Challenges for New Keynesian Models with Sticky Wages∗

Challenges for New Keynesian Models with
Sticky Wages∗
Susanto Basu
Christopher L. House
Boston College and NBER
University of Michigan
and NBER
April 21, 2015
Abstract
Modern monetary business-cycle models lean heavily on price and wage rigidity. While
there is substantial evidence that prices do not adjust frequently, there is much less evidence on whether wage rigidity is an important feature of labor markets in the real world.
While real average hourly earnings are not particularly cyclical, systematic changes in
the composition of employed workers and implicit contracts within employment arrangements make it difficult to draw strong conclusions about the importance of wage rigidity.
We augment a workhorse monetary DSGE model by allowing for endogenous changes
in the composition of employees and also by explicitly allowing for a difference between
allocative wages and remitted wages. We then review the available evidence on the response of wages to a prototypical nominal aggregate disturbance, namely a monetary
policy shock. The data suggest several broad conclusions: (i) effective compensation for
newly hired workers is much more cyclical than average hourly earnings; (ii) both the
wages for newly hired workers and the "user cost" of labor respond strongly to identified
monetary policy innovations; (iii) composition bias plays a modest but noticeable role in
cyclical compensation patterns. Matching these patterns poses problems for a standard
DSGE model and suggests that price rigidity likely plays a substantially more important
role than wage rigidity in governing economic fluctuations.
JEL Codes: E24, E3, E31, E32
Keywords: User Cost of Labor, Composition Bias, Sticky Wages
PRELIMINARY AND INCOMPLETE.
PLEASE DO NOT CITE WITHOUT AUTHORS’ PERMISSION.
∗
Basu: [email protected]; House: [email protected].
1
1
Introduction
Since at least Hume (1752), sluggish adjustment of wages and prices has been thought to
be central to understanding the monetary transmission mechanism. This is certainly true
in modern New Keynesian models, of either the textbook variety or in medium-scale models
that attempt to match the data on a number of dimensions.1 Loosely speaking, models with
nominal rigidities reproduce many of the patterns featured in partial-equilibrium settings,
and can make demand-determined output fluctuations consistent with general equilibrium
and with the basic facts about business cycles and reactions to changes in monetary policy.
Beyond monetary non-neutrality, it is now understood that models with nominal rigidities also
behave differently in response to real shocks. For example, models with nominal rigidities can
create business-cycle comovements in response to intertemporal shocks (such as news about
future technology, uncertainty, or financial frictions that change expected capital returns),
when the flexible-price versions of these models would not display such comovements.2
It is thus important to understand the extent and importance of nominal price and wage
rigidities. While we discuss evidence on both wage and price rigidities in this survey, we
concentrate substantially more heavily on wage rigidity. We do this for several reasons. First,
Christiano, Eichenbaum and Evans (2005; CEE henceforth) and many successors found that
wage rigidity is quantitatively at least as important as price rigidity for explaining the effects of
monetary shocks and for explaining cyclical fluctuations more generally. Second, attempts to
decompose the cyclical behavior of the “labor wedge”–the gap between the marginal product
of labor and the marginal rate of substitution between consumption and leisure–typically find
that sluggish wage adjustment accounts for a large fraction of the observed cyclical behavior
of the total wedge.3 That is, in the language of markups, the wage markup appears much
more cyclical than the price markup. Third, wage rigidity is harder to measure and even to
define, which makes it difficult to understand the phenomenon despite its importance. Fourth,
a large number of recent papers have investigated price rigidity, with the modern literature
inspired by the work of Bils and Klenow (2004). Important empirical and theoretical papers
have discussed issues of interpretation: for example, should the frequency of price changes be
measured inclusive or exclusive of sales? By contrast, there are fewer studies of wage rigidity,
1
For a classic textbook treatment, see Woodford (2003). The canonical medium-scale models are due to
Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2007).
2
See, for example, Basu and Bundick (2012). The basic issues were pointed out by Barro and King (1984).
3
See, for example, Gali, Gertler and Lopez-Salido (2007).
2
and the ones that exist often do not relate their results to macroeconomic models.
While there is a tendency to discuss price and wage rigidity as independent phenomena,
this is incorrect at the macroeconomic level. In some cases, one might be able to take the
microeconomic rates of wage and price rigidity–for example, exogenous Poisson hazard rates
for adjusting wages or prices–as independent parameters. But as macroeconomic models make
clear, the inertia of the aggregate price level–the extent of macroeconomic price rigidity–
depends very much on the rigidity of wages. Since most models assume that target prices are
set as a constant markup on nominal marginal costs, the inertia of the price level depends on
sluggish adjustment of marginal cost. Wages are the largest component of the marginal cost
of producing real value added, and thus wage stickiness reinforces price stickiness. Indeed,
in most medium-scale models, wage stickiness is essential for obtaining price level inertia
and thus, for example, persistent real effects of nominal shocks. (Similarly, wage setting, for
example by monopoly unions, will also be influenced by expectations of future price inflation.)
In another chapter in this Handbook, Taylor (2015) discusses further implications of models
with staggered wage and price setting.
Our main purpose in this survey is to discuss several definitions and measures of wage
stickiness and cyclical wage adjustment and then ask what they imply for wage and price
rigidity in a prototypical medium-scale macroeconomic model. Although this survey concentrates on data, we believe that it is difficult to assess the implications of data without reference
to theory. Thus, we construct a medium-scale DSGE model based on CEE (2005), but with
more extensive modeling of different concepts of wages. The model distinguishes between four
concepts that we discuss below: average earnings, average earnings adjusted for composition,
wages of new hires, and the user cost of labor. These behave in different ways in response
to the monetary shock we study, so we can use the model to predict the behavior of these
different concepts of wages to a monetary policy shock, which is our measure of a prototypical
nominal aggregate shock.
Our use of a model to motivate the measurement has the effect of focusing attention on
the wage and price statistics that we believe are most relevant for macroeconomics. These
are the responses of prices and wages to identified aggregate shocks–that is, conditional
correlations–rather than average business-cycle correlations or the average frequency of wage
or price change. When there are both idiosyncratic and aggregate shocks, micro wages and
prices may change frequently for reasons unrelated to aggregate fluctuations, but they may
change only slowly in response to aggregate shocks. To concentrate attention on the statistics
3
that matter most for macroeconomics, we focus on the responses of nominal wages and prices
to a monetary shock, which is the archetypal nominal aggregate shock.
To preview our findings, we argue that the recent research provides suggestive evidence
that the conceptually correct measure of the allocative wage is strongly procyclical in U.S.
microdata. This finding contrasts sharply with typical estimates in the literature, which
claim that the real wage is roughly acyclical. We discuss the implications of the new facts
about wages for standard models based on nominal rigidities, and find that these models
struggle to explain the empirical facts regarding the effects of monetary policy shocks. We
show that standard DSGE models can be augmented with realistic features to reproduce
many of the wage patterns found in the recent literature but that these features typically
pose serious problems for the ability of DSGE models of the monetary business cycle to
match estimated reactions of other variables to monetary shocks. We argue that additional
evidence on measured adjustment for allocative wages and on propagation mechanisms for
monetary models is needed to reconcile the micro data with our understanding of the monetary
transmission mechanism. The search for this evidence should be a high priority for future
research.
This survey is structured as follows. We first take the workhorse model of CEE (2005),
and extend it along several dimensions. We use the model to draw out implications for several
different concepts of wages, and to some extent prices.
The model enables us to define
several different concepts of wages in a precise manner. Interestingly, the model produces
output corresponding to each wage concept, even though only one is perceived by workers and
firms as the allocative wage. Thus, we are able to use the model to predict the behavior of
both allocative and non-allocative wages in response to a monetary policy shock. We then
relate the model implications to existing micro data studies on wages and prices, and indicate
points where further evidence is needed.
With the concepts from the model in mind, we discuss evidence on wage cyclicality, drawing mostly on research using U.S. micro data. We focus on three issues that are crucial for
interpreting the data but are generally not incorporated into macroeconomic models: composition bias in aggregate wage measures, the distinction between wages of new hires and
wages of workers in continuing employment, and the distinction between spot wages and the
expected time path of wages. We then show how the different measures of real wages respond
to a prototypical aggregate nominal shock, a monetary policy shock. Our main conclusion is
that real wages, correctly defined and measured, are quite procyclical, in contrast to average
4
hourly earnings, which are basically acyclical.
We turn next to a discussion of micro data on price rigidity. We draw on the literature
following Bils and Klenow (2004) on the frequency of price adjustment, and also on the literature that tries to fit a variety of facts about the distribution of micro prices using menu-cost
models. We note that the menu-cost literature emphasizes a different set of “real rigidities”
that help models match the persistence of output fluctuations after a monetary shock than
those that are emphasized on the literature using the Calvo (1983) pricing friction. We suggest
that for robust modeling, macroeconomists should avoid using real rigidities that work in one
class of models but not in the other. We note that the standard forms of the leading theories
for obtaining inflation persistence following a nominal shock imply that micro prices should
be changed every period, and thus are at odds with the data.
2
Defining “the wage”
Macroeconomic models are typically populated by a large number of identical worker-consumers,
who supply labor along the intensive margin in a spot market. In this setting, it is easy to
define the wage: it is the current payment at time t for an extra unit of labor supplied in the
same period.
If the world were as simple as the model, “the” wage would also be easy to
measure. Unfortunately, nearly all of the assumptions about the labor market noted above
are violated in important ways in the data, making the effort to measure wage behavior far
more complicated.
First, workers are not homogeneous.
This obvious fact would not necessarily create a
problem for measuring wage cyclicality if the hours of workers of different types increased and
decreased in synchronized fashion. Then one could define a representative worker as a worker
with human capital equal to the weighted average of the human capital of all workers in the
population, and show that the average wage we observe in the data is also the wage commanded
by the representative worker’s fixed bundle of human capital characteristics. Unfortunately,
the composition of the labor force changes over the cycle. Stockman (1983) conjectured and
Solon, Barsky and Parker (1994) showed that the hours of low-paid workers are more cyclical
than average. Hence, low-paid workers account for a larger share of labor payments in booms
than in recessions. Thus, the cyclical behavior of the aggregate (average) real wage is not the
cyclicality of the wage paid to a representative worker with fixed human capital characteristics,
which is the implicit or explicit concept in almost all business-cycle models. As we shall see,
5
correcting for this composition bias shows that the wage paid to a representative worker with
fixed characteristics is considerably more procyclical than the average wage in the data. This
is also the conclusion of the important early paper by Bils (1985).
Second, since most workers are in long-term relationships with their employers, the labor
market is not a spot market. Thus, their spot wages are not necessarily what the firm perceives
as the marginal cost of labor, which is the key concept for most macroeconomic purposes.
Barro (1977) used the idea of an implicit contract to criticize the “right to hire” model of
wage stickiness, where workers propose a fixed, possibly nominal, wage, and firms choose
employment (or hours) along their labor demand curves. He showed that other contracts
would increase the payoff to both parties in the bargain, and suggested that an efficient
contract would equate workers’ marginal rates of subsitution between consumption and leisure
to firms’ marginal products of labor in every period, with the total compensation for labor
paid out to workers in “installment payments” over the lifetime of the worker-firm association.
An implicit wage contract between optimizing parties without commitment requires adjustment costs, since otherwise the party that is “ahead” in the installment payments would
dissolve the match. The most popular current model of labor adjustment costs is based on
search. Hall (2005) addressed Barro’s (1977) critique of allocative wage stickiness by showing that allocative wage could be history-dependent and hence sticky within the MortensenPissarides model of search in the labor market, as long as the preset wage remains within
the Nash bargaining set. Hall and Milgrom (2008) showed that some wage stickiness could
emerge from alternative-offer bargaining. Pissarides (2009) and Gertler and Trigari (2008)
showed that in the search setting, the key allocative wage is that of new hires. Haefke, Sonntag and van Rens (2013) provide evidence that the wage of new hires is significantly more
procyclical than that of job “stayers.”
Relative to the literature on composition bias, the main contribution of the search-based
papers is to concentrate attention on a subset of wages, namely the wages of new hires. Thus,
for example, Gertler and Trigari (2008) argue that the key statistic is whether new hires
receive the same wages as workers currently employed by the firm they are joining, or whether
they can be hired at different wages that better reflect current economic conditions.
Once it is established that newly-hired workers expect to stay with their current employer
for a significant length of time, it is intuitive that their expected labor compensation consists
of the expected present value of the wages they will receive over the length of the association.
In this case, what matters is actually not even the cyclicality of the spot wage of new hires
6
per se, but the cyclicality of the expected present value of wage payments to new hires.
Thus, one needs to know how the annuity value of the expected present value of wages,
which one might conceptually regard as the “permanent wage,” changes in response to business
cycle shocks. By analogy to the permanent income hypothesis, the behavior of the permanent
wage, not the current wage (even of new hires) is what matters to an optimizing worker or
firm. Much of the search literature implicitly ties the behavior of the permanent wage to that
of the wage for new hires, but to the extent the two differ, the permanent wage matters more.
If workers and firms are in long-term associations and the permanent wage is the correct
measure of the cost of labor input, then it is possible for the observed average wage to appear
insensitive to cyclical fluctuations (“sticky”) even if the correct allocative wage is flexible.
This conclusion was anticipated by Barro (1977, p. 316), who wrote, “In fact, the principal
contribution of the contracting approach to short-run macro-analysis may turn out to be its
implication that some frequently discussed aspects of labor markets are a facade with respect
to employment fluctuations. In this category one can list sticky wages ... .”
3
The Benchmark Model
We begin by extending a standard business cycle model to allow for several real-world features
of wage setting. Our benchmark model is a standard New Keynesian DSGE system built on
the basic framework analyzed in Christiano et al. [2005]. We build on the baseline model by
allowing for (i) endogenous variation in the composition of the workforce and (ii) differences
between the allocative wage and the measured remitted payments to workers. We will spend
more time describing our treatment of labor supply and wage setting and the mapping between
the model variables and data because these are the non-standard features of the model. Many
of the other mechanisms in the model are now common in DSGE literature and the quantitative
New Keynesian literature and so we present them with relatively less detailed discussion.
3.1
Households
Consumers get utility from consumption and real money balances and get disutility from
working. Let  be consumption of a nondurable good, let  be labor supplied at date  and
7
let   be real money balances held at date . Households act to maximize

"∞
X
=0

½
¶¾#
µ
+1
−1



+

+Λ
[+ − +−1 ]  − 
+
−1
+1
+
(1)
subject to the nominal budget constraint
 ( +  +  ( )) +  +  =   +    + −1 (1 + −1 ) + Π + −1
(2)
and the capital accumulation equation
+1 =  (1 − ) +  (  −1 )
(3)
Here  is the nominal prices of the durable and the nondurable,  is the nominal wage rate
and  is the nominal rental price of capital services, which is the product   . Π denotes
profits returned to the household through dividends.  is a lump-sum nominal transfer. 
is the supply of nominal money balances held at time ,  is nominal savings and  is the
nominal interest rate.  is the intertemporal elasticity of substitution,  is the Frisch labor
supply elasticity, Λ () expresses the households valuation for real money balances and  ≥ 0
is a habit persistence term (  0 implies habit persistence in utility). The function  () is an
investment adjustment cost function and  ( ) gives the resource cost of additional utilization
per unit of physical capital. Following Christiano et al. (2005), we assume that
¶¸
∙
µ

 (  −1 ) = 1 − 

−1
with  (1) = 1  0 (1) = 0 and  00 (1) = .
Households choose    ,  ,  and +1 to maximize (1) subject to (2) and (3). The
determination of labor supply  is the key object of interest for this paper and we discuss
this in greater detail below.
3.2
Firms and Price Setting
Following much of the New Keynesian literature we model the production and pricing component of the model as a two-stage process. Final goods are produced from a combination of
intermediate goods. Final goods producers are competitive and have flexible prices. Interme8
diate goods firms are monopolistically competitive and change prices infrequently according
to the Calvo mechanism.
3.2.1
Final Goods Producers
Final goods are produced from intermediates. Specifically, final output is given by the standard
Dixit-Stiglitz aggregator
 =
∙Z
1
−1

 ()
0

¸ −1


(4)
where   1. Final goods producers are perfectly competitive and take the final goods prices
 and intermediate goods prices  () as given. It is straightforward to show that demand
for each intermediate good has the standard isoelastic form
 () = 
µ
 ()

¶−

(5)
Competition among final goods producers ensures that the final goods nominal prices are direct combinations of the nominal prices of the intermediate goods used in production. Specifically,
 =
∙Z
1
1−
 ()
0
3.2.2
1
¸ 1−


(6)
Intermediate Goods Producers
Intermediate goods are produced by monopolistically competitive firms who take the demand
curve (5) as given when they set their prices. Each intermediate goods firm has a constant
returns to scale production function
 () =   ()  ()1−
where  (),  () and  () denote capital, labor and output for intermediate producer  at
time .  is the quantity of capital services inclusive of utilization, and thus is not the firmlevel equivalent of  , which is the capital stock. Similarly,  is number of standardized units
of labor employed by the firm. That is, it is an index of the total labor input the firm derives
from the potentially heterogenous workers it employs, expressed in a common numeraire, such
as the number of high-school-educated workers. This concept of labor, which is relevant for
productivity, should be distinguished from  , which is akin to total employment or the total
9
number of hours worked by all persons, and is the object relevant for utility. Here  is an
aggregate productivity shock common to all firms. While the intermediate goods firms have
some monopoly power in their output markets, they are competitive in the input markets,
and take the nominal input prices  and  as given when making their decisions. Each
period, firms choose their inputs to minimize costs. For any given level of production ¯, the
firm’s cost-minimization problem is min   +  subject to   1− ≥ ¯.
Because the production functions have constant returns to scale, and because capital and
labor can flow freely across firms, firms choose the same capital-to-labor ratios
 ()
 

=
 ()

where we have used the market clearing conditions
R
 ()  =   and
R
 ()  =  .
The nominal marginal cost of production for the intermediate goods producers is common to
all firms (because the firms have constant returns to scale production functions). The date 
nominal marginal cost is
¶1− 1− 
µ ¶ µ
 
1
1
 =


1−

(7)
Price setting for each intermediate good producer is governed by a Calvo mechanism. Let
 be the probability that an intermediate goods producing firm cannot reset its price in a
given period. Thus, each period, 1 −  firms reset their prices as they see fit. The remaining
 firms set their prices according to the backward-looking rule  () = −1 () (1 +  −1 )
where
1 +  =

−1
is the gross nominal inflation rate.
Intermediate goods firms maximize the discounted value of profits for their shareholders
(the households) and thus discount future nominal profits in period  +  by according to the
stochastic discount factor   + (technically,  is the Lagrange multiplier associated with
the nominal constraint (2)). The optimization problem for an intermediate goods firm is to
10
choose a reset price ∗ to maximize the objective
⎡
⎡
⎞⎤⎤
⎛⎡ Y
⎤−
−1
∗
∞
³ Y−1
´
(1 +  + )
⎢X
⎢
⎥⎥
⎜ 
=0
⎦ + ⎟
( ) ⎣  + ∗
(1 + + ) − + ⎝⎣
 ⎣
⎠⎦⎦
=0

+
=0
where is it understood that
Y−1
=0
(1 +  + ) ≡ 1.
Given the reset price ∗ , and using (6), the price of the final good evolves according to
£
¤ 1
 =  ([1 +  −1 ] −1 )1− + (1 −  ) (∗ )1− 1−
Well-known methods show that the optimal reset price together with dynamic evolution of the
aggregate price level together imply that the model satisfies a hybrid New Keynesian Phillips
curve of the form
where   =
3.3
(1− )(1− )


˜ − 
˜ −1 =   
f  +  [˜
 +1 − 
˜]
is the microeconomic rate of price adjustment.4
Labor Supply and Wage Setting
The supply of labor features several mechanisms that are prominent in the empirical literature
on labor supply and the measurement of wages. As in Christiano et al. [2005], we include
nominal wage rigidity in the model. In addition to nominal wage stickiness, we also augment
the model to include two new features: (i) endogenous composition bias and (ii) a difference
between the allocative wage and the remitted wage. Both mechanisms influence the mapping
between model predictions on the one hand and empirical measures of wages and labor supply
on the other. To accommodate these mechanisms we treat the supply of labor as occuring in
two two separate stages within a period. We refer to these simply as stage 1 and stage 2.
In the first stage, the composition bias mechanism allocates workers with differential
productivity to the market. This stage results in a single nominal wage paid for units of
productivity-adjusted labor, and also produces an average wage for employed workers. We

4
We also allow for the possibility for partial indexing  () = −1 () (1 +  −1 ) with  ∈ [0 1]. In this
case, the implied Phillips curve is

˜  − ˜
−1 =   
f  +  [˜
+1 − ˜
]
11
denote the wage for productivity-adjusted labor as 1  the average wage for employed work¯ 1 and the total supply of effective (productivity-adjusted) labor as 1 where the
ers as 
superscript indicates that these variables are determined in stage 1.
In the second stage, an allocative wage is determined. The allocative wage is set according
to the Calvo mechanism taking the wage 1 as the effective marginal cost of supplying units
of effective labor. In addition to the allocative wage which governs actual employment, the
second stage also produces two different observed wages: a new-hire wage  and a wage
for all employed workers that corresponds to average hourly earnings  .
3.3.1
Composition Bias
It is well-understood that the composition of the workforce changes systematically over the
course of the business cycle. Typically, the labor force has a higher fraction of low-wage
workers in booms than in recessions, making the average wage somewhat more countercyclical
than the wage for a representative worker with fixed human capital characteristics, which is
the right concept of the wage in most macroeconomic models. To the extent that composition
fluctuates over the cycle, the changing characteristics of the workforce automatically makes
output per person appear more counter-cyclical than otherwise.
To introduce composition bias into the model, we imagine that labor varies by productivity.
Specifically, we assume that total actual hours of labor (the argument in the utility function
1) is given by
 =
Z

 ()  () 
(8)
0
Here  is an index of productivity and  is the maximum productivity of any individual in
society.  () is the measure of the population with labor productivity  and  () denotes
hours worked per person with productivity . For each type,  () ∈ [0 1]. The total
R
¯ =   () . Each type is paid a nominal wage  ().
population is 
0
Workers supply labor to labor aggregating firms who in turn sell an effective labor aggregate
at a wage 1 . The labor aggregating firms’ maximization problem is to hire different types
of labor to maximize nominal profits.
½
¾
Z
Z
1
1
 ()  −  ()  () 
max 
 ()
12
The first order condition for the choice of  () requires
1 () = 1 
That is, the individual wages are direct reflections of the worker’s productivity.
Consider an increase in  () from the perspective of the representative household. The
utility impact of this increase is
¸
∙
1

1
− +   ()  () ×  ()
where  is the shadow value of money payments to the representative household (i.e.,  is
the Lagrange multiplier on the nominal budget constraint). If the term in brackets is positive
then it is optimal to set  () = 1. If the term in brackets is negative, then it is optimal to
set  () = 0. Using 1 () = 1  we can express the critical productivity cutoff ˆ as
1

=
ˆ 
 1
(9)
For any type   
ˆ it is optimal to work full-time. Types   ˆ remain out of the labor
¯ and effective productivity adjusted labor
force. Total employment is then  = [1 − Φ (ˆ )] 
is
1
=
Z

 () 

ˆ
Equation (9) is essentially a standard labor supply condition except for two differences. First,

ˆ is endogenous and co-varies negatively with aggregate employment  . Secondly, there is
a difference between effective labor  and measured hours of employment  .
¯ 1 of employed workers is simply the ratio of total wage payments to
The average wage 
total hours of work, that is,
¯ 1 =

R
0
1 ()  ()  () 
1  1
=  


In contrast, the composition-adjusted wage in stage 1 is simply 1 . Notice that this implies
that the ratio of total hours worked to effective labor is equal to the ratio of the compositionadjusted wage to the average wage,


=

¯ Using

13
log-linear expressions for  ,  , one can
¯  and  ) satisfies
show that composition bias (the log difference between 
f
¯ −

˜ 1

¸

ˆ 1
˜
=
¯ − 1  

∙
¯ exceeds the wage for the marginal worker 
Moreover, since the average wage 
ˆ 1 , composition bias imparts a negative comovement between the average wage and aggregate hours.
In U.S. data, the cyclical variation in average real wages is negligible, but the compositioncorrected wage is clearly procyclical.
3.3.2
Allocative Wage Rigidity
In addition to the composition bias mechanism presented above, the model features both
nominal wage rigidity as in Christiano, Eichenbaum and Evans [2005] and Erceg, Henderson
and Levine [2000]. Unlike these earlier papers however, we assume that the wage rigidity
applies directly to an “allocative wage” — the relevant wage for determining employment and
work effort. In addition to this wage, the model produces a measured remitted wage which
we discuss below.
We model the workers as infrequently adjusting an allocative wage which we denote by  .
As we did in our treatment of composition bias, we assume that there is a labor aggregating
firm that assembles an aggregate of labor “types.” This aggregating firm supplies effective
labor to the productive firms at flow allocative wage  , but hires labor by type according
to the type-specific allocative wages  . The labor aggregate is given by a CES aggregate of
labor types,
2 =
∙Z
1
( )
−1

0

¸ −1


where the superscript 2 refers to the fact that this labor supply is determined in stage 2. If
the aggregating firm wants to supply labor force   its demand for type  work is given by
the isoelastic function,
2
=
2
µ


¶−

The allocative wages  for each type of labor are set by a monopolist in that type (similar
to a union). The allocative wage for units of the labor aggregate is
 =
∙Z
1
1−
( )
0
14
1
¸ 1−


˜ 1+ ≈ 
˜ 2+ (up to a first order
Note that the labor market clearing condition implies that 
approximation).
As we did with the price setters, we assume that the type-specific wages are set according
to a Calvo mechanism. The probability of adjusting a type-specific wage is 1 −  and the
probability of not adjusting is  . Wage setters who do not get the Calvo draw, instead follow
the wage indexing rule  = −1 (1 +  −1 ).
The union firms try to maximize the discounted value of wage markups  − 1 . An
extra dollar in period  +  is worth   + to the household. Thus, a monopolist who has the
option to set his wage at time  should choose a reset wage ∗ to maximize
⎧ ⎡
⎛ Y−1
⎞− ⎤⎫
⎪
⎪
∗
∞
⎨
⎬
³ Y−1
´
(1 +  + )

⎢X
⎥

∗
1
2 ⎝ 
=0
⎠
 ⎣
( ) + 
(1 +  + ) − + +
max
⎦
=0
∗ ⎪
⎪
+
⎩
⎭
=0
where again is it understood that
Y−1
=0
(1 + + ) = 1. Given the reset wage ∗  the aggregate
allocative wage evolves according to
i 1
h
1−
∗ 1− 1−
+ (1 − ) (˜
 )

 =  {(1 +  −1 ) −1 }
3.3.3
Remitted Wages
Our discussion highlights the difference between the allocative wage — the shadow wage that
governs work effort and employment — and the measured wage which governs the periodic
payments from the employer to the workers. We assume that the remitted wage is a smoothed
function of the allocative wage. Specifically, we assume that workers periodically renegotiate
their contract terms (or separate from their current jobs) and, when they do, they are given
a new remitted wage. Let   be the expected present discounted value of future nominal
allocative wages for a newly employed worker that resets the remitted wage with probability
 ∈ (0 1). That is,
  =  +  (1 − ) 
∙
"∞
#
¸
X
+1
 ++1
 +1 = 
[ (1 − )]
+


=0
Clearly   depends on the reset rate  (even though the reset rate plays no role in allocations). The measured remitted wage for new hires at date  will be a smoothed version of the
15
PDV. Specifically, we assume that the measured wage for new hires will solve
  =  
"∞
X
++1
[ (1 − )]

=0
#
That is,  is a constant wage that will transfer the same expected amount to the workers
given the reset rate  as they would receive by getting the time varying root wage  . For
purposes of comparison with the data,  is a new-hire wage.
We can also track the average outstanding wage for all workers in the model. Let  be
the average hourly earnings of all employed workers. By construction, the average outstanding
wage at time  is the average wage for all workers that did not separate together with the
new-hire wage

 = −1
(1 − ) +  2
where  =  − −1 (1 − ) denotes “new hires” interpreted as all workers who are newly
hired plus those who remain employed but receive new contract terms for their remitted wage.
3.4
Aggregate Conditions and the Steady State
The goods market clearing condition is
 =  +  +   ( ) 
We consider two separate sources of uncertainty. There are random fluctuations in total factor
productivity and also shocks to monetary policy. Monetary policy is given either by a Taylor
rule
#
"
¡
¢
˜

˜  + ˜−1 + 
˜ = 1 −   +  
4
or by a money growth process
 =


= 1 +  −1
+ 

−1

˜ − 
˜ −1 = 1 +  −1
 = 
+ 

For the Taylor rule specification,  and  give the relative reaction of the monetary authority to output and inflation and  is an interest rate smoothing parameter. Innovations
16
to productivity follow a standard autoregressive process
 = (1 −  ) +  −1 + 

We assume that the structural innovations  , 
 and  are uncorrelated.
3.4.1
Non-Stochastic Steady State
We choose parameters to ensure that in the non-stochastic steady state,  =  =  = 1. The
steady state markups are  =

−1
and  =

.
−1
We normalize the steady state productivity
cutoff to 
ˆ = 1. The steady state nominal marginal cost is  = 1 . Since there is no
inflation and no economic growth in the steady state, 1 +  = 1 +  = 1 . It is straightforward
to show that the nominal rental price is  =  +  and we must have  = 0 (1)  Steady state
capital is
=
µ


1
¶ 1−

The remaining details of the steady state are standard and are therefore omitted.
3.4.2
Baseline Calibration
Our calibration for most of the parameters is based on conventional parameter values used
in the DSGE community. In particular, we base most of our baseline parameter settings on
Christiano et al. [2005] and Del Negro et al. [2013]. We set  = 21 which implies a 5
percent markup of the allocative wage over the base wage. We set  = 6 which implies a 20
percent markup of nominal price over nominal marginal cost. The investment adjustment cost
parameter is set to  = 300. The habit weight is  = 065 and the utilization elasticity is
set as
00 (1)
0 (1)
= 001. Our baseline price rigidity parameter is set to  = 090 (quarterly) which
is close to the value estimated by Del Negro et al. [2013] (though this choice implies that
prices change substantially less frequently than observed price changes). We set  = 050
(quarterly). The one parameter that we set in an unconventional way is the Frisch elasticity
. Typical microeconomic estimates of  are quite low reflecting the fact that most people
report that they desire a fixed workweek regardless of their wage rate. In the business cycle
literature, researchers often set values between 0.5 and 1.00. We adopt the higher value of
 = 300. We do this primarily because the mechanisms we emphasize operate primarily at
the so-called “extensive margin” of labor supply which is understood to have a higher supply
17
elasticity than the intensive (hours) margin. We hold these parameters fixed in the numerical
experiments in Section 6. These baseline parameter values are summarized in Table 1.
In addition to the parameters in Table 1, the model also requires values for the parameters
that govern composition bias and the remitted wage. There are three key parameters that
govern these mechanisms: the renegotiation hazard , the steady state ratio of effective labor to
total hours worked

,

and the composition elasticity . For our baseline setting we assume that
neither of these mechanisms is operative and thus we set  = 100 (so the remitted wage is equal
to the allocative wage),  = 100 (so there is no difference in average productivity per hour)
and  = 0 (so hours can be varied without changing the productivity of the marginal worker).
This baseline specification is thus essentially equivalent to a standard sticky-price/stickywage DSGE model. In Section 6, we consider alternative specifications that illustrate how
composition bias and infrequent wage resetting influence the model outcomes. We defer
discussion of these alternate parameter specifications to later.
4
Measuring the wage
Note that in the model above there are several objects that map to measured wages, but only
one, which we have called   , is allocative. For some intuition as to why this is the object
that matters to the firm, consider the following thought experiment. For simplicity, assume
that the probability that an employed worker separates from his employer in a given period
is constant at a positive value s, and is independent of the length of their prior association.
From a firm’s perspective, the opportunity cost of hiring a worker at time t is not hiring him
at t+1 with probability (1 − ). (From t+2 on, the two decisions have identical consequences,
so one needs to compare only the first two periods.) The cost of hiring the worker is thus
the difference between the expected wage payments to a newly-hired worker at time t and a
similar worker hired at t+1. This is exactly the concept that Kudlyak (2014) calls the “user
cost of labor,” which we henceforth abbreviate as UCL.5 Kudlyak finds that unlike the average
wage, the UCL is highly procyclical, even more so than the wage of new hires.
Kudlyak’s finding was foreshadowed in two important early papers by Beaudry and DiNardo (1991, 1995), who found that the permanent wage might be significantly more procyclical than the wage at a point in time. They found that workers hired in recessions received
5
The term is used as an analogy to the “user cost of capital” under adjustment costs, in which case the
decision to add an extra unit of capital is a dynamic decision with long-term consequences.
18
persistently lower wages, even after the economy recovered. Thus, while the spot wage was
cyclical, the present value of the wage fluctuated even more. Beaudry and DiNardo interpreted
their finding as support for the hypothesis of implicit contracts with costly worker mobility.
In a sense, Beaudry and DiNardo approached the problem from the workers’ side, asking why
a worker would take a job in a recession, since the effective (permanent) wage that he or she
receives is so low. We (and Kudlyak) approach the same facts from the firms’ side, asking why
firms do not hire more in recessions, since the effective cost of hiring a worker in a downturn
appears to be low.
Since Kudlyak (2014) attempts to measure exactly the object that is the allocative wage in
our model, we compare the predictions of the model to her estimates. The UCL is a difficult
object to measure, since one needs to follow individual workers over time. Unfortunately, the
two panel data sets for the US, the NLSY and the PSID, have small samples and limited sample
periods. Furthermore, the data are annual, which is not ideal for business-cycle analysis.
Ideally, one would also use panel data to correct for composition bias. Thus, Bils (1985)
uses the NLSY and Solon, Barsky and Parker (1994) use the PSID. The recent papers by
Haefke, Sonntag and van Rens (2013) and Elsby, Shin and Solon (2014) instead use CPS
data. The drawback to using the CPS is that since it is not a true panel, one cannot remove
the effects of unobserved individual effects on the wage by differencing wages over time. The
benefit is that the CPS provides a large and nationally-representative sample, and continuous
data starting in 1979 and going through the Great Recession period and its aftermath.
In parts of our empirical work, we use Kudlyak’s (2014) UCL series, generated using data
and replication files generously shared by Maryanna Kudlyak. Her series uses NLSY79 data,
from 1978-2004. A regression cleans wages of individual- and job-specific characteristics, and
realized future wages are used as expectations. Because jobs last longer than one year, the
User Cost of Labor is the difference between the discounted present value of wages paid to
worker hired in the current period minus that of a worker hired in the next period. Expected
future wages are truncated after 7 years, allowing calculation of the series through 1997. We
tried alternate versions of the series that truncate expectations at 5 years and 9 years and go
through, respectively, 1999 and 1995. These were not noticeably different.
Kudlyak’s original results are annual. We use Chow-Lin interpolation to convert the UCL
from an annual to quarterly frequency. Real GDP, nonfarm payroll employment, average real
wages and the unemployment rate are used as the high-frequency variables. A regression of
the User Cost on that same set of variables projects the User Cost series past 1997 and 1999
19
and before 1978. An alternate Boot-Feibes-Lisman interpolation produces similar results.
Note that all of Kudlyak’s wage series use standard worker characteristics as controls, plus
worker fixed effects to allow for unobserved heterogeneity in worker productivity. As a result,
they also control for composition bias in wages.
Kudlyak also calculates a wage for “new hires,” defined as wages for individuals with less
than one year of job tenure. We plot these data in Figure 1. Even the new-hires wage series,
which does not take into account the dynamics in the UCL, is significantly more procyclical
than average hourly earnings, which are plotted in the figure for comparison. Kudlyak’s results
run from 1978-2004; the remaining sample is filled in by extrapolation using the variables
discussed above. Haefke, Sonntag and van Rens (2013) also estimate a wage series for new
hires; their definition of “new hires” is any worker who was not employed in at least one of
the three months before their wage was observed. Like Kudlyak, they estimate that new hire
wages are significantly more procyclical than average wages.
Figure 2 shows the UCL, based on 7-year expected future wages. The UCL series is much
more procyclical than average real wages, as documented by Kudlyak (2014). The 7-year
UCL declines 29 percent over the 1981-82 recession, and takes four years to return to its prerecession level. After the business cycle peak in 1990, the series gradually declines 14 percent
over four years, and then rebounds in 1994.
The explanatory variables in the projection regression are relatively stable over this period.
Over the 2000-2014 sample, the 7-year series falls 21 percent after the 2001 peak, recovers only
6 percent and then falls 32 percent over the Great Recession, troughing in mid-2011. As of
mid-2014, only 18 percentage points had been recovered. At the end of the 7-year series
(2014:Q2) the user cost is roughly equal to the business cycle peak in 1990. By comparison,
average real wages are up around 30 percent since 1990.
It is not surprising that the two
series differ over this deep and long-lasting recession, since average wages are subject to the
countercyclical composition bias discussed above.
5
Responses of Measured Wage Series to Monetary Shocks
Almost all of the literature on wage cyclicality examines the response of real wages to a
cyclical indicator, typically the unemployment rate. However, the business-cycle literature has
focused in the past two decades on the importance of matching the same impulse responses
between the model and the data. Since we are interested in nominal rigidities, the natural
20
impulse responses to consider are those to a monetary policy shock. The modern literature
on estimating the effects of monetary policy shocks using VARs began with Bernanke and
Blinder (1992). We follow closely CEE (2005), since we wish to follow them in matching the
data impulse responses to those from a model.
The impulse response functions of the different User Cost of Labor measures to a monetary
policy shock follows Christiano, Eichenbaum and Evans (2005). They are generated utilizing
a standard Cholesky decomposition, but in such a way that the monetary policy shock is
unaffected by the different User Cost of Labor series. For each of the three series (UC7, in
which expectations of wages are truncated at 7 years, UC5 and NH which is wages of new
hires) a different VAR is estimated with that series ordered last after the nine variables from
CEE (2005). This means that the innovations of the other variables impact the user cost of
labor contemporaneously, but not vice-versa. The reduced form VAR coefficients of lags of
the user cost of labor are restricted to be zero in the equations for all other variables. Thus,
the current or lagged values of the user cost of labor do not enter enter the equations of the
main block that determines the monetary policy shock, and the impulse response functions
of the different measures are to an identical shock. Since the equations in the VAR do not
have the same set of explanatory variables, the system is estimated with seemingly unrelated
regression to allow for correlated errors.
If the user cost of labor is ordered last, it responds contemporaneously to a monetary
policy shock. This is inconsistent with CEE’s conceptual, ordering as wages come before the
federal funds rate and respond only with a one quarter lag. Thus, we imposed an additional
restriction that the impact effect of the monetary policy shock on UC7, UC5 and NH is
constrained to be zero.
Finally, note that we estimated three separate VAR systems. Each consists of the basic
CEE model, augmented with one new variable at a time. However, given our assumptions, the
new variable does not affect the responses of the other variables in any way. Thus, we report
the constant impulse responses to all other variables to a monetary policy shock, together
with the responses of the three new wage measures.
The CEE variables are ordered as output (OUTNFB), consumption (PCECC96), price
level (IPDNBS), investment (GPDIC96), real compensation per hour (COMPRNFB), labor
productivity (OPHNFB), federal funds rate (FEDFUNDS), corporate profits (CP), and money
growth (M2SL). Consumption, investment and corporate profits come from the NIPA tables,
as in CEE. Unlike CEE, output, price level, compensation and labor productivity are only
21
for the Nonfarm Business sector. We believe that non-farm business is a data concept that
better suits our model of price-setting firms, since the excluded industries (such as utilities
and government production) can hardly be said to optimize with respect to the prices they
set. Corporate profits are deflated by the price level. All variables are in log levels, except
the federal funds rate which is in levels and M2 which is in log differences. All variables were
downloaded from the Federal Reserve Bank of St. Louis FRED web database in March 2015.
The sample period is 1965-1995, which is the sample period of CEE (2005), but our data do
not match theirs exactly, for two reasons. First, of course, the data have been revised by the
statistical agencies. Second, as noted above, we use NFB output and prices rather than GDP
and the GDP deflator.
Figure 3 shows the IRFs of average hourly earnings, the UC and NH wage measures to
a monetary policy shock.
Note that the actual user cost of labor series begins in 1978, so
a similar interpolation and backwards projection is used for the 1965-1977 period as is used
for the later sample that lacks actual data. The UCL responses behave similarly between
regardless of the number of years of truncation, and have a peak response of around one
percent, two years after the shock, roughly on par with the response of investment. The
response of new hire wages is similar to the user cost, but slightly muted. This is in contrast
to wages of all workers, which is an order of magnitude smaller (roughly one basis point) and
statistically significantly different from zero for only two quarters.
We found similar results when estimating IRFs with a sample of 1979Q4-2007Q4. The
beginning of the sample corresponds to the beginning of Paul Volcker’s chairmanship of the
Federal Reserve, but also has the benefit of excluding any backward projection of the user cost
series. In general, the IRFs from the main block of variables are more muted but take longer
to return to trend. Despite this difference, the user cost series still have a peak response near
one percent. The new hire wages oscillate rapidly, but reach a similar peak response after a
similar lag. We conclude that the results are qualitatively unchanged over this shorter sample
period that overlaps significantly with Kudlyak’s data sample and is also the period when the
“modern” era of U.S. monetary policy may be said to have begun.
6
Illustrating the Mechanisms
Here we illustrate how these mechanisms change the quantitative behavior of the DSGE model
presented in Section 3. We begin by illustrating how infrequent renegotiation of the allocative
22
wage is reflected in the measured wages discussed above.
The User Cost of Labor. If the renegotiation rate  is less than 1 then there will be
three distinct wages: wages for the newly hired (  ), average hourly earnings for all workers
(  ) and the user cost of labor (the allocative wage  ). To illustrate this in the context
of the model, we examine the model’s impulse response function following a monetary shock.
For purposes of this experiment, we consider a trend stationary shock to the money supply
 .
Figure 4 presents the reaction of the model for three different parametric specifications:
the baseline specification with  = 1; a “medium” renegotiation probability with  = 0125
and a specification with a long renegotiation horizon with  = 0083. These specifications
correspond to a remitted wage that is reset continually, once every 2 years and once every 3
years respectively. See Table 2 for details of the model parameterization. For convenience,
Figure 4 also includes the empirical impulse response functions for the corresponding measures
discussed above.
Barattieri, Basu and Gottschalk (2014) present data on the frequency of micro wage
changes, which corresponds to the change in the probability of renegotiation in the model.
Their results can be aggregated across different categories of labor (e.g., hourly paid workers
and salaried workers) and used to calibrate the important parameter .
For the baseline specification labeled UC(0) in the figure, the three wages are identical.
All workers are paid the same wage regardless of when they were hired. This specification is
the one used in conventional DSGE models. Clearly, based on the empirical impulse response
functions it will be impossible for any model with a single wage concept to simultaneously
reproduce the three distinct wage paths reported in the figure. For the medium specification
(labeled UC(2)) and the high specification (labeled UC(3)) there is clearly an improvement
over the model without periodic resetting of the allocative wage. In particular, the average
hourly earnings time path reacts only slightly to the monetary impulse. Most remitted wages
are not reset in the immediate aftermath of the shock and thus measured average hourly
earnings are essentially unchanged. Notice also that the user cost (the allocative wage) is
identical for all three specifications.
Cyclical Changes in Labor Composition Figure 5 presents impulse response functions
for the model in which there is no distinction between the remitted wage and the allocative
wage but instead there are endogenous variations in the composition of the labor force. Again,
23
we consider three parameterizations: the baseline specification with no compositional variation (specification 1); a specification with an intermediate degree of compositional changes
( = 3 and  = 010, labeled specification 2 in the figure) and one with a high degree of
compositional changes ( = 5 and  = 020, specification 3). Table 2 reports the parameter
choices for these specifications. All other parameters are set to their baseline values.
Unlike the experiments with the user cost of labor, changes in the degree of labor market
composition have allocative effects. As the composition effects become stronger, the composition corrected wage is revealed to be sharply more procyclical than the average wage which
becomes less and less responsive to the shock. Notice (perhaps paradoxically) that total hours
of employment responds more strongly the greater is the compositional effect. This is because
output (and thus labor) is demand determined. As a consequence, for a given contraction in
total hours, the decline in production is smaller the greater is the compositional effect. Thus
firms lay off even more workers in the composition bias setting. Labor market productivity
measured simply as output per worker rises dramatically in the composition bias case. This
would seem to be a strong prediction of models with composition bias and is clearly at odds
with the data which exhibits declining productivity in the face of a monetary shock.
Comparisons with the Data [To be added]
7
Conclusion
[To be added]
24
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TABLE 1: BASELINE PARAMETER VALUES
Fixed Parameters
Baseline Value
Discount Factor, quarterly (  )
0.99
Frisch Labor Supply Elasticity (  )
3.00
Intertemporal Elasticity of Substitution (  )
1.00
Habit Weight ( h )
0.65
Depreciation Rate, quarterly (  )
0.02
Price Elasticity of Demand (  )
21.00
Wage Elasticity of Demand (  )
6.00
Price Rigidity Parameter (  p )
0.95
Investment Adjustment Cost (  )
3.00
Capital Elasticity (  )
0.36
Utilization Elasticity ( b ''/ b ' )
0.01
TABLE 2: ALTERNATE MODEL SPECIFICATIONS
Baseline
Value
Med. User
Cost
High User
Cost
Med.
Composition
High
Composition
L-N Ratio ( L / N )
1.00
1.00
1.00
3.00
5.00
Composition Elasticity (  )
0.00
0.00
0.00
0.10
0.20
Allocative Wage Rigidity ( w )
0.50
0.50
0.50
0.50
0.50
Renegotiation Rate ( s )
1.00
0.125
0.083
1.00
1.00
Parameters
4.6
4.4
4.2
New Hire Wage
4
Average Hourly Earnings
3.8
3.6
3.4
3.2
1965
1969
1973
1977
1981
1985
1989
1993
1997
2001
2005
2009
2013
FIGURE 1: AVERAGE HOURLY EARNINGS (AHE) AND WAGES FOR NEW HIRES
4.6
4.4
Projected UCL (authors' calculations)
4.2
4
3.8
NLSY77 Original Sample (Kudlyak)
3.6
3.4
3.2
1965
1969
1973
1977
1981
1985
1989
1993
1997
2001
FIGURE 2: THE “USER COST OF LABOR”
2005
2009
2013
User Cost / Allocative Wage
Average Hourly Earnings
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
0
5
10
15
20
25
30
35
25
30
35
-1.5
0
5
10
15
20
New Hire Wage
1
0.5
0
-0.5
-1
-1.5
0
5
10
15
20
FIGURE 3: WAGE RESPONSES TO MONETARY SHOCKS
25
30
35
User Cost / Allocative Wage
Average Hourly Earnings
0.4
0.4
UC(0)
UC(2)
UC(3)
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
-1.2
-1.2
0
5
10
15
20
25
30
35
25
30
35
0
5
10
15
20
25
30
35
New Hire Wage
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
0
5
10
15
20
FIGURE 4: MODEL REACTION FOR DIFFERENT USER COST SPECIFICATIONS
Notes: The figure reports the impulse response function for the DSGE model following a contractionary
monetary shock. The heavy dark lines are the empirical impulse response functions estimated in Section 3.
The parameter values used in the simulation are reported in Tables 1 and 2.
Average Wage
Composition-Corrected Wage
0.2
0.2
specification 1
specification 2
specification 3
0.1
0
0
-0.2
-0.1
-0.2
-0.4
-0.3
-0.6
-0.4
-0.8
-0.5
-1
-0.6
-0.7
0
5
10
15
20
25
30
-1.2
0
5
Employment (Hours)
10
15
20
25
30
25
30
Labor Productivity
0.8
0.2
0.7
0
0.6
-0.2
0.5
-0.4
0.4
0.3
-0.6
0.2
-0.8
0.1
-1
0
-1.2
-1.4
-0.1
0
5
10
15
20
25
30
-0.2
0
5
10
15
20
FIGURE 5: MODEL REACTION FOR DIFFERENT COMPOSITION-BIAS SPECIFICATIONS
Notes: The figure reports the impulse response function for the DSGE model following a contractionary
monetary shock. The parameter values used in the simulation are reported in Tables 1 and 2.