log Wt ’ log Pt = log wt ’ (4.14)

1 + ωp

Substituting this into (4.9) yields

∆ log Wt = κ[log Yt ’ log Wt + log wt ’ log Ytn ] + βEt [∆ log Wt+1 ]

n

ˆ

for nominal wage dynamics given the evolution of nominal GDP and the real disturbances,

where

κ

˜

κ≡

ˆ > 0.

1 + ωp

This equation has the same form as the equation for price dynamics in the model with ¬‚exible

wages and sticky prices, and so our reasoning in section xx above directly applies. It follows

that if we let wt denote E0 log Wt ’ 1, the unique bounded solution is of the form

˜

wt = ’»t+1 ,

˜

by analogy with (2.17), where 0 < » < 1 is the smaller root of (2.18) when κ is substituted

ˆ

for κ. Then noting that (4.14) implies as well that

1

log Yt = log Ytn + (log Yt ’ log Wt + log wt ’ log Ytn ),

n

1 + ωp

96 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

we see that (4.2) implies an output response of the form

E0 log Yt = ’(1 + ωp )’1 wt = (1 + ωp )’1 »t+1 .

˜

The degree of persistence of the output response thus depends upon the root », and

hence upon the size of κ, in the same way as in the sticky-price model. The only possibility

˜

of a di¬erence between the two models as to the likely degree of persistence would be if it

is judged more plausible that κ should be small in the case of the sticky-wage model than

˜

that κ should be small in the case of the sticky-price model. One possible reason for this,

of course, would be if it were observed that wages are in fact adjusted less frequently than

prices. But it is not obvious that this is true to any dramatic extent; in an economy like that

of the U.S., most wages, like most prices, are adjusted annually if not more often. (And this

is in any event not the basis of Andersen™s argument, that assumes two-period commitments

in each case.)

Let us instead consider the likelihood of there existing a “contract multiplier” in the

sense discussed above. We have seen that in the case of the sticky-price model, » > ±p , so

that output e¬ects decay more slowly than the rate at which prices are revised following the

shock, if and only if ζ < 1 (the case of strategic complementarity among pricing decisions),

where ζ is again given (for the present setup) by (1.25). We can similarly show that for the

˜

sticky-wage model, » > ±w if and only if ζ < 1, where

ω + σ ’1

˜

ζ= . (4.15)

(1 + νθw )(1 + ωp )

˜

(Once again, the size of ζ can be interpreted as a measure of the degree of strategic com-

plementarity among the wage-setting decisions of suppliers of di¬erent types of labor.) Our

˜

question then reduces to asking whether it is more plausible that ζ should be less than one

in the sticky-wage model than it is that ζ should be less than one in the sticky-price model.

This turns out to be true in the analysis of Andersen (1998), but his simple model omits

a number of important considerations. His analysis of the sticky-wage model e¬ectively

assumes (in terms of our notation) that ν = σ ’1 = 0 (which also implies that ωw = 0), and

4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 97

hence concludes that

ωp

˜

ζ= < 1,

1 + ωp

implying the existence of a “contract multiplier” regardless of the size of ωp > 0. However,

this result is not as general as it would appear; a contract multiplier less than one is still

theoretically possible in the sticky-wage model (if, for example, one assumes ν = 0 but

σ < 1).

Andersen™s analysis of the sticky-price model similarly e¬ectively assumes that ωp =

σ ’1 = 0 (which also implies that ωw = ν), and hence concludes that ζ = ν, which may or

may not be less than one.69 But this comparison of the two models is misleading. In fact,

˜

as we have seen, the numerators of the expressions for ζ and ζ are identical, both being

equal to ω + σ ’1 ; Andersen obtains the value ωp for the sticky-wage model (which should

be small even if ν is large) only because he has assumed ν = 0 in that model. The most

important di¬erence between the two cases is instead the presence of the factor (1 + νθw )

in the denominator of (4.15), whereas the corresponding factor in the denominator of (1.25)

is (1 + ωp θp ) ” neither of which factors appear in Andersen™s analysis. If one regards ν

as being substantially larger than ωp ,70 and one supposes that there is at least as much

substitutability among types of labor as among di¬erent goods (so that ωw ≥ ωp ), then it

˜

would indeed follow that ζ should be signi¬cantly smaller than ζ, and wage stickiness would

lead to more output persistence than would price stickiness.

This is in fact the basis for the conclusions of Huang and Liu (1998), who also present

a comparison of the consequences of wage and price stickiness, but with explicit micro-

foundations for the assumed wage- and price-setting equations (and hence consistent assump-

69

Andersen suggests that the most plausible assumption about preferences would involve ν > 1; smaller

values are popular in calibrated quantitative business-cycle models in order to account for the relative acycli-

cality of real wages, but he proposes that this re¬‚ects wage stickiness rather than the nature of preferences

regarding labor supply.

70

It is not obvious that this must be true. For example, Hansen (1995) argues for preferences with ν = 0,

as a result of the indivisibility of labor; in his model, variation in hours worked is entirely associated with

variation in the number of workers who work a shift of ¬xed length, rather than variation in the number

of hours worked by each member of the labor force, as assumed in the representative-household model used

here.

98 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

tions about preferences and technologies in the two cases). Their basic model (abstracting

from capital accumulation and endogenous velocity of money) is a special case of the one

just presented, in which the production function f (·) is assumed to be linear, so that ωp = 0

and ωw = ν, and in which u(C) = log C, so that σ = 1. They accordingly ¬nd that

1+ν

˜

ζ= < 1, ζ = 1 + ν > 1,

1 + νθw

so that there is necessarily strategic complementarity in the sticky-wage model and neces-

sarily none in the sticky-price model.

Yet this conclusion depends upon ignoring a number of reasons discussed earlier for a low

value of ζ to be plausible in the model with only sticky prices. In particular, we have shown

above that if we assume that the producers of di¬erent goods hire labor from distinct labor

markets, the value of ζ is instead given by (1.22). In this case, the factor in the denominator

is (1 + ωθp ); this is larger than the factor that appears in (1.25), and especially noteworthy

is the fact that a large value for ν increases the size of this factor in much the same way as

in the case of (4.15). Indeed, under the special parametric assumptions of Huang and Liu,

we would ¬nd

1+ν

ζ= ,

1 + νθp

˜

which is just as small as the value that they obtain for ζ, as long as θp ≥ θw , i.e., as long as

price-setters have no more market power than do wage-setters. Thus there is little reason to

expect that persistence should be greater in the case of a sticky-wage model than in that of

a sticky-price model, once one allows for speci¬c labor markets in the sticky-price model.71

It is nonetheless true that wage stickiness will generally increase the size and persistence

of the real e¬ects of nominal disturbances, holding ¬xed the degree of price stickiness. Note

that if we subtract (4.10) from (4.9) we obtain

∆ log wt = ’(ξw + ξp )(log wt ’ log wt ) + (κw ’ κp )(log Yt ’ log Ytn ) + βEt [∆ log wt+1 ], (4.16)

n

71

Edge (2002) demonstrates this quantitatively, in the case of a complete general-equilibrium model of the

monetary transmission mechanism with endogenous capital accumulation.

4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 99

where wt ≡ Wt /Pt is an aggregate real wage. In the special case that κw = κp (so that wages

are sticky to roughly the same extent as are prices), the output term drops out, and the

only endogenous variable in (4.16) is the real wage. One easily veri¬es that this equation

has a unique bounded solution for the path of the real wage, given a bounded process for

n

the exogenous disturbance wt ; hence in this case, the equilibrium real wage is determined

by this equation alone, independent of monetary policy or any other factors that a¬ect only

the demand side of the model. Let the solution for wt be denoted wt . Then substitution of

¯

this solution into (4.10) gives us an equation for price in¬‚ation of the form

ˆ ˆ

πt = κ(Yt ’ Ytn ) + Et πt+1 + ut , (4.17)

where κ is the common value of κw and κp , and ut is an exogenous term de¬ned as

n

ut ≡ ξp (log wt ’ log wt ).

¯ (4.18)

This is again an aggregate supply relation of the same form as (2.13), except that the

ˆ

exogenous intercept is no longer equal to Ytn . It follows that the e¬ect upon output of a

monetary disturbance is exactly the same as in our previous analysis, and depends upon

the value of κ in the way discussed earlier. (The nature of the exogenous intercept term

does not matter for this question, since in any event it is una¬ected by a purely monetary

disturbance.) We note that in the present case (with both wages and prices sticky), the

slope coe¬cient κ is equal to ξp ωp ,, while in the case of ¬‚exible wages analyzed earlier, it

was equal to ξp (ω + σ ’1 ). Thus κ is a smaller positive quantity in the present case (assuming

that all parameters take the same values, except the parameter ±w determining the degree

of wage stickiness), implying both larger and more persistent output e¬ects of a monetary

disturbance.

Allowing for wage stickiness can also be important in improving the ability of the model

to account for the observed behavior of wages as well as prices. The model with fully ¬‚exible

wages implies that a monetary contraction should lower wages by more than the decline in

prices, for in this limiting case, (4.9) reduces to the equilibrium relation

ˆ ˆ

log Wt ’ log Pt = log wt + (ωw + σ ’1 )(Yt ’ Ytn ).

n

(4.19)

100 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

Thus the predicted decline in real wages must be substantial relative to the decline in output,

unless both ω and σ’1 are quite small in value; in particular, this requires that ν be quite

small (as it is necessary that ω > ν). But if wages are sticky as well as prices, it is possible

for the decline in real wages to be small, or even non-existent, even if ν is of substantial

magnitude. Indeed, we have just seen that if κw = κp , there is no e¬ect of a monetary

disturbance upon the real wage at all, and this condition can hold regardless of the value of

ν. (For any values of the other parameters, it is possible to arrange that κw = κp simply by

72

assigning an appropriate value to ±w .)

The model with both wage and price stickiness also allows for more complicated wage

dynamics than can be achieved in the ¬‚exible-wage model through any choice of the parame-

ters ω and σ, for it ceases to be necessary that the e¬ects of monetary policy on the real wage

be any constant multiple of the e¬ects upon real GDP. For arbitrary coe¬cients ξw , ξp > 0

and bounded processes {Yt , Ytn , wt }, equation (4.16) has a unique bounded solution for the

n

real wage. This is given by

∞

’j’1

’1

log wt = »1 log wt’1 + β »2 Et zt+j , (4.20)

j=0

where 0 < »1 < 1 < »2 are the two roots of the characteristic polynomial

P (») ≡ β»2 ’ (1 + β + ξw + ξp )» + 1 = 0,

and the forcing process {zt } is de¬ned by

zt ≡ (ξw + ξp ) log wt + (κw ’ κp )(log Yt ’ log Ytn ).

n

A monetary disturbance a¬ects the path of zt solely through its e¬ect on Yt , as the other

terms are exogenous. In the case that either wages or prices are perfectly ¬‚exible (so that

either ξw or ξp is unboundedly large), both »1 and »’1 equal zero, and (4.20) implies that

2

log wt is a multiple of zt ; but when both are sticky, so that ξw + ξp is ¬nite, log wt is instead

proportional to a smoothed version of the forcing process.

72

If wages are made su¬ciently sticky, the model would predict an actual increase in the real wage follow-

ing a monetary contraction (the prediction of the classic Keynesian model), though evidence such as that

presented by Christiano et al. (1997) does not indicate that this actually occurs.

4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 101

0.3

ξw=∞

0.25

ξ =3

w

ξw=1

0.2

ξw=0.7

0.15

ξw=0.55

ξw=0.4

0.1

ξw=0.1

0.05

0

’0.05

’0.1

’0.15

’0.2

0 5 10 15 20 25 30 35 40

Figure 3.18: Predicted impulse response of the real wage, for alternative degrees of wage

stickiness. The solid line indicates the response of real GDP.

The e¬ects of wage stickiness are illustrated in Figure 3.18. Here a given impulse response

function for output in response to a monetary disturbance is assumed, and the implied

response of the real wage to the same disturbance is then inferred from (4.20). The implied

real-wage responses are plotted along with the output response for each of several alternative

assumed degrees of nominal wage stickiness, corresponding to di¬erent values of ξw . Here

the assumed response of output is the same as in Figure 3.9, and the values assumed for the

parameters ξp , ωw , ωp , and σ are again those of Rotemberg and Woodford (1997) (see Table

4.1 of chapter 4). The limiting case ξw = ∞ (complete wage ¬‚exibility) corresponds to the

assumptions of Rotemberg and Woodford; the other cases shown each involve some degree

of wage stickiness.

As the degree of wage stickiness increases, the size of the real wage increase associated

with a given size of e¬ect on output declines. When ξw = 0.55, κw falls to a value equal

102 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

to that of κp , and wages and prices are in a sense “equally sticky”; in this case, there is no

longer any real wage response at all to a monetary disturbance that increases output. For

even lower values of ξw , wages are stickier than prices, and the real wage is predicted to move

counter-cyclically.

The estimated real wage response shown in Figure 3.17 suggests that the empirically

relevant case is one in which κw is slightly larger than κp , but not much. Amato and Laubach

(1999) reach a similar conclusion when they estimate a model that allows for wage as well as

price stickiness, as discussed in section xx of chapter 4. Christiano et al. (2001) and Smets

and Wouter (2001) similarly ¬nd that the impulse responses to identi¬ed monetary policy

shocks are best ¬t by a model that incorporates both wage and price stickiness.

Furthermore, allowing for wage as well as price stickiness allows additional ways in which

real disturbances may shift the aggregate supply curve, i.e., the short-run relation between

in¬‚ation and output for given in¬‚ation expectations. In the aggregate supply relation (2.13)

derived under the assumption of sticky prices but ¬‚exible wages, any of a variety of real

disturbances can shift this relation ” variation in the rate of technical progress, government

purchases, and various types of shifts in preferences ” but in each case, the aggregate supply

relation is shifted exactly to the extent that the disturbance in question changes the natural

rate of output (i.e., the equilibrium output level with ¬‚exible wages and prices). But in

(4.17), this is no longer true, unless the term ut is zero.

n

In general this term is not identically zero (i.e., wt = wt ), even when the degree of wage

¯

n

stickiness is exactly that required to make κw = κp . One notes that wt = wt is a solution to

n n

(4.16) in the case that κw = κp only if ∆ log wt = βEt ∆ log wt+1 , which in turn is possible,

n n

if wt is a stationary process, only if wt is a constant ” i.e., if the real disturbances present

in the model would never a¬ect the equilibrium real wage in the case of ¬‚exible wages and

prices. This is true only for extremely special parameter values and/or assumptions about

the kinds of real disturbances that can occur. Under any other assumptions, solving (4.16)

allows us to determine how various types of real disturbances shift the aggregate supply

4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 103

relation in a way that di¬ers from their e¬ect upon the natural rate of output.73 Here we

do not pursue the topic further, except to note that the existence of a non-zero ut term

implies a tension between the goals of stabilizing in¬‚ation and of stabilizing the output gap

ˆ ˆ

Yt ’ Ytn that does not appear in our baseline sticky-price model. The consequences of this

for optimal stabilization policy are taken up in section xxx of chapter 6.

Finally, we note that infrequent reoptimization of wage demands need not mean that

wages remain ¬xed in money terms between the occasions on which they are re-optimized;

instead, wages might be indexed to an aggregate price index in the interim, just as in our

discussion of goods-price indexation in section xxx. Indeed, indexation schemes of this kind

for wages are sometimes a part of multi-year union contracts, so that there is more direct

evidence for the idea of indexation in the case of wages than for prices; and in practice, such

indexation is always to a lagged price index. Suppose we let 0 ¤ γw ¤ 1 be the indexation

rate for wages that are not-reoptimized; that is, if the wage demanded for labor of type j is

not reoptimized in period t, it is adjusted according to the indexation rule

log wt (j) = log wt’1 (j) + γw πt’1 .

This modi¬es the relations for optimal wage-setting in a way that is directly analogous to

our discussion of optimal price-setting with backward-looking indexation in section xxx.

The result is that the system (4.9)“(4.10) becomes instead

ˆ ˆ

πt ’ γw πt’1 = κw (Yt ’ Ytn ) + ξw (log wt ’ log wt ) + βEt [πt+1 ’ γw πt ],

w n w

(4.21)

ˆ ˆ

πt ’ γp πt’1 = κp (Yt ’ Ytn ) + ξp (log wt ’ log wt ) + βEt [πt+1 ’ γp πt ],

n

(4.22)

w

where πt ≡ ∆ log Wt is the rate of wage in¬‚ation, wt ≡ Wt /Pt is the aggregate real wage,

γp is the rate of indexation of price commitments (called simply γ in section xxx), and the

coe¬cients κw , κp , ξw , ξp are again de¬ned exactly as in (4.9)“(4.10). These equations once

73

For general parameter values, of course, we are not able to solve for the equilibrium real wage independent

of the evolution of real activity, as indicated by (4.20). In the general case there is no exogenous term ut for

which an aggregate supply relation of the simple form (4.17) holds. But our more general point, that when

both wages and prices are sticky it is no longer true that in¬‚ation stabilization and output-gap stabilization

imply one another, continues to hold.

104 CHAPTER 3. OPTIMIZING MODELS WITH NOMINAL RIGIDITIES

again provide a pair of coupled equations for the evolution of the aggregate wage and price

indices, given the path of aggregate output and the real disturbances. Note that in the

case that γw = γp = 1, this represents a simpli¬ed version of the aggregate-supply block

of the model of Christiano et al. (2001), in which we abstract from complications such as

endogenous capital accumulation. Smets and Wouters (2001) also estimate a model with an

aggregate-supply block of this kind, but treat γw and γp as free parameters to be estimated.

(Their best-¬tting values are xxxx.) Both groups of authors ¬nd that a model of this type

” incorporating staggered wage-setting as well as staggered price-setting, and automatic

indexation of both wages and prices to recent past in¬‚ation ” can account fairly well for

the joint dynamics of wages, prices, and real activity.

The existence of the additional terms due to indexation results in in¬‚ation inertia much

as has already been discussed in the context of the ¬‚exible-wage model of section xxx. For

example, the in¬‚ation dynamics associated with the Fuhrer-Moore aggregate supply relation

(3.6) can occur as a result of wage rather than price indexation. In the case of fully ¬‚exible

(and hence non-indexed) prices, and the special case of a linear production function (so that

n

ωp = 0), (4.14) reduces to wt = wt , in which case (4.9) implies that

ˆ ˆ

πt ’ γπt’1 = κ(Yt ’ Ytn ) + βEt [πt+1 ’ γπt ] + ut , (4.23)

where γ = γw , κ = κw , and

n n

ut ≡ βEt ∆ log wt+1 ’ ∆ log wt .

Alternatively, similar in¬‚ation dynamics can result from a similar degree of indexation of

wages and prices. For example, in the special case that γw = γp = γ and κw = κp = κ,

subtracting (4.22) from (4.21) again yields an equation that can be solved for the real wage

dynamics independent of monetary policy. Substituting this solution wt = wt into (4.22)

¯

again yields an aggregate supply relation of the form (4.23), but where now ut is de¬ned as

in (4.18).

We thus ¬nd that allowing for wage stickiness does not matter all that much, if our goal

is simply to construct a positive model of the co-movement of in¬‚ation and output, and the

4. CONSEQUENCES OF NOMINAL WAGE STICKINESS 105

way that both can be a¬ected by monetary policy. (Stickiness of wages reduces the slope of

the short-run Phillips curve, but a similar degree of ¬‚atness could alternatively be obtained

by choosing di¬erent values for other parameters in a ¬‚exible-wage model, without any

counterfactual consequences for the dynamics of output and in¬‚ation; stickiness of wages

creates a new way in which real disturbances can shift the short-run Phillips curve, but

similar consequences for the evolution of in¬‚ation and the output gap would be obtained by

simply postulating an exogenous “cost-push shock” as many authors do.) To this extent, the

relative neglect of wage-setting in the early literature on the optimizing models with nominal

rigidities can be given a justi¬cation.

On the other hand, we shall see that there is an important respect in which it does matter

whether one thinks that wages, prices or both are sticky. This has to do with the proper

goals of monetary stabilization policy from a welfare-theoretic point of view. If one takes as

given an ad hoc stabilization goal, de¬ned in terms of the stability of a certain price index

and a certain measure of the output gap, then an adequate model for determining how policy

can best achieve this goal may well be formulated without having to take a stand on the

question of wage stickiness. But if one asks whether it is really appropriate, from the point

of view of economic welfare, to de¬ne the goal of policy in that way, it turns out to matter

after all whether one believes that wages are sticky. Even in the cases discussed above in

which monetary policy is unable to a¬ect the evolution of the real wage (so that the e¬ects

of purely monetary disturbances on wage and price in¬‚ation must be identical), it does not

follow that wage and price in¬‚ation should be indistinguishable: they will be di¬erentially

a¬ected by real disturbances. But it then follows that seeking to stabilize wage in¬‚ation

and seeking to stabilize price in¬‚ation will not be equivalent policies. Which is the more

appropriate goal? The answer depends on the degree to which wages as opposed to prices

are sticky, as we show in chapter 6.