Performance Pay, E¢ ciency Wages andUnemployment Fluctuations 1
Norikazu Tawara2
January 2009
Abstract
This paper argues that introducing worker shirking problems into otherwise standard search models helps generate larger ‡uctuations of employment
and smaller ‡uctuations of wages, than previously thought in the e¢ ciency
wage literature. Unlike in the standard e¢ ciency wage models, I incorporate
performance way into the model, in which compensation consists of two parts:
base wage and performance pay. Two di¤erent wage settings are examined.
In setup I, performance pay is assumed zero and the base wage level is chosen
unilaterally by the employer. This case is similar to the standard shirking
model of e¢ ciency wages. In setup II, both the base wage level, which must
be greater than some given level and the non-negative bonus pay level arechosen by the employer. I …nd that the size of workers’rent is constant under
setup I, but can be countercyclical under setup II, suggesting the promisingrole of performance pay in delivering larger ‡uctuations of unemployment.
Key Words: The business cycle; Employment ‡uctuations; Shirking;Search and matching; workers’match rent
JEL Classi…cations: E24; E32; J41; J63
1

All errors are my own. Financial support from the University of Chicago and the John
Olin Foundation is gratefully acknowledged. The updated version of the paper is posted
at http://nori.tawara.googlepages.com.
2
Department of Economics, Kanto-gakuen University, Japan.
EMail:
nori.tawara@gmail.com

1

1

Introduction

This paper argues that introducing worker e¤ort choices into the MortensenPissarides (MP) search-matching model (Mortensen and Pissarides 1994; Pissarides 1985) makes the model-generated unemployment rate more volatile
than previously thought in the literature. Some previous work has demonstrated that introducing shirking models of e¢ ciency wages (Shapiro andStiglitz 1984) into aggregate business cycle models does not help generating"rigid" wages (Gomme 1999, e.g.). On the other, some authors have argued
that introducing mechanisms of making wages "rigid" into the MP model
may help to generate large ‡uctuations of unemployment and vacancies as
observed in data (Shimer 2004, 2005; Hall 2005). These two sub-literature
seem to imply that introducing shirking models of e¢ ciency wages into the
MP model may not help in delivering large ‡uctuations of vacancies andunemployment as observed in data.
I introduce the performance pay contingent on the worker e¤ort level into
the model. Upon meeting, the …rm chooses the non-negative performance
pay and the base salary level, which must be greater than some exogenously
given lower bound. As the labor market tightness increases, or the unemployment rate decreases, the performance pay needs to be increased. The
workers’gain from employment is decreasing in the labor market tightness,and is increasing in the performance pay level. I show that the workers’gainfrom employment is decreasing in the labor market tightness, i.e., countercyclical in an economy with cyclical shocks to labor productivity. This iscontrasted to that the worker’s rent is acyclic, i.e., independent of the labormarket tightness when performance pay is prohibited as in the standard e¢ ciency wage model. This …nding suggests that introducing performance pay
into the MP model appears promising to deliver larger ‡uctuations of theequilibrium unemployment rate, in light of the …nding by Brugemann andMoscarini (2007). Brugemman and Moscarini argues that countercyclicalworkers’rent is important for the MP model to deliver larger ‡uctuations of2

unemployment.
Literature
I am aware of two papers that combine the MP model with the shirking
model of e¢ ciency wages a la Shapiro and Stiglitz. Rocheteau (2001) examines steady state of such model, in which wages are determined by maximizing
the Nash product so that the no-shirking conditions are satis…ed. Costain
and Jansen (2006) examines a search matching model with aggregate shocks
to productivity in which wage determination is similar to Rocheteau (2001).
In both papers, the no shirking conditions, which say that employees’surplus
has to be su¢ ciently large to motivate employees to exert e¤orts, are binding at some aggregate states and are slack at other aggregate states. In thispaper, I examine a variant of search matching model with aggregate shocks
to labor productivity in which the no-shirking conditions hold with equality
at all aggregate states by assuming that …rms unilaterally determine wages
by taking into account the no-shirking conditions.
A number of papers have assessed matching models of frictional labormarkets. Hall (2003, 2005), Shimer (2005) and others argue that a matchingmodels of Diamond, Mortensen and Pissarides, cannot deliver ‡uctuationsof vacancies and unemployment of an empirically plausible magnitude when
one feeds productivity shocks of the empirical size into the model. Hagedorn
and Manovskii (2008) criticize the calibration practices undertaken by theseauthors, by arguing that important parameters such as workers’bargaining
power and job-seekers leisure value, are not calibrated properly by matchingmodels’predictions to data. My paper is related to their work, in the sensethat I o¤er an alternative way to put disciplines to calibration exercises byusing the binding no-shirking conditions.
The idea that introducing performance pay into the MP model makes themodel-generated unemployment more volatile may be related to Alexopoulos(2004), who develops the real business cycle model with worker shirking
3

problems, in which a shirking employee is punished not by dismissal but by
nonpayment of the deferred wage. Unlike Alexopoulos, both dismissal andnonpayment of performance pay are punishment for shirking workers in mymodel.
The structure of the paper goes as follows. The next section describes anenvironment of the matching model with work incentive problems. Section
3 examines steady state equilibrium. Section 4 examines the model withaggregate shocks to worker productivity. Section 5 concludes.

2

Model Setup

Time is discrete: t 2 f1; 2; 3;
t 1, and its end date t.

g. The beginning of period t is called date

Agents and Preferences. This economy is populated by a continuum ofrisk-neutral workers and …rms, both of whom discount future payo¤s at thecommon discount factor 2 [0; 1). The preference of each worker is given by
E

1
X

t

[ct + (1et )z]

t=1

where ct is consumption, et 2 f0; 1g the fraction of time devoted to work,
which is also called the employee’s e¤ort level. Consumption c is the wage
when the worker is employed and it is the unemployment bene…t b 0 whenunemployed. The preference of each entrepreneur (or …rm) is given by:
E

1
X

t

[ t

Iv k]

t=1

where t is the pro…t, and Iv is the indicator function for the event that the…rm is vacant. Parameter k > 0 is a ‡ow cost of recruiting.
Production Technology.

A pair of a worker and …rm during employ4

ment in tenure t 2 f1; 2; 3; g produces et y, in which et 2 f0; 1g is theworker’s e¤ort level, and y 2 R is match quality. Each match separates withprobability s 2 [0; 1] each period for exogenous reasons.
Search Technology. Let
v=u be the vacancy-unemployment ratio,
where v is a measure of vacant …rms and u unemployed job seekers. Each
unemployed worker contacts a vacant …rm with probability p( ) per period,
where p0 ( ) > 0. Each vacant …rm contacts a job-seeker with probability q( )per period, where q 0 ( ) < 0.
Upon Meeting. Upon meeting, the …rm o¤ers the …xed salary w w 0
and performance pay p
0. Then, the worker decides whether to accept
the o¤er or not. If the worker turns the o¤er down, then the worker and…rm return to the labor market and search for a partner again, in which casepayo¤ U accrues to workers and 0 to …rms. When they choose to match, theworker chooses the e¤ort level e 2 f0; 1g. At this stage, the e¤ort level is notobserved by the …rm. Then, the payment of the …xed salary w 0 is made.
After this, the e¤ort level is revealed to the …rm3 . Then, the …rm chooses
whether or not to pay performance pay p 0. I assume that …rm can commit
to the payment of performance pay contingent on the observed e¤ort level4 .
At the end of each employment period, a ‡ow payo¤ of wt + et pt + (1 et )z
accrues to the worker, and y et wt et pt to the …rm. Note that theworker’s e¤ort level is observed by the …rm after the payment of …xed salaryw is made. Assume that the performance pay p can be contingent on thee¤ort level in an enforceable way. Match quality y is observable and veri…able.
It remains the same during the entire employment duration.
Free Entry of Vacancies. Each vacant …rm incurs recruiting cost c > 0
per period in …nding a worker. Entry into labor markets is free.
3This sequence of events is the same as the setup of Gomme (1999).
Hence, there are no problems examined by Levin (2003) and MacLeod and Malcomson(1998),
45

Determination of Payment: Base wage and performance pay. I assume
that a compensation vector of a …xed salary of w w 0 and a performance
payment of p 0 is chosen unilaterally by the employer upon meeting. To
focus on the interesting cases, I consider two di¤erent restrictions on thespace of compensation levels (w; p). First, I consider the case in which p = 0
and w
0 is chosen by the …rm. This case corresponds to the shirking
model of e¢ ciency wages a la Shapiro and Stiglitz (1984). Second, I consider
0 and p0 is chosen by the employer. The
the case in which w
w
lower bound assumption on the salary level, w
w, is seen capturing one
part of the reality. Moreover, as will be demonstrated below, the model withthis kind of setup will deliver both qualitatively and quantitatively di¤erent
property of equilibrium.

3

Steady State Equilibrium

Asset Equations. Now, I write down a bunch of asset equations in equilibrium, in which every employee exerts high e¤ort. In writing down thecondition in which employees do not shirk, I consider the employee’s choicesof whether to exert high e¤ort.
The value of unemployment U satis…es the following equations:U = z + b + f ( )W + [1f ( )]U(1)

where b is the amount of unemployment bene…ts per period, W the value of
employment for each employee. The value of employment for each employee,
exerting high e¤ort, W must satisfy the following asset equation:
W = w + p + (1s) W + s U(2)where w is the …xed base wage level, and p is the performance pay per period.
The value of each …lled job J must satisfy the following equation:

6

J =y

w

p + (1s) J(3)Generalized No-shirking Conditions. Recall that monitoring employees’ e¤ort is perfect in the sense that the employee’s e¤ort level is observed with no noise, after the …xed salary payment is made. When theworker chooses high e¤ort (e=1), the employment continues unless the match
receives exogenous separation shocks. When the worker chooses low e¤ort(e=0), the employment is terminated. The employee chooses high e¤ort(e=1) if w + p + (1 s)W + sU w + z + U , which is rewritten as:
(1

s)(W

U) + pz(4)

We call this inequality the generalized no-shirking condition, or GNSC in
what follows. The reason why I put generalized is that it is generalizedversion of the no-shirking condition (NSC) of the shirking model of e¢ ciency
wages (Shapiro and Stiglitz, e.g.). Observe that when performance pay p iszero in GNSC, the GNSC becomes NSC, i.e., the no-shirking condition in
the standard shirking model of e¢ ciency wages.

Wage Determination. When the worker and …rm choose to match,
the …rm chooses a wage level of w w, and the performance pay level p 0
so that the generalized no-shirking condition is satis…ed.
Free Entry of Vacancies. Each vacant …rm incurs recruiting cost c > 0
per period in …nding a worker. Entry into the labor market is free. Thisimplies the following equation:
c
=J
q( )

(5)

Now, we can de…ne a steady state equilibrium of a search model withworkers’shirking problems.
7De…nition 1. A steady state equilibrium of a search matching modelwith workers’ shirking problems is de…ned as a list fU; W; J; w; p; ; ug suchthat: (i) U , W , and J satisfy equations (1), (2) and (3); (ii) satis…es
equation (6); (iii) w and p maximize J subject to GNSC; and (iv) u satis…es
the steady state accounting: (1 u)s = uf ( ).
The de…nition given above is very standard except the part (iii), where
two wage components are chosen by the …rm so that they satisfy the GNSC.

3.1

Case of zero performance pay: E¢ ciency wages

I examine both analytical and quantitative properties of the equilibrium of
the model described above, in which performance pay is zero. This casecorresponds to the case of the standard shirking model of e¢ ciency wages ala Shapiro and Stiglitz. I am aware of two papers that introduce the shirking
model of e¢ ciency wages into the MP model: Rocheteau (2001) and Costain
and Jansen (2006). How this section di¤ers from these two papers is describedin Introduction.
Proposition 1. Consider the model developed above in which p = 0. The
workers’rent from match W U is independent of the vacancy unemployment
ratio in equilibrium.
Proof. Since the workers’gain from employment W U equals 1 w[1 zs bf ( )] ,it is increasing in base salary w 0. Given this, the cost-minimizing …rmchooses the …xed base wage w
0 so that the NSC binds, implying thatz
W U = (1 s) . jj
Since the NSC is binding at the …rm’s cost-minimizing solution, the soz
lution will be W U = (1z s) , J = G, where the match surplus G(1 s)
is de…ned by G W + J U . By using equations (2) and (3), one can show
that
8

G=

y
1

(1
(1

)U
s)

(6)

A steady state equilibrium of the model de…ned above, in which performance
pay p is zero, is characterized by a pair ( ; U ) satisfying the following two
equations:U = z + b + f( )
c
= G
q( )

z
(1

s)

+ U

z
(1

(7)
(8)

s)

where the match surplus G is de…ned above. Although my interest in thispaper is aggregate ‡uctuations of labor market aggregates under aggregate
shocks, it is helpful to consider, …rst, the steady state of the model.
Calibration in steady state. My strategy for quantitative exercises
below is to calibrate parameters in steady state and then quantify model
generated moments using those parameter values. One remarkable thing isthat one of important parameters, z, can be calibrated with some disciplines:
Given the observed value of , and the vacancy cost k, the binding no-shirking
conditions o¤er the value of non-working time, z 5 . In steady state without
aggregate shocks, we have:
c
=G
q( )

z
(1

s)

b + (1

s)G

f( )

and
G=y

z

z
(1

s)

These two imply that:
5

Hall and Shimer assumes that z = 0. Hagedorn and Manovskii calibrate z + b = 0:96.

9

y
c
z
+
=
q( )
(1 s)

z
1

b

f( )1z s
(1 s)

(9)

Let f ( ) = 1 and q( ) =. I choose the model period to be
two weeks. I set = exp[log(1=1:05)=26] = 0:9981, so that it is consistent
with the annual interest rate 0.05. Shimer (2005) estimates that the monthly
job …nding rate from 1951 to 2003 on average is 0.45. The corresponding biweekly job-…nding probability is f = 0:2584. The average unemployment ratefrom 1951 to 2003 is 0.0567. This, along with the steady state accounting,
implies that the bi-weekly job separation rate is s = uf =(1 u) = 0:0567
0:2584=(1 0:0567) = 0:0155. I normalize that y = 1. Hall (2005) foundthat the average value of the vacancy unemployment ratio during December
2000 to December 2002 was 0.539 according to JOLTS. But this number
may be too small since the economy was in downturns during that period.
Hence I use = 0:6. This implies that q = f = = 0:2584=0:6 = 0:4307. The
replacement ratio in the U.S. is roughly 0.4. Not all the unemployed quali…ed
for receiving the unemployment bene…t are not receiving the bene…t. So I
choose b = 0:3, instead of 0.4, as has been common in the early literature.
The values of two parameters c and z remain to be chosen. The binding
no-shirking condition gives z as a function of c, which I denote as c(z), withsome abuse of notation. Note that the ‡ow wage w is given by:

1

w b z
z
=
[1 s f ( )]
(1 s)

(10)

Empirically, the value of w is very close to y, which is set to 0.97 for thefollowing reason. Interpret the matching model above as the one with capital, following Pissarides (2000). Then, labor productivity y in my matchingmodel can be interpreted as output minus capital income. The ratio of dividends to the GDP for the U.S. economy from 1946 to 2000 is around 0.02.
I assume that y w in my matching model roughly corresponds to the dividends payouts. This means that (y w)=(y + rk) = 0:02, where y = 1 by

10

normalization. Since capital income rk is about 0.3 of output y +rk = 1+rk,
rk should be 3=7, and output 10=7. Hence, we have (1 w)=(1 + 3=7) = 0:02,
implying that w ' 0:97. 6 I choose the value of z so that the model generated w through equation (10) is 0.97. The value of z = 0:5234 does this job.
Remarkably, this value is close to the value of non-working time calculated
by Hall (2006). It is also slightly below the value calibrated by Hagedorn
and Manovskii. Once the value of z is obtained, one can compute the valueof c satisfying equation (9). The value of c = 0:7424 does this job. Thisvalue is greater than the value of vacancy costs calibrated by Hagedorn andManovskii (2008).
The values of matching function parameters remain to be chosen. Shimer(2005) estimates= 0:72. Mortensen and Nagypal (2007) argue thatshould be smaller than 0.72. Many authors use a value of 0.5. I choose= 0:5. I choose the value of so that the implied f = 0:5 is equal to0.45. = 0:2584=0:60:5 = 0:3336.

TABLE 1: Exogenously given parametersparameter interpretation
value
calibration strategy
discount factor
0.9981 biweekly frequencyylabor productivity
1normalization
matching function parameter 0.5
Petrongolo and Pissaridesbunemployment bene…t
0.25
Hall and Milgrom
Notes: "Exogenous" means that parameter values are selected without
using explict mapping between models and data.

The parameter values exogenously given in Table 1 are the same regardlessof the models setup.
6This number is close to the implied wage productivity ratio in standard DMP models.
For instance, in Shimer’s (2005) calibration, the implied w=y is 0.973.

11

TABLE 2: Calibrated Parameters: Case of zero performance pay
parameter interpretation
value
calibration strategyz
the value of non-working (leisure) 0.5624 w = 0:97
smatch separation prob.
0.0155 u = 0:0567 and f = 0:2584
matching function parameter
0.3336 f = 0:2584 and = 0:6cvacancy cost
0.7424= 0:6
Notes: "Calibrated" here means that the parameter values are selectedwith explicit reference to moment matching between the model and data.

The calibrated values of parameters s and in Table 2 will be also used in
the calibration of the model with positive performance pay below. However,the values of parameters z and c in the same table will be reselected in thecalibration under positive performance pay below.
Now I do some ’numerical’ comparative statics in steady state before
quantifying the model with aggregate shocks below. For y = 1, the model
delivers = 0:6, and w = 0:97. For y = 1:01, the model delivers =0:6698, and w = 0:9783. For y = 0:99, the model delivers = 0:5346, andw = 0:9617. The elasticity of with respect to y is roughly 11.27. The
value of this implied elasticities are larger than 7.56, which is the regressioncoe¢ cient of data of the deviations of log from its HP …ltered trend on dataof deviations of log productivity from its HP trend. The elasticity of wages
with respect to productivity is 0.8557. This number is way greater than 0.449,
which Hagedorn and Manovskii argue to be the empirical elasticity of wages
with respect to productivity. However, Pissarides (2007) recently argues thatthe relevant empirical elasticity of wages with respect to productivity should
be close to 1.

12

TABLE 3: "Comparative Statics" in Steady State: Case of zero
performance pay
y 0.991
1.01
0.5346 0.6 0.6698
w 0.9617 0.97 0.9783
TABLE 4: Data versus Implied Productivity Elasticities
The case of zero performance pay;yModel, e¢ ciency wages
Data
Model (Shimer)
Model (H and M)

w;y

11.27
7.56 (U.S., 1951-2003)
1.75
21.72

0.856
0.449a (BLS, 1951-2004)
0.964
0.449

Note: xy denotes the elasticity of variable x with respect to variable y,where all variables are reported in logs as deviations from an HP trend witha smoothing parameter of either 1600 or 105 . The superscript a indicatesa smoothing parameter of 1600. Data for xy is the regression coe¢ cient ofvariable x on variable y. The value of ;y for Model (Shimer) is what is( )implied by his TABLE 3 ( ;y = ( ;y)= 0:9990:020:035 = 1:75).
(y)
My calibrated search model with e¢ ciency wages delivers a su¢ ciently
large size of productivity elasticity of the v-u ratio. It delivers a productivity
elasticity of wages of 0.856, which is way larger than aggregate data (0.449),
which Hagedorn and Manovskii (2008) estimate. If the goal of introducing
shirking problems of e¢ ciency wages into a search matching model were todeliver sticky wages whose elasticity with respect to productivity is 0.499,
as in some previous papers such as Gomme(1999), then such attempt would
be a failure, although it reduces the elasticity slightly compared with thebenchmark calibration of Shimer(2005). However, Pissarides (2007) has recently argued that what is relevant for assessing a search matching model isbehavior of wages in only new matches. He argues that a plausible empirical13

elasticity of wages in new matches with respect to productivity is close toone. In this respect, the productivity elasticity of wages generated by mycalibrated model is not that di¤erent from what Pissarides (2007) considers
empirically plausible.
Comparison to Shimer and Hagedorn and Manovskii. The key to understanding my calibration is that the binding NSCs o¤ers tight disciplines incalibrating the parameter of the non-working (leisure) time. Comparisonof calibration procedures across three di¤erent papers (Shimer 2005; Hagedorn and Manovskii 2008; this paper) may help in understanding my point.
Start with a standard DMP model with Nash bargaining wage determination. There are two nontrivial key parameters that must be calibrated: thesum of the leisure value and the unemployment bene…t level, and employees’ bargaining power. Shimer implicitly assumes that the leisure value iszero and invokes the empirical replacement ratio of 0.4 to assume that the
unemployment bene…t is 0.4. He invokes Hosios’(Hosios 1990) condition toposit that employees’bargaining power is equal to the elasticity of matcheswith respect to unemployment , which he estimates to be 0.72 by regressing
the measured job …nding probability on the measured vacancy unemployment ratio. Hagedorn and Manovskii calibrated these two parameters by
explicitly matching the model’s implications to data. They choose the valueof the two parameters so that the model generates the empirically observed
productivity elasticity of aggregate wages (0.499) and the pro…t-output ratio(0.03). As argued above, there appears to be controversies about what therelevant empirical moment of productivity elasticity of wages is. It may bedesirable not to use productivity elasticity of wages as a target moment incalibration. My matching model with e¢ ciency wages does not contain aparameter of employees’ bargaining power, since wages are determined by
the binding no-shirking condition. Therefore only the data of pro…t-output
ratio can identify the value of non-working.

14

3.2

Case of positive performance pay

The subsection considers the case in which both the non-negative bonus pay
p 0 and the base salary level w w > 0 are chosen by the employer when
the worker and …rm meet. The model in this section is similar to Alexopoulps(2004), who examines the setup in which the …rm punishes the shirking employee by not paying the deferred wage, and employment continues. My
model in the section di¤ers from Alexopoulos in that employment is terminated when the worker shirks in my model, while it is not in Alexopoulos.
I argue that my setup is more consistent with the pro…t-maximizing …rm,
since the …rm can reduce total payment by letting employment terminatedwhen the worker shirks. The insight in this section is also related to Yang
(2008), who examines the selection of contract forms by the size of turnovercosts.
Proposition 2. Consider the model in which both performance pay
and base wage are chosen by an employer in each period. Assume that thelower bound of the base salary is greater than the unemployment bene…t, i.e.,w > b. In equilibrium, the workers’surplus is decreasing in labor productivityy.
Proof of Proposition 2.
The workers’gain from employment is computed as:
WU=

w+p z b1
[1 s f ( )]Consider the case in which w = w, and p = 0. There are two cases: (A)w z + b; and (B) w < z + b.
Consider, …rst, the case (A) in which w z + b. In this case, the worker’s
participation constraint is satis…ed: W U = 1 w[1 zs bf ( )] 0. If the GNSC
w z b
is satis…ed for w = w and p = 0, i.e., (1z s), then w = w
1 [1 s f ( )]and p = 0 is the …rm’s optimal choice. So, the workers’ match surplus
is 1 w[1 zs bf ( )] , which is decreasing in the market tightness . Unless the15

GNSC is satis…ed for w = w and p = 0, either performance pay p or the…xed salary w needs to increase. The key is that the amount of a payment
increase to satisfy the GNSC is smaller for the performance pay p than for
the …xed salary w7 . Hence, the cost-minimizing …rm chooses to increase theamount of performance pay p until the GNSC binds. Hence, at the …rm’s
optimum, the GNSC is binding, which gives the amount of performance pay:
(1 s)(w b)p = z, which is increasing in , since w > b by assumption.
1+ f ( )Since the binding GNSC implies z = p + (1 s)(WU ), the workers’
match surplus W U is decreasing in .
Next, consider the case (B) in which w < z + b. Consider the paymentof w = w, and p = 0. In this case, the workers’participation constraint is
not satis…ed. The sum of w + p must increase until W = U . In light of the
GNSC and the …rm’s cost minimization, increasing performance pay p is moree¢ cient. So, increase performance pay p until the worker’s participation
constraint binds, W = U , which implies p = z + b w, which is positiveby assumption. The implied NSC of z
p is violated since w > b. The
performance pay p has to increase until the GNSC holds with equality: z =z b(1 s)(W U ) + p, where W U = 1 w+p. This binding GNSC gives[1 s f ( )]the performance pay, which is the same as computed above. Therefore, inthis case, too, the performance pay is increasing in , and the workers’gainfrom employment is decreasing in . Hence, in every case considered, theworkers’gain from employment is decreasing in . The free entry equation
implies that is increasing in aggregate productivity y, this completes theproof of Proposition 2. jj
The result that the workers’gain from employment is decreasing in aggregate productivity y is contrasted with Proposition 1, saying that when
performance pay is prohibited as in the traditional shirking model of e¢ ciency wages (Shapiro and Stiglitz), workers’ gains from employment are
7
Yang (2008) examines the similar mechanism under the relational incentive contracts
in di¤erent contexts.

16independent of the v-u ratio. This result appears promising in light of Brugemann and Moscarini (2007), who demonstrate analytically that countercyclical workers’rents, in stead of its acyclicality, are important for search models
to deliver large ‡uctuations of vacancies and unemployment.
The reason why I consider the case of w > b is that the case of w = b
is not interesting. To see this, suppose w = b. Consider the payment of
w = b(= w) and p = 0. The workers’participation constraint is not satis…ed.
Increase performance pay p until the workers’participation constraint holdswith equality, which implies p = z. The GNSC is weakly satis…ed. Recall
that z > 0 by assumption. Hence, in this case, the workers’ gains from
employment is zero, i.e., W U = 0, and the the value of a …lled job ispositive, since y p w = y z b > 0 by assumption. In this case,
employers captures all of the match rents.
Assume z > b in what follows. In the calibration and simulation exercises
below, I consider a case of w = z + b > b, which falls into the case (A) of
Proposition 1 above.
Calibration in steady state.
Consider the case w = z + b > b. This case corresponds to the case
A of Proposition 2 above. The optimal choice of performance pay is p =z 1+(1 fs)z
and the base wage is w = z + b. The worker’s rent from match is( )W U = 1+ zf ( ) , which is decreasing in . Let
G=

y
1

(1
(1

)U
s)

(11)

The steady state equilibrium in this case is given by ( ; U ) satisfying thefollowing two equations:U = z + b + f( )
c
= G
q( )

z
1 + f( )z
1 + f( )
17

+ U

(12)
(13)

I choose the values of parameters z and c in two di¤erent ways.
CASE A. First, I choose the value of non-working z so that the total
payment
w + p = 2z + b(1 s)z
1 + f( )is targeted to 0:97 when labor productivity y is normalized to 1. Given z, thevalue of c is calibrated using the free entry equation. The values of z = 0:5907
and c = 0:7424 do this job.
CASE B. Second, to make the comparison with the case of zero performance pay clear, the same parameter value of non-working z = 0:5624 as in
the case of zero performance pay is used. Given this value of z, the free entry
equation gives c = 1:5967. In thise case, the implied total wage pay w + p
equals 0.9355.
With the calibrated model at hand, the numerical comparative statics
around the steady state is undertaken here.

TABLE 5: "Comparative Statics" in Steady State: Case of positive
performance pay
y
0.99
1
1.01
(Case A)
0.5089 0.6
0.7018w + p (Case A) 0.9624 0.97
0.9776(Case B)
0.5255 0.6
0.6808w + p (Case B) 0.9296 0.9355 0.9413
Notes: y denotes labor productivity, denotes the vacancy unemployment
ratio, w + p denotes total payment. What Case A and Case B mean is
explained in the main text.

18

TABLE 6: Data versus Implied Productivity Elasticities
The case of positive performance pay;yModel, performance pay, Case A
Model, performance pay, Case B
Model, e¢ ciency wages
Data
Model (Shimer)
Model (H and M)

w;y

16.07
12.95
11.27
7.56 (U.S., 1951-2003)
1.75
21.72

0.78350.62540.85570.449a (BLS, 1951-2004)
0.964
0.449

Note: xy denotes the elasticity of variable x with respect to variable y,where all variables are reported in logs as deviations from an HP trend witha smoothing parameter of either 1600 or 105 . The superscript a indicatesa smoothing parameter of 1600. Data for xy is the regression coe¢ cient ofvariable x on variable y. The value of ;y for Model (Shimer) is what is( )implied by his TABLE 3 ( ;y = ( ;y)= 0:9990:020:035 = 1:75).
(y)4Aggregate Shocks to Labor Productivity[Incomplete]This section adds aggregate shocks to productivity to the model developed
above. Let there be N possible aggregate productivity levels. Labor productivity in state i 2 f1; 2;; N g is denoted by yi , where y1 < y2 < ::: < yN . I
assume that aggregate state follows a N-state Markov chain with transition
matrix y , whose ith row and jth column element ij indicates the probability that next period’s state is j when the current state is i.
The value of being unemployed when the aggregate productivity is y 2
fy1 ; y2 ; :::; yN g, which is denoted by Uy , satis…es:
Uy = z + b + f ( y )Ey (Wy019

Uy0 ) + Ey Uy0(14)where z is the ‡ow value of leisure, b is the unemployment bene…t, andEy is the expectation operator conditional on that the current aggregate
productivity is y. Note that the value of employment for a worker and thevacancy unemployment ratio also depend on aggregate productivity y. y 0 in
the equation indicates the next period’s aggregate productivity. The value of
employment for a worker when aggregate productivity is y, which I denoteas Wy , satis…es:
Wy = wy + pysEy (Wy0(15)

Uy0 ) + Ey Wy0

where it is assumed that the amount of the leisure time for each non-shirking
employee is zero, wy and py are a base wage and performance pay when
productivity is y. The value of a …lled job in state y, which is denoted by Jy ,satis…es:
Jy = ywy

py + (1(16)s) Ey Jy0

Free entry of vacant jobs in state y implies that(17)c = q( y )Ey Jy0

where c > 0 is a ‡ow cost of posting a vacant job. The generalized no-shirking
condition in state y is:(1s)Ey (Wy0Uy0 ) + py

(18)

z

since the employee prefers not to shirk in state y if:
wy + py + (1s)Ey Wy0 + sEy Uy0

wy + z + Ey Uy0

Let the match surplus in state y be de…ned by Gy

W y + JyUy .

Recall that the base wage and performance pay are unilaterally determined by the employer in each employment period after the aggregate pro-20

ductivity level is revealed.8 .
Quantitative Exercises:
The stochastic process for labor productivity is described by the AR(1)process:
log yt+1 = log yt + t+1(19)

where 2 (0; 1) and t+1 is a draw from N (0; 2 ). I follow Hagedorn andManovskii (2008) in choosing = 0:9809 and= 0:0046. The mean of yis normalized to 1. I approximate the AR(1) process by a discrete 7-state
Markov chain, following Tauchen (1986). The grid of log productivity is
log y 2 f 0:0709; 0:0473; 0:0236; 0; 0:0236; 0:0473; 0:0709g. This meansthat y 2 f0:9316; 0:9538; 0:9767; 1; 1:0239; 1:0484; 1:0735g. The transition
matrix is given by:
3
2
0:9886 0:0114
0
0
0
0
0
7
6
60:0028 0:9884 0:0088
0
0
0
0 7
7
6
7
6 0
0:00380:98950:0067
0
0
0
7
6
7
6
=6 0
0
0:0051 0:9898 0:0051
0
0 7
7
6
7
6 0
0
0
0:0067
0:98950:0038
0
7
6
7
6 0
0
0
0
0:00880:98840:0028
5
4
0
0
0
0
0
0:0114 0:9886
The standard deviation of the simulated data of log y, using the transition
matrix above, is 0.03. The standard deviation of log productivity, relative to
its HP …ltered trend, for the U.S. quarterly data is 0.02 when the smoothing
parameter is 105 (Shimer 2005, 1951-2003), and 0.013 when the smoothing
parameter is 1600 (Hagedorn and Manovskii 2008, 1951Q1-2004Q4). Notethat the standard deviation in biweekly frequency should be larger than in
8This assumption is not empirically implausible in light of recent work of Hagedorn
and Manovskii (2008). They calibrate the parameter of employees’ bargaining power tobe close to 0.06.

21

quarterly frequency.

4.1

Case of zero performance pay: E¢ ciency wages

Consider the case in which performance pay py equals zero in every aggregate
state y. Then, one obtains that:
Gy = yzb + (1s)Ey Gy0f ( y)

z
(1

s)

(20)

This equation can be solved numerically for Gy . Using Gy , the free entry
equation in state y is rewritten as:c= Ey Gy0
q( y )

z
(1

s)

(21)

The value of parameters are the same as in steady state. The surplus
vector Gy , the vacancy unemployment ratio vector y , and the wage vector
wy are computed as:= (0:22; 0:3216; 0:4483; 0:6; 0:778; 0:9836; 1:2136)0
G = (1:5761; 1:7957; 2:0248; 2:2596; 2:5; 2:7455; 2:9917)0
w = (0:9158; 0:9332; 0:9514; 0:97; 0:9891; 1:0085; 1:028)0

The simulations are undertaken as follows. Using the transition matrix above, I simulate the Markov chain with 7-state. In order to generate 212 "quarterly" data points, corresponding to quarterly data from 1951
through 2003, I need to generate 1272 "bi-weekly" data points, since mymodel works at biweekly frequency. To do so, for each simulation, I generate 2000 data points, and throw away the …rst 728 data points to eliminate
22

the dependence on the initial value chosen. Then, I aggregate the generated"biweekly-frequency" data to quarterly frequency, resulting in 212 "quarterly" generated data points. Then I detrend the log of the model-generated"quarterly" data using an HP …lter with the smoothing parameter 16009 .
Then I compute the standard deviation of such model-generated data. I
repeat this 10,000 times. The result is summarized by Table 7.

TABLE 7: Data versus Model-generated Moments: Case of zero
performance pay( )= (y)(u)= (y)(v)= (y) Proj(wjy)
Data (Shimer)
19.1(7.56) 9.5 (-3.9) 10.1 (3.67) —Data (H and M)
19.9 (7.83) 9.6 (-2.9) 10.7 (4.9) 0.499
Model, e¢ ciency wages 12.26
5.671
6.071
0.829
Model (Shimer)
1.75
0.45
1.35
0.964
Model (H and M)
22.46
11.15
13.0
0.449
Note: (x) denotes the standard deviation of variable x 2 f ; y; u; vg,
where all variables are reported in logs as deviations from an HP trend withsmoothing parameter mentioned in the Table. For data of ( )= (y), I also
report the regression coe¢ cient of variable on variable y, which is 7.56.
Proj(wjy) is the regression coe¢ cient of variable w on variable y.
My calibrated search model with e¢ ciency wages delivers a su¢ ciently
large ‡uctuations of v-u ratios, the unemployment rate and vacancies, compared with data. TABLE 4 shows that the size of ‡uctuations of variable, u and v, measured by the standard deviation, is greater than the empirical regression coe¢ cient of each variable on productivity (the numbers inparentheses). They are also not way below the empirical standard deviations. Turning to the wage ‡uctuations, my calibrated model with e¢ ciency
wages delivers a standard deviation of wages in relative to productivity of
9This number of the smoothing parameter is standard in the RBC literature when
detrending quarterly data.

23

smoothing parameter
105
1600
1600
105
1600

0.83, which is way larger than data (0.449), which Hagedorn and Manovskii(2008) estimate. If the goal of introducing shirking problems of e¢ ciency
wages into a search matching model were to deliver sticky wages whose elasticity with respect to productivity is 0.499, as in some previous papers such
as Gomme(1999), then such attempt would be a failure, since it reduces thewage ‡uctuations only slightly compared with the benchmark calibration of
Shimer(2005).
However, as Pissarides (2007) has recently raised the issue, what is relevant for assessing a search matching model may be behavior of wages in
only new matches. He argues that a plausible empirical elasticity of wages
in new matches with respect to productivity is close to one. In this respect,
the productivity elasticity of wages generated by my calibrated model is notthat di¤erent from what Pissarides (2007) considers empirically plausible.

4.2

Case of positive performance pay

Assume that w = z + b. Consider the case in which a …rm can choose
performance pay py as well as base wage wy in aggregate state y. Then, one
obtains that:
Gy = yzb + (1s)Ey Gy0f ( y)z b
1 + f ( y)(22)This equation can be solved numerically for Gy . Using Gy , the free entry
equation in state y is rewritten as:k= Ey Gy0
q( y )z b
1 + f ( y)(23)

The surplus vector Gy , the vacancy unemployment ratio vector y , andthe wage vector wy are as follows. In CASE A (z = 0:603 and c = 0:7596),24

= (0:0431; 0:1420; 0:3217; 0:6014; 1:0045; 1:5554; 2:2488)0
G = (0:7528; 1:1262; 1:5473; 2:0098; 2:5149; 3:0625; 3:6296)0
w + p = (0:9275; 0:9415; 0:9556; 0:9694; 0:9828; 0:9956; 1:0074)05

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