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Working Paper Series
Technical Appendix for " Frictional Wage
Dispersion in Search Models: A
Quantitative Assessment"
WP 06-08
Andreas Hornstein
Federal Reserve Bank of Richmond
Per Krusell
Princeton University
Giovanni L. Violante
New York University
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Technical Appendix for
“Frictional Wage Dispersion in Search Models:
A Quantitative Assessment” ∗
Andreas Hornstein†
Per Krusell‡
Giovanni L. Violante§
Federal Reserve Bank of Richmond Working Paper 2006-08
Abstract
In this Technical Appendix to Hornstein, Krusell, and Violante (2006) (HKV,
2006, hereafter) we provide a detailed characterization of the search model with (1)
wage shocks during employment and (2) on-the-job search outlined in Sections 6 and
7 of that paper, and we derive all of the results that are only stated in HKV (2006).
In particular, we derive the expressions for our preferred measure of frictional wage
inequality: the ratio of average wages to the reservation wage, or, the ‘mean-min’
wage ratio.
Keywords: labor market, wage inequality, search frictions, job search
JEL Classification: D83, E24, J31, J41, J63, J64
∗
The views expressed here are those of the authors and do not necessarily reflect those of the
Federal Reserve Bank of Richmond or the Federal Reserve System. Our e-mail addresses are:
andreas.hornstein@rich.frb.org, pkrusell@princeton.edu, and gianluca.violante@nyu.edu
†
Federal Reserve Bank of Richmond
‡
Princeton University, CEPR, and NBER
§
New York University and CEPR
1
Wage shocks during employment
In this section, we characterize the equilibrium of the search model with wage shocks
while employed when wage shocks when employed and unemployed come from the same
distribution. We first state the discrete-time approximation of the search model for a
fixed and finite length of the time period. We then derive the continuous-time representation as the limit of the discrete-time approximation when the length of the time period
becomes arbitrarily small. We show that the Bellman equations for the employment and
unemployment values for the continuous- and discrete-time version are the same. Our
derivation of the mean-min ratio for wage inequality is therefore independent of the time
representation. We then show that in the discrete-time version the first-order autocorrelation coefficient of wages is one minus the arrival rate of wage changes. Finally, we
consider a variation of the baseline model where wage shocks when employed come from
a different distribution than wage shocks when unemployed. In particular, we study the
Mortensen-Pissarides (1994) environment where unemployed workers on meeting a job
always receive the highest wage.
We describe the parameters of the environment in terms of the continuous-time framework. The economy is populated by ex-ante equal, risk-neutral, infinitely lived individuals
who discount the future at rate r. Unemployed agents receive job offers at the instantaneous rate λu , and the wage of employed agents changes at the instantaneous rate δ.
Conditionally on receiving a wage change, the wage is drawn from a well-behaved distribution function F (w) with upper support w max . Draws are i.i.d. over time and across
agents. Note that employed and unemployed agents sample from the same wage distribution. If a job offer w is accepted, the worker is paid a wage w, until the next wage-change
event occurs. While unemployed, the worker receives a utility flow b which includes unemployment benefits and a value of leisure and home production, net of search costs. We
only consider steady-state allocations.
1
1.1
Discrete time versus continuous time
The discrete-time approximations of the Bellman equations for the value of employment,
W (w), and unemployment, U , are
Z
−r∆
W (w) = w∆ + e
δ∆ max {W (z), U } dF (z) + (1 − δ∆) W (w) ,
Z
−r∆
U = b∆ + e
λu ∆ max {W (z), U } dF (z) + (1 − λu ∆) U ,
(1)
(2)
where ∆ is the length of the time interval, and λu ∆ (δ∆) is the probability that an
(un)employed worker receives a wage offer at the end of the interval ∆. Using the definition
of the reservation wage, W (w ∗ ) = U , and rearranging terms the value equations can be
rewritten as
Z wmax
−r∆
−r∆
∗
1−e
W (w) = w∆ + e
δ∆
[W (z) − W (w)] dF (z) − δ∆F (w ) [W (w) − U ] ,
w∗
Z wmax
−r∆
−r∆
1−e
U = b∆ + e
λu ∆
[W (z) − U ] dF (z)
w∗
Dividing by the length of the time interval and taking the limit as ∆ → 0 we get the
continuous-time Bellman equations
Z wmax
rW (w) = w + δ
[W (z) − W (w)] dF (z) − δF (w ∗ ) [W (w) − U ] ,
∗
w
Z wmax
rU = b + λu
[W (z) − U ] dF (z) .
(3)
(4)
w∗
Rather than studying the continuous-time limit of the search model with wage changes,
we can just use the discrete-time approximation and consider a unit length interval, ∆ = 1.
In this case we get the following expressions for the value functions of being employed
and unemployed:
Z wmax
∗
(1 − β) W (w) = w + β δ
[W (z) − W (w)] dF (z) − δF (w ) [W (w) − U ]
w∗
Z wmax
(1 − β) U = b + βλu
[W (z) − U ] dF (z) ,
w∗
where β ≡ e
−r
≡ 1/ (1 + r̄). Note that we can rewrite these discrete-time value equations
as
r̄ W̄ (w) = w + δ
Z
r̄Ū = b + λu
w max
w∗
Z wmax
w∗
W̄ (z) − W̄ (w) dF (z) − δF (w ∗ ) W̄ (w) − Ū
W̄ (z) − Ū dF (z) ,
2
(5)
(6)
where W̄ ≡ W/ (1 + r̄) and Ū ≡ U/ (1 + r̄). Expressions (5) and (6) are formally equivalent to the expressions (3) and (4) for the continuous-time value functions. Therefore,
the results for the mean-min ratio also apply for the discrete-time version of the paper.
1.2
The reservation wage
As a first step towards deriving the mean-min wage ratio for the continuous-time model
we characterize the reservation wage. For this purpose, evaluate the employment value
expression (3) at w ∗ and use the definition of the reservation wage, W (w ∗ ) = U , to get
∗
rU = w + δ
Z
w max
[W (z) − W (w ∗ )] dF (z) .
w∗
Now substitute the unemployment value expression (4) for the left-hand side and solve
for the reservation wage
∗
w = b + (λu − δ)
Z
w max
[W (z) − W (w ∗ )] dF (z) .
(7)
w∗
Integration by parts on the right-hand side yields
Z wmax
Z wmax
w max
∗
∗
[W (z) − W (w )] dF (z) = [(W (z) − W (w )) F (z)]w∗ −
W 0 (z) F (z) dz
∗
∗
w
w
Z wmax
= W (w max ) − W (w ∗ ) −
W 0 (z) F (z) dz
w∗
Z wmax
=
W 0 (z) [1 − F (z)] dz.
(8)
w∗
From the employment value expression (3) it follows that
W 0 (w) =
1
.
r+δ
Hence the reservation wage expression is
Z max
λu − δ w
∗
w = b+
[1 − F (z)] dz
r + δ w∗
Z max
(λu − δ) [1 − F ∗ ] w
1
F (z)
= b+
−
dz
r+δ
1 − F∗ 1 − F∗
w∗
with F ∗ = F (w ∗ ).
3
(9)
(10)
1.3
The equilibrium wage distribution
We now construct the equilibrium wage distribution G (w) implied by the interaction of
the wage-offer distribution and the reservation wage. The measure of agents with wage
below w is (1 − u) G (w) . Agents leave this stock because their wage changes and their
new wage is either less than the reservation wage or higher than the current wage. Agents
enter this stock if they were unemployed and receive an acceptable wage offer below w,
or if they were employed at wage above w and are forced to accept a lower wage; hence
(1 − u) G (w) δ {F (w ∗ ) + [1 − F (w)]}
= {uλu + (1 − u) [1 − G (w)] δ} [F (w) − F (w ∗ )] .
(11)
We can solve this expression for the equilibrium wage distribution as a function of the
wage offer distribution:
λu u
G (w) =
+ 1 [F (w) − F (w ∗ )] .
δ (1 − u)
(12)
In steady state, the inflows and outflows from employment balance:
(1 − u) δF (w ∗ ) = uλu [1 − F (w ∗ )] .
Using the expression for steady-state employment in (12) we get
G (w) =
F (w) − F (w ∗ )
.
1 − F (w ∗ )
(13)
Thus, the equilibrium wage distribution with and without wage shocks during employment
are the same, namely the wage-offer distribution truncated at the reservation wage.
1.4
The mean-min ratio
Based on the equilibrium wage distribution we can calculate the average wage of employed
workers as
w̄ =
Z
w max
w∗
dF (w)
w max − w ∗ F ∗
w
=
−
1 − F∗
1 − F∗
Z
w max
w∗
F (z)
dz.
1 − F∗
(14)
Solving the average-wage expression (14) for the right-hand side integral term and substituting this term for the corresponding integral in the reservation-wage expression (10)
4
yields
w
∗
Z wmax
1
w max − w ∗ F ∗
(λu − δ) [1 − F ∗ ]
= b+
dz + w̄ −
r+δ
1 − F∗
1 − F∗
w∗
(λu − δ) [1 − F ∗ ]
= b+
[w̄ − w ∗ ] .
r+δ
(15)
Using the definition of the replacement rate, b = ρw̄, we can solve equation (15) for the
reservation wage and obtain an expression for the mean-min ratio, that is, the ratio of
average wages to the reservation wage,
w̄
=
w∗
(λu −δ)[1−F (w ∗ )]
r+δ
(λu −δ)[1−F (w ∗ )]
r+δ
+1
+ρ
=
λ∗u −δ+σ ∗
r+δ
λ∗u −δ+σ ∗
r+δ
+1
+ρ
,
(16)
with λ∗u ≡ (1 − F ∗ ) λu and σ ∗ ≡ δF ∗ . This is equation (13) in Section 6 of HKV (2006).
Note that as δ goes to infinity
1.5
w̄
w∗
goes to 1/ρ.
Wage persistence in the discrete time model
It is straightforward to show that the equilibrium wage distribution for the discrete-time
and the continuous-time versions of the model are the same; we now work with the
former. We need the expected value of the cross-product of today’s and tomorrow’s wage,
conditional on being employed in both periods, to calculate the autocorrelation coefficient.
We proceed in two steps: first, we obtain the value conditional on today’s wage, and then
we integrate over today’s wage to get the unconditional expectation.
E [w 0 w|w] = (1 − δ) w 2 + δwE [w̃|w̃ ≥ w ∗ ]
= (1 − δ) w 2 + δww̄
E [w 0 w] = (1 − δ) E w 2 + δ w̄ 2
We can now define the first-order autocorrelation coefficient as
(1 − δ) E [w 2 ] + δ w̄ 2 − w̄ 2
V ar (w)
(1 − δ) (E [w 2 ] − w̄ 2 )
=
V ar (w)
= 1−δ
ρ =
5
1.6
The mean-min ratio for a Mortensen-Pissarides (1994) environment
Suppose now that an unemployed worker who receives a wage offer always receives the
highest wage w max , whereas wage changes of employed workers continue to be drawn from
the distribution F . We will show that the mean-min ratio that we previously derived for
our baseline model, equation (16), represents an upper bound for the mean-min ratio in
this Mortensen-Pissarides (1994) environment.
The value function equation for an employed worker, (3), remains unchanged, but the
value function equation for an unemployed worker is now
rU = b + λu [W (w max ) − U ] .
(17)
Following the same steps as in Section 1.2 we derive the modified expression for the
reservation wage:
λu − δ max
δ
w =b+
[w
− w∗] +
r+δ
r+δ
∗
Z
w max
F (z) dz.
(18)
w∗
The modified steady-state expression characterizing the equilibrium wage distribution is
now
(1 − u) G (w) δ {F (w ∗ ) + [1 − F (w)]}
(19)
= (1 − u) [1 − G (w)] δ [F (w) − F (w ∗ )] for w < w max .
Note that there are no inflows from the pool of unemployed since all unemployed workers
who receive wage offers receive the highest wage. Thus, the equilibrium wage distribution
for w < w max is
G (w) = F (w) − F (w ∗ ) .
(20)
Since all unemployed workers receive the highest wage with probability one there is now
a mass point at w = w max , that is, the cumulative density function is discontinuous at
w max .
Integrating the wage with respect to the equilibrium wage distribution then yields the
average wage
Z wmax
Z
max
∗
max
∗
max
∗
w̄ =
wdF (w) + w F (w ) = w
+ F (w ) (w
− w )−
w∗
6
w max
F (w) dw. (21)
w∗
Solving the average-wage expression (21) for the right-hand side integral term and substituting this term for the corresponding integral in the reservation-wage expression (18)
yields
λu − (1 − F ∗ ) δ
δ
λu + δF ∗ max
∗
1+
w =
ρ−
w̄ +
w
r+δ
r+δ
r+δ
λu − δ (1 − F ∗ )
w̄.
>
ρ+
r+δ
(22)
Note that the last inequality implies that the mean-min ratio for this setup is bounded
above by that of the baseline economy with wage shocks given in (16).
2
On-the-job search
We describe a general version of the on-the-job search model that includes forced jobto-job mobility and wage shocks (the models in Section 7 of HKV, 2006). The Bellman
equations for the employment and unemployment values are
Z
rW (w) = w + λw max {W (z) − W (w) , 0} dF (z) − σ [W (w) − U ]
Z
+φ [max {W (z) , U } − W (w)] dF (z)
Z
rU = b + λu max {W (z) − U, 0} dF (z)
The basic on-the-job search model in HKV (2006), Section 7.1, assumes that a worker
receives outside wage offers at a rate λw . Without loss of generality, and motivated by
what equilibrium firm behavior would dictate, we assume that the wage-offer distribution
is such that unemployed workers accept all wage offers: F (w ∗ ) = 0. A worker can always
reject a wage offer and keep the current wage. The worker may lose the current job at an
exogenous separation rate, σ.
On-the-job search with forced job-to-job mobility in HKV (2006), Section 7.2, assumes
that for some wage offers, the worker just has to take the offer, even if the new job pays
less than the current job. Forced job mobility is reflected in the parameter φ > 0. In the
basic on-the-job search model φ = 0. On-the-job search with wage shocks in HKV (2006),
Section 7.2, is essentially the same as forced job-to-job mobility. The only difference is in
its implications for observable transitions. The arrival of a type φ wage offer now results
7
in a wage cut on the existing job, rather than a separation and transfer to a lower-paying
job.
Finally, we analyze the on-the-job search model of Christensen et al. (2005) where
search effort is endogenous, as discussed in HKV (2006), Section 7.1.
2.1
The reservation wage
Workers continue to follow reservation-wage strategies and the Bellman equations can be
rewritten as
Z
w max
rW (w) = w + λw
[W (z) − W (w)] dF (z) − σ [W (w) − U ]
w
Z wmax
+φ
[W (z) − W (w)] dF (z)
w∗
Z wmax
= w + (λw + φ)
[W (z) − W (w)] dF (z) − σ [W (w) − U ]
w
Z w
+φ
[W (w) − W (z)] dF (z)
w∗
Z wmax
rU = b + λu
[W (z) − U ] dF (z)
(23)
(24)
w∗
Evaluate the employment value equation (23) at w ∗ , using the reservation wage property, W (w ∗ ) = U , and the unemployment value expression (24) to obtain
Z wmax
∗
rU = w + (λw + φ)
[W (z) − W (w ∗ )] dF (z)
∗
w
Z wmax
= b + λu
[W (z) − W (w ∗ )] dF (z) .
w∗
We can solve this expression for the reservation wage
Z wmax
∗
w = b + (λu − λw − φ)
[W (z) − W (w ∗ )] dF (z) .
(25)
w∗
As with equation (7), we can integrate the right-hand side integral by parts, as in (8),
and get the reservation-wage equation
∗
w = b + (λu − λw − φ)
Z
w max
W 0 (z) [1 − F (z)] dz.
(26)
w∗
Note that differentiating the employment value equation (23) with respect to the current
wage yields
W 0 (w) =
1
.
r + σ + φ + λw [1 − F (w)]
8
(27)
Substituting (27) in (26) we can rewrite the reservation wage as
∗
w = b + (λu − λw − φ)
2.2
Z
w max
w∗
1 − F (z)
dz.
r + σ + φ + λw [1 − F (z)]
(28)
The equilibrium wage distribution
We now construct the equilibrium wage distribution G (w) implied by the interaction of
the wage-offer distribution and the reservation wage.
The measure of agents with wage below w is (1 − u) G (w) . Agents leave this stock
because (1) they get separated at rate σ, (2) they receive an outside offer which they
accept at rate λw [1 − F (w)], or (3) they are forced to leave at rate φ but are lucky
enough to get an offer above w. Workers enter this stock if (1) they were unemployed and
receive a wage offer below w, or (2) they were employed at wage above w and are forced
to accept a lower wage. In a steady state the inflows and outflows balance:
(1 − u) G (w) {σ + (λw + φ) [1 − F (w)]}
(29)
= uλu F (w) + φ (1 − u) [1 − G (w)] F (w) .
We can solve this expression for the equilibrium wage distribution as a function of the
wage offer distribution:
G (w) =
λu u + φ (1 − u)
F (w)
·
1−u
σ + φ + λw [1 − F (w)]
(30)
In steady state, if all the job offers are above w ∗ so that F (w ∗ ) = 0,
uλu = (1 − u) σ.
Hence
G (w) =
(σ + φ) F (w)
σ + φ + λw [1 − F (w)]
(31)
and
σ + φ + λw
[1 − F (w)]
σ + φ + λw [1 − F (w)]
r + σ + φ + λw
'
[1 − F (w)]
r + σ + φ + λw [1 − F (w)]
1 − G (w) =
9
(32)
2.3
The mean-min ratio
The average wage is
Z
w̄ =
w max
max
[wG (w)]w
w∗
Z
w max
wdG (z) =
−
w∗
Z wmax
= w max −
G (z) dz
w∗
Z wmax
max
∗
∗
= [w
−w ]+w −
G (z) dz
w∗
Z wmax
∗
= w +
[1 − G (z)] dz.
w∗
G (z) dz
(33)
w∗
Solve the wage distribution expression (32) for 1 − F and use it in the reservation-wage
expression (28) to get
λu − λ w − φ
w 'b+
r + σ + φ + λw
∗
Z
w max
[1 − G (z)] dz.
w∗
Finally substituting for the integral term from the average-wage equation (33) we can
solve for the mean-min ratio:
λu − λ w − φ
(w̄ − w ∗ )
r + σ + φ + λw
λu −λw −φ
+1
r+σ+φ+λw
.
λu −λw −φ
+ρ
r+σ+φ+λw
w ∗ ' ρw̄ +
Mm '
⇒
(34)
Equation (34) corresponds to equations (17) and (20) in Section 7 of HKV (2006).
2.4
Turnover rates in the basic on-the-job search model
In the basic on-the-job search model there are exogenous separations, σ > 0, but no forced
wage changes, φ = 0.
The average completed job tenure in the model is
Z wmax
dG (w)
τ=
.
σ + λw [1 − F (w)]
w∗
(35)
Recall the equilibrium wage distribution from (31) and solve that expression for the wageoffer distribution
1 − F (w) =
σ [1 − G (w)]
.
σ + λw G (w)
10
Substitute this expression in the expression for average job tenure (35) and we get
τ =
=
Note that
Thus
Z
Z
dG (w)
σ+
w∗
1
σ (λw + σ)
w max
w∗
w max
=
w max
[σ + λw G (w)] dG (w)
σ (λw + σ)
G (w) dG (w) .
Z
σ[1−G(w)]
λw σ+λ
w∗
w G(w)
Z wmax
σ + λw
w∗
wmax
G (w) dG (w) = G (w) w∗ −
Z
2
Z
(36)
w max
G (w) dG (w) .
w∗
w max
G (w) dG (w) = 1/2.
w∗
Hence, the average job tenure is
σ + λw /2
τ =
∈
σ (λw + σ)
1 1
,
2σ σ
.
(37)
The (total) separation rate for employed workers, the sum of exogenous separations
and job-to-job transitions, is
sep = σ + λw
Z
w max
[1 − F (w)] dG (w) .
(38)
w∗
Hence, the share of separations attributable to job-to-job separations is
R wmax
λw w∗ [1 − F (w)] dG (w)
jjsep
=
.
R wmax
sep
σ + λw ∗ [1 − F (w)] dG (w)
(39)
w
Following Nagypal (2005), and integrating jjsep by parts, yields
jjsep = λw
Z
w max
[1 − F (w)] dG (w)
Z wmax
= λw − λw
F (w) dG (w)
w∗
w∗
max
(w) G (w)]w
w∗
Z
w max
= λw − λw [F
+ λw
G (w) dF (w)
w∗
Z wmax
= λw
G (w) dF (w)
w∗
Z wmax
F (w)
= λw σ
dF (w) .
σ + λw [1 − F (w)]
w∗
11
(40)
The last step uses equation (31) for G. Now change the variable of integration to z =
F (w), and we get
Z
1
0
λw {z}10 + (λw + σ) {log [σ + λw (1 − z)]}10
z
dz = −
σ + λw [1 − z]
λ2w
λw + (λw + σ) (log σ − log (σ + λw ))
= −
λ2w
w
(λw + σ) log σ+λ
1
σ
=
−
.
λ2w
λw
(41)
Hence, the job-to-job separation rate is
σ (λw + σ) log
jjsep =
λw
σ+λw
σ
−σ
(42)
and the total separation rate is
σ (λw + σ) log
sep = jjsep + σ =
λw
σ+λw
σ
.
(43)
The share of separations attributable to job-to-job transitions is then
jjsep
λw / (σ + λw )
=1−
.
sep
log [(σ + λw ) /σ]
2.5
(44)
Wage cuts in the model with on-the-job wage changes
In this version of the model, not only does a worker receive outside wage offers at the rate
λw , but the wage on the current job is also changing with arrival rate φ. If the new wage
is less than the current wage, the worker has to accept a pay cut.
The average rate at which workers are forced to take a wage cut is
Z wmax
f cut = φ
F (w) dG (w)
w∗
Z wmax
w max
= φ [F (w) G (w)]w∗ − φ
G (w) dF (w)
w∗
Z wmax
= φ−φ
G (w) dF (w) .
(45)
w∗
Now substitute for the equilibrium wage distribution from (31) and we get
Z wmax
F (w)
dF (w) ,
f cut = φ − φσ̂
σ̂ + λw [1 − F (w)]
w∗
12
(46)
where σ̂ ≡ σ + φ. We have previously derived the right-hand side integral: expressions
(40) and (41). Use equation (41) for the integral and we get
"
#
w
(λw + σ̂) log σ̂+λ
1
σ̂
f cut = φ − φσ̂
−
λ2w
λw
"
#
w
σ̂ (λw + σ̂) log σ̂+λ
σ̂
σ̂
= φ 1+
−
.
λw
λ2w
(47)
With f cut being the average arrival rate of a wage cut, the fraction of employed
workers that receive a wage cut in a unit time period is 1 − exp (−f cut).
2.6
Endogenous effort choice in the model with on-the-job search
In the Christensen et al. (2005) model, the utility cost to attain a contact rate λ is c(λ),
with c0 > 0 and c00 > 0. Thus, optimal search choice will deliver a policy function λ(w),
where we use the notation λ∗ ≡ λ(w ∗ ).
The first-order condition at the reservation wage w ∗ reads
Z wmax
1 − F (z)
0 ∗
c (λ ) =
dz.
r + σ + λ(z)(1 − F (z))
w∗
In steady state, flows for the workers employed at wage w or lower satisfy
Z w
(1 − u)G(w)σ + (1 − u) [1 − F (w)]
λ(z)dG(z) = uλ∗ F (w)
(48)
(49)
w∗
and for the unemployment pool the flows satisfy the relation
uλ∗ = (1 − u)σ.
(50)
Combining the last two equations we obtain
σF (w) = G(w)σ + [1 − F (w)]
Z
w
λ(z)dG(z).
w∗
Because λ(z) is decreasing, we have
σF (w) ≤ G(w) [σ + λ∗ (1 − F (w))] ,
so
G(w) ≥
σF (w)
,
σ + λ∗ (1 − F (w))
13
or
1 − G(w) ≤
(σ + λ∗ )(1 − F (w))
.
σ + λ∗ (1 − F (w))
This implies
1 − F (w)
1 − G(w)
σ + λ∗ (1 − F (w))
≥
.
r + σ + λ∗ (1 − F (w))
r + σ + λ∗ (1 − F (w))
σ + λ∗
(51)
Since r is small relative to σ, we have
r + σ + λ∗ (1 − F (w))
σ + λ∗ (1 − F (w))
'
,
σ + λ∗
r + σ + λ∗
and hence, from the inequality in (51)
1 − G(w)
1 − F (w)
≥
∗
r + σ + λ (1 − F (w))
r + σ + λ∗
(52)
almost holds, for all w. Going back to the FOC for effort (48), using the fact that λ(z) is
decreasing, we have
0
∗
c (λ ) ≥
Z
w max
w∗
1 − F (z)
dz,
r + σ + λ∗ (1 − F (z))
and then, using the inequality in (52), we obtain that an approximate inequality for c0 (λ∗ )
is given by
0
∗
c (λ ) ≥
Z
w max
w∗
1 − G(z)
dz =
r + σ + λ∗
R wmax
w∗
(1 − G(z))dz
w̄ − w ∗
=
.
r + σ + λ∗
r + σ + λ∗
Assuming, as done in Christensen et al. (2005), that c(λ) = λγ , we have that c0 (λ) =
γc(λ)/λ so that
c(λ∗ ) ≥
λ∗ (w̄ − w ∗ )
γ(r + σ + λ∗ )
which yields the lower bound for search effort during unemployment discussed in the main
text.
14
References
[1] Christensen, B.J., R. Lentz, D.T. Mortensen, G.R. Neumann, and A. Werwatz (2005).
On-the-Job Search and the Wage Distribution, Journal of Labor Economics, vol. 25(1),
31-58.
[2] Hornstein, Andreas, Per Krusell, and Giovanni L. Violante (2006). Frictional Wage
Dispersion in Search Models: A Quantitative Assessment, Federal Reserve Bank of
Richmond Working Paper 2006-07, available at
http://www.richmondfed.org/publications/economic research/working papers/index.cfm.
[3] Mortensen and Pissarides (1994). Job Creation and Job Destruction in the Theory of
Unemployment, Review of Economic Studies, vol. 61(3), 397-415.
[4] Nagypal, E. (2005). Worker Reallocation Over the Business Cycle: the Importance of
Job-to-Job Transitions, mimeo, Northwestern University.
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