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State-Dependent Pricing and the
Dynamics of Business Cycles

WP 97-02

Michael Dotsey
Federal Reserve Bank of Richmond
Robert G. King
University of Virginia
Alexander L. Wolman
Federal Reserve Bank of Richmond

This paper can be downloaded without charge from:
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Working Paper 97-2

State-Dependent Pricing and
the Dynamics of Business Cycles *
Michael Dotseyt

Robert G. King’

Alexander L. Wolman§

February 13, 1997

Abstract
The nature of price dynamics has long been thought important for the
origin and duration of business cycles. To investigate this topic, we construct a dynamic stochastic general equilibrium macroeconomic model in
which monopolistically competitive f?rms face fixed costs of changing the
nominal prices of final goods. These prices are thus changed infrequently
and discretely. The framework captures major features of the price dynamics stressed by the New Keynesian research program, particularly work
on (s,S) pricing rules. However, by treating firms as heterogeneous with
respect to the size of fixed costs of price adjustment, we are able to study
a wider range of issues than in the prior literature. For example, we explore how the nature of optimal price-setting depends on (i) the extent of
persistence of variations in the money stock and (ii) the interest elasticity
of money demand. Further, our model can be used to study a wide range
of aspects of the positive and normative economics of monetary policy. We
illustrate these topics by considering the consequences of changing the rate
of inflation and by evaluating alternative policy rules.
‘The authors have benefitted from consultation with Marianne Baxter, Marvin Goodfriend,
Michael Woodford, David Romer, and Julio Rotemberg, and from seminar participants’ comments at Yale University,the Universityof Pennsylvania’sWharton School, the Federal Reserve
Bank of San l+anciscc, and the University of California at Berkeley. The views expressed here
do not necessarilyreflect those of the Federal Reserve Bank of Richmond or the Federal Reserve
System.
tFederal Reserve Bank of Richmond, P.O. Box 27622, Richmond, VA 23261-7622.
XUniversityof Virginia, Federal Reserve Bank of Richmond, and NBER.
8FederalReserve Bank of Richmond.

The nature of price dynamics has been long thought important for the origin
and duration of business cycles. Recent work in New Keynesian macroeconomics
has reemphasized the empirical observation that the prices of individual firms frequently remain fixed for substantial periods of time. This literature has attributed
price stickiness to costs of changing prices at the level of the firm-sometimes
called menu costs-which lead individual firms to adjust prices only when there
are sufficiently large variations in costs or demand.r
The standard development of dynamically optimal pricing policy with menu
costs follows an inventory theoretic approach, yielding price adjustment rules that
are (s,S) . Early work by Barre [1972], and She&in&i and Weiss [1983] on (s,S)
policies has been followed by a large number of recent studies, most notably those
of Caplin and Spulber [1987] and Caplin and Leahy [1991]. Yet, there is as yet
no work that incorporates optimal (s,S) p o h‘ties into a fully articulated dynamic
macroeconomic model. This shortcoming occurs for two reasons. First, price adjustments in a menu cost setting are state-dependent.
Second, the individual &m’s
price is adjusted in a discrete manner in response to the state of the economy.
This discreteness makes it difficult to characterize optimal aggregate price dynamics in a way that permits integration into a complete macroeconomic model. For
example, in response to a shock of a given size, the simplest (s,S) model would
predict that no firm (or ah firms) would adjust their prices. Richer (s,S) modelssuch as those in Caplin and Spulber [1987], Caplin and Leahy [1991] or Caballero
and Engel [1991]- moderate this prediction, by introducing heterogeneity in the
circumstances of individual firms. Yet, -these richer models typically involve a
“curse of dimensionality”, in the sense of Bellman [1957], so that they can be
solved under very restrictive conditions (random walk driving processes and interest inelastic money demand). These limitations mean that it is very difficult
to perform dynamic studies similar to those undertaken by real business cycle analysts. One cannot, for example, explore the consequences of alternative driving
processes for the money stock or alternative rules for monetary policy, which are
necessary ingredients to modern business cycle analysis.
‘Recent surveys that discuss the empirical evidence for infrequent price adjustment
and
its macroeconomic
consequences
are Rotemberg [1987] and Weiss [1993]. The twin volumes
edited by Mankiw and Romer [1991] contain core references in New Keynesian macroeconomics,
including some of the work on price adjustment that is most closely related to the topic of this
paper.

Thus, most recent work on the role of price dynamics in business cycles has
used an alternative approach, which is called time-dependent pricing. The most
attractive version is due to Calve [1983], w h o deve1o p s a model in which the timing
of price adjustment by an individual fhm is governed by an exogenous mechanism
which specifies that the probability of adjustment is a constant, independent of
calendar time and also of the length of time elapsed since the last adjustment. An
attractive feature of Calvo’s model is that it leads to a very simple representation
of aggregate price dynamics. 2 While the circumstances of individual firms are
random and price adjustments discrete, there is a sufficient number of firms that
aggregate price dynamics are described by a simple, low order expectational difference equation. Thus, in contrast to the complexity of the state-dependent pricing
rules that arise from the (s,S) literature, it is easy to incorporate time-dependent
price dynamics into standard macroeconomic models.3
In this paper, we derive a similarly tractable representation of state-dependent
pricing and imbed it in a small-scale macroeconomic model. We then contrast the
nature of business cycles with state-dependent pricing to those arising with an
identical steady-state pattern of time-dependent price adjustment. We thus undertake two extensions of an earlier literature on forward-looking pricing that uses
the Calvo setup, including work of Buiter and Miller [1985], Ball [1994] and others. On the basis of that time-dependent pricing model, one can reach three major
conclusions about the influence of changes in monetary policy on economic activity. First, if changes in the money stock are perceived to be temporary by f?rms,
then there is little response of prices. Second, if changes in the money stock are
perceived to be permanent, then there is a much larger change by fums adjusting
prices: they would choose to adjust most of the way toward the proportionately
higher price level that would prevail in the long run. However, the overall price
level would still behave sluggishly because many f?rms would not adjust. Consequently, there would be important aggregate effects of the monetary change on
real economic activity when the shift was temporary or permanent, but there is
some presumption that the effect is larger when the shock is temporary because
of more sluggish price adjustment. Third, if the monetary authority permanently
2However,there is an importantlimitation of Calve’s specification. As shown by King and
Wolman [1996], if the marginal probability of nonadjustment
in a quarterly model is greater
than .9, then firms will choose not to operate in the Calvo setup when there are inflation rates
of 10 percentor more. These difficultiesare not shared by the extensionof the Calvo model
that we develop below.
3Fkcent examples include Yun[1994], King and Wolman[1996], and King and Watson [1996].
3

increases the inflation rate in a credible manner, then there is little real effect or,
at least, little departure from the real effects that would prevail in a flexible price
model.
We explore two modifications of the Calvo framework below. In the first, we
allow the probability of price adjustment to be lower for firms that have recently
adjusted their price and higher for those that have not adjusted their price for
many periods. That is, we study the effects of a richer pattern of time-dependent
price adjustment. Looking at the dynamic response of an economy to some standard monetary changes, we find that there are some similarities with the basic
setup, but also some important differences. Notably, with a richer probability
structure, there is no longer a short-run super-neutrality: a permanent increase in
the inflation rate temporarily increases real activity.
In the second of these modifications, we permit a response of the pattern of
adjustment to the state of the economy: more firms undertake costly adjustments
when there is a larger present value of benefits to adjustment. Relative to a timedependent model with the same steady state patterns of adjustment, there are
many important differences. Some of these are relatively simple and intuitive.
For example, when there is a permanent increase in the quantity of money, there
is a faster pattern of adjustment in the price level because a larger portion of
firms adjust. Other implications are more complicated and less intuitive: we
find that there is a tendency for the dynamic responses of prices and output to
display oscillatory paths. Overall, though, we find that state-dependent pricing
is important for the analysis of inflation/disinflation policy: for permanent and
credible changes in the inflation rate, it restores approximate superneutrality.
The organization of the paper is as follows. Section 1 reports on some empirical aspects of price dynamics at the micro and macro levels that we want our
theory to capture. To undertake a comparison between the two models of pricing,
we must begin by extending the prior literature in two directions. In section 2,
we extend the Calvo [1983] fr amework to allow for a richer pattern of time dependence in price adjustment. In the original setup, the conditional probability
of an individual firm’s being able to adjust its price in any period is independent of how long it had been since its last adjustment. The extension that we
consider permits this conditional probability to vary in an arbitrary manner. In
section 3, we develop an analogous representation of the economy when there is
costly price adjustment. To do so, we begin by studying the price setting problem of a firm which faces a fixed cost of price adjustment that is random, rather
4

than certain as in much of the (s,S) literature. We derive a discrete individual
choice rule for the f&m: it will choose to adjust its price only if the gains from
doing so are su.fEciently large to warrant payment of the fixed cost. Thus, our
model captures price inflexibility at the level of the individual firm. However,
we also assume that there are many firms in the economy (technically a continuum) and that each faces a different level of fixed costs. Thus, there is always
a marginal firm that is indifferent between adjustment and nonadjustment even
when there are no differences in demand conditions across firms. We also impose
the restrictions that the randomness in adjustment costs is independent over time
implying that all Erms that adjust price choose the same price. This common
price outcome means that there is a low dimension of the state of the economy.
With such smoothness and dimensionality conditions satisfied, it is direct to use
conventional linear approximation model solution procedures.4 In section 4, we
discuss the structure of the rest of the small-scale macroeconomic model used in
this investigation. 5 In section 5, we develop the steady state of this model and
discuss issues of calibration, including the structure of adjustment costs.
Section 6 reports on how the state-dependent model of price adjustment works
with respect to basic changes in the quantity of money. To provide a reference
point for this discussion, we also consider a time-dependent model with an identical steady-state pattern of adjustment probabilities. Our discussion considers the
effects of three basic monetary policy shocks, assuming that the monetary authority makes the money stock its instrument. We contrast the effects of a temporary
increase in the quantity of money, a permanent increase in the quantity of money,
and a persistent increase in the money growth rate. We End that the nature of
the monetary policy shock determines whether there are quantitatively important
differences between state-dependent and time-dependent model responses: for a
purely temporary change in money, there is little difference but there are very
important differences in other cases, which generally increase the responsiveness
of the price level to monetary shocks.
Section 6 also reports on the interaction between optimal price adjustment and
the structure of the\rest of the macroeconomic model, specifically on the specifi*That is, we use linear approximation
methods 8s in the real business cycle analyses of
Kydland and Prescott [1982] or King, Plosser and Fkbelo [1988]. Our specific implementation
of
this approach draws on the model solution theory and algorithms of King and Watson [1995a,b].
‘This model is developedfrom those employed in King and Watson [1995c] and King and
Wohnan (19961.

cation of money demand which is typically assumed to be a quantity equation in
most standard (s,S) price adjustment analyses. We demonstrate that the impact
effects of money on output depend in a quantitatively important manner on the
specification of money demand. When we employ a money demand specification
that captures long-run interest sensitivity of real balances (based on the shopping
time specification estimated in Wohnan [1996]), we fmd that there are only minor effects of temporary monetary changes on output and prices: a one percent
increase in the money stock has about a .2 percentage point effect on output and
a negligible effect on the price level. When the persistence of monetary variations
is increased, there are larger effects on both output and on the price level. By
contrast, with a constant velocity specification imposed, the effect of money on
output is roughly one-for-one when the shock is temporary, but is increasingly
dissipated by price adjustment as the monetary changes are assumed to be more
persistent.
The discussion then turns to the implications of the state-dependent pricing
model for various alternative representations of monetary policy. In section 7,
we show that our pricing structure can be used with both interest rate rules
and with price level rules. In terms of the former, we trace out how an interest
rate shock would affect macroeconomic activity under state and time-dependent
pricing policies. In terms of the latter, we consider the effects of a permanent
productivity disturbance on the path of output under a policy of stabilizing the
price level, paralleling the prior investigation of King and Wolman [1996]. We
f.?nd that the time-dependent and state-dependent pricing models give responses
that are essentially identical to each other and also to the responses produced in a
frictionless price adjustment (real business cycle) model. In section 8, we consider
the transition between alternative rates of inflation in various sticky price models.
Our reference point for this discussion is the prior work by Buiter and Miller [1985],
Ball [1994] and King and Wohnan [1996], w h’lch sh ow two major results for the
Calvo model. The first of these is that there is no stimulative effect of an increase
in the inflation rate if it is simply increased in an unexpected manner. This
fmding arises from a combination of the price adjustment mechanism and the fact
that there is an accommodation of the changing level of real demand for money
by the monetary authority.6 The second of these is that a substantial expansion
6Indeed, in the King and Wohnan [1996] version of this experiment, there is a modest decline
in output that occurs as individuals substitute out of market activity into nonmarket substitutes
for monetized exchange.

occurs for several quarters if there is a permanent increase in the growth rate of
money. Reconsidering these inflation experiments, we find three major results in
this section. First, our more general time-dependent model does not share the
implication of the Calvo model for unexpected changes in the inflation rate: an
expansion arises when the inflation rate increases. This result is traced to the fact
that our timedependent model assigns a small marginal probability of adjustment
to 6rms that have recently adjusted price. Second, this expansion is virtually
eliminated by state-dependent pricing . Third, in both state-dependent and timedependent setups, there continues to be a quantitatively important difference
between increases in the money growth rate and increases in the inflation rate.
Section 9 provides a brief summary, reports our main conclusions, and discusses
directions for future research.

1. Price Dynamics

In this section, we begin by describing six facts about the dynamics of prices that
we think any macroeconomic model should be able to capture. We use these facts
as a basis for evaluating some existing models of price dynamics and as a rationale
for our current work.
1.1. Stylized Facts
The first four of the stylized facts stressed by the New Keynesian macroeconomics
apply to the behavior of individual prices and the price level; the latter two involve
the behavior of individual prices and the inflation rate.
Four facts concerning individual prices and the price level: Figure 1 displays
some hypothetical examples of paths of price adjustment that display four sets of
facts that motivate our investigation. These sorts of paths could be selected, for
example, from the Stigler and Kindahl [1970] data studied by Carlton [1986] or
the Israeli data that is discussed in Weiss’s [1993] recent summary of the case for
sticky prices. More specifically, these paths capture the range of price adjustment
facts produced in the recent set of studies by Lath and Tsiddon [1992, 19961.
First, as is shown in panel A of Figure 1, the paths of individual prices are
adjusted in ways that include infrequent adjustments, irregularly timed adjustments and changes of differing sizes (including many very small changes).
Second, as is shown in panel B of Figure 1, the adjustments of individual i?rms

are sufficiently imperfectly correlated that the path of adjustment for the price
level is relatively smooth.
Two more facts about injbtionay
situations: In situations of inflation, it is
necessary to add two additional facts to this list, namely that the changes in individual firm prices become larger and they also occur with greater frequency.
1.2. Implications for model building
Macroeconomists have developed models of price dynamics that seek to explain
these stylized facts. One approach is to simply assume that a representative f%m
faces quadratic costs of adjusting its price (as in Rotemberg [1982]): this approach
captures the dynamics of the price level well, but is inconsistent with the dynamics
of individual prices. Another approach is to assume that an individual firm has
an exogenously timed, random pattern of opportunities to adjust its price (as in
Calvo [1983]): this app roach can capture five of the six stylized facts, but it cannot
explain the greater frequency of price change in inflationary settings. Each of these
approaches has been criticized, for example by Blanchard and Fischer [1989], for
being too mechanical. However, each is very tractable and can be used to evaluate
the macroeconomic consequences of alternative monetary policies in line with the
general methodological recommendations of Lucas ([1976], [1980]).
Most recent work has followed Barre [1972] and Sheshinski and Weiss [1983] in
building models in which firms face constant real costs of changing nominal prices.
These models can also capture five of the six five stylized facts, but a different
subset than is produced by the Calvo setup. The (s,S) models can readily explain
the response of the frequency of price adjustment to inflation, but they cannot
easily rationalize the existence of many small price changes.
Further, despite a great deal of hard work-by Caballero and Engel [1991]
among others--’ rt has not proved possible to mold the (s,S) model into a useful tool. There are two main reasons. First, to avoid the implication that all
firms adjust simultaneously, it is necessary to introduce heterogeneity in the initial conditions (prices) or in the demand or cost conditions that firms face. This
heterogeneity has been introduced in a manner that requires development of extensive aggregation technology, which is itself unwieldy. Second, the optimality of
the simple (s,S) policy depends on assumptions about the driving processes of the
economy which are highly restrictive, specifically that micro and macro shocks are
continuous time random walks, as well as the absence of any effects of inflation on

the demand for money. The aggregate results in the (s,S) literature thus typically
require strong restrictions on forcing processes and behavior. These methods also
preclude building complete macroeconomic models. For these reasons, business
cycle researchers have mainly turned to the time-dependent pricing approach that
we will describe in the next section.7
The state-dependent pricing approach that we will develop in this paper can
capture all six of the facts discussed above, as well as being suflkiently simple that
it can be incorporated into a dynamic stochastic general equilibrium model that
can be solve in a rapid manner using linear systems methods. After we develop
the approach in the next two sections of the paper, we will use it to learn about
the consequences of discrete and occasional price setting for various aspects of
business cycles.

2. Time-dependent

Pricing

We begin by exploring the rational pricing practices of a monopolistically competitive firm that is required to hold its nominal price fixed for an interval of
random length that is exogenously determined. This firm is also assumed to
satisfy all demand at the posted price, which is the conventional assumption in
sticky price models. New Keynesian macroeconomists have stressed that as long
as price exceeds marginal cost, a positive response of quantity to demand will be
an optimal policy for a monopolistic competitor with a fixed price. Initially, we
will focus on expositing the adjustment structure and we then describe the nature
of dynamically optimal pricing policies.
7T~o natural questions that arise when one looks at Figure 1 are as follows. What type
of economic activity would not be subject to the discrete and infrequent adjustment
at the
microeconomiclevel? In what senseare stickypricesspecial, relative to these other categories
of economicactivity? King and Thomas [1996] provide a detailed discussion of the continuum
economy strategy that we use in this paper, together with a set of examples from labor economics, suggesting that it can be applied in many other contexts.
It is not clear to us that
prices are indeed special in terms of microeconomic
stickiness; Hamermesh [1989] argues that
employment
is discretely and infrequently adjusted at the firm level but that even relatively
coarse aggregation
mechanisms (less than 10 firms) produce series that one might plausibly
model as smooth.

2.1. Adjustment

structure

Shortly after a firm enters a period of our discrete time setup, there is a realization
of a random “signal” that determines whether it will be able to adjust its price. We
will let cyi be the conditional probability that a firm whose price has been fixed for
i - 1 periods will be allowed to adjust its price in the ith period. Correspondingly,
we will define qi as the probability of nonadjustment, qi = 1 - cyi. We assume that
there is a maximum time period, J, at which adjustment takes place for certain.
The adjustment process is displayed in Figure 2: it shows the pattern of flows
of firms of each type within each period of the discrete time structure and the
manner in which these flows alter the stock of firms of each type within the next
period. At the beginning of a given period of time t, a fraction of firms Sj, has not
adjusted its price for j periods, for j=1,2,...J. Subsequently, a fraction pi’ of each
type of firms receives the adjustment signal and a fraction of firms 17.jreceives the
nonadjustment signal. There is thus a fraction of firms, equal to ‘&
(Y& which
adjusts its price within period t. Comparably, there is a fraction of firms qjOj,,
in each category j = 1,2, . .. J , which continues to charge the nominal price set j
periods ago.
Thus, the fractions of firms are governed by a system of linear difference equations:
8.3+1,t+1 = qjejt for j = 1,2, . ..J - 1

(2-i)

J
e 1,t+1

C
j=l

Tieit

(2.2)

It is easy to calculate stationary values of the 6’ (the stationary distribution of
price-setters in terms of duration of price fix&y),

The Calvo [1983] version of this time-dependent price adjustment specification is
that aj = LYand that J + 00. Under this assumption, it follows that qj = rlj-’
and that Oj = (1- ~)77j-~. Thi s case will form a benchmark for some of our later
discussion.
With stationary probabilities, the time-dependent pricing structure makes it
easy to describe the evolution of firms through time and easy to create aggregates.
10

For example, if we let I$:-,, be the nominal price that was set by all adjusting firms
h periods ago, then a fixed weight price index could be calculated as

since fraction r],,& of firms is stuck with prices that they set prices h periods ago.
2.2. Optimal price setting with exogenous adjustment
The optimal pricing policy of a fum that is a monopolistic competitor can be
developed as follows. We define Vh(P&, St) as the nominal market value of a firm
that set its price h periods ago and A(St+h, St) as the nominal discount factor
for contingent cash flows in state S at date t+h. S is the state of the aggregate
economy, which includes all the factors governing the general price level P. We
define l&,(Pt-,,,St) as the flow profits accruing to a fim that last adjusted its
price h periods ago and set the price P&.
2.2.1. Firms that are not adjusting
For &ms that have not received the price adjustment signal, there is a dynamic
programming recursion of the form,

+a+lE[A(St+,,
+vh+dW(St+l,

St)vO(St+l)llSt
St)Vh+l(&,

(24

St+l>llSt)

where the customary max operation is omitted because we are specializing our
discussion to the case in which there are no decisions that the firm must make
other than price setting. This specification reflects the fact that the firm will
adjust next period with probability crh+l, in which case it will have nominal value
Vo(&+l)lSt, and will be unable to adjust with probability r]h+r, in which case it
will have nominal value Vh+l(Pt;l-h, St+,)lSt. Notice that we are assuming that
there is no effect of the length of the interval of price f!ixity on the value of the
firm if it adjust. The value recursions imply an “envelope theorem” condition for

each vintage,

(2.5)

dK+l(Pt++l-h,
St-l>IL
+qh+dWSt+l>St)
ap;-,
which will play an important role below.
2.2.2. Firms that are adjusting
For a firm that is capable of adjusting its price, we have the dynamic programming
recursion,
VOW

= =q@0(P,*,

St>

cw
+ ql~[A(St+l,St)~(P,*,St+l>]lSt}.

+alE[A(St+l,St)Vo(St+l)]lSt

Assuming differentiability of the value functions Vh for h = 1, . .. J - 1, it follows
that efficient price setting satisfies the first-order condition

0 = ab(p,‘,st)
apt’

+ qlEIA(&+l,

&)“(p,”

apt’

st+‘)],s t

Using updated versions of the envelope theorem conditions together with this
fist-order condition, it follows that we can write
0= 2

E{cp,,[A(S,+,, St) “‘(;;

;

St+h)] 1st

This condition indicates that the firm must be equating probability weighted
discounted marginal costs and revenues from a change in price.
2.2.3. A dynamic markup equation
We can combine the expressions of the model to generate an equation that is
a “dynamic markup equation”, which indicates how fkms adjust their prices in
response to interest rates and to their expectations about future costs and demand.
12

Assuming that there is a constant elasticity demand for its product, with --t being
the elasticity of demand, it follows that the optimal price satisfies

pt*

E C;z E{dA(St+h,
St>W+dd(P;,
St+dllSt
e-1
*
Cc; ~{p&Wt+,,
St)d(P;, St+~)llSt

(2.9)

In this expression, \E(St+h) is marginal cost at date t+h and d(P,*,St+h) =
is the level of real demand for the firm’s product at date t + h,
(p,*/pt+h)-%+h
given its choice of price today and the future values of the price levei and aggregate
demand. (Notice that the terms involving P: can be dropped from the right hand
side of the expression, since they enter in the numerator and denominator in the
same fashion). If adjustment were immediate or if costs were constant over time,
the price would be set as a simple markup over cost, P: = p\k, with p = 5.
Accordingly, within the time-dependent pricing setup, there is a distribution of
6rms in terms of prices and markups. King and Wolman [1996] follow Calvo [1983]
in studying the special case of this price adjustment specification in which (oh = qh.
They show that a higher rate of inflation increases the marginal markup, i.e. that
which an adjusting firm chooses. For small inflations, they show that there is
little effect on the average markup that is charged by all firms in the economy
(including those that are adjusting and those that are not). At higher rates of
inflation (those in excess of 5%)) by contrast, further increases in expected inflation
raise the marginal markup sufficiently so that the average markup actually rises.
However, the time-dependent price adjustment structure makes it impossible to
examine the effects of inflation on the frequency of price adjustment, since that
is specified exogenously.

3. State-dependent

pricing

For the purpose of studying the effects of steady state inflation and the dynamics
of business cycles, we now develop a closely related model in which the frequency
of price adjustment is endogenous. This model will imply that the fractions of
firms in the various “bins” in Figure 2 evolves through time and becomes part
of the state of the economy. Within this setting, for example, an increase in
the average inflation rate will mean there will be higher values of oj for every
j and potentially a smaller value of J. Inflation will make it more likely that
an individual firm will find it worthwhile to pay the (fixed) costs of adjusting
its price. However, our state-dependent pricing model will also have implications
13

for the dynamics of business cycles, as similar trade-offs emerge in response to
business cycle developments.
We want to stay as close as possible to the model of the previous section.
To produce a similarly tractable pattern of state-dependent adjustment, we need
three alterations in the framework above. First, we need to specify the nature of
the (fixed) costs of price adjustment that are present in our model. Second, we
need to determine the nature of the optimal adjustment decisions of firms which
govern the evolution of the fractions oit. Third, we need to explore the nature of
optimal price setting on the part of firms.
3.1. Adjustment

costs and adjustment rates

The key alteration of the previous model is that we explicitly allow for heterogeneity among firms in terms of discrete costs of adjustment, although the previous
model may be reinterpreted as one for which each firm learns the realization of a
random variable that implies either zero or infinite costs of adjustment. In contrast to this reinterpretation, we model the size of these fixed costs as a continuous
function of the fraction of firms that are adjusting.
To see how this structure works, let’s focus momentarily on a specific firm
in bin j. We assume that there is a f?xed labor cost of {jt hours that must be
paid if this Crm chooses to adjust its price. Since the nominal commodity cost is
IV(&)&, there will be variations due to changes in real wages and in the general
price level. These f&d costs will mean that our individual firm will choose either
to adjust or not to adjust, so that individual actions will be discrete.
While the individual decisions are discrete, we assume that-at the level of
the jth bin-there is a continuum of firms differentiated by the level of&: there
is a continuous function on the unit interval, 0 5 ojt 5 1, such that the real
labor cost of the marginal firm is [j(ojt) if the fraction of firms ajt is adjusting.
This function is graphed in Figure 3: the key properties of this function are (i)
that it originates at the origin (so that there is always some firm with no cost of
adjustment) and (ii) that it is continuous and everywhere increasing.
We continue to use the same notation as in the previous section of the paper
to denote the value of a firm that has not adjusted price for j periods, ~(Pt’-j, St).
For consistency we view this as describing the value of a Erm that currently does
not incur any fixed cost of adjustment, but will face future adjustment costs.
Firms of this type will face a continuum of adjustment costs and it follows that

there will be a critical value of ajt such that a firm will just be indifferent between
adjusting and not,
W&(Qjt)

= vO(St)

K(p,‘_j,&),

(3.1)

or there will be full adjustment (as with the firms in the Jth bin) if
w&(l)

< %(st) - &(p;-J,

St)-

Thus (3.1) describes the endogenous determination of the fraction of firms that
are adjusting price. Increases in wages reduce the fraction of firms adjusting;
increases in the value difference Vo(St) - K (Pt’_j, St) raise the fraction of firms
adjusting within the jth bin.
3.2. Values of firms and optimal pricing decisions
We next want to make this pattern of adjustment consistent with the rest of the
model developed in the previous section. There are two main issues here.
Uniformity of action levels: We want all firms that adjust to take the same
price setting action, i.e., to select the same P,‘, so that it is necessary for us
to carry along only a single price for each “bin” rather than a distribution of
prices. For this reason, we assume that firms face a fixed cost that is a serially
independent random variable.
Eflects of expected future adjustment costs on firm value: We also need to
modify the value function recursions above to introduce expected future costs
of adjustment. We assume that these costs are entirely born by the firms that
undertake the adjustment. Consequently, conditional on adjustment, the expected
fixed cost is Sj(Qjt) = vtj’ <j(Z)dZ]/(ajt).
The value function recursions are as follows. First, for firms in h = 1,2, . . J,
vh(Pt*-h, St) = Wh(PL,

+Eh+l,t+lWt+l,
+Eh+l,t+lwt+l,

St)

St)(vO(St+l)- ~t+,~,(~h+l,t+l))]lSt
st>vh+l(pt+_~,

(3.2)

St+1)]lSt}

BY the notation, ah+l,t+i and qh+l,t+r, we mean to indicate that these adjustment
rates are functions of the firm’s price and the general state of the economy, i.e.,
CYh+l(PLh,St+l) and ~h+l(&,&+f).
It is the essence of (3.1) that adjustment
is state-dependent in precisely this manner.
15

For a firm that chooses to adjust its price, we have the dynamic programming
recursion,
VOW

= mw+0(p,‘,

St)

+Ebl,t+lA(St+l> St)(vO(St+l) - W+l=&l,t+l))]lSt
+Eh,t+l Wt+1, Wi

(3.3)

(P:, St,,)] I&>

A nice feature of this model is that the efficiency condition for price-setting is
very close to that in the time-dependent setup, which is a special case when the
adjustment rates are exogenous. The partial derivative of the right hand side with
respect to price is:
1s

+lA(s,,l,s,)a~(p,t~st+l)],s
aP;

(3.4

It may appear that there should be additional terms in this expression, which take
into account the effect that the price has on the probability of future adjustment.
However, these additional terms are zero when we impose the requirement that
there is zero value for the marginal adjusting firm in (3.1).8 This expression can
be iterated to reproduce a version of (2.9), with the only modification being that
the conditional probabilities of maintaining price stickiness for h periods are now
functions of the future state of the economy. In this sense, the time-dependent
model is an apprtimation to the more general state-dependent setup in which
there is small variation in the probabilities.
3.3. Dynamics and aggregation
There are now a new set of endogenous state variables for our aggregate model,
the fractions of &ms that enter each period in the J bins of the economy. These
fractions of firms are governed by a system of linear difference equations:
‘The

additionalterms are as follows,

from straightforward
<(crl,t+l).
bracketed

differentiation.

From the definition

It follows that there is no contribution
term is zero.

of 8, it follows that a(pl*t~~~~llPt+l))

from this area, since (3.1) imp&

that the

8.3+1,t+1 = t@jt

for j = 1,2, .. . J - 1

P-5)

J
e 1,t+1

Qjtejt

(3.6)

j=l

One major difficulty with state-dependent pricing models is that they are very
difficult to aggregate, which Blanchard and Fischer [1989] stress as a key reason
that there has not been more work on the business cycle implications of these
models. Our setup makes aggregation almost as easy as in the time-dependent
pricing model framework of the previous section, because our model also has the
implication that there is a f&rite number of lags and there is a single price set by
all adjusting firms. The fixed weight price index described above then is
f~bhtw]Ptf
h=l

Jz(llhtsht)
h=l

q-h,

since fraction qhtehtof firms chooses to maintain the fixed price set h periods ago.
However, in line with the underlying monopolistic competition structure of the
model, we employ an alternative price level aggregate in our analysis below,
J-l

pt = {[~(~hteht)]Ppe)
h=l

+ J~(~hteht)(P~-h)‘l-~‘~c~‘.
h=l

(3.7)

in which,
as above,
-E is the (constant) elasticity of demand for each firm’s
product. Like the simpler fixed weight price index, this price level is affected by
both the prices set by adjusting firms and the fractions of firms in various “bins.“g

4. The rest of the model
We comment only very briefly on the structure of the rest of the model, since it
has been extensively discussed in King and Wolman [1996].“.
‘This
this class
“King
but they
rule used

price index is a natural outcome

of the Dixit-Stiglitz
preferences commonly used in
of models. For more detail, see Blanchard and Kiyotaki [19xX].
and Watson [1995c] provide a very detailed discussion of the real side of the model,
use a specification of time dependent pricing that is an ad hoc approximation
to the
by King and Wolman [1996].

Households: The households in our model are infinitely lived representative
agents, selecting contingency plans for consumption, labor supply and real balances. These plans are chosen to maximize the expected value of a discounted,
time separable utility function subject to an intertemporal budget constraint, a
time constraint, and a “shopping time” technology that specifies that money reduces the time that one would otherwise need to devote to transactions activity.
Firms: The firms in our households choose contingency plans for labor demand, investment, and prices so as to maximize their expected discounted value.
Marlcets: The .labor market in our model is perfectly competitive and the
commodity market is imperfectly competitive. A full set of markets in state
contingent claims is presumed to exist.
Government: There is no fiscal policy in our model economy. The decisions
of the monetary authority are described by a policy rule. We consider some
alternative policy rules in the discussion below.

5. The steady state and calibration
The model economy that we are studying has a nonstochastic steady state that
is relatively complex when compared to other models in the literature. We need
to understand this steady state because we are going to study the near steadystate dynamics of the model using linear approximation methods. In this section,
we briefly outline the nature of the steady state computations that we have undertaken. We then discuss other issues of calibration. Finally, we make some
comparisons across inflationary steady states of our model.
5.1. Computing the stationary distribution
The steady state of our model economy involves a stationary distribution of firms
in terms of the time since the date of last price adjustment. For given value of
the wage rate, the nominal interest rate, and the adjustment cost function C(Q),
we can describe how the steady state will operate and how it must be computed.
There is a fixed point problem in this economy because it contains is what
Bertsekas [1976] and Rust 119851call a “regenerative optimal stopping problem.”
To begin, the problem is an optimal stopping problem because there is an unknown
horizon J (the last bin in Figure 2) which must be determined according to the
rule that it is always optimal to pay the costs of fully adjusting. As discussed

above, this requires that
W[J(l)

< % - vJ(P*).

(5.1)

In contrast to many optimal stopping problems, this stopping point is influenced
by the value Vo that is associated with restarting the process. It is in this sense
that it is regeneragw.
For arbitrary V. and P, it is easy to determine optimal stopping time and
indeed to then construct (via backward induction) the remainder of the value
functions. Taking the future adjustment policy as given (from the prior step in
the dynamic programming recursions), it follows that:

In this latter expression, yp is the inflation rate in the steady-state that we are
studying, which is introduced when we make the problem stationary, and A is the
discount factor on one period nominal cash flows. Accordingly, the value function
recursions involve a real discount factor Arp. Implicitly, in these expressions, we
are treating the price as set at an arbitrary at an earlier date 0.
Given the value functions, it is then direct to compute the optimal policy using

w&h) = G - v&(P)

(5.3)

simply by “inverting” the [ function.
Proceeding through the value recursions, we can then determine a maximum
discounted profit at the initial date by optimizing over the various values of P to
find the value funztion (Vo) and the policy function (P*) for a firm which takes
the choke value Vo as exogenously speci@d. We then must find a fixed point,
i.e., a value of Vi that is optimal when Vo = I&,. The outcome of this process is
a pricing rule (value of P’) and a set of adjustment fractions that we can use as
an ingredient to our study of business cycles. The pricing rule is in the form of a
markup over marginal cost that depends on the likelihood of future adjustments.
The steady state has basic homogeneity properties. The value functions, wage
rate and price are all homogenous of degree one in the price level; the optimal
adjustment policy is unaffected by the general level of prices.

5.2. Interaction with the rest of the steady state
We specify a parametric representation of the rest of our steady state as is commonly done in real business cycle models. We can solve for the stationary state
of the rest of the model economy in a fairly straight-forward manner, conditional
on the outcome of the previous section. As it happens, we must iterate between
these two tasks. To see why, recognize that the real wage rate in our model is
given by UJ= W/P = fi[ay],
where jZ IS
’ an exogenously specified value of the
] is the marginal product of labor. Accordingly,
steady state markup and [athe material of section 5.1 may be viewed as determining an optimal markup p
conditional on a hypothesized value of the markup (6, which enters in the wage
rate). Again, we must seek a fixed point.
The computation of the steady state is thus somewhat involved, taking a few
minutes on a Pentium PC. This contrasts with the seconds that are typically
involved in either solving for the steady state of a typical RBC model or in computing the dynamic outcomes that we’ll consider further below.
5.3. The effects of inflation on adjustment frequency
We now use the model economy to provide a sample discussion of the effects
of inflation on the pattern of price adjustment. We specify an adjustment cost
function of the form
c = By

(5.4)

with B = .07 and b = 1. This value of B means that if all Erms adjusted fully
within the period, then there would be labor costs of 35% of market time (market
time is .20 of total time in the economy that we construct). However, because
firms choose to adjust only infrequently, there will be much smaller costs in the
calibrated steady state.ll
Figure 4 shows the steady-state distribution of firms by duration of price adjustment in three models. First, there is the result of computing the steady state
this specification governs the “marginal” fixed costs and is linear, our model has a form
adjustment costs” that may explain why it produces price dynamicssomewhat
similar to those of Fkkemberg [1982), in which the individual
firm faces quadratic oosts of
adjusting prices. However, our model economy is not exactly the same as the quadratic cost-ofchangesmodel, yieldingadditionalstate variablesthat describethe distributionof firmsacross
the “bins” of Figure 2 that make for more complicated dynamics.
“Since

of “quadratic

of an economy as described above. This economy involves 5% annual inflation:
we call it our benchmark model. Second, there is a Calvo model with the same
expected duration of price fixity (7.9 quarters).12 In our benchmark case, the
structure of adjustment shown in Figure 4 implies that only .54% of total market
labor time is spent in price adjustment: it is thus consistent with the frequently
expressed view that small “menu costs” can produce a relatively protracted average pattern of adjustment (as suggested, for example, by Mankiw [1985] and
Rotemberg [1987]) . Third, we compute an alternative steady state with 10%
inflation under our given price adjustment structure (5.4): we call this our high
in6ation model. Higher inAation results in more time allocated to price adjustment; 1.05% with 10% inflation. By way of reference, this increase in time cost
is smaller than the time cost associated with economizing on transactions costs
that result from the same increase in inflation, which Ring and Wolman [1996]
estimate as about one percent of market time.
The main points to be made about this figure are as follows. First, the
first panel of the figure shows that the steady state of the benchmark model involves virtually no chance that a firm will adjust its price within the first quarter
( a1 = .005) and roughly 63% chance that price fixity will last for a year or more.
However, the model also implies that these probabilities rise sharply through time
after the fist year and the maximum lag is nine quarters. At the same time, as
shown in panel B, the steady state also implies that 12.7% of the firms are adjusting each quarter. Second, to have the same expected duration of price 6xity,
the Calvo model is characterized by very different overall patterns of adjustment.
The Calvo model, therefore, appears to be a poor approximation to an economic
environment characterized by menu-costs and state-dependent pricing. The more
general time-dependent models that we developed above may be more useful approximations, especially if one is primarily concerned with issues involving steady
states. Third, there are quantitatively important effects of changing the average
rate of inflation on the average pattern of price adjustment. Increasing the inflation rate from 5% to 10% sharply increases the frequency of price adjustment for
the parameters that we employ: the expected duration of price fixity drops from
7.9 to 5.6 quarters and the maximum lag drops from 20 to 12.
The choice of these two parameter values, b = 1 and B = .07, is meant to
help us illustrate the nature of the modeling approach. We make no pretense that
12This was obtained by
the benchmark model.

choosinga value of 77to producethe sameexpected

durationas in

these parameters are calibrated to match any aspect of actual price dynamics.
However, the exercise of this section makes it clear that one could use data on
how the steady state pattern of price adjustment depends on the inflation rate to
select these parameter values.

6. Implications of state-dependent

adjustment

We now explore how our model economy with state-dependent pricing responds
to a basic set of monetary policy shocks. We think that these experiments shed
light on four important issues. The first issue concerns the size of the price
changes in response to various types of innovations in monetary policy. The
second issue pertains to the varying degrees of price level sluggishness and output
responsiveness associated with different types of price adjustment mechanisms.
The third issue involves the type of information that is useful for forecasting the
behavior of prices. The fourth issues involves the extent to which optimal pricing
is related to other aspects of the macroeconomic model, especially the money
demand function.
6.1. Dynamic effects of monetary expansions
To study the dynamic effects of monetary expansions, we have conducted a set of
experiments; Figures 5-8 report some summary some summary results on these.
We use the convention that the state-dependent model results are given by (o’s)
and the time-dependent model results are given by (+‘s), which we also use later
in the paper.
In Figures 5 and 6, we study the effects of a one percent monetary expansion
at date t = 1: the two panels of Figure 5 make the monetary expansion temporary
in alternative ways and the two figures of Figure 6 make the monetary expansion
permanent in alternative ways. By looking across these two figures, we can thus
determine how the price and output effects of a monetary expansion depends on
the nature of the monetary policy rule.
6.1.1. Temporary expansions
Figure 5 shows the effects of a one percent increase in the quantity of money
under two alternative scenarios that make the increase temporary. In the “fully
transitory” case of Figure 5A, the money stock is increased for one quarter of a year
22

(at date 1) and then returns to its normal level at date 2 and all future periods. In
Figure 5B, the money stock is increased by one percent in quarter one and then is
slowly brought back to path (the money stock rule is log(M)t = .8 *log(Mt-1) +~t
so that the effect of et = 1 on log(M)t is .8 in quarter 2, .64 in quarter 3 etc.)
In the fully transitory case in Figure 5A, there is essentially no effect of money
on the price level. The one percent change in the quantity of money has less
than a .Ol percent effect on the price level in both the state-dependent and timedependent models. There is also little difference between the two models in terms
of effects on output, the nominal interest rate or on the markup. In Figure 5B,
the disturbance is more persistent, but ultimately transitory.13 The effect on the
price level is larger than in Figure 5A, but continues to be small (less than .1
percent) in both the state-dependent and time-dependent cases.
It is noteworthy that when the monetary change is more persistent, there are
larger output effects at date 1 in the time-dependent model: investment and consumption respond more to the sustained monetary change. Investment responds
more dramatically because increased persistence implies future real demand will
be higher (a “rational expectations accelerator” effect); consumption responds
more dramatically because increased persistence implies that there are changes in
income of larger present value (a “permanent income” effect).
However, with increased persistence, we also begin to see a difference between
time-dependent and state-dependent pricing cases, which generally works to make
output less responsive to demand and prices more responsive to demand.
6.1.2. Permanent increases in the quantity of money
Figure 6 shows the dynamic response of the two model economies to permanent
increases in the quantity of money: panel A concerns a one-time change in the
quantity of money and panel B concerns a sustained, but ultimately temporary
increase in the money growth rate. In this Figure, the benchmark results for the
time-dependent model are again given by the +‘s and o’s are the results for the
state-dependent price adjustment.
A once-and-for-all increase in money: There are three basic findings in panel
A of Figure 6. The first is a continuation of a finding in Figure 5: increasing the
13The total effect on the stock of money is 1 + .8-t .64 + . . . = &
= 5 so that “mean lag”
reasoningwould suggest that the effects are approximately equivalent to the money stock being
one percent higher for 5 quarters.

persistence of the monetary disturbance all the way to a unit root continues to
increase the effect of a monetary shock on date 1 output. By way of reference,
the response of output to money is roughly one-for-one in the time-dependent
model of Figure 6A, while it was about .2 in panel B of Figure 5. The second is
a recurrent finding about how sticky price macroeconomic models with rational
expectations respond to permanent monetary shocks: nominal interest rates rise
as a result of a monetary expansion. The finding is recurrent because the price
level is largely predetermined in the short-run, but will ultimately rise in the
long-run. There is thus an increase in expected inflation and a necessary rise in
the nominal interest rate. The third lesson concerns the difference between statedependent and time-dependent models: a major effect of state-dependent pricing
is to mitigate the reai effects of the monetary expansion. For example, in panel A
of Figure 6, we see that there is an impact effect on output of about one-half the
size of the time-dependent model. Firms with low real price and consequent high
demand to are willing to pay to adjust. Typically, it is the firms that have not
adjusted for several quarters who fmd it most desirable to make the adjustment
so that the monetary shock induces a change in the distribution away from its
steady state levels.14
‘Ib-ansition to a higher path of money. Panel B of Figure 6 displays the effect
of a shock to money when the driving process is log(M,) - log(M,-1) = .67 *
PodM-1) -10&w-2)1
+ it, which is a form suggested by estimates of univariate
autoregressive models of Ml money growth over the post-war period. According
to this model, a positive one percent shock (Ed= .Ol) will raise the level of money
by three percent in the long-run. Such a shock thus induces a gradual transition
from an initial money supply path to a higher one. The three basic findings of
panel A are continued in panel B: (i) given that future money increases by more
than current money, there is a larger output effect at the impact date; (ii) nominal
interest rates rise rather than fall in response to a monetary injection; and (iii)
state-dependent pricing substantially mitigates the effects of money on output.
14With state dependent pricing, there is also a recurrent tendency for the price level and
real quantities to display oscillatory responses to the new steady-state,
i.e., for a period of
overshooting
to arise. More generally, oscillatory patterns for the adjustment
fractions (the
a’s and the O’s) are a recurrent part of smooth macroeconomic
models with discrete individual
chokes and, in principle, thee ascillations can therefore be translated to other variables. We
returnlaterto considerationof the extent of variation in the cr’s.

6.2. Factors affecting the price level
In this section, we explore the factors affecting the general level of prices, when
there is a once-and-for-all increase in the general level of prices: the influence of
these factors is shown graphically in Figure 7. The central message of this figure
is that changing patterns of adjustment are quantitatively very important for the
price level, particularly so a short horizons. Our analysis proceeds in two stages.
First, we recall that the price level is given as

h=l

in (3.7) above. Thus, in response to a permanent increase in the quantity of
money, there will be an important influence of lagged prices (PC-h). However,
the extent of this influence depends on the fraction of firms that choose to hold
prices fixed: with P: higher than P’feh, the price level will increase if more firms
choose to adjust. On impact, in panel A of Figure 7, this is precisely what
happens: the price level increases by about one-half of its long run increase,
with most of it stemming from increases in ah,&=1relative to their steady state
levels. More specifically, panel A of Figure 7 shows the decomposition of Pt into
components attributable to changing probabilities (the dashed line, representing
the effects of variations in 8,q and cr) and changes in these probabilities and in
the prices chosen by firms (the solid line). In the short-run, shifting weights are
dominant. In the long-run, the increases in prices are entirely due to shifts in
chosen prices, with the Q’S, q’s and B’s returning to their steady state values.
One implication of this decomposition is that empirical distributed lags “price
equations” would be subject to important econometric misspecification if they
omitted the determinants of the a’s and 8’s.
Second, we turn to the prices chosen by adjusting firms. In the state-dependent
pricing model, this takes the form,

C-1

~~E(cph[A(s,+h,St)~(s~+h)(P(St+h))l+"y(St+h)llSt
(6.2)
cl~E{(ph(A(St+h,St)(P(St+h))Ey(St+h)jlSt

In this expression, (Ph is the probability of nonadjustment between t and t + h,
(Ph =rlh(St+h)rlh-l(St+h-l)...rll(St+l), $(&+h)

P(St+h)

is ml

marginal

cost at date

t+h,

is the price level at date t + h and y(&+h) is aggregate output at date
25

t + h. The form of (6.2) thus suggests a decomposition of movements in P
into components attributable to the discount factors applied to future cash flows
(including both market discounting (A) and probability discounting (9)) and the
factors affecting cash flows: real marginal cost; the price level; and real output.
Figure 7, panel B, produces this decomposition for the case of a once-andfor-all increase in the quantity of money. This diagram shows that in the “long
run” of about twenty quarters P* increases by one per cent: all of this “long
run” variation is due to an increase in the general level of prices. However, on
impact, matters are substantially different. Increases in real marginal cost raise
P* by about .2% since there is an expansion of real economic activity and a rising
real wage is necessary to clear the labor market (this effect is presented with the
solid line in the diagram). Changes in the adjustment probabilities push up P*
by an additional 0.1% on impact. This reflects the fact that since many firms
adjust on impact, there is a slight decrease in the likelihood of adjustment in the
near future, meaning firms set a higher price today. Finally, the rising general
price level contributes a firlll% upward impetus to P: in our calibrated model,
this effect operates nearly entirely through nominal marginal cost, although there
is also a “demand switching” effect (represented by P(&+h)‘) that is present
in theory. Similarly, there are effects of market discounting (A) and aggregate
demand (y) that are present in theory but are not quantitatively important in
our calibrated model.
6.3. Price and output dynamics with a constant velocity specification
A notable feature of Figures 5 and 6 is that increased persistence of the monetary
driving process raises the responsiveness of both the price level and real output
to a monetary shock. This finding is sensitive to the assumed form of the money
demand function.
As a result of the “shopping time” model, there is a money demand function
which takes the approximate form near our steady state:
log(A4t) = log@)

+ vlog(ct) + (I - v> log(W) - d&s

That is, money demand depends positively on real consumption and the real
wage rate, with coefficients summing to unity (this restriction arises from our
requirement that there be constant velocity of money constant along a steady
state growth path driven by technical progress). The nominal interest rate has a
26

negative effect on the demand for money, as it induces substitution from money
into alternative time-consuming activities. Our choices of the money demand
parameters are based on a nonlinear shopping time model estimated on long U.S.
time series data, as in King and Wolman [1996], and are u = .29 and z9 = 24.
In terms of interpreting the semi-elasticity, we assume that interest rates are
measured as decimals, not percentages, and at quarterly rates. The implied semielasticity for an annual, percentage rate, is thus .06, which is not too different from
the .lO estimate that Stock and Watson [1993] derive similar long-term U.S. data.
It is nevertheless large relative to many estimates of the interest semi-elasticity of
the demand for money applicable over the horizons studied here.
An alternative money demand model-utilized in the studies of Caplin and
Spulber [1987] and Caplin and Leahy [1991]-is the constant velocity specification:

log(M) = log(P,)+ lo&t).
Under that alternative specification, there must be trade-off between the response
of prices to money and the real output effects of money.
In this subsection, we describe a variant of our model that replaces the shopping time demand for money with the constant velocity specification, while otherwise leaving the rest of the model unaltered. The results are presented in Figure 8:
panel A-shows the response to a purely temporary monetary increase and panel B
shows the response to a once-and-for-all increase in the quantity of money. There
are three features that are worth stressing. First, an increase in the quantity of
money has a smaller impact effect under state-dependent pricing than under timedependent pricing, in line with our earlier 6ndings in Figures 5 and 6. Second,
increased persistence of money raises the responsiveness of prices and thus cuts
the responsiveness of output to the monetary shock. Third, the permanent monetary disturbance raises the nominal interest rate, in line with our earlier results
in Figures 5 and 6.

7. Alternative

monetary policy rules

In this brief section, we discuss how state-dependent pricing is important for the
consequences of monetary and real disturbances under alternative policy rules.
While we treat it only briefly in the current paper, we think that it is of substantial
practical interest that the choice of the policy rule can have an important influence
on the structure of price adjustment.
27

7.1. Interest rate shocks
Some monetary economists have argued that Federal Reserve policy is better
modeled as an interest rate rule rather than a money stock rule (see, for example,
Bernanke and Blinder [1992], Goodf’riend [1987] and Sims [1989]). Our statedependent pricing framework is capable of modeling macroeconomic activity under
such interest rate rules; we assume that there is a mild, positive dependence of
the interest rate on the price level so that there is a unique stable equilibrium.15
Figure 9 shows the dynamic response of two model economies to a persistent,
but ultimately transitory, policy-induced variation in the nominal interest rate.
In this Figure, the benchmark results are given by the +‘s and o’s are the results
for the state-dependent price adjustment.
The major effect of state-dependent pricing is again to mitigate the real effects
of the expansionary monetary disturbance (a decline in the nominal interest rate)
in line with our earlier Endings. Looking again at the output responses in panel
A of Figure 9, we see that there is an impact effect on output of about one-half
the size of that in the corresponding time-dependent model. We also again see
some tendency for oscillatory dynamics in real and nominal variables.
7.2. Targeting the price level
In the Calvo version of the time-dependent model of price adjustment, King and
Wolman [1996] study the effects of conducting monetary policy so as to target a
path for the price level. In this subsection, we assume that there is a permanent
productivity improvement under a policy rule that adjusts the money stock so that
there is a fixed path for the price level. Figure 10 shows the consequences of a one
percentage point permanent increase in productivity for macroeconomic activity
when there is a P rule in place. There is an equivalence of three sets of real outcomes: those under flexible prices (a monopolistically competitive RBC model),
those under time-dependent prices and those under state-dependent prices.16As
15The interest

rate rule is Rt - R = f(log(Pt)

- log(P))

+ xt with f = .l.

The analysis
in many
macroeconomic
models. Our chosen value off means that the monetary authority will raise the
nominalinterestrate by 40 basis points if the price level is one percentabove its target level.
We choose a relatively small, positive value of f to achieve determinacy
while making a great
deal of the response of interest rates exogenous at short horizons.
“This equivalence
is not exact, but is accurate to the extent that one cannot distinguish
visually between the outcomes when graphed as in Figure 10.

of Kerr and King [1995] suggests that any f > 0 will produce a unique equilibrium

stressed by King and Wolman [1996], by targeting the path of the price level, the
central bank makes the quantity of money move in ways that eliminate a potential demand-deficiency that would occur if an M rule was alternatively used.17The
results of this section thus reinforce the previous conclusions concerning the desirability of targeting the path of the price level.
An novel element of the price level targeting rule is that it makes state and
time-dependent pricing equivalent.. With no change in the inflation rate, there
is little incentive for individual agents to change the time pattern of adjustment.
This contrasts sharply with the differences documented for other policy rules
above.

8. Changing

the rate of inflation

We now use our framework to explore the implications of permanently increasing the rate of inflation. We take as background to our analysis the results of
comparable experiments in King and Wolman [1996, section 6.21 that concern
the effects of such permanent changes in inflation within the Calvo version of the
time-dependent model, although we do not explicitly reproduce those results here.
The findings of that earlier analysis and those of Buiter and Miller [1985] and Ball
[1994] were that disinflations introduced by a decline in the money growth rate
could cause large recessions, but that a disinflation undertaken with an initial
monetary accommodation could be costless.
8.1. A permanent increase in money growth
Figure 11 shows the effect of increasing the money growth rate by two percentage
points, so that infiation must also by two percent in the long run. In both the timedependent model and the state-dependent model, there is a large initial increase
in the level of real economic activity, although the initial “boom” is substantially
smaller in the model with state-dependent pricing (the peak effect on output is
about 12 percent relative to trend with time-dependent pricing, while it is about
9 percent with state-dependent pricing). Inflation must temporarily exceed the
171n our setting, an M rule would lead to an initial contraction of labor input in response to
a productivity
disturbance,althoughthe magnitudeof this declinewould be somewhatsmaller

with state dependentratherthan time dependentpricing.

long run level, as the economy transits to a new lower level of real balances in the
new steady-state situation.
These Sndings are broadly in line with the results of similar experiments in
King and Wolman [1996], with the Calvo form of time-dependent pricing, which
in turn are similar to the earlier findings of Buiter and Miller [1985] and Ball
[1994].
8.2. A permanent increase in inflation
Figure 12 shows that the effects of unexpected, permanent increase in the inflation
rate from 5% to 7% per annum. As shown in panel B, this requires a complicated
pattern of money stock adjustment as the central bank accommodates the timevarying demand for money that arises as a result of this shock. Notably, in contrast
to earlier experiments with the Calvo form of time-dependent pricing, there is a
temporary expansion of real economic activity (a peak effect of about 4% on
output, which is significantly smaller than that which arises with the sustained
monetary in Figure 11). This result can be traced to the fact that our timedependent model assigns a small marginal probability of adjustment to firms that
have recently adjusted price and large marginal probabilities to who have not
recently adjusted price. Accordingly, to increase the inflation rate, the average
markup of price over marginal cost must fall, resulting in an expansion of economic
activity.
However, the. initial expansion is eliminated by the introduction of statedependent pricing. Essentially, firms with a lengthy interval since their last price
adjustment choose to adjust in response to the change in inflation, so that the
average markup and real activity respond minimally to the policy shift.

9. Summary and conclusions
In this paper, we derived a tractable representation of state-dependent pricing,
which we used to explore the dynamics of business cycles. It is flexible enough
that we were able to study the impact of real and nominal shocks under a wide
variety of alternative monetary policies and general equilibrium specifications.
More specifically, we illustrated the use of this approach with a range of “quantitative theory” experiments, contrasting it to a time-dependent price adjustment
mechanism with the same steady state properties. Based on this comparison, we
30

conclude that there is an important potential difference between time and statedependent pricing: the average (steady state) pattern of price rigidity may be a
poor description of the marginal pattern of price rigidity in response to specific
shocks.
Our initial set of experiments with this framework also highlighted the role that
expectations about the future play in the pace and pattern of price adjustment.
Under some scenarios, we found that there can be little real effect of changes
in inflation if there are state-dependent prices. Thus, in our future research, we
plan an analysis of disinflations that are viewed as imperfectly credible by private
agents, who must learn about the true state of the monetary authority’s objectives
by studying a path of money growth outcomes.

Figure 1. Representative price dynamics

A. Individual price dynamics

’ 0

100

time

B. Individual and aggregate price dynamics
lo
4

.,, - 1
’
I
I
,

I-\

+?-

i___,
I

,-’
I
I
I
I

‘1
I

OD0

‘:

I
t
I
I
\-,

I
11

,
L--

,
,

,
I
I
I

‘9
0I

I
I

--\,

I
I
--

’

I-1
‘I
‘!

I
I

------I
8
I

‘-

N
,I

---mm-

\
’
I
I

-I
\

1
I

‘,I

tn
i
’

time

‘00

Figure2. Evolutionof “vintages” of price-setters

J
c j=laj,t
,

ej t

Adjustment at t

(

%,t

e2,t

8 24

6 2,t+l

.
.
.

Date t
initial conditions

Date t+l
initial conditions

Figure 3. Deter&x&ion of the marginal firm

costs and
benefits

CD
dF

Vo’Vj > C

Cadiustmentj
\---,-.--~---.,

V,-Vj < Costs

(no
adiustmentl
\
4
,

0.3

0.4

0.5

0.6

fiaction of lkms (alpha)

0.7

Figure 4. Steady State Distribution of Firm
10% inflation (0) Calvo (-)
Benchmark (0 )

A Conditional Adjustment Probabilities

Quarters

B. Fraction of Fii

at Each P’,

j (quarters)

Figure 5. Temporary Increase in the Quantity of Money
Tie Dependent (+)
Pricing: State Dependent (0)
B. rho =.8
A. rho = 0
output

3.
2

j-k
,O

InterestRata

Nominol

I
2

OUWtWS

10
0uort.n

Figure 6. Integrated Driving Processes for M
Time Dependent (+)
Pricing: State Dependent (0)
B. serially correlated money growth shock
A random walk M shock
output
Output

Price lmrsl (and

Nominal

lntsrost

mmsy

supply)

Auto

,.,......

. . ..~.

ouortarr

f
In
'0

10
OUWtU*

02468

10
OUOI-MS

Figure 7. Price Level Decompositions

A
% deviations

Decomposition of Pt

.\
\\
\\
.

contribution of changing
‘\ . _ probabilities
------___,
---__ __----14

QGrt er s’2

B. Decomposition of P:
% deviations

-? -

\
.\
\

_c----_____--------

real

---

marginal

cost

plus

probabilities

marginal

cast

plus

probabilities

i
j-

,
0

marginal

cost

alone
plus

price

level

Figure 8. Interest Inelastic Money Demand (M/P=Y)
Pricing: State Dependent (0)
Time Dependent (+)
A. white noise M shock
B. random walk M shock
outwt

output

“!
0
I0

Prim

xNI
5: *;O

Lsval (ond money supply)

. ,
6

22
10

Markup

‘9
‘;O

Figure 9. Persistent Interest Rate Shock (rho=.9)
(Weak Feedback Rule)
Pricing: State Dependent (0)
Time Dependent (+)

8’
do

C: Nominal

Interest

Rote

(and

shock)

0
72
lo

-‘.‘-‘..
2

D: Horkup

Figure 10. Permanent Productivity Shock
Price Level Targeting
Pricing: State Dependent (0)
Time Dependent (+)
A:outpur
dG
1

!!I
2

C: Nomlnol

Interest

Rote

10
12
auortsrs

10
12
ouarters

Figure 11. Permanent Increase in Money Growth
A: Output

1.10

1.06

/
1.02

/
/

0.98
0

Quarters

B. Money Stock

Quarters

C. Price Level

Figure 12. P&manent Increase in hflation
A: Output
1.14

1.06

1.02

0.98
0

Quarters

B. Money Stock

0.90 1
0

Quarters

C. Price Level

I
20

Quarters

References
[l] Ball, Lawrence, “Credible Disinflation with Staggered Price Setting,” American Economic Review, vol. 84, (1994), 282-289.
[2] Ball, Lawrence, “Disinflation with Imperfect Credibility,” Journal of Monetary Economics, vol. 35, (1995), 5-23.
[3] Barre, Robert J., “A Theory of Monopolistic Price Adjustment,” Review of
Economic Studies, vol. 34, no. 1 (January 1972), 17-26.
[4] Bellman, Richard, Dynamic
versity Press, 1957

Programming,

Princeton, N.J.: Princeton Uni-

[5] Bemanke, Ben and Alan Blinder, “The Federal Funds Rate and the Channels
of Monetary Transmission,” American Economic Review, vol. 82, (September
1992)) 901-921.
[6] Bertsekas, Dimitrius P., Dynamic Progmmming
York: Academic Press, 1976.

and Stochastic

Control, New

[7] Blanchard, O.J. and N. Kiyotaki, “Monopolistic Competition and the Effects
of Aggregate Demand,” American Economic Review, 77 (September 1987),
647-666.
[S] Blanchard, Oli vier J. and Stanley Fischer, Lectures on Macroeconomics,
MIT Press

1989,

[9] Buiter, Willem H., and Marcus Miller, “Cost and Benefits of an AntiInAationary Policy: Questions and Issues,” in Victor E. Argy and John W.
Neville, eds., In.tion
and Unemployment:
Theory, Eqverience and Policy
Making, London: AIIen and Unwin, 1985, pages.
[lo] Caballero, Ricardo J. and Eduardo M.A. Engel, “Dynamic (S,s) Economies”,
Econometrica, vol. 59, (November 1991): 1659-1686.
[ll] CaIvo, Guillermo A., “Staggered Prices in a Utility-Maximizing namework,”
Journal of Monetary Economics, ~01.12, (1983): 383-398.
32

WI Caplin,

Andrew, and Daniel Spulber, “Menu Costs and the Neutrality of
Money,” Quarterly Journal of Ewnomics, vol CII, (1987): 703-725.

WI Caplin,

Andrew, and John Leahy, “State Dependent Pricing and the Dynamics of Money and Output,” Quarterly Journal of Economics, vol. 106, (1991):
683-708.

Carlton, Dennis, “The Rigidity of Prices,” American
no. 4 (September 1986), 637-658.

Economic

Review, 76,

[=I Chirinko,

Robert, “Business Fixed Investment Spending: A Critical Survey
of Modelling Strategies, Empirical Results and Policy Implications,” Journal
of Economic Literature, vol. 31, (1993): 1875-1911.

WI Dotsey,

Michael and Robert G. King, “Rational Expectations Business Cycle
Models,” Federal Reserve Bank of Richmond Economic Review, 74, (1988),
3-15.

P-1 E&stein,
Economic

Otto and Gary Fromm(1968), “The Price Equation,”
Review, vol. 58, (1968), 1159-1183.

American

WI Goodfriend,

Marvin S., “Interest Rate Smoothing and Price Level TrendStationarity,” Journal of Monetary Economics, vol. 19, (1987): 335-348.

Hamerrnesh,, Daniel, “Aggregate Employment Dynamics and Lumpy Adjustment Costs,” Carnegie Rochester Conference Series on Public Policy, vol. 33,
(1990)) 93-129.

PO1Hayashi,

Fumio J., “Tobin’s Marginal q and Average q: A Neoclassical Interpretation,” Econometrica, vol. 50, (1982)) 215224.

WI Kerr,

William, and Robert G. King, “Limits on Interest Rate Rules in the
IS Model,” Federal Reserve Bank of Richmond Economic Review, 82, (1996),
l-20.

WI King,

Robert G., Charles I. Plosser and Sergio T. Rebelo, “Production,
Growth and Business Cycles,” Journal of Monetary Economics, vol. 21,
(1988), 195-232.

[23] King, Robert G. and Mark W. Watson (1995a), “The Solution of Singular
Linear Difference Models Under Rational Expectations,” manuscript, the
University of Virginia.
[24] King, Robert G.and Mark W. Watson (1995b), “System Reduction and Solution Algorithms for Singular Linear Difference Systems Under Rational
Expectations,” manuscript, the University of Virginia.
[25] King, Robert G. and Mark W. Watson (1995c), “Money, Prices, Interest
Rates and the Business Cycle,” Federal Reserve Bank of Chicago, working
paper, (version of an article in Review of Economics and Statistics, February
1996).
[26] King, Robert G., and Alexander L. Wolman (1996), “Inflation Targeting in
a St. Louis Model of the 21st Century,” Federal Reserve Bank of St. Louis
Review, 78, (1996), 83-107.
[27] King, Robert G. and Julia-K. Thomas, “Smooth Macroeconomic Models with
Discrete Individual Choices,” manuscript in progress, 1996.
[28] Kydland, Finn, and Edward C. Prescott, “Time to Build and Aggregate
Fluctuations,” Econometrica, vol. 50, (1982): 1345-1370.
[29] Lath, Saul, and Daniel Tsiddon, “Staggering and Synchronization in PriceSetting: Evidence from Multiproduct Firms,” American Economic Review,
forthcoming.
[30] Lath, Saul, and Daniel Tsiddon, “The Effects of Expected and Unexpected
Inflation on the Variability of Relative Prices,” Economics Letters, vol. 41,
(1993), 53-56.
[31] Leeper, Eric, and Christopher A. Sims,“Towards a Macroeconomic Model
Usable for Policy Analysis”, NBER Macroeconomics Annual, 1994.
[32] Lucas, Robert E. Jr., “Econometric Policy Evaluation: A Critique,”
Carnegie-Rochester Conference Series on Public Policy, 1976, 1: 19-46.
[33] Lucas, Robert E. Jr., “Methods and Problems in Business Cycle Theory,”
J~urnaZ of Money, Credit and Banking, vol. 12, (1980).

[34] Mar.&w, N. Gregory and David Romer, New Keynesian Macroeconomics, 2
volumes, Cambridge, MA: MIT Press, 1991.
[35] McCaIlum, Bennett T., “Price Level Determinacy with an Interest Rate
Policy and Rational Expectations,” Journal of Monetary Economics, vol.
8, (1975), 319-329.
[36] McCallum, Bennett T. and Marvin S. Goodfriend, “Theoretical Studies of
the Demand for Money,” in The New Palgmve: A Dictionary of Ewnomics,
John Eatwell, Murray Milgate and Peter Newman eds., Macmillan: London;
Stockton Press: New York, 775-781.
[37] Rogerson, Richard, “Indivisible Labor, Lotteries and Equilibrium” Journal
of Monetary Economics, vol. 21, (1988): 3-16.
[38] Rotemberg, Julio, Monopolistic Price Adjustment and Aggregate Output,”
Review of Economic Studies, 44, (1982), 517-531.
[39] Rotemberg, Julio, “The New Keynesian Microeconomic
NBER Macmeconomics Annual, 1987, 69-114.

Foundations,”

[40] Rust, John, “Stationary Equilibrium in a Market for Durable Assets”, Econometrica, vol. 55, no. 4 (July 1985): 783-805.
[41] Scarf, Herbert, “The Optimality of (s,S) Policies in the Dynamic Inventory
Problem,” in K.J. Arrow, S. Karlin and P. Suppes, eds., Mathematical Models
for the Social Sciences, 1959, Sanford Univesity Press, chapter 13.
[42] Sheshinki, Eytan, and Yoram Weiss, “Optimum Pricing Under Stochastic
Inflation,” Review of Economic Studies, vol. 50, (1983), 513-529.
[43] Sims, Christopher A., “Comparison of Interwar and Postwar Business Cycles:
Monetarism Reconsidered,” American Economic Review, vol. 70, (1980):
250-257.
[44] Sims, Christopher A., “Models and their Uses,” American
cultuml Economics, vol. 71, (1989), 489-494.

Journal of Agri-

[45] Sims, Christopher A., “Interpreting the Macroeconomic Time Series Facts:
The Effects of Monetary Policy, European Economic Review, vol. 36, (1992),
975-1012.
35

[46] Stigler, George, and James Kindahl, The Behavior of Industrial Prices,
NBER General Series, no. 90, New York: Columbia University Press, 1970.
[47] Weiss, Yoram, “Inflation and Price Adjustment: A Survey of Findings from
Micro-Data,” chapter 1 in Optimal Pricing, Inflation and Aggregate Dynamics, E. Sheshinki and Y. Weiss, eds., MIT Press, 1993.
[48] Yun, Tack (1994), “N ominal Price Rigidity, Money Supply Endogeneity, and
Business Cycles,” Journal of Monetary Economics, vol. 37, (1996), 345-370.

Full text of Working Papers (Federal Reserve Bank of Richmond) : State-Dependent Pricing and the Dynamics of Business Cycles, Working Paper 97-02

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