Michael Woodford

Princeton University

Revised December 2002

Preliminary and Incomplete

c Michael Woodford 2002

Contents

3 Optimizing Models with Nominal Rigidities 1

1 A Basic Sticky-Price Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1 Price-Setting and Endogenous Output . . . . . . . . . . . . . . . . . 6

1.2 Consequences of Prices Fixed in Advance . . . . . . . . . . . . . . . . 17

1.3 A “New Classical” Phillips Curve . . . . . . . . . . . . . . . . . . . . 20

1.4 Sources of Strategic Complementarity . . . . . . . . . . . . . . . . . . 26

2 In¬‚ation Dynamics with Staggered Price-Setting . . . . . . . . . . . . . . . . 38

2.1 The Calvo Model of Price-Setting . . . . . . . . . . . . . . . . . . . . 42

2.2 A “New Keynesian” Phillips Curve . . . . . . . . . . . . . . . . . . . 51

2.3 Persistent Real E¬ects of Nominal Disturbances . . . . . . . . . . . . 52

2.4 Consequences of Persistence in the Growth of Nominal Spending . . . 62

2.5 Consequences of Sectoral Asymmetries . . . . . . . . . . . . . . . . . 65

3 Delayed E¬ects of Nominal Disturbances on In¬‚ation . . . . . . . . . . . . . 71

3.1 Staggered Pricing with Delayed Price Changes . . . . . . . . . . . . . 74

3.2 Consequences of Indexation to Past In¬‚ation . . . . . . . . . . . . . . 80

4 Consequences of Nominal Wage Stickiness . . . . . . . . . . . . . . . . . . . 87

4.1 A Model of Staggered Wage-Setting . . . . . . . . . . . . . . . . . . . 89

4.2 Sticky Wages and the Real E¬ects of Nominal Disturbances . . . . . 94

Chapter 3

Optimizing Models with Nominal

Rigidities

We turn now to the analysis of models in which monetary policy a¬ects the level of real

economic activity, and not just the level of money prices of goods and services. This requires

us to go beyond the analysis of endowment economies, as in the previous chapter, and allow

instead for endogenous supply decisions. But as is well-known, even when we allow for en-

dogenous supply, monetary policy can have only small e¬ects on the equilibrium allocation

of resources in an environment with perfect wage and price ¬‚exibility (and identical informa-

tion on the part of all decision-makers). Hence we also extend our analytical framework to

allow for delays in the adjustment of prices and/or wages to changing aggregate conditions.

In this way, we allow for non-trivial real e¬ects of monetary policy.

This extension of our framework is also important for a more realistic discussion of

central-bank interest-rate policy. In the basic model of chapter 2, the equilibrium real rate

of return is completely independent of monetary policy. This means that a central bank can

have no e¬ect on nominal interest rates except insofar as it can shift in¬‚ation expectations.

In our analysis above, we have assumed that it is able to do so, as long as the change in

expectations that is called for involves no violation of the postulate of rational expectations

on the part of the private sector; but our analysis may give the appearance of assuming precise

central-bank control of something upon which banks have little direct in¬‚uence in reality.

The introduction of price stickiness will make our assumption that the central bank can set

1

2 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

a short-term nominal interest rate less paradoxical. If private-sector in¬‚ation expectations

do not change when the central bank seeks to adjust the nominal interest rate, this does

not prevent the central bank from achieving its operating target; it simply means that the

private sector will perceive the real interest rate to have changed, which will a¬ect desired

expenditure and hence the degree of utilization of existing productive capacity. (We re-

examine the topic of in¬‚ation determination under an interest-rate rule in the context of

models with nominal rigidities in chapter 4.)

We shall give primary emphasis to models with sticky prices, though we also discuss the

consequences of wage stickiness as well. In this we follow most of the literature of the past

twenty years, but the choice perhaps deserves brief comment. One reason for emphasizing

price stickiness, at least for pedagogical purposes, is simply that models with only sticky

prices provide a simpler framework for the consideration of basic issues regarding the nature

of in¬‚ation determination. If we are to talk about the determinants of in¬‚ation (which in

the context of current policy debate, almost invariably means the rate of increase in goods

prices rather than wages), we must model the goods market; and if we are to consider such

central issues as the relation between interest rates and expenditure decisions, we need to

adjust our nominal interest rate for the expected rate of price in¬‚ation as well. On the other

hand, it is not equally essential to explicitly consider wage determination. It is possible to

model endogenous supply decisions without any reference to a labor market at all, as in

the familiar “yeoman farmer” models; and so the simplest models developed here will be of

this form, or will be equivalent in their implications to such a model, even if a (completely

frictionless) labor market is represented.

It is also often argued that there is more reason to believe that the stickiness of prices

matters for the allocation of resources. A well-known criticism of the models of nominal

wage rigidity popular in the 1970s was that the mere observation of infrequent changes in

individual nominal wages did not in itself prove the existence of a nominal rigidity with

any allocative consequences (Barro, 1977; Hall, 1980). Because employment is an ongoing

relationship rather than a spot-market transaction, the e¬ective cost to a ¬rm of increased

3

employment of labor inputs at a point in time need not equal the wage paid per hour of work

at that time; and under an e¬cient implicit contract, wages might well be smoother than the

e¬ective cost of labor, owing to a preference of workers for a smoother income stream. On

the other hand, it is less plausible that the observed rigidity of consumer goods prices should

not have allocative consequences, given the absence of a similar kind of ongoing relationship

between the suppliers of consumer goods and their customers (Rotemberg, 1987). However,

no convincing evidence has ever been o¬ered that the stickiness of nominal wages does not

result in stickiness of the e¬ective nominal cost of labor inputs; and evidence described below

” indicating that the evolution of U.S. in¬‚ation can be well explained by the evolution of

unit labor costs ” suggests that a model of supply costs that treats the reported wage as the

true marginal cost of additional hours of labor is not too inaccurate. Thus a more empirically

realistic model is likely to involve both wage and price stickiness. We treat models of that

kind in section xx below.

We shall give particular attention to the derivation of models with sticky prices and/or

wages in which prices and wages are nonetheless set optimally on the occasions when they

are adjusted. Allowing for optimal price- and wage-setting is important for several reasons.

One is that it allows us to highlight the importance of expectations for wage and price

dynamics. As we shall see in later chapters, forward-looking private sector behavior, in this

and other respects, has profound consequences for the optimal conduct of monetary policy.

It would thus be a serious mistake to simply assume mechanical wage- and price-adjustment

equations (perhaps drawn from the econometric literature), and treat these as structural for

purposes of an analysis of optimal monetary policy, as Lucas (1976) so forcefully argued.

Another reason for modeling optimal price- and wage-setting is that we are interested

in the welfare evaluation of alternative monetary policies. An especially appealing basis for

such evaluation is to ask how alternative possible equilibria compare from the point of view

of the private-sector objectives that underlie the behavior assumed in one™s model of the

e¬ects of alternative policies; but this is only possible insofar as the structural equations of

our model of the monetary transmission mechanism are derived from optimizing foundations.

4 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

As we shall see in chapter 6, alternative assumptions about the nature of price and wage

stickiness imply that alternative stabilization objectives for monetary policy are appropriate.

While we shall give detailed attention to the consequences of assumed delays in the ad-

justment of prices and/or wages, we shall not attempt here to say anything new about the

underlying reasons for these delays. Our assumptions about the frequency with which ¬rms

adjust their prices, or the time lag that may be involved between the decision about a price

and the time that the new price takes e¬ect, are treated as structural features of the environ-

ment in which ¬rms sell their products, with the same status as their production functions.

How reasonable this is depends on the question that one intends to ask of one™s model.

The “endogenous growth” has emphasized that, when thinking about the determinants of

economies™ long-run growth prospects, it is probably a mistake to ignore the endogeneity of

production functions ” for changes in economic conditions can change the incentives that

private parties have to devote resources to research and development, to introduce new prod-

ucts, and so on. On the other hand, for purposes of a comparison of alternative monetary

policy rules, it may not be a bad approximation to assume given production functions; the

real e¬ects of alternative monetary policies are relatively short-lived, and over this short

horizon production possibilities are unlikely to be much a¬ected by the temporary alteration

of incentives to innovate that may have occurred.

Similarly, if we wished to analyze the consequences of highly in¬‚ationary policies, we

would surely not want to treat as given the frequency of price adjustment, the degree of

indexation of wage contracts, or even the currency in terms of which prices are quoted;

we know that practices adjust in all of these respects (and for reasons that are easy to

understand) in economies that su¬er from sustained high in¬‚ation. But our interest in

the present study is in the identi¬cation of better monetary policies within the class of

policies under which in¬‚ation is never very great; in fact, we shall make extensive use of

approximations that are expected to be accurate only for the analysis of policies of that

kind. (It will perhaps not be giving away too much to divulge at this point that, according

to our analysis, optimal policy will indeed involve low and stable in¬‚ation!) For this purpose,

5

treating the delays involved in price and wage adjustment as structural may not be a bad

approximation. The sizes of wage and price increases do clearly vary from year to year, in

response to changes in perceived market conditions; practices with regard to the times at

which prices or wages are reconsidered, or the units in which they are announced, occur

much less often and only in response to more drastic changes in the economic environment.

Finally, we freely grant that the simple models presented here should be viewed only as

crude approximations to the actual monetary transmission mechanism. A realistic quan-

titative model would need to incorporate a large number of complications from which we

abstract here, in order to clarify basic concepts. One may wish to add endogenization of the

timing of price and wage adjustments to the list of re¬nements that one would like to incor-

porate into an eventual, truly accurate model. It is not clear, however, that this particular

re¬nement should be placed too high on the list of re¬nements when ranked in terms of their

likely quantitative importance for the analysis of monetary policy.

Nor is it even clear that any of the models with endogenous timing of price changes

that currently exist should be regarded as more realistic than the models presented below,

quite apart from the question of complexity. Some feel that models of “state-dependent

pricing”, such as those of Caplin and Leahy (1991) or Dotsey et al. (1999), have “better

microfoundations” than do the sorts of models presented here, that assume a given timing

for price and wage changes. But this is not obvious. These models assume that ¬rms are

constantly re-evaluating the price that they would adopt if they were to change their price,

and the expected bene¬ts from the change, and then weighing these bene¬ts against the

current “menu cost” of a price change to decide whether to actually change their price or

not. Yet in reality, the main bene¬t of infrequent price changes is not lower “menu costs”, but

reduction of the costs associated with information collection and decision-making (Zbaracki

et al., 1999). Obtaining this bene¬t necessarily means that the timing of the occasions upon

which prices are reconsidered will be largely independent of current market conditions; for

example, ¬rms often reconsider pricing policy at a particular time of year.

We begin in section 1 with a basic model of monopolistic competition, in which the prices

6 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

of some goods must be determined a period in advance. This very simple example of price

stickiness is useful for introducing a number of basic concepts. It also provides optimizing

foundations for a familiar aggregate-supply speci¬cation, the “New Classical Phillips curve”

used in many well-known analyses of optimal monetary policy, such as those of Kydland and

Prescott (1977) and Barro and Gordon (1983). While the literature using this speci¬cation

has produced a number of insights of more general importance, this relation is quite inade-

quate as a realistic account of the co-movement of real and nominal variables; it allows, for

example, no persistent e¬ects of monetary policy on real activity, and no e¬ects of antici-

pated policy. These strong conclusions are not general consequences of optimal price-setting,

as we show in section 2 through the analysis of a slightly more complex speci¬cation, the

Calvo (1983) model of staggered price-setting. While still very simple, this model implies

an aggregate-supply relation, sometimes called the “New Keynesian Phillips curve”, that

has proven capable of explaining at least some of the more gross features of in¬‚ation dy-

namics in the U.S. and elsewhere. Section 3 discusses still more complex speci¬cations with

increased empirical realism, that introduce delays in the e¬ects of monetary policy changes

on in¬‚ation. Finally, section 4 discusses models in which nominal wages are sticky as well

as prices.

1 A Basic Sticky-Price Model

We begin by displaying the structure of a very basic model, in which monetary policy has

real e¬ects as a result of some goods prices being ¬xed in advance. A number of issues

that are easy to analyze in this simple context will turn out also to be relevant to the more

realistic models to be developed in later sections.

1.1 Price-Setting and Endogenous Output

In order to be able to model price-setting, we must ¬rst extend the representative-household

model introduced in the previous chapter in certain respects that are quite distinct from the

issue of whether prices are assumed to be sticky. In particular, we must allow for endogenous

1. A BASIC STICKY-PRICE MODEL 7

goods supply, rather than simply assuming a given endowment of goods. This requires that

we introduce a production technology and at least one variable factor of production (which

is labor, in this basic model). We shall be concerned to understand the determinants of

the costs of supplying goods, as supply costs are a prime determinant of optimal pricing.

We shall furthermore introduce di¬erentiated goods, and monopolistic competition among

the suppliers of these goods, as in the “New Keynesian” literature originated by Rotemberg

(1982), Mankiw (1985), Svensson (1986) and Blanchard and Kiyotaki (1987), rather than

assuming a single good in competitive supply. This last device, which is now quite com-

monplace, allows individual suppliers a degree of market power, and hence a decision about

how to set their prices. It also implies that a supplier that fails to immediately adjust its

price in response to a change in demand conditions does not su¬er an unboundedly large

(percentage) change in its sales, so that it becomes more plausible that prices should not be

constantly adjusted.1

We thus now assume that the representative household seeks to maximize a discounted

sum of utilities of the form

∞ 1

t

E0 β u(Ct , Mt /Pt ; ξt ) ’ v(ht (i); ξt )di . (1.1)

0

t=0

Here Ct is now an index of the household™s consumption of each of the individual goods

that are supplied, and Pt is a corresponding index of the prices of these goods, while ht (i)

is the quantity of labor of type i supplied. We assume that each of the di¬erentiated goods

(indexed by i over the unit interval) uses a specialized labor input in its production (and

in this chapter, this will be the only variable input); labor of type i is used to produce

di¬erentiated good i.

The introduction of di¬erentiated labor inputs is not necessary in order to allow us to

1

The size of the “menu costs” required to rationalize the failure of suppliers to adjust their prices imme-

diately has been the subject of an extensive literature. We shall not pursue this issue here, though we note

that the same sorts of “strategic complementarities” in price-setting that increase the degree of stickiness of

the general price level (as discussed below) when price adjustment is asynchronous also tend to reduce the

size of the costs of price changes to individual suppliers that are required to rationalize a failure to adjust

prices in response to an aggregate demand disturbance. On this latter point, see in particular Ball and

Romer (19xx).

8 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

analyze monopolistically competitive goods supply. But as we shall see, it is convenient to

do so. One reason is that we are able, in this case, to derive a model with factor markets

that is equivalent to the frequently-used “yeoman farmer” model, in which households are

assumed to directly supply goods. A more important reason is that it turns out that the

“strategic complementarity” between di¬erent suppliers™ pricing decisions is greater when we

assume that they do not hire labor from a single homogeneous (competitive) labor market.

Because we regard the conclusion obtained in the case of di¬erentiated labor inputs as the

more realistic one, we choose this speci¬cation as our baseline model. (The case of a single

homogeneous labor market is discussed in section 1.4 below.)

The term v(ht (i); ξt ) represents the disutility of supplying labor of type i; we assume that

for each possible value of ξ, v(·; ξ) is an increasing, convex function.2 We have written (1.1)

as if the representative household simultaneously supplies all of the types of labor. However,

we might equally well assume that each household specializes in the supply of only one type

of labor, but that there are an equal number of households supplying each type. In this case,

a household that supplies labor of type i seeks to maximize

∞

β t [u(Ct , Mt /Pt ; ξt ) ’ v(ht (i); ξt )] .

E0

t=0

When not all goods prices are set at the same time, households™ wage incomes will be

di¬erent, depending upon the type of labor they supply. But we may assume that there

exist competitive ¬nancial markets in which these risks are e¬ciently shared.

In this case, and if all households start with initial ¬nancial assets that give them the same

initial intertemporal budget constraints,3 then since households value consumption streams

(and money balances) identically and face the same prices, all households will choose identical

consumption and real balances in all states. (Note that while we have allowed for preference

2

Once again, ξt is a vector, so that the use of the same notation for exogenous disturbances to the functions

u and v involves no assumption about statistical dependence between the shifts in these two functions.

3

This means that if any households face di¬erent present values of their expected wage incomes as of date

zero, they hold initial ¬nancial claims W0 that di¬er in exactly the way necessary to o¬set the di¬erence in

their expected wages. Note that if this condition has ever held, then optimization in the presence of complete

¬nancial markets implies that it will hold forever after, regardless of which values may be realized for the

exogenous disturbances.

1. A BASIC STICKY-PRICE MODEL 9

shocks ξt in (1.1), we assume that these are the same for all households “ we here contemplate

only aggregate shocks.) They will also choose portfolios of ¬nancial assets that insure that

they continue to have identical intertemporal budget constraints at all subsequent dates.4

The common intertemporal budget constraint in each state will in turn be exactly that of a

household that supplies all of the types of labor, and pools the wage income received.

Because each household chooses exactly the same state-contingent consumption plan,

the ¬rst-order conditions for optimal supply of each type of labor are exactly the same as

when a single household type supplies all types of labor so as to maximize (1.1). Thus the

conditions that determine equilibrium prices and quantities are the same in the two models.

Furthermore, if our welfare criterion in the specialized-labor model is the average level of

utility of all households, the level of social welfare associated with a given equilibrium will be

measured by the value of (1.1). Thus it makes no di¬erence to our conclusions which version

of the model we assume. The ¬ction that each household directly supplies all types of labor,

and so receives its pro rata share of the aggregate wage bill of the entire economy, simpli¬es

the exposition in that it allows us to dispense with explicit discussion of the risk-sharing

arrangements just referred to.

Following Dixit and Stiglitz (1977), we shall assume that the index Ct is a constant-

elasticity-of-substitution aggregator

θ

1 θ’1

θ’1

Ct ≡ ct (i) di (1.2)

θ

0

with θ > 1, and that Pt is the corresponding price index

1

1 1’θ

pt (i)1’θ di

Pt ≡ . (1.3)

0

Note that (1.3) de¬nes the minimum cost of a unit of the aggregate de¬ned by (1.2), given

the individual goods prices {pt (i)}; since a household cares only about the number of units

4

Here our argument relies upon the assumption in (1.1) that the disutility of labor supply is additively

separable from the other terms. This implies that even if households expect to work di¬erent amounts in

particular states, they equalize their marginal utility of income in each state by holding assets that allow

them to a¬ord to consume exactly the same amount and hold exactly the same money balances as one

another. The more complicated case of non-separable preferences is treated in section xx below.

10 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

of this aggregate that it can purchase, de¬‚ation by Pt is an appropriate measure of the

purchasing power of nominal money balances Mt .

The household™s budget constraints are then as in chapter 2, except that the term pt ct

1

for nominal consumption expenditure must now be replaced by pt (i)ct (i)di, and the term

0

pt yt for income from the sale of goods must now be replaced by

1 1

wt (i)ht (i)di + Πt (i)di, (1.4)

0 0

where wt (i) is the nominal wage of labor of type i in period t, and Πt (i) represents the

nominal pro¬ts from sales of good i. In writing this last expression, we assume that each

household owns an equal share of all of the ¬rms that produce the various goods. Again,

given our assumption of complete ¬nancial markets, this assumption of distributed ownership

is irrelevant. We could also introduce trading in the shares of the ¬rms, without any change

in the conditions for a rational expectations equilibrium, except that then equilibrium share

prices would also be determined. As these extensions of the framework have no consequences

for the equilibrium evolution of goods prices or the quantities of goods supplied, we omit

further discussion of them.

As in chapter 2, each household then faces a single intertemporal budget constraint. Op-

timal (price-taking) household behavior is then described by the conjunction of three sets

of requirements. First, the household™s consumption spending must be optimally allocated

across di¬erentiated goods at each point in time, taking as given the overall level of expen-

diture. Thus the relative expenditures on di¬erent goods must be such as to maximize the

index (1.2) given the level of total expenditure. As in other applications of the Dixit-Stiglitz

model, this requires that purchases of each good i satisfy

’θ

pt (i)

ct (i) = Ct . (1.5)

Pt

This rule for distributing expenditure is easily seen to imply that total expenditure will

equal Pt Ct . Using this substitution, we can write both the household™s utility and its budget

constraints solely in terms of Pt and Ct , without any reference to quantities or prices of the

individual goods that are purchased.

1. A BASIC STICKY-PRICE MODEL 11

Second, taking as given the optimal allocation of consumption expenditure at each date

(just described) and the amount of labor supplied (considered below), the household must

choose optimal levels of total consumption expenditure at each date, an optimal level of

money balances to hold at each date, an optimal amount of ¬nancial wealth to accumulate,

and an optimal portfolio allocation across the various types of state-contingent bonds that

are available. Necessary and su¬cient conditions for optimization in this respect are given by

exactly the same conditions as in chapter 2 “ namely, conditions (1.2), (1.12), (1.13), (1.15)

and (1.16) of that chapter must again hold at all times, where however Pt now refers to the

price index (1.3), ct is replaced by the index Ct de¬ned in (1.2), and yt is replaced by Yt ,

a similarly de¬ned aggregate of the quantities supplied of the various di¬erentiated goods.5

This is because both preferences over alternative streams of the consumption aggregate and

budget constraints written in terms of a¬ordable paths for the consumption aggregate are

exactly the same as in chapter 2.6

And ¬nally, the household must choose an optimal quantity of each kind of labor to

supply, given the wages that it faces and the value to it of additional income (determined by

the consumption-allocation problem just described). The ¬rst-order condition for optimal

supply of labor of type i at date t is given by

vh (ht (i); ξt ) wt (i)

= . (1.6)

uc (Ct , mt ; ξt ) Pt

These conditions, together with those listed earlier, comprise a complete set of necessary and

su¬cient conditions for household optimization.

We turn next to the speci¬cation of production possibilities. We assume that each good

i has a production function

yt (i) = At f (ht (i)), (1.7)

5

The relative quantities supplied of the various goods must be distributed in the same way as the relative

demands implied by (1.5). (When we allow below for government purchases, we assume that the government

seeks to maximize a similar aggregate of its purchases, and so distributes its purchases in the same manner.)

It then follows that total non-¬nancial income (1.4), which must equal total sales revenues of all of the ¬rms,

can be written as Pt Yt .

6

Here the additive separability of the disutility-of-labor terms in (1.1) is again crucial, allowing us to

obtain preferences over paths for the consumption aggregate and real balances that are the same as in our

earlier model with no labor supply decision.

12 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

where At > 0 is a time-varying exogenous technology factor, and f is an increasing, concave

function. Here labor is represented as the only factor of production (with one speci¬c type of

labor being used in the production of each good). We may think of capital as being allocated

to each ¬rm in a ¬xed amount, with capital goods never depreciating, never being produced,

and (because they are speci¬c to the ¬rm that uses them) never being reallocated among

¬rms; in this case, the additional argument of the production function may be suppressed.

(An extension of the model to allow for endogenous capital accumulation is presented in the

next chapter.)

It follows that the variable cost of supplying a quantity yt (i) of good i is given by

wt (i)f ’1 (yt (i)/At ).

Di¬erentiating this, we ¬nd that the (nominal) marginal cost of supplying good i is equal

to7

wt (i)

St (i) = Ψ(yt (i)/At ),

At

where

1

Ψ(y) ≡ (1.8)

f (f ’1 (y))

is an increasing positive function. Substituting the labor supply function (1.6) for the wage,

we obtain a relation between the real marginal supply cost and the quantity supplied:

˜

st (i) ≡ St (i)/Pt = s(yt (i), Yt ; ξt ),

where the real marginal cost function is de¬ned by

’1

˜ = vh (f (y/A); ξ) Ψ(y/A).

s(y, Y ; ξ) (1.9)

uc (Y ; ξ)A

7

Here we assume that the producer is a wage-taker, even though we have supposed each good uses a

di¬erent type of labor with its own market. But our assumption of di¬erentiated labor inputs need not

imply that each producer is a monopsonist in its labor market. The only assumption that is important for

our results below is that producers that change their prices at di¬erent times also hire labor inputs from

distinct markets. We might, for example, assume a double continuum of di¬erentiated goods, indexed by

(i, j), with an elasticity of substitution of θ between any two goods, as above. We might then assume that

all goods with the same index i change their prices at the same time (and so always charge the same price),

and are also all produced using the same type of labor (type i labor). The degree of market power of each

producer in its product market would then be as assumed here, but the fact that a continuum of producers

all bid for type i labor would eliminate any market power in their labor market.

1. A BASIC STICKY-PRICE MODEL 13

˜

In this last expression, ξt represents the complete vector of exogenous disturbances, in which

the preference shocks ξt have been augmented by the technology factor At , we have substi-

tuted into the labor supply function the sectoral labor requirement as a function of sectoral

output, and we have used the fact that in equilibrium, the index of aggregate consumption

Ct must at all times equal the index of output Yt .8

Note also that we have suppressed real balances as an argument of uc (and hence as an

argument of the real marginal cost function) in the denominator. Abstracting from such

“real balance e¬ects” can be justi¬ed along any of several grounds discussed in chapter 2.

It is simplest to suppose that the economy considered in this chapter is a “cashless” one, in

which monetary policy is implemented in the way considered in section 1 of chapter 2.9 As in

chapter 2, we shall here assume a cashless economy as our baseline model; the consequences

of real balance e¬ects are considered in section 3.2 of chapter 4.

This model of production costs might alternatively be derived from a “yeoman farmer”

model, in which households directly supply goods, seeking to maximize

∞ 1

˜

t

E0 β u(Ct , Mt /Pt ; ξt ) ’ v (yt (i); ξt )di

˜ . (1.10)

0

t=0

If we convert the marginal disutility of supply of good i into units of an equivalent quantity

of the consumption aggregate, we obtain a “real marginal cost” of good i equal to

˜

vy (yt (i); ξt )

˜

.

uc (Yt ; ξt )

8

When we allow for exogenous variation in government purchases, we can still apply this equation, if

˜

we understand u(Y ; ξ) to mean the function u(Yt ; ξt ) introduced in section xx of chapter 2, measuring

˜

household utility ¬‚ow as a function of aggregate demand rather than consumption expenditure. Under this

interpretation, the level of government purchases is just one element of the vector of exogenous disturbances

˜

ξ that shifts this relation. When we allow for endogenous variations in investment spending, matters are

more complex; in such a case, it is important to remember that it is really uc (C; ξ) rather than uc (Y ; ξ) that

belongs in the denominator of (1.9).

9

Alternatively, our results apply to a Sidrauski-Brock model in which utility is separable, as discussed

in section xx of chapter 2; to a cash-in-advance model of the special type discussed in section xx of the

appendix to chapter 2; or to a “cashless limiting economy” of the sort discussed in section xx of chapter 2.

They also apply to a much broader class of models with transactions frictions, in the case that monetary

policy is implemented through a procedure under which the interest-rate di¬erential ∆t is held constant,

ˆ

under a suitable reinterpretation of the parameter σ and the disturbance gt , so that ’σ ’1 (Yt ’ gt ) is the

deviation of the log marginal utility of real income from its steady-state level, in the case of a constant

interest-rate di¬erential (rather than a constant level of real balances). For discussion of this last case, see

section 3.2 of chapter 4, especially footnote xx.

14 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

This is in fact identical to (1.9) if the disutility of output supply is given by

˜˜

v (y; ξ) ≡ v(f ’1 (y/A); ξ).

This concept of “real marginal cost” plays exactly the same role in optimal pricing in the

yeoman farmer model as does the more conventional concept in the case of supply by ¬rms

that purchase inputs, and the results that we obtain below are identical to those that one

would obtain from a yeoman farmer model. As noted above, this is one reason for interest

in the model with di¬erentiated labor inputs assumed here. However, explicitly modeling

the labor market has the advantage of allowing us to derive additional implications of the

model. It will also make the extension, below, to a model with sticky wages as well as prices

more straightforward.

With our theory of marginal supply costs in place, we now turn to the question of optimal

pricing. We shall ¬rst examine the case of perfectly ¬‚exible prices; that is, we shall assume

that the supplier of each good chooses a price for it each period, not constrained in any way

by the price that has been charged for the good in the past, and with full information about

current demand and cost conditions. As usual in a model of monopolistic competition, we

assume that each supplier understands that his sales depend upon the price charged for his

good, according to the demand function

’θ

pt (i)

yt (i) = Yt . (1.11)

Pt

(The form of the demand curve assumed here follows from (1.5); when all purchases are for

private consumption, the index of aggregate demand Yt corresponds simply to the represen-

tative household™s choice of the index Ct .) Because good i accounts for only an in¬nitesimal

contribution to households™ budgets and their utility from consumption, the supplier of an

individual good does not believe that his pricing decision can a¬ect the evolution of either

the index of aggregate demand Yt or the price index Pt ; thus pt (i) is chosen taking the latter

two quantities as given. Optimization by the supplier of good i then involves setting a price

pt (i) = µSt (i), where µ ≡ θ/(θ ’ 1) > 1 is the seller™s desired markup, determined by the

usual Lerner formula.

1. A BASIC STICKY-PRICE MODEL 15

It follows that each supplier will wish to charge a relative price satisfying

pt (i) ˜

= µs(yt (i), Yt ; ξt ). (1.12)

Pt

It then follows from (1.11) that the relative supply of good i must satisfy

’1/θ

yt (i) ˜

= µs(yt (i), Yt ; ξt ).

Yt

Because s is increasing in its ¬rst argument, this equation must have a unique solution for

yt (i) given Yt . It follows that in equilibrium, the same quantity must be supplied of each

good, and that common quantity must equal Yt . Equilibrium output must then be given by

˜

Yt = Y n (ξt ), where the latter function indicates the solution to the equation

˜

s(Ytn , Ytn ; ξt ) = µ’1 . (1.13)

Because s is also increasing in its second argument, this equation as well must have a unique

˜

solution for each speci¬cation of the exogenous shocks ξt .

We thus ¬nd that in the case of fully ¬‚exible prices, equilibrium output is completely

independent of monetary policy. Given this solution for aggregate output as a function of

the exogenous shocks, our model of price-level determination then reduces to exactly the

model analyzed in chapter 2 (where an exogenous supply of goods was simply assumed).

Thus neither our introduction of endogenous supply nor our assumption of monopolistic

competition has any necessary consequences for the e¬ects of monetary policy. But they

now make it possible for us to consider other assumptions about pricing behavior, and we

shall see that in the case of sticky prices our conclusions are di¬erent.

The solution to equation (1.13) “ which we call the natural rate of output following

Friedman (1968) “ continues to be a useful construct even in the case of sticky prices (though

it no longer need equal the equilibrium10 level of output at all times). This is because a log-

linear approximation to the real marginal cost function (1.9) is given by

ˆ ˆ

st (i) = ω yt (i) + σ ’1 Yt ’ (ω + σ ’1 )Ytn ,

ˆ ˆ (1.14)

10

Some authors would say that it is the “equilibrium” level of output, but that the level of output actually

observed as a result of sticky prices is instead a “disequilibrium” level of output. We shall avoid this

terminology in this study; for us, “equilibrium” always refers to the prediction of our model, whether it

involves fully ¬‚exible prices or not.

16 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

where ω > 0 represents the elasticity of s with respect to its ¬rst argument, and σ > 0 is

the intertemporal elasticity of substitution of private expenditure, as in chapter 2. (Here

˜

we log-linearize around the steady-state equilibrium in the case of ¬‚exible prices and ξt = 0

¯

at all times.11 Letting Y be the constant level of output in this steady state, we de¬ne

ˆ ¯ˆ ¯

Yt as in chapter 2, and correspondingly de¬ne yt (i) ≡ log(yt (i)/Y ), Ytn ≡ log(Ytn /Y ), and

ˆ

st (i) ≡ log(µst (i)).) Thus the natural rate of output provides a useful summary of the way

ˆ

in which disturbances shift the real marginal cost function, whether prices are constantly

adjusted or not.

For later purposes it is useful to note that in (1.14), the elasticity ω can be decomposed

as

ω = ωw + ωp , (1.15)

where ωw > 0 is the elasticity of the marginal disutility of work with respect to output

increases, and ωp > 0 is the elasticity of the function Ψ de¬ned in (1.8). Thus ωw indicates

the elasticity of real wage demands with respect to the level of output, holding ¬xed the

marginal utility of income, while ωp indicates the negative of the elasticity of the marginal

product of labor with respect to the level of output.12

Of course, our result here that monetary policy is completely irrelevant to the determi-

nation of real activity is rather special. If, for example, we allow for real balance e¬ects,

we shall ¬nd that monetary policy can a¬ect equilibrium output even under ¬‚exible prices,

owing to the e¬ects of expected in¬‚ation upon equilibrium real balances. If we furthermore

allow for endogenous capital accumulation, we shall ¬nd that the natural rate of output de-

pends upon the capital stock, and insofar as real balance e¬ects are able to a¬ect equilibrium

capital accumulation, they may have a further e¬ect upon equilibrium output under ¬‚exible

prices through this channel as well. However, these e¬ects are not plausibly very large in

quantitative terms, as studies such as that of Cooley and Hansen (1989) have shown. Thus

˜

11

The element of ξt that measures aggregate technology is here taken to be at ≡ log At rather than At

itself. Thus the steady-state value of the technology factor At is normalized as one.

12

In a model with wage and price-setting, ωw indicates the degree to which higher economic activity

increases workers™ desired wages given prices, while ωp indicates the degree to which higher economic activity

increases producers™ desired prices given wages.

1. A BASIC STICKY-PRICE MODEL 17

the conclusion from our simple model remains essentially correct.

1.2 Consequences of Prices Fixed in Advance

We now contrast these results to those that we obtain under a simple form of price stickiness.

Let us suppose that all prices pt (i) must be ¬xed a period in advance; that is, when pt (i) is

chosen, the exogenous disturbances (including possible random variation in monetary policy)

realized in periods t ’ 1 or earlier are known, but not any of the disturbances that are to

be realized only in period t. (Whether the stickiness of prices results because the price that

applies in period t has to be announced at an earlier time, or simply because price-setters

make their decision on the basis of old information, does not matter for our conclusions.) We

shall suppose that the supplier of good i is committed to supply whatever quantity buyers

may wish to purchase at the predetermined price pt (i), and hence to purchase whatever

quantity of inputs may turn out to be necessary to ¬ll orders.

When the price pt (i) is chosen, in period t ’ 1, the consequences for sales and pro¬ts in

period t are not yet known with certainty. Hence we assume that the ¬rm seeks to maximize

the present value of period t pro¬ts, given by

Et’1 [Qt’1,t Πt (i)], (1.16)

where Qt’1,t is the stochastic discount factor introduced in chapter 2, and pro¬ts from the

sale of good i are given by

Πt (i) = pt (i)yt (i) ’ wt (i)ht (i).

Using (1.7) and (1.11), we can write this objective as a function of the choice variable pt (i),

obtaining

Et’1 Qt’1,t [Yt Ptθ pt (i)1’θ ’ wt (i)f ’1 (Yt Ptθ pt (i)’θ /At )] . (1.17)

The supplier of good i then chooses pt (i) on the basis of information available at date

t ’ 1 so as to maximize this expression, given the expected state-contingent values of the

random variables Qt’1,t , Yt , Pt , wt (i) and At .

18 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

The expression inside the square brackets in (1.17) is easily seen to be a concave function

of pt (i)’θ (which we might alternatively choose as the choice variable), so that the entire

expression is similarly concave. It follows that expected pro¬ts have a unique maximum

at the price that satis¬es the ¬rst-order condition obtained by di¬erentiating (1.17) with

respect to pt (i). This may be written

Et’1 Qt’1,t Yt Ptθ [pt (i) ’ µSt (i)] = 0. (1.18)

This way of writing the ¬rst-order condition indicates that the price is set to equal µ times

a weighted average of the values of the nominal marginal cost of supplying good i that are

expected to occur in di¬erent possible states at date t.

Substituting the demand function (1.11) into the real marginal cost function (1.9), one

sees that the marginal cost of supplying good i is a decreasing function of the price pt (i)

(given the values of the variables that the producer cannot a¬ect) in each state. Hence

(1.18) has a unique solution for the optimal price pt (i), that is the same for each good i. In

equilibrium, each producer sets an identical price, and this common price will be Pt . Using

the result that pt (i) = Pt , and substituting the solution for the stochastic discount factor

from equation (xx) of chapter 2, we ¬nd that (1.18) requires that

˜

Et’1 uc (Yt ; ξt )Yt [µ’1 ’ s(Yt , Yt ; ξt )] = 0. (1.19)

This is a restriction that must be satis¬ed by the joint distribution of Yt and the exogenous

˜

disturbances ξt , conditional upon information at date t ’ 1; note that it involves no nominal

variables. It is a weaker version of our result in the case of ¬‚exible prices, that Yt = Ytn at all

˜

times. Output equal to the natural rate is equivalent to requiring that s(Yt , Yt ; ξt ) = µ’1 at

all times; instead, (1.19) requires only that this hold “on average” (where the “average” in

question does not involve weights exactly equal to the probability of each state™s occurrence).

Together with the stipulation that Pt is predetermined, (1.19) represents an aggregate

supply relation for this model. We can examine its implications, without needing to specify

the rest of the model, if we assume that monetary policy is used to achieve an exogenous

1. A BASIC STICKY-PRICE MODEL 19

target path for nominal GDP, Yt = Pt Yt . This sort of aggregate demand speci¬cation is

very commonly assumed in the literature on sticky-price models, usually by stipulating

that nominal GDP is proportional to the money supply, and then that monetary policy is

speci¬ed by an exogenous target path for the money supply. It is not attractive, for our

purposes, to assume either a constant velocity of money or monetary targeting. But we may

nonetheless examine an equivalent aggregate demand speci¬cation, by assuming that policy

is speci¬ed in terms of a target path for nominal GDP, that is then achieved by adjusting

the interest-rate instrument as necessary. In addition to allowing comparisons with familiar

literature, this assumption allows us to examine the consequences of alternative aggregate

supply speci¬cations without needing to specify the way in which monetary policy a¬ects

aggregate spending. Accordingly, we specify aggregate demand in terms of an exogenous

process {Yt } throughout the present chapter. (The interest-rate adjustments required in

order to control aggregate spending are then taken up in chapter 4.)

Substituting Yt = Yt /Pt into (1.19), we observe that the equilibrium price level is given

by the solution to the equation

˜

Et’1 uc (Yt /Pt ; ξt )Yt [µ’1 ’ s(Yt /Pt , Yt /Pt ; ξt )] = 0. (1.20)

˜

This implies that Pt is a function solely of the joint distribution of {Yt , ξt }, conditional

upon information at date t ’ 1, and that this function is homogeneous of degree one in the

distribution of values anticipated for Yt . Given the value of Pt determined by this ex ante

distribution, the level of output Yt is then determined by the ex post realization of Yt . The

homogeneity property just referred to implies that Yt depends only upon the level of Yt

relative to the distribution of levels of nominal spending that were regarded as possible at

date t ’ 1.

This result can be stated more simply if we make use of a log-linear approximation to the

˜

solution (1.20). We log-linearize around the steady-state equilibrium in which ξt = 0 and

Yt /Yt’1 = 1 at all times, and obtain an expression that approximates the exact solution as

˜

long as ξt and Yt /Yt’1 are always su¬ciently close to these values. Making use of the log-

20 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

linear approximation to the real marginal cost function (1.14), we ¬nd that the equilibrium

price level is approximately given by

log Pt = Et’1 log Yt ’ Et’1 log Ytn ,

from which it follows that

log Yt = Et’1 log Ytn + [log Yt ’ Et’1 log Yt ].

We then observe that the component of output that can be forecasted a period in advance

is always equal to the forecast of the natural rate,

Et’1 log Yt = Et’1 log Ytn ,

and hence is independent of monetary policy. The unexpected component of output ¬‚uctu-

ations, by contrast, is equal to the unexpected component of nominal GDP (or of nominal

GDP growth):

log Yt ’ Et’1 log Yt = log Yt ’ Et’1 log Yt .

Thus monetary policy a¬ects real activity in this model only insofar as it causes unexpected

variation in nominal spending, and the resulting variations in output must themselves be

purely unexpected.

1.3 A “New Classical” Phillips Curve

The above model can be generalized by allowing some prices to be ¬‚exible, though others

are ¬xed in advance. This allows us to consider the robustness of our previous conclusions

to allowing some prices to be ¬‚exible, even in the very short run (as we do in fact observe).

It also allows us to derive a “Phillips curve” relation between price movements and output

movements, of a kind familiar from the “New Classical” literature of the 1970s.

Suppose now that a fraction 0 < ι < 1 of the goods prices are fully ¬‚exible “ which is to

say, set each period on the basis of full information about current demand and cost conditions

“ while the remaining 1 ’ ι are set a period in advance, as in the previous subsection. The

1. A BASIC STICKY-PRICE MODEL 21

supplier of each ¬‚exible-price good will then set its price each period according to (1.12),

while the supplier of each sticky-price good will set its price in advance according to (1.18).

The marginal costs of supplying di¬erent goods i will di¬er only insofar as the quantities

supplied di¬er, and these in turn will di¬er only insofar as the prices of the goods di¬er. It

follows that all ¬‚exible-price goods will have a common price p1t , and all sticky-price goods

will similarly have a common price p2t . We similarly let y1t denote the common equilibrium

output of all ¬‚exible-price goods, and y2t the output of sticky-price goods.

Taking a log-linear approximation to the two pricing equations, we obtain

ˆ ˆ

log p1t = log Pt + ω y1t + σ ’1 Yt ’ (ω + σ ’1 )Ytn ,

ˆ

ˆ ˆ

log p2t = Et’1 [log Pt + ω y2t + σ ’1 Yt ’ (ω + σ ’1 )Ytn ],

ˆ

¯

de¬ning yit ≡ log(yit /Y ) for i = 1, 2, and again using (1.14) to approximate the real marginal

ˆ

cost function. These approximations apply as long as the ¬‚uctuations in pit /Pt , Yt , and Ytn

¯ ¯

around the values (1, Y , and Y respectively) near which we log-linearize are small enough.

Note that up to this log-linear approximation, (log) predetermined prices are set at a value

equal to a constant markup over the conditional expectation, at the time that the price is

set, of (log) marginal cost. Substituting the demand function (1.11) to eliminate the yit

variables, we obtain more simply

ˆ ˆ

log p1t = log Pt + ζ(Yt ’ Ytn ), (1.21)

ˆ ˆ

log p2t = Et’1 log Pt + ζ(Yt ’ Ytn ) ,

where

ω + σ ’1

ζ≡ > 0. (1.22)

1 + ωθ

These relations imply that up to our log-linear approximation,

log p2t = Et’1 log p1t .

But a corresponding log-linear approximation to the aggregate price index (1.3) yields

log Pt = ι log p1t + (1 ’ ι) log p2t .

22 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

It follows that

ι

πt ’ Et’1 πt = log Pt ’ Et’1 log Pt = (log p1t ’ log Pt ).

1’ι

Then using (1.21), we obtain the “New Classical” aggregate supply relation

ˆ ˆ

πt = κ(Yt ’ Ytn ) + Et’1 πt , (1.23)

where

ι

κ≡ ζ.

1’ι

Note that this relation has the form of an “expectations-augmented Phillips curve” of the

kind hypothesized by Phelps (1967) and Friedman (1968), in which the speci¬c in¬‚ation

expectation that is relevant is the expectation at the time at which current predetermined

prices were ¬xed regarding in¬‚ation over the interval until the present. This particular form

of dependence of aggregate supply upon in¬‚ation expectations was stressed in the “New

Classical” literature of the 1970s (e.g., Sargent and Wallace, 1975).

This aggregate supply relation implies once again that

Et’1 log Yt = Et’1 log Ytn ,

so that the component of output that can be forecasted a period in advance is still indepen-

dent of monetary policy. However, unexpected variations in aggregate demand now give rise

to in¬‚ation variation as well, rather than a¬ecting only output. Again taking the stochastic

process for aggregate nominal spending Yt as given, (1.23) implies that aggregate output

will equal

log Yt = log Ytn + (1 + κ)’1 (log Yt ’ Et’1 log Y t ), (1.24)

so that the aggregate price level will equal

κ

log Pt = (Et’1 log Y t ’ log Ytn ) + (log Yt ’ Et’1 log Yt ).

1+κ

How does the degree of price ¬‚exibility a¬ect the impact of ¬‚uctuations in nominal

spending upon real activity? Not surprisingly, a larger number of ¬‚exible prices (larger ι)

1. A BASIC STICKY-PRICE MODEL 23

implies a higher value of κ (steeper short-run Phillips curve), and hence a smaller value of

the elasticity of output with respect to unexpected variations in nominal spending in (1.24).

In the limit as ι approaches one (all prices ¬‚exible), κ becomes unboundedly large, and

the output e¬ects of variations in nominal spending approach zero, as found earlier. More

interesting is the question whether intermediate values of ι result in e¬ects more like those

of the fully-¬‚exible limit or the fully-sticky limit. This turns out to depend upon the size of

ζ.

Figure 3.1 plots the elasticity 1/(1 + κ) occurring in (1.24) as a function of ι, for each

of several values of ζ. In each case, the function is monotonically decreasing, as one would

expect, and takes the same values in the two limiting cases of ι = 0 and ι = 1. However, when

ζ > 1, the function is convex (and more so the larger is ζ), while when ζ < 1, the function

is concave (and more so the smaller is ζ). Thus if ζ is large, that even some goods have

¬‚exible prices is enough to prevent variations in nominal spending from having much e¬ect

on aggregate output. In such a case, the ¬‚exible-price model would provide a reasonable

approximation to the evolution of aggregate prices and quantities. But if ζ is small, that

even some prices are sticky is enough to result in a substantial e¬ect of variations in nominal

spending upon output. Indeed, in such a case, the overall price index responds very little to

unexpected variations in nominal spending, and the simple model in which all prices were

assumed to be predetermined is a reasonably good guide to aggregate outcomes.

There is a simple intuition for the signi¬cance of the parameter ζ: it describes the degree

of strategic complementarity between the price-setting decisions of the suppliers of di¬erent

goods. Let us consider the following simple game among price-setters: aggregate nominal

spending Yt is given, and the suppliers of individual goods i are to simultaneously choose the

prices pt (i) for their goods. In what way does a given supplier™s optimal price depend upon

the level of the prices chosen by the other suppliers? We shall say that the pricing decisions

are strategic complements if an increase in the prices charged for other goods increases the

price that it is optimal to charge for one™s own good. Correspondingly, one may speak of

strategic substitutes if an increase in the other prices makes it optimal for one to reduce the

24 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

1

0.9

0.8

ζ = 0.1

0.7

ζ = 0.5

0.6

ζ=1

1/1+k

0.5

ζ=2

0.4

0.3

ζ = 10

0.2

0.1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ι

Figure 3.1: Real e¬ects of variation in nominal spending, for alternative degrees of strategic

complementarity in price-setting.

price of one™s own good.13

To determine the answer to this question, we must ¬rst consider what one may call the

notional short-run aggregate supply (SRAS) curve,

p— (i) ˜

t

= „¦(Yt ; ξt ),

Pt

which indicates how each supplier™s desired relative price varies with the level of aggregate

activity. Here the “desired” price p— (i) is the price that the supplier of good i would prefer if

t

the price of good i could be freely set on the basis of demand and cost conditions at date t,

independent of any prices charged or announced in the past, and with no consequences for

the prices that may be charged at any later dates as well. This is a purely notional supply

13

See Bulow, Geanakoplos and Klemperer (1985) for the general notion of strategic complementarity, and

Haltiwanger and Waldman (1989) for discussion of it in the context of adjustment to aggregate shocks more

generally.

1. A BASIC STICKY-PRICE MODEL 25

curve in that it need not indicate how the prices of any goods are actually set, in an economy

where prices are sticky; nonetheless, we shall see that the concept is a useful one, even when

no prices are perfectly ¬‚exible. In the context of our present framework, the notional SRAS

curve is given by (1.12). When log-linearized, it takes the form (1.21); thus the coe¬cient ζ

is just the elasticity of the function „¦ with respect to Y . Ball and Romer (1990) refer to an

economy in which this elasticity is small as being characterized by real rigidities.14

Best-response curves for individual price-setters can then be derived by substituting the

identity Yt = Yt /Pt into the notional SRAS curve, to obtain

˜

p— (i) = Pt „¦(Yt /Pt ; ξt ).

t

The degree of strategic complementarity is then indicated by the partial derivative of the

right-hand side with respect to Pt . We thus ¬nd that strategic complementarity exists15 (a

positive partial derivative) if and only if ζ < 1, while the pricing decisions of the separate

¬rms are strategic substitutes if ζ > 1.

Thus it is the existence or not of strategic complementarity that determines whether

the fraction of the suppliers with sticky prices exert a disproportionate e¬ect upon the

degree of adjustment of the aggregate price index. The intuition for this is simple: if prices

are strategic complements, then the fraction of prices that do not adjust in response to a

disturbance to nominal spending lead even the ¬‚exible-price suppliers to adjust their prices

by less than they otherwise would. If the strategic complementarity is strong enough, there

will be little aggregate price adjustment (and so a large output e¬ect) even if the fraction

14

The analogy is with “nominal rigidity”, a situation in which the nominal price charged by a given supplier

(rather than its relative price) is insensitive to the quantity that it must supply. Because the nominal SRAS

as de¬ned here is independent of any speci¬cation of the speed at which prices may be adjusted, the degree

to which an economy is characterized by “real rigidities” depends solely upon real factors, the structure of

production costs and of demand. The term, though widely used, is somewhat unfortunate, as it suggests

that there are costs of adjusting real quantities, whereas in fact it means a situation in which (notional)

supply is very elastic. We shall thus mainly speak here of “elastic supply” or “strategic complementarity in

price-setting” instead.

15

Note that the question of whether di¬erent ¬rms™ pricing decisions are strategic complements is only

de¬ned relative to a particular pricing game, in which aggregate nominal spending is taken as given, in-

dependent of the prices chosen by any of the ¬rms. This is a useful question to consider, however, when

analyzing the dynamics of prices and output under an arbitrarily speci¬ed stochastic process for nominal

spending, as we shall repeatedly do in this chapter.

26 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

of ¬‚exible-price suppliers is large (though less than one). On the other hand, if prices are

strategic substitutes, then the aggregate price level may adjust nearly in proportion to the

unexpected change in nominal spending, even when many prices are sticky. For in this case,

the fact that some prices do not adjust causes the ¬‚exible-price suppliers to compensate for

this by adjusting their prices even more than they would in a ¬‚exible-price equilibrium. In

the case of a high degree of strategic substitutability (large ζ), this can result in the aggregate

price index adjusting nearly in proportion to the change in nominal spending; but it will not

adjust by quite that amount, else there would be no change in real aggregate demand, and

so no desire on the part of ¬‚exible-price suppliers to change their relative prices.

The possibility of strategic substitutability explains why some authors, such Ohanian et

al. (1996) and Christiano and Eichenbaum (1996), ¬nd in models with sticky prices for only

some goods that the partial price stickiness has little e¬ect upon the aggregate e¬ects of

monetary shocks. These papers assume parameter values that imply an elasticity ζ greater

than one, the case of strategic substitutes. At the same time, the “New Keynesian” literature

of the 1980™s routinely assumed an elasticity ζ < 1, and hence found that pricing decisions

should be strategic complements. This result favored the conclusion that it was plausible to

suppose that nominal rigidities mattered a great deal for the character of short-run responses

to shocks. It is thus worth considering in further detail possible determinants of the degree

of strategic complementarity in pricing, and the plausible size of the elasticity ζ.

1.4 Sources of Strategic Complementarity

In the simple model introduced above, the elasticity of the notional SRAS curve is given by

(1.22). Most early “New Keynesian” literature16 assumes linear utility of consumption. This

corresponds in our notation to the limiting case σ ’1 = 0, in which case (1.22) reduces to

ω

ζ= .

1 + ωθ

Given that we must have θ > 1, this expression implies that ζ < 1, so that the pricing

decisions of di¬erent suppliers are necessarily strategic complements “ the conclusion relied

16

See, for example, the presentation in Blanchard and Fischer (1989, sec. 8.1),

1. A BASIC STICKY-PRICE MODEL 27

upon in that literature. However, this conclusion is less obviously true if, more realistically,

we assume a ¬nite value for the intertemporal elasticity of substitution σ.

Indeed, many of the early authors in the 1990™s seeking to incorporate sticky prices

into otherwise standard real business cycle models adopted numerical calibrations (derived

from the RBC literature) that implied strategic substitutability. One reason for this was

the widespread assumption, unlike what we have assumed above, of common economy-wide

factor markets, so that the marginal cost of supply would be equal for all goods i at any

point in time.17 In this case the marginal cost of supplying any good is given by

wt

St = ,

At f (ht )

where wt is the wage paid by all producers for the homogeneous labor input, and ht is the

common labor-capital ratio chosen by all producers. Capital is reallocated among ¬rms

so as to allow all ¬rms to use the same e¬cient labor-capital ratio, even if they produce

di¬erent quantities of their respective goods; this common labor-capital ratio will be given

by ht = f ’1 (Xt /At ), where the common output-capital ratio Xt satis¬es

1

Xt = yt (i)di.

0

Finally, the economy-wide wage wt satis¬es (1.6), where the representative household™s labor

supply must equal ht . Marginal supply cost is therefore given by

˜

St (i) = Pt s(Xt , Yt ; ξt ),

independent of the quantity produced of the individual good i, where s is again the function

de¬ned in (1.9).

Using this alternative expression for marginal cost in the derivation of our pricing equa-

tions, we again obtain equations such as (1.21), but in which we now de¬ne

ζ ≡ ω + σ ’1 > 0.

17

The consequences of this issue for the degree of strategic complementarity in pricing are stressed by

Kimball (1995).

28 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

(Note that up to a log-linear approximation, percentage ¬‚uctuations in Xt equal percentage

¬‚uctuations in Yt .) In the real business cycle literature, it is standard to assume that σ = 1.

(See, e.g., Cooley and Prescott, 1995.) Under this assumption, ζ > 1, and pricing decisions

are strategic substitutes, regardless of the (necessarily positive) value assigned to ω.

On the other hand, even if one were to assume linear utility from consumption (the most

extreme possible value for σ), one might easily conclude that pricing decisions should be

strategic substitutes, under the assumption of common factor markets. For example, the

baseline calibration of Chari et al. (2000) implies that ω = 1.25, while σ = 1. As they

assume common factor markets, their parameterization implies a value of ζ = 2.25, and thus

a considerable degree of strategic substitutability. But even if they were to assume linear

utility of consumption (σ ’1 = 0), despite the implausibility of such an extreme value, the

assumed degree of diminishing returns to labor in the production function and the assumed

degree of increasing marginal disutility of work (which together imply their value of ω) would

still imply ζ = 1.25, so that there would still be strategic substitutability. On the other hand,

even with their assumed values for ω and σ, if these authors were to assume speci¬c factor

markets (as in our analysis above), then (given that they also assign the value θ = 10) they

would have found that ζ = .17, using (1.22). This would imply a great degree of strategic

complementarity. Thus it is the assumption of common factor markets that makes the most

crucial di¬erence for the ¬nding of these and other authors that pricing decisions are strategic

substitutes.

The assumption of common factor markets is thus far from innocuous. It is also far from

realistic. Of course, it makes sense that high wages in one part of the economy eventually raise

wages in the rest, as workers migrate from low-wage to high-wage labor markets. Similarly,

it makes sense that high returns to capital in one part of the economy eventually raise the

rental rate for capital services throughout the economy, as capital is shifted to higher-return

uses (if only through the allocation of new investment). A failure to allow for these factor-

price equalization mechanisms makes our model of speci¬c factor markets unrealistic as a

model of the long-run e¬ects upon sectoral marginal costs of a permanent failure of prices

1. A BASIC STICKY-PRICE MODEL 29

in one sector to adjust.

Nonetheless, the opposite extreme assumption ” that all factor prices are instantaneously

equalized across the suppliers of di¬erent goods ” is also unrealistic. In the short run, it

is not easy for workers to migrate to regions, specialties, or even ¬rms that happen to have

temporarily higher labor demand; and it is even less easy for capital goods, once installed,

to be reassigned to ¬rms with a temporarily high rate of utilization of their capital stock.

This “quasi-¬xed” character of factor inputs allows equilibrium factor prices to vary across

suppliers for some period following a shock that a¬ects them asymmetrically; and it is

mainly these short-run dynamics of factor prices that matter for determining the short-run

dynamics of price adjustment in response to shocks, which are what matter for comparing

the real allocation of resources under alternative monetary policies.18

Factor speci¬city need not be an all-or-nothing assumption. For example, one might

assume (as in Sbordone, 1998) that all suppliers hire the same kind of labor inputs in a

single economy-wide labor market, but that each ¬rm has a production function of the form

yt (i) = At f (ht (i)), where f is strictly concave. This amounts to assuming that each ¬rm™s

allocation of capital goods remains ¬xed, rather than letting capital be reallocated to equalize

its rental rate. In this case, real marginal costs are given by

ˆ ˆ

st (i) = ωp yt (i) + (ωw + σ ’1 )Yt ’ (ω + σ ’1 )Ytn ,

ˆ ˆ

where ωp , ωw > 0 are the two components of ω introduced in (1.15). It then follows that

ω + σ ’1

ζ= > 0. (1.25)

1 + ωp θ

This value of ζ is higher than that implied by (1.22), so that there is less strategic comple-

mentarity than in the case of speci¬c labor markets as well (given a ¬nite Frisch elasticity

of labor supply, so that ωw > 0). But it remains a lower value than is obtained under the

assumption of common factor markets, i.e., instantaneous equalization of the rental price of

18

In chapter 4, we show how endogenous reallocation of capital across sectors can be added to the model,

so that the rental price of capital must be equalized across sectors in the long run. We nonetheless ¬nd that

in the presence of adjustment costs for investment, the short-run dynamics of marginal supply costs are not

too di¬erent from those predicted by our simple model with a ¬xed allocation of capital to ¬rms.

30 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

capital services. For example, Chari et al. (2000) assume a Cobb-Douglas production func-

tion with a labor share of .67, which implies ωp = .5. Thus under their baseline calibration,

if they were to assume a common labor market but speci¬c capital inputs, they would have

obtained ζ = .38, and so a substantial degree of strategic complementarity in price-setting.

There are other reasons as well to believe that the simple model with common fac-

tor markets set out above underestimates the likely degree of strategic complementarity in

price-setting. One that has received some attention in the literature is the possibility that

preferences over di¬erentiated goods need not be of the constant-elasticity (Dixit-Stiglitz)

form (1.2). The result is that the elasticity of demand, and hence the desired markup of

price over marginal cost, need not be independent of the relative quantities produced of

di¬erent goods. In particular, Kimball (1995) shows that if the elasticity of demand is lower

for the products of suppliers who sell more (because they have relatively low prices), this

will increase strategic complementarity, and hence the real e¬ects of variations in nominal

spending. For in the event of a decline in nominal spending, price cuts by ¬‚exible-price

suppliers will lead to an increase in their relative sales (though a reduction in the absolute

quantity sold), because the sticky-price suppliers fail to cut their prices. If this increase in

relative sales leads to less elastic demand for their products, their desired markups will rise,

mitigating the e¬ect upon their desired relative price of the decline in their sales (and hence

reduction in their marginal supply cost); this will cause them to cut their prices less, so that

the decline in their output is greater.

Following Kimball, let us generalize (1.2) by assuming that the consumption aggregate

Ct is implicitly de¬ned by a relation of the form

1

ψ(ct (i)/Ct )di = 1, (1.26)

0

where ψ(x) is an increasing, strictly concave function satisfying ψ(1) = 1. (Note that this

reduces to (1.2) if ψ(x) = xθ’1/θ .) The demand curve for good i is then implicitly de¬ned by

yt (i) pt (i)

ψ = ψ (1) , (1.27)

Yt Pt

1. A BASIC STICKY-PRICE MODEL 31

where the price index Pt for aggregate (1.26) is implicitly de¬ned by

pt (i) ’1 pt (i)

1

ψ ψ (1) di = 1. (1.28)

Pt Pt

0

The elasticity of the demand curve (1.27) faced by supplier i is then equal to θ(yt (i)/Yt ),

where

ψ (x)

θ(x) ≡ ’ . (1.29)

xψ (x)

It follows that the desired markup of price over marginal cost for a ¬‚exible-price supplier is

equal to µ(yt (i)/Yt ), where

θ(x)

µ(x) ≡ . (1.30)

θ(x) ’ 1

Note that in general, neither θ nor µ is a constant as assumed before; but we continue to

assume a function ψ(x) such that θ(x) > 1 for all x in a neighborhood of 1, so that µ(x) > 1

is well-de¬ned, at least in that neighborhood.

The desired relative price of a ¬‚exible-price supplier is then given by

pt (i) ˜

= µ(yt (i)/Yt )s(yt (i), Yt ; ξt ), (1.31)

Pt

generalizing (1.12). Log-linearization of this relation in turn yields

ˆ ˆ ˆ ˆ

’ Yt ) + sy (ˆt (i) ’ Ytn ) + sY (Yt ’ Ytn ),

log pt (i) = log Pt + µ (ˆt (i)

y y

where is the elasticity of µ(x) at the value x = 1, and sy and sY are the elasticities of

µ

the real marginal cost function with respect to its ¬rst two arguments respectively. Using a

log-linear approximation to (1.27),

ˆ

yt (i) = Yt ’ θ(log pt (i) ’ log Pt ),

ˆ

to eliminate yt (i), and then solving for log pt (i), we obtain a log-linear notional SRAS curve,

ˆ

and observe that now

sy + sY

ζ= . (1.32)

1 + θ( µ + sy )

(Here the coe¬cient θ refers to θ(1).) For given values of the other parameters, we observe

that a positive ” which means θ(x) decreasing in x, as discussed above ” lowers the

µ

value of ζ, thus increasing the degree of strategic complementarity.

32 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

In principle, as Kimball shows, could be an arbitrarily large quantity, and so this factor

µ

could result in ζ being arbitrarily small, regardless of the size of the coe¬cients sy , sY and θ.

(Note that ζ ’ 0 as ’ ∞.) Bergin and Feenstra (2000), however, argue that a plausible

µ

calibration would assign a value of approximately θ = 1, so that is positive but not

µ µ

extremely high. If the desired markup depends upon relative demand for di¬erent goods as

above, one can show that the desired price of supplier i satis¬es

1 θµ

log pt (i) = log St (i) + log Pt ,

1+θ 1+θ

µ µ

regardless of the determinants of nominal marginal cost St (i). Bergin and Feenstra note

that the literature on exchange-rate pass-through typically ¬nds that a devaluation raises

the domestic-currency price of imported goods by only about 0.5 of the percentage of the

devaluation. If one interprets this is as measuring the elasticity of pt (i) with respect to

changes in St (i) (on the assumption that the devaluation does not a¬ect the foreign-currency

marginal cost of supplying the imported goods, or the domestic-currency prices of most goods

equals approximately 1.19

in the domestic price index), then one may conclude that θ µ

In the case of homogeneous factor markets (sy = 0, sY = ω + σ ’1 ), assumed by Bergin

and Feenstra, (1.32) reduces to

ω + σ ’1

ζ= .

1+θ µ

In this case, their suggested value of reduces ζ by a factor of 2 relative to what one would

µ

obtain under the assumption of Dixit-Stiglitz preferences. This is not enough to radically

change one™s views about the real e¬ects of monetary policy (as they conclude), but it is

nonetheless a non-trivial correction. However, in the case of speci¬c factor markets (sy = ω,

sY = σ ’1 ), the correction matters much less. In this case, (1.32) reduces to

ω + σ ’1

ζ= .

1 + θ( µ + ω)

Here setting θ equal to 1 instead of zero does not make such a great di¬erence, since the

µ

term θω in the denominator is now likely to be much greater than one. For example, in the

19

Bergin and Feenstra also note that this size of markup elasticity would be implied by the translog

speci¬cation of preferences that has been popular in econometric studies of demand.

1. A BASIC STICKY-PRICE MODEL 33

case of the Chari et al. values for ω, σ and θ discussed above, allowing for the variation in

desired markups only reduces the predicted ζ from 0.17 to 0.16.

Another reason for greater strategic complementarity than is indicated by the baseline

analysis above is the economy™s input-output structure, stressed by Basu (1995). Our above

analysis of marginal supply costs assumes that labor is the only variable factor of production,

ignoring the role of intermediate inputs. While this is a familiar assumption in equilibrium

business-cycle models (it is, for example, routine in the real business cycle literature), it

is far from being literally correct. The production function for “output” as a function of

capital and labor inputs that one typically encounters in such models must be interpreted

as a functional relation between the value added in production (GDP) and primary factor

inputs, rather than a relation between gross output and all factors of production (including

those that are themselves produced). Under certain conditions, and for some purposes, it

su¬ces to model the economy as if this “value-added production function” were the actual

production function of individual producers ” for example, for purposes of predicting the

evolution of real GDP in the context of a ¬‚exible-price, perfectly competitive model, as in

RBC theory. But as Rotemberg and Woodford (1995) note, it is important not to con¬‚ate

gross-output and value-added production functions in the case of a model with imperfect

competition and prices that do not co-move perfectly with marginal cost.

Rotemberg and Woodford (1995) propose a simple gross-output production function of

the form20

At f (ht (i)) mt (i)

yt (i) = min , , (1.33)

1 ’ sm sm

where At f (ht (i)) is the value-added production function (as above), mt (i) is the quantity of

materials inputs used by ¬rm i, and 0 ¤ sm < 1 is a parameter of the production technology

(that can be identi¬ed, for purposes of calibration, with the share of materials costs in the

20

Basu (1995) and Bergin and Feenstra (2000) instead assume that gross output is a Cobb-Douglas function

of labor and materials inputs. The simple analysis here, however, which has the advantage of nesting our

previous speci¬cation as a limiting case, allows us to reach quite similar conclusions about the consequences

of the input-output structure for the degree of strategic complementarity in pricing in a highly transparent

way. There seems in any event to be no argument other than tractability o¬ered for the Cobb-Douglas

speci¬cation used in these other papers.

34 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

value of gross output). The argument mt (i) refers to the number of units of the aggregate

de¬ned in (1.2) ” or more generally in (1.26) ” that are purchased for use in producing good

i; it is thus a composite of all of the goods produced in the economy, which are all assumed

equally to be both ¬nal goods and intermediate inputs. If ¬rms allocate their intermediate

input purchases across goods j in a cost-minimizing way, the quantity purchased of good j

for use in producing good i will equal mt (i)(pt (j)/Pt )’θ (or the corresponding generalization

in the case of a non-Dixit-Stiglitz aggregator), and the cost per unit of materials inputs will

be the same price index Pt as for consumption purposes. Total demand for ¬rm i™s output,

given by the sum of ¬nal demand and intermediate-input demand from other producers, will

still be given by (1.5) ” or more generally by (1.27) ” where now aggregate demand Yt is

equal to

1

Yt = Ct + mt (i)di. (1.34)

0

It follows from the production function (1.33) that real marginal cost for ¬rm i will equal

st (i) = (1 ’ sm )sV A (i) + sm , (1.35)

t

where sV A (i) is the real marginal cost of an additional unit of theoretical value-added (the

t

function of primary input use given by At f (ht (i))), and we have used the fact that the price

of the materials aggregate is Pt . The marginal cost of supplying value-added is given by the

same real marginal cost function derived earlier; for example, in the case of speci¬c factor

markets, it is given by

1

˜

sV A (i) = s((1 ’ sm )yt (i), Yt ’ sm yt (i)di; ξt ),

t

0

˜

where s(y, Y ; ξ) is again de¬ned as in (1.9). The existence of a symmetric steady state then

requires that sm < µ’1 , where the latter quantity represents the steady-state level of real

marginal cost, given ¬rms™ desired markup of µ > 1.21 Such a steady state involves a constant

¯

level of ¬nal-goods demand (or real value added) Y implicitly de¬ned by22

1 ’ µsm

¯¯

s(Y , Y ; 0) = . (1.36)

µ(1 ’ sm )

21

In the case that the desired markup is variable, we continue to use the coe¬cient µ to refer to the

steady-state value, µ(1).

1. A BASIC STICKY-PRICE MODEL 35

Regardless of the factor market structure, we ¬nd that a log-linear approximation to the

real marginal cost function for value added is given by

ˆ

sV A (i) = sy yt (i) + sY Yt ,

ˆt ˆ

where the elasticities sy and sY are the same as earlier. Log-linearization of (1.35) then

implies that

ˆ

st (i) = (1 ’ µsm )(sy yt (i) + sY Yt ).

ˆ ˆ

It is the appearance of the multiplicative factor 1 ’ µsm < 1 here that explains how the

economy™s input-output structure gives rise to “real rigidity” of the sort that increases the

strategic complementarity among pricing decisions. Substitution of the demand curve for

yt (i) as before then allows us to solve once again for ¬rm i™s desired relative price; we now

ˆ

¬nd that

(1 ’ µsm )(sy + sY )

ζ= , (1.37)

1 + θ[ µ + (1 ’ µsm )sy ]

generalizing (1.32).

We observe that allowing for a positive materials share lowers ζ, for given values of

the other parameters; and once again, this e¬ect could in principle result in an arbitrarily

small value for ζ, regardless of the values of the other parameters. (Note that ζ ’ 0 as

sm ’ µ’1 < 1.) However, the share of materials costs in total costs for U.S. manufacturing

sectors is typically on the order of 50 or 60 percent; this suggests that a reasonable calibration

would be on the order of µsm = 0.6.23 In the case of homogeneous factor markets (sy = 0),