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Working Paper Series
Productivity Insurance: The Role of
Unemployment Benefits in a Multi-Sector
Model
WP 13-11
David L. Fuller
Concordia University and CIREQ
Marianna Kudlyak
Federal Reserve Bank of Richmond
Damba Lkhagvasuren
Concordia University and CIREQ
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Productivity Insurance: The Role of Unemployment
Benefits in a Multi-Sector Model∗
David L. Fuller†
Concordia University and CIREQ
Marianna Kudlyak‡
The Federal Reserve Bank of Richmond
Damba Lkhagvasuren§
Concordia University and CIREQ
Working Paper No. 13-11
August 8, 2013
Abstract
We construct a multi-sector search and matching model where the unemployed
receive idiosyncratic productivity shocks that make working in certain sectors more
productive than in the others. Agents must decide which sector to search in and face
moving costs when leaving their current sector for another. In this environment, unemployment is associated with an additional risk: low future wages if mobility costs
preclude search in the appropriate sector. This introduces a new role for unemployment
benefits – productivity insurance while unemployed. Analytically, we characterize two
competing effects of benefits on productivity, a moral hazard effect and a consumption effect. In a stylized quantitative analysis, we show that the consumption effect
dominates, so that unemployment benefits increase per-worker productivity. We also
analyze the welfare-maximizing benefit level and find that it decreases as moving costs
increase.
Keywords: unemployment insurance, search, mobility, productivity
JEL classification: J62, J63, J64, J65
∗
We thank Mark Bils, Yongsung Chang, Daniel Coen-Piranni and participants of the 2012 Alan Stockman
Conference at the University of Rochester for helpful comments. The views expressed here are those of the
authors and do not reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve System.
†
Department of Economics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec,
H3G 1M8 Canada; E-mail: david.fuller@concordia.ca.
‡
Research Department, The Federal Reserve Bank of Richmond, 701 E Byrd St., Richmond, VA, 23219.
E-mail: marianna.kudlyak@rich.frb.org.
§
Department of Economics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec,
H3G 1M8 Canada; E-mail: damba.lkhagvasuren@concordia.ca.
1
1
Introduction
The existing literature on the provision of unemployment insurance (hereafter UI) has focused
on the role of UI in smoothing consumption between employment states. At the center of
such analyses is a fundamental trade-off between insurance and incentives (e.g., Hopenhayn
and Nicolini, 1997). More insurance implies reduced output as the duration and incidence of
unemployment increases. The relationship between UI and output, however, can change if
one recognizes that UI also encourages unemployed workers to seek higher productivity jobs.
Allowing the composition of jobs to respond endogenously, Acemoglu and Shimer (1999,
2000) show that UI benefits encourage firms to create higher productivity jobs, which in
turn might lead to an increase in aggregate output.
In this paper, we introduce an additional role for unemployment benefits: insurance
against idiosyncratic sector-specific productivity shocks while unemployed. Similarly to Acemoglu and Shimer (1999, 2000), we allow the composition of jobs to be endogenously determined. In particular, we consider an environment in which, upon becoming unemployed,
individuals are subject to idiosyncratic shocks that render their current skills less suitable
for their most recent sector of employment. If a move from one sector to another is costless,
such a shock poses no additional risk to the unemployed; they simply move to the most productive sector for their particular skills. If, however, the move requires paying moving costs,
then unemployment poses a risk to future wage prospects if unemployed workers search in
the relatively less productive sector. In such an environment, unemployment benefits may
help insure individuals against this risk by effectively reducing the costs of moving.
Specifically, we analyze a directed search model with matching frictions, multiple sectors,
and risk-averse agents. An unemployed agent receives an idiosyncratic productivity shock
that makes the agent more productive in one sector relative to the other sectors. The
unemployed agent must decide which sector to search in, and she faces moving costs when
leaving her current sector for another. We consider two alternative ways of modeling moving
costs: a utility cost of moving and a pecuniary one (i.e., the cost in terms of consumption
2
goods). Mobility between sectors is directed. Workers know their productivity in another
sector before leaving their current sector (as in Roy (1951) and Heckman and Taber (2008)).
In each sector, firms post wages and agents direct their search to a specific job as in Moen
(1997) and Rogerson, Shimer, and Wright (2005). Therefore, the model represents a blend
of a dynamic Roy model and a competitive search model.
In contrast to standard sectoral reallocation theory (e.g., Lucas and Prescott, 1974),
the model allows for explicit distinction between inter-sectoral mobility and within-sector
trading frictions. Such distinction is essential for examining the link between UI and sectoral
mobility.1
We first analyze how unemployment benefits affect equilibrium outcomes, focusing on the
productivity effects. We show that the mobility decision is characterized by a reservation
rule for productivity shocks. For idiosyncratic shocks above the reservation value, the agent
moves sectors. For idiosyncratic shocks below the reservation value, the agent remains in
the current sector. This feature implies that the effect of benefits on productivity depends
on how changes in benefits affect the reservation value. Analytically, we show that benefits
have two main effects on the reservation value, which we refer to as a “moral hazard effect”
and a “consumption effect.”
The moral hazard effect occurs because the benefit acts as a subsidy to search. Increasing
benefits increases the value of unemployment, reducing the gain from moving to higher productivity sectors. Agents require larger idiosyncratic shocks to be willing to move sectors.
The moral hazard effect increases the reservation productivity, decreasing per-worker productivity. This effect resembles the moral hazard effects in a McCall search model (McCall,
1970): a higher value of unemployment implies workers become more selective, resulting in
longer unemployment durations. The difference between the effect in McCall (1970) and in
our model is that in our multi-sector model, the increased selectiveness of when to switch
1
Lkhagvasuren (2012a) shows that the interaction of between-sector mobility and within-sector trading
frictions might be key to accounting for the negative correlation between unemployment and sectoral mobility
reported by Moscarini and Thomsson (2007) and Kambourov and Manovskii (2009).
3
sectors implies a decrease in the productivity of matches that do occur. In the McCall
search model, in contrast, workers being more selective implies higher wages/productivity
once employed.
An increase in unemployment benefits may also decrease the reservation productivity,
which we refer to as the consumption effect. When benefits increase, the marginal flow utility
of consumption from moving increases faster than the marginal flow utility of remaining in
the current sector. That is, the lifetime utility of moving is closer to the lifetime utility of
remaining in the current sector, for all values of the idiosyncratic productivity shock. This
necessarily implies that the reservation productivity decreases, which increases per-worker
productivity.
We analytically characterize the moral hazard and consumption effects. The combined
effect of benefits on productivity remains ambiguous. To quantitatively illustrate our theory
and determine which effect dominates, we calibrate the model to the U.S. economy. We
find that the consumption effect dominates, so that increasing benefits increases per-worker
productivity. The dominant consumption effect implies that workers move more frequently
in response to idiosyncratic productivity shocks. Unemployment increase, while vacancies
decrease.
Quantitatively, we find that with additively separable moving costs, the overall effect
on per-worker productivity is relatively small. A 25% increase in the benefit level increases
per-worker productivity by 0.1%. With a pecuniary moving cost (which directly reduces
the consumption of the unemployed upon moving), we find that the effect on productivity
may be quite large. In the calibrated version, a 12.5% increase in benefits implies a nearly
2% increase in per-worker productivity. When the moving costs are initially prohibitive
(so that mobility is zero), a 12.5% increase in benefits implies a 9% increase in per-worker
productivity.
We also analyze the welfare-maximizing unemployment benefit in our multi-sector setting
and find that the optimal level of benefits depends on the size of the moving cost. As moving
4
costs increase, the optimal benefit decreases. Determining the optimal benefit requires managing the aforementioned trade-off between the moral hazard and consumption effects. As
the moving cost increases, the moral hazard effect becomes stronger, which puts downward
pressure on the optimal benefit level. We find that in response to a 1 percent increase in
the moving cost, the benefit decreases by 6 percent. Thus, while in the case of additively
separable moving costs the productivity effects are relatively small, the overall role of UI as
productivity insurance may be quite important.
The productivity results in our model relate to the efficient UI literature, most notably,
Acemoglu and Shimer (1999, 2000). In their model, there exists an endogenous trade-off
between productivity and the arrival rate of job offers: more productive jobs arrive less
frequently than lower productivity jobs. Since the unemployment risk consists of the duration
of unemployment (i.e., low income during a longer period), benefits help workers endure the
longer durations necessary to search for higher productivity jobs. Alvarez-Parra and Sanchez
(2009) show similar effects in a model of optimal unemployment insurance (without firms)
where the unemployed may work in an informal sector.
In contrast, in our model, higher wage jobs are offered more frequently, given the agent
searches in the appropriate sector. More importantly, we introduce another role for UI beyond smoothing income shocks. In particular, UI serves as insurance against the risk of lower
future wages if the search is in a “wrong” sector. Per-worker productivity increases because
benefits help workers incur the costs of moving. That is, the benefits provide insurance
against the productivity shocks associated with unemployment.
The remainder of the paper is as follows. Section 2 describes the environment and
agent decisions. Section 3 characterizes the equilibrium of the model economy. Section 4
characterizes the productivity effects of benefits, and Section 5 presents our quantitative
analysis. We discuss the pecuniary moving cost in Section 6, and Section 7 concludes. All
proofs are contained in Appendix A.
5
2
Model
2.1
Environment
The economy is composed of two sectors, denoted by 0 and 1, populated by a measure one
of risk-averse workers and a continuum of risk-neutral firms. Individuals are either employed
or unemployed. Employed workers are matched with a firm. In each period an unemployed
worker chooses to search in the current sector or move to the other sector to search. While
employed, workers do not engage in on-the-job search. Therefore, every mover is unemployed,
while not all unemployed workers are movers.2
Each period, firms search for workers by creating vacancies. The flow cost of a vacancy
is k. Free entry drives the expected present value of an open vacancy to zero. Vacant jobs
and unemployed workers are matched according to a matching technology. Without loss of
generality, we assume that each firm employs at most one worker. All matches are dissolved
exogenously with probability λ.
Let b denote per-period income of a worker while unemployed. Flow utility of an unemployed worker searching for a job within his own sector is log(b), while that of an unemployed
worker moving across sectors is log(b) − c, where c > 0 is the utility cost of moving. The
flow utility of a worker is log(w), where w denotes the wage. Workers and firms discount the
future at the same rate, β.
2.2
Production Technology
Let yi (x) be the production function describing per-period output produced by a firm that
employs a worker with productivity x and operates in sector i ∈ {0, 1}. We assume that
y0 (x) = 1 − x
2
(1)
This is not inconsistent with the fact that unemployment among recent movers is much higher than
that among local workers, even after controlling for age and education of the labor force. See Lkhagvasuren
(2012a) for details.
6
and
y1 (x) = 1 + x.
(2)
Equations (1) and (2) imply that individual productivity is perfectly negatively correlated
across sectors: the best workers of sector 0 are the worst workers of sector 1.3 We further
assume that x remains unobservable to the UI agency.
We present the model in terms of two sectors, i ∈ {0, 1}. The model, however, can be
generalized to an economy with N sectors by interpreting yi (x) as the agent’s productivity
shock in the current sector, and y1−i (x) as the highest of the N − 1 productivity shocks from
the remaining N − 1 sectors.
By construction, idiosyncratic productivity does not change within a given match. If an
employed worker becomes unemployed, she draws her new productivity from the uniform
distribution on the interval [−ω, ω].4 For relatively high values of the productivity shock x,
the unemployed worker prefers to search in the current sector; for relatively low values of x,
the worker prefers to move and search in another sector. We also assume that 0 < b < 1 − ω.
If upon unemployment the worker decides to search in another sector, she incurs the moving
cost c.
2.3
Wages
We assume that wages are determined through competitive search, as in Moen (1997) and
Rogerson et al. (2005). A firm decides whether or not to post a vacancy. A vacancy is fully
characterized by the productivity level, x, the wage, w, and the sector, i. If a firm decides
3
See Moscarini and Vella (2008) and Lkhagvasuren (2012b) for related dynamic extensions of Roy’s (1951)
framework. One can consider labor income shocks that are not perfectly negatively correlated across sectors.
For example, suppose that e0 and e1 are productivity of a worker in sectors 0 and 1, respectively. Further
suppose that these two shocks are not perfectly negatively correlated; i.e., Corr(0 , 1 ) > −1. Then, consider
the following decomposition: 0 = z + x and 1 = z − x, where z and x are uncorrelated shocks. Since z is
common across locations and not affected by mobility, it is not essential for the productivity distribution.
4
To increase the tractability of the model, we adopt this specification of the persistent idiosyncratic shock
from Andolfatto and Gomme (1996) and Merz (1999). Note that when a worker draws an idiosyncratic
shock upon separation, this can also be reinterpreted as an all-in-one shock capturing the wage risk over the
expected employment duration.
7
to post a vacancy, it chooses these three variables in order to maximize its expected profits.
An unemployed worker directs her search towards the most attractive job. Let Wi (x) denote
the set of wages posted at the productivity level x in sector i.
2.4
Matching Technology
Let nτ denote the number of unemployed workers searching for a job of type τ = (w, x, i),
and vτ denote the number of vacant type τ jobs. The total number of type τ matches is
given by
Mτ = µnητ vτ1−η
where 0 < η < 1 and µ > 0. Let qτ = nτ /vτ . We refer to qτ as the queue length for a
vacancy of type τ . A type τ vacant job is filled with probability α(qτ ) = µqτη , and any of the
nτ workers finds a job with the probability f (qτ ) = µ/qτ1−η .
2.5
Timing of the Events
Figure 1 displays the timing of events. Each period consists of four stages. At the beginning
of each period, fraction λ of the existing matches is dissolved. The workers separated from
these matches become unemployed, observe their idiosyncratic shock, x, and decide which
sector to search in. In the second stage, moving takes place, i.e., the unemployed workers who
have decided to search in other than their current sector move. In the third stage, production
takes place in the surviving matches, firms post vacancies, and unemployed workers search
for jobs. In the last stage, new matches are formed.
2.6
Discussion
We have added two elements to the textbook model described in Rogerson et al. (2005):
sector-specific productivity and moving costs. Specifically, if there is no sector-specific productivity dispersion (i.e., ω = 0), the economy converges to the standard one-sector model
8
Figure 1: Timing of the Events
some old
out-migration
matches are
dissolved
production
6
t
A
K
At
t
A
AU
t
?
idiosyncratic
shocks: x
in-migration
new matches
are formed
t
-
t+1
job search and
vacancy creation
described in Rogerson et al. (2005). Moreover, if moving across sectors is costless (i.e.,
c = 0) or prohibitive (c = ∞), the economy is equivalent to a one-sector model with exogenous idiosyncratic productivity (e.g., Bils, Chang, and Kim, 2011). Therefore, endogenous
idiosyncratic productivity in the presence of costly mobility is the key equilibrium channel
considered in this paper.
In this economy, unemployment imposes two risks. First, a worker loses her employment
earnings; i.e. income drops from w to b. Second, an unemployed worker also risks an idiosyncratic productivity shock that renders her skills unsuitable for the sector the worker is
currently in. Since there exist moving costs (c > 0), the unemployed worker may prefer to
continue searching in the relatively less productive sector. Below we show that the productivity risk affects the future lifetime earnings of an unemployed worker in two ways: a lower
future wage and a lower job-finding probability.
2.7
2.7.1
Value Functions
Workers
Consider a worker who is unemployed at the beginning of the current period, in sector i with
productivity x. Let Si (x) denote the lifetime utility value to the worker of searching for a
job in the current sector, i. Let Mi (x) denote the lifetime value to the worker of moving
9
from sector i to sector 1 − i. The value function of the unemployed worker is
Ui (x) = max {Si (x), Mi (x)}.
(3)
Given the moving cost, c, and the timing of mobility, the value of moving from sector i to
sector 1 − i is given by
Mi (x) = log(b) − c + βS1−i (x).
(4)
Let Wi (w, x) denote the value of being employed, in sector i with productivity x, by a
firm who pays wage w:
Z
ω
Wi (w, x) = log(w) + β(1 − λ)Wi (w, x) + βλ
Ui (x0 )dG(x0 ),
(5)
−ω
where G denotes the uniform distribution function on the interval [−ω, ω]. Then, the expected lifetime utility value of searching for a job in sector i is given by
Si (x) =
max
w∈W(x,i)
log(b) + βf (qw,x,i )Wi (w, x) + β 1 − f (qw,x,i ) Si (x) .
(6)
As in Rogerson et al. (2005), a worker takes qτ as given.
2.7.2
Firms
Now consider a matched firm operating at productivity level x in sector i. Given the wage
w, the value of the match to the firm is given by
Ji (w, x) = yi (x) − w + β(1 − λ)Ji (w, x).
(7)
Let Vi (x) denote the value of posting a vacancy at productivity level x in sector i. Vi (x) is
defined by
Vi (x) = max{−k + βα(qw,x,i )Ji (w, x)}
w
10
(8)
Due to free entry and profit maximization, all rents from vacancy creation are exhausted in
the economy. Thus, for any pair (x, i):
Vi (x) = 0.
2.8
(9)
Unemployment and Mobility
Let ψiu (x) denote the number of unemployed workers searching for jobs in sector i at productivity level x. Similarly, let ψie (x) denote the number of workers employed at productivity
level x in sector i. Since the total population is normalized to one,
XZ
i
ω
(ψiu (x) + ψie (x))dx = 1.
(10)
−ω
The economy-wide unemployment rate is given by
u=
XZ
ω
ψiu (x)dx.
(11)
−ω
i
Let Ωi denote the decision rule governing whether an unemployed worker in sector i stays
in her current sector:
1 if Si (x) ≥ Mi (x),
Ωi (x) =
0 otherwise.
(12)
Then, the measure of workers moving from sector i to 1−i is given by ψim = (1−Ωi (x))ψiu (x).
Therefore, the economy-wide mobility rate is given by
m=
XZ
i
ω
ψim (x)dG(x).
−ω
11
(13)
2.9
Definition of the Equilibrium
The equilibrium consists of a set of value functions {Ui , Si , Wi , Ji , Vi }, a decision rule Ωi ,
sets of posted wages Wi for any i ∈ {0, 1}, and measures {n, v} such that
1. Given (S0 , S1 ), the decision rule Ωi (x) and the value function Ui (x) solve (3);
2. Given Ui , the value function Wi (w, x) solves (5);
3. Given qi , Ui and Wi , the value function Si (x) solves (6) for each w ∈ Wi (x);
4. The value function Ji (w, x) solves (7);
5. Given Ji , n and v, the value function Vi (x) solves (8) for each w ∈ W(x); and
6. Free entry:
v(w, x, i) > 0 and Vi (x) = 0 if w ∈ Wi (x),
v(w, x, i) = 0 and Vi (x) = 0 if w 6∈ Wi (x) or Wi (x) is an empty set.
3
Equilibrium Characterization
We solve the model in two steps. First, we find the local labor market equilibrium, treating
R
U i = Ui (x)dG(x) as a parameter. After obtaining workers’ and firms’ decisions within a
local market, we determine U i using equation (3).
3.1
Queue Length and Wages
Taking U i as given, equation (5) can be re-written as
Wi (w, x) =
log(w) + βλU i
,
A
(14)
where A = 1 − β(1 − λ). Inserting the latter into (6), we have
log(w) =
A(1 − β)Si (x) − A log(b)
+ ASi (x) − βλU i
βf (qw,x,i )
12
(15)
Using equations (7) and (8), a firm’s problem can be written as:
max {α(qw,x,i ) (yi (x) − w)} .
(16)
qw,x,i
A firm posting a vacancy at the productivity level xi takes Si (x) and U i as given. Therefore,
combining equations (15) and (16), a firm’s problem becomes
A(1 − β)Si (x) − A log(b)
max α(qw,x,i ) yi (x) − exp
+ ASi (x) − βλU i
.
qw,x,i
βf (qw,x,i )
(17)
Taking the FOC in (17) and combining the result with the free entry condition, it can be
shown that
qw,x,i =
ηk exp
1−η
i (x)−A log(b)
− A(1−β)S
βf (qw,x,i )
− ASi (x) + βλU i
(1 − β)Si (x) − log(b)
.
(18)
Proposition 1 (Queue length). All firms creating a vacancy at the productivity level x in
sector i choose the same queue length qi (x).
Corollary 1 (Wage). The free entry condition, Vi (x) = 0, and Proposition 1 imply that the
wage is also unique for each pair (x, i) and is given by
wi (x) = yi (x) −
kA
.
βα(qi (x))
(19)
Therefore, each job is fully characterized by the productivity level, x, and the sector, i.
To summarize, given U 0 and U 1 , the local labor market equilibrium is given by (15), (18)
and (19). In Appendix, we show that the wage, wi (x), and the value of searching for a job
in the current sector, Si (x), increase with productivity, yi (x), while the queue length, qi (x),
and the value of moving, Mi (x), decrease with productivity for each i.
These results also imply the following two corollaries:
Corollary 2 (Queue length). The queue length qx,i decreases with productivity yi (x) for
13
each i.
Corollary 3 (Wage). The wage wx,i increases with productivity yi (x) for each i.
These two corollaries highlight the productivity insurance aspect of UI. Specifically, a
shock x that implies higher productivity in sector i, yi (x), also implies a higher wage and a
higher job finding probability. Notice, this represents a different trade-off from the models
of Acemoglu and Shimer (1999, 2000). There, the trade-off is between higher wages and
higher job-finding probabilities. In that sense, unemployment benefits allow the worker to
endure longer durations to achieve a more productive match. In contrast, in our model,
higher wages are associated with higher job-finding rates, provided the worker searches in
the appropriate sector.
Given the results above, we also characterize the effects of productivity on the value
of staying in the current sector or moving, respectively. These results are important for
understanding the impact of unemployment benefits on mobility (and thus on productivity).
Corollary 4 (Value of staying). The value of searching in the current sector, Si (x),
increases with productivity yi (x) for each i.
Corollary 5 (Value of moving). The value of moving from sector i to sector 1 − i, Mi (x),
decreases with productivity yi (x) for each i.
3.2
Mobility Decision
Next we characterize a worker’s mobility decision and determine the equilibrium values of
U 0 and U 1 . If the moving cost is too high, mobility is zero. Let c denote the lowest moving
cost that prohibits mobility. For moving costs below c, each period a certain fraction (but
not all) of unemployed workers move between the two sectors. Thus, for each i, there exists
a minimum productivity level x̂i such that
Si (x̂i ) = Mi (x̂i ).
14
(20)
Productivity level x̂i represents a reservation value for the mobility decision. Depending on
the current sector, for values of x above (sector i = 0) or below (sector i = 1) x̂i , the agent
prefers to search in the other sector. In a frictionless environment (i.e. without moving
costs), the workers switch sectors in response to any positive (sector i = 0) x; however, with
positive moving costs, workers do not always move to the most productive sector.
Figure 2 shows the determination of x̂i , for i = 0, 1. Note that 0 ≤ x̂0 < ω and −ω <
x̂1 ≤ 0. Given the symmetry of the productivity shock in equations (1) and (2), the curve
S1 (x) is a reflection of the curve S0 (x) with respect to a vertical line x = 0:
S1 (x) = S0 (−x).
(21)
Moreover, M1 (x) is also a reflection of M0 (x) with respect to the same line. Consequently, as
shown in Figure 2, the decision rule for moving across sectors is symmetric with respect to
0: x̂0 = −x̂1 . The minimum per-period match output is also the same between the sectors,
i.e., ymin = 1 − |x̂1 | = 1 − x̂0 . In the event of a transition from employment to unemployment,
the probability of moving to another sector upon job separation is p = (ω − |x̂1 |)/(2ω) =
(ω − x̂0 )/(2ω).5
Given x̂0 and x̂1 , the continuation values U 0 and U 1 are given by
Z
x̂0
U0 =
ω
Z
S0 (x)dG(x) +
−ω
M0 (x)dG(x)
(22)
S1 (x)dG(x).
(23)
x̂0
and
Z
x̂1
U1 =
Z
ω
M1 (x)dG(x) +
−ω
x̂1
Equilibrium is fully characterized by equations (15), (18) to (20), (22) and (23). Unemployment benefits affect equilibrium outcomes primarily through two channels. First, as in a
standard one-sector search and matching model, the benefit level b affects the queue length
qi (x) for each (x, i), and thus also affects the job-finding rate. Second, the level of b affects
5
Recall that G(x) is a uniform distribution.
15
Figure 2: Mobility Decision
utility
6
HH
HH
H
HH
HH
H
H
HH
HH
HH
HH
H
H
H
HH
H
HH
H
HH
−ω
x̂1
0
x̂0
S1 (x)
M0 (x)
S0 (x)
M1 (x)
ω
x
Notes: Si (x) denotes the lifetime utility value to the worker of searching for a job on her
current island i when her productivity level is x ∈ [−ω, ω]. Mi (x) denotes the lifetime value
to the worker of moving from sector i to sector 1−i when her productivity level is x ∈ [−ω, ω].
The value of searching for a job in the current sector Si (x) increases with productivity yi (x),
while Mi (x) decreases with productivity for each i (see Corollaries 4 and 5). Therefore,
an unemployed worker of sector 1 moves to sector 0 if her productivity shock is below x̂1 .
Analogously, an unemployed worker of sector 0 moves to sector 1 if her productivity shock
is above x̂0 .
x̂i for each i, which determines the mobility decision. Below we characterize the role of each
factor to determine the impact of benefits on productivity.
4
Impact of Benefits on Productivity
Mobility in response to idiosyncratic productivity shocks represents the key addition of
our model, relative to the standard one-sector model. Thus, to determine whether or not
unemployment benefits can insure unemployed workers against these shocks, we need to
understand how the mobility decision (i.e. x̂i ) responds to benefits.
Benefits affect mobility through two channels. To highlight these effects, we combine
equations (4) and (20) to get, for each i,
Si (x̂i ; b) = log(b) − c + βS1−i (x̂i ; b).
16
(24)
Then, using the symmetry property in equation (21), for each i,
Si (x̂i ; b) = log(b) − c + βSi (−x̂i ; b).
(25)
Using equation (25), we characterize how an increase in b affects the mobility decision x̂i .
Without loss of generality, we consider the mobility decision of workers in sector 1, as
0
(x; b) denote the derivative of S1 (x; b)
the case of a sector 0 worker is symmetric. Let S1,x
with respect to x. Then, taking the first order Taylor approximation, we have,
S1 (x; b) ' S1 (0; b) + Γ(b)x,
(26)
where Γ(b) denotes the slope of S1 (x, b) with respect to x at x = 0:
0
Γ(b) = S1,x
(0; b).
(27)
It can be seen that Γ(b) > 0 (see Corollary 4 in Appendix).
Combining equations (25) and (26), we have:
|x̂1 | '
Π(b)
,
(1 + β)Γ(b)
(28)
where
Π(b) = (1 − β)S1 (0, b) − (log(b) − c).
(29)
What does Π(b) measure? Recall that log(b) − c is the flow utility of a mover. Given
the discount factor β, (1 − β)S1 (0, b) is the average flow utility associated with searching
at the average productivity level 0, S1 (0, b). Therefore, Π(b) measures the value of staying
in the current sector relative to the value of moving across sectors. Notice that Π(b) is
also positive, since the value of searching for a job, S1 (0, b), is higher than the value of not
searching, log(b)/(1 − β).
17
Using equation (28), taking the log of each side, we have
log |x̂1 | ' − log(1 + β) − log(Γ(b)) + log(Π(b)).
(30)
In equation (30), clearly the benefit b affects x̂1 .6 Whether an increase in benefits works
to increase mobility, and thus help insure the unemployed against the risk of productivity
shocks, depends on the relative size of the two effects. We characterize these effects below.
4.1
Moral Hazard Effect
The first channel concerns within-sector trading frictions. Specifically, unemployment benefits affect Γ(b), the slope of the value function S1 (x; b) with respect to x: higher benefits
lower the utility differences between high and low productivity jobs, making S1 (x; b) flatter
with respect to x. In other words, |∂Si (x; b)/∂x| and |∂Mi (x; b)/∂x| decrease for each (x, i).
Figure 3 displays this effect.
This effect occurs because the attractiveness of high wage jobs diminishes relative to
low wage jobs. Consequently, moving across sectors become less rewarding, and individuals
become more selective. Thus, as benefits increase, the slope Γ(b) goes down, putting upward
pressure on |x̂1 | (see equation (30)). Indeed, Figure 3 shows that a decrease in the slope
lowers x̂1 (raises x̂0 ).
When x̂1 decreases, mobility also decreases. This implies that unemployed workers become less responsive to the idiosyncratic productivity shocks. That is, more unemployed
remain in a relatively unproductive sector for their particular skills; as a result, average
productivity decreases. We refer to this as the moral hazard effect.
6
Using the symmetry property, one can show that x̂0 = |x̂1 |.
18
Figure 3: Moral Hazard Effect
Panel A. Before the benefit increase
utility
6
H
S1 (x)
H
HH
HH
HH
H
H
HH
HH
HH
HH
H
H
H
HH
H
HH
H
HH
H
−ω
x̂1
0
x̂0
M0 (x)
S0 (x)
M1 (x)
ω
x
Panel B. After the benefit increase
utility
6
XXX
XXX
X
XXX
XXX
X
XXX
X
X
XXX
XXX
XXX
X
X
−ω x̂1
0
S1 (x)
M0 (x)
S0 (x)
M1 (x)
x̂0 ω
-
x
Notes: A higher benefit level lowers the value of searching for high productivity jobs relative
to that for low productivity jobs (i.e., it makes the curves Si (x) and Mi (x) flatter for each x
and i), putting downward pressure on productivity. We refer to this as the the moral hazard
effect.
4.2
Consumption Effect
We now describe the potential role unemployment benefits can play in insuring workers
against the idiosyncratic shocks. This second effect works through the marginal utility of
consumption, and we refer to it as the consumption effect.
Specifically, benefits affect Π(b), the value of staying in the current sector relative to
the value of moving across sectors. As benefits increase, the flow utility of moving across
sectors, log(b), grows faster than the constant flow utility associated with staying in the
19
−β
.
b
Thus, the relative value of staying in the current sector, Π(b), goes down. This reduces |x̂1 |
current sector, (1 − β)S1 (0, b). That is, differentiating Π(b) with respect to b gives
(see Figure 4), increasing mobility, and thus average productivity.
Analytically the total effect on productivity remains ambiguous, depending on whether
the moral hazard or consumption effect dominates. For the remainder of the paper, we
evaluate these effects quantitatively.
Figure 4: Consumption Effect
Panel A. Before the benefit increase
utility
6
H
S1 (x)
HH
H
HH
HH
H
HH HH
H
H
HH
HH
H
H
HH
H
HH
HH
H
H
−ω
x̂1
0
x̂0
M0 (x)
S0 (x)
M1 (x)
ω
x
Panel B. After the benefit increase
utility
6
HH
S1 (x)
HH
H
M0 (x)
H
HH HH
H
H
HH
HH
H
H
HH HH
H
H
HH H
H
H
H
−ω
x̂1 0 x̂0
S0 (x)
M1 (x)
ω
-
x
Notes: Higher benefits reduce the gap between Si (x) and Mi (x) for each x and i, putting
upward pressure on productivity.
20
5
Quantitative Evaluation
Below we first describe our baseline parametrization, and then we present our quantitative
results.
5.1
Calibration
The time period is one month. We set the discount factor β = 1/1.051/12 , consistent
with an annual interest rate of 5%, and the separation rate to λ = 0.033, consistent with
Shimer (2005). The elasticity of the matching function, η, is set to 0.5, which is in the range
of estimates in Petrongolo and Pissarides (2001). The flow utility of unemployed workers
staying in their current sector is b = 0.4 (Shimer, 2005; Mortensen and Nagypál, 2007). The
volatility of the idiosyncratic shocks is set to ω = 0.2. This value gives us approximately 10
percent wage variation.
Empirically, sectors can be thought of in terms of geographical locations, industries,
occupations, or a combination of these. The cost of moving across sectors is chosen to target
an annual mobility rate of 10 percent.7 The moving cost is c = 4.66, while the average wage
P Rω
1
e
defined as w = 1−u
i −ω wi (x)ψi (x)dx is slightly greater than one. This implies a moving
cost equal to approximately four months of labor income.
Given the rest of the parameter values, the coefficient of the matching function µ is set to
0.4875 by targeting an economy-wide unemployment rate of 6 percent (Shimer, 2005). The
vacancy cost is set to 1.27, targeting overall labor market tightness of 0.6, which is between
the values obtained by Hall (2005) and Hagedorn and Manovskii (2008). Table 1 summarizes
the key parameters. The column labeled benchmark in Table 2 displays the predictions of
the baseline model.
7
In terms of the empirical mobility rates, Murphy and Topel (1987) report industry annual mobility
rates at 6-10 percent, Jovanovic and Moffitt (1990) report industry bi-annual mobility rates at 15 percent,
Kambourov and Manovskii (2009) report annual occupational mobility rate at 16 percent in the early 1970s
and at 21 percent in the mid-1990s. For geographical labor markets, Ihrke, Faber, and Koerber (2011)
estimate that the annual inter-county mobility rate is approximately 6 percent.
21
Table 1: Parameters of the Benchmark Model
Parameters
Values
Description
β
λ
η
b
µ
k
ω
c
0.9959
0.033
0.50
0.4
0.4875
1.2661
0.2
4.6574
the time-discount factor
the job separation rate
the elasticity of the matching technology
flow utility of unemployment
the efficiency of the matching technology
the vacancy cost
volatility of the sector-specific shock
moving cost
Notes: This table summarizes the key parameters of the model.
Table 2: Prediction of the Model
Variables
Benchmark
b = 0.40
u
m
v/u
ymin
y
w
γ
π
0.0586
0.0990
0.6198
0.9052
1.0770
1.0004
n/a
n/a
Higher benefit levels
b = 0.45 b = 0.50
Description
0.0642
0.0996
0.5398
0.9065
1.0776
1.0061
-0.0062
-0.0197
unemployment
annual mobility
the vacancy-unemployment ratio
minimum observed productivity
per-worker output
the average wage
the moral hazard effect
the consumption effect
0.0703
0.1000
0.4679
0.9076
1.0781
1.0115
-0.0122
-0.0375
22
5.2
Policy Experiments
We first explore if unemployment benefits can insure workers against the idiosyncratic productivity shocks that they receive upon becoming unemployed. The answer to this question
depends on which effect dominates: moral hazard or consumption. If the consumption effect
dominates, then unemployment benefits work to increase average productivity by encouraging workers to search in the relatively more productive sector. To analyze these effects, we
first simulate the benchmark model for different levels of b. Table 2 summarizes the results.
5.2.1
Productivity
Table 2 indicates that benefits lower |x̂j |. As a result, higher benefits raise both the minimum
P Rω
1
e
productivity ymin = 1 − |x̂1 | and average productivity y = 1−u
i −ω yi (x)ψi (x)dx. Therefore, the results imply that the consumption effect Π(b) dominates the moral hazard effect
Γ(b) (see Figure 5). To quantify the relative magnitudes of these two effects, we calculate
the responses of Γ(b) and Π(b) to b. For this purpose, we consider the following two values:
γ = log(Γ(b)) − log(Γ(bBM ))
(31)
π = log(Π(b)) − log(Π(bBM )),
(32)
and
where bBM denotes the benchmark value of unemployment benefits. These two variables are
summarized in the last two rows of Table 2. The Table implies that the consumption effect
(π) is three times larger than the moral hazard effect (γ).
5.2.2
Unemployment
As can be seen in Table 2, as benefits increase, unemployment increases, the ratio of vacancies
to unemployment decreases and the average productivity increases. While qualitatively
similar to the results in Acemoglu and Shimer (1999, 2000), the mechanism driving the
23
Figure 5: Moral Hazard and Consumption Effects
Panel A. Before the benefit increase
utility
6
H
S1 (x)
H
HH
HH
HH
H
H
HH
HH
HH
HH
H
H
H
HH
H
HH
H
−ω
HH
H
x̂1
0
x̂0
M0 (x)
S0 (x)
M1 (x)
ω
x
Panel B. After the benefit increase
utility
6
XX
S1 (x)
X
XXX
XX
XXX
Y
HM (x)
H
X
X
X
X
0
X
X
X
X
XX XX
XX X
X
XX
X S0 (x)
I
@
@M (x)
1
−ω
x̂1 0 x̂0
ω
-
x
Notes: Case where the consumption effect dominates the moral hazard effect.
results in this paper is quite different.
As |x̂j | decreases with benefits, unemployed workers become more selective and, on average, search for a job at a higher productivity level. This puts upward pressure on the
probability of finding a job. As |x̂j | decreases, the probability of moving, given a transition
from employment to unemployment, increases. Since unemployed workers move to a sector
where their productivity is higher, job offers are more likely to occur. This represents an
important distinction between the mechanism in our paper relative to Acemoglu and Shimer
(1999, 2000). In Acemoglu and Shimer (1999, 2000), more productive jobs take longer to
find. Thus, the fundamental role of unemployment benefits in their environment is to help
24
smooth consumption and thus encourage workers to endure longer durations of unemployment in favor of higher productivity jobs. In contrast, in our model, the unemployed workers
who move to their higher productivity sector have a higher probability of finding job. The
risk to the unemployed in searching in this sector is that it requires mobility across sectors.
Such mobility is costly. Unemployment benefits help insure agents against this risk, thus
increasing average productivity.
As in standard one-sector search and matching models (Pissarides, 2000), as benefits increase the vacancy-unemployment ratio decreases at each productivity level, exerting downward pressure on the job finding rate. Table 2 shows that as the benefits rise, unemployment
rises, indicating that the effect through the vacancy-unemployment ratio is much stronger
than the other two effects. This is not surprising since the fraction of unemployed workers
searching for a job in their own sector is much larger than those moving across sectors and,
thus, unemployment is mainly determined by within-market search frictions.
5.2.3
Mobility
What is the impact of an increase in benefits on mobility? Using the accounting equations
of labor market flows and stocks in Section 2.8, the mobility rate is given by
m = λ(1 − u)p,
(33)
where 1−u is employment and p = (ω −|x̂1 |)/(2ω) is the probability of moving across sectors
(given a transition from employment to unemployment).
The results in Table 2 show that higher benefits lower employment, 1 − u. At the same
time, workers become more selective (i.e., |x̂j | decreases), which increases p, the probability of
moving upon separation. Therefore, the impact of benefits on overall mobility is analytically
uncertain. Table 2 shows that benefits raise overall mobility, indicating that the probability
of moving across sectors, given a transition from employment to unemployment, responds to
benefits more than employment, in percentage terms.
25
Table 3: Optimal Benefit Level by Moving Cost
Moving cost, c
Optimal Benefit Level
Tax, τ ,
Welfare Gain, %
0
1
3
7
12
0.6909
0.6788
0.6655
0.6509
0.6148
0.0284
0.0268
0.0253
0.0237
0.0189
0.0058
0.0057
0.0055
0.0049
0.0040
Notes: The welfare gain refers to the percentage increase in the average flow utility measured
by exp (1 − β)(U(b) − U(bBM )) − 1 where U(b) is given by equation (34).
5.3
Moving Costs and Optimal Benefit Level
In this section, we analyze the relationship between moving costs and the welfare-maximizing
benefit level. Determining the optimal benefit level is an exercise in balancing the traditional
insurance vs. incentives (i.e. moral hazard effect), and the additional insurance motive
provided by the consumption effect. We thus determine the optimal benefit level and how
it changes with moving costs. In the welfare comparisons, we consider benefits above the
benefit level in the benchmark economy, bBM , that are financed by a lump sum tax τ . Then,
the aggregate welfare is given by:
U = u log(b − τ ) − mc +
XZ
i
ω
log(wi (x) − τ )ψie (x)dx.
(34)
−ω
The optimal benefit level is determined by finding the combination of τ and b that maximizes
U, subject to the budget constraint τ = u(b − bBM ).
Table 3 describes how the optimal benefit level varies with moving costs. Our main
finding from this experiment is that the optimal benefit level is decreasing in moving costs.
Moreover, the total welfare gains from adopting the optimal benefit level are also decreasing
in moving costs. There are many factors behind this result. The basic idea, however, can be
explained by comparing the two extreme cases: 1) zero moving cost and 2) prohibitive moving
cost (i.e. no mobility). In both economies, unemployment benefits do not affect mobility.
26
In the case of zero moving costs, agents continue to move to the most productive sector,
while in the case of prohibitive moving costs, no one moves regardless of their productivity
shock. With zero moving costs, the insurance vs. incentives trade-off becomes relatively
tilted in favor of insurance. Since agents are moving to more productive sectors, often with
more vacancies available per searcher, the optimal benefit can tolerate more insurance before
the moral hazard problem begins to reduce welfare. In contrast, in the economy with high
moving costs and no mobility, the moral hazard problem is more troublesome, as agents
simply take more time to find a job in a relatively unproductive sector.
6
Alternative Moving Cost
In the analysis above, we model moving costs as an additively separable flow utility. We now
consider an alternative specification. Specifically, we consider the possibility that switching
industries or moving across locations may result in a loss of consumption, i.e., flow utility of
a mover is given by log(b − c) rather than log(b) − c.
An example of such costs is the costs associated with selling or buying a house when
moving across locations. Such costs can be especially sizeable if the house price falls below
the amount owed on the mortgage. A worker whose house price falls below the amount he
owes on the mortgage may choose to remain in his current sector instead of moving to the
sector where he is more productive. When a sector refers to an industry or an occupation,
then pecuniary costs may include costs associated with schooling or other training programs.
6.1
Within-Sector Frictions
The characterization of the local labor market equilibrium follows that in Section 3. The
reason is that conditional on U 0 and U 1 , the wage and the queue length are characterized as
in 3.1. Therefore, the uniqueness results in Proposition 1 and Corollary 1 also hold in this
case.
27
6.2
Mobility
The mobility decision is characterized analogously to that in the previous cases. However,
now the value of moving from sector i to sector 1 − i is given by
Mi (x) = log(b − c) + βS1−i (x).
(35)
As in the previous case, the impact of benefits on productivity is given by equation (30).
The only difference is that now we have
Π(b) = (1 − β)S1 (0, b) − log(b − c)
(36)
instead of equation (29). Comparing equations (29) and (36), the consumption effect Π(b)
becomes even stronger, as an increase in benefits will have a much higher marginal effect
on movers than on stayers because the impact of a marginal increase in benefits on the flow
utility of a mover is proportional to
6.3
1
b−c
instead of 1b .
Numerical Results
We consider the case where all the parameters except for the moving cost are fixed at their
benchmark values. The moving cost is adjusted to target the annual mobility rate of 10%.
The moving cost is then c = 0.396. The model predictions are summarized in Table 4. It
shows that the benefits have a much larger effect on unemployment than in the previous case.
Moreover, the response of productivity to benefits is now much larger. The consumption
effect is now 15 times higher than the moral hazard effect.
Clearly, in equation (35), if the benefit level b is very close to the moving cost c, there
is no mobility. Under such circumstances, a small increase in the benefit level b can sharply
raise productivity. To illustrate this point numerically, we set the moving cost to c = b−10−6
and analyze the impact of higher benefits on mobility and productivity. The results are in
28
Table 4: Economy with Alternative Moving Cost
Variables
u
m
v/u
ymin
y
w
γ
π
Benefit levels
0.40
0.45
0.50
0.059
0.065
0.100 0.1407
0.6206 0.5695
0.9062 0.9505
1.0775 1.0933
1.0008 1.0199
n/a
-0.0047
n/a
-0.0724
0.071
0.1500
0.5028
0.9615
1.0957
1.0267
-0.0102
-0.1560
Description
unemployment
annual mobility
the vacancy-unemployment ratio
minimum productivity, 1 + x̂1
average productivity
the average wage
the moral hazard effect
the consumption effect
Table 5: Mobility-Inducing Benefits
Variables
u
m
v/u
ymin
y
w
γ
π
Benefit levels
0.40
0.45
0.50
0.0558 0.0652
0
0.1394
0.5469 0.5686
0.8
0.9492
1.000 1.0929
0.9278 1.0196
n/a
-0.0034
n/a
-0.0519
Description
0.0711
0.1494
0.5023
0.9608
1.0955
1.0266
-0.0088
-0.1348
unemployment
annual mobility
the vacancy-unemployment ratio
the lowest productivity level, 1 + x̂1
per-worker output /productivity
the average wage
the moral hazard effect
the consumption effect
Table 5. They show that annual mobility increases from 0 to 14 percent as the benefit level
increases from 0.4 to 0.45. What is more remarkable is that economy-wide productivity
increases by almost 10 percent. Thus, a moderate change in unemployment benefits can
have a large positive impact on both productivity and mobility.
7
Conclusion
We construct a search-matching model with sectoral mobility and analyze the provision
of unemployment benefits. Unemployment in this environment poses an additional risk
29
because the unemployed workers are subject to idiosyncratic productivity shocks that affect
future wage prospects. Unemployment benefits increase mobility, which reduces the risks of
idiosyncratic productivity shocks the unemployed face. Our results show that unemployment
benefits can have a substantial impact on productivity through higher mobility. Ignoring
costly sectoral mobility leads to a downward bias in the impact of benefits on unemployment,
relative to a standard one-sector model. The optimal replacement ratio decreases with the
costs of moving, and the welfare gains decrease from 0.6% when moving is costless to 0.4%
when moving is prohibitively costly.
Our model and results have implications beyond the direct relationship between unemployment and mobility. In the model, an increase in moving costs is observationally
equivalent to a decrease in productivity. Since mobility is driven by idiosyncratic productivity shocks, higher mobility implies higher average productivity. Thus, higher moving
costs imply lower average productivity. Interpreted in this manner, our model potentially
has implications for how unemployment benefits should respond to business cycles, an issue
that has received some attention recently (see for example Mitman and Rabinovich (2011)).
Specifically, our analysis suggests that in response to a productivity shock (in our model an
increase in moving costs) unemployment benefits should decrease.
The 2007-2009 recession in the U.S. and its aftermath have brought increased attention
to both the provision of unemployment insurance and to sectoral reallocation. The period
is characterized by (1) unprecedented extensions of the duration of UI benefits, and (2) a
coexistence of a persistently high unemployment rate and an increase in vacancies. The latter
raises the question to what extent job seekers’ skills do not correspond to the requirements
of jobs they search for. The former raises the question to what extent the extensions of
UI benefits limit the reallocation of the unemployed to their best-match vacancies. The
model in this paper offers a fitting laboratory to study such phenomena, which represents
an interesting direction for future research.
30
A
Analytical Results
Here we prove a set of claims made in the text.
A.1
Wages, Queue Length and Productivity
Proof of Proposition 1. Recall that f is a strictly decreasing function of qw,x,i . Moreover, for the value of search to exceed the value of unemployment, it must be that Si (x) >
log(b)/(1 − β). Thus, the right-hand side of equation (18) strictly decreases with qw,x,i , while
the left-hand side increases. Thus, for each (x, i) there is a unique queue length, qw,x,i , which
we denote by qi (x).
Proof of Corollary 1. See Proof of Proposition 1.
Proof of Corollary 2. Rewrite equation (15) using the uniqueness result,
1−β
log(wx,i ) = C(b) + 1 +
βf (qx,i )
Ki (x),
(A.1)
where
C(b) =
A log(b)
− βλU i
1−β
(A.2)
and
log(b)
Ki (x) = A Si (x) −
.
1−β
(A.3)
Note that the term C(b) is common across different productivity levels and different sectors. It is important to keep this in mind in the analysis below. On the other hand, using
equation (18),
Ki (x) =
where r =
ηkA
.
(1−η)(1−β)
r
,
wx,i qx,i
(A.4)
Inserting equation (A.4) for Ki (x) and then equation (19) for w into
31
equation (A.1), respectively, one can get
log yi (x) −
kA
βα(qx,i )
= C(b) +
r
r(1 − β)
+
qx,i βα(qx,i (x))
yi (x) −
kA
βα(qx,i )
−1
.
(A.5)
Without loss of generality let i = 1. Also, notice that the left-hand side of equation A.5
increases with qx,i , while the right-hand side decreases with qx,i . Since y1 (x) = 1 + x, an
increase in x raises the left-hand side of equation A.5 while lowering its right-hand side.
Therefore, the equilibrium queue length qx,1 decreases with x. Using the symmetric production function, it can be seen that the equilibrium queue length qx,0 increases with x.
Therefore, an increase in productivity yi (x) lowers the queue length.
Proof of Corollary 3. Combining equations (A.1) and (A.4), one can write
log(wx,i ) −
1−β
1
+
qx,i βα(qx,i )
r
= C(b).
wx,i
(A.6)
Recall that α is a strictly increasing function. Therefore, the left hand side of equation (A.6)
is an increasing function of qx,i . Thus, wx,i and qx,i are negatively related between different
values of x. Therefore, since the queue length qx,i decreases with productivity, the wage wx,i
increases with productivity.
Proof of Corollaries 4 and 5. Combine equation (4) with Corollaries 2 and 3.
A.2
Impact of Benefits
In Section 5.2.2 we discuss the effects of within sector frictions on the average duration of
unemployment. The following two results summarize the effects of unemployment benefits
in a particular sector.
Proposition A.1 (Queue length and benefits). Benefits raise the queue length at each
32
productivity level.
Proof. Recall that the left-hand side of equation A.5 increases with qx,i , while the right-hand
side decreases with qx,i . Moreover, the benefit level b affects the right-hand side through the
term C(b). Specifically, an increase in b raises C(b), since the first term on the right-hand
side of equation (A.2), A log(b)/(1 − β), dominates the second term βλU i . Therefore, the
benefit level will also raise the equilibrium queue length qx,i for each pair (x, i).
Proposition A.2 (Wage and benefits). Benefits raise the wage at each productivity level.
Proof. The productivity specific wage increases with the queue length (see equation (19)).
Then using Corollary 4, it can be seen that the wage wi,x increases with the benefit level b
for each pair (i, x).
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