Monopolistic Competition and Unemployment
Fluctuations∗
Ha Dao†
UQAM
Job Market Paper
October 2009
Abstract
This paper contributes to the recent literature on the unemployment volatility
puzzle through developing a very simple monopolistic competition model à la
Dixit-Stiglitz. Firms having monopoly power charge a price that is higher than their
marginal cost by an amount equal to the markup. When competition increases, the
markup decreases in each firm yet the industry demand for labor increases as firms
need more workers to expand output. Market entry continues to respect the free entry
condition, leaving little profit for each firm, thus implying even a small change in
labor productivity will significantly amplify job creation rates. I find that the model
parameterization that helps pin down the weak procyclicality of real wage explains
nearly 70% of volatility and all cyclical behavior of labor market key variables found
in the data. Moreover, the intra-firm bargaining mechanism to reflect the overhiring
incentive in large firms is shown to be negligible.
Keywords: small profit, free entry, endogenous competition, markup
JEL codes: D42, E24, E32, J64
∗
I would like to thank my thesis advisers Alain Delacroix and Étienne Wasmer for their helpful comments
and suggestions.
†
Ph.D. candidate, Department of Economics, University of Quebec at Montreal, P.O Box 8888, Downtown Station, Montreal (QC) H3C-3P8.
Phone: 001-514-987-3000 Ext. 2745#
E-mail: dao.ngoc [email protected]
URL: http://www.er.uqam.ca/nobel/m132224/
1
1
Introduction
Unemployment fluctuations deserve intense study not only because they are associated with
human capital depreciation but also because of the related productive and social externalities.
The standard matching model presented by Mortensen and Pissarides (1994) (hereafter MP
model) represents one of the main theories used to study the labor market, even though it
fails to reach empirical labor market volatility levels, known as the unemployment volatility
puzzle. Shimer (2005) initiates the debate by simulating a version of this model, finding
that unemployment and vacancies are as volatile as labor productivity, even though the
aggregate data reveals that they are much more volatile - by a factor of 20. He points out
that the Nash bargaining solution used to determine wage is responsible for this. In the
model, wage is negotiated according to the rule for surplus sharing, whereby an increase in
labor productivity will increase wage by the same proportion. As such, firms’ profits will
remain unchanged over the business cycle, meaning they have little incentive to create jobs
and thus the model exhibits a moderate level of labor market volatility.
This paper starts with the small profit concept put forward by Hagedorn and Manovskii
(2008), whereby the solution would be to have a high unemployment income representing
95% of total output, which typically induces small profits left for firms and guarantees that
their net payoff would be more responsive to small changes in productivity, thus resulting in
higher job creation rates. This strategy has been subject to a great deal of criticism because,
as demonstrated by Costain and Reiter (2007), significant increases in aggregate productivity
would cause unrealistic fluctuations in unemployment rate. Moreover, obtaining adequate
measures of non-market value is a difficult task in itself.
In this paper, I explore a different mechanism through the development of a very simple
model of monopolistic competition à la Dixit-Stiglitz. The originality is the introduction of
competition to all sectors where each sector has a certain number of competing firms that
respect the free entry condition. My model allows the markup to be variable instead of constant as in the Dixit-Stiglitz model. The intuition of why adding monopolistic competition
could help amplifying the unemployment volatility is that monopolistic firms charge a price
2
that is higher than their marginal cost by an amount equal to the markup. When competition increases, the markup decreases in each firm since it has now less incentive to restrict
output level and at the same time the industry demand for labor increases as firms need
more workers to expand output. Market entry continues to respect the free entry condition,
leaving little profit for each firm. More specifically, consider now an increase in labor productivity. In the short run - defined by taking an exogenous given number of monopolistic
firms who can restrict the output level - the output demand will expand involving a higher
demand for labor given the price of goods is unchanged. Since the amount of increase in
labor demand depends on the set of parameters used, I show that within a plausible set of
parameters, significant amplification takes places. Overtime, higher industry profits attract
more firms to go into business and therefore increases in the degree of competition will result
in smaller markup since each firm now has an incentive and alternatively further increases in
job creation, in order to meet the increased demand for output. As a result, as firms go into
business many more jobs will be available in the long run, until industry profits are driven
to zero.
My contribution to the literature on the unemployment volatility is straightforward. In
this paper I develop the framework inspired by that of Ebell and Haefke (2009) and Delacroix
(2006), yet it differs in the following characteristics. First, I focus on the cyclical behavior of
the labor market variables while these two papers study the effect of policies on the steadystate labor market outcomes. More specifically, the first paper studies how the Carter-Reagan
deregulation in the early 1980s affect the US labor market and shows that barriers preventing
a new firm from entering the market lead to a long-run decline in industry size equilibrium,
and thus a decline in the degree of competition. In the long run, the effect on unemployment equilibrium is modest and the real wage significantly decreases. The second paper
applies the Ebell-Haefke framework but especially focuses on the impact of monopolistic
competition on unionized sectors and on non-unionized sectors. As such the author studies
the interactions between unions and unemployment insurance, and his results are consistent with the unemployment benefit in Europe, which are more generous when compared
3
to those in the US, yet also characterized by more extensive unionization and lower payroll
taxes. This paper was designed for a different purpose: that of solving the unemployment
volatility puzzle. The second main distinguishing feature of the model is that neglects intrafirm bargaining (IFB) because in the business cycle study this mechanism is quantitatively
negligible. IFB was first introduced by Stole and Zwiebel (1996) and further developed in a
matching model context by Cahuc and Wasmer (2001) or Cahuc et al. (2000) to reflect the
overhiring incentive in large firms, whereby they hire more labor to weaken the bargaining
position of existing workers and therefore their salaries. In other words, and in contrast
with traditional theory, with more workers each worker’s pay is reduced, resulting in firms
having an incentive to overhire. The IFB thus helps Ebell and Haefke (2009) in identifying
significant decreases in real wage when competition increases and Delacroix (2006) in finding
overhiring incentive in non-unionized sectors as compared to unionized sectors. My model
however ignores IFB for the very simple reason that in long-run equilibrium, the number of
firms entering business is much greater than in short-run equilibrium. More firms entering
the sector means more vacancy positions are available and the markup becomes relatively
small, especially when the number of firms approaches infinity. Meanwhile, the overhiring
effect also reduces markups yet in the long run, small markups are already guaranteed by
the free entry condition, and as such the overhiring effect becomes negligible. Although intuitively IFB results in greater amplification, it does not help improve the model’s outcome.
The quantitative results are consistent with the findings of Krause and Lubik (2007), who
inspect the IFB mechanism within a RBC model context. They study the dynamic behavior
of the labor market key variables on which Shimer (2005) focused as well as the implications
for business cycle statistics, with and without the IFB version. Both model versions thus
produce virtually the same results, with unemployment volatility being as moderate as in the
standard MP model. In their papers however, the authors are not able to explain why they
obtain such results while in my model I do so successfully. Finally, while the two original
papers only focus on the effects that policies have on the steady-state outcome of the labor
market, this paper allows aggregate shocks to be stochastic. From the quantitative point
4
of view, the model performs really well through assigning a plausible set of parameters. I
find that the model parameterization that helps pin down the weak procyclicality of real
wage reaches nearly 70% of volatility and all cyclical behavior of labor market key variables
found in the data. When the model is calibrated by using the standard parameterization, it
explains twice greater than the standard matching model.
Apart from the paper by Hagedorn and Manovskii (2008), numerous others also attempt
to solve this puzzle, and a great deal of progress has been made. Shimer (2004) and Hall
(2005) share the idea that to improve volatility in the labor market a sticky wage model
is needed. Shimer in fact examines a constant wage model while Hall proposes a fixed
bargaining wage set, and both conclude that a sticky wage model may result in substantial
volatility. In the real world however a constant wage is implausible, yet the source of the
bargaining set is still an unanswered question. They thus suggest that future research should
focus on building a rigid wage model, and that a richer model would feature either wages
affect worker turnover rates or asymmetric information formats. Certain studies attempt to
endogenize wage rigidity by applying asymmetric information to match-specific productivity,
and two of them, namely Guerrieri (2007) and Brügemann and Moscarini (2008), find tiny
fluctuations in labor market variables. In one of my other papers, I develop a different
type of asymmetric information introduced by Delacroix and Wasmer (2006). The idea is
that workers and firms bargain over a wage by means of take-it-or-leave-it offers. Through
possessing private information, each party does not know whether his offer is accepted for sure
by the other, such that when aggregate productivity increases, low productivity firms make
more generous offers than those with high productivity, while high amenity workers require
more than the low ones, implying that average wages closely follow aggregate productivity,
meaning there is little job creation.
Moreover,
other papers attempt to make profit size small,
such as that
by Mortensen and Nagypal (2007) for example who include turnover costs,
or
Elsby and Michaels (2008), who introduce downward-sloped labor demand. Compared to
the models in these papers, the one I developed has some specific and important distinguish-
5
ing features. First, both papers only analyze the steady-state comparative statics. Although
obtaining similar steady-state analysis is possible, the dynamic behavior might be quite different. Obviously including turnover costs (the costs of training and hiring workers) would
be another way of guaranteeing small profit size, and represents a model completely different
from the monopolistic competition model. Mortensen and Nagypal (2007) are able to explain up to two third of the volatility found in the data. I find two problems with this paper.
First, it’s quite possible that the relative standard deviation between a particular variable,
say the labor market tightness, and the labor productivity is small, while the correlation
coefficient between them is relatively high. Second, the model turnover costs deserve further
investigation due to the empirical questions that surface. Their strategy is to chose a value
to target the cyclical variation in tightness and then imply the relative standard deviation,
which is inconvincible.1 Indeed it is Silva and Toledo (2008) who report evidence on turnover
costs and simulate the model, and thus are able to attain up to one fourth of the volatility
observed in labor market tightness. Downward-sloped labor demand means that output is
a concave function of labor input, yet when firms demand more labor, the marginal return
of labor decreases, resulting in a small marginal surplus for a match. Elsby and Michaels
(2008) reach one third of the cyclical variation in labor market tightness.2 Monopolistic
competition differs because even with the linear production function, the key feature here
is the endogenous degree of competition as measured by the number of firms competing in
long-run equilibrium, allowing each firm in a particular sector to attain a small markup.
In the remainder of this paper, Section 2 describes the model and discuss the analytical
results. In Section 3, a few numerical exercises are carried out and the model’s results are
produced. Section 4 reveals my conclusions.
1
Elasticity of tightness with respect to labor productivity is equal to the ratio of the standard deviation
of first over the second’s one, multiplied by the correlation coefficient between the two time series.
2
Elsby and Michaels (2008) in fact obtains the full cyclical variation in the job finding rate but this latter
is a direct function of the tightness generated from the matching function, resulting in virtually the same
cyclical variation in both variables. The elasticity of tightness with respect to labor productivity as calculated
using Shimer (2005)’s summary statistics is three times greater than that found by Elsby and Michaels (2008).
That explains my data translation.
6
2
The model
In the economy there are two markets: goods and labor. The former is characterized by a
monopolistic competition according to Dixit-Stiglitz and the latter by a standard matching
model (e.g. Pissarides (2000)). Monopolistic firms enter an industry by paying an entry
cost and once this has taken place, they take households’ demand for goods as given and
choose the number of vacancies needed to maximize the discounted value of future profits.
Households sell their units of labor supply to the labor market and buy goods from the
goods market. A fraction of workers are unemployed and searching for a job while the rest
are employed and face a certain probability of losing their jobs. All agents in the economy
are risk neutral and discount their future payoffs at the rate r. To obtain analytical results
and to gain some intuitions, I first consider the case of deterministic aggregate productivity
and then later the dynamic stochastic case. I apply Delacroix (2006)’s formulation, which
is somewhat different from Ebell and Haefke (2009), and allows me to develop a simpler
model representation and save calculation steps without causing any changes to the model’s
outcome.
2.1
The goods market
There are H households in the economy and each is indexed by the subscript h and has a
Dixit-Stiglitz preference for g differentiated goods. The economy is composed of g sectors
and each specializes in a typical good i. Let Ci,h be the quantity of goods i consumed by
household h not surpassing its real income Ih . Let pi and P be the price of goods i and the
price index for all goods respectively. The household h’s utility maximization problem can
be described as follows
max
Ci,h
subject to the budget constraint
g
X
1
σ
αi Ci,h
i=1
g
X
pi
i=1
P
σ−1
σ
σ
! σ−1
Ci,h = Ih .
7
,
In the utility function for household h, σ represents the elasticity of substitution across
consumption goods and αi the weight assigned to goods i. In the symmetric equilibrium all
goods are identical and thus αi = 1/g. The aggregate demand for goods i is obtained by
solving the aforesaid problem
YiD =
1 pi −σ
I,
g P
(1)
P
where I = H
Ih is the aggregate real income, and the composite price index P is defined
P h=1
1
g
1 1−σ 1−σ
by P =
.
i=1 g pi
Suppose that all firms in one sector play the Cournot game, i.e, a firm takes other firms’
output in its sector as given. Let Ni be the number of firms competing in sector i. The
demand function for a particular firm, indexed by j, in this sector is therefore
− 1
pi
Yi,j + (Ni − 1)Y i,−j σ
= g
,
P
I
(2)
where Yi,j is the output of firm j and Y i,−j the average output for all Ni − 1 other firms.
In symmetric equilibrium, all firms are identical so that Yi,j = Y i,−j and the subscript j be
dropped giving the elasticity of demand faced by a firm in sector i as
ǫi = σNi .
(3)
The elasticity of demand is the product of the elasticity of substitution across goods and the
number of competing firms. Increase either σ or Ni will lead to an increase in the monopoly
power ǫi . The former characterizes the degree of competition among sectors while the latter
the degree of competition within each sector. In the literature however σ is often treated
as a preference parameter, and thus throughout this paper the possibility of it becoming a
measure of competition would be ruled out, thus the degree of competition is determined
by Ni . As the number of firms specializing in a good increases, then the greater number of
households able to choose to buy this good from different firms.
8
2.2
The labor market
Matching frictions
First, it is assumed that due to specialized skills households can only work in one sector.
Frictions in the labor market however do not allow instantaneous meetings between firms and
workers so that firms have to use its resources to find workers in the labor market. There are
numerous sources of friction, as for example Pissarides (2000) tells us lack of coordination,
asymmetric information, and heterogeneity of vacancies and workers, and all these factors
can make matching costly. If a sector comprises a number of unemployed workers ui and a
number of vacancy positions vi , then the number of hirings, denote by m, is a combination
of these two variables. The matching technology is defined by a Cobb-Douglas function, as
is the production function, which combines labor and capital input to produce output
m(ui , vi ) = µuηi vi1−η ,
where µ measures the matching efficiency and η the unemployment elasticity with respect
to match. The matching function is a reduced form representation of the frictions. Notice
that m is (i) increasing and strictly concave in each separate argument, and (ii) constant
returns to scale (CRS) in both arguments. As such, there are likely more matches when
more workers and firms are searching, yet when holding an input constant, matching results
in diminishing marginal returns. Let define labor market tightness as the ratio of vacancies
to unemployment, θi = vi /ui . It is convenient to define θi , as will be seen in the equilibrium,
all model variables will depend only on it. Given the Cobb-Douglas form of the matching
function, workers find new jobs at rate f (θi ) =
at rate q(θi ) =
m(ui ,vi )
vi
m(ui ,vi )
u
= µθi1−η while a vacancy is filled
= µθi−η . When labor market tightness increases - either vacancy
increases or unemployment decreases - it is easier for workers to find jobs but more difficult
for firms to find workers.
A match between firm and worker can be broken either due to an unproductive match
pair or a ”death shock”, which induces a firm to exit the industry sector. This model
9
only considers the exogenous separation case. If for one period of time the firm’s survival
probability is δe , and the probability of a unproductive match pair is δs , then the average
separation rate is δ = δe + (1 − δe )δs . Let Li be the current employment level in sector i,
then Li accumulates according to the following equation
L′i = q(θi )vi + (1 − δs )Li ,
(4)
where the ”prime” superscript indicates the next period. Eq. (4) states that next-period
employment level is the sum of two flows: those who have just been hired and those who
stay in the previous employment pool, except that one part of them moves out of that pool.
Steady-state value functions
Since the worker’s optimization problem is found only in the goods market (the demand
function for goods (2.3)), defining her value function is straightforward. The firm’s optimization problem is however a bit more complicated as it will be based on the employment
dynamic (4), and thus the problem will be written in dynamic form before being simplifying
into a steady-state form. Let U and W be the values of being unemployed and employed
respectively. In flow terms, they satisfy the following equations
rU = b + f (θi )[W − U]
(5)
rW = wi + δ[U − W ].
(6)
Eq. (5) states that a unemployed worker receives a flow income b which is common across
sectors. At the rate f (θi ), she meets an employment opportunity and, if taken, realizes a
capital gain W − U. Likewise in Eq. (6), an employed worker in her sector is paid a wage wi ,
and might lose the job at an exogenous probability δ associated with a utility loss U − W .
The value function for a firm is not habitually applied since as opposed to the MP model
this model allows for multiple workers in each firm. Suppose for instance that a firm has
already paid the entry costs for business entry. After this procedure, the firm must decide
10
to post a certain number of vacancy positions vi at the real unit cost κ in order to attract
new workers for the next period. If any of these positions is filled then production starts
with the technology given by
Yi = y · Li ,
(7)
where y represents labor productivity. Let V (Li ) be the firm’s value function which depends
on the current level of employment as this latter is a state variable. From the theoretical
point of view, wage should be a function of Li , given that its determination is based on the
marginal surplus of each worker, the so-called IFB. As Krause and Lubik (2007) have shown,
IFB has no impact on the dynamic behavior of labor market variables and the implications
for business cycle statistics. In Section 2.4, I show that in the context of this model the
IFB can be ignored, and alternatively in Section 3, a numerical exercise shows that both
with and without IFB versions of the model, there is virtually no difference in results. In
my model, I neglect IFB, based on the argument used by Krause and Lubik (2007): ”firms
behave myopically by taking the wage of their incumbent workforce as given when choosing
employment”. Accordingly, the firm’s optimization problem can be written as follows
o
1 n pi
′
Yi − wiLi − κvi + (1 − δe )V (Li ) ,
V (Li ) = max′
vi ,Li 1 + r
P
(8)
subject to
the demand function (2),
the employment dynamic (4),
the production function (7).
The first-order condition with respect to vacancy is
κ
∂V (L′i )
= (1 − δe )
,
q(θi )
∂L′i
(9)
which means that the total cost of posting a vacant position equals the expected value of
11
filling it.3 Appendix A shows that the steady-state version of (9) is equivalent to the following
typical monopolistic price
σNi 1 r + δ κ
pi
=
+ wi .
P
σNi − 1 y 1 − δe q(θi )
(10)
Eq. (10) is a well known proposition: monopolistic firms charge a price that is higher than
their marginal cost. Here the marginal cost is the sum of a vacancy cost and a wage paid
to the worker. The term
σNi
σNi −1
reflects the markup over the total marginal cost. As will
be shown in the equilibrium section, in the context of solving the unemployment volatility
puzzle Eq. (10) is the key equation.
Wage bargaining
When a firm and a worker sit down to bargain over a wage paid, both parties make alternative
offers until the agreement is concluded. The Nash product, which is based on the principle of
sharing the total match surplus, is the solution to this type of negotiation game. With β being
equal to the worker’s bargaining power, wage is the outcome of the following optimization
problem
max [W − U]β [∂V (Li )/∂Li ]1−β ,
wi
The first component in the maximization problem stands for the worker’s surplus while the
second the firm’s marginal surplus of an additional worker. The first-order condition with
respect to wage is given by
wi = β
pi
σNi
y
+ (1 − β)rU,
P σNi − 1
which is quite intuitive. Real wage is the weighted average of the marginal benefit of adding
a worker and her outside option. Going one step further, Appendix B gives an explicit
3
Given q(θi ) is the Poisson arrival rate of a new match for a firm, 1/q(θi ) is thus the average duration of
a vacant position.
12
solution for real wage, as a function of labor market tightness
wi = b +
β[(r + δ + f (θi )] κ
,
(1 − β)(1 − δe ) q(θi )
(11)
Beveridge curve
The Beveridge curve describes the link between unemployment level and vacancy level in the
steady state. If the flow out of unemployment f (θi ) remains constant for a sufficiently long
period and given the constant separation rate for all sectors then the sectorial unemployment
rate converges to the steady-state rate
ui =
δ
.
δ + f (θi )
(12)
This is called the Beveridge curve in the v − u space. Through normalizing the economy’s
total labor force to 1, for each sector the labor force becomes 1/g for reasons of symmetry.
The sectorial employment level is thus the product of firm-level employment and the number
of firms competing in the sector
Ni Li =
1 f (θi )
.
g δ + f (θi )
(13)
Eq. (13) is another presentation of the traditional Beveridge curve.
2.3
Equilibrium
The model is now ready to be closed. Let define a short-run general equilibrium and a
long-run general equilibrium separately. The former is characterized by taking the degree of
competition as given while the later by endogenizing the number of competing firms.
13
Short-run general equilibrium
Definition 1 A short-run general equilibrium (SRGE) consists of a set of three endogenous
variables {θi , wi, pi /P }, for a given number of firms specializing in one good Ni , that satisfy
the following system of equations:
1. the monopolistic price (10),
2. the wage equation (11),
3. the aggregate resource constraint
I=
g
X
i=1
Ni
pi
Yi .
P
(14)
For reasons of symmetry, i.e,, all goods are identical, the relative price is equal to unity so
that I = gNi Yi and the aggregate demand function () is equal to the aggregate income and
to the aggregate production. Expanding the SRGE system of equations makes it possible to
obtain one equation, linking labor productivity y to labor market tightness θi
σNi
r + δ + βf (θi ) κ
y=
b+
.
σNi − 1
(1 − β)(1 − δe ) q(θi )
(15)
Eq. (15) is called the job creation equation because it was directly derived from the firm’s
optimization problem.
Proposition 1 If y >
σNi
b
σNi −1
then the short-run general equilibrium is unique.
Proof 1 The right hand side of (15) is strictly increasing in θi while its left hand side is a
constant. The inequality is a necessarily condition to ensure that the constant line and the
curve (the increasing function of θi ) meet.
Given the uniqueness of the equilibrium θi for a given Ni , once it has been solved, all other
model variables can be found because they all depend on θi . The following proposition is
essential:
14
Θ
E '2
E '1
Job Creation
Hat y ' > yL
E2
Job Creation
E1
H at yL
Degree of
Competition
N
N1
N2
Figure 1: Short-run equilibrium and the effect of a positive aggregate shock
Proposition 2 In the SRGE,
1. the labor market tightness θi is strictly increasing in the degree of competition Ni ,
2. under the condition y >
σNi
b,
σNi −1
increase the degree of competition will further amplify
the cyclical variation in θi .
Proof 2 The proof of the first part of Proposition 2 can be directly obtained from (15). The
second part is thus a corollary of the first part and is illustrated in Figure 1.
In Figure 1, the job creation curve describes θi as is a strictly increasing function of Ni . It is
now straightforward to analyze the short-run cycle behavior of the labor market tightness θi .
Consider the economy is initially at the equilibrium point E1 which is unique as Proposition
1 proves. If the labor productivity increases from y to a higher level y ′ then the equilibrium
′
moves from E1 to E1 followed by an increase in the labor market tightness. If the degree of
competition however increases from N1 to N2 , then the equilibrium will move toward point
′
E2 , leading further increase in θi . The intuition behind this is that in the short run, an
15
increase in labor productivity increase output demand. Then, given that the price of goods
and the markup remain unchanged, firms must hire more workers to meet the increased
demand for goods. Overtime, higher industry profits attract more firms to enter the business
and thus increase the degree of competition, leaving a smaller markup and alternatively, a
further increase in job creation.
Readers might like to further examine the issue of variations in θi relative to changes in
y. From Eq. (15), it is possible to calculate the elasticity of θi with respect to y
Elasticityθi ,y =
(1 − β)(1 − δe )q(θi )b + κ(r + δ + βf (θi )) 1
× .
(β(1 − η)f (θi ) + η(r + δ + βf (θi ))
κ
By taking the standard parameterization used in the literature: r = 0.012, δ = 0.1, δe =
0.024, f (θi ) = 1.35, q(θi ) = 0.71, β = η = 0.5, κ = 0.156, and b = 0.6, I obtain an elasticity
value of 2.90, roughly better than the value 1.03 computed by Shimer (2005). Note however
that the highest elasticity value Shimer (2005) obtained is 1.39, when setting the value of
worker’s bargaining power β to zero, while the elasticity I obtain here simply results from
the short-run equilibrium and in long-run equilibrium it will be improved.
Long-run general equilibrium
The long-run general equilibrium (LRGE) is defined such that it allows Ni to be endogenized.
In the short run, monopolistic competitors may either earn positive profits and attract new
entrants, or operate at a loss, resulting in an industry shakeout. More particularly, the
number of firms entering into business will have to respect the free entry condition, i.e.
entry is allowed once the expected industry profits are driven to zero. The cut-off point, i.e.
the maximum number of firms allowed to enter, is determined by equalizing the expected
net present value of average profits from engaging in business and the costs of establishing a
standard firm. In doing so, firms must take into account the exogenous probability of exiting
16
from their sector δe . As such, the free entry condition is defined as
ci =
t
∞ X
1 − δe
t=1
1+r
πi ,
where ci is the entry costs which is taken as fixed costs because data may provide this kind of
information. The term πi =
pi
Y
P i
− wi Li − κvi is thus the current profit. After some algebra,
the free entry condition is given by
κδs
1 + r f (θi )
y − wi −
,
ci gNi =
r + δe δ + f (θi )
q(θi )
(16)
Eq. (16) along with Eq. (10) determine the endogenous degree of competition and characterize the long-run equilibrium defined below:
Definition 2 A long-run general equilibrium (LRGE) consists of a set of four endogenous
variables {θi , wi , pi /P, Ni} that satisfy the following system of equations:
1. the monopolistic price (10),
2. the wage equation (11),
3. the aggregate resource constraint (14),
4. the free entry condition (16).
Proposition 3 If y > b then the long-run general equilibrium is unique.
Proof 3 Given wi in (11), the long-run equilibrium system can be reduced to two equations
with two unknowns (θi , Ni ). The first equation is the job creation (15) which gives a positive
relation between θi and Ni . The second equation is the free entry condition (16) which clearly
shows a negative relation between θi and Ni . The condition for the two curves cut at a point
is obtained by taking the limit of both curves when Ni → ∞ and imposing the limit of the
first equation higher than the second’s one.
17
Θ
Free Entry (y' > y)
Job Creation
(y' > y)
Free Entry (y)
E 'LR
Job Creation (y)
E 'SR
ELR
ESR
Long-run
Equilibrium
Short-run
Equilibrium
N
Figure 2: The effect of a positive aggregate shock: SRGE vs. LRGE
Given Proposition 3, the following proposition is considered the main result of this paper.
Proposition 4 Under the condition y > b,
1. the labor market tightness in the long run is always higher than in the short run,
2. a positive aggregate shock will increase the labor market tightness θi more in the long
run than in the short run.
Proof 4 See Figure 2.
Consider the long-run equilibrium system in the θi −Ni space shown in Figure 2. The proof of
the first part of Proposition 4 is straightforward because if the short-run equilibrium ESR lies
on the right of the long-run equilibrium ELR then the free entry condition is not respected,
18
i.e., firms’ profit is negative leading them to shakeout. The second part of Proposition 4 is
just a corollary of the first part along with the second part of Proposition 2. The intuition
behind this result is that suppose a positive aggregate productivity shock hits the economy.
In the short run, output demand will increase and because the number of competing firms
stays unchanged, firms must post more vacancies to hire workers and thus meet rise in output
demand, which is translated by the upward shift of the job creation curve. Labor market
tightness then increases as the number of unemployed workers remains constant. At the
same time, a positive shock allows firms to earn higher profits and given a fixed level of Ni ,
this implies a parallel shift upward of the free entry curve. As second part of Proposition
2 proves, the amplification is guaranteed because when N increases, the creation job curves
diverge so that the variation in θi in the long run is higher than in the short run.
2.4
Does intra-firm bargaining matter for the business cycle
study?
The intra-firm bargaining (IBF) mechanism was introduced by Stole and Zwiebel (1996)
and developed in a matching model context by Cahuc and Wasmer (2001) and Cahuc et al.
(2000). The idea is that firms that hire additional workers then existing workers may find
themselves in uncomfortable positions when negotiating wages with firms individually and
this leads to an overhiring incentive. My question here is similar to the one asked by
Krause and Lubik (2007) even though they do so in a RBC model context. Indeed they
find there are negligible effects on labor market dynamics and volatility. To learn why this
paper ignores this mechanism, I now consider that the wage received by each worker depends
on the firm’s current level of employment, wi = w(Li ). After redoing all calculation steps4 ,
the only change here is Eq. (15) which now becomes
r + δ + βf (θi ) κ
σNi − β
b+
.
y=
σNi − 1
(1 − β)(1 − δe ) q(θi )
4
See Ebell and Haefke (2009) for details on calculations.
19
It is obvious that only the markup changes. Since the new markup,
old one,
σNi
,
σNi −1
σNi −β
,
σNi −1
is smaller than the
meaning that the overhiring incentive weakens the firms’ monopoly power.
From a quantitative point of view however, β is relatively small compared to σNi , especially
when Ni → ∞ the overhiring effect is negligible. The intuition behind this result is that
in long-run equilibrium, the number of firms entering business is much more higher than in
the short run equilibrium. More firms entering means more vacancy positions are available
and the markup becomes relatively small, especially when the number of firms approaches
infinity. Meanwhile, the overhiring incentive also reduces the markup yet in the long run, the
small markup is already guaranteed by the free entry condition, and as such the overhiring
effect becomes negligible.
2.5
Dynamic version
To rewrite the model in a dynamic version and in continuous time, I follow the method
used by Shimer (2005). Suppose now that y is subject to aggregate shocks arriving at a
Poisson probability λ which then change y to a new level y ′ . This new productivity is drawn
from a first-order Markov process in continuous time. Let Ey Xy′ be the expected value of
an arbitrary variable X, following the next aggregate shock, and conditional on the current
state y. All variables in the model become jump variables when the economy changes to a
new state.
Now, given the state of the economy as it changes into a new one with probability λ, all
value functions should be subject to this feature. The goods market however is unaffected
by the latter because households’ decision is a one-period optimization problem. The value
functions for two types of workers are now changed to
rUy = b + f (θiy )[Wy − Uy ] + λ[Ey Uy′ − Uy ],
rWy = wiy + δ[Uy − Wy ] + λ[Ey Wy′ − Wy ].
20
For a firm, its value function satisfies
o
1 n piy
′
′
′
Vy (Li ) = max′
Yiy − wiy Li − κvi + (1 − δe )[λEy Vy (Li ) + (1 − λ)Vy (Li )] ,
vi ,Li 1 + r
P
subject to (2), (4) and (7). Denoted by Sy =
∂Vy (Li )
∂Li
+ Wy − Uy be the total surplus of a
worker-firm match pair, Appendix C shows that it must satisfy
rSy =
ǫiy − 1 piy
y
− b − βf (θiy )Sy − δSy + λ[Ey Sy′ − Sy ].
ǫiy
P
(17)
Therefore, the steady-state first-order condition with respect to vacancy is now given by
κ
= λEy q(θiy′ )Sy′ + (1 − λ)q(θiy )Sy .
(1 − δe )(1 − β)
(18)
Rearranging (17) to yield
y=
ǫiy
[b + (r + δ + βf (θiy ) + λ)Sy − λEy Sy′ ] ,
ǫiy − 1
(19)
which is somewhat similar to the key equation (15), yet it now includes a new term resulted
from the aggregate shocks. With a y vector and given Ni is defined by the free entry condition
(16), the system of equations (18) and (19) can be solved recursively to obtain a θi vector
and a S vector. The dynamic version then appears to be new to the original paper by
Ebell and Haefke (2009).
3
Numerical exercises
I first examine the steady-state comparative statics to quantify the effect of a positive shock
on three aggregate variables: the unemployment rate u, the vacancy v and the labor market
tightness θ. Apart from the model parameterization, the standard parameterization used
by literature will also be considered to see how well the model performs relative to the
literature. In what follows, I simulate the model dynamic version and report the summary
21
statistics as in Shimer (2005). I will only solve for the LRGE because the SRGE requires
setting a fixed number of competing firms. That said, the SRGE is only used for analytical
analysis. The LRGE in particular is a more interesting case due to the endogenous degree
of competition involved and when used to study business cycle facts, the mechanism results
in more amplification.
3.1
Steady-state comparative statics
Calibration strategy
The model’s parameters are calibrated on a quarterly basis and the model parameterization
is summarized in Table 1. The deterministic aggregate productivity y = 1 is taken for
normalization. The number of sectors g is also normalized to 1 without any loss of generality,
and consequently the elasticity of substitution across goods σ = 1 and the subscript i will
be deleted hereafter. The standard quarterly interest rate selected is r = 0.012, providing a
consistent annual U.S. real interest rate of around 5%.
The most critical parameter is unemployment income b. As Hagedorn and Manovskii
(2008) show labor market volatility depends mostly on this parameter yet the value they
chose, b = 0.95, is spectacularly implausible. Various values of b have been used as for
example: Shimer (2005) sets b = 0.4, Hall (2005) and Mortensen and Nagypal (2007) 0.73,
Elsby and Michaels (2008) 0.622, and Silva and Toledo (2008) 0.677. In the present model,
I choose b = 0.6 as in Delacroix (2006) who refers to the OECD Unemployment Benefit Entitlements and Replacement Rates database (1997) as a database to evaluate unemployment
income. According to the latter, 30% of the average output is devoted to unemployment
insurance replacement rate and home production also accounts for 30% of average output.
Adding these two sources yields a unemployment income of 0.6 which falls below the average
of the range used by the literature.
To calibrate the parameters in the matching function, I refer to the monthly
unemployment-to-employment flow estimated by Shimer (2005), 0.45, as the job finding rate
f (θ) = µθ1−η which is consistent with the average unemployment duration of 2.2 months.
22
Table 1: Quarterly parameters
Parameter
Meaning
Source
Normalization
y=1
g=1
σ=1
aggregate productivity
number of sectors
elasticity of substitution
Observed parameters
r = 0.012
interest rate
Shimer (2005)
Choosing parameters
η = 0.2
b = 0.6
ρ = 0.068
δ = 0.1
δe = 0.024
unemployment elasticity
unemployment income
entry costs
average separation rate
firm’s survival probability
Anderson and Burgess (2000)
Delacroix (2006)
Djankov et al. (2002)
Shimer (2005)
Ebell and Haefke (2009)
Calibrated parameters
β = 0.01
µ = 0.86
κ = 2.08
workers’ bargaining power
matching efficiency
vacancy cost
Calibration target
f (θ) = 1.35
θ = 1.90
ǫw,p = 0.5
job finding rate
labor market tightness
real wage’s cyclicality
Shimer (2005)
den Haan et al. (2000)
Hagedorn and Manovskii (2008)
The average quarterly unemployment-to-employment flow should be approximately equal to
the probability times the length of time, which is 3×0.45 = 1.35. I then choose the job filling
rate reported by den Haan et al. (2000), q(θ) = 0.71, implying θ = f (θ)/q(θ) = 1.9. I take
the elasticity of unemployment with respect to match estimated by Anderson and Burgess
(2000), η = 0.2, for the matching function. The authors obtain such a value by regressing the
log of number of state-level matches on the log of state unemployment rate, derived from the
Continuous Wage and Benefit History (CWBH) project over the period 1978 to 1984. This
estimate falls outside and far from the range [0.5; 0.7] estimated by Petrongolo and Pissarides
(2001) and the value 0.72 by Shimer (2005). In the literature, the value of η is not sensitive
to model outcomes while in the present model it is. Given η = 0.2, the matching efficiency
parameter is then µ = f (θ)/θ1−η = 0.86.
23
I choose the average quarterly separation rate δ = 0.1 as in Shimer (2005). Following
Ebell and Haefke (2009), I set the probability that a firm will leave the market each quarter
δe = 0.024, which give me the probability of an unproductive match pair δs = (δ − δe )/(1 −
δe ) = 0.08.
I follow the study done by Djankov et al. (2002) to take a fraction ρ of the annual per
capita GDP devoting to entry costs.5 On a quarterly basis, the fraction should approximately
be ρ = 6.8%. Since all the population is available to work, the per capita GDP is then equal
to the aggregate output divided by the labor force, y(1 − u). Hence
ci = ρy(1 − u)
(20)
The literature on matching models often imposes β = η to satisfy the Hosios’ efficient
condition (Hosios (1990)). In this model, β = 0.01 is chosen to replicate the weak cyclicality
of real wage as measured by the elasticity of real wage with respect to aggregate productivity. The observed real wage is calculated by multiplying the labor share with output per
worker (both time series are obtained from the Bureau of Labor Statistics website). The
elasticity reported by Hagedorn and Manovskii (2008) is 0.5 requiring them to set β = 0.052
while in the present model β = 0.01. Notice that within the current debate, a fairly low
value of β may not be very convincing, but it is quite consistent with the range [0.01;0.08]
estimated by Blanchfower et al. (1996) and Hildreth and Oswald (1997). Other study also
finds low β as for example Delacroix (2006) who finds β = 0.045 to represent the 7% of
the relative difference between union and non-union wages. Finally, in order to target the
average tightness θ = 1.9 I need to set κ = 2.05. Whatever the value of κ takes in the model
is meaningless because it is only used to meet a calibration target.
5
In Ebell and Haefke (2009), the initial hiring costs or the total costs of finding just Li employees needed
to start the production process should be added to the entry
h costs.
Including these costs
i will not change
f (θ)
κδs
1+r
κ
my results analytically as Eq. (16) now becomes ci gNi = r+δ
y
−
w
−
−
i
q(θ)
q(θ) δ+f (θ) . It is clear
e
that this latter still results in a negative relation between θ and Ni and the model produces virtually similar
results.
24
Table 2: Elasticities w.r.t to a 1% increase in labor productivity
Variable
Data
Standard model
Monopolistic competition model
u
-3.88
-0.43
-5.1
v
3.68
1.34
2.38
θ
7.56
1.75
7.88
w
0.5
0.97
0.5
Model results
After solving the model equilibrium, I can then proceed to quantify the cyclical variation in
unemployment, vacancy and tightness by calculating the elasticities of these variables with
respect to a 1% increase in labor productivity. Although Shimer (2005) did not report these
elasticities, yet they can be indeed obtained from his summary statistics. As seen in Table
5, the elasticity of θ with respect to y - the most attractive one - equals the relative standard
deviation times the correlation between these two variables: ǫθ,y =
target is ǫθ,y =
0.382
0.020
std(θ)
std(y)
× corr(θ, y). Its
× 0.396 = 7.56 in the sample he constructs.
Table 2 shows the model’s results and compares them with those of the standard matching
model and data. The second column indicates what the data tells us while the third column
presents the MP model’s prediction, revealing very moderate labor market cyclicality. In
the third column, my monopolistic competition model explains the data’s more cyclical
variations in unemployment and tightness, while the vacancy is less well explained.
To compare the model’s performance relative to that of the standard model, in Table 3 I
report how sensitive the model’s results are to variations in β and η. First, I set η = 0.2 as
in the previous exercise and set β = η = 0.2 to satisfy Hosios’ efficient condition. The first
column shows that now unemployment is as countercyclical as in the data, while tightness
is somewhat less procyclical, and that there is improvement in the vacancy’s cyclicality yet
real wage is strongly procyclical. Second, I take the standard unemployment elasticity value
estimated by Petrongolo and Pissarides (2001), η = 0.5, yet set β = 0.02 to target the
25
Table 3: Sensibility analysis
Variable
Monopolistic competition model
β = η = 0.2 β = 0.02, η = 0.5 β = η = 0.5
u
-3.45
-2.00
-1.33
v
2.91
2.35
1.54
θ
6.58
4.44
2.91
w
0.97
0.5
0.99
cyclical behavior of real wage. Finally, I set β = η = 0.5, the same value often used in
the literature, and when compared to the standard MP model this produces three times
more contercyclicality in unemployment, almost twice as much as tightness and for vacancy
is somewhat more procyclical, yet real wage is as procyclical as in the MP model. These
results suggest that labor market cyclicality depends not only on real wage rigidity but also
on the elasticity of unemployment relative to match.
Does intra-firm bargaining matter?
Remember that the present model neglects the IFB mechanism, the reason for which I discussed in Section 2.4. I now perform an exercise similar to that used in the previous section,
and thus determine how the IFB might have differ from both standard parameterization
(β = η = 0.5) and model parameterization (β = 0.01, η = 0.2). The results in Table 4 show
that ignoring IFB results in almost no changes in cyclical variations. That said, complicating
the model would not help improve the results, compared to those obtained using the simple
form of the model described previously.
3.2
Simulation
Given the model’s positive results in term of business cycle analysis, it is essential to examine
how well the model performs in term of labor market volatility. Before doing so, Shimer
26
Table 4: Intra-firm bargaining vs. without IFB
Model parameterization
β = 0.01, η = 0.2
Standard parameterization
β = η = 0.5
IFB
without IFB
IFB
without IFB
u
-5.1
-5.1
-1.30
-1.33
v
2.37
2.38
1.51
1.54
θ
7.88
7.88
2.84
2.91
w
0.5
0.99
0.5
0.99
(2005)’s summary statistics of quarterly U.S. data and his simulation results of the MP model
as reported in Table 5 should be noted, and thanks to them the business cycle facts reported
in Table 2 could be obtained. Compare the second panel to the first panel of Table 5, it thus
becomes clear that the standard model fails to obtain the labor market volatility seen in the
data. In the third panel of this table, I include the results of Hagedorn and Manovskii (2008)
who simulate the standard model with high unemployment income (b=0.95) and this strategy
explains all or even more volatility in the data since
Std.(θ)
Std.(y)
=
0.292
0.013
= 22.46 > 19.10 =
0.382
).
0.02
In the fourth column, Silva and Toledo (2008) simulate the turnover costs model allowing
them to reach 27% of the volatility ( 0.105
= 0.27).
0.382
Stochastic process
To simulate the dynamic version of the present model, consider now the stochastic process
by which labor productivity follows as described in Shimer (2005)
y = b + ex (y ∗ − b),
where y ∗ measures the long-run average productivity, which is equal to 1, and x is an
Ornstain-Uhlenbeck process with persistence parameter γ and standard deviation ξ. The
27
Table 5: Summary statistics: Data vs. literature
u
v
θ
y
Data: 1951I-2003IV
Standard deviation
Autocorrelation
Correlation matrix
u
v
θ
y
0.190
0.936
0.202
0.940
0.382
0.941
0.020
0.878
1
-0.894
1
-0.971
0.975
1
-0.408
0.364
0.396
1
Standard MP model
Standard deviation
Autocorrelation
Correlation matrix
u
v
θ
y
0.009
0.939
0.027
0.835
0.035
0.878
0.020
0.878
1
-0.927
1
-0.958
0.996
1
-0.958
0.995
0.999
1
Hagedorn and Manovskii (2008)
Standard deviation
Autocorrelation
Correlation matrix
u
v
θ
y
a
0.145
0.830
0.169
0.575
0.292
0.751
0.013
0.765
1
-0.724
1
-0.916
0.940
1
-0.892
0.904
0.967
1
Silva and Toledo (2008)
Standard deviation
Autocorrelation
Correlation matrix
u
v
θ
y
0.049
0.954
0.062
0.738
0.105
0.883
0.020
0.883
1
-0.808
1
-0.938
0.962
1
-0.959
0.938
0.995
1
Note: Hagedorn and Manovskii (2008) use the standard smoothing parameter for quarterly data (1600)
which is much less smooth than the one used by Shimer (2005) and Silva and Toledo (2008) (105 ) so that
why they obtain lower standard deviation but the relative standard deviation is unchanged.
a
28
realization of x takes place on a discrete grid
x ∈ {−n∆. − (n − 1)∆, ..., 0, ..., (n − 1)∆, n∆},
where ∆ > 0 is the step size and n ≥ 1 is an integer number to ensure that the number of
grid points is at least 3. When a Poisson shock λ hits the economy, x changes to a new level
x′ by one grid point where

 x + ∆ with probability
′
x =
 x − ∆ with probability
1
2
1
2
x
n∆
,
x
1 + n∆
.
1−
That means the chance of moving down to a bad state is higher than the chance of moving
√
up to a good state. To complete the process’ characteristics, define γ = λ/n and ξ = λ∆.
These two parameters also characterize the behavior of y and setting γ = 0.004 and ξ = 0.025
will pin down the autocorrelation and standard deviation of y, as shown in the first part of
Table 5.
I solve the system of equations (18) and (19) recursively and obtain a vector of θ. I then
simulate the model by starting with an initial θ in a given aggregate state and compute the
initial unemployment rate given by the Beveridge curve (13). In the following step, 1212
levels of unemployment rate are randomly generated and the first 1000 observations are discarded to eliminate any initial effect. The sample remains at 212 observations, corresponding
to the number of quarters from 1951 to 2003. I log the data generated by the model and and
detrend it, using an HP filter with a smoothing parameter of 105 . This procedure is then
repeated 10,000 times. I will only use the model parameterization reported in Table 1 and
will not consider again the sensibility analysis. The IFB is however compared to the study
by Krause and Lubik (2007).
29
Table 6: Simulation results for β = 0.01, η = 0.2
Standard deviation
Autocorrelation
Correlation matrix
u
v
θ
y
u
v
θ
y
0.134
0.915
0.118
0.498
0.225
0.802
0.020
0.878
1
-0.651
1
-0.918
0.888
1
-0.729
0.657
0.754
1
Results
Table 6 shows the simulation results I obtained using the model’s dynamic version. In terms
of the labor market volatility, the model’s vacancy and tightness explain up to 60% of the
data while unemployment explains 70% of the data. In terms of quarterly autocorrelation
and with the exception of unemployment and tightness, vacancy does not fit the data well.
The correlation between unemployment and vacancy - the Beveridge curve is weaker | −
0.6| < | − 0.89|, yet the average correlation between model variables and labor productivity,
0.7, is closer to the data. Neither the standard model nor Hagedorn and Manovskii (2008)
nor Silva and Toledo (2008) could however improve the correlation matrix results as labor
market variables are strongly correlated with with y, 0.95 on average, while the correlation
coefficients provided by the data are relatively small, 0.4. This result, although it remains
unexplained, still represents a significant improvement for the model. The simulation results
correspond very well with the steady-state comparative statistics as does the elasticity value
of θi with respect to y derived from Table 6, which becomes ǫθi ,y =
0.225
0.020
× 0.754 = 8.48. This
value is somewhat higher than the 7.88 appearing in Table 3. Note also that the correlation
value between θi and y (0.754) derived from the modelis higher than the 0.396 for the data,
thus explaining why the model provides better a better approximation of the data in terms
of cyclical variation than it does for labor market volatility. In other words, if the correlation
between θi and y derived from the model were equal to 0.396, then the model would fit the
data’s value of labor market volatility very well.
30
Table 7: The effect of intra-firm bargaining
u
v
θ
y
Model with IFB
Standard deviation
Autocorrelation
Correlation matrix
u
v
θ
y
0.131
0.914
0.118
0.501
0.222
0.799
0.020
0.878
1
-0.654
1
-0.916
0.893
1
-0.723
0.656
0.748
1
Krause and Lubik (2007): RBC with IFB
Standard deviation
0.78
0.95
1.55
1.62
Krause and Lubik (2007): RBC without IFB
Standard deviation
Correlation matrix
(similar for both)
u
v
θ
y
0.68
0.84
1.36
1.62
1
-0.58
1
-0.85
0.91
1
-0.86
0.91
0.99
1
Does intra-firm bargaining matter?
Table 7 reports both the simulation results as well as those found inKrause and Lubik (2007),
which quantify the effect that IFB has on the dynamics and volatility of a real business cycle
model involving large firms and search frictions. They find that when comparing the RBC
model with IFB, the RBC without IFB produced roughly 10% less volatility, yet the relative
standard deviation between labor market variables and productivity shock is quite close to
the standard MP model, and both versions produce virtually the same correlation matrix.
In my simulation, I find that adding IFB to the model does not improve the dynamics and
volatility of labor market variables, a result that is also consistent with the steady-state
comparative statics calculated in Section 3.1.
31
4
Conclusion
In this research a simple matching model with monopolistic competition has been developed,
with the main purpose being to solve the unemployment volatility problem. The model
described in this paper appears to be simpler than the original one of Ebell and Haefke
(2009) as it has shown that the intra-firm bargaining to reflect the overhiring incentive is
negligible yet it has been extended by adding a new feature that allows stochastic aggregate
shocks to be included. Although the intra-firm bargaining mechanism makes sense in a
multiple-worker environment and in monopolistic competition, generally seems to be far too
complicated. From the analytical point of view, the monopolistic competition mechanism
serves an efficient method for understanding the extent to which labor market variables react
to aggregate shocks. As such, the key point is the long-run number of firms that enter into
business or the model’s degree of competition making markup small. This means that when
the goods market is very competitive, the economy’s aggregate number of vacant positions
available in the labor market increases substantially, simply due to the large number of firms
entering into profitable sectors. From the quantitative point of view, the real advantage of
this mechanism is its significant degree of amplification in both the cyclical variation study
and labor market volatility. The only problem that I am concerned is the low calibrated value
of workers’ bargaining power β = 0.01, yet literature on matching models has never paid any
attention to the cyclical behavior of real wage which is weak in the data and relatively high in
models. Real wage rigidity is effectively a very intuitive reason to gain further amplification
as argued by Shimer.
32
APPENDIX
A
Proof of equation (10)
Omitting the subscript i, denoting the firm’s output as a function of L by F (L) =
p(Y (L))
Y
P
(L)
and applying the envelope theorem to the firm’s value function to get
1
∂V (L′ )
∂V (L)
′
=
F (L) − w + (1 − δe )(1 − δs )
.
∂L
1+r
∂L′
In the steady state, the required condition is that L′ = L so that
∂V (L)
1
=
[F ′ (L) − w],
∂L
r+δ
where F ′ (L) =
∂p(Y )/P
∂L
Y +
p(Y ) ∂Y
P ∂L
=
p(Y ) ǫ−1
y ǫ .
P
(21)
Inserting (21) into the first-order condition
with respect to vacancy yields the monopolistic price (10).
B
Proof of equation (11)
Since the Nash solution provides the usual equality β ∂V∂L(L) = (1 − β)(W − U) and given (9),
the flow value of a unemployed worker is given by
rU = b + f (θ)
β
κ
.
1 − β (1 − δe )q(θ)
(22)
Combining the last equation with (10) and (21) concludes the proof of the wage equation
(11).
33
C
Proof of equation (17)
Following the same procedure as in the case of deterministic productivity, i.e, applying the
envelope theorem to the dynamic version of the firm’s value function to get
∂Vy (L)
1
∂Vy′ (L′ )
∂Vy (L′ )
′
=
F (L) − w + (1 − δe )(1 − δs ) λEy
+ (1 − λ
) .
∂L
1+r y
∂L′
∂L′
For the sake of simplification, suppose that λδ is relatively small as compared to λ, then
∂Vy (L)
∂Vy (L)
∂Vy′ (L) ∂Vy (L)
′
= Fy (L) − wy − δ
+ λ Ey
−
.
r
∂L
∂L
∂L
∂L
Given the flow values of a unemployed worker and an employed worker are expressed in
section (2.5) and the Nash sharing rule
∂Vy (L)
∂L
= (1 − β)Sy , the proof of equation (17) is
therefore done.
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