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22
Note that the right-hand side of (1.36) is the the reciprocal of the steady-state вЂњvalue-added markupвЂќ
or ineп¬ѓciency wedge resulting from market power that Rotemberg and Woodford (1995) contrast with
the steady-state вЂњgross-output markupвЂќ or п¬Ѓrm-level markup of price over marginal cost Вµ. 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 Оё = 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 Оё = 10
would imply that О¶ = 0. But such a calibration is not easy to interpret. If, for example, one proposes that a
large fraction of primary input purchases represent п¬Ѓxed or вЂњoverheadвЂќ 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 О¶ under alternative assumptions.

Пѓ в€’1 = 1, П‰ = 1.25 Пѓ в€’1 = .16, П‰ = .47
ОёВµ Вµsm 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 О¶ is reduced by a factor of 2.5 (i.e., it is multiplied by 0.4). This is again
a signiп¬Ѓcant reduction, and if this eп¬Ђect is combined with the degree of variation in the
desired markup argued for above, the value of О¶ is only one-п¬Ѓfth the size indicated by the
baseline calculation of Chari et al. вЂ” 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 speciп¬Ѓc factor markets, this correction
matters much less, unless sm is assumed to be much closer than Вµв€’1 than seems realistic.
Using the calibration discussed above, assuming Вµsm = 0.6 rather than zero reduces О¶ 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 diп¬Ђerence in oneвЂ™s conclusions about the degree
of strategic complementarity.

Our conclusions about the eп¬Ђects 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 О¶ under the assumption that П‰ = 1.25, Пѓ = 1, and Оё = 10, as assumed by Chari et al.,
under eight diп¬Ђerent cases, representing three binary choices. These are the assumption of
homogeneous factor markets (sy = 0, sY = П‰ + Пѓ в€’1 ) versus speciп¬Ѓc factor markets (sy = П‰,
sY = Пѓ в€’1 ); the assumption of a constant desired markup ( = 0) versus a realistic degree
Вµ

of markup variation (Оё mu = 1); and the assumption of no intermediate inputs (sm = 0)
versus a realistic intermediate input share in costs (Вµsm = 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 О¶ substantially
1. A BASIC STICKY-PRICE MODEL 37

relative to their baseline case. However, it is the allowance for factor speciп¬Ѓcity that matters
most, if one accepts that this is the more realistic assumption. For while each of the other
two corrections makes a signiп¬Ѓcant diп¬Ђerence in the case of homogeneous factor markets, and
together they make an even larger diп¬Ђerence (as stressed by Bergin and Feenstra), in the
case of speciп¬Ѓc factor markets even their combined eп¬Ђect is not at all dramatic. Furthermore,
factor speciп¬Ѓcity alone reduces the value of О¶ 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 oп¬Ђers a similar comparison, when instead lower values are
assumed for both Пѓ в€’1 and П‰. The values suggested here are those obtained by Rotemberg
and Woodford (1997) when a slightly more complicated version of this pricing model is
п¬Ѓt to U.S. time series, as discussed further in chapter 4. The higher value of Пѓ (implying
much greater interest-sensitivity of private expenditure) is needed in order to account for the
observed size of the eп¬Ђects of an identiп¬Ѓed monetary policy shock on real aggregate demand;
the lower value of П‰ (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, О¶ < 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 О¶ = .13,
if we make realistic assumptions about the other two sources of real rigidity. Thus a value
of О¶ 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 Inп¬‚ation Dynamics with Staggered Price-Setting

An unsatisfactory feature of the вЂњNew ClassicalвЂќ aggregate supply relation derived above
is its implication that only unanticipated п¬‚uctuations in nominal spending have any eп¬Ђect
upon real activity, and that equilibrium п¬‚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., Sheп¬Ђrin, 1996, chap. 2). They imply
that only the immediate eп¬Ђects of a monetary policy shock upon nominal expenditure should
have any consequences for real activity; delayed eп¬Ђects (eп¬Ђects on nominal spending after
the вЂњperiodвЂќ in which the shock occurs) should not aп¬Ђect output at all, only the price level.
And such real eп¬Ђects of a monetary policy shock as occur must be purely transitory, i.e.,
must last no longer than the вЂњperiodвЂќ for which the sticky prices are п¬Ѓxed in advance.
These predictions are quite inconsistent with the eп¬Ђects identiп¬Ѓed in the вЂњstructural VARвЂќ
literature. As an example, Figure 3.2 plots the impulse response of nominal GDP to an
identiп¬Ѓed monetary policy shock, according to the structural VAR model of Christiano et al.
(2001).24 There is practically no measurable eп¬Ђect 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 вЂњNew ClassicalвЂќ aggregate-supply relation, monetary shocks should
have no eп¬Ђect upon real activity at all, unless the вЂњperiodвЂќ for which sticky prices are п¬Ѓxed
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 inп¬‚ation and real GDP responses that are reported there. I thank Charlie Evans for supplying
the data plotted in this and subsequent п¬Ѓgures. 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 eп¬Ђect upon nominal GDP in quarter 0 is an artifact of the authorsвЂ™ iden-
tiп¬Ѓcation scheme, which assumes that any contemporaneous correlation between interest-rate innovations
and innovations in either real GDP or inп¬‚ation is due to feedback from the latter variables to the current
interest-rate operating target, as under a Taylor rule. However, the estimated eп¬Ђect in quarter 1 is in no
way constrained by the identiп¬Ѓcation scheme; the fact that neither output nor inп¬‚ation is estimated to be
signiп¬Ѓcantly aп¬Ђected in quarter 1 provides some support for the assumption relied upon in the identiп¬Ѓcation
scheme.
2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 39

3.5

3

2.5

2

1.5

1

0.5

0

в€’0.5

в€’1

в€’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 eп¬Ђect 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 eп¬Ђect
of the shock. (Note that in both п¬Ѓgures, the monetary policy shock is normalized so that
the long-run eп¬Ђect on nominal GDP is an increase of one percentage point.) Furthermore,
the eп¬Ђect occurs with a substantial delay; there is essentially no eп¬Ђect on output until the
second quarter following the policy shock,26 and the peak output eп¬Ђect occurs only in the
sixth quarter following the shock. The estimated eп¬Ђect 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 eп¬Ђect 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 eп¬Ђect 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

в€’0.5

в€’1

в€’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 eп¬Ђect on log real GDP in percentage points. Source:
Christiano, Eichenbaum and Evans (2001).

Such long-lasting eп¬Ђects are inconsistent with the вЂњNew ClassicalвЂќ aggregate-supply relation,
unless the вЂњperiodвЂќ for which prices are п¬Ѓxed in advance is longer than two years. But survey
evidence on price changes (e.g., Blinder et al., 1998) п¬Ѓnds that the majority of п¬Ѓrms 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 п¬Ѓrst two quarters following a shock.
This undesirable feature of the model can be avoided, however вЂ” 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 вЂ” by
assuming that the intervals over which the prices of diп¬Ђerent goods remain п¬Ѓxed overlap,
rather than being perfectly synchronized. As Phelps (1978) and Taylor (1979a, 1980) п¬Ѓrst
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 suppliersвЂ™ 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 signiп¬Ѓcant eп¬Ђect 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 eп¬Ђects 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 вЂњexpectations-augmented Phillips-curveвЂќ relation. However, in this
variant case, the kind of inп¬‚ation expectations that determine the location of the short-run
Phillips curve are current expectations regarding future inп¬‚ation, rather than past expecta-
tions regarding current inп¬‚ation. This might seem a small diп¬Ђerence (given the degree of
serial correlation in inп¬‚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 modiп¬Ѓcation, the model
becomes consistent with the occurrence of prolonged п¬‚uctuations in real activity following a
monetary policy shock, of at least roughly the kind estimated in the VAR literature.

27
These п¬Ѓrst applications of the idea assumed that wages, rather than prices, were п¬Ѓxed for a period
of time (owing to wage contracts), and thus concerned the eп¬Ђects 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 < О± < 1 of goods prices remain unchanged each period, while new prices are chosen for
the other 1 в€’ О± of the goods. For simplicity, the probability that any given price will be
adjusted in any given period is assumed to be 1 в€’ О±, independent of the length of time since
the price was set, and independent of what the particular goodвЂ™s current price may be.
These last assumptions are plainly unrealistic, but they are very convenient in simplifying
the analysis of equilibrium inп¬‚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 pв€—
t

is the same for each of them, and so in equilibrium, all prices that are chosen in period t
have the common value pв€— . The remaining fraction О± of prices charged in period t are simply
t

a subset of the prices charged in period t в€’ 1, with each price appearing in the period t
distribution of unchanged prices with the same relative frequency as in the period t в€’ 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
satisп¬Ѓes
1 1
Pt1в€’Оё в‰Ў pt (i)1в€’Оё di = (1 в€’ О±)pв€—1в€’Оё + О± ptв€’1 (i)1в€’Оё di,
t
0 0

so that
1

О±)pв€—1в€’Оё 1в€’Оё 1в€’Оё
Pt = (1 в€’ + О±Ptв€’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 pв€— 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 п¬Ѓrst use of a discrete-time version
of CalvoвЂ™s model of price-setting, in the context of a complete intertemporal equilibrium model of aggregate
п¬‚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

pв€— , in turn, depends upon current and expected future demand conditions for the individual
t

good, but (1.11) implies that other prices aп¬Ђect 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 periodвЂ™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
A supplier that changes its price in period t chooses its new price pt (i) to maximize
в€ћ
О±T в€’t Qt,T [О i (pt (i))] ,
Et (2.2)
T
T =t

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

О i (p) = [Yt PtОё p1в€’Оё в€’ wt (i)f в€’1 (Yt PtОё pв€’Оё /At ). (2.3)
t

Here the factor О±T в€’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 в‰Ґ t.
Corresponding to (1.18) we obtain in this case the п¬Ѓrst-order condition
в€ћ
О±T в€’t Qt,T YT PT [pt (i) в€’ ВµST (i)] = 0.
Оё
Et (2.4)
T =t

Thus there is again a sense in which the price pt (i) is set equal to Вµ 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 speciп¬Ѓc 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),
Лњ
aggregate output Yt , and the vector of aggregate shocks Оѕt . 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 О±T в€’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 deп¬Ѓned 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
в€ћ
Лњ
(О±ОІ)T в€’t uc (YT ; ОѕT )YT PT [pв€— в€’ ВµPT s(YT PT pв€—в€’Оё , YT ; ОѕT )] = 0
Оёв€’1 Оё
Et (2.5)
t t
T =t

as an equation that implicitly deп¬Ѓnes the optimal price pв€— . Note that the monotonicity of s
t

in its п¬Ѓrst argument implies that the left-hand side of this expression is increasing in pв€— , so
t

that the optimal price is uniquely deп¬Ѓned by this condition. The value of pв€— then determines
t

the evolution of the price index Pt through (2.1).
Once again, we may usefully approximate the equilibrium dynamics of inп¬‚ation in the case
Лњ
of small enough disturbances Оѕt by considering a log-linear approximation to these equations.
Лњ ВЇ
If Оѕt = 0 and Yt = Y at all times, equations (2.1) and (2.5) have a solution with zero inп¬‚ation,
Лњ
in which Pt = pв€— = Ptв€’1 each period. In the case of small enough п¬‚uctuations in Оѕt and Yt
t

around these values, we accordingly look for a solution in which Pt /Ptв€’1 and pв€— /Pt remain
t

always close to 1, though the (log) price level may contain a unit root.32 The п¬‚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 pв€— 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 simpliп¬Ѓed by log-linearizing around a steady state with a zero inп¬‚ation
rate, rather than some other constant inп¬‚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 inп¬‚ation
near zero at all times. This does not mean, however, that we can consider only policy rules that make the
average inп¬‚ation rate exactly zero, or that involve a вЂњtargetвЂќ inп¬‚ation rate of zero. It is only necessary
that the average inп¬‚ation rate be suп¬ѓciently small (technically, of order O(||Оѕ||), if we want the error in
our characterization of the evolution of inп¬‚ation and other variables to be of order O(||Оѕ||2 ), where ||Оѕ|| is a
bound on the size of the disturbances). As we shall argue (see chapter 6) that desirable policies do imply
a low average inп¬‚ation rate, this does not seem an inconvenient restriction, from the standpoint of our goal
of characterizing optimal policy. Characterization of the eп¬Ђects of policy in a high-inп¬‚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),
1в€’О± в€—
ПЂt = p,
Л† (2.6)
О±t
where pв€— в‰Ў log(pв€— /Pt ). They must also approximately satisfy a log-linear approximation to
Л†t t

(2.5), given by пЈ± пЈј
пЈІ пЈЅ
в€ћ T
(О±ОІ)T в€’t пЈіpв€— в€’ [Л†t,T +
Et Л†t s ПЂП„ ]пЈѕ = 0, (2.7)
П„ =t+1
T =t

where st,T denotes the deviation (from its steady-state value) of the log of real marginal cost
Л†
in period T for a supplier whose price is pв€— . (The term in square brackets in (2.7) is thus
t

the deviation of the log of
Лњ
(PT /Pt )s(YT PT pв€—в€’Оё , YT ; ОѕT )
Оё
t

from its steady-state value of Вµв€’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
producerвЂ™s own level of output, the variable st,T can furthermore be approximated by
Л†
T
П‰Оё[Л†в€—
st,T = sT в€’
Л† Л† pt в€’ ПЂП„ ],
П„ =t+1

where sT denotes the deviation of the log of the average level of real marginal cost (real
Л†
marginal cost for a good with output yT (i) = YT ) from its steady-state value. Substituting
this into (2.7), we can solve for
в€ћ T
pв€— T в€’t в€’1
Л†t = (1 в€’ О±ОІ)Et (О±ОІ) [(1 + П‰Оё) sT +
Л† ПЂП„ ]
П„ =t+1
T =t
в€ћ в€ћ
1 в€’ О±ОІ
(О±ОІ)T в€’t Et sT + (О±ОІ)T в€’t Et ПЂT .
= Л†
1 + П‰Оё 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-diп¬Ђerenced to yield

1 в€’ О±ОІ
pв€— = st + О±ОІEt ПЂt+1 + О±ОІEt pв€— .
Л†t Л† Л†t+1
1 + П‰Оё
46 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

Then using (2.6) to substitute for pв€— on both sides of the above, we obtain a stochastic
Л†t
diп¬Ђerence equation for the inп¬‚ation rate,

ПЂt = ОѕЛ†t + ОІEt ПЂt+1 ,
s (2.8)

where
1 в€’ О± 1 в€’ О±ОІ
Оѕв‰Ў > 0.
О± 1 + П‰Оё
Writing this alternatively as

в€† log Pt = Оѕ[log St в€’ log Pt + log Вµ] + ОІEt в€† 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 . Speciп¬Ѓcally,
if {log St } is a diп¬Ђerence-stationary process, there is a unique solution for the price index
process such that {log Pt } is also diп¬Ђerence-stationary (i.e., inп¬‚ation is a stationary process),
given by
в€ћ
О»в€’j [log Вµ + Et log St+j ],
О»в€’1 )
log Pt = О»1 log Ptв€’1 + (1 в€’ О»1 )(1 в€’ (2.10)
2
2
j=0

where 0 < О»1 < 1 < ОІ в€’1 < О»2 are the two roots33 of the characteristic polynomial

ОІО»2 в€’ (1 + ОІ + Оѕ)О» + 1 = 0. (2.11)

This prediction of the Calvo pricing model is in fact independent of any speciп¬Ѓcation 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 supplierвЂ™s marginal
cost with respect to its own output is equal to П‰ > 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 вЂњNew KeynesianвЂќ aggregate supply relation derived below (which
depends as well upon the details of oneвЂ™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 < ОІ < 1 and
Оѕ > 0.
2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 47

15

inflation
ULC growth

10

5

0

в€’5

в€’10

1965 1970 1975 1980 1985 1990 1995

Figure 3.4: U.S. inп¬‚ation (quarterly change in GDP deп¬‚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 ОІ and Оѕ
(which then imply values for О»1 and О»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 speciп¬Ѓcation). 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

в€’0.01

в€’0.02

actual
в€’0.03 predicted

в€’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 п¬‚exible prices (the Оѕ в†’ в€ћ limit of the above model), the predicted
price series would simply be given by log Pt = log Вµ + log St , so that predicted inп¬‚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. inп¬‚ation; in
particular, one observes that inп¬‚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 (Оѕ п¬Ѓnite), the model implies less volatile
inп¬‚ation, and in fact, for a certain value of Оѕ, 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 modelвЂ™s prediction in the case that Оѕ = .055.35 The п¬Ѓt 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. inп¬‚ation (measured by growth in the GDP
deп¬‚ator) with the path predicted by the model. The п¬Ѓt 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 п¬Ѓts U.S. inп¬‚ation dynamics is
perhaps surprising, given that the assumption of a п¬Ѓxed 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 reп¬‚ects the fact that, given the small value of Оѕ implied
by SbordoneвЂ™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 п¬Ѓt of the model does clearly depend upon the existence of
staggered price changes of the kind assumed by Calvo. The вЂњNew ClassicalвЂќ pricing model
considered above, for example, would imply that п¬‚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 signiп¬Ѓcant 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 modelвЂ™s predicted path for log Pt , when ОІ is assigned a value of 1. Note that she actually reports the
value of Оѕ в€’1 = 18.3.
36
Gali and Gertler (1999) obtain similar results for U.S. inп¬‚ation, using an alternative, instrumental-
variables strategy for estimating equation (2.8). Gali and Gertler п¬Ѓnd that they can statistically reject this
baseline pricing model in favor of a generalization in which some price-setters use a backward-looking вЂњrule of
thumbвЂќ, discussed further below in section xx. However, they п¬Ѓnd that the baseline model already explains
historical inп¬‚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 signiп¬Ѓcant might also be questioned. Batini
et al. (2000) similarly п¬Ѓnd that the Calvo pricing model can explain U.K. inп¬‚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

в€’2

в€’4

в€’6
1965 1970 1975 1980 1985 1990 1995

Figure 3.6: Actual path of inп¬‚ation (quarterly U.S. data) compared to prediction of the
Calvo pricing model. Source: Sbordone (2002).

model,37 suп¬ѓces 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 п¬Ѓt nearly as well by any purely backward-looking moving average of
unit labor costs, either; the kind of smoothing reп¬‚ected in the п¬Ѓgures depends importantly
upon the presence of the forward-looking terms on the right-hand side of (2.10), and not
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 О»1 and О»в€’1 in (2.10), without imposing
38
2
the restriction that they correspond to the two roots of (2.11). She п¬Ѓnds not only that both coeп¬ѓcients are
signiп¬Ѓcantly positive, but that the values that yield the best п¬Ѓt are nearly equal in size, as the Calvo pricing
model would imply. (Note that the two roots of (2.11) necessarily satisfy О»1 О»2 = ОІ в€’1 .)
2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 51

But it seems fortuitous that such a simple model of price-setting can explain actual inп¬‚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 п¬Ѓnds that none of
them improves the п¬Ѓt of the pricing model. One might also wonder whether forecasts based
on a VAR model with constant coeп¬ѓcients п¬Ѓt to the entire period 1960-97 should correspond
very closely to peopleвЂ™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 inп¬‚ation dynamics have been substantially
diп¬Ђerent since the disinп¬‚ation of the early 1980s; but this would imply that unit labor cost
dynamics should have been diп¬Ђerent as well.) But, once again, the close п¬Ѓt of the model,
and the fact that it п¬Ѓts 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 вЂњNew KeynesianвЂќ Phillips Curve

We can use the model of optimal price-setting just derived to obtain an aggregate supply
relation вЂ” i.e., a structural relation between inп¬‚ation dynamics and the level of real activity
вЂ” 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
Л† Л†
st = (П‰ + Пѓ в€’1 )(Yt в€’ Ytn ).
Л† (2.12)

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

Л† Л†
ПЂt = Оє(Yt в€’ Ytn ) + ОІEt ПЂt+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 eп¬Ђect upon the predicted path for prices. And Figure 3.5 itself indicates at
least a small amount of low-frequency speciп¬Ѓcation 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 в€’ О±)(1 в€’ О±ОІ) П‰ + Пѓ в€’1
в€’1
Оє в‰Ў (П‰ + Пѓ )Оѕ = > 0.
О± 1 + П‰Оё
Alternatively, the slope coeп¬ѓcient can be written as

(1 в€’ О±)(1 в€’ О±ОІ)
Оє= О¶, (2.14)
О±

where О¶ is deп¬Ѓned in (1.22). This is what Roberts (1995) calls the вЂњNew Keynesian Phillips
Curve,вЂќ as this speciп¬Ѓcation, 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 п¬‚atter (for any given inп¬‚ation expectations) the smaller the value of О¶, and thus the
greater the degree of strategic complementarity in price-setting. It is also п¬‚atter the larger
is О±, 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 inп¬‚ation expectations that shift the curve are current expectations of
future inп¬‚ation, rather than past expectations of the current inп¬‚ation rate, as in (1.23). The
diп¬Ђerence turns out to be crucial for the modelвЂ™s ability to allow for forecastable п¬‚uctuations
Л†Л†
in the output gap. It is now possible for non-zero values of Yt в€’ Ytn to be forecasted at some
earlier date t в€’ j; this simply requires that Etв€’j ПЂt not equal ОІEtв€’j ПЂt+1 .

2.3 Persistent Real Eп¬Ђects of Nominal Disturbances

We return now to the question of the persistence of the real eп¬Ђects 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 RobertsвЂ™
presentation of the вЂњNew Keynesian Phillips Curve,вЂќ the discount factor ОІ is set equal to one. This simpli-
п¬Ѓcation may seem appealing, in that it implies a vertical вЂњlong-runвЂќ inп¬‚ation-output tradeoп¬Ђ. 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 в‰Ґ 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 Pв€’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 в€’1, the extent to which the expected price-level response in period
Лњ
t diп¬Ђers from exact proportionality to the permanent increase in nominal spending. Then
we must have E0 log Yt = в€’Лњt for each horizon t в‰Ґ 0; substituting this into (2.13), we п¬Ѓnd
p
that the price-level responses must satisfy

ptв€’1 в€’ (1 + ОІ + Оє)Лњt + ОІ pt+1 = 0
Лњ p Лњ (2.16)

for each t в‰Ґ 0, starting from the initial condition pв€’1 = в€’1. The unique bounded solution is
Лњ
easily seen to be given by
pt = в€’О»t+1 ,
Лњ (2.17)

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

P (О») в‰Ў ОІО»2 в€’ (1 + ОІ + Оє)О» + 1 = 0. (2.18)

Since E0 log Pt = 1 + pt , we observe that the log price level is expected to rise monotonically,
Лњ
asymptotically reaching a level proportional to the increase in nominal spending. Since
E0 log Yt = в€’Лњt , 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 ОІ = .99
and Оє = .024.42 )
54 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

1.5

1

0.5

0
nominal GDP
real output
price level

в€’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 eп¬Ђect upon real activity is thus dependent upon the size
of the root О»; the closer О» is to one (its theoretical upper bound), the longer the time it takes
for real activity to return to its вЂњpotentialвЂќ level following a nominal disturbance. (Note that
О» also determines the size of the initial eп¬Ђect upon real activity in period zero, as well. Thus
the same factors that increase the amplitude of the eп¬Ђect upon real activity вЂ” by making
aggregate price adjustment more sluggish вЂ” also make the eп¬Ђect more persistent.) It is
easily seen, in turn, that the smaller root of P (О») varies inversely with the size of Оє, with a
value that approaches zero in the case of very large Оє, and a value that approaches one for Оє
near zero. Thus a small value of Оє (a п¬‚at short-run Phillips curve) is required for signiп¬Ѓcant
persistence. This in turn could occur either as a result of О± being near one (infrequent price

41
Since P (0) > 0, P (1) < 0, P (ОІ в€’1 ) < 0, and P (О») > 0 for all large enough О» > 0, it is evident that the
equation has two real roots, 0 < О» < 1 and another root greater than ОІ в€’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
О¶ = 0.1

0.6
О¶ = 0.5

0.5 О¶=1
О»

О¶=2

0.4

О¶ = 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
О±

Figure 3.8: Persistence of real eп¬Ђects 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 О¶ being small (strong strategic complementarity).

It is perhaps most interesting to observe that the degree of strategic complementarity
can have a considerable eп¬Ђect upon the degree of persistence, holding п¬Ѓxed the frequency of
price changes. Note that for any value of О±, the value of Оє can be made arbitrarily small,
or arbitrarily large, through assignment of an appropriate value to О¶. The eп¬Ђects of О± and
О¶ upon the degree of persistence are shown quantitatively in Figure 3.8, where the implied
value of О» is plotted as a function of these two parameters, assuming the value ОІ = .99.

One observes that О» > О± if and only if О¶ < 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 (О±) = (1 в€’ О±)(1 в€’ О±ОІ)(1 в€’ О¶),
56 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

where we have used (2.14) to substitute for Оє. Thus P (О±) > 0, so that О± < О» < 1, if and
only if О¶ < 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 О±t .)
In this case, Taylor (1980) speaks of the existence of a вЂњcontract multiplierвЂќ as a result of the
staggering of price adjustments.43 Such a multiplier depends upon the existence of strategic
complementarity among diп¬Ђerent suppliersвЂ™ 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 (О¶ > 1), there is actually even less persistence than one would
expect for purely mechanical reasons (О» < О±), because the п¬Ѓrst 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 О¶ and the value of Оє indicated by (2.14)
continues to hold when additional sources of strategic complementarity are introduced, such
as non-CES preferences over diп¬Ђerentiated 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 proп¬Ѓts are instead of the form

О i (p) = [pYt d(p/Pt ) в€’ wt (i)f в€’1 ((1 в€’ sm )d(p/Pt )Yt /At ) в€’ 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 п¬Ѓrst-order condition for optimal price-setting by a п¬Ѓrm that chooses its price
at date t is then
в€ћ
О±T в€’t Qt,T YT d(pt (i)/PT )[pt (i) в€’ ВµT (i)ST (i)] = 0,
Et
T =t

generalizing (2.4). Here the desired markup is given by ВµT (i) = Вµ(d(pt (i)/PT )), where the
function Вµ(x) is again deп¬Ѓned by (1.30), and marginal cost is now given by (1.35). As before,
43
In TaylorвЂ™s original formulation, it is actually the staggered negotiation of wage contracts that gives rise
to the вЂњmultiplierвЂќ.
2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 57

we may log-linearize this to obtain
в€ћ
(О±ОІ)T в€’t {Л†t,T в€’ [Л†t,T + st,T ]} = 0,
Et p Вµ Л† (2.19)
T =t

generalizing (2.7). Here we introduce the notation
T
pв€—
pt,T =
Л† Л†t в€’ ПЂП„
П„ =t+1

for the (log) relative price at date T of a п¬Ѓrm that has set its price at date t, yt,T for its (log)
Л†
output relative to steady state, and

Л†
Вµt,T =
Л† Вµ (Л†t,T
y в€’ YT )

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

Л† Л† Л†
+ sy )(Л†t,T в€’ YT ) в€’ (sy + sY )(YT в€’ YTn )
pt,T в€’ [Л†t,T + st,T ] = pt,T в€’ (
Л† Вµ Л† Л† y
Вµ

Л† Л†
+ sy )]Л†t,T в€’ (sy + sY )(YT в€’ YTn ),
= [1 + Оё( p
Вµ

we observe that (2.19) can equivalently be written
в€ћ
Л† Л†
(О±ОІ)T в€’t pв€— в€’ О¶(Yt в€’ Ytn ) = 0,
Et Л†t (2.20)
T =t

where О¶ is now given by (1.37). This in turn allows once again to derive an aggregate supply
relation of the form (2.13), where Оє 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 eп¬Ђects 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 вЂњcontract multiplierвЂќ that would be needed, and an under-
estimate of the empirically plausible degree of strategic complementarity. They deп¬Ѓne the
58 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

contract multiplier as вЂњthe ratio of the half-life of output deviations after a monetary shock
with staggered price-setting to the corresponding half-life with synchronized price-settingвЂќ
(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 О»t , where О»
is the smaller root of (2.18). Thus the вЂњhalf-lifeвЂќ with staggered price setting would equal
log 2/ log О»в€’1 periods.

By the corresponding half-life вЂњwith synchronized price-settingвЂќ, 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 О±k+1 , so that the вЂњhalf-life with synchronized price-settingвЂќ would equal
log 2/ log О±в€’1 .45 The contract multiplier implied by the Calvo pricing model is then equal to
log О±в€’1 / log О»в€’1 . It follows from our results above that the multiplier is greater than one if
and only if О¶ < 1, which is to say if and only if there is strategic complementarity in the
pricing decisions of diп¬Ђerent 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 вЂњstripped-down versionвЂќ 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 п¬Ѓxed-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 п¬Ѓxed.
46
As noted in the previous two footnotes, Chari et al. consider a diп¬Ђerent model of staggered pricing, but
obtain this same conclusion; their п¬Ѓnding that the multiplier is necessarily less than one in their вЂњstripped-
downвЂќ model follows from the fact that О¶ is necessarily greater than one, for the reasons that we have
discussed above. (They call this parameter Оі; 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 О¶ 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

2aв€—
, (2.21)
[4aв€— (aв€— + 1)О¶ + 1]1/2 в€’ 1

where aв€— 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
aв€— >> 1, this is approximately О¶ в€’1/2 , regardless of the size of aв€— ; this provides an analytical
explanation for the п¬Ѓnding 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 п¬Ѓt a univariate ARMA model to de-
trended real GDP for the postwar U.S., and conclude that the half-life of output п¬‚uctuations
around trend is approximately 10 quarters. At the same time, they argue that a reasonable
length of time to assume that prices are п¬Ѓxed 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 О¶ were quite small (.004 in the
case of our continuous-time limit).48 Instead, they п¬Ѓnd that their baseline parameter values
imply a value of О¶ 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 п¬Ѓxed
for many years in order to account for observed persistence.
However, this way of identifying the persistence of the output eп¬Ђects of monetary shocks
assumes that all п¬‚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 п¬Ѓx О¶ and let О± and ОІ decrease with в€†, the period length, so that as в€† в†’ 0, log ОІ в€’1 /в€†
and log О±в€’1 /в€† both approach positive constants, ПЃ and a respectively. Then, expanding the characteristic
polynomial (2.18) in powers of в€†, one can show that log О»в€’1 /в€† approaches a positive constant n as well,
which depends upon О¶ and the ratio aв€— в‰Ў a/ПЃ. The multiplier (2.21) is then equal to a/n.
48
This value follows from (2.21) under the assumption that aв€— = 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 О¶ is required in the case of any large value for aв€— .
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 eп¬Ђects 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
diп¬Ђerent paths of real as well as nominal variables.
One needs, then, to identify the real eп¬Ђects 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
п¬‚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
вЂњhalf-lifeвЂќ 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 п¬Ѓrst substantially aп¬Ђected; this response, shown in Figure
xx below, exhibits a вЂњhalf-lifeвЂќ 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. п¬Ѓrms) of 9 months.50 This
49
As shown in Figure xx, the peak (and п¬Ѓrst signiп¬Ѓcant) output contraction occurs two quarters following
a contractionary policy shock; log output has returned about half-way to its trend level by the п¬Ѓfth 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. п¬Ѓnd to be possible. In the continuous-time limit
of the Calvo pricing model derived above,51 this requires only that О¶ = 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 speciп¬Ѓcity.52
A larger value of Пѓ makes this even easier, and as we shall argue in chapter 4, it is most
reasonable to calibrate this model with a value of Пѓ much larger than one.
In fact, our discussion above has indicated that a variety of plausible assumptions can
justify a value of О¶ 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 вЂњhalf-lifeвЂќ 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 eп¬Ђects 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 eп¬Ђects of identiп¬Ѓed monetary shocks.53

51
Here we use (2.21), assuming that a = 0.33/quarter and ПЃ = 0.01/quarter, so that aв€— = 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 О¶,
approximately 0.13. The diп¬Ђerence is obtained because their model is a discrete-time model with вЂњperiodsвЂќ of
length three months (which results in a degree of persistence somewhat diп¬Ђerent than that in the continuous-
time limit), because they set О± = 2/3 (so that exactly one-third of all prices are changed each quarter, rather
than the fraction 1 в€’ eв€’a = 0.28 implied by integration of the continuous-time model), and because their
identiп¬Ѓed monetary policy shock does not result in an immediate, permanent increase in nominal GDP as
assumed above.
53
The real eп¬Ђects of the identiп¬Ѓed 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 (п¬Ѓrst column of their Figure 2), the peak contraction
in real GDP following an unexpected monetary tightening occurs only п¬Ѓve 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 вЂњhalf-lifeвЂќ of п¬Ѓve 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
deп¬Ѓnition 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 identiп¬Ѓed in the VAR literature; thus the modelвЂ™s predictions for this case
cannot be directly compared to any empirical estimates. Identiп¬Ѓed monetary policy shocks
tend to aп¬Ђect nominal spending only slightly (if at all) in the п¬Ѓrst few months, with an eп¬Ђect
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 п¬Ѓrst-order autoregressive process,

в€† log Yt = ПЃв€† log Ytв€’1 + (2.22)
t

with 0 < ПЃ < 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 в€’ ПЃ)в€’1 (1 в€’ ПЃj+1 ) t which increases monotonically with j, asymp-
totically approaching a permanent eп¬Ђect that is (1 в€’ ПЃ)в€’1 > 1 times as large as the initial
eп¬Ђect.54 Let us consider the impulse response to a positive innovation = 1 в€’ ПЃ, which
0

results in a unit increase in the expected long-run level of nominal spending. Again letting
pt denote E0 log Pt в€’ 1, we п¬Ѓnd that the price-level responses must satisfy
Лњ

ptв€’1 в€’ (1 + ОІ + Оє)Лњt + ОІ pt+1 = ОєПЃt+1 ,
Лњ p Лњ (2.23)

a generalization of (2.16), again starting from initial condition pв€’1 = в€’1. (Because nominal
Лњ
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 в€’ 1] в€’ pt = в€’(Лњt + ПЃt+1 ).) The unique bounded solution to (2.23) is given by the
Лњ p
solution to the п¬Ѓrst-order equation,
Оє
ПЃt+1 ,
pt = О»Лњtв€’1 в€’
Лњ p в€’1 в€’ ОІПЃ
О»
again starting from the given value for pв€’1 , where 0 < О» < 1 is again deп¬Ѓned as in (2.17).
Лњ
(Note that in the limiting case ПЃ = 0, this reduces to the same solution (2.17) as before.)
Then using the relations

E0 log Pt = pt + 1,
Лњ

E0 log Yt = в€’(Лњt + ПЃt+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 ОІ = .99, Оє = .024, and ПЃ = .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 вЂњhump-shapedвЂќ output response is predicted in the case of any large enough value of
ПЃ, speciп¬Ѓcally, for any value such that О» + ПЃ > 1. The responses yt в‰Ў E0 log Yt satisfy
Лњ

(1 в€’ ПЃL)(1 в€’ О»L)Лњt = 0
y

for all t в‰Ґ 1, starting from initial values yв€’1 = 0 and
Лњ
(1 в€’ ПЃ)(1 в€’ ОІПЃ)
y0 =
Лњ > 0.
(О»в€’1 в€’ ОІПЃ)
A second-order diп¬Ђerence 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 п¬Ѓrst increases to a peak,
and then decays monotonically back to zero. Since y2 = (О» + ПЃ)Лњ1 , the second case occurs if
Лњ y
and only if О» + ПЃ > 1.
55
The п¬Ѓgure also shows the impulse response of the inп¬‚ation rate, here measured in annual percentage
points, so that the inп¬‚ation rate plotted is deп¬Ѓned as 4 в€† log Pt . The dynamics of the inп¬‚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

в€’0.2
в€’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 вЂњhump-shapedвЂќ 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) п¬Ѓnds that a slightly modiп¬Ѓed version of the вЂњNew KeynesianвЂќ 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 speciп¬Ѓcation is that it can allow signiп¬Ѓcant output eп¬Ђects 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 eп¬Ђect upon nominal spending.
This can be seen by considering the above calculation in the case that ПЃ is close to 1. In
such a case, there is little increase in nominal spending in period zero (log Y0 increases only
by 1 в€’ ПЃ, yet the peak output eп¬Ђect may be an arbitrarily large fraction of 1. (For small
enough Оє, the value of О» 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 вЂњNew ClassicalвЂќ aggregate supply
relation (1.23), which implies that the peak output eп¬Ђect (which must occur in period zero)
is bounded above by 1 в€’ ПЃ, no matter how п¬‚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
diп¬Ђerent goods. In particular, we have assumed that all real disturbances aп¬Ђect 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 diп¬Ђerentially aп¬Ђect various sectors of the economy. Here
we brieп¬‚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 вЂњNew KeynesianвЂќ 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
aп¬Ђect 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
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