Note that the right-hand side of (1.36) is the the reciprocal of the steady-state “value-added markup”

or ine¬ciency wedge resulting from market power that Rotemberg and Woodford (1995) contrast with

the steady-state “gross-output markup” or ¬rm-level markup of price over marginal cost µ. Distinguishing

between the two is essential in calibrating an imperfectly competitive model that abstracts from the existence

of intermediate inputs.

23

This would correspond to a materials share of 54 percent, if we assume θ = 10 as do Chari et al. Note

that Basu (1995) suggests that a share parameter as high as sm = 0.9 could be reasonable, on the ground

that the share of intermediate inputs in marginal cost could be substantially higher than their share in

average cost. Under such an assumption this correction would be much more important than under the

calibration suggested here; for example, the assumption sm = 0.9 combined with an assumption θ = 10

would imply that ζ = 0. But such a calibration is not easy to interpret. If, for example, one proposes that a

large fraction of primary input purchases represent ¬xed or “overhead” costs, this would imply substantial

increasing returns (average cost much higher than marginal cost), which would in turn not be consistent

with equilibrium in the absence of an implausibly high degree of market power.

36 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

Table 3.1: The value of ζ under alternative assumptions.

σ ’1 = 1, ω = 1.25 σ ’1 = .16, ω = .47

θµ µsm homo. factor spec. factor homo. factor spec. factor

0 0 2.25 0.17 0.63 0.11

1 0 1.13 0.16 0.32 0.09

0 .6 0.90 0.15 0.25 0.09

1 .6 0.45 0.13 0.13 0.06

this implies that ζ is reduced by a factor of 2.5 (i.e., it is multiplied by 0.4). This is again

a signi¬cant reduction, and if this e¬ect is combined with the degree of variation in the

desired markup argued for above, the value of ζ is only one-¬fth the size indicated by the

baseline calculation of Chari et al. ” 0.45 instead of 2.25, which is to say well below 1

(implying strategic complementarity) rather than being well above 1 (implying strategic

substitutability). On the other hand, in the case of speci¬c factor markets, this correction

matters much less, unless sm is assumed to be much closer than µ’1 than seems realistic.

Using the calibration discussed above, assuming µsm = 0.6 rather than zero reduces ζ from

0.17 to 0.15 (in the constant-markup case), or from 0.16 to 0.13 (in the variable-markup

case). These reductions make no dramatic di¬erence in one™s conclusions about the degree

of strategic complementarity.

Our conclusions about the e¬ects of these several corrections, in the case of the calibra-

tions just discussed, are summarized in Table 3.1. The left side of the table gives the value

of ζ under the assumption that ω = 1.25, σ = 1, and θ = 10, as assumed by Chari et al.,

under eight di¬erent cases, representing three binary choices. These are the assumption of

homogeneous factor markets (sy = 0, sY = ω + σ ’1 ) versus speci¬c factor markets (sy = ω,

sY = σ ’1 ); the assumption of a constant desired markup ( = 0) versus a realistic degree

µ

of markup variation (θ mu = 1); and the assumption of no intermediate inputs (sm = 0)

versus a realistic intermediate input share in costs (µsm = 0.6). In each case, the simple

assumption made in the baseline case of Chari et al. is the one least favorable to strategic

complementarity. Changing any of these assumptions individually reduces ζ substantially

1. A BASIC STICKY-PRICE MODEL 37

relative to their baseline case. However, it is the allowance for factor speci¬city that matters

most, if one accepts that this is the more realistic assumption. For while each of the other

two corrections makes a signi¬cant di¬erence in the case of homogeneous factor markets, and

together they make an even larger di¬erence (as stressed by Bergin and Feenstra), in the

case of speci¬c factor markets even their combined e¬ect is not at all dramatic. Furthermore,

factor speci¬city alone reduces the value of ζ much more than the other two factors combined

do, if the importance of these latter factors is calibrated in what seems a reasonable way.

The right side of the table o¬ers a similar comparison, when instead lower values are

assumed for both σ ’1 and ω. The values suggested here are those obtained by Rotemberg

and Woodford (1997) when a slightly more complicated version of this pricing model is

¬t to U.S. time series, as discussed further in chapter 4. The higher value of σ (implying

much greater interest-sensitivity of private expenditure) is needed in order to account for the

observed size of the e¬ects of an identi¬ed monetary policy shock on real aggregate demand;

the lower value of ω (implying much more elastic labor supply) is needed in order to account

for the observed modest declines in real wages that accompany such a large decline in output.

Under these alternative assumptions regarding preferences, ζ < 1, implying a modest degree

of strategic complementarity, even in the case of homogeneous factor markets, Dixit-Stiglitz

preferences, and no intermediate inputs. But once again the degree of real rigidity is increased

by modifying any of these last three assumptions. It is interesting to observe in this case

that even under the assumption of homogenous factor markets, we obtain a value of ζ = .13,

if we make realistic assumptions about the other two sources of real rigidity. Thus a value

of ζ in the range between 0.10 and 0.15 does not require implausible assumptions. This is

a value that implies substantial strategic complementarity, and as we shall see, enough to

explain roughly the observed degree of sluggishness of aggregate price adjustment in response

to variations in nominal expenditure, given the observed frequency of price adjustment in

economies like that of the U.S.

38 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

2 In¬‚ation Dynamics with Staggered Price-Setting

An unsatisfactory feature of the “New Classical” aggregate supply relation derived above

is its implication that only unanticipated ¬‚uctuations in nominal spending have any e¬ect

upon real activity, and that equilibrium ¬‚uctuations in the output gap must be completely

unforecastable. These strong predictions were the occasion of a great deal of discussion and

criticism during the 1970s and early 1980s (see, e.g., She¬rin, 1996, chap. 2). They imply

that only the immediate e¬ects of a monetary policy shock upon nominal expenditure should

have any consequences for real activity; delayed e¬ects (e¬ects on nominal spending after

the “period” in which the shock occurs) should not a¬ect output at all, only the price level.

And such real e¬ects of a monetary policy shock as occur must be purely transitory, i.e.,

must last no longer than the “period” for which the sticky prices are ¬xed in advance.

These predictions are quite inconsistent with the e¬ects identi¬ed in the “structural VAR”

literature. As an example, Figure 3.2 plots the impulse response of nominal GDP to an

identi¬ed monetary policy shock, according to the structural VAR model of Christiano et al.

(2001).24 There is practically no measurable e¬ect of an unexpected interest-rate reduction

in quarter zero upon nominal GDP until the second quarter following the monetary policy

shock, though there is a strong increase in nominal GDP at that time.25 But this means

that according to the “New Classical” aggregate-supply relation, monetary shocks should

have no e¬ect upon real activity at all, unless the “period” for which sticky prices are ¬xed

in advance is longer than six months.

Figure 3.3 next shows the estimated impulse response of real GDP to the same shock in

24

This particular impulse response is not reported in their paper, though the point estimates here are

implied by the in¬‚ation and real GDP responses that are reported there. I thank Charlie Evans for supplying

the data plotted in this and subsequent ¬gures. The impulse responses implied by this particular VAR study

are representative of those found in many others; see Christiano et al. (1999) for a review.

25

The fact that there is zero e¬ect upon nominal GDP in quarter 0 is an artifact of the authors™ iden-

ti¬cation scheme, which assumes that any contemporaneous correlation between interest-rate innovations

and innovations in either real GDP or in¬‚ation is due to feedback from the latter variables to the current

interest-rate operating target, as under a Taylor rule. However, the estimated e¬ect in quarter 1 is in no

way constrained by the identi¬cation scheme; the fact that neither output nor in¬‚ation is estimated to be

signi¬cantly a¬ected in quarter 1 provides some support for the assumption relied upon in the identi¬cation

scheme.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 39

3.5

3

2.5

2

1.5

1

0.5

0

’0.5

’1

’1.5

0 5 10 15 20 25 30

Figure 3.2: Impulse response of nominal GDP to an unanticipated interest-rate reduction

in quarter 0. Periods on horizontal axis represent quarters, while the vertical axis measures

the e¬ect on log nominal GDP in percentage points. Source: Christiano, Eichenbaum and

Evans (2001).

quarter zero. Contrary to the suggestion just mentioned, there is a substantial real e¬ect

of the shock. (Note that in both ¬gures, the monetary policy shock is normalized so that

the long-run e¬ect on nominal GDP is an increase of one percentage point.) Furthermore,

the e¬ect occurs with a substantial delay; there is essentially no e¬ect on output until the

second quarter following the policy shock,26 and the peak output e¬ect occurs only in the

sixth quarter following the shock. The estimated e¬ect is still more than two standard errors

greater than zero two full years after the shock, and (at least according to the point estimate)

the e¬ect is still at more than a third of its maximum level ten quarters after the shock.

26

It may be wondered why we assert that the shock actually occurs in quarter zero, given that there is

no e¬ect on nominal expenditure until quarter two. The occurrence of the policy shock is indicated by the

substantial decline in the federal funds rate in quarter zero; see Christiano et al. (2001) for a plot of this

response.

40 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

2.5

2

1.5

1

0.5

0

’0.5

’1

’1.5

0 5 10 15 20 25 30

Figure 3.3: Impulse response of real GDP to the same monetary policy shock as in Figure

3.2. The vertical axis measures the e¬ect on log real GDP in percentage points. Source:

Christiano, Eichenbaum and Evans (2001).

Such long-lasting e¬ects are inconsistent with the “New Classical” aggregate-supply relation,

unless the “period” for which prices are ¬xed in advance is longer than two years. But survey

evidence on price changes (e.g., Blinder et al., 1998) ¬nds that the majority of ¬rms change

their prices more frequently than once per year, so that more than half of all prices should

have been adjusted at least once within the ¬rst two quarters following a shock.

This undesirable feature of the model can be avoided, however ” without abandoning

the assumption that prices are set optimally (under rational expectations) when they are

adjusted, and without assuming counterfactually long intervals between price changes ” by

assuming that the intervals over which the prices of di¬erent goods remain ¬xed overlap,

rather than being perfectly synchronized. As Phelps (1978) and Taylor (1979a, 1980) ¬rst

pointed out,27 such staggering of price changes can lead to a sluggish process of adjustment

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 41

of the overall price level, even when individual prices are adjusted relatively frequently.

This is because, if individual suppliers™ pricing decisions are strategic complements, then

the fact that some prices are not yet being adjusted will restrain the degree to which prices

are changed by those suppliers that do adjust their prices. At a later date, the adjustment of

the prices that earlier were sticky will in turn be restrained by the fact that the prices that

earlier were adjusted were not changed very much. As a result of this process, the level of

prices prior to a shock can continue to have a signi¬cant e¬ect on the general level of prices

even after most prices have been adjusted at least once since the shock. If the strategic

complementarity between pricing decisions is great enough, the adjustment of the general

level of prices can be quite slow, even though individual price adjustments are frequent. The

consequence will be prolonged e¬ects on real activity of a sustained change in the level of

nominal spending.

As we shall see, this theory of pricing can again justify an aggregate supply relation that

takes the form of an “expectations-augmented Phillips-curve” relation. However, in this

variant case, the kind of in¬‚ation expectations that determine the location of the short-run

Phillips curve are current expectations regarding future in¬‚ation, rather than past expecta-

tions regarding current in¬‚ation. This might seem a small di¬erence (given the degree of

serial correlation in in¬‚ation expectations, at least in recent decades), but it is a crucial one;

for the aggregate supply relation, together with rational expectations, no longer exclude the

possibility of forecastable variations in the output gap. With this modi¬cation, the model

becomes consistent with the occurrence of prolonged ¬‚uctuations in real activity following a

monetary policy shock, of at least roughly the kind estimated in the VAR literature.

27

These ¬rst applications of the idea assumed that wages, rather than prices, were ¬xed for a period

of time (owing to wage contracts), and thus concerned the e¬ects of staggered wage negotiations rather

than staggered price-setting. But as Blanchard (1983) pointed out, the same principle can be applied to

price-setting.

42 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

2.1 The Calvo Model of Price-Setting

Here we present a particular example of a model with staggered price-setting, a discrete-

time variant of a model proposed by Guillermo Calvo (1983).28 In this model, a fraction

0 < ± < 1 of goods prices remain unchanged each period, while new prices are chosen for

the other 1 ’ ± of the goods. For simplicity, the probability that any given price will be

adjusted in any given period is assumed to be 1 ’ ±, independent of the length of time since

the price was set, and independent of what the particular good™s current price may be.

These last assumptions are plainly unrealistic, but they are very convenient in simplifying

the analysis of equilibrium in¬‚ation dynamics, as they greatly reduce the size of the state

space required to characterize those dynamics. Because each supplier that chooses a new

price for its good in period t faces exactly the same decision problem, the optimal price p—

t

is the same for each of them, and so in equilibrium, all prices that are chosen in period t

have the common value p— . The remaining fraction ± of prices charged in period t are simply

t

a subset of the prices charged in period t ’ 1, with each price appearing in the period t

distribution of unchanged prices with the same relative frequency as in the period t ’ 1 price

distribution. (For this last argument it is crucial that each price has an equal probability

of being adjusted in a given period.) Then the Dixit-Stiglitz price index (1.3) in period t

satis¬es

1 1

Pt1’θ ≡ pt (i)1’θ di = (1 ’ ±)p—1’θ + ± pt’1 (i)1’θ di,

t

0 0

so that

1

±)p—1’θ 1’θ 1’θ

Pt = (1 ’ + ±Pt’1 . (2.1)

t

It follows that in order to determine the evolution of this price index, we need only know

its initial value, and the single new price p— that is chosen each period. The determination of

t

28

See also Rotemberg (1987) for discussion of this model, and the similarity of its implications for aggregate

dynamics to those of a model with convex costs of price adjustment. The ¬rst use of a discrete-time version

of Calvo™s model of price-setting, in the context of a complete intertemporal equilibrium model of aggregate

¬‚uctuations, was in the work of Yun (1996). Other early applications of the same device include Woodford

(1996), King and Watson (1996), King and Wolman (1996), and Goodfriend and King (1997). Kimball (1995)

also assumes Calvo pricing, albeit in continuous time, in another important early study of an intertemporal

equilibrium model with sticky prices.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 43

p— , in turn, depends upon current and expected future demand conditions for the individual

t

good, but (1.11) implies that other prices a¬ect the demand curve for good i only through

the value of the price index Pt . Thus we can determine the equilibrium value of the index Pt

as a function of its previous period™s value, the expected future path of this same index, and

current and expected future values of aggregate real variables. There is no need for reference

to additional information about past prices.29

A supplier that changes its price in period t chooses its new price pt (i) to maximize

∞

±T ’t Qt,T [Πi (pt (i))] ,

Et (2.2)

T

T =t

by analogy with (1.16), where once again nominal pro¬ts each period are given by

Πi (p) = [Yt Ptθ p1’θ ’ wt (i)f ’1 (Yt Ptθ p’θ /At ). (2.3)

t

Here the factor ±T ’t multiplying the stochastic discount factor indicates the probability

that price pt (i) will still be charged in period T .30 The price pt (i) is chosen on the basis

of information available at date t so as to maximize this expression, given the expected

state-contingent values of the random variables Qt,T , YT , PT , wT (i) and AT for all dates

T ≥ t.

Corresponding to (1.18) we obtain in this case the ¬rst-order condition

∞

±T ’t Qt,T YT PT [pt (i) ’ µST (i)] = 0.

θ

Et (2.4)

T =t

Thus there is again a sense in which the price pt (i) is set equal to µ times a weighted average

of the levels of marginal costs expected to prevail in the various future states, and at the

29

This is only precisely true in the case of our baseline model, with speci¬c factor markets, since in this case

the real marginal cost of supplying a given good depends only upon the quantity supplied of that good yt (i),

˜

aggregate output Yt , and the vector of aggregate shocks ξt . If we instead assume common factor markets,

as in Yun (1996), real marginal cost depends upon an alternative output aggregate (the one that determines

aggregate demand for factors), and not solely upon the Dixit-Stiglitz aggregate Yt (that determines the

utility value of output). This means that the equilibrium conditions for the evolution of Pt also involve a

second price index, as shown by Yun. However, even in that case, up to a log-linear approximation the

equilibrium conditions can be written solely in terms of the index Pt .

30

Note that (2.2) reduces to (1.16) if the factor ±T ’t is replaced by one that takes the value 1 if T = t + 1

and the value zero for all other T .

44 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

various future dates, at which the price pt (i) applies. But once again, (2.4) does not give a

closed-form expression for the optimal price, since ST (i) will depend upon the price of good

i.31 Substituting the demand function (1.11) into the real marginal cost function de¬ned in

(1.9), and using this to replace ST (i) in (2.4), and again substituting the solution for the

stochastic discount factor from equation (xx) of chapter 2, we obtain

∞

˜

(±β)T ’t uc (YT ; ξT )YT PT [p— ’ µPT s(YT PT p—’θ , YT ; ξT )] = 0

θ’1 θ

Et (2.5)

t t

T =t

as an equation that implicitly de¬nes the optimal price p— . Note that the monotonicity of s

t

in its ¬rst argument implies that the left-hand side of this expression is increasing in p— , so

t

that the optimal price is uniquely de¬ned by this condition. The value of p— then determines

t

the evolution of the price index Pt through (2.1).

Once again, we may usefully approximate the equilibrium dynamics of in¬‚ation in the case

˜

of small enough disturbances ξt by considering a log-linear approximation to these equations.

˜ ¯

If ξt = 0 and Yt = Y at all times, equations (2.1) and (2.5) have a solution with zero in¬‚ation,

˜

in which Pt = p— = Pt’1 each period. In the case of small enough ¬‚uctuations in ξt and Yt

t

around these values, we accordingly look for a solution in which Pt /Pt’1 and p— /Pt remain

t

always close to 1, though the (log) price level may contain a unit root.32 The ¬‚uctuations

in these two stationary variables must approximately satisfy a log-linear approximation to

31

Authors such as Yun (1996) or Goodfriend and King (1997), who assume common factor markets, are

instead able to solve for the optimal price p— in closed form. However, the assumption of common factor

t

markets results in an unrealistic estimate of the likely degree of strategic complementarity in price-setting,

as discussed above.

32

Our algebra at this point is simpli¬ed by log-linearizing around a steady state with a zero in¬‚ation

rate, rather than some other constant in¬‚ation rate. The resulting log-linear structural equations thus apply

only to the determination of equilibrium in the case of a policy rule that does in fact generate in¬‚ation

near zero at all times. This does not mean, however, that we can consider only policy rules that make the

average in¬‚ation rate exactly zero, or that involve a “target” in¬‚ation rate of zero. It is only necessary

that the average in¬‚ation rate be su¬ciently small (technically, of order O(||ξ||), if we want the error in

our characterization of the evolution of in¬‚ation and other variables to be of order O(||ξ||2 ), where ||ξ|| is a

bound on the size of the disturbances). As we shall argue (see chapter 6) that desirable policies do imply

a low average in¬‚ation rate, this does not seem an inconvenient restriction, from the standpoint of our goal

of characterizing optimal policy. Characterization of the e¬ects of policy in a high-in¬‚ation economy might

require a more accurate approximation. But the Calvo pricing model itself is implausible as a model of

pricing under such circumstances, as many individual prices are likely to be indexed to some broader price

index (or to an exchange rate). The model with backward-looking price indexation, discussed in section xx

below, would likely be more realistic for such purposes.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 45

(2.1),

1’± —

πt = p,

ˆ (2.6)

±t

where p— ≡ log(p— /Pt ). They must also approximately satisfy a log-linear approximation to

ˆt t

(2.5), given by ±

∞ T

(±β)T ’t p— ’ [ˆt,T +

Et ˆt s π„ ] = 0, (2.7)

„ =t+1

T =t

where st,T denotes the deviation (from its steady-state value) of the log of real marginal cost

ˆ

in period T for a supplier whose price is p— . (The term in square brackets in (2.7) is thus

t

the deviation of the log of

˜

(PT /Pt )s(YT PT p—’θ , YT ; ξT )

θ

t

from its steady-state value of µ’1 . There are no terms in (2.7) corresponding to stochastic

variation in the discount factors in (2.5), because the steady-state value of the term in square

brackets in that equation is zero.)

Under our characterization (1.14) of the dependence of marginal supply cost upon a

producer™s own level of output, the variable st,T can furthermore be approximated by

ˆ

T

ωθ[ˆ—

st,T = sT ’

ˆ ˆ pt ’ π„ ],

„ =t+1

where sT denotes the deviation of the log of the average level of real marginal cost (real

ˆ

marginal cost for a good with output yT (i) = YT ) from its steady-state value. Substituting

this into (2.7), we can solve for

∞ T

p— T ’t ’1

ˆt = (1 ’ ±β)Et (±β) [(1 + ωθ) sT +

ˆ π„ ]

„ =t+1

T =t

∞ ∞

1 ’ ±β

(±β)T ’t Et sT + (±β)T ’t Et πT .

= ˆ

1 + ωθ T =t T =t+1

Thus the relative price chosen by those suppliers who adjust their prices in period t is a

purely forward-looking function of aggregate conditions at that date.

This last expression in turn can be quasi-di¬erenced to yield

1 ’ ±β

p— = st + ±βEt πt+1 + ±βEt p— .

ˆt ˆ ˆt+1

1 + ωθ

46 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

Then using (2.6) to substitute for p— on both sides of the above, we obtain a stochastic

ˆt

di¬erence equation for the in¬‚ation rate,

πt = ξˆt + βEt πt+1 ,

s (2.8)

where

1 ’ ± 1 ’ ±β

ξ≡ > 0.

± 1 + ωθ

Writing this alternatively as

∆ log Pt = ξ[log St ’ log Pt + log µ] + βEt ∆ log Pt+1 , (2.9)

we see that we have an equation that can be solved for the predicted path of the price

index Pt , given the evolution of the average level of nominal marginal cost St . Speci¬cally,

if {log St } is a di¬erence-stationary process, there is a unique solution for the price index

process such that {log Pt } is also di¬erence-stationary (i.e., in¬‚ation is a stationary process),

given by

∞

»’j [log µ + Et log St+j ],

»’1 )

log Pt = »1 log Pt’1 + (1 ’ »1 )(1 ’ (2.10)

2

2

j=0

where 0 < »1 < 1 < β ’1 < »2 are the two roots33 of the characteristic polynomial

β»2 ’ (1 + β + ξ)» + 1 = 0. (2.11)

This prediction of the Calvo pricing model is in fact independent of any speci¬cation of

the determinants of average marginal costs; the only feature of the marginal cost function

used in deriving (2.8) was the fact that the elasticity of an individual supplier™s marginal

cost with respect to its own output is equal to ω > 0. Tests of the empirical adequacy of

this theory of pricing are therefore appropriately focused upon this prediction, rather than

upon the accuracy of the “New Keynesian” aggregate supply relation derived below (which

depends as well upon the details of one™s theory of supply costs). This is the approach

33

The roots can be shown to be real and to satisfy the inequalities just stated, as long as 0 < β < 1 and

ξ > 0.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 47

15

inflation

ULC growth

10

5

0

’5

’10

1965 1970 1975 1980 1985 1990 1995

Figure 3.4: U.S. in¬‚ation (quarterly change in GDP de¬‚ator, in percentage points of equiv-

alent annual rate) compared to growth rate of unit labor cost. Source: Sbordone (2002).

taken by Sbordone (1998, 2002), who uses data on the average level of unit labor cost in the

U.S. economy as a measure of nominal marginal cost.34 She estimates a small atheoretical

VAR model with which to forecast the future path of unit labor costs, using quarterly U.S.

data over the period 1960-1997. This then allows one to construct an implied series for the

forward-looking terms on the right-hand side of (2.10), for any assumed values of β and ξ

(which then imply values for »1 and »2 ). Starting from an initial condition for the price level

(given by its historical value in the initial quarter), one can then simulate (2.10) to obtain

a predicted time path for the price level, given the observed path of unit labor costs (and of

forecasted future labor costs). One can then compare this prediction to the actual path of

34

Note that according to the model developed here, average marginal supply cost should be proportional

to average unit labor cost under the assumption of a production technology with a constant elasticity of

output with respect to the labor input (e.g., the familiar Cobb-Douglas speci¬cation). See Rotemberg and

Woodford (1999b) for further discussion of this common measure of marginal cost and alternatives.

48 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

0.04

0.03

0.02

0.01

0

’0.01

’0.02

actual

’0.03 predicted

’0.04

1965 1970 1975 1980 1985 1990 1995

Figure 3.5: Actual path of price/ULC ratio (quarterly U.S. data, reported as log deviation

from mean) compared to prediction of the Calvo pricing model. Source: Sbordone (2002).

the aggregate price level over the same period.

Note that in the case of ¬‚exible prices (the ξ ’ ∞ limit of the above model), the predicted

price series would simply be given by log Pt = log µ + log St , so that predicted in¬‚ation would

be given by the percentage growth in nominal unit labor costs from one quarter to the next.

As shown in Figure 3.4, this would be quite a poor explanation of actual U.S. in¬‚ation; in

particular, one observes that in¬‚ation has been much less volatile than the growth rate of

unit labor costs. Alternatively, as shown in Figure 3.5, there has been substantial variation

in the ratio of price to unit labor cost from year to year, whereas the model implies that

this should be constant. In the case of sticky prices (ξ ¬nite), the model implies less volatile

in¬‚ation, and in fact, for a certain value of ξ, it predicts a path for the price level quite

similar to its actual path. Figure 3.5 compares the actual path of the (demeaned, log)

price/ULC ratio to the model™s prediction in the case that ξ = .055.35 The ¬t is quite good;

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 49

the mean squared error of the predicted price/ULC series is only 12 percent as large as the

variance of the actual price/ULC series. And even this statistic fails to emphasize the extent

to which the model succeeds in explaining the timing of quarterly changes in the ratio; the

discrepancy between the predicted and actual series is mainly at quite low frequencies. Figure

3.6 similarly compares the actual path of U.S. in¬‚ation (measured by growth in the GDP

de¬‚ator) with the path predicted by the model. The ¬t is again quite good (which is another

way of seeing that the discrepancy in Figure 3.5 is almost entirely at low frequencies).36

These results provide persuasive evidence for price stickiness of roughly the sort implied

by the Calvo pricing model. The degree to which the model ¬ts U.S. in¬‚ation dynamics is

perhaps surprising, given that the assumption of a ¬xed probability of price change for all

suppliers has been chosen for analytical convenience rather than out of any belief that it

ought to be realistic. Probably this re¬‚ects the fact that, given the small value of ξ implied

by Sbordone™s estimates (and hence the high degree of smoothing of marginal cost in the

price dynamics), the details of the distribution of intervals between price changes does not

matter much for the evolution of the aggregate price index, but only the average rate at

which prices are revised. But the ¬t of the model does clearly depend upon the existence of

staggered price changes of the kind assumed by Calvo. The “New Classical” pricing model

considered above, for example, would imply that ¬‚uctuations in the ratio of price to marginal

cost should be unforecastable, so that the predicted P/ULC series would necessarily exhibit

no serial correlation. The signi¬cant serial correlation of the P/ULC series shown in Figure

3.5, and the closely similar degree of serial correlation of the series predicted by the Calvo

35

Sbordone selects this value as her estimate, on the ground that it minimizes the mean-squared error of

the model™s predicted path for log Pt , when β is assigned a value of 1. Note that she actually reports the

value of ξ ’1 = 18.3.

36

Gali and Gertler (1999) obtain similar results for U.S. in¬‚ation, using an alternative, instrumental-

variables strategy for estimating equation (2.8). Gali and Gertler ¬nd that they can statistically reject this

baseline pricing model in favor of a generalization in which some price-setters use a backward-looking “rule of

thumb”, discussed further below in section xx. However, they ¬nd that the baseline model already explains

historical in¬‚ation dynamics quite well, as Figure 3.6 shows; and the validity of the standard errors used to

determine that the rejection of the baseline model is statistically signi¬cant might also be questioned. Batini

et al. (2000) similarly ¬nd that the Calvo pricing model can explain U.K. in¬‚ation dynamics, while Gali et

al. (2001) obtain similar results for several European countries.

50 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

10

actual

8 predicted

6

4

2

0

’2

’4

’6

1965 1970 1975 1980 1985 1990 1995

Figure 3.6: Actual path of in¬‚ation (quarterly U.S. data) compared to prediction of the

Calvo pricing model. Source: Sbordone (2002).

model,37 su¬ces to show that the kind of price stickiness allowed in that model cannot

similarly explain the U.S. data. Perhaps more surprisingly, Sbordone shows that the U.S.

price index cannot be ¬t nearly as well by any purely backward-looking moving average of

unit labor costs, either; the kind of smoothing re¬‚ected in the ¬gures depends importantly

upon the presence of the forward-looking terms on the right-hand side of (2.10), and not

simply upon the partial-adjustment term.38

One might, of course, doubt the accuracy of the simple unit-labor-cost measure of

marginal cost, for various reasons that are reviewed by Rotemberg and Woodford (1999b).

37

The autocorrelation functions of both series, with standard errors for the data series, are presented in

Sbordone (1998, 2002).

Sbordone shows this formally by separately estimating the weights »1 and »’1 in (2.10), without imposing

38

2

the restriction that they correspond to the two roots of (2.11). She ¬nds not only that both coe¬cients are

signi¬cantly positive, but that the values that yield the best ¬t are nearly equal in size, as the Calvo pricing

model would imply. (Note that the two roots of (2.11) necessarily satisfy »1 »2 = β ’1 .)

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 51

But it seems fortuitous that such a simple model of price-setting can explain actual in¬‚ation

dynamics so well, if unit labor cost is not in fact a fairly accurate measure of variations

in marginal cost, at least at moderate frequencies.39 In fact, Sbordone experiments with a

number of possible corrections to her simple measure of marginal cost, and ¬nds that none of

them improves the ¬t of the pricing model. One might also wonder whether forecasts based

on a VAR model with constant coe¬cients ¬t to the entire period 1960-97 should correspond

very closely to people™s expectations, even assuming that their expectations were rational,

on the ground that the dynamics of unit labor costs need not have been constant over this

period. (One might think, for example, that in¬‚ation dynamics have been substantially

di¬erent since the disin¬‚ation of the early 1980s; but this would imply that unit labor cost

dynamics should have been di¬erent as well.) But, once again, the close ¬t of the model,

and the fact that it ¬ts equally well both before and after the 1979-82 period, suggest that

the hypothesis of a common (and at least roughly unbiased) forecasting rule over the entire

period is not too inaccurate.

2.2 A “New Keynesian” Phillips Curve

We can use the model of optimal price-setting just derived to obtain an aggregate supply

relation ” i.e., a structural relation between in¬‚ation dynamics and the level of real activity

” by adjoining to our above pricing equation a theory of how real marginal costs depend

upon the level of real activity. The simple model used earlier in this chapter (recall equation

(1.14)) implies that

ˆ ˆ

st = (ω + σ ’1 )(Yt ’ Ytn ).

ˆ (2.12)

Substituting this into (2.8) then yields an aggregate supply relation of the form

ˆ ˆ

πt = κ(Yt ’ Ytn ) + βEt πt+1 , (2.13)

39

Because of the degree of smoothing involved in (2.10), high-frequency error in the measure of marginal

cost will have relatively little e¬ect upon the predicted path for prices. And Figure 3.5 itself indicates at

least a small amount of low-frequency speci¬cation error, which might be due to low-frequency error in the

measure of marginal cost.

52 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

where

(1 ’ ±)(1 ’ ±β) ω + σ ’1

’1

κ ≡ (ω + σ )ξ = > 0.

± 1 + ωθ

Alternatively, the slope coe¬cient can be written as

(1 ’ ±)(1 ’ ±β)

κ= ζ, (2.14)

±

where ζ is de¬ned in (1.22). This is what Roberts (1995) calls the “New Keynesian Phillips

Curve,” as this speci¬cation, or one similar to it, is implied by a variety of simple models

of optimal price-setting.40 From (2.14) we observe once again that the short-run Phillips

curve is ¬‚atter (for any given in¬‚ation expectations) the smaller the value of ζ, and thus the

greater the degree of strategic complementarity in price-setting. It is also ¬‚atter the larger

is ±, which is to say, the longer the average time interval between price changes.

Our aggregate supply relation again has the form of an expectations-augmented Phillips

curve, but now the in¬‚ation expectations that shift the curve are current expectations of

future in¬‚ation, rather than past expectations of the current in¬‚ation rate, as in (1.23). The

di¬erence turns out to be crucial for the model™s ability to allow for forecastable ¬‚uctuations

ˆˆ

in the output gap. It is now possible for non-zero values of Yt ’ Ytn to be forecasted at some

earlier date t ’ j; this simply requires that Et’j πt not equal βEt’j πt+1 .

2.3 Persistent Real E¬ects of Nominal Disturbances

We return now to the question of the persistence of the real e¬ects of disturbances to nominal

spending. Once again, we assume a given stochastic process for aggregate nominal spending

Yt , and consider what processes for Pt and Yt are then implied by the aggregate supply

relation (2.13). As a simple example, suppose that an unexpected disturbance permanently

increases log Y by a unit amount at date zero, which is then expected to maintain the higher

40

Notably, the same form of aggregate supply relation, up to our log-linear approximation, is implied

by a model with convex costs of price adjustment, as shown by Rotemberg (1987). Note that in Roberts™

presentation of the “New Keynesian Phillips Curve,” the discount factor β is set equal to one. This simpli-

¬cation may seem appealing, in that it implies a vertical “long-run” in¬‚ation-output tradeo¬. But correctly

accounting for the presence of the discount factor in (2.13) has important consequences for the analysis of

optimal policy, as is shown in chapters 6 and 7.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 53

value. Thus

E0 log Yt = 1 (2.15)

for all t ≥ 0, where log Yt is measured as a deviation from the constant level that would

have been expected in the absence of the disturbance. Let us consider the expected paths

E0 log Pt and E0 log Yt that are consistent with (2.13), given the initial condition log P’1 = 0,

where log Pt and log Yt are also measured as deviations from the levels that would have been

maintained but for the shock. We assume that the shock (a purely nominal disturbance)

involves no change in the constant expected level for the natural rate of output.

Let pt denote E0 log Pt ’1, the extent to which the expected price-level response in period

˜

t di¬ers from exact proportionality to the permanent increase in nominal spending. Then

we must have E0 log Yt = ’˜t for each horizon t ≥ 0; substituting this into (2.13), we ¬nd

p

that the price-level responses must satisfy

pt’1 ’ (1 + β + κ)˜t + β pt+1 = 0

˜ p ˜ (2.16)

for each t ≥ 0, starting from the initial condition p’1 = ’1. The unique bounded solution is

˜

easily seen to be given by

pt = ’»t+1 ,

˜ (2.17)

where 0 < » < 1 is the smaller of the two real roots41 of the characteristic equation

P (») ≡ β»2 ’ (1 + β + κ)» + 1 = 0. (2.18)

Since E0 log Pt = 1 + pt , we observe that the log price level is expected to rise monotonically,

˜

asymptotically reaching a level proportional to the increase in nominal spending. Since

E0 log Yt = ’˜t , we observe that output increases in the period of the disturbance, by an

p

amount less than (though possibly close to) proportional to the increase in nominal spending,

then decays monotonically back to its original level. (These impulse responses of the price

level and of output are plotted in Figure 3.7, for the illustrative parameter values β = .99

and κ = .024.42 )

54 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

1.5

1

0.5

0

nominal GDP

real output

price level

’0.5

0 5 10 15 20

Figure 3.7: Impulse responses to an immediate permanent increase in nominal GDP.

The degree of persistence of the e¬ect upon real activity is thus dependent upon the size

of the root »; the closer » is to one (its theoretical upper bound), the longer the time it takes

for real activity to return to its “potential” level following a nominal disturbance. (Note that

» also determines the size of the initial e¬ect upon real activity in period zero, as well. Thus

the same factors that increase the amplitude of the e¬ect upon real activity ” by making

aggregate price adjustment more sluggish ” also make the e¬ect more persistent.) It is

easily seen, in turn, that the smaller root of P (») varies inversely with the size of κ, with a

value that approaches zero in the case of very large κ, and a value that approaches one for κ

near zero. Thus a small value of κ (a ¬‚at short-run Phillips curve) is required for signi¬cant

persistence. This in turn could occur either as a result of ± being near one (infrequent price

41

Since P (0) > 0, P (1) < 0, P (β ’1 ) < 0, and P (») > 0 for all large enough » > 0, it is evident that the

equation has two real roots, 0 < » < 1 and another root greater than β ’1 .

42

These values are taken from the estimates of Rotemberg and Woodford (1997), discussed further in

chapter 4.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 55

1

0.9

0.8

0.7

ζ = 0.1

0.6

ζ = 0.5

0.5 ζ=1

»

ζ=2

0.4

ζ = 10

0.3

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

±

Figure 3.8: Persistence of real e¬ects of an increase in nominal spending as a function of

the frequency of price adjustment, for alternative degrees of strategic complementarity in

price-setting.

changes) or ζ being small (strong strategic complementarity).

It is perhaps most interesting to observe that the degree of strategic complementarity

can have a considerable e¬ect upon the degree of persistence, holding ¬xed the frequency of

price changes. Note that for any value of ±, the value of κ can be made arbitrarily small,

or arbitrarily large, through assignment of an appropriate value to ζ. The e¬ects of ± and

ζ upon the degree of persistence are shown quantitatively in Figure 3.8, where the implied

value of » is plotted as a function of these two parameters, assuming the value β = .99.

One observes that » > ± if and only if ζ < 1, i.e., if and only if pricing decisions are

strategic complements in the sense discussed above. This can be shown analytically by

observing that

P (±) = (1 ’ ±)(1 ’ ±β)(1 ’ ζ),

56 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

where we have used (2.14) to substitute for κ. Thus P (±) > 0, so that ± < » < 1, if and

only if ζ < 1. It is in this case that the rate of adjustment of the aggregate price index is

slower than would be expected mechanically as a result of the fact that not all prices have

yet had an opportunity to be updated since the occurrence of the shock. (The fraction of

prices in period t that have not yet been adjusted even once since the shock is given by ±t .)

In this case, Taylor (1980) speaks of the existence of a “contract multiplier” as a result of the

staggering of price adjustments.43 Such a multiplier depends upon the existence of strategic

complementarity among di¬erent suppliers™ pricing decisions, so that the fact that other

prices have not fully adjusted makes an individual supplier adjust its own price less. In the

case of strategic substitutes (ζ > 1), there is actually even less persistence than one would

expect for purely mechanical reasons (» < ±), because the ¬rst prices that are adjusted

actually over-adjust in reaction to the failure of other prices to adjust in proportion to the

increase in nominal spending.

In fact, the connection between the value of ζ and the value of κ indicated by (2.14)

continues to hold when additional sources of strategic complementarity are introduced, such

as non-CES preferences over di¬erentiated goods, or intermediate inputs. If we generalize

our assumptions about demand and production costs in both of these directions, as in section

xx above, nominal pro¬ts are instead of the form

Πi (p) = [pYt d(p/Pt ) ’ wt (i)f ’1 ((1 ’ sm )d(p/Pt )Yt /At ) ’ sm Pt Yt d(p/Pt ),

t

generalizing (2.3). Here d(pt (i)/Pt ) is the relative demand ct (i)/Ct implied by (1.27). The

corresponding ¬rst-order condition for optimal price-setting by a ¬rm that chooses its price

at date t is then

∞

±T ’t Qt,T YT d(pt (i)/PT )[pt (i) ’ µT (i)ST (i)] = 0,

Et

T =t

generalizing (2.4). Here the desired markup is given by µT (i) = µ(d(pt (i)/PT )), where the

function µ(x) is again de¬ned by (1.30), and marginal cost is now given by (1.35). As before,

43

In Taylor™s original formulation, it is actually the staggered negotiation of wage contracts that gives rise

to the “multiplier”.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 57

we may log-linearize this to obtain

∞

(±β)T ’t {ˆt,T ’ [ˆt,T + st,T ]} = 0,

Et p µ ˆ (2.19)

T =t

generalizing (2.7). Here we introduce the notation

T

p—

pt,T =

ˆ ˆt ’ π„

„ =t+1

for the (log) relative price at date T of a ¬rm that has set its price at date t, yt,T for its (log)

ˆ

output relative to steady state, and

ˆ

µt,T =

ˆ µ (ˆt,T

y ’ YT )

for its (log) desired markup relative to steady state.

Noting that

ˆ ˆ ˆ

+ sy )(ˆt,T ’ YT ) ’ (sy + sY )(YT ’ YTn )

pt,T ’ [ˆt,T + st,T ] = pt,T ’ (

ˆ µ ˆ ˆ y

µ

ˆ ˆ

+ sy )]ˆt,T ’ (sy + sY )(YT ’ YTn ),

= [1 + θ( p

µ

we observe that (2.19) can equivalently be written

∞

ˆ ˆ

(±β)T ’t p— ’ ζ(Yt ’ Ytn ) = 0,

Et ˆt (2.20)

T =t

where ζ is now given by (1.37). This in turn allows once again to derive an aggregate supply

relation of the form (2.13), where κ is again given by (2.14). Thus it continues to be the

case that the degree of strategic complementarity is a crucial determinant of the degree of

persistence of the real e¬ects of an increase in nominal spending; each of the factors discussed

above that increase strategic complementarity increases persistence.

Can staggering of price changes give rise to an empirically realistic degree of persistence,

assuming an empirically realistic average interval between price changes? Chari, Kehoe and

McGrattan (2000) argue that it cannot. However, their conclusion depends both upon an

exaggeration of the size of the “contract multiplier” that would be needed, and an under-

estimate of the empirically plausible degree of strategic complementarity. They de¬ne the

58 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

contract multiplier as “the ratio of the half-life of output deviations after a monetary shock

with staggered price-setting to the corresponding half-life with synchronized price-setting”

(p. 1152). In our analysis above of the response of output under Calvo price-setting to

a monetary shock that permanently increases the level of nominal GDP,44 we have shown

that the level of output returns to its steady-state level following a shock as »t , where »

is the smaller root of (2.18). Thus the “half-life” with staggered price setting would equal

log 2/ log »’1 periods.

By the corresponding half-life “with synchronized price-setting”, Chari et al. mean the

case in which each price that is revised after the shock occurs is immediately adjusted

all the way to the new expected long-run price level, so that the fraction of the eventual

aggregate price adjustment that has occurred at any time is equal to the fraction of prices

that have been revised at least once since the occurrence of the monetary shock. In the case

of the Calvo pricing model, the fraction of prices that have not yet been adjusted k periods

following a shock is ±k+1 , so that the “half-life with synchronized price-setting” would equal

log 2/ log ±’1 .45 The contract multiplier implied by the Calvo pricing model is then equal to

log ±’1 / log »’1 . It follows from our results above that the multiplier is greater than one if

and only if ζ < 1, which is to say if and only if there is strategic complementarity in the

pricing decisions of di¬erent suppliers.46 In fact, in the continuous-time limit of the Calvo

44

This is not exactly the form of monetary shock considered by Chari et al., who instead assume an

exogenous process for the money supply, and consider its implications within a complete general-equilibrium

model of the monetary transmission mechanism. However, in their analysis of a “stripped-down version” of

their baseline model, they assume a static money demand function according to which nominal GDP is at

all times proportional to the money supply, and also assume a random walk for the (log) money supply; this

is thus an experiment of exactly the kind we consider here.

45

Chari et al. instead assume ¬xed-length price commitments, as a result of which this half life is equal

to N/2 periods, where is the number of periods that each price remains ¬xed.

46

As noted in the previous two footnotes, Chari et al. consider a di¬erent model of staggered pricing, but

obtain this same conclusion; their ¬nding that the multiplier is necessarily less than one in their “stripped-

down” model follows from the fact that ζ is necessarily greater than one, for the reasons that we have

discussed above. (They call this parameter γ; their equation (34) shows that it must exceed one.) In their

analysis, as here, the question of whether a contract multiplier much greater than one is possible amounts

largely to a consideration of whether ζ can plausibly be much less than one.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 59

model, one can show that the contract multiplier is equal to

2a—

, (2.21)

[4a— (a— + 1)ζ + 1]1/2 ’ 1

where a— is the ratio of the (continuous) rate of price adjustment to the (continuous) rate

of time preference, independent of the values of the other parameters.47 In the case that

a— >> 1, this is approximately ζ ’1/2 , regardless of the size of a— ; this provides an analytical

explanation for the ¬nding of Chari et al. that the predicted contract multiplier in a model

with staggered price-setting depends little upon the assumed length of time between price

changes.

Chari et al. argue that a very large multiplier would be needed in order for staggered

price-setting to account for observed persistence. They ¬t a univariate ARMA model to de-

trended real GDP for the postwar U.S., and conclude that the half-life of output ¬‚uctuations

around trend is approximately 10 quarters. At the same time, they argue that a reasonable

length of time to assume that prices are ¬xed would be only one quarter, implying a half-life

with synchronized price-setting of only half a quarter. Thus they argue that a contract mul-

tiplier of 20 would be needed; this would be possible only if ζ were quite small (.004 in the

case of our continuous-time limit).48 Instead, they ¬nd that their baseline parameter values

imply a value of ζ well above one, and so a contract multiplier less than one; and while

alternative parameter values can raise the multiplier somewhat, they argue that it cannot

plausibly be greater than two. They conclude that one must believe that prices remain ¬xed

for many years in order to account for observed persistence.

However, this way of identifying the persistence of the output e¬ects of monetary shocks

assumes that all ¬‚uctuations of output around a deterministic trend path are due to monetary

shocks. There is no reason to assume this; indeed, a central contention of this study is

47

Here we ¬x ζ and let ± and β decrease with ∆, the period length, so that as ∆ ’ 0, log β ’1 /∆

and log ±’1 /∆ both approach positive constants, ρ and a respectively. Then, expanding the characteristic

polynomial (2.18) in powers of ∆, one can show that log »’1 /∆ approaches a positive constant n as well,

which depends upon ζ and the ratio a— ≡ a/ρ. The multiplier (2.21) is then equal to a/n.

48

This value follows from (2.21) under the assumption that a— = 33, a value suggested by survey evidence

on the frequency of price change in the U.S. economy, as discussed below. Note, however, that a similarly

small value of ζ is required in the case of any large value for a— .

60 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

that the task of monetary policy is to respond appropriately to a large variety of types of

real disturbances to which economies are subject. One may attribute great importance to

monetary policy without holding that any large fraction of the overall variability of output

is due (in the sense of ultimate causation) to the random component of monetary policy; for

systematic monetary policy can greatly change the e¬ects of real disturbances. For example,

the estimated model of Rotemberg and Woodford (1997), discussed in section xx of chapter

4, attributes only a few percent of the overall variance of real GDP around trend over the

sample period to monetary policy shocks; yet the counter-factual simulations reported in

that paper show that alternative systematic monetary policies would have implied greatly

di¬erent paths of real as well as nominal variables.

One needs, then, to identify the real e¬ects of monetary policy shocks alone in order

to determine the relevant half-life. This may be much shorter than the one estimated by

Chari et al., without any implication that monetary policy is unimportant for business

¬‚uctuations. For example, the impulse response reported in Figure 3.3 above shows that

only three quarters after the peak output response, the level of (log) output has already

returned halfway to the level that would have been expected prior to the shock; and two

quarters after that, the response has fallen to only 20 percent of its peak level. This suggest a

“half-life” of only 2.5 to 3 quarters. The structual VAR of Rotemberg and Woodford (1997),

discussed in chapter 4, yields an output response that involves more nearly exponential decay

after the quarter in which output is ¬rst substantially a¬ected; this response, shown in Figure

xx below, exhibits a “half-life” of only about 3 quarters.49

Furthermore, survey evidence indicates that many prices remain unchanged for longer

than a quarter on average. For example, the survey of Blinder et al. (1998) indicates an aver-

age time between price changes (for a representative sample of U.S. ¬rms) of 9 months.50 This

49

As shown in Figure xx, the peak (and ¬rst signi¬cant) output contraction occurs two quarters following

a contractionary policy shock; log output has returned about half-way to its trend level by the ¬fth quarter

following the shock. Nonetheless, the model of Rotemberg and Woodford is fully consistent with the ob-

served degree of persistence of the deviations of output from trend; compare the predicted and estimated

autocorrelation functions reproduced in Figure xx below.

50

See Rotemberg and Woodford (1997) for discussion of other survey evidence from studies of smaller

sectors of the economy.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 61

suggests that the continuous rate of price change should be parameterized as approximately

0.33/quarter, implying a half-life of 2.08 quarters. Thus the required contract multiplier is

only 1.44, within the range that Chari et al. ¬nd to be possible. In the continuous-time limit

of the Calvo pricing model derived above,51 this requires only that ζ = 0.49, and (as shown

in Table 3.1 above) a value this low or even lower is easily consistent with the assumptions

about preferences and technology made by Chari et al., once we allow for non-CES prefer-

ences and intermediate inputs, or alternatively for at least some degree of factor speci¬city.52

A larger value of σ makes this even easier, and as we shall argue in chapter 4, it is most

reasonable to calibrate this model with a value of σ much larger than one.

In fact, our discussion above has indicated that a variety of plausible assumptions can

justify a value of ζ in the range of 0.10-0.15. This would imply a contract multiplier (in

the continuous-time limit of our model) in the range of 2.6 to 3.3, or a “half-life” for the

output response (assuming the rate of price adjustment indicated by the survey evidence)

between 5.5 and 6.8 quarters. This is a degree of persistence of the e¬ects on real activity of

an increase in nominal expenditure considerably greater than the one indicated by the VAR

evidence cited by Rotemberg and Woodford, but is comparable to the degree of persistence

indicated by some other studies of the e¬ects of identi¬ed monetary shocks.53

51

Here we use (2.21), assuming that a = 0.33/quarter and ρ = 0.01/quarter, so that a— = 33.

52

In fact, the model of Rotemberg and Woodford is not able to reproduce their estimated impulse response

function for real GDP in response to a monetary policy shock without a considerably smaller value of ζ,

approximately 0.13. The di¬erence is obtained because their model is a discrete-time model with “periods” of

length three months (which results in a degree of persistence somewhat di¬erent than that in the continuous-

time limit), because they set ± = 2/3 (so that exactly one-third of all prices are changed each quarter, rather

than the fraction 1 ’ e’a = 0.28 implied by integration of the continuous-time model), and because their

identi¬ed monetary policy shock does not result in an immediate, permanent increase in nominal GDP as

assumed above.

53

The real e¬ects of the identi¬ed monetary policy shocks obtained by Rotemberg and Woodford (1997)

are less persistent than those obtained in some other VAR studies, mainly using longer sample periods. For

example, in the baseline results of Christiano et al. (1999), using quarterly data and identifying monetary

policy shocks with innovations in the federal funds rate (¬rst column of their Figure 2), the peak contraction

in real GDP following an unexpected monetary tightening occurs only ¬ve to six quarters after the shock,

while output has returned half-way to its original trend path by the eleventh quarter following the shock;

this would indicate a “half-life” of ¬ve to six quarters. The later results of Christiano et al. (2001) indicate

a half-life of less than three quarters, however, as noted above.

62 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

2.4 Consequences of Persistence in the Growth of Nominal Spend-

ing

We have thus far considered only the response of real activity to an unexpected, permanent

increase in nominal spending. While this particular thought experiment allows us a clear

de¬nition of the degree of persistence, such a perturbation of the expected path of nominal

spending has little similarity to the estimated responses of nominal spending to the monetary

policy shocks identi¬ed in the VAR literature; thus the model™s predictions for this case

cannot be directly compared to any empirical estimates. Identi¬ed monetary policy shocks

tend to a¬ect nominal spending only slightly (if at all) in the ¬rst few months, with an e¬ect

that increases cumulatively over a period of several quarters, eventually bringing expected

future nominal spending to a new permanent level.

As a simple case that allows for shocks of this kind, let us assume that the growth rate

of nominal spending follows a ¬rst-order autoregressive process,

∆ log Yt = ρ∆ log Yt’1 + (2.22)

t

with 0 < ρ < 1, where is an i.i.d. random variable, assumed for simplicity to have

t

mean zero. This process implies that an innovation increases the conditional expectation

t

Et log Yt+j by an amount (1 ’ ρ)’1 (1 ’ ρj+1 ) t which increases monotonically with j, asymp-

totically approaching a permanent e¬ect that is (1 ’ ρ)’1 > 1 times as large as the initial

e¬ect.54 Let us consider the impulse response to a positive innovation = 1 ’ ρ, which

0

results in a unit increase in the expected long-run level of nominal spending. Again letting

pt denote E0 log Pt ’ 1, we ¬nd that the price-level responses must satisfy

˜

pt’1 ’ (1 + β + κ)˜t + β pt+1 = κρt+1 ,

˜ p ˜ (2.23)

a generalization of (2.16), again starting from initial condition p’1 = ’1. (Because nominal

˜

spending does not immediately jump to its expected long-run value, we now have E0 log Yt =

54

This form of stochastic process is used as a rough approximation of actual U.S. time series in Rotemberg

(1996).

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 63

[E0 log Yt ’ 1] ’ pt = ’(˜t + ρt+1 ).) The unique bounded solution to (2.23) is given by the

˜ p

solution to the ¬rst-order equation,

κ

ρt+1 ,

pt = »˜t’1 ’

˜ p ’1 ’ βρ

»

again starting from the given value for p’1 , where 0 < » < 1 is again de¬ned as in (2.17).

˜

(Note that in the limiting case ρ = 0, this reduces to the same solution (2.17) as before.)

Then using the relations

E0 log Pt = pt + 1,

˜

E0 log Yt = ’(˜t + ρt+1 ),

p

we can compute the impulse responses of the price level and of aggregate output to an inno-

vation in the nominal spending process. These are plotted in Figure 3.9, for the illustrative

parameter values β = .99, κ = .024, and ρ = .4, along with the impulse response of nominal

spending itself.55 We observe that in this case innovation in nominal spending results in an

increase in real activity, that continues to grow larger, reaching its peak only in the second

and third quarters after the shock, and then declines monotonically back to its original level.

This “hump-shaped” output response is predicted in the case of any large enough value of

ρ, speci¬cally, for any value such that » + ρ > 1. The responses yt ≡ E0 log Yt satisfy

˜

(1 ’ ρL)(1 ’ »L)˜t = 0

y

for all t ≥ 1, starting from initial values y’1 = 0 and

˜

(1 ’ ρ)(1 ’ βρ)

y0 =

˜ > 0.

(»’1 ’ βρ)

A second-order di¬erence equation of this kind has a solution that either decays monotoni-

cally back to zero after a peak in the period of the shock, or that ¬rst increases to a peak,

and then decays monotonically back to zero. Since y2 = (» + ρ)˜1 , the second case occurs if

˜ y

and only if » + ρ > 1.

55

The ¬gure also shows the impulse response of the in¬‚ation rate, here measured in annual percentage

points, so that the in¬‚ation rate plotted is de¬ned as 4 ∆ log Pt . The dynamics of the in¬‚ation rate are

discussed further in section xx below.

64 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

1

0.8

nominal GDP

real GDP

price level

0.6

inflation

0.4

0.2

0

’0.2

’2 0 2 4 6 8 10 12 14 16 18 20

Figure 3.9: Impulse responses to an innovation in nominal GDP, in the case of persistence

in nominal GDP growth.

The reason for such an equilibrium response is simple. Even though prices do not change

much immediately following the shock, the increase in output is not initially large, because

nominal spending has not yet increased much. As nominal spending increases further, real

output increases as well. Eventually, prices adjust, and real activity falls back to its original

level (or, in a model with trend output growth, back to its original trend).

The “hump-shaped” output response shown in Figure 3.9 is at least roughly of the kind

typically estimated using structural VAR methodology (see, e.g., Figure 3.3 above). Indeed,

Cochrane (1996) ¬nds that a slightly modi¬ed version of the “New Keynesian” aggregate

supply relation (2.13) can more closely match the estimated response of output to a monetary

policy shock than any of the other simple aggregate supply relations that he considers.56 A

particular advantage of this speci¬cation is that it can allow signi¬cant output e¬ects of

56

The form used by Cochrane is actually equation (3.2), with a delay d = 1 quarter.

2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 65

a nominal disturbance even when there is little immediate e¬ect upon nominal spending.

This can be seen by considering the above calculation in the case that ρ is close to 1. In

such a case, there is little increase in nominal spending in period zero (log Y0 increases only

by 1 ’ ρ, yet the peak output e¬ect may be an arbitrarily large fraction of 1. (For small

enough κ, the value of » may be made arbitrarily close to 1, so that the adjustment of prices

is slow compared to the rate at which expected nominal spending approaches its long-run

value.) This contrasts sharply with the prediction of the “New Classical” aggregate supply

relation (1.23), which implies that the peak output e¬ect (which must occur in period zero)

is bounded above by 1 ’ ρ, no matter how ¬‚at the short-run Phillips curve may be.

2.5 Consequences of Sectoral Asymmetries

Thus far we have considered only a completely symmetric model, in the sense that all pref-

erences and production relations would remain the same if we interchanged the labels of

di¬erent goods. In particular, we have assumed that all real disturbances a¬ect supply and

demand conditions for all goods in exactly the same way: technical progress lowers the cost

of producing each good in exactly the same proportion, and so on. In reality, of course, there

are many kinds of disturbances that di¬erentially a¬ect various sectors of the economy. Here

we brie¬‚y consider an extension of the basic model with staggered pricing that allows for

several kinds of asymmetries. One reason for interest in this extension is that in the presence

of such asymmetries, it is no longer generally the case that stabilization of an aggregate price

index and stabilization of an aggregate output gap are equivalent policies (an important but

obviously special implication of the “New Keynesian” Phillips curve (2.13)). Allowing for

sectoral asymmetries is also especially important in analyzing monetary policy for an open

economy (where, for example, one will want to consider the consequences of shocks that

a¬ect the terms of trade), though we do not develop that extension here.57

Instead of assuming that the consumption index Ct that enters the utility function of the