An Estimated Monetary DSGE Modelwith
Unemployment and Staggered Nominal Wage Bargaining∗Mark Gertler* , Luca Sala** and Antonella Trigari***New York University, ** Bocconi University, IGIER
October, 2007
(first draft November, 2006)

Abstract
We develop and estimate a medium scale macroeconomic model that allows for unemployment and staggered nominal wage contracting. In contrast to most existing quantitative models,
employment adjustment is on the extensive margain and the employment of existing workers is
eﬃcient. Wage rigidity, however, aﬀects the hiring of new workers. The former is introduced
via the staggered Nash bargaing setup of Gertler and Trigari (2006). A robust finding is thatthe model with wage rigidity provides a better description of the data than does a flexible wage
version. Overall, the model fits the data roughly as well as existing quantitative macroeconomic
models, such as Smets and Wouters (2007) or Christiano, Eichenbaum and Evans (2005). Morework is necessary, however, to ensure a robust identification of the key labor market parameters.

∗

Prepared for the JMCB conference in honor of Ernst Baltensperger, Bern, Switzerland, 2007. Mark Gertler
thanks the National Science Foundation and the Guggenheim Foundation for financial support.

1

1

Introduction

This paper develops and estimates a quantitative macroeconomic framework that incorporates labormarket frictions. Our starting point is the now conventional monetary DSGE model developed by
Christiano, Eichenbaum and Evans (CEE, 2005), Smets and Wouters (SW, 2007) and others. We
introduce labor market frictions with a variant of the Mortensen and Pissarides search and matchingframework. This variant allows for staggered Nash wage bargaining, as in Gertler and Trigari (GT,2006).
Our motivation is twofold. First, there are some compelling theoretical considerations. The
recent vintage of monetary DSGE models typically has employment adjusting on the intensive
margin along with staggered nominal wage contracting. The latter feature, further, is important for
the quantitative performance of the model: The wage stickiness helps accounting for the volatilityof hours. However, as a consequence, these frameworks are susceptible to Barro’s (1977) argumentthat wages may not be allocational in this kind of environment, given that firm’s and workers havean on-going relationship. If wages are not allocational then wage rigidity does not influence model
dynamics. By contrast, in the model we present, firms adjust employment along the extensive
margin. In this instance, wage rigidity aﬀects employment by influencing the rate at which firms
add new workers to their respective labor forces. As emphasized by Hall (2005a), since new workers
have yet to form on-going relationships with firms, in this kind of setting the Barro’s critique does
not apply.
Second, within the search and matching a literature, there is a debate over how well the baseline
Mortensen and Pissarides framework can account for labor market volatility, or whether it may benecessary to introduce additional features such as wage rigidity, etc. (e.g. Shimer, 2005, Hall, 2005,
Hagedorn and Manovskii, 2006, Mortensen and Nagypal, 2007). A typical approach in the literature
has been to develop a calibrated model, subject the model to productivity shocks, and then examine
model moments against moments of the data, with various features such as wage rigidity or on-thejob search shut on and then shut oﬀ. We instead estimate a complete macroeconomic frameworkusing Bayesian methods. Doing so allows us to formally evaluate the significance of diﬀerent
mechanisms such as wage rigidity to overall model performance. In addition, our full information
procedure permits us to account for the complete range of shocks that hit the economy.
In section 2 we develop the model. The basic framework follows CEE and SW closely. The
only significant diﬀerence involves the treatment of the labor market. As in GT, we incorporate avariation of Mortensen/Pissarides that retains the empirically appealing feature of Nash bargaining,
but replace the assumption of period-by-period wage negotiations to allow for staggered multiperiod wage contracting. Each period, only a subset of firms and workers negotiate a wage contract.
Each wage bargain, further, is between a firm and its existing workforce: Workers hired in-between
contract settlements receive the existing wage.1 In the language of Hall (2005), the existing contract
1

A number of authors argue that the wages of new hires are more flexible that those of existing workers. See, for2

wage provides a wage norm for the workers hired in-between contracting periods. We restrict theform of the wage contract to call for a fixed wage per period over an exogenously given horizon.
Though it surely would be preferable to fully endogenize the contract structure, the payoﬀ fromour shortcut is a simple empirically appealing wage equation that is an intuitive generalization of
the standard Nash bargaining outcome. In this instance, a key primitive parameter of the model isthe average frequency of wage adjustment. Whereas in GT this parameter is calibrated to existing
evidence on wage contract length, here we are able to estimate it.
Another significant diﬀerence from GT, which is a purely real model, is that wage contracting
is in nominal terms. However, as in CEE and SW, we allow for indexing of wages to past inflationand estimate the degree of indexing. This consideration is important for the following reasons.
As indexing to past inflation becomes complete, nominal wage rigidity begins to approximate real
wage rigidity. As Blanchard and Galí (2006) emphasize, real wage rigidity complicates the shortrun output/inflation trade-oﬀ that the central bank faces, beyond what would arise from simple
nominal rigidities.
In section 4 we describe our estimation procedure and then present the model estimates. We use
Bayesian methods, following closely SW and Primiceri, Schaumburg and Tambalotti (PST, 2006).
We present a variety of diagnostics to evaluate the overall model performance and in particular therole of wage rigidity in our framework.
Before proceeding, we emphasize that there have been a number of papers related to ours.
Trigari (2004) and Walsh (2005) were among the first to integrate a search and matching setup
within a monetary DSGE model with nominal price rigidities. Blanchard and Galí (2006) develop
a qualitative version of this model with a simple form of real wage rigidities. Christoﬀel et al.
(2006) have also estimated a monetary DSGE model with labor market frictions and wage rigidity.
They employ a setup with right-to-manage bargaining as in Trigari (2006), where ex post hours
may be ineﬃcient. Since part our interest is to address the Barro critique, we employ the setup
of GT, which has eﬃcient bargaining along with staggered wage setting. Thus, within our setup
the employment of existing workers is fully eﬃcient: wage rigidity aﬀects hiring at the extensive
margin. In addition, while Christoﬀel et al. (2006) model wage rigidity by introducing adjustment
costs of wage changes for a representative firm, we do so by having staggered contracting.2 We
also diﬀer in the details of the precise model we estimate, as well as the exact estimation procedureand data.

example, the survey by Pissarides (2007). In Gertler, Huckfeld and Trigari (2007), however, we show that by more
carefully controlling for job tenure and other forms of heterogenity, the evidence that new workes wages are more
flexible disappears. In this study we use data from the CPS and are currently extending the analysis to other data
sets.
2
More recently, Christiano, Motto and Rostagno (2007) have also incorporated the GT variation of the MP model
into the CEE/SW framework to study stock market boom bust cycles. The focus of our paper is instead in parameter
estimation and model comparison.

3

2

The Model

As we discussed, the model is a variant of the conventional monetary DSGE framework. It has the
key features that many have found useful for capturing the data. These include habit formation,
costs of adjusting the flow of investment, variable capital utilization, nominal price and wage
rigidities, and so on. The key changes involve the labor market. Rather than having hours vary on
the intensive margin, we introduce variation on the extensive margin and unemployment. We do soby introducing search and matching in the spirit of Mortensen and Pissarides and others. Further,to introduce nominal wage rigidity, we use the staggered Nash bargaining approach of GT.
We note that in an earlier version of this paper we also allowed for variation in hours on theintensive margin. We drop this feature for two reasons. First, as Figure 1 shows, most of thecyclical variation in hours in the U.S. is on the extensive margin. Second, our earlier estimatesconfirmed that the intensive margin was unimportant to cyclical variation. We estimated a Frischelasticity close to zero, which is certainly in line with the microeconomic evidence.
Our model has three types of agents: households, wholesale firms, and retail firms. We use a
representative family construct, similar to Merz (1995) in order to introduce complete consumption
insurance. Production takes place at wholesale firms. These firms are competitive. They hire
workers and negotiate wage contracts with them. Retail firms buy goods from wholesalers andthen repackage them as final goods. Retailers are monopolistic competitors and set prices ona staggered basis. We separate retailers from wholesalers to keep the wage bargaining problem
tractable.3
Finally, following SW, we introduce a number of exogenous shocks that correspond exactly tothe number of data series we consider in our estimation.

2.1

Households

There is a representative household with a continuum of members of measure unity. The numberof family members currently employed is nt . Employment is determined through a search andmatching process that we describe shortly.
Except for the treatment of labor supply, the household’s decision problem is identical to thatin CEE and SW. In the latter, the household receives utility from leisure and it varies labor only on
the intensive margin. Here there is no utility gain from leisure. Individuals not currently working
are searching for jobs.
Accordingly, conditional on nt , the household chooses consumption ct , government bonds Bt ,3

To keep the bargaining problem tractable, it is necessary to have constant returns at the firm level. This is tomake the average and marginal worker the same, thus avoiding bargaining spillovers among workers. Introducing
staggered price setting requires that firms face downward sloping demand curves, implying diﬀerences between averageand marginal.

4

capital utilization zt , investment it , and physical capital ktp to maximize the utility function
Et

∞
P

s=0

β s εbt+s log (ct+s − hct+s−1 ) ,(1)where h is the degree of habit persistence in consumption preferences and where εbt is a preference
shock with mean unity that obeys
log εbt = ρb log εbt−1 + ς bt ,(2)

and where all primitive innovations, including ς bt , are zero-mean i.i.d. random variables.
Let Πt be lump sum profits, Tt lump sum transfers, pt the nominal price level, and rt the one
period nominal interest rate (specifically, the central bank policy instrument). Then the household’sbudget constraint is
ct + it +

Bt
Bt−1
p
p
= wt nt + (1 − nt ) bt + rtk zt kt−1
+ Πt + Tt − a (zt ) kt−1
+
.
pt rt
pt

(3)

Households own capital and choose the capital utilization rate, zt , which transforms physicalcapital into eﬀective capital according top.
kt = zt kt−1

(4)

Eﬀective capital is rented to the firms at the rate rtk . The cost of capital utilization per unit ofphysical capital is a (zt ). We assume that zt = 1 in the steady state and a(1) = 0.
The physical capital accumulation equation is
µ
µ
¶¶
it
p
p
i
it ,
(5)
kt = (1 − δ) kt−1 + εt 1 − s
it−1where we assume s (γ a ) = s0 (γ a ) = 0 and s00 (γ a ) > 0 where γ a is the economy’s steady state
growth rate.
εit is an investment specific technological shock with mean unity aﬀecting the eﬃciency withwhich consumption goods are transformed into capital. We assume εit follows the exogenous stochastic process(6)log εit = ρi log εit−1 + ς it .
The first order necessary conditions yield:
(ct )
εbt+1
εbt
− βhEt,λt =ct − hct−1
ct+1 − hct(Bt )λt = rt βEtµ5

λt+1 pt
pt+1

¶

,

(7)

(8)

(zt )
rtk = a0 (zt ) ,

(9)

µ
µ
¶¶
µ
¶
µ
¶µ
¶
λt+1 0 it+1
it
it
it
it+1 2
k i 0
k
i
1−s
= qt εt s− βEt qt+1 εt+1
s
+ 1,
it−1
it−1 it−1
λt
it
it

(10)

(it )
qtk εit(ktp )
qtk = βEt´λt+1 ³kk(1 − δ) qt+1
+ rt+1
zt+1 − a (zt+1 ) ,
λt

(11)

where λt = u0 (ct ) and qtk is the value of installed capital in consumption units.
Again, except for the treatment of the labor supply decision, the household sector is conventional.

2.2

Unemployment, Vacancies and Matching

At time t, each firm posts vt (i) vacancies in order to attract new workers and employs nt (i) workers.
R1
R1
The total number of vacancies and employed workers are vt = 0 vt (i)di and nt = 0 nt (i)di. All
unemployed workers at t look for jobs. Our timing assumptions are such that unemployed workers
who find a match go to work immediately within the period. Accordingly, the pool of unemployedworkers searching for a job at t, ut , is given by the diﬀerence between unity (the total populationof workers) and the number of employed workers at the end of period t − 1, nt−1 :ut = 1 − nt−1 .

(12)

The number of new hires or “matches”, mt , is a function of searching workers and vacancies,
as follows:(13)
mt = σ m uσt vt1−σ .
The probability a firm fills a vacancy in period t, qt , is given by
mt
qt =
.
vt
Similarly, the probability a searching worker finds a job, st , is given by
mt
st =
.
ut

(14)

(15)

Both firms and workers take qt and st as given.
Finally, each period, firms exogenously separate from a fraction 1 − ρ of their existing workforce
nt−1 (i). Workers losing their job at time t are not allowed to search until next period. Accordingly, within our framework fluctuations in unemployment are due to cyclical variation in hiring as
opposed to separations. Both Hall (2005) and Shimer (2005, 2007) present evidence in support ofthis phenomenon.
62.3

Wholesale Firms

Each period, wholesale firms produce output yt (i) using capital, kt (i), and labor, nt (i):yt (i) = [kt (i)]α [at nt (i)]1−α ,(16)where at is a common labor-augmenting productivity factor. We assume εat = at /at−1 obeys thefollowing exogenous stochastic process
log εat = (1 − ρa ) log εa + ρa log εat−1 + ς at .

(17)

Note that the steady state value εa corresponds to the economy’s growth rate γ a . Thus, following
PST, we are allowing technology be non-stationary in levels, though stationary in growth rates.
For simplicity, we assume that capital is perfectly mobile across firms and that there is acompetitive rental market in capital. These assumptions ensure constant returns to scale at thefirm level, which greatly simplifies the wage bargaining problem (see the discussion in GT).
It is useful to define the hiring rate xt (i) as the ratio of new hires qt vt (i) to the existing
workforce nt−1 (i):qt vt (i).
(18)
xt (i) =nt−1 (i)Observe that due to the law of large numbers the firm knows xt (i) with certainty at time t since it
knows the likelihood qt that each vacancy it posts will be filled. The hiring rate is thus eﬀectively
the firm’s control variable.
The total workforce, in turn, is the sum of the number of surviving workers ρnt−1 (i) and newhires xt (i) nt−1 (i):(19)
nt (i) = ρnt−1 (i) + xt (i) nt−1 (i) .
Equation (19) reflects the timing assumption that new hires go to work immediately.4k
Let pwt be the relative price of intermediate goods, wt (i) the wage, rt the rental rate of capital,and βEt Λt,t+1 be the firm’s discount rate, where the parameter β is the household’s subjective
discount factor and where Λt,t+1 = λt+1 /λt . Then, the value of the firm, Ft (i), may be expressedas:κt 2
x (i) nt−1 (i) − rtk kt (i) + βEt Λt,t+1 Ft+1 (i) ,(20)
Ft (i) = pw
t yt (i) − wt (i) nt (i) −
2 t
with
(21)
κt = κat .
As wage dispersion is present, we replace the standard assumption of fixed costs of posting a vacancy
with quadratic labor adjustment costs, given here by κ2t x2t (i) nt−1 (i) . We allow adjustment costs
to drift proportionately with productivity in order to maintain a balanced steady state growth path(otherwise adjustment costs become relatively less important as the economy grows.)
4

Blanchard and Gali (2006) use a similar timing.

7

At any time, the firm maximizes its value by choosing the hiring rate (by posting vacancies)
and its capital stock, given its existing employment stock, the probability of filling a vacancy, therental rate on capital and the current and expected path of wages. If it is a firm that is able torenegotiate the wage, it bargains with its workforce over a new contract. If it is not renegotiating, ittakes as given the wage at the previous period’s level, as well the likelihood it will be renegotiatingin the future.
We next consider the firm’s hiring and capital rental decisions, and defer a bit the descriptionof the wage bargain. The first order condition for capital is simply:
rtk = pwt αyt (i)
yt
= pw
t α .
kt (i)
kt(22)

Given Cobb-Douglas technology and perfect capital mobility, all firms choose the same capital/output ratio and, in turn, the same capital/labor and labor/output ratios
Firms choose nt (i) by setting xt (i) or, equivalently, vt (i). The firm’s hiring decision yields:
κt xt (i) = pw
t fnt (i) − wt (i) + βEt Λt,t+1 ∂Ft+1 (i) /∂nt (i) ,with
fnt (i) = (1 − α)ytyt (i)= (1 − α)= fnt .
nt (i)
nt

(23)

(24)

By making use of the envelope theorem to obtain ∂Ft (i) /∂nt−1 (i) and combining equations,
we obtain
κt xt (i) = pw
t fnt − wt (i) + βEt Λt,t+1

κt+1 2x (i) + ρβEt Λt,t+1 κt+1 xt+1 (i) .
2 t+1

(25)

The hiring rate thus depends on the discounted stream of earnings and savings on adjustment costs.
Finally, for the purpose of the wage bargain it is useful to define Jt (i) , the value to the firm of
having another worker at time t after new workers have joined the firm, i.e., after adjustment costs
are sunk. Diﬀerentiating Ft (i) w.r.t. nt (i) , taking xt (i) as given yields:
Jt (i) = pw
t fnt − wt (i) + βEt Λt,t+1 ∂Ft+1 (i) /∂nt (i) .

(26)

By making use of the hiring rate condition and the relation for the evolution of the workforce, Jt (i)may be expressed as expected average profits per worker net of the first period adjustment costs,
with the discount factor accounting for future changes in workforce size:
Jt (i) = pw
t fnt − wt (i) − βEt Λt,t+1

κt+1
nt+1
xt+1 (i)2 + Et(i) βΛt,t+1 Jt+1 (i) .
2
nt

8

(27)

2.4

Workers

In this sub-section we develop an expression for a worker’s surplus from employment, which is a
critical determinant of the outcome of the wage bargain.
Let Vt (i) be the value to a worker of employment at firm i and let Ut be the value of unemployment. These values are defined after hiring decisions at time t have been made and are in units ofconsumption goods. Vt (i) is given by
Vt (i) = wt (i) + βEt Λt,t+1 [ρVt+1 (i) + (1 − ρ) Ut+1 ] .

(28)

To construct the value of unemployment, we first define Vx,t as the average value of employment
conditional on being a new worker at t:
Z 1
xt (i)nt−1 (i)Vt (i)
di,(29)
Vx,t =
xt nt−10where xt (i)nt−1 (i) is total new workers at firm i and xt nt−1 is total new workers at t.5 Next, let
bt be the flow value from unemployment, including unemployment benefits, as well as other factorsthat can be measured in units of consumption goods. As before, let st be the probability of findinga job for the subsequent period. Then Ut may be expressed as
Ut = bt + βEt Λt,t+1 [st+1 Vx,t+1 + (1 − st+1 ) Ut+1 ] ,

(30)

with
bt = bktp ,and where ktp is the economy-wide capital stock. We assume that bt grows proportionately to ktp inorder to maintain a balanced growth; otherwise bt would become a smaller fraction of labor income
as the economy grows. The value of unemployment thus depends on the current flow value bt andthe likelihood of being employed versus unemployed next period. Note that the value of finding a
job next period for a worker that is currently unemployed is Vx,t+1 , the average value of working
next period conditional on being a new worker. That is, unemployed workers do not have a priori
knowledge of which firms might be paying higher wages next period. They instead just randomly
flock to firms posting vacancies.6
Ht (i) and Hx,t are given by:(31)
Ht (i) = Vt (i) − Ut ,andHx,t = Vx,t − Ut .

(32)
U1

Vx,t is thus distinct from the unconditional average value of employment Vt = 0 Vt (i) nnt (i)di.
tThere is accordingly no directed search. Note, however, that wage diﬀerentials across firms are only due to thediﬀerential timing of contracts, which is transitory. Thus, because a worker who arrives at a firm in the midst of an
existing contract may expect a new one reasonably soon, the payoﬀ from directed search may not be large.
56

9

It follows that:
Ht (i) = wt (i) − bt + βEt Λt,t+1 [ρHt+1 (i) − st+1 Hx,t+1 ] .

2.5

(33)

Nash Bargaining and Wage Dynamics

As we noted earlier, we introduce staggered Nash wage bargaining as in GT, but with the following
diﬀerence: Because we have a monetary model, we allow for nominal wage contracting, though we
also allow for the possibility of indexing to past inflation.
We introduce staggered multi-period wage contracting in a way that simplifies aggregation. In
particular, each period a firm has a fixed probability 1 − λ that it may re-negotiate the wage.7
This adjustment probability is independent of its history, making it unnecessary to keep track ofindividual firms’ wage histories. Thus, while how long an individual wage contract lasts is uncertain,
the average duration is fixed at 1/(1 − λ). The coeﬃcient λ is thus a measure of the degree of wage
stickiness. In GT the parameter λ is calibrated. Here we are able to estimate it.
Since we allow for the possibility of indexing to past inflation, π t−1 , the fraction λ of firms thatcannot renegotiate their contract set their nominal wages wtn (i) following the indexation rule:n(i) π γt−1 ,wtn (i) = γwt−1(34)where π t = pt /pt−1 , γ = γ a π 1−γ and where γ [0, 1] reflects the degree of indexing to past inflation.
We also estimate the parameter γ. The term γ in the indexing rule provides an adjustment fortrend productivity growth and trend inflation.
Firms that enter a new wage agreement at t negotiate with the existing workforce, including
the recent new hires. Due to constant returns, all workers are the same at the margin. The
wage is chosen so that the negotiating firm and the marginal worker share the surplus from themarginal match. Given the symmetry to which we just alluded, all workers employed at the firm
receive the same newly-negotiated wage.8 When firms are not allowed to renegotiate the wage, allexisting and newly hired workers employed at the firm receive the existing contracting wage (i.e.
last period nominal wage adjusted for possibly indexing.)9 As we discussed earlier, we appeal toscale economies in bargaining to rule out separate negotiations for worker who arrive in between
7This kind of Poisson adjustment process is widely used in macroeconomic models with staggered price setting,
beginning with Calvo (1983).
8
To be clear, with constant returns, one could either think of the firm bargaining with each marginal worker
individually or bargaining with a union that wishes to maximixe average worker surplus.
9
As we noted earlier, in Gertler, Huckfeld and Trigari (2007) we present evidence in support of this assumption.
Bewley (1999) also presents survey evidence that the wages of new hires are linked to those of existing workers. Other
studies of the cyclical behavior of wages for new hires (e.g. Bils, 1985) do not examine the link with existing workers
wages (due to data limitations) and thus do not speak to our hypothesis. We think that explaining the facts in thesestudies will require introducing heterogeneity into our framework.

10contracting periods.10 Of course, the newly hired workers recognize that they will be able tore-negotiate the wage at the next round of contracting.
Let wtn∗ denote the wage of a firm that renegotiates at t. Given constant returns, all sets ofrenegotiating firms and workers at time t face the same problem, and thus set the same wage. As
we noted earlier, the firm negotiates with the marginal worker over the surplus from the marginal
match. We assume Nash bargaining, which implies that the contract wage wtn∗ is chosen to solve
max Ht (i)ηt Jt (i)1−ηt ,
s.t.
n
(i) =
wt+j

(

(35)

n
γwt+j−1(i) π γt+j−1 with probability λn∗ with probability 1 − λwt+j∀ j ≥ 1, where ηt [0, 1] is the worker’s relative bargaining power, and where Jt (i) and Ht (i) are
given by equations (27) and (33).
In the conventional search and matching framework the bargaining power parameter is constant.
Here, in order to allow for an error term in the wage equation we allow this parameter to evolve
exogenously according to:(36)
η t = ηεηt ,where εηt is a mean-unity bargaining power shock that follows the stochastic process:log εηt = ρη log εηt−1 + ς ηt .

(37)

The first order necessary condition for the Nash bargaining solution is given byη∂Jt (i)∂Ht (i)
Jt (i) + (1 − η)
Ht (i) = 0.
n∂wt (i)∂wtn (i)(38)

Letting ∆t = pt ∂Ht (i) /∂wtn (i) and Σt (i) = −pt ∂Jt (i) /∂wtn (i), we have:∆t = 1 + Et Λt,t+1 (ρλβ)
andΣt (i) = 1 + Et Λt,t+1pt
γπ γt ∆t+1 ,
pt+1

nt+1
pt
(i) (λβ)γπ γ Σt+1 (i) .
nt
pt+1 t

10

(39)

(40)

In addition to scale economies in bargaining, there are several complementary justifications for why hires in
between contracts receive the existing contract wage. First, as we noted earlier, Bewley (1999) argues that internal
equity constrains workers of similar productivity to receive similar wages. Second, Menzio and Moen (2006) show
how the trade-oﬀ between eﬃcient provision of insurance to senior workers and eﬃcient recruitment of junior ones
links the wages of new and existing workers in response to small and negative productivity shocks. Third, consistent
with Hall (2005a), one might interpret the existing contract wage as the “wage norm” for workers hired in between
contracts.

11

Observe that ∆t and −Σt (i) are the derivatives of the worker and the firm surpluses with respect
to the real wage.
The first order condition for wages, then, can be rewritten as:
χt (i) Jt (i) = (1 − χt (i)) Ht (i) ,

(41)

ηt
.
ηt + (1 − η t ) Σt (i) /∆t

(42)

with
χt (i) =One diﬀerence between our solution and the period-by-period Nash solution is that the relative
weight χt (i) depends not only on the worker’s bargaining power η t , but also on Σt (i) and ∆t ,the firm and worker’s respective cumulative discount factors. This diﬀerence arises because thetwo agents have diﬀerent horizons: unlike the worker, the firm cares about the implications of thecontract for new workers as well as existing ones, whereas the worker only cares about his or her
tenure at the job. Observe that the firm’s cumulative discount factor depends on the expectation of(i), thethe product of five factors: the firm’s employment size at time t + 1 relative to time t, nnt+1
t
probability the contract survives to t + 1, λ, the inverse of inflation at t + 1, pt /pt+1 , a correction
for indexing to past inflation and growth, γπ γt , and the households’ discount factor, βΛt,t+1 . It issimilar for the worker, except the survival probability ρ replaces the relative employment size. Since(i) exceeds ρ, the firm places relatively more weight on the future than does theon average nnt+1
t
worker. Since Σt (i) /∆t exceeds unity on average, χt (i) is less than η t on average. Intuitively, since
movements in the contract wage have a larger impact on discounted firm surplus than on worker
surplus, the “horizon eﬀect” works to raise the eﬀective bargaining power of firms from 1 − η to1 − χ (i) on average. On the other hand, while the horizon eﬀect (arising from the diﬀerence
between η t and χt (i)) may be of interest from a theoretical perspective, it will turn out not to beof quantitative importance in this case.
Following GT and converting nominal to real wages, the solution to the bargaining problem
leads to the following diﬀerence equation for the real wage wt∗ = wtn∗ /pt 11 :∗,∆t wt∗ = wto (i) + ρλβEt Λt,t+1 ∆t+1 wt+1

(43)

where the forcing variable wto (i) can be thought of as the real “target” wage and is given by³´
κt+1 2
xt+1 (i)(44)
wto (i) = χ pw
t fnt + βEt Λt,t+1
2+ (1 − χ) (bt + st+1 βEt Λt,t+1 Hx,t+1 ) + Φt (i) ,withΦt = φt (i) − ρβEt Λt,t+1 φt+1 (i) ,φt (i) = (χt (i) − χ) [Jt (i) + Ht (i)] .
11

See the appendix for details and derivations.

12In contrast to the standard case of period-by-period Nash bargaining, the wage set in the current
period may apply in the future. Hence in this instance the contract wage wt∗ depends not only on
the current target wage, but on the expected sequence of future target wages (as iterating forward
equation (43) suggests.)
Observe that the target compensation, wto (i), has the same form that would emerge underperiod-by-period Nash bargaining. In particular, it is a convex combination of what a worker
contributes to the match and what the worker loses by accepting a job, where the weights depend(indirectly) on the worker’s relative bargaining power, η. The worker’s contribution is the marginal
product of labor plus the saving on adjustment costs. With our quadratic cost formulation, this2saving is measured by κt+1
2 xt+1 (i). The foregone benefit from unemployment, in turn, is the flow
value of unemployment, bt , plus the expected discounted gain of moving from unemployment thisperiod to employment next period, st βΛt,t+1 Hx,t+1 . The additional term Φt (i) reflects the horizon
eﬀect, as well as exogenous movements in the bargaining power.
In the limiting case of period-by-period wage contracting, λ = 0, the solution corresponds
to the conventional Nash bargaining outcome: wt∗ = wto (i), with χt (i) = η t . With multi-period
contracting, however, wt∗ depends on a discounted stream of future values of wto (i), where the latter
depend on horizon adjusted bargaining weights.
Finally, the average nominal wage across workers is given by
Z 1
nt (i)nwtn (i)
di.
(45)
wt =
nt
0

2.6

Retailers

There is a continuum of monopolistically competitive retailers indexed by j on the unit interval.
Retailers buy intermediate goods from the wholesale firms described in the previous section. Theyin turn diﬀerentiate them with a technology that transforms one unit of intermediate goods into
one unit of retail goods, then re-sell them to the households. In addition, they set prices on a
staggered basis.
Following SW, we permit each firm’s elasticity to depend inversely on its relative market share,using a formulation due to Kimball (1995). The endogenous elasticity introduces a strategic complementarity in price setting (or real rigidity) that makes it easier for the model to match the micro
evidence on price adjustment (see, e.g., the discussion in Woodford, 2003.)
The Kimball formulation is a based on a generalization of the Dixit/Stiglitz aggregator ofindividual goods into a composite that allows for a varying elasticity of substitution between goods.
Let yt (j) be the quantity of output sold by retailer j and let pt (j) be the nominal sale price. Under
the Kimball formulation, accordingly, final goods, denoted with yt , are a composite of individual
retail goods, defined as follows:¶
Z 1 µyt (j) pG, εt dj = 1
(46)
yt
0
13

where the function G is increasing and strictly concave, with G(1) = 1. εpt in an exogenous nonnegative random variable that influences the elasticity of demand and thus the frictionless pricemarkup. We assume that εpt evolves as follows:
log εpt = (1 − ρp ) log εp + ρp log εpt−1 + ς pt .
Cost minimization then leads to the follow demand curve facing each retailer j:
¸
∙
¶
Z 1 µ
yt (j)
0−1 pt (j)0 yt (j)G
dj yt ,
yt (j) = G
pt
yt
yt
0

(47)

(48)

where pt is the aggregate price index.
We assume “Calvo” staggering applies to price setting and we also allow for indexing. Let 1−λpbe the probability a firm adjusts its price. Firms not adjusting their target price obey the following
indexing rule:p(49)
pt (j) = γ p pt−1 (j) π γt−1 ,where, in analogy to the case of wage setting, both γ p and λp are parameters that we estimate, andpwhere γ p = π 1−γ is an adjustment for trend inflation.
Firms that are able to adjust their price at t, set the price to maximize expected discounted
profits subject to the constraints on the frequency of price adjustment. Since all firms that are
resetting price are identical ex ante, they all choose the price, p∗t . It follows that by combining
equations (46) and (48) and applying the law of large numbers, we can express the price index as
µ ∗ ¶
µ p
¶
γ
γ
p ∗ 0−1 pt
p p
0−1 γ pt−1 π t−1(50)
τ t + λ (γ pt−1 π t−1 )Gτt ,pt = (1 − λ )pt G
pt
pt
´
R1 ³
yt (j)
where τ t = 0 G0 yty(j)yt dj.
t
Re-optimizing retailers choose a target price, p∗t , to maximize the following discounted streamof future profits:
∙ ∗ ³
¸
´
∞
P
pt Qs
p s
p γp
w
(λ β) Λt,t+s(51)
Et
k=1 γ π t+k−1 − pt+s yt+s (j) ,
pt+s
s=0

subject to the demand curve given by equation (48). Note that the sequence of discount factors
depend on λp , the probability that the price remains fixed in the subsequent period.
The first order condition for the target price is given by
∙
¸
´
∞
P
p∗t ³Qs
p s
p γp
w
Et
(λ β) Λt,t+s (1 + Θt )(52)
k=1 γ π t+k−1 − Θt pt+s yt+s (j) = 0,
pt+s
s=0
with

£
¤−1 © 0 £ 0−1¤¤ª£
G G (τ t p∗t /pt+s ) /G00 G0−1 (τ t p∗t /pt+s ) .
Θt = G0−1 (τ t p∗t /pt+s )14

By loglinearizing this condition, one can show that p∗t depends on an expected discounted streamof the retailers nominal marginal cost, given by the nominal wholesale price pt pw
t .
By inverting the hiring condition derived earlier, one can obtain an expression for the retailers
real marginal cost, pw
t :
pw
t =

i
κt+1 21 h
wt (i) + κt xt (i) − βEt Λt,t+1
xt+1 (i) − βEt Λt,t+1 ρκt+1 xt+1 (i) .
fnt
2

(53)

Real marginal costs thus depend on unit labor costs, plus terms that correct for the adjustment
costs of hiring workers.
Observe that since we have normalized the relative price of final output at unity, the retailer’s
markup is given by µpt = 1/pw
t . Since final goods prices are sticky and wholesale prices are flexible,
this markup will in general exhibit cyclical behavior, with the direction depending on the natureof the disturbances hitting the economy, as well as other features of the model.
Finally, as we show in the appendix, by loglinearizing expressions for the price index and for the
optimal reset price, equations (50) and (52), one can obtain a relation for consumer price inflationthat is a variation of the conventional hybrid New Keynesian Phillips curve relating inflation to
movements in real marginal cost, expected future inflation and lagged inflation. In this instancereal marginal cost is simply the relative price of wholesale goods, pw
t . The slope coeﬃcient onreal marginal cost, further, depends inversely on the degree of price stickiness, measured by λp .
It also depends inversely on ξ, the percent change in the firm’s price elasticity with respect to a
one percent change in its relative price.12 Everything else equal, a positive value of ξ reduces thefirm’s desired adjustment of its relative price. As discussed in Woodford (2003), this introduces apricing complementarity that induces price adjusters to keep their relative prices closer to those of
non-adjusters. This in turn reduces the sensitivity of inflation to movements in real marginal cost.

2.7

Government

Monetary policy obeys the following simple Taylor rule:∙µ ¶ ¸(1−ρs )rt ³ rt−1 ´ρs ³ π t ´rπ yt ry
=
εrt ,
r
r
π
ynt

(54)

log εrt = ρr log εrt−1 + ς rt .

(55)

¶
µ
1
gt = 1 − g yt ,
εt

(56)

where
Government spending obeys:

1200

Let η = −G0 (yt (j) /yt ) / yty(j)G (yt (j) /yt ) be the firm’s elasticity of demand with respect to shifts in its markettpt (j)/pt∂ηshare. Then ξ = ∂(pt (j)/p
.
η
t)

15

wherelog εgt = (1 − ρg ) ln εg + ρg log εgt−1 + ς gt .

2.8

(57)

Resource Constraint

The resource constraint divides output between consumption, investment and adjustment and utilization costs:
R1
p
.
(58)
yt = ct + it + gt + (κt /2) 0 x2t (i) nt−1 (i) di + a (zt ) kt−1
This completes the description of the model.

3

Wage and Hiring Dynamics

The key features of the model that diﬀerentiates it from the conventional monetary DSGE model
involve wage and hiring dynamics. In this section, accordingly, we derive loglinear relationships forthese variables.13 The analysis closely follows GT.
Loglinearizing the expression for the contract wage, we obtain the following loglinear diﬀerence
equation for real contract wage, w
bt∗ :
¡
¢
∗
w
bt∗ = [(1 − ρλβ) w
bto (i) + ρλβEt ππt + bεat+1 ] + ρλβEt w
bt+1.
bt+1 − γb(59)

The contract wage depends on the current and expected future path of the target wage w
bto (i) andterms that reflect adjustments for indexing.
Let w
bto be the “spillover free” target wage. Absent the horizon eﬀect, it corresponds to the real
wage that would arise if all firms were doing period-by-period Nash bargaining, i.e., the case of
perfectly flexible wages. It is possible to show that:
¡
¢
τ1
τ2
∗
Et w
(w
bt − w
bto +bt+1
bt∗ ) ,
(60)
bt+1 − w+w
bto (i) = w
1 − ρλβ1 − ρλβ
with

with

³
´
bnt + (ϕ + ϕ ) Et x+f
bt+1 + ϕs Et sbt+1 + ϕbbbt
w
bto = ϕf n pbw
x
s
t
´
³
¡
¢
bt+1 − λbt + ϕχ χb t − β (ρ − s) χεw
+ (ϕs + ϕx /2) Et λb t+1 + bt ,= (κx)−1w = (w/a)
ϕf n = χpw (fn /a) w−1

ϕx = χβ (1 − ρ) −1 w−1

ϕb = (1 − χ) b (k/a) w−1

ϕs = χβs −1 w−1

ϕχ = χ (1 − χ)−1 −1 w−1

13

See the appendix for details and derivations.

16

(61)

and
= ϕη (1 − βρη (ρ − s)) b
εηt
b
εw
t
ϕη = χ (1 − η)−1 −1 w−1

and whereτ 1 = ϕs Γ (1 − ρλβ) ,£¤τ 2 = ϕx λ − ϕχ (1 − χ) (1 − ρ) Ψ (1 − ρλβ) Σw̄ (1 − ρλβ) .

As in GT, the parameters τ 1 and τ 2 in equation (60) reflect the influence of “spillovers” on
the bargaining process: The second term on the right is a direct spillover. If, everything else
∗ , opportunities are unusually good for workers expecting to move into
bt+1 exceeds Et w
bt+1
equal, Et w
∗ . By influencing the worker’s
employment next period, and vice-versa if Et w
bt+1 is below Et w
bt+1
outside option in this way, the expected average market wage at t + 1 induces a direct spillover
eﬀect on the wage bargain. An indirect spillover emerges in the third term because the hiring rateof the renegotiating firm aﬀects the bargaining outcome. It does so by influencing both the firm’ssaving in adjustment costs and the horizon-adjusted bargaining weight (because it aﬀects the firm’s
cumulative discount factor.) The diﬀerence between the hiring rate x
bt (i) and average hiring rate
bt and the contract wagex
bt depends positively on the diﬀerence between the average market wage w∗
w
bt . Both spillover eﬀects work to increase the degree of wage rigidity.
The loglinearized real wage index is in turn given bybt∗ + λ (wbt−1 − πbt + γbπ t−1 − bεat ) .
w
bt = (1 − λ) w

Combining these equations along with the relation for w
bto (i) then yields:¡¢bt−1 − πbt + γbπ t−1 − bεat ) + γ o w
bto + γ f w
bt+1 − γbπt + bεat+1 ,
bt+1 + πw
bt = γ b (w

(62)

(63)

γ b = (1 + τ 2 ) φ−1 ,γ o = ςφ−1 ,γf= (ρβ − τ 1 ) φ−1 ,φ = 1 + τ 2 + ς + ρβ − τ 1 ,

ς = (1 − λ) (1 − ρλβ) λ−1 .

with γ b + γ + γ f = 1. Note the forcing variable in the diﬀerence equation is the “spillover free”
target wage w
bto .
bt−1 as well as the expected future
Due to staggered contracting, w
bt depends on the lagged wage w
bt+1 . Solving out for the reduced form will yield an expression that relates the wage to thewage Et w
lagged wage and a discounted stream of expected future values of w
bto . Note that the spillover eﬀects,
17

measured by τ 1 and τ 2 work to raise the relative importance of the lagged wage (by raising γ b ) andreduce the importance of the expected future wage (by reducing γ f ). In this way, the spillovers
work to raise the inertia in the evolution of the wage. In this respect, the spillover eﬀects work ina similar (though not identical) way as to how real relative price rigidities enhance nominal pricestickiness in monetary models with time-dependent pricing (see, for example, Woodford, 2003).
Note also that as we converge to λ = 0 (the case of period by period wage bargaining), both γ b
and γ f go to zero), implying that w
bt simply tracks w
bto in this instance. Further, as we noted earlier,w
bto , becomes identical to the wage in the flexible case. The model thus nests the conventional
period-by-period wage bargaining setup.
Next, loglinearizing the equation for the hiring rate and aggregating yields:
³
³
´
´
b
bt .
bt+1 − λbt+1 + (ρ + x/2) β Et λx
bt = pw (fn /a) pbw
bt + βEt x(64)
t + fnt − (w/a) w

The hiring rate thus depends on current and expected movements of the marginal product of labor
relative to the wage. The stickiness in the wage due to staggered contracting, everything else equal,
implies that current and expected movement in the marginal product of labor will have a greater
impact on the hiring rate than would have been the case otherwise.

4

Model Estimation

4.1

Estimation Procedure

We consider seven variables in our estimation. To facilitate comparison with the literature, we
employ the seven quarterly series used in CEE, SW and PST.14 Thus, the variables we use include:(1) per capita real GDP; (2) per capita real personal consumption expenditures of nondurables; (3)per capita real investment equal to the sum of per capita real private investment plus per capita
real personal consumption of durables; (4) hours of all persons in the non-farm business sector
divided by the population times the ratio of total employment to employment in non-farm businesssector; (5) the real wage (compensation per hour in the nonfarm business sector); (6) inflation as
measured by the quarter to quarter growth rate of the GDP deflator; and (7) the Federal Funds
rate expressed in quarterly terms. To convert any nominal variable to real terms, we always use
the GDP deflator. The sample goes from 1960Q1 to 2005Q1.
We first log-linearize the model around a deterministic balanced growth steady state. The
appendix contains the complete log-linear model, as well as the steady state. The coeﬃcients of
the log-linear model depend on the primitive parameters of the model, as well as steady statevalues of variables. We use the steady state conditions of the model to solve out for a number of
14

There are some slight diﬀerences in the series used in SW versus CEE and PST. As in CEE and PST, we include
consumer durables in investment. As in SW, we use an economy-wide measure of hours based on an adjustment of
non-farm business hours.

18parameters. The model also contains seven exogenous shocks, one corresponding to each variable.
In our estimation, further, we allow for the fact that the quantity variables (output, investment,
consumption, etc.) are non-stationary due to the presence of a unit root in the technology shock.
There are twenty-two parameters, not including the parameters that characterize the exogenous
shocks. Of the twenty-two, there are five new parameters that arise from our modification of thelabor market. These include: the steady state flow value of unemployment as fraction of thecontribution of the worker to the job, b, the worker’s relative bargaining power, η, the elasticityof new matches with respect to labor market tightness, σ, the job survival rate, ρ, and the steadystate job finding rate, s.15
Because we are not adding any new variables but are adding new parameters, in this firstpass at the data we calibrate three of the five labor market parameters, for which there exists
independent evidence. In particular, we choose the average monthly separation rate 1 − ρ basedon the observation that jobs last about two years and a half. Therefore, we set ρ = 1 − 0.105.
We choose the elasticity of matches to unemployment, σ, to be equal to 0.5, the midpoint of theevidence typically cited in the literature.16 In addition, this choice is within the range of plausible
values of 0.5 to 0.7 reported by Petrongolo and Pissarides (2001) in their survey of the literatureon the estimation of the matching function. We then set s = 0.95 to match recent estimates of the
U.S. average monthly job finding rate (Shimer, 2005).
The two labor market parameters we estimate are b and η. Both these parameters are critical
determinants of the eﬀective elasticity of labor supply along the extensive margin in the flexible
wage case. As emphasized by Hagedorn and Manovskii (2006) and others, the closer b is to unity, thebetter able is the period-by-period Nash bargaining framework to capture unemployment and wage
dynamics. For example, when b is very close to unity, there is little diﬀerence between the valueof employment versus unemployment to the worker. Eﬀectively, labor supply along the extensive
margin is very elastic in this instance. The response of wages to a shift in the value of a worker tothe firm is dampened because in this case a small change in the wage has a large percentage eﬀect
on the relative gains to the worker from employment versus unemployment. Indeed, Hagedorn
and Manovskii (2006) show that a model calibrated with b close to unity can capture the relative
volatilities of labor market variables. This calibration however is quite controversial: Shimer (2005)
argues in favor of 0.4 based on the interpretation of b as unemployment insurance, while Hall (2006)
suggest 0.7, based on a broader interpretation that permits utility from leisure. Given the criticalrole of this parameter, it seems to be a prime candidate to estimate. The worker’s bargaining
parameter η is similarly important. The smaller is η, the less sensitive are wages to movements inthe shadow value of labor, and thus the more sensitive is employment. Indeed, the Hagedorn and
The relative flow benefit from unemployement is given by b = b(κ/a)/ pw (fn /a) + (β/γ a ) (κ/2) x2 .
The values for σ used in the literature are: 0.24 in Hall (2005a), 0.4 in Blanchard and Diamond (1989), Andolfatto(1994) and Merz (1995), 0.45 in Mortensen and Nagypal (2006), 0.5 in Hagedorn and Manovskii (2006), 0.5 in Farmer(2004), 0.72 in Shimer (2005a). See also a brief discussion in Mortensen and Nagypal (2006), p. 10, comparing theirvalue of 0.45 to Shimer’s one.
15
16

19

Manovskii calculation requires not only a high value of b but also a low value of η.
There are four “conventional” parameters that we calibrate: the discount factor, β, the depreciation rate, δ, the “share” parameter on capital in the Cobb-Douglas production function, α, thesteady state ratio of government consumption to output, yg , and the sensitivity of firm demand
elasticity to market share (the Kimball aggregator), ξ. We use conventional values for all theseparameters: β = 0.99, δ = 0.025, α = 0.33, yg = 0.2 and ξ = 10.17
Table 1: Calibrated parameters
β
0.99

δ
0.025

α
0.33

g/y
0.2

ξ
10

σ
0.5

ρ
0.895

s
0.95

The conventional parameters we estimate include: the elasticity of the utilization rate to therental rate of capital, η z 18 ; the elasticity of the capital adjustment cost function, η k ; the habit
persistence parameter, h; the steady state price markup εP ; the wage and price rigidity parameters,
λ and λp ; the wage and price indexing parameters, γ and γ p ; and the Taylor rule parameters, rπ , ry
and ρs . In addition, we estimate the first order autocorrelations of all the exogenous disturbances,
as well as their respective standard deviations.
We estimate the model with Bayesian methods (see An and Schorfheide, 2007, for a comprehensive survey). We combine the likelihood function of the model, L(θ, Y ), with priors for theparameters to be estimated, p(θ), to obtain the posterior distribution, L(θ, Y )p(θ). Draws fromthe posterior distribution are generated with the Random-Walk Metropolis Hastings (RWMH)
algorithm.

4.2

Estimation Results

Table 2 reports the prior and the posterior estimates for each parameter. The table also reports
the parameter configuration which maximizes the posterior (called Max), along with the mean andthe values at the 5 and 95 percent tails. Similarly, Table 3 presents the estimates of the prior andposterior distribution of the shock processes.
For the conventional parameters, for the most part we use the same priors as in PST, which inturn follow closely those employed by SW. We proceed this way in order to facilitate comparison
with the literature. Note that we propose uniforms priors for the two indexing parameters γ and
17

Note in contrast to the frictionless labor market model, the term 1 − α does not correspond to the labor share andwill depend on the outcome of the bargaining process. However, because a wide rage of values of the bargaining power
imply a labor share just below 1 − α, here we simply follow convention by setting 1 − α = 2/3. In our calculations,
1 − α equals 0.667 and the labor share 0.666.
18z
We follow SW, define ψz such that η z = 1−ψand estimate ψ z . When ψ z = 1, it is very costly to change theψzcapital utilization rate and the utilization rate does not vary. When ψz = 0, the marginal cost of changing the capitalutilization rate is constant and, as a result, the rental rate of capital does not vary.

20

γ p , based on the view that existing theory and evidence oﬀer no guidance for the appropriate valuesof these parameters. Further, a priori, values of zero and unity for these parameters seem equallyplausible. Finally, we note that in all instances the priors that we choose are reasonably loose.

Table 2: Prior and posterior distribution of structural parameters
Parameter
ψz
ηk
h
η
b̄
γ
γP
λ
λP
εP
rπ
ry
ρs
γa

Prior
Distribution
Beta (0.5,0.1)
Normal (4,1.5)
Beta (0.5,0.1)
Beta (0.5,0.1)
Beta (0.5,0.1)
Uniform (0,1)
Uniform (0,1)
Beta (0.75,0.1)
Beta (0.66,0.1)
Normal (1.15,0.05)
Normal (1.7,0.3)
Gamma (0.125,0.1)
Beta (0.75,0.1)
Uniform (1,1.5)

Max

Posterior
Mean
5%

95%

0.695
2.425
0.727
0.907
0.726
0.816
0.000
0.717
0.848
1.405
2.015
0.333
0.773
1.004

0.700
2.375
0.708
0.907
0.723
0.815
0.018
0.725
0.846
1.408
2.006
0.332
0.772
1.004

0.775
3.007
0.757
0.941
0.786
0.918
0.051
0.780
0.884
1.454
2.125
0.405
0.811
1.005

0.623
1.767
0.652
0.862
0.664
0.713
0.001
0.670
0.803
1.362
1.893
0.264
0.731
1.003

Notes: For the Uniform distribution the two numbers in parenthesis are the lower andthe upper bound. Otherwise, the two numbers are the mean and the standard deviation.

As noted, we estimate two new labor market parameters, η and b. There is little direct evidence
on the worker’s bargaining power parameter η. In their survey paper, Mortensen and Nagypal
propose a value of 0.5, which appears to reflect conventional thinking in the literature. Accordingly
we set the mean of the prior for this variable at 0.5, with a standard deviation of unity. We choose
a similar prior for b. As we noted earlier, Shimer proposes 0.4 as a "generous" value for b̄, while
Hall suggests 0.7 if one permits a broader interpretation of this variable. Both these estimates fit
within a ninety-five percent confidence interval of our prior.
Since the estimates of the “conventional” parameters are consistent with other studies, we focus
on the new parameters we consider here.19 In particular, we estimate a very reasonable degree ofwage rigidity. The estimate of λ is 0.72, which suggests a mean of just over three quarters between
19

In Appendix D we present estimates of the SW model, which has been shown to fit the data nearly as well as a
VAR. The precise version of the model we estimate is due to PST. Their formulation diﬀers from SW only in someminor details.

21

wage contracting periods.20 The evidence from micro-data (Gottshalk, 2006), suggests a modal
adjustment time of one year, though is silent (to our knowledge) about medians and means. In
addition, the estimates suggest a high degree of indexing of wages to past inflation: The estimateof the indexing parameter is almost 1, which suggests a high degree of eﬀective real wage rigidity.
The estimate of the key parameter b, the flow value of unemployment, is 0.72. What it suggests
is that in units of consumption goods, the flow value of unemployment is seventy percent of theworker’s marginal flow value to the firm. This percentage is close to the value proposed by Hall(2006), who as we suggested earlier motivates b as reflecting not only unemployment insurance
benefits but also utility gains from leisure. We also note that this estimate is well below the near
unity value required for the conventional flexible model to account for the data. Thus, the data
seem to prefer a combination of highly sticky wages and (eﬀectively) inelastic labor supply along
the extensive margin.
Table 3: Prior and posterior distribution of shock processes
Parameter
ρa
ρr
ρb
ρi
ρp
ρw
ρg
σa
σr
σb
σi
σp
σw
σg

Prior
Distribution
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)

Max

Posterior
Mean
5%

95%

0.140
0.207
0.713
0.605
0.808
0.264
0.991
1.039
0.224
0.362
0.166
0.062
0.578
0.357

0.096
0.179
0.724
0.599
0.814
0.261
0.993
1.025
0.226
0.334
0.165
0.06
0.586
0.358

0.167
0.266
0.788
0.675
0.863
0.322
0.998
1.084
0.248
0.436
0.207
0.076
0.650
0.391

0.036
0.096
0.660
0.520
0.762
0.199
0.987
0.968
0.206
0.251
0.126
0.046
0.528
0.327

.

Notes: The two numbers in parenthesis are the mean and the st. dev. of the distribution.
20

It is true that our estimate of the degree of price rigidity (λp = 0.85) is somehwat higher than that for wage
rigidity. Several points: First, our estimate is similar to what one obtains with the SW model (see Appendix D).
Second, our measure of inflation is based on the GDP deflator, which consists of producer prices, which are stickier
than consumer prices. In this regard, our estimate of λp suggests a median duration of price changes of roughly four
quarters, which is not too far above Steinsson and Nakamura (2007) estimates of three quarters for producer prices.
Note that the Poisson process for price adjustment leads to the mean exceeding the median, since the constant hazard
process suggests that some prices may not be adjusted for a long time.

22

Finally, the estimate of the worker’s bargaining power parameter is 0.9. This value lies above therange considered in the literature, typically 0.5 - 0.7. As noted earlier, however, there is virtually no
direct evidence on what an appropriate value of this parameter should be. One possibility, is thatwithin our framework it is very diﬃcult to separately identify b and η. Both parameters enter the
loglinear system via their respective impact on the steady state wage (see the Appendix). It may
be that to achieve a good identification of these parameters, we may need to introduce additional
labor market information.
It is also worth noting that within our framework, the lion’s share of the serial correlation in thereal wages is accounted for by the wage contracting structure. The exogenous shock to the wage
equation (modeled as a shock to bargaining power) has a first order serial coeﬃcient of only 0.26.
We next consider the model without wage rigidity. Table 4 presents the parameter estimatesand Table 5 presents the estimates of the shock processes in this case.

Table 4: Prior and posterior distribution of structural parameters, λ = 0
Parameter
ψz
ηk
h
η
b
γ
γP
λ
λP
εP
rπ
ry
ρs
γa

Prior
Distribution
Beta (0.5,0.1)
Normal (4,1.5)
Beta (0.5,0.1)
Beta (0.5,0.1)
Beta (0.5,0.1)
Uniform (0,1)
Uniform (0,1)
Beta (0.75,0.1)
Beta (0.66,0.1)
Normal (1.15,0.05)
Normal (1.7,0.3)
Gamma (0.125,0.1)
Beta (0.75,0.1)
Uniform (1,1.5)

Max

Posterior
Mean
5%

95%

0.861
1.023
0.801
0.616
0.983
–
0.000
–
0.574
1.347
1.927
0.013
0.685
1.003

0.852
1.179
0.803
0.589
0.982
–
0.037
–
0.575
1.351
1.999
0.019
0.700
1.003

0.911
1.635
0.840
0.726
0.987
–
0.090
–
0.630
1.407
2.297
0.043
0.746
1.004

0.783
0.803
0.760
0.451
0.975
–
0.003
–
0.512
1.298
1.748
0.003
0.648
1.001

Notes: For the Uniform distribution the two numbers in parenthesis are the lower andthe upper bound. Otherwise, the two numbers are the mean and the standard deviation.

The estimates of the conventional parameters do not change much. There is however now alarge change is the estimates of the two key labor market parameters: b increases to 0.98 and η falls
to 0.6. The former is close to the value we described earlier that Hagedorn and Manovskii used to23

argue that a flexible wage model could account for labor market volatility.21 Our estimates confirm
that absent wage rigidity, it is necessary to have highly elastic labor supply along the extensive
margin to account for the facts.22

Table 5: Prior and posterior distribution of shock processes, λ = 0
Parameter
ρa
ρr
ρb
ρi
ρp
ρw
ρg
σa
σr
σb
σi
σp
σw
σg

Prior
Distribution
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)

Max

Posterior
Mean
5%

95%

0.287
0.249
0.351
0.865
0.916
0.984
0.987
1.071
0.237
0.686
0.066
0.093
0.250
0.352

0.282
0.250
0.363
0.852
0.909
0.984
0.987
1.083
0.241
0.738
0.074
0.093
0.274
0.355

0.378
0.334
0.507
0.891
0.946
0.990
0.992
1.188
0.267
1.146
0.094
0.115
0.369
0.389

0.193
0.162
0.231
0.812
0.866
0.977
0.981
0.992
0.218
0.436
0.059
0.076
0.183
0.324

Notes: The two numbers in parenthesis are the mean and the st. dev. of the distribution.

One virtue of the Bayesian approach is that it is straightforward to compare the fit of thebaseline model versus the model without wage rigidity Table 6 reports the log marginal likelihoods
for the two models.
Table 6: Log Marginal Likelihood
Baseline Model-1191

Flex Wage Model
-1234

21

Hagedorn and Manvoski employ a slightly smaller value of b, 0.95, as opposed to 0.98, and much smaller value of
η, 0.05, as opposed to 0.62. Note that our prior on η is suﬃciently loose so as not to exclude values well below 0.5.
22
The large value of b suggests a huge response of employment to changes in unemployment insurance benefits, as
Hornstein, Krusell and Gianluca Violante (2005) and others have noted.

24

The baseline model clearly is preferred to the flex wage model. The diﬀerence in marginal
likelihood is forty loglikelihood points, which is a significant diﬀerence.
Another way to assess how the model captures the data is to portray the autocovariance functions of the model variables against the data. Figure 2 reports this information. The solid line ineach panel reports the autocovariance function of the data. The dashed lines are ninety-five percent
posterior intervals of the model autocovariances.23 Overall, the baseline model does well. For themost part, the empirical autocovariances functions lie within the model standard error bands. In
this regard, the model does particularly well in capturing the reduced form dynamics of wages andhours.
As a further check on the model, we explore how well it is able to account for the dynamics of
unemployment. As the bottom row shows, the empirical autocovariance function for unemployment
lies within the standard error bands of the model generated sample moments. We interpret this asa strong test of the model given that we did not use information on unemployment to fit model
parameters.
Figure 3 presents the autocovariance functions for the model without wage rigidity. Overall
this model doesn’t do as well. For a number of key variables, including output, wages, hours, andunemployment, the empirical autocovariances in places lie well outside the model-generated bands.
This is particularly true for wages, hours, output and unemployment.
The sharp diﬀerences in the autocovariance functions suggest why the model with wage rigidity
appears to outperform the one without. Roughly speaking, the flexible wage model is not suﬃciently
flexible to permit the framework to capture all the various moments of the data.
We next illustrate the properties of the model economy by simulating the response to several key
shocks. We analyze the role of wage rigidity, in particular, by examining both our benchmark modeland the same model with staggered contracting replaced by period by period wage negotiations.
As Table 7 shows, the estimates suggest that the main driving force is the investment shock which,strictly speaking is interpretable as shock to investment-specific technological change. It accounts
for more than half the variation in output growth on impact and more than forty percent at all
horizons. This finding is consistent with both SW and PST. Next in importance is the disembodied
productivity shock which accounts for roughly seventeen percent of the variation at horizons of ayear or greater.
23

The posterior intervals are computed as follows. We sample 500 points from the posterior and for each of them
we generate 160 observations, which is the length of the data sample, 500 times. Then, for each draw we computeautocovariances and we report the fifth and the ninety-fifth percentile.

25

Table 7: Variance decomposition for output (∆ log y) at diﬀerent horizons(in percentage)

Shocksa shock - technology
r shock - monetaryb shock - preferences
i shock - investmentg shock - government
p shock - price markup
w shock - bargaining poweron impact
16.7
6.1
11.1
54.8
9.4
1.9
0.0

Horizons
1 year
4 years
32.5
5.0
9.2
41.9
8.7
2.5
0.2

31.0
5.4
9.5
42.4
8.2
3.2
0.3

long run
31.0
5.4
9.5
42.4
8.2
3.2
0.3

The recent literature on unemployment fluctuations that we alluded to in the introduction
almost uniformly treats productivity shocks as the main driving force. Thus for purposes of comparison, we begin with this disturbance. In particular, Figure 4 illustrates the response of the model
economy to a productivity shock. The thick line is the model with wage rigidity. The dotted line
has wage rigidity turned oﬀ.24 Notice that the response of output and employment is significantly
greater with wage rigidity than without. Conversely, due to the staggered contracting the responseof wages is much smoother. The smooth response of wages, of course, implies a larger responseof profits to the technology shock than otherwise. This leads to a stronger response of output
and employment relative to the flexible wage case. Note also that there is an immediate drop in
inflation following the productivity shock, which is in line with the evidence in Altig, Christiano,
Eichenbaum and Evans (2004). This does not come as puzzle in this framework: with wage rigidity
the rise in productivity reduces marginal costs and hence inflation.
We next turn to the investment shock, as portrayed in Figure 5. In contrast to the case of the
productivity shock, in this instance the response of output and employment is very similar acrossthe two models. Note, however, that the responses of wages, w, and inflation, π, are quite diﬀerent.
In the case with wage rigidity shut oﬀ, the responses of these variables appear counterfactuallylarge. In absolute value, the response of wages is three times as large as the response of output
and the response of inflation (annualized) is nearly half as large. To our knowledge there do notexist series for wages or inflation that display this kind of volatility relative to output.25 Intuitively,
the investment shock shifts output demand without directly aﬀecting factor productivities. With24

To shut oﬀ wage rigidity we simply set the probability that wages do not adjust, λ, equal to zero.
It is true that the experiment here is conditional on the investment shock and the evidence to which we arealluding consists of unconditional moments. However, the investment shock accounts for nearly fifty percent of thevariation in output growth within the model. If the flex wage model were true then we should observe relatively
volatile behavior of real wages and price markups.
25

26

nominal price rigidities, markups decline and employment adjusts to meet demand. In the staggered
contracting model, because labor costs are sticky, the employment response is associated with onlya modest change in real wages and inflation. With wage rigidity shut oﬀ, however, there is a sharp
increase in wages which boosts up real marginal costs and hence inflation. In sum, in the case ofinvestment demand shocks, the wage rigidity smooths out the response of wages and inflation to
the disturbances.
Finally, we consider how well our model fits the data as compared to SW. In appendix D wereport the parameter estimates for the SW model. By and large, the estimates for the “non-labormarket” parameters are similar across the two models. To explore relative fit, one possibility would
be to compare the marginal likelihoods, as we did we the sticky and flex wage versions of our model.
In what we did earlier, however, one model (the sticky wage) nested the other (the flex wage) andthe priors used in the estimation are identical. Neither of these conditions applies in this instance.
While in principle this should not be a problem, in practice it may be, given the sensitivity of themarginal likelihood to the priors. Accordingly, to get a sense of relative fit we simply compare theautocovariance functions of each model. As Figure 6 shows, the model standard error bands of ourmodel overlap closely with those of the SW model. We conclude that from a practical standpoint,
the ability of our model to characterize the data is very similar to that of SW.

5

Concluding Remarks

We have developed and estimate a medium scale macroeconomic model that allows for unemployment and staggered nominal wage contracting. In contrast to most existing quantitative models,the employment of existing workers is eﬃcient. Thus, the model is immune to the Barro’s (1977)
critique that models relying on wage rigidity to have allocative eﬀects in situations where firms
and workers have on-going relationships ignore mutual gains from trade. In our model, in contrast,
wage rigidity aﬀects the hiring of new workers. The former is introduced via the staggered Nashbargaining setup of Gertler and Trigari (2006). A robust finding is that the model with rigidity
provides a better description of the data than does a flexible wage version. Further, our model
appears to capture the moments of the data as well as Smets and Wouters (2007). In addition,
while the conventional model is silent about the behavior of unemployment, our model generates
dynamics for this variable that are in line with the data.
More work is necessary, however, to ensure a robust identification of the key labor market
parameters. Our preliminary estimates of the degree of wage rigidity and the flow value of unemployment appear to be quite reasonable. The estimate of worker’s bargaining power lies above
conventional wisdom, though there is little direct evidence on what this parameter should be. One
possibility is that it may be diﬃcult to separately identify some of the key labor market parameters that influence employment volatility. Accordingly, it may be necessary to introduce additional
labor market information.

27

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[30] Mortensen, Dale and Eva Nagypal (2007). “More on Unemployment and Vacancy Fluctuations,” Review of Economic Dynamics, 10(3), 327-347.
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30

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31

APPENDIX A
A1. The contract wage
• The worker and the firm surpluses are
Ht (i) = wt (i) − bt + ρβEt Λt,t+1 Ht+1 (i) − st+1 βEt Λt,t+1 Hx,t+1∞
P= Wt (i) − Et(ρβ)s Λt,t+s [bt+s + st+s+1 βΛt+s,t+s+1 Hx,t+s+1 ]
s=0

κt+1
xt+1 (i)2 + ρβEt Λt,t+1 Jt+1 (i)Jt (i) = pw
t fnt − wt (i) + βEt Λt,t+1
2
h
i
∞
P
κt+s+1
2− Wt (i)x(ρβ)s Λt,t+s pw
f
+
βΛ
(i)
= Et
nt+s
t+s,t+s+1
t+s+1
t+s
2
s=0

where Wt (i) denotes the discounted sum of expected future wages to be received by a worker
over the life of the relationship at the firm:
Wt (i) = Et∞P(ρβ)s Λt,t+s wt+s (i)s=0• Consider a renegotiating firm. The current and future expected wages are given by:
wt (i) =Et wt+1 (i) = Et
Et wt+2 (i) = Et

wt∗n
pt

¤
1 £
∗n
λγπ γt wt∗n + (1 − λ) wt+1

pt+1
£
¤
1
∗n
∗n
+ (1 − λ) wt+2λ2 γπ γt γπ γt+1 wt∗n + λ (1 − λ) γπ γt+1 wt+1

pt+2

and so on....

Using these expressions, we can write:
Wt (i) =wt∗npt+ (ρβ) Et Λt,t+1¤
1 £ ∗n γ∗nλwt γπ t + (1 − λ) wt+1

pt+1¤
1 £ 2 ∗n γ γ∗n∗n
λ wt γπ t γπ t+1 + λ (1 − λ) wt+1
γπ γt+1 + (1 − λ) wt+2
+ (ρβ)2 Et Λt,t+2
pt+2
+...

32

Wt (i) = wt∗

∙
¸
γ
∗ pt
∗
+ (ρβ) Et Λt,t+1 λwt
γπ + (1 − λ) wt+1
pt+1 t∙¸γγγ
22 ∗ pt∗ pt+1∗+ (ρβ) Et Λt,t+2 λ wt
γπ γπ
+ λ (1 − λ) wt+1
γπ
+ (1 − λ) wt+2
pt+2 t t+1
pt+2 t+1+...

• Collecting terms:

∙
¸
pt
pt
γ
γ
γ
2
Wt (i) = 1 + (ρλβ) Λt,t+1
γπ + (ρλβ) Λt,t+2γπ γπ+ ... wt∗
pt+1 tpt+2 t t+1¸∙
pt+1 γ∗+ (1 − λ) (ρβ) Et Λt,t+1 1 + (ρλβ) Λt+1,t+2
γπ t+1 + ... wt+1
pt+2¸∙
pt+2 γ2∗+ (1 − λ) (ρβ) Et Λt,t+2 1 + (ρλβ) Λt+2,t+3
γπ+ .... wt+2
pt+3 t+2• Given that∆t = Et∞P(ρλβ)s Λt,t+s

s=0

we have¢pt ¡Qsγ
k=1 γπ t+k−1
pt+s

Wt (i) = ∆t wt∗∗+ (1 − λ) (ρβ) Et Λt,t+1 ∆t+1 wt+1∗
+ (1 − λ) (ρβ)2 Et Λt,t+2 ∆t+2 wt+2+...
• Finally, rearranging:

Wt (i) = ∆t wt∗ + (1 − λ) Et

∞
P

s=1

∗
(ρβ)s Λt,t+s ∆t+s wt+s

• Substituting now the expression for Wt (i) in the worker and the firm surpluses we get∞P(ρβ)s Λt,t+s [bt+s + st+s+1 βΛt+s,t+s+1 Hx,t+s+1¤∗− (1 − λ) (ρβ) Λt+s,t+s+1 ∆t+s+1 wt+s+1

Ht (i) = ∆t wt∗ − Ets=0hκt+s+1xt+s+1 (i)2(ρβ)s Λt,t+s pw
t+s fnt+s + βΛt+s,t+s+1
2
s=0
¤
∗
− (1 − λ) (ρβ) Λt+s,t+s+1 ∆t+s+1 wt+s+1− ∆t wt∗

Jt (i) = Et

∞
P

33

• The Nash first-order condition isχt (i) Jt (i) = (1 − χt (i)) Ht (i)• This can be rewritten as

χJt (i) = (1 − χ) Ht (i) − φt (i)withφt (i) = (χt (i) − χ) [Jt (i) + Ht (i)]• Substituting the expression of Jt (i) and Ht (i) for a renegotiating firm:nh ³´
κt+s+1
2
x
f
+
βΛ
(i)
χ pw
+
t+s,t+s+1
t+s+1
t+s nt+s2
s=0(1 − χ) (bt+s + st+s+1 βΛt+s,t+s+1 Hx,t+s+1 )]ª∗− (1 − λ) ρβΛt+s,t+s+1 ∆t+s+1 wt+s+1∆t wt∗ − φt (i) = Et∞P(ρβ)s Λt,t+s• The above equation can be written in a recursive form in the following way:
´
³
κt+1
2
+ (1 − χ) (bt + st+1 βEt Λt,t+1 Hx,t+1 )∆t wt∗ − φt (i) = χ pw
x
f
+
βE
Λ
(i)
nt
t
t,t+1
t+1
t
2
¢
¡
∗
∗
− (1 − λ) ρβEt Λt,t+1 ∆t+1 wt+1
+ ρβEt Λt,t+1 ∆t+1 wt+1
− φt+1 (i)• Simplifying, we obtain∗∆t wt∗ = wto (i) + ρλβEt Λt,t+1 ∆t+1 wt+1

with wto (i) denoting the target wage:´³
κt+1
xt+1 (i)2
wto (i) = χ pw
t fnt + βEt Λt,t+1
2+ (1 − χ) (bt + st+1 βEt Λt,t+1 Hx,t+1 ) + Φt
andΦt = φt (i) − ρβEt Λt,t+1 φt+1 (i)A2. The loglinearized target wage
• Loglinearizing the target wage then yields
³
´
´
³
b
bt +
bt+1 − λwbto (i) = ϕf n pbw
bt+1 (i) + (ϕx /2) Et λt + fnt + ϕx Et x³´¤£
bt+1 − λbt + ϕ χb t+1 + λϕbbbt + ϕs Et sbt+1 + H
bt+1 (i)χ b t (i) − ρβEt χ34wherew = (w/a)
ϕf n = χpw (fn /a) w−1

ϕx = χβ −1 (1 − ρ) w−1

ϕb = (1 − χ) b (k/a) w−1

ϕs = χβs −1 w−1

ϕχ = χ (1 − χ)−1 −1 w−1= (κx)−1
and wherebt
b x,t = H
H

• Finally, loglinearizing the weight in the target wage gives³´
b t (i) − ∆b t + (1 − χ) (1 − η)−1 bεηtχb t (i) = − (1 − χ) Σ
with

³
´
bt+1 − λbt − πb t+1 − bb t = ρλβEt λbt+1 + γbπt + ∆
εat+1∆³´
bt+1 − λbt − πb t+1 (i) − bb t (i) = xλβEt x
bt+1 (i) + λβEt λbt+1 + γbπt + Σ
εat+1
Σ
A3. Spillover eﬀects• In order to illustrate the spillover eﬀects, we derive expressions for x
bt+1 (i) , χbt (i) and χ
bt+1 (i)∗bbt , as well as an expression for Ht+1 as a function of the
as a function of the wage gap w
bt − w∗bt+1 .
wage gap w
bt+1 − w• The loglinear version of the real wage index isbt∗ + λ (wbt−1 − πbt + γbπ t−1 − bεat )wbt = (1 − λ) w

Using the loglinear versions for the expected future wages at a renegotiated firm, we obtain:bt (i) = w
bt − wbt∗
w
bt − w

Et (wbt+1 − w
bt+1 (i)) = λ (wbt − wbt∗ )Et (w
bt+2 − w
bt+2 (i)) = λ2 (wbt − wbt∗ )and so on....

35

• The loglinear version of the job creation condition, equation (25), is:´³bb t,t+1xbt (i) = pw (fn /a) pbw
bt (i) + βEt x
bt+1 (i) + (ρ + x/2) βEt Λt + fnt − w w

Substracting the loglinear version of the aggregate job creation condition, we can write:x
bt (i) − x
bt = − w (wbt (i) − w
bt ) + βEt (bxt+1 (i) − x
bt+1 )

This can be iterated forward to give:bt =x
bt (i) − xw (wbt − wbt (i))+β wEt (wbt+1 − w
bt+1 (i))+β 2 wEt (wbt+2 − w
bt+2 (i))+...

Substituting the previous expressions and rearranging yields:xbt (i) = x
bt + Σw (wbt − w
bt∗ )

with

Σ=

1
1 − λβMore generally, we havexbt (i) = x
bt + Σw (wbt − wbt∗ )Et x
bt+1 (i) = Et x
bt+1 + Σwλ (wbt − wbt∗ )bt+2 (i) = Et x
bt+2 + Σwλ2 (wbt − wbt∗ )Et x
and so on....

• The loglinear horizon-adjusted bargaining weight can be written as:´³b t + (1 − χ) (1 − η)−1 bb t (i) − Σ
b t − (1 − χ) Σεηtχb t (i) = χ
with

´
³
bt
bt − ∆
χb t = − (1 − χ) Σ

and

³
´
b t (i) − Σ
b t = (xλβ) Et (bb t+1
b t+1 (i) − ΣΣ
xt+1 (i) − x
bt+1 ) + λβEt Σ
xt+1 (i) − x
bt+1 )= (xλβ) Et (bxt+2 (i) − x
bt+2 ) ++ (xλβ) λβEt (b+ (xλβ) (λβ)2 Et (bxt+3 (i) − x
bt+3 ) +...

36

Substituting the expressions for the future hiring rates and collecting terms, yields:b t = (1 − ρ) Ψ Σw (wb t (i) − Σ
bt − w
bt∗ )
Σ

with

Ψ=

λ2 β
1 − λ2 βso thatandbt − (1 − χ) (1 − ρ) Ψ Σw (wbt − wbt∗ ) + (1 − χ) (1 − η)−1 bεηtχbt (i) = χ
bt+1 (i) = Et χ
bt+1 − (1 − χ) (1 − ρ) Ψ Σwλ (wbt − wbt∗ ) + (1 − χ) (1 − η)−1 Etb
εηt+1
Et χ

• The loglinear version of the worker’s surplus, equation (33), is:´³b t (i) = (1 − χ) χ−1 ww
bt (i) − b (k/a) bbt +
H
³
³
´
´
bt+1 − λbt + Hbt+1 − λbt + Hb t+1 (i) − sβEt sbt+1 + λb t+1ρβEt λSubstracting the loglinear version of the average worker’s surplus, we obtain:³´b t (i) − H
b t = (1 − χ) χ−1 w (wb t+1
b t+1 (i) − H
Hbt (i) − w
bt ) + ρβEt HThis can be iterated forward to give:b t (i) − H
b t = (1 − χ) χ−1 w (wH
bt (i) − w
bt )+ (1 − χ) χ−1 w (ρβ) Et (wbt+1 (i) − w
bt+1 )

bt+2 (i) − w
bt+2 )+ (1 − χ) χ−1 w (ρβ)2 Et (w+...

Substituting and rearranging

withb t (i) = H
b t − (1 − χ) χ−1 ∆w (wH
bt − wbt∗ )∆=1
1 − ρλβThe loglinear version of the Nash bargaining condition, equation (41), isb t (i)Jbt (i) + (1 − χ)−1 χ
b t (i) = H
37

b t (i) yields:
where Jbt (i) = xbt (i). Substituting the expressions for x
bt (i), χbt (i) and H
bt = xbt + Γ (wbt − wbt∗ ) + (1 − χ)−1 χ
bt + (1 − η)−1 b
εηt
H

with

Γ = (1 − η (1 − ρ) Ψ)−1 η −1 Σw

We then have:¡¢∗b t+1 = Et x
bt+1 + ΓEt w
bt+1
b t+1 + (1 − η)−1 Etb
εηt+1
Et H
bt+1 − w+ (1 − χ)−1 Et χb t+1 into the loglinear target• Substitute now the expressions for x
bt+1 (i) , χbt (i), χb t+1 (i) and H
wage, to obtain

with

and

bto +
w
bto (i) = w

¡
¢
τ2
τ1
∗
bt+1 − w+
Et w(w
bt − wbt+1bt∗ )1 − ρλβ1 − ρλβ³´bnt + (ϕ + ϕ ) Et xw
bto = ϕf n pbw+
fbt+1 + ϕs Et sbt+1 + ϕbbbt
x
s
t
´
³
¡
¢
bt + ϕχ χ
bt+1 − λbt − β (ρ − s) χεw
+ (ϕs + ϕx /2) Et λbt+1 + bt= ϕη (1 − βρη (ρ − s)) b
εηt
b
εw
t
ϕη = χ (1 − η)−1 −1 w−1andτ 1 = ϕs Γ (1 − ρλβ)£¤τ 2 = ϕx λ − ϕχ (1 − χ) (1 − ρ) Ψ (1 − ρβλ) Σw (1 − ρλβ)38

APPENDIX B
Steady state calculation• First obtain

s
1−ρ+s
u=1−n
su
x=
n

n=

and
χ=
• Then getηη + (1 − η) Σ/∆

rk = 1/β̄ − 1 + δ
β̄ = β/γ a
k/a
αpw
= k
y/a
r
µ
¶
1−δ
i/a
k/a
= 1−
γa
y/a
γa
y/a
µ
¶ 1
k/a 1−α
k/a
=
n
y/a
µ
¶
k/a α
y/a
=
n
n
fn /a = (1 − α)• We also havey/anz = qk = 1• Then κ and w solve the following system(
κx = pw (fn /a) − (w/a) + β (κ/2) x2 + βρκx
¤
£ ¡
£
¢¤
(w/a) = χ pw (fn /a) + β (κ/2) x2 + βsκx + (1 − χ) b pw (fn /a) + β (κ/2) x2
where

b=
• Finallyb (k/a)pw (fn /a) + β (κ/2) x2

g/ai/aκ x2 n
g/a
i/a
κ
x2
c/a
=1−
−
−
=1−
−
−
y/a
y/a y/a 2 y/ay/a y/a 2 (y/a) /n
39

APPENDIX C
The complete loglinear GST model• Technology

kt + (1 − α) nbt
ybt = αb(E1)ybt = cyb
ct + iybit + gyb
gt + zyb
zt + xy (2bxt + nbt−1 )(E2)m
b t = σb
ut + (1 − σ) vbt(E3)• Resource constraint

2

c/a
i/a
g/a
κx n
where cy = y/a, iy = y/a, gy = y/a, zy = rk k/ay/a and xy = 2 y/a .

• Matching• Employment dynamics• Transition probabilities• Unemployment• Eﬀective capital• Physical capital dynamics

where ξ = 1−δγ

nt−1 + (1 − ρ) mbtnbt = ρbb t − vbt
qbt = mbt − ubt
sbt = mnbt−1ubt = − nupbεat = zbt + b
kt−1
kt + b

´
³
´
³
p
b
kt−1
−b
εat + (1 − ξ) bit + b
εitktp = ξ b

(E4)

(E5)

(E6)

(E7)

(E8)

(E9)

a

• Aggregate vacancies• Consumption-saving

bt−1x
bt = qbt + vbt − nbt+1 + (bb t = Et λrt − Et π
bt+1 ) − Etb
εat+1
λ
40

(E10)

(E11)

• Marginal utility

¡
¡
¢¡
¢
¢
bt = h̄ (bct−1 − bεat ) − 1 + β h̄2 b
1 − h̄ 1 − β h̄ λ
ct +
´
¡
¢
¡
¢³ b
εat+1 + 1 − h̄ b
εbt+1
ct+1 + bεt − β h̄Etb
β h̄Et b(E12)where h measures the degree of habit persistence in consumption and where h̄ = γha

• Capital utilizationz
where η z = a0 (z) /a00 (z) = 1−ψψz• Investment

zbt = η z rbtk

´ 1/ ¡η γ 2 ¢ ³
´
³
´
³
1
β
k a
a
bit =
bit−1 − b
Et bit+1 + b
εt +
εit +
εat+1
qbtk + b
1+β
1+β
1+β

(E13)

(E14)

where η k = s00• Capital renting• Tobin’s q

bt − bkt = rbtk
pbw
t +y

£
¤
k
k
+ 1 − β̄ (1 − δ) Et rbt+1− (brt − Et π
bt+1 )
qbtk = β̄ (1 − δ) Et qbt+1• Aggregate hiring rate
´
´
³
³
b
bt
bt+1 − λbt + βEt x
bt+1 + (ρ + x/2) β Et λx
bt = pw (fn /a) pbw
t + fnt − (w/a) w

(E15)

(E16)

(E17)

where

= (κx)−1• Marginal product of labor• Weight in Nash bargaining
withbt
fbnt = ybt − n´³bt
bt − ∆
χb t = − (1 − χ) Σ³´
bt+1 − λbt − πb t+1 − bb t = ρλβEt λbt+1 + γbπt + ∆
εat+1∆³´
bt+1 − λbt − πb t+1 − bb t = (1 − ρ) λβEt x
bt+1 + λβEt λbt+1 + γbπt + Σ
εat+1
Σ
41

(E18)

(E19)
(E20)
(E21)

• Spillover-free target wage³´
bw
bto = ϕf n pbw
bt+1 + ϕs Et sbt+1 + ϕbbbt
t + fnt + (ϕx + ϕs ) Et x´³¡¢
bt + ϕ χ
bt+1 − λ+ (ϕs + ϕx /2) Et λbt+1 + bεw
χ b t − β (ρ − s) χ
t

(E22)

where

w = (w/a)w

ϕf n = χp (fn /a) w−1

ϕx = χβ (1 − ρ) −1 w−1

ϕb = (1 − χ) b (k/a) w−1

ϕs = χβs −1 w−1

ϕχ = χ (1 − χ)−1 −1 w−1• Aggregate wage
where

¡
¢
w
bt = γ b (wbt−1 − πbt + γbπ t−1 − bεat ) + γ o w
bto + γ f w
bt+1 − γbπt + bεat+1
bt+1 + π
γ b = (1 + τ 2 ) φ−1

γ o = ςφ−1

φ = 1 + τ 2 + ς + ρβ − τ 1(E23)

γ f = (ρβ − τ 1 ) φ−1

ς = (1 − λ) (1 − ρλβ) λ−1

τ 1 = ϕs Γ (1 − ρλβ)£¤τ 2 = ϕx λ − ϕχ (1 − χ) (1 − ρ) Ψ∆−1 Σw (1 − ρλβ)¡¢Γ = (1 − η (1 − ρ) Ψ) η −1 Σw
Ψ = βλ2 / 1 − βλ2Σ = (1 − βλ)−1• Phillips curvewherebt−1 + ιo (bpwεpt ) + ιf Eb
π t+1
πbt = ιb π
t +bιb = γ p (φp )−1φp = 1 + βγ p
• Tayor rule
• Government spending• Market tightness• Benefitsιo = (ς p /τ p ) (φp )−1

ς p = (1 − λp ) (1 − λp β) (λp )−1(E24)

ιf = β (φp )−1¡¢
τ p = 1 + εP − 1 ξ

bt + ry (byt − ybnt )] + b
εrtrbt = ρs rbt−1 + (1 − ρs ) [rπ π

(E25)

b
bt
θt = vbt − u(E27)

gbt = ybt +1 − gy gbεgy t

bbt = b
ktp

42

(E26)

(E28)

APPENDIX D
Estimates of the SW model
Note that the parameters ω and εw are, respectively, the Frisch elasticity of labor supply andthe gross steady state wage markup. The parameters ρw and σ w refer to a wage markup shock.
Table A1: Prior and posterior distribution of structural parameters, SW modelParameter
ψz
ηk
h
ω
γ
γP
λ
λP
εP
εw
rπ
ry
ρs
γa

Prior
Distribution

Max

Posterior
Mean
5%

95%

Beta (0.5,0.1)
Normal (4,1.5)
Beta (0.5,0.1)
Gamma(2,0.75)
Uniform (0,1)
Uniform (0,1)
Beta (0.75,0.1)
Beta (0.66,0.1)
Normal (1.15,0.05)
Normal (1.15,0.05)
Normal (1.7,0.3)
Gamma (0.125,0.1)
Beta (0.75,0.1)
Uniform (1,1.5)

0.667
2.922
0.746
3.910
0.796
0.000
0.881
0.856
1.392
1.138
2.057
0.307
0.807
1.004

0.657
3.593
0.772
4.041
0.763
0.025
0.865
0.854
1.391
1.127
2.053
0.320
0.813
1.004

0.765
5.297
0.839
5.452
0.938
0.078
0.925
0.890
1.448
1.210
2.413
0.437
0.855
1.005

0.546
2.286
0.703
2.829
0.574
0.001
0.783
0.816
1.336
1.042
1.725
0.224
0.765
1.003

Notes: For the Uniform distribution the two numbers in parenthesis are the lower andthe upper bound. Otherwise, the two numbers are the mean and the standard deviation.

43

Table A2: Prior and posterior distribution of shock processes, SW modelParameter
ρa
ρr
ρb
ρi
ρp
ρw
ρg
σa
σr
σb
σi
σp
σw
σg

Prior
Distribution
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
Beta (0.5,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)
IGamma (0.15,0.15)

Max

Posterior
Mean
5%

95%

0.129
0.241
0.698
0.531
0.810
0.296
0.991
1.022
0.228
0.420
0.195
0.060
0.197
0.359

0.143
0.254
0.662
0.508
0.799
0.309
0.989
1.041
0.231
0.583
0.241
0.063
0.197
0.362

0.226
0.359
0.777
0.631
0.851
0.420
0.995
1.141
0.253
0.999
0.354
0.080
0.231
0.396

0.067
0.155
0.527
0.385
0.746
0.195
0.982
0.952
0.211
0.337
0.154
0.048
0.166
0.331

.

Notes: The two numbers in parenthesis are the mean and the st. dev. of the distribution.

44

Figure 1: Total hours and employment
6
Employment
Hours

4

2

0

−2

−4

−6

−8
1960

1965

1970

1975

1980

1985

45

1990

1995

2000

2005

Figure 2: Autocovariance function of GST model vs dataY(t) − Y(t+k) ,Y(t) − W(t+k) ,

2

0.3

Y

1.5

0

5

0.3

W

0

5

−0.1
5

−0.1
−0.1
−0.2
−0.2

0
5

0

H(t) − W(t−k) ,5
H(t) − H(t+k) ,

1

20

0

H(t) − π(t+k) ,

5

−1

0

π(t) − Y(t−k) ,

5

0.2

−100
−1

π(t) − W(t−k) ,

5

−2
0

π(t) − H(t−k) ,50π(t) − π(t+k) ,

5

−200
0

π(t) − i(t+k) ,

5

−0.1
−0.2

0.4

15

0.2

0.2

10

0

5

0

i(t) − Y(t−k) ,

i

5

−0.2

−0.1

u(t) − W(t−k) ,

10
0
0

0

u(t) − i(t−k) ,5u(t) − u(t+k) ,

2000
1500

10

1000

10
0

500

−200

−10

5

20

−150

5

u(t) − π(t−k) ,

5

−100

5
0

0

u(t) − H(t−k) ,

−50

−5

0

20

20

0

15

u

5

5
i(t) − u(t+k) ,

0.2
0

0
30

0.4

−3
5

5
i(t) − i(t+k) ,

0.6

0.2
0

0
1

−2

u(t) − Y(t−k) ,

10

5

0.8
0.4

−0.2

20

i(t) − π(t−k) ,

−1

−0.6
5

0

0.6

−0.15

0

5
i(t) − H(t−k) ,

0

0
−0.05

−0.4

0

i(t) − W(t−k) ,

0

5

5

−3
0.2

π(t) − u(t+k) ,

20

0.4
−2

−0.2

0
25

0.6
0.6

−1

π

−3

0.8

0

0

−150

−2

5
0

5
H(t) − u(t+k) ,

−50

0

−1

10

0

−2

0
H(t) − i(t+k) ,

15
0.5

5
W(t) − u(t+k) ,

0

25

1

0

0

−2
−4
−6
−8
−10
−12
5 0

0

0.5

0

H(t) − Y(t−k) ,5
W(t) − i(t+k) ,000

W(t) − π(t+k) ,

−20

−0.6
5 0

1

0.2

0

0

W(t) − H(t+k) ,

0.4

0.1

H

5

W(t) − W(t+k) ,

−10

−0.4

−0.3

0.6

0.2

−0.2

−0.2
1
0

W(t) − Y(t−k) ,

0

2

0

0

Y(t) − u(t+k) ,

10

0

0.1

0.5

Y(t) − i(t+k) ,

0
−0.1

0.2

1

Y(t) − π(t+k) ,Y(t) − H(t+k) ,

3

0
0

5

0

5

46

0

5

0

5

0

5

Figure 3: Autocovariance function of flex W model vs dataY(t) − Y(t+k) ,Y(t) − W(t+k) ,

0.4

10
1.5

3

Y

Y(t) − π(t+k) ,Y(t) − H(t+k) ,

4

6

0.5

4

0

2

2
1
0
0

8

1

5

0

W(t) − Y(t−k) ,

5

1.5

−0.2
5 0

5

0

W

H(t) − Y(t−k) ,

5

0

0

−0.1
5 0

−0.1
5 0

3

80

2

60

1

2

40

π(t) − Y(t−k) ,50π(t) − W(t−k) ,50π(t) − H(t−k) ,

5

π

0.2

2

0

0

−0.2
−0.4
0

5
i(t) − Y(t−k) ,50i(t) − W(t−k) ,

5

−600
0
0

0

5
H(t) − u(t+k) ,

−400

2

π(t) − π(t+k) ,

−800

5

0

−1000
5 0

π(t) − i(t+k) ,0i(t) − H(t−k) ,i(t) − π(t−k) ,5π(t) − u(t+k) ,

5

20

0.5
0.4
0.3
0.2
0.1

0.2

−2
0

0

−200

0.4

−0.2

5

4

0.6

0.2
0

−40

−1

0
0

−20

H(t) − i(t+k) ,

0

20

−5

H(t) − π(t+k) ,5
W(t) − u(t+k) ,

0

0.1

100

0

W(t) − i(t+k) ,0.3

H(t) − H(t+k) ,

4

H

0

H(t) − W(t−k) ,

5

−100
5 0

0.1

0
0

−0.5
5 0

0.2

2
0

−50

0.2

2

0
−0.5

W(t) − π(t+k) ,

0.3

4

1
0.5

0
0

W(t) − H(t+k) ,4

Y(t) − u(t+k) ,

0

0.2

0

W(t) − W(t+k) ,Y(t) − i(t+k) ,

0.5

10
0
−10
−20
0

5

0

i(t) − i(t+k) ,5i(t) − u(t+k) ,

1
0

0.3

2

i
−0.4

0.1

0

0

−2

0
u(t) − Y(t−k) ,

0.2

0.2

5

0

u(t) − W(t−k) ,

0

20

5

0

u(t) − H(t−k) ,u(t) − π(t−k) ,

5

−600

0

−800

−20
0

−40
5 0

−1000
5 0

4000

−20
5

47

2000

−30
0

5

5
u(t) − u(t+k) ,

6000

−10

−20

0

8000

0

−10

5
10000

10

u

−20

−20
0
u(t) − i(t−k) ,

−200
−400

0

0

0.4

10
40

20

0.6

0

−0.1
5 0

−0.6

0.8

0.4

0.2

−0.2

0

5

0

5

Figure 4: Impulse response functions to a technology shock
Y

I

C

1.4

2.2

1.6

1.2

2

1.4

1.8

1

1.2

1.6
0.8

1
1.4

0.6

0.8
1.2

0.4

0.6

1

0.2

0.4

0.8
flex W

0

0

20

40

0.6

0

W

20

40

0.2

0

40

π

NT

1.5

0.6

0.02

1

0.4

0

0.2

0.5

20

−0.02

0
0

−0.04
−0.2

−0.5
−1
−1.5

−0.06

−0.4

−0.08

−0.6
0

20

40

−0.8

0

R

20

40

a shock

0.05

1.4
1.2

0

1
0.8

−0.05
0.6
0.4

−0.1

0.2
−0.15

0

20

40

0

0

20

48

40

−0.1

0

20

40

Figure 5: Impulse response functions to an investment specific shock

Y

I

1

C

4

0.3

flex W

0.8

0.2
3
0.1

0.6
2

0

0.4
−0.1

1
0.2

−0.2
0

0
−0.2

−0.3
0

20

40

−1

0

W

20

40

−0.4

0

20

40

π

NT

3.5

1

0.04

3

0.8

0.03

2.5

0.6

2

0.4

1.5

0.2

1

0

0.5

−0.2

0.02
0.01

0

0

20

40

−0.4

0
−0.01

0

R

20

40

i shock

0.2

0.2

0.15

0.15

0.1
0.1
0.05
0.05

0

−0.05

0

20

40

0

0

20

49

40

−0.02

0

20

40

Figure 6: Autocovariance functions of GST and SW vs dataY(t) − Y(t+k) ,

Data
GST
GST
SW
SW

2
1.5

Y

Y(t) − W(t+k) ,

1

2

0

5

0

W(t) − W(t+k) ,0.3

5W(t) − π(t+k) ,

0.2
0.4

0.5

0.2

−0.1

0

0
0

5

0

H(t) − Y(t−k) ,

5

−0.05

0

−0.1

−0.05

−0.15

−0.1

−0.2

−0.15

H(t) − W(t−k) ,

5

1

H

5

0

5

−1

−2

5

π(t) − Y(t−k) ,

0

−1.5

10

0

5
H(t) − i(t+k) ,

−1

15

0

0

H(t) − π(t+k) ,

0

0.5

−2

5

−0.5

20

0
−1

0

H(t) − H(t+k) ,

25

1

−0.2

−0.25
0

5
W(t) − i(t+k) ,

0.05

0
1

0

−0.6
5
0

0

W(t) − H(t+k) ,

0.6

0.1

−0.4

−0.3

0.5

W(t) − Y(t−k) ,

W

−0.2
−0.2

1

0
5

−0.1

1.5

0.1

0.5

0

0

2.5
0.2

Y(t) − i(t+k) ,

0

3

0.3

0

Y(t) − π(t+k) ,Y(t) − H(t+k) ,

−2

−2.5
0

π(t) − W(t−k) ,50π(t) − H(t−k) ,50π(t) − π(t+k) ,5π(t) − i(t+k) ,

0.2
0

−0.5

−0.05

−1

0.1

π

0
−0.1
−0.2
−0.3

−0.1

−1.5

−0.15

−2

−0.2

−2.5

−0.25
0

5

0.6
0.6

0.2

0.2

−3
0

i(t) − Y(t−k) ,50i(t) − W(t−k) ,

5

0

5

0

i(t) − π(t−k) ,i(t) − H(t−k) ,5i(t) − i(t+k) ,

0

0.2
0
0

i

0.4

0.4

0.8

−1

−0.05

0.4

−0.1

−0.2

1

0.6

0.6

−2
−0.15

−0.4

−0.2

0.2

−3

0

−0.6
0

5

0.4

0.2

0

5

0

5

50

0

5

0

5