to rise under an inflation peg. These tax movements will alter the responses of
20
employment and investment to productivity shocks relative to the RBC responses.
The simulations put some quantitative detail on this qualitative frame.
The model is calibrated as follows. Preferences are given by U(c,1-L) = ln(c) –
AL, where A is chosen to imply that steady-state work effort is 1/3. Assuming the time
period to be one quarter, we set β = 0.99 implying a 4% annual real rate of interest. The
capital coefficient is set at α = 0.34, and the quarterly rate of depreciation is δ = 0.02.
The steady-state level of quarterly money growth is G = 1.01 in all simulations,
implying a steady-state annual nominal rate of 8%. The steady-state mark-up is set at
10% implying η = 11. As for the degree of stickiness in the interest rate peg economies,
figure 1 reports results for ν = 0.20, ν = 0.80, and ν = 1 (the RBC economy). Figure 1
also shows the impulse response for the inflation peg economy. The shock is a 1%
productivity shock with ρ = .95.
A key result is that the two monetary policies lead to differing movements of the
implicit taxes. Depending on the degree of stickiness, the tax movement under an
inflation peg can be higher or lower than the tax movement under an interest rate peg.
With ν=1 the wage tax is constant. With ν = 0, in order to keep consumption constant
the wage tax must increase from 9.1% to 12.5%! In contrast, under an inflation peg, to
keep consumption constant the consumption tax need only rise from 1.98% to 2.3%.
This demonstrates clearly the difference between a consumption tax and a wage tax in a
model with investment.
The intermediate values of ν illustrate an important non-linearity. A little
flexibility goes a long way. With ν=0 the wage tax rises by 3.4%, vs. 0.9% for ν=0.2
and 0.1% for ν=0.8. This non-linearity also appears in the employment response. With
21
ν = 0, employment falls by 9.1%. At the other extreme, with ν = 1, employment rises
by 0.97%. With ν = 0.2, the employment response is –1.6% (3/4 of the way between
–9.1% and 0.97%) while the employment response is 0.76% (99% of the way between
–9.1% and 0.97%) with ν=0.8.
In the case of an inflation peg, the temporary consumption tax increase
discourages work effort but causes an investment surge (relative to an RBC economy)
which tends to stimulate work effort. Taken together, the inflation peg’s employment
response is only slightly below that of the RBC model, while investment is amplified
considerably relative to RBC (6.2% vs. 5.1%). Consumption does not respond
contemporaneously, but then jumps up in the second period.
Comparing the different employment responses for the ν = 0.80 case and the
inflation peg case reveals the differing distortions at work here. The shock causes the
implicit consumption tax (in the inflation peg economy) to rise by more than the
implicit wage tax (in the interest rate peg economy). But since one is a wage tax, and
the other is a consumption tax, the effect on employment is much different because of
the possibility of altering investment behavior.
The earlier 100% depreciation example emphasized that under an inflation peg a
shock leads to a persistent deviation of the nominal rate (or implicit consumption tax)
from the steady state. In contrast, under an interest rate peg the movement of marginal
cost from steady state ends after one period (as soon as all prices can adjust). With
plausible calibration, however, this qualitative difference in the two monetary policies
seems to be quantitatively unimportant. The deviation of the nominal rate from steady
state is trivial in the periods after the shock. Thus, in the period after a shock, the two
22
economies behave essentially as an RBC economy with a constant wage tax and a
constant consumption tax. For example the capital-on-capital coefficients are identical
while the consumption-on-capital coefficient rises from 0.5 in the interest rate peg
economy to 0.52 in the inflation peg economy.
A Welfare Analysis
We will exploit this approximate RBC-equivalence (in the period following the
shock) in conducting a simple welfare experiment. Consider two economies: (1) an
economy operating under an inflation-peg, and (2) an economy operating under an
interest-rate peg. Suppose both economies begin at the non-stochastic steady-state and
are hit by a one-time productivity innovation (with ρ=0.95) in, say, period 1. The
previous discussion implies that from period 2 onwards the economies (essentially)
follow RBC dynamics. To solve for the initial response to the shock, we will use the
period 1 Euler equations and the log-linear decision rule for RBC consumption in period
2.9 We can then solve this nonlinear system that describes period 1 behavior without
resorting to any further linear approximations. This is particularly useful as the
fluctuating taxes have important nonlinear effects.10 Once we have calculated period 1
consumption, employment, and investment, we can then compare utility across the two
economies by turning period 2 capital into utility units using period 2’s RBC marginal
utility of consumption. Since the period 1 responses are non-linear, we perform this
9
In the case of the interest rate peg, we also need an expression for period 2 employment and thus add
period 2’s (nonlinear) employment Euler equation to the system.
10
For example, suppose that utility were log-log and we used the log-linear decision rules for period 1.
Then the productivity shock would enter the welfare comparison linearly so that the expected welfare
23
calculation for a positive and a negative shock, and then average the two together. That
is, we are interested in the welfare consequences of either a positive or negative
productivity innovation, each of which occurs with equal probability.
The results of the welfare experiments are reported in Figures 2 and 3 (measured
in utility differences). For very low levels of price flexibility (ν < 0.18), the inflation
peg dominates the interest rate peg. However, if the economy is more flexible than this
(ν > 0.18), an interest rate peg dominates although the welfare gain is relatively small.
This basic result is largely insensitive to alternative values for the substitution
elasticity (η) and money growth rate (G). These values do reveal an interesting
nonlinearity. Under an interest rate rule, for large values of η (η = 51), stickiness is
preferred to flexibility. That is, welfare is higher for 0.09 < ν < 1 than an economy
with perfect price flexibility (ν = 1). This implies that a variable wage tax is preferred
to a constant wage tax. This surprising result occurs because in the former case the
average value of z is larger than the non-stochastic steady-state value of z (recall that z =
1 is the first-best).
This nonlinearity can best be understood by recalling the pricing equation (3)
and the binding cash constraint (9). With a predetermined money stock and sticky
prices, movements in consumption can occur only if marginal cost (z) moves in the
opposite direction. This implicit relationship between consumption and marginal cost is
convex. Hence, fluctuating consumption will lead to an average level of z that is higher
than the steady-state z. For benchmark parameters (G = 1.01, and η = 11), this effect
differences would all be zero.
24
peaks at ν = 0.05, where the average wage tax (1-z) is 9.03% vs. a steady-state wage tax
of 9.09%. With greater convexity, this effect is magnified: with η = 51 (a 2% markup), the average wage tax is 1.71%, while the steady-state is 2.0%.
There are a few caveats to this welfare comparison. First, by using the log-linear
decision rule for period 2 consumption we ignore a potential nonlinearity. It is unclear
which way this biases the results. However, given that we start the economy at the
steady-state capital stock and consider a small productivity shock, this bias is likely to
be small. Second, as already noted, the estimate assumes that under the inflation rate
peg the nominal rate is back to the steady-state for period 2 and onwards. This
assumption probably biases the results in favor of an inflation peg.
Third, and most obviously, these are welfare comparisons along an impulse
response from a one-time shock, and not lifetime utility measures in a stochastic
economy. In particular, we are ignoring all future nonlinearities that would arise in an
economy subject to repeated shocks. For example, as noted above the expected future
value of marginal cost is greater than the non-stochastic steady state. A similar condition
holds for the consumption tax in the inflation peg case. This discrepancy is very small:
for the baseline calibration, the average consumption tax (1-1/R) is 1.9805% vs.
1.9801% for the non-stochastic steady state. These omissions tend to bias the results in
favor of the inflation peg.
6. Conclusion
This paper is a tale of distortions. The model economy is distorted relative to
RBC behavior by marginal cost and the nominal rate of interest. These distortions are
25
fundamental, arising from the assumption of sticky prices and the need to use money to
facilitate transactions.
The monetary policy rule in place alters the economy’s response to shocks by
altering the endogenous movement of these distortions. A price level peg stabilizes
marginal cost, but allows the nominal rate to vary. An interest rate peg has the exact
opposite effect.
As a result, these two different monetary policies have different positive
implications for the real economy’s response to a shock. Under an interest rate peg, the
employment and investment responses are “too small” (relative to RBC). Under an
inflation peg, the investment response is “too large” while the consumption response is
“too small”.
On the normative side, these very different responses lead to a ranking of
policies. For small degrees of price flexibility, the marginal cost distortion is potentially
quite variable so that an inflation peg dominates. If more than 20% of firms have price
flexibility, this dominance is reversed and the interest rate peg is preferred. However,
the welfare differences of the two policies are quite small.
This near-welfare-invariance to cyclical monetary policy is similar to the results
of Dow (1995), Carlstrom and Fuerst (1995), and Ireland (1998). All of these analyses
ignore the potential welfare cost of a monetary policy that introduces sunspot equilibria
into the economy. These costs could be quite large. From this perspective, the most
useful policy suggestion is to do no harm: be somewhat agnostic in the choice of
cyclical monetary policy, but make sure that one chooses a policy rule that does not
introduce sunspot equilibria. This latter goal is achieved by using a policy rule that
26
severely restricts the immediate policy response to innovations in the real environment.
27
References
Blanchard, Olivier, J., and Nobuhiro Kiyotaki, "Monopolistic Competition and
the Effects of Aggregate Demand," American Economic Review (77), 1987, 647-666.
Calvo, Guillermo A., "Staggered Prices in a Utility-Maximizing Framework,"
Journal of Monetary Economics 12 (3), September 1983, 383-398.
Carlstrom, Charles T, and Timothy S. Fuerst, "Interest Rate Rules vs. Money
Growth Rules: A Welfare Comparison in a Cash-in-Advance Economy," Journal of
Monetary Economics (36), 1995, 247-267.
Carlstrom, Charles T, and Timothy S. Fuerst, "Real Indeterminacy under Inflation
Rate Targeting" 1997 working paper.
Carlstrom, Charles T, and Timothy S. Fuerst, "A Comment on The Role of
Countercyclical Monetary Policy," 1998, forthcoming Journal of Political
Economy.
Chari, V.V., Patrick Kehoe, and Ellen McGrattan, "Sticky Price Models of the
Business Cycle: Can the Contract Multiplier Solve The Persistence Problem," FRB
Minneapolis working paper #217, 1996.
Christiano, Larry, and Martin Eichenbaum, "Liquidity Effects and the Monetary
Transmission Mechanism," American Economic Review 82, 1992, 346-353.
Fuerst, Timothy S., "Liquidity, Loanable Funds, and Real Activity," Journal of
Monetary Economics 29, 1992, 3-24.
Goodfriend, Marvin and Robert G. King, "The New Neoclassical Synthesis and
the Role of Monetary Policy," 1997 NBER Macroeconomics Annual.
Ireland, Peter N., "The Role of Countercyclical Monetary Policy," Journal of
Political Economy (104), 1996, 704-723.
Ireland, Peter N., “A small, structural, quarterly model for monetary policy
evaluation,” (1998), forthcoming, Carnegie-Rochester Conference Series, 47.
Lucas, Robert E., Jr., "Liquidity and Interest Rates," Journal of Economic Theory
50, 1990, 237-264.
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