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57
In fact, the two-sector model presented here closely resembles the treatment of a two-country monetary
union in Benigno (1999).
66 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

representative household is de¬ned by the CES index (1.2), let us suppose that it is a CES
aggregate of two sub-indices,
·
·’1 ·’1 ·’1
1 1
· ·
Ct ≡ (n1 •1t ) C1t + (n2 •2t ) C2t , (2.24)
· ·




for some elasticity of substitution · > 0. The sub-indices are in turn CES aggregates of the
quantities purchased of the continuum of di¬erentiated goods in each of the two sectors,
θ
θ’1
θ’1
Cjt ≡ ct (i) di ,
θ
Nj

for j = 1, 2, where the intervals of goods belonging to the two sectors are respectively
N1 ≡ [0, n1 ] and N2 ≡ (n1 , 1], and once again θ > 1.58 In the aggregator (2.24), nj is the
number of goods of each type (n2 ≡ 1 ’ n1 ), and the random coe¬cients •jt are at all times
positive and satisfy the identity n1 •1t +n2 •2t = 1. (The variation in the •jt thus represents a
single disturbance each period, a shift in the relative demand for the two sectors™ products.)
It follows from this speci¬cation of preferences that the minimum cost of obtaining a unit
of the sectoral composite good Cjt will be given by the sectoral price index
1
1’θ
pt (i)1’θ di
Pjt ≡
Nj

for j = 1, 2, and that the minimum cost of obtaining a unit of Ct will correspondingly be
given by the overall price index
1
1’· 1’· 1’·
Pt ≡ n1 •1t P1t + n2 •2t P2t .

The optimal allocation of demand across the various di¬erentiated goods by price-taking
consumers will satisfy
ct (i) = Cjt (pt (i)/Pjt )’θ

for each good i in sector j, and the index of sectoral demand will satisfy

Cjt = nj •jt Ct (Pjt /Pt )’·
58
We need not assume that · > 1 in order for there to be a well-behaved equilibrium of the two-sector
model under monopolistic competition. In fact, the limiting case in which · ’ 1 and the aggregator (2.24)
becomes Cobb-Douglas is frequently assumed; see, e.g., Benigno (1999).
2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 67

for each sector, regardless of total consumption expenditure in a given period. Note that the
random factors •jt appear as multiplicative disturbances in the sectoral demand functions.
The aggregators have been normalized so that in the event of a common price p for all
goods, the price indices each are equal to that common price, and the demands are equal to
ct (i) = •jt Ct for each good i in sector j.
We furthermore assume a disutility of supplying labor of type i equal to v(ht (i); ξjt ) in
the case of each good i in sector j; thus we allow for a sector-speci¬c (though not good-
speci¬c) disturbance to preferences regarding labor supply. The production function for this
good is assumed to be of the form

yt (i) = Ajt f (ht (i)),

where the function f (·) is common to all goods as before, but we now allow for a sector-
speci¬c technology disturbance as well. Assuming that each ¬rm is a wage-taker in a ¬rm-
speci¬c labor market as before, nominal pro¬ts for ¬rm i in sector j are given by

Πij (p) = Yjt Pjt p1’θ ’ wt (i) f ’1 (A’1 Yjt Pjt p’θ ),
θ θ
t jt


generalizing (2.3).
The real marginal cost of supplying any good i in sector j will be given by
’1
˜t ) ≡ vh (f (yt (i)/Ajt ); ξjt ) Ψ(yt (i)/Ajt ),
j
s (yt (i), Yt ; ξ
uc (Yt ; ξt )Ajt
a function of the quantity supplied of the individual good, aggregate output, and sector-
speci¬c real disturbances. Because each ¬rm faces a demand curve with constant elasticity
θ as before, the desired markup will again be µ. We can then de¬ne a natural rate of output
n
Yjt for each sector j as the common equilibrium output of each good i in that sector in the
case of ¬‚exible prices. These levels satisfy
1
’·
n
Yjt
˜
µ sj (Yjt , Ytn ; ξt ) =
n
,
nj •jt Ytn
for j = 1, 2, where the right-hand side indicates the relative price Pjt /Pt required to induce
the relative demand Yjt /Ytn , and the natural rate of aggregate output, Ytn , aggregates Y1t
n n
68 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

˜
n
and Y2t using (2.24). In the case that ξt = 0 for all t, and in addition •jt = 1 for all t and
¯
both sectors, then the ¬‚exible-price equilibrium involves a common level of output Y for all
goods, de¬ned in the same way as before. Log-linearizing around this allocation, the real
marginal cost function in sector j can be approximated by

ˆn ˆ ˆ ˆ ˆn
sj (i) = ω(ˆt (i) ’ Yjt ) + σ ’1 (Yt ’ Ytn ) + ·(•jt + Ytn ’ Yjt ),
ˆt y ˆ (2.25)

generalizing (1.14). (Here •jt ≡ log •jt , and the hatted output variables all represent log
ˆ
¯
deviations from Y .)
Let us now suppose that there is Calvo-style staggered pricing in each of the two sectors,
with ±j the fraction of goods prices that remain unchanged each period in sector j. A
supplier in sector j that changes its price in period t chooses its new price pt (i) to maximize

±j ’t Qt,T [Πij (pt (i))] ,
T
Et (2.26)
T
T =t


generalizing (2.2). This implies a ¬rst-order condition that, when log-linearized as before, is
of the form ± 
 
∞ T
(±j β)T ’t p— ’ [ˆj ’ pjT +
Et ˆjt st,T ˆ πj„ ] = 0, (2.27)
„ =t+1
T =t

where p— ≡ log(p— /Pjt ) is the relative price at date t (relative to other ¬rms in the same
ˆjt jt

sector) of the ¬rms in sector j that have newly revised their prices, pjt ≡ log(Pjt /Pt ) is
ˆ
the relative price index for sector j (relative to goods prices in the entire economy), πjt ≡
log(Pjt /Pj,t’1 ) is the rate of in¬‚ation in sector j, and sj is the value of sj (i) for ¬rms in sector
ˆt,T ˆT
j that last revised their prices at date t. (This generalizes (2.7) above.) Log-linearization of
the equation de¬ning the sectoral price index further implies that

1 ’ ±j —
πjt = pjt ,
ˆ (2.28)
±j

generalizing (2.6).
Equations (2.27) “ (2.28) then determine the evolution of the sectoral rate of in¬‚ation as
a function of expected future in¬‚ation in that sector and expectations regarding sj ’ pj at
ˆt,T ˆT
2. INFLATION DYNAMICS WITH STAGGERED PRICE-SETTING 69

all future dates T . It further follows from (2.25) that
T
sj sj ωθ[ˆ—
ˆt,T = ˆT ’ pjt ’ πj„ ],
„ =t+1


where sj denotes the deviation of the log of the average level of real marginal cost in sector
ˆt
j from its steady-state value. A series of manipulations analogous to those discussed earlier
then yields a sectoral in¬‚ation equation of the form

πjt = ξj (ˆj ’ pjt ) + βEt πj,t+1 ,
st ˆ (2.29)

where
1 ’ ±j 1 ’ ±j β
ξj ≡ > 0.
±j 1 + ωθ
This generalizes our in¬‚ation equation (2.8) for the one-sector model.
Finally, (2.25) implies that

ˆ ˆn ˆ ˆ ˆ ˆn
sj = ω(Yjt ’ Yjt ) + σ ’1 (Yt ’ Ytn ) + ·(•jt + Ytn ’ Yjt ).
ˆt ˆ

Substituting
ˆ ˆ
Yjt = •jt + Yt ’ · pjt
ˆ ˆ

from the sectoral demand equation, we can alternatively express this as

ˆ ˆ
sj ’ pjt = (ω + σ ’1 )(Yt ’ Ytn ) ’ (1 + ω·)(ˆjt ’ pn ),
ˆt ˆ p ˆjt

where pn is the log relative price index for sector j in a ¬‚exible-price equilibrium (a function
ˆjt
solely of the exogenous real disturbances). Substituting this into (2.29), we obtain

ˆ ˆ
πjt = κj (Yt ’ Ytn ) + γj (ˆRt ’ pn ) + βEt πj,t+1 ,
p ˆRt (2.30)

generalizing (2.13). Here we have written each of the sectoral relative prices as multiples of
the single relative price pRt ≡ log(P2t /P1t ) using the identities
ˆ

p1t ≡ ’n2 pRt ,
ˆ ˆ p2t ≡ n1 pRt ,
ˆ ˆ
70 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

pn = n’1 pn is the “natural” level of this relative price, and the coe¬cients are given by
ˆRt 1 ˆ2t


κj ≡ ξj (ω + σ ’1 ) > 0

for j = 1, 2, and

γ1 ≡ n2 ξ1 (1 + ω·) > 0, γ2 ≡ ’n1 ξ2 (1 + ω·) < 0.

In¬‚ationary pressure in each sector is thus a function of the aggregate level of real ac-
tivity (relative to its “natural” level) and of the sector™s relative price index (relative to the
“natural” value of this relative price). High aggregate output increases in¬‚ationary pressure
in both sectors, while a high relative price in one sector reduces in¬‚ationary pressure in that
sector. Combining equations (2.30) for j = 1, 2 with the identity

pRt = pR,t’1 + π2t ’ π1t ,
ˆ ˆ (2.31)

one has a complete system of equations for the evolution of the price indices for both sectors
ˆ
given the evolution of aggregate real activity and the two composite real disturbances Ytn
and pR . The latter term re¬‚ects the various ways in which the real disturbances di¬erentially
ˆt
a¬ect the two sectors, and is given by

1 ˆn ˆn
pn ≡
ˆRt [(•2t ’ •1t ) ’ (Y2t ’ Y1t )].
ˆ ˆ
·

In general, this model implies that in¬‚ation in both sectors, and hence aggregate in¬‚ation
as well, depends on a lagged endogenous variable, pR,t’1 ; thus in¬‚ation is not so purely
ˆ
forward-looking in this theory as in the fully symmetric (one-sector) case. However, in the
case that prices are equally sticky in both sectors (±1 = ±2 ), ξ1 = ξ2 , and hence

n1 ξ1 + n2 ξ2 = 0.

It follows that if we average the relation (2.30) over the two sectors, weighting each relation by
the size of the corresponding sector, we obtain once again equation (2.13) for the evolution
of aggregate in¬‚ation. Thus we ¬nd that the existence of an equilibrium relation of the
3. DELAYED EFFECTS OF NOMINAL DISTURBANCES ON INFLATION 71

form (2.13) ” implying that there is no incompatibility between stabilization of the overall
ˆ ˆ
price index and stabilization of the “output gap” Yt ’ Ytn ” does not require that all
real disturbances a¬ect the demand for and cost of production of each good identically, as
assumed earlier. In the case of a model that is otherwise symmetrical (the same degree
of price stickiness for all goods, the same form of production function for all goods up
to, and so on), a relation of this kind is obtained even in the presence of several types of
asymmetric disturbances: disturbances to the relative disutility of supplying di¬erent types
of labor, disturbances to the relative productivity of labor in di¬erent sectors, and shifts in
the relative preferences of households for di¬erent goods.

On the other hand, if prices are not equally sticky for di¬erent types of goods, it ceases
to be true that the same policy can simultaneously stabilize the aggregate output gap and
the overall in¬‚ation rate. The consequences of the resulting tradeo¬ for optimal stabilization
policy are considered in chapter 6.




3 Delayed E¬ects of Nominal Disturbances on In¬‚a-
tion

We have seen that the assumption of staggered price-setting (as interpreted by Calvo in
particular) gives rise to an aggregate-supply relation that ¬ts much better with basic facts
about the e¬ects of monetary disturbances than did the simple “New Classical Phillips
curve”. For this reason, the “New Keynesian Phillips curve” has been employed in many
recent discussions of monetary policy that seek to take account of forward-looking private-
sector behavior. Nonetheless, even this model has been subject to a good bit of criticism
as not ¬tting too well with econometric evidence regarding the comovements of real and
nominal variables.

A central criticism has been that the model implies that in¬‚ation should be a more
purely forward-looking process than it seems to be in reality. Note that (2.13) can be
72 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

“solved forward” to yield

ˆ ˆn
β j Et [Yt+j ’ Yt+j ].
πt = κ (3.1)
j=0

Thus the predicted rate of in¬‚ation at any time should depend solely upon the predicted
output gaps at that time and later, in a way that is completely independent of either output
gaps or in¬‚ation in the past. This does not square well with many economists™ intuitive view
of the in¬‚ation process, or with the in¬‚ation dynamics implied by the models currently used
in most central banks; these models instead assume a substantial degree of inertia in the
in¬‚ation process, so that recent past in¬‚ation ¬gures as an important determinant of current
in¬‚ation.
Clear evidence that the New Keynesian Phillips curve cannot account for the co-movement
that is observed between real activity and in¬‚ation, or for the persistence of in¬‚ation dy-
namics themselves, is not as easy to obtain as often seems to be assumed. Claims that the
equation is grossly at odds with the facts are often based upon the use of one or another
ˆ ˆ
conventional “output gap” series as a proxy for Yt ’ Ytn . For example, (2.13) implies that
the series πt+1 ’ β ’1 πt , which is essentially the rate of acceleration of in¬‚ation, should be
ˆ ˆ
negatively correlated with Yt ’ Ytn .59 Instead, conventional series for the U.S. “output gap”
(which subtract one relatively smooth trend or another from a log real GDP series) are gen-
erally found to be positively correlated with the subsequent acceleration of in¬‚ation. But
ˆ
our model gives us good reason to suppose that Ytn may not be a smooth trend; it should be
a¬ected immediately by changes in government purchases or other “autonomous” compo-
nents of expenditure, and by variations in household impatience to consume or in attitudes
toward work, in addition to such slower-moving factors as capital accumulation, technical
progress and growth in the labor force. If there are relatively high-frequency variations in
ˆ ˆ ˆ
Ytn , traditional “gap” measures could easily be negatively correlated with Yt ’ Ytn , owing
to policies tending to stabilize output around a smooth trend rather than around the time-
varying natural rate (Gali, 1999). The fact that average real unit labor costs ” which should
ˆ ˆ
59
To be precise, it implies that this quasi-acceleration statistic should equal ’β ’1 κ (Yt ’ Ytn ), plus a
ˆ ˆ
forecast error that should be uncorrelated with all period t information, including Yt ’ Ytn .
3. DELAYED EFFECTS OF NOMINAL DISTURBANCES ON INFLATION 73

ˆ ˆ
correspond to variations in Yt ’ Ytn under fairly general assumptions about the nature of
the disturbances, as discussed above ” are negatively correlated with detrended real GDP
suggests that this may well be the case. In fact, real unit labor costs are negatively correlated
with in¬‚ation acceleration in U.S. data, and the New Keynesian Phillips curve accounts quite
ˆ ˆ
well for U.S. in¬‚ation dynamics when real unit labor costs are used to measure Yt ’ Ytn , as
discussed above.
However, evidence that the simple New Keynesian Phillips curve may indeed be too
forward-looking can be found by considering the dynamics of output and in¬‚ation in response
to identi¬ed monetary policy shocks. If the identi¬cation of monetary shocks in the VAR
studies mentioned earlier is correct, then the estimated impulse responses for real GDP
should also correspond to the impulse response for the theoretically correct gap measure,
ˆˆ ˆ
Yt ’ Ytn , since the path of Ytn should be una¬ected.60 Hence the estimated impulse responses
for real GDP and in¬‚ation should satisfy (2.13), or equivalently (3.1).
Typically, they do not. In particular, one generally observes that the main e¬ect of a
monetary policy shock on in¬‚ation occurs in the quarters following those in which the output
response is strongest. (For example, Figure 3.10 shows the responses obtained in the study
of Christiano et al. (2001) discussed earlier.) But this is inconsistent with (3.1), regardless of
the assumed parameter values, because this equation states that the in¬‚ation response each
quarter should be an increasing function of the output responses that are expected in that
quarter and later. Thus the e¬ect on in¬‚ation should precede the e¬ect on output, insofar as
the latter e¬ect is predictable in advance (as the output impulse response indicates to be the
case); and it should peak earlier than does the e¬ect on output, since once the peak output
e¬ect is reached, the output gaps that can be anticipated from then on are smaller than
those that could still be expected a short while earlier. This is not what the VAR studies
60
This is no longer exactly true once we allow for endogenous capital accumulation, as we do in chapter 4.
ˆ
But even so, the e¬ect upon Ytn owing to endogenous variation in the capital stock should not be large during
the ¬rst few quarters. Furthermore, taking account of this e¬ect will only exacerbate the problem sketched
here: the true output gap really returns to its previously expected level even faster than does real GDP,
because increased investment during the early quarters following an interest-rate reduction should raise the
natural rate of output. This would only make the tendency of in¬‚ation to peak after the main e¬ect on the
output gap even more dramatic than it appears to be in Figure 3.10.
74 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES


1.5

real GDP /4
inflation
conf.interval
1




0.5




0




’0.5




’1




’1.5
0 5 10 15 20 25 30



Figure 3.10: Estimated impulse responses of real GDP and in¬‚ation to an unexpected
interest-rate reduction. Source: Christiano et al. (2001).

indicate. Instead, the e¬ects of monetary disturbances on in¬‚ation are delayed, and more
persistent than would be predicted by our simple model of staggered pricing.
The model of staggered pricing can be extended, however, in a number of ways that
make it more realistic in this respect, while continuing to derive in¬‚ation dynamics from
optimal pricing decisions, subject to certain assumed constraints. Here we give particular
attention to two approaches that have been useful in reconciling optimizing models with
VAR evidence.


3.1 Staggered Pricing with Delayed Price Changes

One simple way of avoiding the counterfactual prediction that ” because of the forward-
looking character of in¬‚ation in the Calvo model ” the rate of in¬‚ation should respond
immediately to a monetary disturbance that is expected to a¬ect nominal expenditure even-
3. DELAYED EFFECTS OF NOMINAL DISTURBANCES ON INFLATION 75

tually, is to modify the assumption that newly chosen prices take e¬ect immediately. We
have assumed in our derivation above that the fraction 1 ’ ± of prices that change in a given
period are chosen optimally, given aggregate conditions in the period in which the new price
takes e¬ect. One reason for this assumption was that it allowed us to nest the case of fully
¬‚exible prices within our speci¬cation (as the limiting case in which ± = 0). However, the
literature on staggered wage- and price-setting has often assumed that new wage contracts
and/or price commitments are chosen at a date before they ¬rst take e¬ect ” and as a
result, are optimal only conditional upon the information that was available at that earlier
date. Allowing for a delay before newly chosen prices take e¬ect will obviously have the
consequence that a monetary policy shock will not a¬ect prices until after this delay. It
could then also a¬ect output sooner than its e¬ect on in¬‚ation.
We can easily modify our presentation above of the discrete-time Calvo model to allow
for such delays in the introduction of new prices. Suppose that each of the new prices chosen
in period t takes e¬ect only in period t + d, for some integer d ≥ 0. Conditional upon a
new price being chosen for a given good in period t, that price applies in periods prior to
period t + d with probability zero, in period t + d with probability 1, in period t + d + 1 with
probability ±, and more generally in period t + d + k with probability ±k , for any k ≥ 0. A
supplier i that chooses a new price in period t chooses that new price pt+d (i) to maximize
± 
∞ 
±T ’t’d Qt,T [Πi (pt+d (i))] ,
Et  T
T =t+d


generalizing (2.2), where once again nominal pro¬ts each period are given by (2.3). Corre-
sponding to (2.5) we obtain in this case the ¬rst-order condition
± 
∞ 
˜
(±β)T ’t’d uc (YT ; ξT )YT PT [p— ’ µPT s(YT PT p—’θ , YT ; ξT )]
θ θ
Et =0
t+d t+d
 
T =t+d


to implicitly de¬ne the optimal price p— , that is the same for all suppliers choosing a new
t+d

price at date t. The value of p— then determines the evolution of the price index Pt through
t

(2.1), just as before.
76 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

Through a series of manipulations similar to those presented above, log-linearization of
the above ¬rst-order condition yields a log-linear aggregate supply relation of the form

ˆ ˆ
πt = κEt’d (Yt ’ Ytn ) + βEt’d πt+1 , ) (3.2)

generalizing (2.13). (All variables and coe¬cients in this equation have the same de¬nitions
as before.) In the case that d = 1, this is the form of aggregate supply relation used by
Cochrane (1996) and by Bernanke and Woodford (1997). Note that the right-hand side of
(3.2) consists entirely of terms that are a function of period t’d information; thus this model
implies that in¬‚ation πt is a predetermined variable, depending only upon disturbances in
ˆ
period t ’ d or earlier. In this case, only ¬‚uctuations in Et’d Ytn , the forecastable component
of the natural rate of output, matter for in¬‚ation and output determination.
An alternative interpretation of this model would be not that price changes must actually
be determined in advance (say, because advance notice to customers is expected), but rather
that when an opportunity to change price arises at date t, the new price (that applies
beginning in period t) is chosen on the basis of old information, namely, the state of the world
as of period t ’ d. This assumption would result in exactly the same optimality criterion
for new prices as above, and hence exactly the same aggregate supply relation (3.2). Thus
the hypothesis may alternatively be described as one involving information delays, as for
example in the work of Mankiw and Reis (2001a, 2001b).61
Under either interpretation, the model implies that a monetary policy shock in period
t has no e¬ect on in¬‚ation before period t + d. This eliminates an embarrassing feature of
the basic “New Keynesian” speci¬cation that is especially evident when we recognize that,
according to the VAR studies, monetary disturbances a¬ect aggregate nominal expenditure
only with a delay (as in Figure 3.2). Suppose that nominal GDP evolves according to a
61
One advantage of the hypothesis of information delays is that we need not assume the same delay in
the case of all types of news. In order to obtain the result that monetary policy shocks do not a¬ect
in¬‚ation within the ¬rst year, for example, it would be necessary to assume a delay of a year in the receipt
of information about changes in interest rates; but one might simultaneously assume that other kinds of
disturbances are observed by suppliers much more quickly, so that one would not have to assert that all
price changes are determined entirely by conditions in the previous year and earlier. We do not pursue this
extension here, however.
3. DELAYED EFFECTS OF NOMINAL DISTURBANCES ON INFLATION 77




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 3.11: Impulse responses to a monetary disturbance with a delayed e¬ect on nominal
expenditure (s = 2), according to the basic Calvo model.

stochastic process of the form

∆ log Yt = ρ∆ log Yt’1 + t’s , (3.3)

where the integer s ≥ 0 indicates the lag between the time at which the monetary policy
shock occurs and the ¬rst time at which it a¬ects nominal GDP. (As before, is assumed
t

to be a mean-zero disturbance realized at date t, completely unforecastable before that
date.) We can once again solve for the equilibrium paths of in¬‚ation and output given the
aggregate supply relation (3.2). We now wish to consider the consequences of alternative
assumed delays d in the latter relation.
Figure 3.11 shows the implied impulse responses of in¬‚ation and output to a monetary
disturbance at date zero according to the standard “New Keynesian” Phillips curve, in the
case that the e¬ect of the disturbance on nominal GDP is delayed. Our shock is once again
78 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

an unexpected loosening of policy that implies an eventual increase in nominal GDP of one
percentage point, and the assumed value of ρ is equal to 0.4, as in Figure 3.9; but we now
assume that s = 2, to match the delay in the e¬ect of an interest-rate reduction on nominal
GDP shown in Figure 3.2. (The values assumed for β and κ are the same as in the case of
Figure 3.9.)
In the case of the basic “New Keynesian” speci¬cation, we obtain the embarrassing
prediction that a monetary disturbance that is expected to increase nominal expenditure
beginning two periods later should contract real activity in the short run. This is because
an expectation of higher real activity and/or in¬‚ation two periods from now implies that
those suppliers who change prices sooner than that (but after learning of the shock) should
already raise their prices at a higher than normal rate, in anticipation of high demand
and high competitors™ prices in the future. In¬‚ation should thus increase immediately in
response to the expectation of higher nominal expenditure in the near future; but since
nominal expenditure does not increase immediately (by hypothesis), this implies a temporary
contraction of real activity. This is not, of course, at all what estimated output response is
like (recall Figure 3.3).
The problem can be solved by assuming a delay d = 2 quarters before newly chosen
prices take e¬ect (replacing (2.13) by (3.2)). The corresponding impulse responses in this
case are shown in Figure 3.12. The predicted responses of all variables are exactly the same
as in Figure 3.9, except that the entire impulse response functions are shifted to the right
by 2 quarters. Note that the hypothesis of delayed price changes also implies no e¬ect of
the monetary shock on output until two quarters later, given the assumption of a delay in
the e¬ect of the shock on nominal GDP.62 The implied response of output in this case is
qualitatively fairly similar to what the VAR studies estimate ” an e¬ect that is delayed for
two quarters, “hump-shaped” thereafter, persistent, and never signi¬cantly negative.
This ¬gure still di¬ers from the responses shown in Figure 3.10, though, in that the
62
Of course, the question remains why an interest-rate reduction should have no e¬ect on nominal GDP
until two quarters later. This is a question about the aggregate-demand block of our model, to be deferred
until the next chapter.
3. DELAYED EFFECTS OF NOMINAL DISTURBANCES ON INFLATION 79




1




0.8

nominal GDP
real GDP
0.6 price level
inflation


0.4




0.2




0




’0.2
’2 0 2 4 6 8 10 12 14 16 18 20



Figure 3.12: Impulse responses to the same monetary disturbance when d = 2 quarters in
AS relation (3.2).


in¬‚ation e¬ect peaks earlier than the output e¬ect. This problem can be ameliorated by
assuming an even longer delay d before price changes take e¬ect. (We note that the response
shown in Figure 3.10 implies that the price level is no higher than it would have been in the
absence of the shock, until six quarters following the shock; so these estimated responses are
consistent with d being as long as 6 quarters.) Figure 3.13 shows the corresponding impulse
responses in the case that d = 4 quarters. We observe now that the e¬ect of the shock on real
output peaks before there is any e¬ect on in¬‚ation at all (i.e., in the third quarter following
the shock).
Even so, the e¬ects on in¬‚ation, once the delay d is past, are predicted to appear abruptly;
the peak e¬ect on in¬‚ation is very clearly in the ¬rst quarter in which any e¬ect on prices
can occur, with a sharp decline in the in¬‚ation e¬ect thereafter. Thus the predicted in¬‚ation
response still does not exhibit the kind of persistence seen (at least according to the point
80 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES




1




0.8



nominal GDP
0.6
real GDP
price level
inflation
0.4




0.2




0




’0.2
’2 0 2 4 6 8 10 12 14 16 18 20



Figure 3.13: Impulse responses to the same monetary disturbance when d = 4 quarters.

estimates) in Figure 3.10. A further modi¬cation of the Calvo model can help with this.

3.2 Consequences of Indexation to Past In¬‚ation

In the classic Calvo model, it is assumed that prices remain ¬xed in money terms between
those occasions upon which they are re-optimized. While this is a simple way of resolving
the question of what to do about prices between revisions, and the assumption conforms
with apparent practice at many ¬rms, we have certainly not shown that there is anything
optimal about this aspect of the Calvo pricing model; we have instead analyzed optimal
pricing policy taking this feature of it as a constraint. One might instead assume that prices
are automatically raised in accordance with some mechanical rule between the occasions on
which they are reconsidered. If the rule is simple enough, the fact that ¬rms refrain from
reconsidering the optimality of their prices for intervals of months at a time will still result
in substantial savings on managerial costs, and so it may be plausible that such an interim
3. DELAYED EFFECTS OF NOMINAL DISTURBANCES ON INFLATION 81

“rule of thumb” would be used.
One obvious type of more sophisticated interim rule than simply ¬xing prices in terms of
money is one that seeks to correct, in at least a simple way, for increases in the general price
index. As an example, Yun (1996) assumes that prices are automatically increased at some
rate π between occasions on which they are reconsidered, where π is the actual long-run
¯ ¯
average rate of in¬‚ation in the economy. This results in an aggregate supply relation of the
form
ˆ ˆ
πt = (1 ’ β)¯ + κ (Yt ’ Ytn ) + β Et πt+1 ,
π

generalizing (2.13). In comparing the expected consequences of monetary policy rules that
imply di¬erent long-run average rates of in¬‚ation, Yun assumes that the parameter π of
¯
¬rms™ pricing policies should change accordingly; this results in a vertical “long-run Phillips
curve” relation, unlike the classic Calvo model.
In practice, in economies where in¬‚ation has been enough of a problem for indexation of
long-term monetary commitments to be worth undertaking, indexation schemes are generally
based on a measure of in¬‚ation over some relatively short recent time interval, as there is
no presumption that in¬‚ation can be expected to remain always near some non-zero steady-
state value. (Of course, this is related to the fact that there has been little experience of
stable commitment to a ¬xed in¬‚ation target in any country, before the past decade!) This
suggests that it may be more plausible to assume automatic indexation of price commitments
(or wages, as discussed in the next section) to the change in the overall price index over some
recent past period. Note, however, that it is not realistic to assume that it should be possible
to index individual prices to the current price index. Apart from the simultaneity problem
that this would create, the assumption that this is possible would not be in the spirit of our
assumption that continual monitoring of current conditions in order to maintain a constantly
optimal price is too costly to be worthwhile. It is far more plausible, then, to imagine a policy
of automatic indexation of one™s price (between the occasions on which a full review of the
optimality of the price is undertaken) to the change in an overall price index over some past
time interval.
82 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

Christiano et al. (2001) and Smets and Wouter (2001) assume partial or full indexation
of this kind for both wages and prices, and argue that this extension of the Calvo pricing
model improves the empirical ¬t of their models. Here we examine the consequences of
backward-looking indexation for in¬‚ation dynamics, continuing for now to assume e¬cient
labor-market contracting. Let us suppose once again that each period a randomly chosen
fraction 1 ’ ± of all prices are reconsidered, and that these are set optimally; but the price
of each good i that is not reconsidered is adjusted according to the indexation rule

log pt (i) = log pt’1 (i) + γ πt’1 , (3.4)

where 0 ¤ γ ¤ 1 measures the degree of indexation to the most recently available in¬‚ation
measure. (Note that even when γ = 1, as assumed by Christiano et al., nominal rigidities
still matter for the e¬ects of aggregate disturbances, because of the one-quarter lag in the
indexation.)
This assumption about how prices are adjusted in the interim between re-optimizations
a¬ects the way in which prices should be set when they are reconsidered. If we assume, as
in the basic Calvo model, that newly-optimized prices take e¬ect immediately, then a new
price pt (i) chosen in period t should be selected to maximize

±T ’t Qt,T [Πi (pt (i) (PT ’1 /Pt’1 )γ )] .
Et T
T =t

This results in a ¬rst-order condition
± 
γ(1’θ)
∞ 
PT ’1 ˜
(±β)T ’t uc (YT ; ξT ) YT PT
θ
p— ’ µPT s(YT (p— /PT )’θ (PT ’1 /Pt’1 )’γθ , YT ; ξT )
Et =0
t t
 
Pt’1
T =t


to implicitly de¬ne the optimal price p— , once again the same for all suppliers choosing a
t

new price at date t. Given the choice of p— each period, the overall price index then evolves
t

according to
®  1
1’θ
γ 1’θ
Pt’1
Pt = °(1 ’ ±)p—1’θ + ± Pt’1 » , (3.5)
t
Pt’2

generalizing (2.1).
3. DELAYED EFFECTS OF NOMINAL DISTURBANCES ON INFLATION 83

Log-linearization of the ¬rst-order condition and of the law of motion (3.5) for the Dixit-
Stiglitz price index, together with a series of manipulations analogous to those presented
earlier, then yields a log-linear aggregate supply relation of the form

ˆ ˆ
πt ’ γπt’1 = κ (Yt ’ Ytn ) + β Et (πt+1 ’ γπt ), ) (3.6)

ˆ
where β, κ, and Ytn all have the same de¬nitions as in (2.13). The allowance for backward-
looking indexation generalizes the New Keynesian Phillips curve in a fairly straightforward
way: it is now the quasi-di¬erenced in¬‚ation rate, πt ’ γπt’1 , rather than the in¬‚ation rate
itself, that is related to the output gap in the way indicated by the previous relation.
In particular, it is still possible to solve (3.6) forward to obtain

ˆ ˆn
β j Et [Yt+j ’ Yt+j ],
πt = γπt’1 + κ
j=0

generalizing (3.1). The quasi-di¬erenced in¬‚ation rate is still a purely forward-looking func-
tion of the expected path of the output gap; but now the in¬‚ation rate predicted for periods
t and later will depend not only upon the predicted path of the output gap in those peri-
ods, but also upon the initial in¬‚ation rate πt’1 . Thus the extended theory implies in¬‚ation
inertia, to an extent that is greater the larger is the indexation parameter γ.
The di¬erence that is made by a substantial degree of indexation can be illustrated by
again considering the predicted impulse responses of in¬‚ation and output to a monetary
policy shock that results in a persistent increase in the growth rate of nominal GDP. Figure
3.14 shows the predicted impulse responses in the case of aggregate supply relation (3.6), in
the case γ = 1, when (as in Figure 3.9) there is no delay in the e¬ect of the shock on nominal
expenditure. We see now that even without the hypothesis of delay before new prices can
take e¬ect, it is possible to explain the observed delay in the e¬ect of a monetary disturbance
on in¬‚ation, relative to its e¬ect on output. We also observe that in this model, the in¬‚ation
response is “hump-shaped,” rather than immediately declining sharply after the quarter of
the disturbance, as is predicted by the basic Calvo model (see Figure 3.9).
A number of authors have argued that this kind of modi¬cation of the basic Calvo model
results in a more realistic speci¬cation. Christiano et al. (2001) argue that a model with
84 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
0 4 8 12 16 20 24 28



Figure 3.14: Impulse responses to the same monetary disturbance as in Figure 9 (s = 0), in
the case of backward-looking indexation of prices (γ = 1).


γ = 1 better ¬ts their estimated impulse responses than does the standard model with
γ = 0. Smets and Wouter (2001) treat γ as a free parameter (in addition to using a di¬erent
estimation strategy), and conclude that the best-¬tting value of γ is an intermediate value,
approximately 0.7. Boivin and Giannoni (2001) also let γ be a free parameter, and estimate
(using a strategy based on matching impulse responses, but for a di¬erent sample period
than that used by Christiano et al.) a value of xx.

It should also be noted that when γ = 1, relation (3.6) is essentially identical to the
aggregate supply relation of Fuhrer and Moore (1995a, 1995b), that has been popular in
econometric work.63 Relation (3.6) is also of essentially the same form as the generalization
of the Calvo model proposed by Gali and Gertler (1999). Gali and Gertler assume that

63
In the limiting case β = 1, the relation is identical to that of Fuhrer and Moore, although the derivation
that they o¬er for their relation is di¬erent from the one given here.
3. DELAYED EFFECTS OF NOMINAL DISTURBANCES ON INFLATION 85

prices remain ¬xed in monetary terms for stochastic intervals of time, as in the Calvo model;
but when prices are adjusted, some prices are chosen optimally (as in the Calvo model),
while others are adjusted according to a backward-looking rule of thumb that introduces
dependence upon lagged in¬‚ation. The fraction of suppliers who are backward-looking is
treated as a free parameter; variation in this parameter has essentially the same e¬ect as
variation in γ in the indexation model. In the limiting case β = 1, the two models have
identical implications; relation (3.6) for any given value of γ can be obtained from the Gali-
Gertler model through an appropriate choice of the fraction of backward-looking price-setters.
While Gali and Gertler ¬nd that the basic Calvo model ¬ts U.S. in¬‚ation dynamics fairly
well (once real unit labor cost is used as a proxy for the output gap, rather than a traditional
output-based measure), their instrumental variables estimates indicate signi¬cant rejection
of the hypothesis of no backward-looking price-setters. Their point estimates for U.S. data
since 1980 indicate a fraction of backward-looking price-setters that would imply (in terms
of our notation) a value of γ on the order of 0.6. Gali, Gertler and Lopez-Salido (2000)
obtain similar results using European data. Thus there exists a fair amount of consensus ”
using a variety of empirical proxies for the output gap, and a variety of estimation strategies
” that the relation (3.6) better characterizes U.S. in¬‚ation dynamics when an indexation
parameter between 0.5 and 1.0 is included.
The backward-looking indexation model retains one unfortunate feature of the basic
Calvo model, however, even when γ is large. That is that if a monetary disturbance increases
nominal GDP only with a lag, as indicated in Figure 3.2, the model predicts that output
should initially contract in response to such a shock. The reason is that the expectation of
a future output increase implies a desire to increase πt ’ γπt’1 immediately; so there should
be an increase in in¬‚ation even before nominal GDP begins to increase.
This problem can be solved, once again, by assuming a delay of d quarters before a newly
chosen price takes e¬ect. (In the meantime, a ¬rm™s price continues to be adjusted using
(3.4).) In this case, the aggregate supply relation instead becomes

ˆ ˆ
πt ’ γπt’1 = κ Et’d (Yt ’ Ytn ) + β Et’d (πt+1 ’ γπt ).) (3.7)
86 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
0 4 8 12 16 20 24 28



Figure 3.15: Impulse responses to the same monetary disturbance as in Figures 11-13 (s = 2),
in the case of backward-looking indexation (γ = 1) and a delay of d = 2 quarters in price
changes.

Figure 3.15 shows the dynamic responses of output and in¬‚ation to an innovation in the
process (3.3) for nominal GDP, in the case that s = 2 quarters for conformity with Figure
3.2, and in the case of aggregate supply relation (3.7) with γ = 1 and d = 2. In this
case there is no e¬ect on output until two quarters following the shock, and then a “hump-
shaped” response, as in Figure 3.3. In addition, in¬‚ation exhibits a “hump-shaped” response
of its own, which peaks later than the output response; in this respect, note that this ¬gure
is much more similar to Figure 3.10 than was Figure 3.12 (the case γ = 0). Figure 3.16
shows the responses to the same disturbance in the case that d = 4. This results in a
stronger output response and a further delay in the in¬‚ation response. This ¬gure results in
predicted responses that are most similar to those implied by the VAR estimates shown in
Figure 3.10.64
4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 87




1




0.8
nominal GDP
real GDP
price level
0.6
inflation


0.4




0.2




0




’0.2
0 4 8 12 16 20 24 28



Figure 3.16: Impulse responses to the same monetary disturbance as in Figure 15, in the
case that γ = 1 and d = 4 quarters.


4 Consequences of Nominal Wage Stickiness

Thus far the only form of nominal rigidity considered is a delay of one sort or another
in the adjustment of the money prices at which goods are o¬ered for sale. In particular,
we have not assumed any corresponding stickiness of nominal wages, though this is another
familiar explanation for the real e¬ects of monetary policy, emphasized for example in Keynes
(1936). We have instead assumed that wages are fully ¬‚exible, or equivalently (as far as the
derivation of our aggregate supply relations is concerned) that there is e¬cient contracting
between ¬rms and their workers.65
64
We could do better at quantitatively matching the responses shown in Figure 3.10 if we were to treat κ
as a free parameter. In the experiments reported here, we have ¬xed the value of κ at the value estimated
by Rotemberg and Woodford (1997). However, their estimates are for a model without backward-looking
indexation, in addition to being based on a VAR and a sample period di¬erent from those of Christiano et
al.
88 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

Our exclusive emphasis upon price stickiness has allowed our models to take a particularly
simple form; in particular, it has been possible, for many purposes, to analyze in¬‚ation and
output dynamics without any reference to the labor market. Furthermore, as between the
two simple hypotheses (only sticky prices or only sticky wages), the hypothesis of sticky
prices is often regarded as more compelling on both theoretical and empirical grounds. We
have mentioned in the introduction to this chapter the objection that nominal wages might
be constant in the face of disturbances even when the e¬ective cost of marginal hours of labor
to ¬rms changes, owing to the existence of implicit contracts between ¬rms and workers. The
hypothesis of pure wage stickiness has also often been criticized on account of its implication
that real wages should move countercyclically (a criticism of Keynes™ model ¬rst raised by
Dunlop, 19xx, and Tarshis, 19xx). More relevant than overall business-cycle correlations is
the ¬nding, in VAR studies such as that of Christiano et al. (1997, 2001), of mildly pro-
cyclical real wage movements in response to identi¬ed monetary policy disturbances. (See,
for example, Figure 3.17, which shows the estimated impulse response of the average real
wage from the latter study. Here the impulse response of real GDP to the same type of
disturbance is also shown for purposes of comparison.) Since these responses ought to be
uncorrelated with changes in technology, the failure of the real wage to sharply decline at the
time of the increase in real activity is di¬cult to reconcile with a sticky-wage/¬‚exible-price
model.

But even if a model with only sticky wages is unappealing, it may be desirable to allow
for stickiness of wages as well as prices. Indeed, the study by Christiano et al. (1997)
criticizes sticky-price (but ¬‚exible-wage) models of the monetary transmission mechanism
on the ground that they imply too sharp a real wage decline in response to a tightening of
monetary policy ” one so strong that producers™ pro¬ts ought actually to increase, despite

65
As far as the relation between price changes and output is concerned, it is enough that goods suppliers
face a cost of marginal labor input at each point in time that is equal to the marginal disutility of labor
supply expressed in the monetary unit of account. It does not matter whether this occurs because this is the
wage that clears a competitive spot market for labor (as assumed explicitly above in our discussion of the
relation between unit labor costs and prices), or because an e¬cient labor contract leads ¬rms to internalize
the cost to their workers of requiring additional hours of work.
4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 89


1.5

real GDP
real wage
conf.interval
1




0.5




0




’0.5




’1
0 5 10 15 20 25 30



Figure 3.17: Estimated impulse response of the real wage to an unexpected interest-rate
reduction. Source: Christiano et al. (2001).

their reduced sales (and contrary to fact). This problem can be ameliorated by assuming
preference parameters that imply more elastic labor supply, as in the model of Rotemberg
and Woodford (1997). But slow adjustment of wages to changes in labor demand may be a
more plausible explanation for the relatively modest response of real wages seen in Figure
3.17, and, as we discuss in chapter 6, the choice among these two explanations matters for
welfare analysis. Hence we develop here an extension of our baseline model that incorporates
wage as well as price stickiness.


4.1 A Model of Staggered Wage-Setting

Here we follow Erceg et al. (1999) and model wage-setting in a way that is directly analogous
to the model of staggered pricing introduced by Calvo (1983). We introduce wage-setting
agents by assuming monopolistic competition among the suppliers of di¬erentiated types
90 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

of labor, analogous to our treatment above of the goods market. The exposition will be
simplest if we follow Erceg et al. in assuming a single economy-wide labor market, with the
producers of all goods hiring the same kinds of labor and facing the same wages.66 However,
we assume that the labor used to produce each good is a CES aggregate of the continuum
of individual types of labor supplied by the representative household, de¬ned by
θw
1 θw ’1
θw ’1
Ht ≡ ht (j) dj (4.1)
θw
0

for some elasticity of substitution θw > 1. Here ht (j) is the labor of type j that is hired; note
that the continuum of di¬erentiated types of labor is no longer identi¬ed with the continuum
of di¬erentiated goods indexed by i, as labor of all types is used in producing each good. It
follows that the demand for labor of type j on the part of wage-taking ¬rms will be given by
’θw
wt (j)
ht (j) = Ht , (4.2)
Wt
where wt (j) is the (nominal) wage demanded for labor of type j and Wt is a wage index
de¬ned analogously with (1.3).
We assume that the wage for each type of labor is set by the monopoly supplier of that
type, who then stands ready to supply as many hours of work as turn out to be demanded
at that wage. As in our model of monopolistic competition in the goods market, we assume
an independent wage-setting decision for each type j, made under the assumption that the
choice of that individual wage has no non-negligible e¬ect upon the the wage index Wt or
upon the demand Ht for the labor aggregate.67 We furthermore assume, as in the Calvo
66
The model presented here di¬ers, however, from that of Erceg et al. in that we do not assume that capital
can be instantaneously reallocated among ¬rms so as to equalize the return to capital services across ¬rms
that change their prices at di¬erent times. Instead, as in the baseline model of the previous section, capital
is here assumed to be completely immobile. We also adopt less parametric speci¬cations of preferences and
technology than in the paper of Erceg et al.
67
Both the assumption of a monopoly supplier of each type of labor and that the supplier of an individual
type of labor has no power to a¬ect the wage index Wt indicate that we can no longer assume a continuum of
identical households, each supplying the same continuum of types of di¬erentiated labor. Instead, we must
assume that households specialize in supplying a particular type of labor, the disutility of which is additively
separable from the utility of consumption as before. Risk-sharing among the households that supply types
of labor that set their wages at di¬erent dates can then result in a common budget constraint for each of
the households, which is the same as if each household were to receive its pro rata share of the economy™s
total wage bill, as earlier.
4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 91

model of staggered pricing, that each of the wages is adjusted with only a probability 1 ’ ±w
each period, for some 0 < ±w < 1, which probability is independent of the time since a given
wage was last adjusted or the current level of that wage.
It follows that a wage wt (j) that is adjusted in period t should be chosen to maximize

(±w β)T ’t [ΛT wt (j)hT (wt (j)) ’ v(hT (wt (j)); ξT )] ,
Et (4.3)
T =t

where ΛT is the representative household™s marginal utility of nominal income68 in period
T and the dependence of labor demand hT (j) upon the wage is given by (4.2). (We have
omitted a j in our notation for this last function because the function hT (w) is the same for
all j.) The solution to this problem satis¬es the ¬rst-order condition

θ
±w’t Qt,T HT WT w [wt (j) ’ µw V(hT (j), CT ; ξT )PT ] = 0,
T
Et (4.4)
T =t

which has a form analogous to (2.4). Here
vh (h; ξ)
V(h, C; ξ) ≡
uc (C; ξ)
is the marginal rate of substitution between work and consumption for the supplier of a given
type of labor, and µw ≡ θw /(θw ’ 1) > 1 is the desired markup of a household™s real wage
demand over its marginal rate of substitution owing to its monopoly power. If we substitute
(4.2) for hT (j) in (4.4), we obtain a relation that implicitly de¬nes the optimal wage choice
— —
wt , which is the same for all wages j that are adjusted at date t. The choice of wt then
determines the evolution of the wage index Wt , through a law of motion analogous to (2.1).
Again it is useful to approximate equilibrium wage dynamics in the case of small dis-
turbances using a log-linear approximation, computed under the assumption that Wt /Pt ,

Pt /Pt’1 and wt /Wt all remain close to their steady state-values (w, 1 and 1 respectively) at
¯
all times. For the law of motion of the wage index we obtain
1 ’ ±w —
w
πt ≡ ∆ log Wt = wt ,
ˆ (4.5)
±w
68
Note that although there is technically not a representative household in the present model, in the sense
of a household whose trades are equal to its per capita share of all trades in the economy (owing to labor
specialization), it remains true that all households have the same consumption budget and choose the same
consumption. Hence they have the same marginal utility of income Λt at all times, equal to uc (Ct ; ξt )/Pt .
92 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

ˆ— —
where wt ≡ log(wt /Wt ), analogous to the approximate relation (2.6) for the price index. As
an approximation to (4.4) we obtain
± 
 
∞ T
(±w β)T ’t wt ’ [ˆt,T ’ wT +
ˆ— w
Et v ˆ π„ ] = 0, (4.6)
„ =t+1
T =t

where vt,T denotes the deviation (from its steady-state value) of the log of the marginal rate
ˆ
of substitution between labor and consumption in period T for a type of labor with wage

wt , and wT denotes deviation of the log of the real wage WT /PT from log w, analogous to
ˆ ¯
relation (2.7) for optimal price-setting. Noting furthermore that
T
ˆ— w
vt,T = vT ’
ˆ ˆ νθw [wt ’ π„ ],
„ =t+1

where vT is the average value of vt,T over di¬erent types of labor, and
ˆ ˆ
¯
vh h(h; 0)
ν≡¯ ¯ > 0,
hvh (h; 0)
we can solve (4.6) for
∞ T
ˆ— T ’t ’1 w
wt = (1 ’ ±w β)Et (±w β) [(1 + νθw ) (ˆT ’ wT ) +
v ˆ π„ ].
„ =t+1
T =t

This allows us to express the relative wage of types of labor that adjust their wage at date
t as a forward-looking function of aggregate conditions.
Manipulations directly analogous to those used in our analysis of staggered pricing allow
us to obtain a relation of the form

∆ log Wt = ξw [ˆt + log w + log Pt ’ log Wt ] + βEt [∆ log Wt+1 ],
v ¯ (4.7)

where
(1 ’ ±w )(1 ’ ±w β)
ξw ≡ > 0.
±w (1 + νθw )
If we continue to assume a production function of the form (1.7) for each good (i.e., a ¬xed
allocation of capital across ¬rms, even though there is a single economy-wide labor market),
the corresponding equation for the price dynamics is

ˆ
∆ log Pt = ξp [ψt ’ log w + log Wt ’ log Pt ] + βEt [∆ log Pt+1 ],
¯ (4.8)
4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 93

ˆ
where ψt denotes minus the average deviation of the log marginal product of labor from its
steady-state value. Here
(1 ’ ±p )(1 ’ ±p β)
ξp ≡ > 0,
±p (1 + ωp θp )
where the parameters previously denoted ± and θ are now denoted ±p and θp , and ωp is
de¬ned as in (1.15). Note that equation (4.8) is just an alternative way of writing (2.8),
given that
ˆ
st = log Wt ’ log Pt + ψt .
ˆ

(The only di¬erence is that the expression for ξp is di¬erent now owing to the assumption of
an economy-wide labor market.)
Our assumed forms for the utility and production functions furthermore imply that

ˆ ˆ
ψt = ωp Yt ’ (1 + ωp )at ,
ˆ ˆ
vt = (ωw + σ ’1 )Yt ’ (ω + σ ’1 )Ytn + (1 + ωp )at ,
ˆ

where ωw , ωp represent the decomposition presented in (1.15) and at ≡ log At . In deriving
the second equation, we use the fact that ωw ≡ νφ, where
¯
f (h)
φ≡ ¯ ¯ >1
hf (h)
is the elasticity of the representative ¬rm™s labor requirement with respect to its level of
production. Note that log w + vt represents the equilibrium log real wage in a model with
¯ˆ
¬‚exible wages, so that the above expressions imply expression (2.12) for the case of ¬‚exible
wages. Substituting these into (4.7) and (4.8), we obtain a pair of coupled equations for
evolution of wages and prices given the path of real activity and the exogenous disturbances.
These can be written in the form

ˆ ˆ
∆ log Wt = κw (Yt ’ Ytn ) + ξw (log wt + log Pt ’ log Wt ) + βEt [∆ log Wt+1 ],
n
(4.9)

ˆ ˆ
∆ log Pt = κp (Yt ’ Ytn ) + ξp (log Wt ’ log Pt ’ log wt ) + βEt [∆ log Pt+1 ],
n
(4.10)

where
κw ≡ ξw (ωw + σ ’1 ) > 0, κp ≡ ξp ωp > 0,
94 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

and
ˆ
log wt ≡ log w + (1 + ωp )at ’ ωp Ytn
n
¯

represents the “natural real wage”, i.e., the equilibrium real wage when both wages and
prices are fully ¬‚exible.
These equations generalize the aggregate supply relation derived earlier for the ¬‚exible-
wage model in an obvious respect. Note that if we de¬ne a particular weighted average of
wage and price in¬‚ation,
’1 ’1
ξp πt + ξw πwt
πt ≡
¯ , (4.11)
’1 ’1
ξp + ξw
then the corresponding weighted average of (4.9) and (4.10) reduces to

ˆ ˆ
πt = κ(Yt ’ Ytn ) + βEt πt+1 ,
¯ ¯ (4.12)

where
σ ’1 + ω
κ ≡ ’1 >0 (4.13)
’1
ξp + ξw
is a coe¬cient that is smaller the greater the degree of rigidity of either wages or prices. In
the limit as ±w ’ 0, ξw ’ ∞, πt simply measures price in¬‚ation, and (4.12) is again the
¯
aggregate supply relation (2.13) obtained earlier, in which (4.13) corresponds once more to
(2.14) using the value (1.25) for ζ. More generally, we ¬nd that an alternative in¬‚ation index
is the same kind of purely forward-looking function of the output gap, but this index involves
wage in¬‚ation with a weight that is greater the greater the relative stickiness of wages. In
the limiting case that only wages are sticky, (4.12) becomes a “Phillips curve” for wages.


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

We turn now to the implications of wage stickiness for the real e¬ects of ¬‚uctuations in
nominal expenditure due to a purely monetary disturbance. As is well-known, nominal wage
stickiness implies that such disturbances should have temporary e¬ects on real activity,
during the time that it takes for wages to adjust, even in the case of fully ¬‚exible goods
prices. A more subtle question is whether this mechanism should result in real e¬ects that
4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 95

are more or less persistent than those that would result from sticky prices. Some authors
(e.g., Andersen, 1998; Huang and Liu, 1998) have argued that sticky wages result in more
persistent e¬ects than do sticky prices, and that the assumption of sticky prices (as in our
treatment in the earlier part of this chapter) therefore underestimates the likely importance
of the real e¬ects of monetary policy.
In order to compare the degree of persistence resulting from wage stickiness with that
resulting from price stickiness, it is useful to consider once again the e¬ects of an unexpected
permanent increase in nominal GDP, unrelated to any real disturbance (that could a¬ect
ˆ
Ytn or wt ). Let us again assume a shock that results in (2.15) holding for all t ≥ 0, and
n


consider the expected paths E0 log Pt , E0 log Wt and E0 log Yt that are consistent with the
system (4.9) “ (4.10), given initial conditions log W’1 = log P’1 = 0.
In the case of sticky wages but purely ¬‚exible prices, (4.10) reduces to an equilibrium
relation between the real wage and the output gap, that can be written in the form
ωp
n
(log Yt ’ log Wt + log wt ’ log Ytn );

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