Three Essays on Unemployment, Self-Selection and Wage
Differentials
by
Tal Regev
B.A., Tel-Aviv University (1994)
Submitted to the Department of Economics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2006
( Tal Regev. All rights reserved.
The author hereby grants to Massachusetts Institute of Technology permission to
reproduce and
to distribute copies of this thesis document in whole or in part.
Signature of Author
.......................................
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A
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Department of Economics
19 May 2006
Certifiedby........
....................... y/ ... ....
\\
f6harles P. Kindleberg
.......................
Daron Acemoglu
Professor of Applied Economics
Thesis Supervisor
Certifiedby........
Ivan Werning
Assistant Professor, Macroeconomics
Thesis Supervisor
Accepted by ................................
Peter Temin
Elisha Gray II Professor of Economics
Chairman, Departmental Committee on Graduate Studies
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Three Essays on Unemployment, Self-Selection and Wage Differentials
by
Tal Regev
Submitted to the Department of Economics
on 19 May 2006, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
This thesis is a collection of three essays on labor economics from a macroeconomic prospective.
Chapter 1 discusses imperfect information, self-selection and the market for higher education. It explores how the steady trends in increased tuition costs, collegeenrollment and returns
to education might be related to the quality of college graduates. The model shows that the
signaling role of education might be an important, yet largely neglected ingredient in these
recent changes. In a special signaling model, workers face the same costs, but can expect different returns from college. Allocation of ability into skill is determined by the equilibrium skill
premium. Incorporating a production of higher education, the properties of the college market
equilibrium are discussed. A skill biased technical change initially decreases self-selection into
college, but the general equilibrium effect can overturn the initial decline, since increased enrollment and rising tuition costs increase selection. Higher initial human capital has an external
effect on subsequent investment: all agents increase their schooling investment, and the higher
equilibrium tuition costs increase self-selection and the college premium.
Chapter 2 is about unemployment insurance and the uninsured. Under Federal-State law
workers who quit a job are not entitled to unemployment insurance. How does the existence
of the uninsured affect wages and employment? An equilibrium search model is extended to
account for two types of unemployed workers. In addition to the unemployed who are currently
receiving unemployment benefits and for whom an increase in unemployment benefits reduces
the incentive to work, there are also unemployed who are currently not insured. For these, work
provides an added value in the form of future eligibility, and an increase in unemployment benefits increases their willingness to work. Incorporating both types into a search model permits
solving analytically for the endogenous wage dispersion and insurance rate in the economy. It
is shown that in general equilibrium, when firms adjust their job creation margin, the wage
dispersion is reduced and the overall effect of benefits can be signed: higher unemployment
benefits increase average wages and decrease the vacancy-to-unemployment ratio.
Chapter 3 explores the optimal provision of unemployment insurance within a search model.
Adding risk aversion to the standard search and matching model allows for an analytic discussion of the optimal provision of unemployment insurance. The government's capacity to insure
workers is limited by the market wage setting, which gives workers a share in the employment
surplus. When the government provides higher unemployment benefits, the bargained wages increase, and unemployment rises. These equilibrium responses have a negative effect on workers'
welfare if workers' bargaining power is above a certain point, which is lower than the matching
2
elasticity. As risk aversion increases, workers' share in the wage bargain is smaller, and thus the
equilibrium effects are attenuated. The constrained optimal provision of unemployment benefits
is a modification of the Hosios condition for efficient unemployment insurance and highlights
the roles of bargaining and risk aversion. The optimal level of insurance increases with risk
aversion, with the costs of creating a vacancy and with workers' higher bargaining power.
Thesis Supervisor: Daron Acemoglu
Title: Charles P. Kindleberger Professor of Applied Economics
Thesis Supervisor: Ivan Werning
Title: Assistant Professor, Macroeconomics
3
I would like to thank my advisors who have provided support and guidance. I thank Daron
Acemoglu for always being there, for believing in me and for insisting on the finer points.
I
have greatly benefited from his advice, and his energy and generosity have been truly inspiring.
I am also indebted to Ivan Werning for his insightful questions and for prodding me on. I feel
lucky to have had such great teachers, and I look forward to continue learning from them in
the years to follow.
I am thankful for the many people who lent an ear and contributed valuable comments:
to Josh Angrist, David Autor, Olivier Blanchard, Ricardo Caballero, Peter Diamond, Glenn
Ellison, Jordi Gali, Mike Golosov, Guido Lorenzoni and Muhamet Yildiz.
For teaching me
the power of economics, I thank the many dedicated teachers throughout the years and, in
particular,
my first teachers, Yoram Weiss and Ariel Rubinstein.
This journey would not have been possible without the support of many caring individuals.
I was honored to share these years with my classmates and friends Veronica Rappoport, Karna
Basu, Ruben Segura-Cayuela and Ivan Fernandez-Val. I wish to thank my family: my sisters
for their endless support; Mor, the love of my life, for making it all happen; and Carmel, Omer
and Yotam for their patience and the joy they bring to my life. Finally, my gratitude goes to
my mother and father, to whom I dedicate these essays.
4
Contents
1 Imperfect Information, Self-Selection and the Market for Higher Education
.......................................
7
1.1
Introduction
1.2
Model ..........................................
11
1.2.1
Setup
12
1.2.2
Equilibrium Definition .............................
13
1.2.3
Types of Equilibria ...............................
13
7
......................................
1.3 Equilibrium Analysis ..................................
16
1.3.1
A Change of Variables: from Quantities to Qualities ............
17
1.3.2
Mixed Strategy Equilibrium .
17
. . . . . . . . . .
. . .......
1.4 Results: Self-Selection, Skill Premium, and Human Capital Investment ......
1.4.1
Self-Selection
1.4.2
Skill Premium
...........
.......................
. . . . . . . . . . . . . . . . ...
19
21
1.4.3 Investment in Human Capital .........................
22
1.4.4
25
Welfare
.....................................
1.5 Endogenous Cost of College ..............................
1.5.1
College Production
...............................
1.5.2 Equilibrium in the College Market ......................
1.6 Conclusion ...................
1.7 Appendix ...................
26
27
. 28
.....................
29
Introduction . . . . . . . . . . . . . . . . . . . ...................
5
25
...................
2 Unemployment Insurance and the Unemployed
2.1
. . . . . . .. . .. . . .
19
38
. 38
2.2
Simple Two-Type Model ................................
41
2.2.1
Model.
41
2.2.2
Equilibrium: Definition and Characterization ................
45
2.2.3
Total and Differential Effects of Benefits ...................
50
2.3 Endogenous Entry Decision ..............................
2.3.1
Solution.
2.3.2
Discussion.
51
54
...................................
56
2.4 Conclusion .......................................
58
2.5 Appendix ........................................
59
2.5.1
Proofs for Section 2.2 .............................
59
2.5.2
Proofs and Derivations for Section 2.3 ....................
62
3 Optimal Unemployment Insurance with Search
67
3.1 The Economy.
69
3.2 Equilibrium .......................................
71
..........
3.3 Unconstrained Optimum ...........
3.3.1
Decentralizing the Unconstrained Optimum .................
3.4 Government Constrained Optimal Policy .......................
3.4.1
Constrained Optimum with Outside Financing ...............
3.4.2 Interpretation .
3.5
.................................
.
74
76
78
79
85
... . 86
Conclusion
6
Chapter 1
Imperfect Information,
Self-Selection and the Market for
Higher Education
Summary 1 This chapter explores how the steady trends in increased tuition costs, college
enrollment and returns to education might be related to the quality of college graduates. The
model shows that the signaling role of education might be an important, yet largely neglected
ingredient in these recent changes. In a special signaling model, workers face the same costs,
but can expect different returns from college. Allocation of ability into skill is determined by the
equilibrium skill premium. Incorporating a production of higher education, the properties of the
collegemarket equilibrium are discussed. A skill biased technical change initially decreasesselfselection into college, but the general equilibrium effect can overturn the initial decline, since
increased enrollment and rising tuition costs increase selection. Higher initial human capital
has an external effect on subsequent investment: all agents increase their schooling investment,
and the higher equilibrium tuition costs increase self-selection and the collegepremium.
1.1
Introduction
Since mid-century there has been a steady increase in the returns to college, which was interrupted between 1971 to 1979, and then became quite steep after the late 70[ (see figure[1-1]).
7
This change reflects an increase in collegegraduates' wages and at times a decline in high-school
graduates wages. 1 The number of college graduates followed this trend closely, rising for 24
years and older from 8% in 1960 to 25% in 1997 (see figure[1-2]). 2 The third phenomenon is
an increase in college costs, which have increased more than two fold during the past thirty
years (see figure.[1-3]) This paper explores how these changes in the college market are related
to the quality and self-selection of collegegraduates. While issues of selection of heterogeneous
workers across educational levels and occupations has received much attention, these studies
typically assume that workers select higher education according to an exogenous distribution
of costs and returns. The growing body of evidence suggesting a large signal component to
education has been largely ignored.3 Nevertheless, the signal role of college education might
be an important ingredient in the recent changes in the demand for college education and the
structure of the college market. This paper constructs a signaling model of the demand for
skills, skill premium and college market equilibrium.
Perhaps one reason the original Spence (1973) model was not fully applied to the college
market is the assumption that workers' ability is negatively correlated with their costs of attending college. It seems unlikely that low-ability workers face substantially higher costs than
able workers. On the contrary, if a large part of the cost of going to collegeare forgone earnings,
the correlation between ability and costs could even be positive.4 This paper departs from the
standard signalling framework by assuming workers face the same costs but can expect different
returns from college, due to the fact that an able worker is more likely to be perceived as such.5
Allowing some information on a worker's ability reintroduces a form of single crossing property,
which is responsible for the resulting self-selection. When the information available to firms is
'According to Juhn, Murphy and pierce (1993) the college to high school wage ratio was 1.45 in 1979, and
soared to 1.9 in 1986. Katz and Murphy (1992) show a decline in the wages of low skill workers between 1979
and 1987. See also Goldin and Katz (2000), Katz and Autor (1999), Autor, Katz and Kearny (2005) and Card
(1999).
2
US Bureu of the Census, 1998
3Tyler, Murnane and Willett (2000) find that a General Educational Development signal increases wages by
10-19% net of human capital effects. Lang and Kropps (1986) provide more direct evidence on signaling as an
equilibrium phenomenon, and show compulsory schooling laws affect attendance decisions even for non-marginal
agents. In the same spirit Bedard (2001) shows that high school dropout rates increase when the pool of high
school graduates deteriorates.
4A point stressed by Griliches (1977) for instance. Altonji (1993) provides evidence that the returns from
post-secondary education are higher for more able workers.
5See also Arrow (1973b) for educational signaling. See Weiss (1983) and Hvide (2003) for a setup similar to
mine, but where workers don't know their abilities.
8
limited, they still need to infer a worker's ability using the composition of his skill group, thus
preserving the value of education as a signal.
We explore the properties of this special signaling equilibrium, and show that a separating
equilibrium does not exist. The fact that a Riley (1979) critique does not apply provides
some justification for focusing on the totally mixed equilibrium.6 This equilibrium also seems
plausible from an empirical viewpoint, and is more interesting because the resulting distribution
of abilities within skill-levles is non-trivial. We show that an equilibrium with positive ability
selection into skill arises when it is socially inefficient for the low-ability worker to invest in
schooling. We proceed to show that self-selection increases when the cost of college is higher
or when productivity of college graduates is lower. Finally we show that total investment in
education may actually increase with college costs. Behind this result are strategic externalities.
Lower net returns to college make schooling less attractive for all workers. When low-ability
workers shift away from college, the signaling value of going to college increases and the value
of remaining unskilled simultaneously declines. If the relative increase in the value of skill
dominates the original increase in tuition costs, enrollment rates can be higher. In such a case
the demand for college will be upward sloping.
The model also predicts human capital externalities. All workers invest more in their human
capital when the average human capital is initially high. The equilibrium composition of each
skill level is given such that workers are indifferent between the skill choices, and it is invariant
with the initial level of ability. When there are more able workers in the population, all workers
must increase their investments in education to keep the same equilibrium proportion intact.
Groups with higher initial human capital will invest more in education, a fact which has a
dynamic implication of increasing divergence between groups. This result is reminiscent of the
statistical discrimination literature. 7 However, the multiple equilibria and coordination failure
assumption driving these results is not present here. Closer in spirit is Acemoglu (1996), where
a different mechanism results in increasing returns to human capital.8
To close the college market, a college supply function is introduced.
We assume college
6In the standard setup, the only equilibrium surviving refinements, such as the Cho Kreps (1987) critirion, is
the Riley equilibrium which is the best separating equilibrium.
7
Arrow (1973a), Phelps(1972), Coate and Loury (1993)
8In Acemoglu (1996) the increasing returns stem from ex-ante investment and costly search.
9
production uses some scientists who are in limited supply, and whose wage is determined in
equilibrium. Combining this with the demand for college, we solve for the equilibrium in the
college market. We use this framework to describe the recent changes in the college market
with the augmented selection prediction. An increase in initial human capital has no first order
effect on selection, but through increased investment the price of college increases, resulting
in higher self-selection, lower wages for unskilled workers and a higher college premium. More
complex are the effects of a skill-biased technical change (SBTC), which is widely believed
to have taken place during this period.9 The direct increase in the skill premium following
a SBTC is undermined by the lower quality of workers who now choose to become skilled.
Within-skill (residual) inequality increases, while the collegepremium declines, as has been the
case throughout the 70.
When enrollment and tuition increase, the general equilibrium can
overturn the initial decline in selection, resulting in an increase in the skill premium. Such
higher self-selection can possibly account not only for the wage increase of high-skill workers,
but also for the reduced wages of the uneducated, a fact which otherwise remains a puzzle.10
Almost all studies find a positive selection bias. 1l l More controversial is the evidence regarding the dynamic change in selection over time. Cameron and Heckman (1998) report a decline
in the quality of college graduates. Juhn Kim and Vella (2005) suggest a smaller decline in the
quality of younger more educated cohorts. Card and Lemieux (2001) find new cohorts to have
higher returns, but do not interpret this as an increase in the ability component. Murnane,
Willett and Levy (1995) find an increase in the ability composition of educated workers.l2 How9
See Acemoglu (2002) for a comprehensive study. See also Autor, Katz and Krueger (1998) and Autor,
Levy and Murnane (2003) on the skill content of computerization. Krusell and al. (2000) use capital-skill
complementarities to estimate the skill biased change. Berman Bound and Griliches (1994) bring independent
evidence from the manufacturing sector pointing to an increase in SBTC in the 70l[.
l°See Autor Katz and Kearny (2005) for an estimate of the declining prices for low skill workers. Autor, Levy
and Murnane (2003) suggest possible explanations for the polarization of the labor market can be computerization. See Acemoglu (1999) for an alternative explanation of how a SBTC changed the composition of jobs,
reducing the wages of low skill workers.
1 These conclusions aggregate over findings of large selection bias (Blackburn, 1995), and a small negative bias
(Angrist and Krueger, 1991). The measure of selection is usually derived from a comparison of the OLS estimate
with the unbiased IV estimate of the returns to skill, and the size of the estimated bias depends on institutional
change used as the instrument. A different identitification is given in Ashenfelter and Rouse (1998) who use a
sample of identical twins to estimate a small upward ability bias. See Card (2002) for a complete survey.
12Cameron and Heckman (1998) find that the location of average ability of graduates in the baseline distribution
has steadily declined from .92 to .85 during the course of fifty years (for the cohorts born in 1916 to those born
in 1963). Card and Lemieux (2001) interpret their findings as arrising from complementarities between cohorts.
Murnane Willet and Levy (1995) use direct test scores measures to control for the ability bias.
10
ever, these studies are primarily concerned with the returns to skill and hence estimate only the
composition of the skilled labor force. Nevertheless, this model predicts changes in the relative
composition of the skilled and unskilled pools. The strongest evidence in favor of increased
ability selection into skill is given in Steinberger (2005). Using direct new data on test scores in
1979 and 1999 he reports a 4% rise in ability for male graduates with a simultaneous decline in
the ability of high school graduates. A direct test of the mechanisms leading to such increased
selection is not available yet.
Treatments of the whole college market are few. Hoxby (1997) investigates the supply side of
college and its increased differentiation and competition. Rothschild and White (1995) present
a pricing model of college. There is ongoing interest in the demand side and, in particular,
in the effect of tuition subsidies (Feldstein (1995)). Hendel et al. (2005) look at the effect of
subsidies on inequality in a signaling equilibrium with credit constraints. There has not been
an attempt to look at the college market equilibrium within a signaling perspective.
The rest of the paper proceeds as follows. Section 1 presents the model and solves for the
signaling equilibria. Section 2 analyses the mixed strategy equilibria. Section 3 discusses selfselection, the skill premium, the investment decision of workers and welfare. Section 4 extends
to college production and the equilibrium in the college market. Section 5 concludes.
1.2
Model
Workers of heterogenous abilities decide whether to acquire costly education which is productivity augmenting. Firms observe the worker's schooling choice and an additional proxy on a
worker's ability. Wages are determined in a competitive labor market. We assume for simplicity that workers of various abilities and skill levels are perfect substitutes. This simplifying
assumption makes equilibrium wages depend, in effect, only on the quality of workers with the
same observed skill level, and leaves out the standard quantity effects.13
' 3 A natural extention would be to allow for some complementarity between skill levels, and also incorporate
the quantity effects. See Moro and Norman (2004) for a general equilibrium model of missing information and
production complementarities.
11
1.2.1
Setup
The economy consists of a continuum of mass M of firms, and a unit mass of risk neutral
workers with heterogenous abilities. We identify each type of worker i E {h, l} by his ability
ai E
=
ah, al}. A worker's type is private information. The prior distribution of types in
the population is common knowledge and is given by (p(ah),p(al)) = (f, 1 - f).
A worker of type i chooses his skill level e E E =
probability distribution over the set of actions, ai : E
-
L, H). A strategy for worker i is a
[0,1].
All firms observe each worker's skill choice e and an additional noisy signal s E S = {h, I}
on the worker's true ability. Let the likelihood of a worker i emitting a signal s be given by the
distribution Pi S
-
[0, 1] which is also common knowledge. A firm's strategy is a wage offer
We(S) where w E x S
R+.
A firm that hires l workers of type i and skill level e produces output via a linear technology
Y
=
ji ai(Al + L) where
A > 1. That is, each worker's marginal productivity
is his ability,
enhanced by a multiplicative premium of ) if the worker is skilled.l4
Each firm's payoff from hiring a worker with observable (e, s) is given by the quadratic loss
function ir(e, s, w) = (we(s)-AIE (ale, ))2 where Ai = Aif e = H and 1 otherwise, which is the
standard shortcut to replicate a competitive labor market outcome. Worker's i payoff from any
pure action e is given by ui(e, s, w) = we(s)- C(e), where we normalize C _ C(H) > C(L) = 0
Note workers of different types get the same utility. Their payoffs only differ in expectation,
Eui(e,s, w) = E pi(s)We(s) - C(e).
We assume the signal is informative in the sense that the likelihood of a good signal for the
high-ability worker is larger than the likelihood of a good signal for a low-ability worker:
Assumption Al (Monotone Likelihood): PhI increases with s.
The assumption that signals depend only on the worker's ability and not on his skill choice
is made for simplicity. 1 5 Throughout we use lower case {h, l} to denote the signal on abilities,
while skill levels are denoted by upper case {H, L}. Whenever there is no ambiguity we will use
14Wecould allow skill to affect workers' productivities differentially, i.e., have Al 5 Ah without any substantial
changes to the results.
' 5A realistic modification will assume that the signal is more precise for high skilled workers, that is: PhlH >
PhIL and PIIH< plL.This will result in different within skill variance.
12
the shortcut notation (ai, 1 - ai) - (ai(H), ai(L)).
1.2.2 Equilibrium Definition
Let p(. e, s) denote the posterior distribution over types {ah, al} after observing (e, s).
Definition 1 A Perfect Bayesian Equilibrium of this game is a tuple {a*, w*(s), t*(. le,s)} of
workers' strategies and firms' wage offers and beliefs such that:
1. Workers maximize their expected payoffs:
Vi,4 ( ) = argmax 8sp(sai) [i(H) (wH(s) - C) + (1-
i (H))wL(s)]
2. Firms pay the workers their expected productivity:
w*(s) = AI Z
si .*(aie s)ai.
3. Posterior beliefs are Bayesian wherever possible:
ip*(aie,
s) = p(ai)a*(e)p(sLai)/(Ei, p(ai,)ai*(e)p(sIai,))
if E, p(ai,)a (e)p(sai,) > 0, and any probability distribution over {ah, al) otherwise.
Denote the expected wage a worker of type ai gets if he chooses skill level e by Ewe(ai)
Esp(slai)w*(s).
A worker's expected wage from a skill choice e depends on his own type
through the probability term p(slai), and on the equilibrium composition of his skill group
through the equilibrium wage term w (s). This is the feature of the model which allows workers' returns to increase with ability, reintroducing a type of single crossing property, while
keeping the information externality intact. The inferred ability of a worker still depends on the
equilibrium composition. If in equilibrium the posterior probability of finding a high-ability
worker is higher in the skilled than in the unskilled group, then acquiring skill has an additional
signaling value.
1.2.3
Types of Equilibria
A worker of ability ai compares his expected payoff when he acquires skill, EwH(ai) - C, and
his expected payoff when he does not, EwL(ai), and chooses the education level with the higher
returns given equilibrium play of all other agents. If he is indifferent between the choices then
13
any mixed strategy is (weakly) optimal. Each type's strategy represents the fraction of the
population of that type that plays the pure strategy skill choice. Because we are interested in a
non-trivial composition of skills, we are mainly interested in the interior (fully mixed strategy)
solution. An additional theoretical justification for studying the interior equilibrium is that
there is no separating equilibrium, as shown below.
We briefly characterize the full equilibrium possibilities. We begin with two Lemmas:
Lemma 1 w*(s) increases with s.
Proof. we(s)
AI Eai
*(aile, s)ai is an increasing function of beliefs and by the monotone
likely hood assumption (A1), beliefs are an increasing function of the signal, s, i.e., for all equilibrium beliefs formed by Bayes rule, *(ahIH, h) > *(ahIH, 1) and y*(ahIL, h)
with strict inequality for interior beliefs y*(ahl.)
,*(ahL, I)
{0, 1}. ·
Lemma 2 Ewe(ah) > Ewe(al).
Proof. Follows from previous lemma by the monotone likelihood assumption. ·
We use these to prove:
Proposition 1 There is no equilibrium in which the high-ability worker reveals himself.
Proof. Assume to the contrary that ah reveals himself in skill level e. By Bayes rule
p*(ahje,
h) = op*(ahIe,1) = 1 so that we(h) = we(l) = we. By the previous Lemma, in the other
sector e we have Ew(al)
Ew(ah) • Ewe(ah) = Ewe(al) = we where the second inequality
follows from ah's choice and the equality from the beliefs being *(ahle, ) = 1. This contradicts
Ew-(al) > Ewe(al).
There can be no separating or semi-separating equilibrium in which the high-type reveals
himself. The proof provides the intuition: if there was an equilibrium in which the high-type
reveals himself in e, then Bayes rule would dictate that firms believe a worker is of high-ability
when they observe skill choice e, regardless of the ability signal. But if the ability signal has
no power, there is nothing keeping the low-ability type from imitating the high-ability type. In
14
fact, he will weakly prefer to do so, since he always does worse than the high-ability type by
choosing e where the ability signal has power. 1 6
For the sake of completeness we now characterize the full set of equilibria. In what follows
let A_ C-al(A-1)
ah-al
'-
Ph
b-
--Ph
Proposition 2 Characterizationof equilibriain terms of workers'strategies.
(i) (Fully mixed strategy): An equilibrium with (l,
4ab
(a+b) 2
h) E (0,1)2 exists only if
i(-A+) A
>
'
(ii) (Poolingon H): An equilibriumwith (,
`ah)= (1, 1)
exists if f
P P + (1-Ph)(1-Pl)
1-ph+l-pl
Ph+Pl
>
A
A'
(iii) (PoolingonL): An equilibrium
with(l,h)
= (0,0) existsif f - (Ph
+ (--p)+(1-p)
A.
(iv) (al reveals himself in L): An equilibrium with a E (0,1), ah
=
1 exist iff
there is a
solution to PhPlPh
+
( 1 --p)(-Ph)
= A
+aLl
p
(1-ph)+al(l-nP)
A
(v) (al reveals himself in H):: An equilibrium with cl E (0, 1),ah = 0 exist if f there is a
solution to -h-pl
(1- )(-Ph)
= A (whichcan happenonly If A < 0).
ph+(-al)pl -_Ph(1-ph)+(1-a1)(1-pl]
And there is no other equilibrium.
Proof. See Appendix. ·
There are basically three types of equilibria: the fully mixed strategy, two pooling equilibria, and two equilibria where the high-ability worker plays a pure strategy and the low-ability
worker mixes (and hence reveals himself). These equilibria exist in different regions (not mutually exclusive) of the parameter space. The four dimensional parameter space consists of the
information probabilities Ph and pi, the skill technology term A, and the term A, interpreted
below as the social cost of having low-types invest in school.
Both types can pool on skill if the social cost of low-types investing in skill is small enough.
On the other hand, pooling on L exists if there's a cost associated with low-type investing in
skill, or if the gains from investing are small enough. However, these two pooling equilibria are
not very robust to refinements that use forward-induction type arguments.
16
This result might be viewed as a weakness because it implies a discontinuity of the solution in the neighborhood of perfect information. We address this issue in the Appendix.
15
<
To be concrete, the pooling equilibria fail the divinity criterion (Banks and Sobel, 1987).17
According to the divinity criterion, we can eliminate an equilibrium if we can show that there
are beliefs regarding off-the-equilibrium-path
skill choice for which only one type of worker
would like to deviate. To adapt the divinity criterion to this setting with minimal notations,
we define D(ai, e) to be the set of beliefs which makes type ai strictly prefer deviating to e
over his equilibrium strategy a*. Define D°(ai, e) the set of beliefs for which type ai is exactly
indifferent.
Definition 2 A type ai is deleted for strategy e under the divinity criterion if there is another
type aj s.t. D(ai, e) U D°(ai, e) C D(aj, e).
In other words, if type ai is willing to deviate for a strictly smaller set of beliefs, then firms
should believe that type aj is the one deviating. The pooling equilibrium is thus destroyed.
Proposition 3 The pooling equilibria do not survive the divinity criterion.
Proof. See Appendix.
These refinements, however, can only eliminate an equilibrium which has off-equilibrium
belief assignments. They do not apply to the semi-separating and fully mixed equilibrium
where beliefs are set by Bayes rule. Consider the semi-separating equilibrium. If A > 0 there's
a social cost associated with the low-ability worker getting skill, and the corresponding semiseparating equilibrium has all the high-ability workers investing in skill. If it is socially efficient
for the low-ability worker to invest in skill (A < 0), then the corresponding semi-separating
equilibrium has all the high-ability workers not getting any skill.
We now turn to the fully mixed strategy.
1.3
Equilibrium Analysis
In the fully mixed strategy equilibrium, each worker type is indifferent between the skill choices.
To gain more intuition regarding the forces at work, we think about the solution in terms of the
' 7Note that a pooling equilibrium does not fail the slightly weaker Cho-Kreps (1987) intuitive criterion. A
pooling equilibrium fails the intuitive criterion if deviating is equilibrium-dominated for the low-type but the
high-type would prefer to deviate once the firm's beliefs assign probability zero to a low-type deviating. In our
case, a low-type will like to deviate if the firm strongly believes a deviator is high-ability. Hence the first part of
the criterion is never satisfied.
16
resulting quality in each skill level instead of the investment strategies ai. Explicitly writing
the two equilibrium equations in terms of the quality variables we see there are at most two
mixed-strategy equilibria. We then discuss the properties of these equilibria
1.3.1
A Change of Variables: from Quantities to Qualities
Consider the following change of variables. Let 0 denote the probability of finding a highability worker in the skilled pool, and similarly let
be the fraction of high-ability workers in
the unskilled group. That is
0p(a = ahjH) fh + (1-f)
(1.1)
forh - (1 - f)o'l
?J These proportions
p(a =ahIL)
-
) (1-f)(
of able workers in each skill group can be interpreted
as the endogenous
quality of the two groups. When firms make up their wage decision, they take into account
these variables, indicative of the group's composition. The they update this 'interim-prior'
based on the additional individual ability signal. We will.see how this change of variables turns
out to be quite useful, as the predictions regarding investment are limited, but the equilibrium
forces are more easily interpreted through these quality variables.
1.3.2
Mixed Strategy Equilibrium
To solve for the equilibrium, we assume firms use workers' equilibrium mixing v* to form their
correct beliefs p*(.le, s) = p(ai)a(e)p(slai)/
(ai
p(ai,)o*(e)p(slai,))
.
can construct expected wages for each skill level
leveland ability signal, we(s)
Given these beliefs, we
=
A
i *(aie, s)ai.
Finally, workers take these wages as given when making their skill decision. In the fully mixed
strategy equilibrium, the expected returns from a worker's choices must be equalized. This
boils down to two equilibrium equations, one for each type of worker i,
Zp(sai)wH(s) - C = Zp(s ai)4L(s).
S
s
17
(1.2)
Plugging in the expressions for wages, these two equations solve for (ah, al) provided they are
between zero and one. If they are not, then a fully mixing equilibrium does not exist. Using
our change of variables, the equilibrium equations simplify. After some algebra we get the
equilibrium conditions as
l
Cph
ph + (1 - )PI =
(1l-Ph)
)b(l-Ph)+(1-)(1-pl)
'Ph
Ph+(1 -
(C- al - 1))
(ah - al)
__(1.3)
2k(1-Ph)
(1-ph)+(1-)(1-pl)
(C-all -1))
(ah-al)
(1.4)
Each equation corresponds to an ability signal: the first equation equates the rewards for a
high-signal (s = h) from getting skill or remaining unskilled, and the second equation does the
same for a low signal (s = 1). In other words, the composition of the abilities in each skill-level,
X and
0b,must be such that the returns to an ability signal are the same across skill levels. Both
conditions impose a positive relationship between the quality of workers in the H and L skill
levels. This is because the investment decision of high-ability workers has a positive external
effect on the value of the skill group. To equate the returns across skill choices, 0 and Obmust
move together.
These expressions also disentangle the real value of skill from the informational value. Looking at each equation separately, the last term on the right, A-
(-a(A-1))),
is the (normalized)
(ah-al)
net cost of having low-ability workers invest in skill. It can be thought of as the social cost of
having imperfect information . The other two terms of the form F(q) = qpl_l
)p
are the in-
ference terms. The distribution of abilities must be such that the value added from information
just compensates for the cost.
This representation also highlights the impact of having some information available to firms.
Suppose they had no information regarding workers' ability. In such a case, with Ph = pI, the
two conditions collapse to one and the equilibrium cannot be pinned down. Adding some
information introduces a way to differentiate the returns of the two types, and substantially
reduces the number of equilibria.
In fact, since the first equation is increasing and concave in (4, ) space, and the second
equation is increasing and convex in (4', q~)and since workers' mixed strategies uniquely define
the Bayesian beliefs, this proves:
18
Proposition 4 There are at most two mixed-strategy equilibria.
1.4
Results: Self-Selection, Skill Premium, and Human Capital
Investment
In this section we explore the implication of the equilibrium allocation and wages for our objects
of interest. We show that self-selection arises if and only if it is inefficient for the low-ability
type to invest in skill. As for the comparative statics, a higher cost of education increases selfselection, while a skill-biased technical change reduces it. We show how self-selection translates
directly to the skill premium, look at investment and how it responds to price changes in the
economy, show how investment increases in the initial level of human capital, and finally end
with a short discussion of welfare.
1.4.1
Self-Selection
Any solution (*, 0*) can be characterized by the degree to which high-ability persons are more
concentrated in the H sector. We define an index of self-selection as the difference between
the proportion of high-ability persons in the H sector and their proportion in the L sector.
Definition 3 Let the measure of self-selection be X -
7.
We say that there is "self-selection" or "positive sorting" when the fraction of high-ability
workers is higher in the H sector than in the L sector, that is, if
> O or alternatively ah > o0.
A necessary and sufficient condition for self-selection to arise is the following condition, which
we will hereafter assume,
Assumption A2: It is inefficient for the low-abilitytype to invest in skill (C-al(A-1)
< 0).
This represents the net-normalized cost of having low-ability workers pretend to be highability. To make the model interesting, we want to assume this cost is positive, that is, if
information were perfect, low-ability workers would not find it rewarding to invest in skill. This
assumption turns out to be crucial for self-selection to arise, as we now prove:
19
Proposition 5 Self-selection arises if f
that is,
>
Proof.
X
it is inefficient for the lowest type to invest in skill,
C > al( - 1).
Assume C > al(A - 1) : C-ah-l1) > 0. Together with the first equilibrium
equation [1.3] we have
[1.4] imply
+(--)
1 1)Pj(
)p > 'h
ap+(
1--
+h )p The two equilibrium equations [1.3] and
+(
)p
-(-
P1 = lkPh+ 1-0 P1 l1PPh- 1
holds iff (1-ph+(1-0)PI) <I)
< (1--kph- 1'-)p>)
P1
>
·
)p,
P1
So that the last inequality
0
-A X q >> + (or
(or h >
> sl )- a
Assumption A2 basically assures us that selection goes the right way. The assumption
serves the same function as the standard single crossing assumption that costs decline with
ability. However, it is not its natural analogue. A first guess would have been that Assumption
1 (monotone likelihood assumption)
is sufficient for self-selection since it assures us that the
high-ability worker gets higher returns from skill. However, recall that Assumption 1 implies
that the high-ability worker gets higher returns on any skill level (see Lemma 3).
We could also assume that it is efficient for the high-ability type to invest in skill (ah <
,ah - C). However, we do not need this assumption. There can be a mixed equilibrium with
some fraction of both-types investing in skill even if it inefficient for both of them to invest.
For our next result on the comparative statics of self-selection, we need the following condition placed on the parameters:
Condition 1 T-:;Phpl
2
(fPh+{1-f)P ) 2
Condition
(1-Ph)(-pl)
1 < ((-f)ph-(1-f)pl)
Proposition 6 self-selection increases and the quality of unskilled workers declines ((0 - ~b)T
and
/
1) with:
(i) increased costs (C T );
(ii) decreased returns to skill (A 1);
(iii) decreasedproductivity of high-ability worker(ah );
(iv) increased productivity of low-ability worker (al T), provided it is efficient for the highability type to invest in skill,
if Condition(2) holds.18
18This condition is only a convenient sufficient condition to prove this result. Rigorous simulation suggest the
result holds under much narrower restrictions.
20
Proof. See Appendix.
Anything that increases the costs of skill for the low-ability worker reduces his relative
investment and increases self-selection. The empirical implications are straightforward. If we
believe that there has been an increase in the real skill premium, we should expect the relative
quality of high-skill workers to decrease, as there will be relatively more low-ability workers
trying to gain from the higher returns. The quality of low-skill workers should decline in
absolute terms. On the other hand, the on going increase in education costs increases selfselection, that is, the demand for college by quality applicants increases with tuition costs.
1.4.2
Skill Premium
Wages in this economy depend on the composition of the skill group through the information
externality. In equilibrium wages are constructed as the expected productivity of an individual
with a certain skill level and ability signal. Workers benefit from an increase in the quality of
their skill group regardless of the specific signal they eventually emit:
Lemma 3 w*(s) increases with p(ahIe) for all s.
Proof. By construction.
All workers within a skill level benefit from its quality. Average wages increase with the
quality of the group not simply as an averaging result. Rather, the higher average wage truly
comes from increased wages for all workers in the skill group. The composition of the group
actually affects prices and is not a phantom "composition effect" we try to control for in our
wage regressions.
Keeping this in mind, it is natural to define the skill premium as the difference between
expected (or average) wages of a skilled and unskilled worker, where expected wages are given
by Ewe - Z. p(aile) Esp(sai)w*(s)
= Eai p(aiJe)ai . The actual realization of the wage is
just some noise around these means.
Definition 4 The skill premium is defined as EWH - EWL.
And so we have:
Proposition 7 The skill premium increases with selection.
21
Proof.
Writing out the expressions for expected wages we have EWH
and EWL =
Oah +
(1 -
=
A (bah + (1 - O)al)
)al so that the skill premium, EWH - EWL = (AX - Vb))(ah
- al),
increases with 0 - 0.
Any parameter which increases selection, increases the skill premium. From proposition (6)
we know what these are under some condition. However, we can prove a stronger unconditional
result:
Proposition 8 (i) An increase in the real returns to skill, A , or an increase in the ability
gap, (ah - al)T, directly increases the skill premium but has an indirect dampening effect on the
skill premium through decreasedselection.
(ii) Higher investment costs increase the skill premium through the indirect increase in
selection.
Proof. See Appendix. ·
The first result suggests that a skill-biased technical change would have created larger wage
dispersion without the endogenous selection adjustment. The relative shift of low-ability workers into education reduces the relative quality of the skilled. Hence, a firm's willingness to pay
for the higher productivity, A,is diminished. The second result says that an increase in education costs improves the quality of college graduates and thus indirectly increases their wages.
Costs, which are uncorrelated with abilities or wages, turn out to have an effect on wages. This
result highlights an empirical implication of the model. We cannot control for selection when
looking for the skill premium because selection is part of what drives wages.
1.4.3 Investment in Human Capital
The comparative statics results for investment are not as clean. To see this we can back out our
investment variables al and h, which mechanically decrease with each of the quality variables
rh
VI)(1.5)
=
1-y
22
Total investment in education is therefore
I - fah + (1- f)a=
(1.6)
which has an ambiguous sign when we take derivatives with respect to cost, productivities and
even the "real" skill premium A.
This ambiguity is interesting nevertheless. The implication is that an increase in the cost of
education might actually increase the demand for education. Why would this happen? When
costs increase, skill becomes less attractive for both types of workers. However, when the lowability types retract from school, the quality of skilled workers improves. Firms are willing to
pay a higher wage for a worker of higher expected ability. This increase in the value of skill is
due to an increase in its value as a signal on ability. This increase in the relative value of skill
could potentially dominate the absolute increase in the cost of skill. Figure [1-5]presents such
a case. Wherever costs increase investment, we have an upward sloping demand curve.
The one parameter that unambiguously affects investment is f, the ability prevalence in the
population, which represents the initial endowment of human capital. This is an important parameter of the economy and has a central role in an environment with asymmetric information.
Any inference on behalf of the ignorant party takes this prior ability distribution as the basis for
subsequent updating. To see how a worker's choice depends on this initial ability distribution,
we begin with the following Lemma:
Lemma 4 Initial human capital endowment does not affect self-selection or wages.
Proof. The problem stated in terms of the probability parameters X and b does not involve
the fraction of able-to-unable persons, f.
·
Workers sort themselves into skill levels to make the returns (per signal) equal across skills.
This implies some relationship between the quality of workers in each skill level regardless of
the initial distribution of abilities. The effect on wages follows, since wages depend on the
endogenous compositions and not on the original underlying distribution.
Investment, however, is affected by f.
Proposition 9 Investment of both types of workers increases with initial human capital, f.
23
and gh,we only need to consider the direct
Proof. Since f does not affect equilibrium
effect of f on investment. Differentiating the expressions for investment [1.5] with respect to f
we have dh= = f2
(_ >
0 and
df =
(f) -
) >
2
0.
There are externalities to human capital. A population endowed with more human capital
will choose to invest even further in its human capital. Underlying this result are complementarities between workers' choices. Consider an increase in the population's ability. Since the
equilibrium fraction of able-to-unable workers has to be the same to keep returns equal, then
high-ability workers must increase their investment in skill. However, this entails an increase
in investment of low-ability types due to the complementarities.
This result is extremely relevant when discussing the welfare of disadvantaged groups. Even
a high-ability individual will invest less in education if there are fewer able individuals in his
group. Since any identifiable group is subject to a separate market, we can compare what will
happen to such groups that differ along the ability dimension. In a more dynamic setting,
where investment in education today affects the ability of the next generation, we are heading
toward wage dispersion and increased inequality between groups. To break away from this
course of events we need to invest in disadvantaged groups early to increase their ability to be
productive participants in the labor market. In addition to the direct value added, there will
be the additional positive information externality just discussed.
We haven't referred to any feature such as gender, race or ethenicity as the identifying feature
of a group. However, this is undue caution; what we call ability is really the productivity of
a worker employed in a labor market with some specific technology. It is hardly controversial
to claim that some groups are less productive than others: new immigrants might have some
cultural or language barriers, females might have skills not suited for a predominantly male
industry, etc. The implication of the model for these more concrete examples would be that
the increase in female educational attainment may have also been exacerbated by the increased
share of females who were now well prepared to take part in market production. Their increased
investment pulled into school females who were initially less prepared for market work.
24
1.4.4
Welfare
Welfare, too, has an ambiguous response to changes in prices and productivity. This follows
directly from the ambiguity of investment. The welfare loss (DWL) results from the inefficiency
imposed by the information friction. It is equal to the weighted sum of efficiency-lossfrom
workers investing when they should not, or not investing when they should. While we have
assumed it is inefficient for the low-ability workers to invest in skill (Assumption A2), only now
do we have to specify whether the high-type's investment is efficient or not,
DWL = (1 - f)ol(C - al(A- 1)) + f(1 - ah)((A - l)ah - C) if Aah- C > ah (1.7)
= (1 - f)al(C - al(A- 1)) + forh(C - ( - 1)ah)
if Aah - C < ah.
We could have an interior equilibrium where both types of workers should not invest (and would
not if information was complete,) but in equilibrium they do.1 9
Note also that an increase in human capital investment is not always welfare improving.
We have assumed that the investment of low-ability workers is inefficient, so any investment
on their part reduces welfare. Even if the investment of the high-ability worker is efficient, we
would still need to weigh the relative loss and gain.
1.5
Endogenous Cost of College
We now put the model in context of the college market and think about the general equilibrium
consequences of changing market conditions. Thus far, the signaling equilibrium provided the
demand for education, taking the cost of college as given. The possibility that human capital
investment increases with the cost of investment can create non-standard results in the market.
To see the full effects we need to see how the cost of college is determined
in equilibrium.
We therefore specify how production of education takes place, and solve for the equilibrium
tuition and quantity of students. We then discuss how the market will react to an increase in
skill-biased technology, a change in the college market structure, etc.
' 9 This is the same as in the standard signaling environment.
25
1.5.1
College Production
Tuition cost is endogenized by specifying that the production of skill uses scientists who are
in limited supply. In particular, we will assume that a constant fraction of expenditures is
spent on these scarce resources. This natural assumption allows for a supply curve which is
not perfectly elastic. College expenditure data suggests that college production is highly labor
intensive, with the share of expenditures on research and instruction going around 0.4.20
We therefore add a higher education sector in the following straightforward way. Assume
that production of college graduates, LH, takes as inputs a general aggregate good, Y, whose
price is normalized to one, and some scientists, S, who are in limited supply and earn a competitive wage, w. Production of college graduates, LH, takes the Cobb-Douglas form with the
share of scientists being oa,
LH = SaY
1- a .
(1.8)
Competitive firms sell college education to students at the market tuition rate of C and
therefore face the standard maximization problem
maxCS'Y
-1 '
S,Y
- wS - pyY.
(1.9)
From the first order conditions we can solve for the cost of college, C,
C= X(W)a.
(1.10)
Where X= a- (1 -c ) - ( - " )
The scientists' wage, w, is given by the equilibrium in the scientists' market. From the
firms' maximization, we have the demand for scientists given by
S d = LHW-(1-
)a-1
( 1 . 11)
This must be equal to the fixed supply, S. Substituting for the wage, w, from (1.10) we have
20
The Integrated Postsecondary Education Data System (IPEDS) data.
26
the supply of college given by
1-
1--a
L
(
) -
(
S.
(1.12)
This is a standard upward sloping supply curve which increases with the price, C. It exhibits
economies of scale if the share of scientists is smaller than half (a < 0.5).
1.5.2 Equilibrium in the College Market
We are ready to solve for the college market equilibrium.
Definition 5 The college market equilibrium is given by {a*,w;(s), ,*(. e,s)} U {C}, which
satisfies the signaling equilibrium conditions above and the additional college market clearing
condition,
Zp(aj)ui
= L.
i
Rewriting the clearing market condition using our parameters (, l)), we have
1-a
(C)- 'b(C)
O(C)
-10( (1-)aC
-
1--C,
(-)
.
S,
(1.13)
where the solutions for ((C), ?,(C)) are given by the signaling equilibrium. The demand for
education generally has an ambiguous slope, as we saw in our discussion of investment. If it is
upward sloping, there is a potential for multiple equilibria; however, only one of them is stable.
In the stable equilibrium, the elasticity of demand must be greater than the elasticity of supply.
With the equilibrium in place, we can now look at the comparative statics, and show how
the model explains the recent trends, and what the predictions are for selection. Consider
first a skill-biased technical change (A). Selection first unambiguously declines, with a likely
increase in investment. The increase in college demand increases tuition costs, which, through
the general equilibrium effect, increase selection and counteract the initial decline. The skill
premium likely increases, both from the initial skill-biased change and the second order increase
in selection. Next, consider an increase in initial human capital (f). Investment increases, with
no first order effect on selection. The increase in tuition fees due to increased demand increases
selection and the skill premium unambiguously.
27
Both of these explanations fit the broad facts of the college market: increased tuition,
increased enrollment, and an increased premium for education. These explanations differ along
the new dimension that the model introduces: self-selection. A SBTC has a complex effect on
selection, which can be negative if the direct and general equilibrium effects are strong enough.
An increase in human capital endowment will entail an increase in selection.
The model is consistent with the college market facts. It provides a possible mechanism that
takes into account that workers have heterogenous abilities, and that their choices may cause
an additional selection effect. While there is reasonable consensus that the major changes in
the labor market over the past few years are due to a skill-biased technical change, this model
offers an alternative trigger. An exogenous increase in human capital can lead to the same
observed consequences. The likelihood of such an increase in human capital is left for future
research.
1.6
Conclusion
An equilibrium model of the college market was presented in which demand for education is part
of a special signaling equilibrium of the labor market and in which the production of education
uses scarce scientists as factors.
This paper contributes to the signaling literature by exploring the possibility of self-selection
that is due to differential returns and not differential costs. It shows that an equilibrium with
positive selection arises if it is socially inefficient for low-ability workers to invest in education.
Solving the problem in terms of the quality variables instead of the standard quantity ones turns
out to be very useful, since quality affects prices in an environment of imperfect information.
Stating the problem in this way allows us to derive clean comparative statics, which do not
exist for the investment variables: self-selection increases with the net cost of low-ability types
investing in education.
Finally, the model takes a step beyond the signaling framework by looking at the general
equilibrium implication of the signaling equilibrium. When the production of collegeeducation
depends on scientists in limited supply, the equilibrium wages feed back into collegetuition (and
back into the equilibrium selection). An initial exogenous technological advance which biases
28
skill will result in decreased self-selection.
The increase in high-skill wages, while somewhat
mitigated by the decline in quality, still increases the cost of supplying college education. This
increase in costs works to reverse the original decline in student quality.
The two main results of the paper have important implication for inequality and social
policy. The first concerns the debate around the ongoing expanded financial assistance for
education at state and federal levels. The above analysis suggests that an increase in grants
increases the relative investment of low-ability workers in education. While it may be inefficient
in the short run, it will mitigate inequality. This is in stark contrast to Hendel et al. (2005), who
use a similar framework, augmented with credit constraints, to reach an opposite conclusion.
The second result compares the education decisions of identifiable groups that differ in their
average human capital. The likelihood of going to college increases for all agents in the group
who have an initial higher productivity potential. Individuals with initial low-ability will be
pulled up to earn an education because there are more able individuals in their group. This
suggests that early intervention programs that increase potential market productivity have yet
another benefit. These programs create a positive externality on agents in the same group that
have not been treated by the policy.
The quality of skilled and unskilled labor is an evasive empirical entity. Nevertheless, this
work suggests that we may want to get a better understanding of the ability composition of
skill-groups: not only to correct for such compositional bias of the "real" skill premium, but as
an object in itself. It would be valuable to see how the quality of the workforce is endogenously
determined by the wages rewarded and the tuition costs, and, in turn, how the quality of
workers affects those same prices.
1.7
Appendix
The result that no separating equilibrium exists might be viewed as a weakness because it
implies a discontinuity of the solution in the neighborhood of perfect information. To see
this, assume the parameters are such that workers' optimal choice under full information is
separation. As information gets better the equilibrium will converge to the fully separating
equilibrium, but the limit will not exist. We solve this discontinuity and restore the existence
29
of the separating equilibrium by small behavioral perturbations. In this way beliefs will never
ignore new information completely.
Lemma 5 (Robustness of Proposition 1) Proposition 1 is not robust to small behavioral trembles: If separation is optimal when information is complete then for any small fraction 2e of
workers of each type who randomize between the two skill levels there exists Pho and Plo such
that for any Ph > pho and any PI < plo there exists a separating equilibrium.
Proof.
For any interior beliefs ]p(aile, s) E (0, 1) taking the limits as ph
-.
1 and pl
-
0
we have limwe(h) = Aah and limwe(l) = al so that limEwH(ah) - C > limEwL(ah) and
limEwH(al) - C < limEwL(al) and separation is optimal. To sustain this as an equilibrium
we must have the Bayesian beliefs be interior. But this is always the case with a fraction 2e
of agents randomizing since posteriors are updates on the interior 'interim-priors' given by:
H)=
p = p(ahI
(1-e)f
and p = p(ahIL) =
Ef+(l-EZf
Recall the definition of the likelihood variables ('interim-priors'):
fdh
(
- _ p(ahH) =
fah + (1 - &I
p(aL)= f(1 -ah)
f (1- h)+ (1- f)(1- )'
We can fully express our equilibrium in terms of the four posteriors,
X
-
(ahH, s = h) =
~l -
/a(ahIH,
s--l)
-h -
(ahL,s=h)
s = l)
~1 - y(ahlL,
=
bph+(1- )pl
O(1 - ph)
(1 - Ph) + (1 -I)(1-Pl)
4'Ph
ph +(1-POb)pl
(
h) + ( pl )(
)
Proof. of Proposition 2: Types of Equilibria.
(i) In a completely mixed solution both types must be indifferent, Vi, EwH(ai) - C =
EwL(ai). As shown in the text (section 2) these two equations can be rewritten as (1.3) and
(1.4). These are two equations in (,
4)in the second degree. Using brute-force and explicitly
30
solving we get that a necessary condition for existence is \-(A+1)
A > (.+b)y
(a
. We still need
A-A A+1
to make sure when these solutions translate to probabilities between (0,1).
(ii) We will show that (h, a)* = (1, 1) can be a part of an equilibrium iff
(-Ph)(l-P)
> A Compatible beliefs with these strategies are kh = PhP;
Assign off equilibrium path beliefs to be
Oh
=
PhP
-Ph)+(1-P).
= 0 and AI' = O0.If al prefers H, so does ah, because
he always has higher probability of good signal. Finally, al prefers H if f PlAh + (1 -pl)A
A
+
>
-=PhPhP
+ (-ph)(1-pl)
> A
A
+PI
1--ph+l--pl
(iii) We will show that
beliefs are h
(h,
al)*
=
(0,0) can always a part of an equilibrium. Compatible
+1zhp
Assign off path
Ph+Pl
' = (1--ph)1(--pl)
l-h(-p)
beliefs to be
Oh
= 0 and
1=
O0.
If al
prefers H, so does ah, because he always has higher probability of good signal. al prefers H if
P1(-Oh)+ (1 - pI)(-ilJ) < A, X=Ph+P1
Ph+ + (l-ph)+(l-P)
(l-ph)(+-P) > -A (whichis alwaystrue if A > 0).
(iv) We will show that ah = 1; a* E (0, 1) can be a part of an equilibrium if there is a solution
al E (0,1) whichsolvesp
by
+ (1 -
aP
(1P-h)+¢
,ph)
=
X. The compatiblebeliefsare given
(1-Ph)
01 = (+a
-p); Oh =
; I = 0 If al is indifferent, ah will surely prefer
Ph+all; qYl
(1-ph)+l(l)
H. We therefore only require as indifference condition to hold, p (Ah) + (1 - P) ( 1) = A
h =
which is the condition for al given.
(v) We will show that at E (0,1), a* = 0 can be a part of an equilibrium only if A < 0.
The compatible beliefs are h = 0, ~1 = 0, yh
al indifferent requires pi (-oh)
=
+ (1 - Pl) (-)
pa
tI =
To have
(1-p a)
= A which can only be true for A < 0.
Proof. of Proposition 3 (Pooling is not Divine): We will show that pooling on L fails the
divine criterion if both workers prefer skill over the equilibrium pooling, when firms believe
only able persons acquire skill. For i = h, 1 define gi : [0,1] (1 -Pi) (Ph)+(
(1
[0,1], gi(x) =Pixph+(lh
)p +
Note that gi(x) is increasing and monotone in x, with gi(0) = 0 and
gi(1) = 1, and that gh(x) > gl(x) because of monotone likelihood assumption. The conditions
of the proposition state that Agi(1) > gi(f)+A. Pooling on L implies Agi(0) < g(f)+A.
By the
intermediate value theorem there exist xh and xl s.t. Agi(xi ) = gi(f)+A. Because gh(f) > gl(f)
we have gh(xh) > gl(xl). Except for non generic payoffs this implies xh Z: x l . Assume x i > xj
then the for all xo E (xi, x i ) we have Agi(xo) < gi(f) + A but Ag(xo) > g(f)
+ A. Hence
D(ai, H) UD°(ai, H) = [xi, 1] and D(aj, H) = (xj, 1] with D(ai, H) UD°(ai,H) C D(aj, H).
31
Similarly, pooling on H fails the divine criterion if both workers prefer no-skill over pooling
on H, when firms believe only able persons don't acquire skill. ·
For the proof of proposition 6 we will use the shortcut notation p for the probability of
having a high signal conditional on being in the H group, and q for the probability of having a
high signal conditional on being in the L group, and finally f as the unconditional probability,
That is
= p(s = hH) = ph + (1-)p
= p(s= hL) = ph+ (1-)P
f = p(s = h) = fph + (1- f)pl.
= p=
>f > .
Lemma 6 A2 = 0 > f >
Proof. of Lemma: By Proportions 2: A2 ==
fand
= f(1-ah)+(1-f1-a1)
f(-a
<f
bph+ (1 - q)Pl since Ph
> b
From < f <
'=.
ah
> al. Hence
.Oph + (1-)p
=
fh
>
frah+(1-f)a
>
< fph + (1- f)p<
Pl U
·
Proof. of Proposition 6 (self-selection): By implicitly differentiating the equilibrium equa-
dC
= -
P
1
A(ah--az)PhP
(1-Ph)(I -P) ( )((~)--
nominator is negative by the Lemma. The Nominator is negative since
assumption and (
wehave
)2
>(O
O
<
--
(
l)
The de-
(X)
21
Ph)(Pl
<
(
(1-ph)(1-pl) < (--f)2
by
by the Lemma. Similarly differentiating for 0 and subtracting
(h)()
(l-PP
_ ()+
(-h)
haveFd
+-i >u ))-
P)_
Lemmathis expressionis smaller than ((I--Ph)Il)
pp-
MPh,
(1-ph)(-
(i-p
A
tions (1.3) and (1.4) we get
(f)
-P)
+ (i)
< 0. Butbythe
(1(l--ph)(--p)) =
(1-fPh)-P)2) (A - 1) < 0 by assumption. All of the other parameters, except A, have
exactly the same expression with only the leading term changing a bit with the appropriate
signs. So we only need to check the derivative with respect to A. This is messy, but turns out
the same condition is sufficient for d(O~-) < 0
·
Proof. of Proposition 7 (skill premium): Adding the appropriate A to the proof of propo-
(
>0 *
(AL) (-1P)2
+
(-ph)(
<
dCwhich
Lemma
is proceeding
truebythe Proposition(6). There is no need for condition(1-
sition(6)we have d(+)
which is true by the Lemma proceeding Proposition(6). There is no need for condition1. ·
32
o
College Premium
1.7
1.6
-i-
--
1.5 -: -
---
:
:
/
":
1.2i -------/---1.3 -
~1965~
'-'
.''- 7 .. : "i:.'.-.'.'1-
.-6
'
'--9 2--0
!0
1
Figure 1-1:
Source: National Center for Education Statistics.
(14-1). Median annual earnings (in
constant 2002 dollars) of all full-time, full-year wage and salary workers ages 25[B4, by sex and
educational level: 197172002.
33
College Enrollment
36.0
34.0
32.0
-"..-- ,
22.0
30.0
cE
28.0
26.0
--
24.0
'...
- ,,
20.0
20.0
...............
, . ;-
, .,.
.
-
;
Figure 1-2:
Source: CPS, Percent of the population 14 to 24 years old enrolled in college
34
,
Tuition (2004 Dollars)
25000
20000
15000
. ..........
... .......... .. .................. .......... .................................... ........... ....
I
10000
- Private College
- .. Public College
5000
0
<0
q
t
V
so,
4"
49D
4'
caV
bo
A caei
Qs
qb
c
>YeaO
Q
Academic Year
Figure 1-3:
Source: 1987-88 to 2004-05: data from Annual Survey of Colleges, The College Board,
New York, NY, weighted by full-time undergraduate enrollment; 1976-77 to 1986-87: data
from Integrated Postsecondary Education Data System (IPEDS), U.S. Department of Education, National Center for Education Statistics, weighted by full time equivalent undergraduate
enrollment
35
Wages
I.L
-
-
1.1
1
--- I----
0.9
0.8
1:
'I'
O) 0.7
0D.6
.......
-Z-'_;_>
I~~~~~~~~
0.5
--_
II
I
o~~~~~~~~
++~~~~~~
* +
-+~~~~.
0.4
+~~~~~~~~~~~~
+
+
At
|
s
|
|
t~~~~~~~~~~~
0.3
n
0.3
0.35
0.4
0.45
0.5
0.55
Cost of Skill
Figure 1-4:
36
0.6
0.65
0.7
Investment and Selection
1
.
..
..
.
:1.9
C
0.0
9
[1.7
4
(,O
a)
.
1
crs
~*
~
~
~~
N
0.6
*
~
~
~*
0.5
0[.4
I
+,
,,
,
,,
r
I
I
I>,
.
*.
l
SelfSelsction=4)-
..
."..
0.3
4~~~~~~~~~~.
+
II
v~~~~~~~~
,
*a
·
+
C.2
0.1
In
0. 3
0.35
·
·
·
·
·
·
0.4
0.45
0.5
0.55
0.6
0.65
Cost of Skill
Figure 1-5:
37
0.7
Chapter 2
Unemployment Insurance and the
Unemployed
Summary 2 Under Federal-State law workers who quit a job are not entitled to unemployment insurance. How does the existence of the uninsured affect wages and employment? An
equilibrium search model is extended to account for two types of unemployed workers. In addition to the unemployed who are currently receiving unemployment benefits and for whom an
increase in unemployment benefits reduces the incentive to work, there are also unemployed
who are currently not insured. For these, work provides an added value in the form of future
eligibility, and an increase in unemployment benefits increases their willingness to work. Incorporating both types into a search model permits solving analyticallyfor the endogenous wage
dispersion and insurance rate in the economy. It is shown that in general equilibrium, when
firms adjust their job creation margin, the wage dispersion is reduced and the overall effect of
benefits can be signed: higher unemployment benefits increase average wages and decrease the
vacancy-to-unemployment ratio.
2.1
Introduction
The empirical literature on unemployment insurance has stressed the disincentive to work that
unemployment benefits provide. Meyer (1990) discusses the effect on prolonged unemployment
spells, Cullen and Gruber (2000) show the reduction in spousal labor supply and Feldstein
38
(1978) points to the increase in temporary layoffs.' The concern in these is that the increase
in the value of unemployment will tilt the decision of the marginal worker to opt for the higher
paid unemployment. Raising unemployment benefits reduces labor supply, or alternatively,
increases the wages workers demand.
However, there is also an effect of unemployment insurance (UI) on the unemployed who are
not insured or non-eligible. New entrants to the labor force, unemployed who have exhausted
their benefits, unemployed who had non-stable employment, and re-entrants to the labor force
constitute more than half of the unemployed and are all non-recipients of benefits.2 For them
work has an added value in the form of future eligibility and unemployment rents. Work
by Green and Riddell (1993) and Levine (1993) suggest that this is not simply a theoretical
curiosity. In the first case, disentitlement of the elderly from UI resulted in their withdrawal
from the labor force.3 In the second case, an increase in benefits was shown to reduce the
unemployment duration of the uninsured.4
Given the two opposing incentive effects of unemployment insurance on eligible and noneligible unemployed, can we know what are the total and distributional effects of unemployment
benefits on wages, layoffs and unemployment? Mortensen (1977) pointed out these two opposite
incentive effects of unemployment insurance and analyzed the differential job search of the two
types. His analysis considered only the labor supply decision and concluded that "the predicted
sign of the effect of an increase in benefits on unemployment duration is ambiguous."5 Recently
Krueger and Meyer (2002) restated the ambiguity result and suggested general equilibrium
could possibly magnify the adverse employment effects of UI.
Here we show that neither of these conjectures is robust to general equilibrium. By integrating the two effects into a general equilibrium search model a la Diamond-Mortensen-Pissarides
'See also Topel (1983) for the effects on layoffs and Krueger and Meyer (2002) on a survey of labor supply
effects of Unemployment Insurance.
2Blank and Card (1991) estimate the fraction unemployed who are non-eligible to be 0.6. See also Anderson
and Meyer (1997).
3The authors focus on adverse selection issues, but in fact their evidence is very relevant here, as the removal
of benefits reduced the value of employment, which resulted in the subsequent decline in labor supply.
4Levine attributes this to the substitution between the insured and the uninsured. Given the increased
benefits, the search of the insured decreases and facilitates job finding by the uninsured. However, his hypothesis
is observationally equivalent to the one posed here, that the uninsured search harder (and are more likely to
work) because of the added value in employment.
5Mortensen (1977) pg. 506
39
(Diamond 1982, Mortensen and Pissarides 1994, Pissarides 2000) we verify that the differential effects on the two groups still exist in general equilibrium, but are muted due to some
form of cross-subsidization. We also find that the total effect is not ambiguous, and is the
standard one: increasing generosity of UI increases wages and unemployment. An additional
outcome of the analysis is the derived rate of insurance in the economy. Since workers who
are eligible for unemployment insurance tend to have longer unemployment durations, they are
disproportionately represented among the unemployed.
The driving force creating wage dispersion is that benefit eligibility determines the Nash
bargaining outcome. Given complete contracts, negotiators' bargaining position is determined
by their initial conditions.6 Eligible negotiators are entitled to benefits and can bargain to a
higher wage. This is the "pure entitlement" effect, which non-eligible negotiators lack. When
bargaining takes place over future flows, an additional "future entitlement" effect comes into
play. Non-eligible negotiators experience an additional gain from work in the form of future UI
entitlement and are therefore willing to accept even lower wages.
When the two unemployed types coexist, the differential effects are reduced. A searching
firm is expecting to meet an average type. If a worker who demands lower wages is actually
matched, the firm experiences a gain relative to its expectations. Sharing these gains with the
low wage worker increases his wage somewhat, but leaves the overall effect of benefits on his
wages negative.
The total (or average or expected) effects of benefits on wages and employment, turn out to
be the standard ones.7 The effect of increasing benefits sums up to the pure entitlement effect
on the insured group. In a steady-state equilibrium, the wage differences across the two types
are achieved through cross-subsidization between the groups. The only "real" effect of benefit
increase is that expected wages increase in proportion to the expected benefit collection, which
depends on the size of the insured group.
This increase in expected wages is accompanied by a decline in market tightness. Taking
into account this general equilibrium effect of deteriorating market tightness tends to reduce
the wages of both types, reducing the wages of the uninsured even further, and mitigating the
6
These being the only relevant payoffs in case of disagreement.
are the average wages of workers in the economy, and are the wages a firm opening a vacancy is
expecting to pay.
7These
40
wage increase of the insured.
US unemployment policy provides a strong case for the analysis. Under the Federal-State
UI program, unemployment benefits are provided to qualified workers who are unemployed
through no fault of their own. An unemployed worker is eligible for benefits if he was laid
off, but not if he quit or was fired. Insurance is conditional on its being the firm's decision to
separate. The model uses this conditional feature of US policy to distinguish between eligible
(unemployed who were laid off) and non-eligible (unemployed who quit their previous job). 8
We first present the extension of the basic search model to two types of unemployed workers
and explore the sources of the differential effects of unemployment benefits on wages, the total
(average or expected) effects and the general equilibrium effects. To see the actual incentive
effect, the next section introduces heterogenous productivity and accordingly allows workers
to decide on an entry productivity. The lower wage of uninsured workers translates to their
willingness to work for lower productivities.
Other implications, such as endogenizing the
insurance rate, are further discussed. We conclude with a few open questions.
2.2
2.2.1
Simple Two-Type Model
Model
We extend a standard search model (Pissarides 2000) to allow for two types of unemployed:
those currently receiving unemployment benefits, and those who are currently not covered. The
unemployment eligibility status depends on how separation and the fall into unemployment
occurred. If separation occurred because the worker quit, he is subsequently not eligible for UI.
If the fall into unemployment was caused by the firm's decision to lay off the worker, then the
unemployed is eligible. The contract is completely specified at the time of engagement. This
pins down the original disagreement point as the only relevant one for surplus sharing. Search
frictions create quasi rents that can be split with both types, allowing for two wages to coexist.
Initially we consider a model where separations are exogenous.
Time is continuous. There is a measure one of workers and a larger measure of firms, both
8Albrecht
and Vroman (2005) use the limited duration of unemployment benefits to similarly justify equilibrium wage dispersion.
41
risk neutral. Let uo + ul be the measure of unemployed workers, where uo is the measure of
the unemployed who are not receiving benefits, and ul is the measure of UI recipients. Denote
by g2o=
uO the fraction of non-recipients among the unemployed. The measure of vacant
firms is denoted by v , and define the market tightness as 0 = o
Assume workers and
firms meet via a constant returns to scale matching technology, where the rate of job matches
per unit of time is given by m(uo + ul, v). The rate at which vacant firms meet workers is then
q(O) - m(uo + ul, v)/v , and the rate at which workers find jobs is Oq(O).Note that, in this
specification search is not directed.
When a firm decides to open a vacancy it incurs a flow
search cost 'yo while looking for a worker.
When a worker and a firm meet, they can jointly produce y. Upon being matched the
uninsured negotiate a wage contract wo, and the insured negotiate a wage contract wl. These
wages are the Nash solutions to the bargaining problems over the present discounted value of
future flows where the bargaining power of workers is given by 0 < 3 < 1. Since it will turn
out that wl > wo we will also use the terms "high-wage" and "low-wage."
Productivity is subject to bad shocks which arrive at a Poisson rate of A and are followed
by layoffs, after which the discharged worker is eligible for a flow unemployment benefit of z.
For now, assume that y > z so that it is always jointly efficient to produce.
There is also an
exogenous probability of workers quitting at a Poisson rate of s after which the ex-worker is not
covered by UI, since the separation was his fault. The following diagram illustrates the flow of
workers in and out of the four different states, where we denote by lo and 11the fraction of the
labor force who are employed at low wages, and those who are employed at high wages,
Denote by Jji the present discounted value of being in state ji, where j E {U, E, V, F}
{Unemployed, Employed, Vacant, Filled} and i E {0, 1 } -{Uninsured,
production
Insured}. When joint
occurs these values depend on the realized level of production,
y. Firms values
possibly differ depending on whether they match with an uninsured or an insured worker (JEO(y)
or JEl(y)), since the bargaining position of an insured worker is higher than the bargaining
position of an uninsured.
The value of a vacant firm, however, is simply JV. The values for a
worker are similarly Juo, Jul , JEO(y) and JE1(y).
The economy can be described by a series of Bellman equations:
A vacant firm incurs the flow cost of opening a vacancy -
42
0,
and finds a match at the rate
eq(e)
.
Uo
v
..\ lo ,'
.-
UI,>
A
i
.
I
I .. /
I
i
_%./
.................
'\,
' x
eq(e)
Figure 2-1: Flows
q(O). Since search is not directed, with probability uo the worker is not receiving insurance,
and hence the firm transitions to have a value of JFO(y); or alternatively, the firm meets an
insured unemployed worker, and transitions to have a value of JFl(y),
rJV = -o + q(O)[uo(JFO(y)
- JV) + (1 -
0o)(JFl(y) _
JV)]
(2.1)
A filled firm which employs a previously uninsured worker gets a flow of y - wo, and is
either hit by a productivity shock A or a taste / relocation shock for the worker s. In both
cases, the relationship is destroyed and the firm returns to its vacant state. If the firm employs
a previously insured worker, we simply need to adjust the wages contracted to be wl and the
firm value which reflects this change is
jF1.
We therefore have the values of a firm with a filled
position, for i = 0, 1
rJFi(y))]
+ S[jV _ Fi(y)].
(2.2)
We can similarly describe the worker's value in each state. When a worker is unemployed
and not eligible for UI, his flow value is simply the expected gain from future matches. He will
meet a firm at a rate of Oq(O),at which point he'll switch to have value JEO(y). An insured
worker in addition receives a flow z while unemployed.
43
For i = 0, 1 we have the unemployed
values given by
rJUi = zi + Oq(9)(JEi(y)
- Ui).
(2.3)
Once a worker of previous status i is matched with a firm, he earns the flownegotiated wage
wi and can separate from a firm for two reasons. Either a shock hits the firm (A), or he quits
through his own fault (s),
rJEi(y) = wi +_[Jul _ jEi(y)] + [JUO_ jEi(y)].
(2.4)
Note in equation (3.4) that whether a worker was eligible or not for benefits, he might be
subject to a shock A in which case he'll be laid off and consequently receive unemployment
benefits, transitioning to state Jul; or, might become non-eligible if he gets hit by a shock s
which forces him to quit and move to state JUO. These future deviations from the original
bargaining positions will be taken into account in the subsequent Nash bargain, resulting in the
"future entitlement" effects we will soon discuss.
The free entry condition for firms is given by
JV = 0.
(2.5)
Wage determination is given by the Nash bargaining solution over the match surplus. The
share of a worker being matched with a vacant firm is given by (for i = 0, 1)
JEi(Y)_ JUi
=
(JFi(y)_
+V
JE(y)
_i - JUi)=
1-15
(JFi(y)- JV).
(2.6)
In addition, in a steady-state equilibrium we must have the following flow equations hold.
The flows into and out of uninsured unemployment must be equal
uoOq(O) = (1 - uo - ui)s.
(2.7)
The flow into and out-of insured unemployment must be equal,
ulq(0) = (1 - uo - Ul)A.
44
(2.8)
The flow into and out-of low wage employment must be equal,
uoq(O) = lo(s + A).
(2.9)
These equations are the extension of the Beveridge Curve to our two-type (four states) environment.
2.2.2
Equilibrium: Definition and Characterization
A steady-state equilibrium is a solution to the wages, the flow variables and the state values
{WO, W1, i O,
lo
0 0, UO U, 10,
U Jji)
satisfying
(i) the flow equations (2.7) to (2.9)
(ii) the Bellman equations (2.1) to (2.4)
(iii) the free entry condition (2.5) and
(iv) the wage equations (2.6).
We can characterize the equilibrium as follows. Define the average (or expected) wages
in the economy as Ew
uiiowo+ (1 - iuo)wl. Then equilibrium and average wages are given
by the intersection of two curves: the Creation curve and the Expected wages curve. The
equilibrium differential wages w0 and wl are then calculated using the equilibrium market
tightness. Consider first the Creation curve9,
Ew = y - (r + A + s) q-)
(2.10)
This can also be thought of as the free entry condition. For expected profits to be zero, the
expected wage bill must be equal to productivity minus flow search costs. As search costs
increase when there are many firms (high market tightness), the expected wages must be lower.
Thus, the Creation curve is downward sloping in (0, Ew) space, and it can be interpreted as a
labor demand curve.
The two individual wage equation are derived from the Nash solution using the above
9
Plugging 2.5 into 2.1 , and replacing with 2.2.
45
1
Ln
Q)
CM
wO
0.1
0.2
0.03
0.4
0.5
0.6
0.7
8 - market tightness
Figure 2-2: Equilibrium
46
0.8
0.9
1
Creation curve (2.10)10
wo(y) = /y + (1 -
)rJU -
(1 - B)(JUl - JUO)
= P3[y+ oyO+ (1 - iio) (1 -
)Oq(O)z] - A(1 -/)
r + /30q(O)
w1(y) =-
y + (1 - 3)rJUl+ s(l -
(2.11)
r+
z
q(O)
J)(JUl
- JUO)
[y+-too
- o r +-00qiO-)
()z]+(1- P)z
lz++s1-(O)
(- r +430q(O)
Note that the wage gap between workers who were previously UI recipients and those who
were not eligible is given by
Aw -
wl(y)- wo(y)
(2.12)
(1 - )(r + A+ s)z
r + 00q(9)
Benefits, z, are the reason the wage gap exists in the first place. The wage bargain gives insured
workers a share 1 - F of these benefits to correct for future changes in worker status and the
consequent loss of (1 - /)z to employers. The PDV of this share of benefits is discounted by
r +/?q(O). Effectively the wage gap decreases as vacancies increase. Higher vacancies imply
a higher exit rate from unemployment, so that the gains from benefits are given for a shorter
period of time.
Consider next the individual wage equations. In the uninsured wage, wo(y) = 3y + (1 -
3)rJU - A(1 -
)(Jul _ JUO), we see the usual Nash bargaining terms 8/y+ (1 -
)rJUO
which correspond to the worker's share from flow output above his bargaining position. The
additional term -A(1 - /)(JU1 - JUO) corrects for the fact that in the future the worker might
actually receive more than he is currently entitled to from his bargaining position. There's a
probability rate of A that the firm will lay him off, and he'll end up unemployed and insured.
He needs to pay the firm a share of 1 -
from this gain in advance, through lower wages. This
is the future entitlement effect. Note that the wage of the insured, wl, has a similar future
l°using 3.5 , 3.4 and 2.2 we get w(y) = y + (1 - P)rJU - A(1 - p)(JL1 - JU). The unemployment values
substituted into this equation are found by replacing 2.2 into 3.3 and using the creation equation 2.10 to replace
y - wo . Similarly for wl.
47
entitlement effect +s(1 -,3)(JU1 _ JUO)which is working in the opposite direction. An insured
worker is entitled to his unemployment benefits, but might loose them if he quits at the rate s.
He is compensated in advance for this contingency by having his wages increase further, above
the standard entitlement (or bargaining position) effect of (1 - /)z.
When the creation relation is used in the unemployment values and then substituted into the
wage equations, we can track down another effect resulting from the existence of the two workertypes together. The uninsured experience a gain of /3(1 - 2o)(1-P)qW)z while the insured have
their wages adjusted by -
2
(1o-)0q() . These are basically pecuniary externalities resulting
from the fact that a firm expects to pay an average wage, but ends up paying an uninsured
worker (for instance) a lower wage. The firm's gain from paying lower wages than expected is
shared with the (uninsured) worker, thus increasing his wages slightly.
The expected wage equation is derived as the average of these two individual wage equations,
Ew -
owo(y)+ (1- o)wl(y)
/(Y+ -o) + (1- o)(. - )z + (1-)
)[(1-o)s-oA].
(2.13)
It has three terms. The first is the usual share / of employment gain (y + -yoO).The second
term arises from the worker's entitlement to benefits z but only by the fraction of unemployed
who are actually entitled to it, (1 - uo). The last term keeps track of the wage corrections due
to expected deviations from initial bargaining position.
Finally, from the steady-state flow equations we get that
S
-
A+s'
or,
Aio = s(1- uo).
A steady-state equilibrium requires we have the measure of uninsured who later become insured
equal to the measure of insured who become uninsured. This means that the wage corrections
for future deviations from the initial bargaining position are washed out across the two groups.
Now the last term in the expected wage equation is zero. This also implies that the expected
48
wage curve is unambiguously increasing in market tightness, Ew = 0l(y+-o)
+ (1- io)(1- -)z1
To solve for uo and ul we imposea steady-state condition on the flow equations, and get
s
uO= Oq(O)+ A+ s
and
A
= U01U
s
=
A
Oq(O)+ A+ s
Note that the solution for the insurance rate, u2o=
+, is solely determined by the exogenous
rates of separations. When we extend the model in the next section to allow for worker's entry
decisions we will endogenize this insurance rate of the unemployed. Finally, solving for lo and
11,we can get the following useful ratio of unemployment-to-employmentl1
U
uo + ul
E
lo +11
uO
s+ A
lo
Oq(O)
Using the solution for Gowe can now write the second equilibrium equation, the Expected Wage
equation, as
Ew = P(y + yo0)+ A
(1 -
)z.
(2.14)
Proposition 10 An equilibrium exists if y > - z and it is unique.
Proof. An equilibrium exists if there is a solution to the Creation (2.10) and Expected
Wage equation (2.14). Substituting out Ew , an equilibrium exists if there is a solution to
I3Yo0 + (r + A+ )
-
(1 - ,3)(Y - A
z). Sincethe F(0) P
- 700+ (r + A+ s)q
and increasing in 0 and has lim o 0 F(O) = 0 and lime,,
is positive
F(O) = oo there exists a unique
solution for 0 as long as y - W+iz > 0. Uniqueness of the equilibrium follows since the rest of
the equilibrium variables are uniquely determined from .
11
=
11 =
uq(O)
^
s+A
A= (1
s
49
Oq(O)
Oq(O)+A+s
)o
q(O)
q(O + A+
2.2.3
Total and Differential Effects of Benefits
Our comparative static results on the effect of benefits on the differential wages, on expected
wages and on the equilibrium market tightness are as follows:
Proposition 11 The total effect of benefits on Expected wages and market tightness
(i)
Expected wages increase with benefits.
(ii)
Market tightness decreases with benefits.
Proof. See appendix.
An increase in unemployment benefits increases average wages in proportion to the fraction
of unemployed workers who are insured, (1 -
o). Since this leaves the firm with lower profits,
the creation margin suffers, with a smaller number of firms entering the market. We further
have that
Proposition 12 Benefits increase the wage gap.
Proof. See appendix.
Recall that the wage gap is driven by the discounted present value of benefits which insured
workers have, Aw = (1-q(
)z. An increase in benefits directly increases the wage gap, and
this increase is magnified through the lower creation margin. Since there are fewer vacancies,
unemployed workers have a harder time matching with firms, and stay unemployed longer. The
gain to the insured is thus multiplied.
Since benefits increase average wages and the wage gap, we have the following corollary,
Corollary 1 Benefits increase the wages of the insured.
Higher benefits increase the wage of the insured directly through the higher bargaining
position of workers (or the 'entitlement effect'). However, there is an additional effect through
lower market tightness. The complex contribution of the lower discount term q(0) does not
overturn the initial wage increase. Finally, we have the effects of benefits on the wages of the
uninsured:
Proposition 13 Benefits decrease the wages of the uninsured
50
Proof. See appendix.
An increase in unemployment insurance directly decreases the wages uninsured workers
get. While their bargaining position is not affected by the increase in benefits, they stand to
gain higher unemployment benefits when they will be laid off in the future. In the bargaining
outcome, this future gain is deducted from their wage, since currently they are not entitled to
any benefits. In other words, workers are willing to take lower wages because of the prospects
of future insurance.
We can then think of the wage curves as the wages workers demand for a given labor
supplied. 12 It other words, it is the labor supply curve. Thus, if higher unemployment benefits
reduce the wage demanded by the uninsured, they are in fact willing to work for lower wages.
Their incentive to work has increased. By next adding an entry decision we permit workers to
change their behavior in response to changes in market conditions. We can then explicitly talk
about the differential incentive effects of unemployment insurance.
2.3
Endogenous Entry Decision
In the simple model above workers didn't make any employment decision since productivity
was such that they always preferred working to unemployment.
Since workers didn't make any
work decision, all effects went through the wage bargain and the adjustment of creation by
firms. To be able to speak about an incentive effect we must give workers a choice to make.
We do this by introducing a stochastic matching productivity, and the worker's entry decision
or his reservation wage.
Adding stochastic productivity and an entry decision produces two more worthwhile results.
First, the economy's insurance rate is now endogenous. In the steady-state, the fraction of
insured out of the unemployed (o) is not simply the exogenous rate of quits to total separations
(s-x) as before, but is determined in equilibrium through both types' decision to enter the
employment relationship. This takes away the convenient but artificial feature of the previous
model where the wage compensations across the two groups disappear. Also, adding uncertainty
to the model introduces an amplification of the wage/entry gap through a term resembling an
' 2 When the creation is not substituted in it, as in the appendix.
51
option value. We discuss this later. Note that the entry decision can be also interpreted as a
search decision, whereby worker can decide to prolong search and unemployment duration.
13
We now extend the model to allow for stochastic job matches. Productivity y is realized
after the match is made, and then the worker decides whether to continue searching or to
take up employment (we will see that his decision also corresponds to the firm's decision).
Since a worker's value increases with productivity, this decision defines entry productivities (or
reservation productivities) Ro for the uninsured and R1 for the insured.
Formally we introduce the two entry conditions (for i = 0, 1)
JEi(Ri) = JUi
(2.15)
and modify all employment values and wages to depend on productivity and all non-employment
values to incorporate the appropriate expectation given the decision variables Ri.
The Bellman equation for a vacant firm is
rJV = -o + q(9) [uo (JFO() - JV)dF(s)+ (1- o)
The profits JFi(y) _-j
(JFl (s) - JV)dF(s)
(2.16)
now depend on the realized productivity, y, for all matches that proceed
to the production phase, y E [Ri,y].
The value for a firm employing a previously uninsured worker (i = 0) or previously insured
worker (i = 1) also depends on realized productivity, y,
rJFi(y) = y - wi(y) + A[V _ JFi(y)] + s[Jv - jFi(y)].
(2.17)
Similarly for workers, we have the value for an unemployed worker depends on his eligibility
status i = 0, 1 and his expected gains from future employment,
rJUi = zi + Oq(O)
JEi(s)- JUi)dF(s).
13
(2.18)
However, the results are different than if we add a search intensity choice, which will be an ex-ante choice
by workers before they meet with a firm, and hence will not allign the worker's and the firm's interests.
52
The value for an employed worker again depends on his previous insurance status, i = 0, 1
and the realized productivity, y,
rJEi(y) = wi(y) + A[JU1- JEi(y)]+ s[JUO_ jEi(y)].
(2.19)
The wage bargaining solutions satisfy
Ei() _ jUi = (Fi(y) _ V + JEi(y)_ J) =
1-0
(JF()
V).
(2.20)
With 0 and nuodefined as before.
The flow equations are also modified. Now the flows out of unemployment depend on the
entry variables Ri. A worker becomes employed if a match takes place (at a rate Oq(O))and if
furthermore the realized productivity is high enough (which occurs with probability 1- F(Ri)).
This gives the followingsteady-state flow equations. Equating the flows in and out of uninsured
unemployment
we have,
uo0q(O)[1 - F(Ro)] = (lo + lI)s.
(2.21)
Equating the flows in and out of insured unemployment we have,
ulq(O)[1 - F(R1)] = (lo + 11)A
(2.22)
The flow in and out of low-wage employment is given by
uoq(O)[1 - F(Ro)] = lo(s + A)
(2.23)
Finally we have the accounting equality,
lo + 11= 1 - o - ul.
53
(2.24)
Figure 2-3:
2.3.1
Solution
The solution for 0 , Ro , R 1 and io are given by the four equations: ERfirm (The firms'
creation equation), the two Entry equations for R0 and R 1 by which we construct ERW° k (the
average entry by workers), and the equation for iio derived from the condition for a steady-state
equilibrium.
The creation equation is
oR0 + (1-;o)R 1 =
J
(2.25)
F(s)ds = Oq(O)/3-- AR1.
(2.26)
(+ + s) Yo -o i-(
F(s)ds - (1 - o) | F(s)ds.
The uninsured entry equation is
(r + s + q(O)P)Ro+ Oq(O)/
The insured entry equation is
J
(r + A+ Oq(O))/)Ri
+ Oq(O)
F(s)ds = (r + A+ s)z + Oq(O)/3
- sRo.
R1
54
(2.27)
Steady-state insurance rate is given by
s(1 - F(R1))
(2.28)
s(1- F(R1))+ A(1- F(Ro))
From the two entry equations (2.26) and (2.27) we constructed the average entry, ERwork,
and the entry gap, AR,
TioRo+ (1 - io)Rl = (1 - o)z +
_
(r +
0 -t [
1 - '8
+ s)
-
Oq(O)
R1 - Ro(=( + q()/3 r+(),1
+
(1-
[RI
o] (RI
()s - Ro)
r+A+s
1
F(s)ds]
(2.29)
(2.30)
Wages are given by
wo(y)
=
3y+(1 - )Ro
wl(y)
=
+y( -)R1,
(2.31)
so that
wI(y) - Wo(y) =
(1 - )(R - Ro)
(r + A+ s)
) r + q(t)6
(2.32)
(1- )q()/3
r + q(t),8
R]
The first term is the same as in the model without stochastic matching. The second (new)
term is due to uncertainty. It reflects the higher option value that non-recipients get from a
relationship because they begin employment sooner (Ro < R1 ). They pay (1 - /) of this in
their wages, reducing their wages by this extra term.
The unique equilibrium can be depicted by the intersection of the ERfirm (expected creation) and ERw ° k (average entry) curves. This gives the equilibrium market tightness *. The
equilibrium entry productivities Ro and R 1 are then found given this market tightness.
55
Entry Productivity
RI*
Ro*
[ion
e*
O=v/u
Figure 2-4:
2.3.2
Discussion
The firm's creation decision is basically unchanged. As before, free entry implies that vacancies
will be opened to the point where the costs of opening a vacancy 'Yoare covered by the expected
value of a producing firm. The difference is that now the firm will be producing only above the
threshold productivities Ro and R1 . The Expected creation curve is thus given by
Yo)-
u
q(9)
JFO
°(
F(s)dF(s)
(2.33)
(s)
= Uor + A + S
- Ro)dF(s)+ (1-
) r + A+
(s - R)dF(s).
That is, firms choose market tightness (their vacancy opening) such that the expected cost of a
vacancy is equal to the expected productivity. Firms take the expected entry decisions (Ro, R1
and uo) as given, so that this curve does not shift with benefits, z. Increasing unemployment
insurance is represented by a shift along this curve to a new equilibrium, due to the shift of the
worker's entry curve.
56
The model's value-added is the additional entry decisions which allow us to speak of the
incentive effect of UI. Workers decide to enter the productive relationship with the matched
firm if the value they get is higher than the value of staying unemployed. Higher values of unemployment induce workers to postpone entry by increasing the threshold productivity defined
by JEO(Ro) = JUO. Note that given Nash Bargaining, this corresponds to the point where
the firm is indifferent between staying vacant and producing. This in turn means that at the
threshold, productivity is equal to wages (Ro = wo(Ro)). Using this fact in the wage bargaining
solution we find that
JUo)
Ro = rJ U ° - A(JU1
(2.34)
R1 = rJUl +s(JUl -_JU).
In their entry decision workers realize there's an additional value to employment due to the expected change in eligibility. Since uninsured expect a future entitlement of UI benefits through
work, they are willing to enter earlier, at a point where productivity equals their current unemployment value minus the future gain due to eligibility change, and similarly, for the insured,
the entry decision is postponed due to the expected loss of benefit eligibility in case of a quit.
Wages simply track the entry productivities in a very simple way: wi(y) = By + (1 - O)Ri.
Moving on to the entry/wage gap R1 - Ro = ('
+
+q(6)
[Ofo F(s)ds] we note the
new term which resembles an option value term. Uninsured workers are now willing to enter
the relationship for lower productivities, not only because of the increased value of employment
due to future unemployment benefits, but there is an amplifying effect of this entry gap. Given
the earlier entry, uninsured workers' option value from work is larger, which increases the value
from employment even further, reducing entry productivity.
Endogenous entry also implies that the fraction of uninsured to insured (o) is now endogenously determined. Relative to the case where the steady-state flows were exogenously
determined, now a larger fraction of uninsured are working , and a larger fraction of insured
are unemployed: o0 =
s(1-F(Ri))
s(1-F(R))±A(1-F(Ro))
s.
s-+A
57
This also implies that (1 - io)s - iiA > 0.
Inspecting the worker's average productivity,
/3
(R - Ro)
ioRo+ (1 - o)R1= (1 - o)z+ 1 _ y07o
+ [(1- o)s- Ao] r + +
(2.35)
we note that now the last term does not vanish. The effect of unemployment benefits on
worker's average threshold productivity is increasing not only because of the benefits given to
the insured, (1 - iio)z, but this effect is magnified by allowing the unemployed to adjust their
behavior. Since the fraction of insured to uninsured is higher than when no entry decision was
allowed, there is now a positive discrepancy across the two groups which increases the average
reservation wage and leads to higher unemployment.
The comparative static results concerning the effect of increased benefits on the equilibrium
outcomes are similar to those we derived when productivity took only two values. We have an
additional prediction regarding the effect on the differential and total reservation productivities.
These effects followthe effects on wages. In particular, higher unemployment benefits increases
the average entry productivity, increases the entry productivity for the insured worker and
decreases the entry productivity for the uninsured worker. Proofs are provided in the appendix.
2.4
Conclusion
We augmented a search model to account for the existence of unemployed workers who are
not currently receiving benefits and investigated whether their increased incentive to work can
dominate the lower incentives of the insured unemployed. We have shown that the conjecture
in the literature is false, and that in equilibrium the total effect of unemployment benefits is
driven by the fraction of workers who are insured and receiving benefits. Higher unemployment
benefits have the standard average effect. They decrease the vacancy/unemployment rate and
increases expected wages. However, benefits have a different effect on the uninsured than on
the insured. When benefits increase, the uninsured value employment more because of the
possibility of future benefits receipt. They are therefore willing to take lower wages. The
insured experience a wage increase due to the direct increase in their bargaining position, and
an additional increase to compensate them for the possibility that they will need to quit the
job and loose their UI eligibility.
58
In equilibrium, these partial effects are attenuated by the existence of two types of unemployed through a pecuniary externality. Because firms' entry decision is targeted at an average
employee, there is a form of cross-subsidization between the two types of workers. General
equilibrium also works to reduce wages of all workers through the lower market tightness. This
results in a further reduction of the uninsured wages, and a smaller increase in the wages of the
insured. When we allow the unemployed to respond endogenously to employment incentives
these results are intensified. Given the stronger incentive to work (through future entitlement),
the uninsured leave unemployment at a higher rate than the insured, resulting in an insurance
rate of the economy which is higher than before.
There are two lessons to be drawn from the discussion in the paper. In the context of the
benchmark search model, ultimately what matters for welfare is the net increase in insurance.
The fact that there is another population that does not directly benefit from the program has
distributional consequences but no real effects. Furthermore, while the non-eligible population
suffers from reduces wages in the short run, they too experience an increase in their expected
lifetime utility.
The model and its results rely on the bargaining assumption embedded in the original
search model. There are two steps that logically follow. The first is to use this model with
its bargaining mechanism to think about more specific features of the unemployment insurance
policy, in the spirit of the Atkinson and Micklewright (1991) program. In particular, there are
interesting effects on the bargaining positions of workers and firms in the presence of experience
rated taxes. 4 Since workers have limited liability, it is also natural to ask what happens when
bargaining is subject to incomplete contracts. Finally it would be interesting to verify these
results in a setting with competitive search. When search is directed, at the very least we don't
expect to find the same distributional results.
2.5
2.5.1
Appendix
Proofs for Section 2.2
Claim 1 Total Effect of benefits on Expected wages and market tightness
14
Blanchard and Tirole (2004) address some of these questions.
59
(i)
Expected wages increase with benefits.
(ii)
Market tightness decreases with benefits.
Proof. By implicit differentiation of the equilibrium equations. Creation (2.10) and Expected Wages (2.14) give us a system in (0, Ew):
F 1 : Ew-
(y+ 0) + (1
- )( -)z]
=
F 2 : Ew- [N(Y+ -o 9) ( - io)(1- )z =
:Implicitly differentiating, we have,
(i)
(1 - iio)(l
-/)(r
dEw
dz
dz
+ A + s)q4q'
-/?o + (r + + s) q'
Both numerator and denominator are negative since q'(0) < 0. Also
dEw
dz < 0
d
(ii) similarly,
dO
(1-
)( -
)
d<0
dz -/yo +(r+A+s)~q
and
dO
dz
.
The total effect is therefore only due to the actual increase in benefits, which is only by the
fraction who are insured, (1 - o).
Claim 2 Benefits increase the wage gap.
Proof. Totally differentiating the wage gap,
A
wl(y) - wo(y)
_ (1-/3)(r +A +s)z
r +/3q(0)
60
dA
dz
-
OA
dz
a dO
O dz
+ A + s)
(1 -)(r
(1 -3)(r
r + Oq(O)
(1 -
o)(1 -
+ A + s)z/?(q + Oq')
(r + 0q() )2
)
*(-3o+(r+As)q)
O
since q < O
Claim 3 Benefits decrease the wages of the uninsured
Proof. To derive the total effect of Benefits on the uninsured it is useful to present the
wage equations without substitution of the creation equation. In this alternative representation
of the equilibrium the wage equations (which can now be interpreted as labor supply curves)
are given by15
wo(y)
w(y==
r+A+
+ q(-))
_r
++(A+ss+Oq(O)
(1--
+ +s+s
+ q(O)
r +r+
r+/0q(O)
(
r+s+39q(O)r +A+s )
(
+ q(O) + + + 9 q(O)
r+-9
r +s
[r+A+s+
r+
q(O)]
X [ + + s+3 q(O)J+ (1- iio)(1-)Zr + + s +/0q(O)
r+A+s+q( )
1(Y) = /~r + A+s + /q(O)
Ew =
_
r + +s + q(O)) [(1- o)s- oA]
+r+ O0q(O)
We can then show
dwo
awo
awo dO
dz
Oz
O dz
(-) + (+)(-) < o.
' 5To derive these, we proceed as before to derive w(y) = /,y + (1 - )rJU - A(1- P)(Jul - JU). However, the
solution for rJU we use here does not use the creation relation. Rather, the expression above is used to replace
the term y - w.
61
2.5.2
Proofs and Derivations for Section 2.3
We first present a brief guide to deriving the equilibrium equations. Using the creation equation
(2.25) with the firm vacancy value (2.16) we get
o=
o
oJF°(s)dF(s).
(2.36)
The value for unemployed workers is derived by plugging the wage equation (2.20) into the
unemployment values (2.18) to get
°
rJUO
f
-
q(O)
(2.37)
F°(s)dF(s)
rJUl = Z+p1 _q(s)X
JFl(s)dF(s)
In the wage bargain equation (2.20) we replace jEi and jFi from (2.19) and (2.17) to get wages
in terms of productivity and unemployment values,
wo(y) =' fly + (1 -
)rJ O° - A(1-_ 3)(JUl - JUO)
wl(y) = O/y+ (1 - f)rJUl + s(1 -
(2.38)
)(jU1 _ JUOj
Next we solve for the entry decision by using the wage equation (2.20) at y = Ri. Noting that
jEi(Ri) = Jui we find that the firms must also be indifferent between producing or not at the
entry level, or JFi(Ri) = 0. Using this with (2.17) jFi(y) =
substituting
Xw4) we have Ri =
wi(Ri). When
for the above wage equations (2.38) at Ri we have
Ro
=
rJUO - A(JU 1 - JUo)
(2.39)
R1 = rJUl+ s(Ju _ JUo).
Which gives us Jul _ JUo = R-Ro
r+A+s and the linear relationship between wages and entry
productivity, wi(y) = fy + (1 - P)Ri. By plugging this wage into the (2.17) we can solve for
the firm values
JFi(y) =(1 -
)(y-Ri)
r+A+s
62
(2.40)
Plug these into the unemployment values derived in (2.37) to have (after integrating by parts),
rj-zi
+ Oq(O)/
r + A+ S
-R - -
F(s)ds
This also gives the difference between unemployment values as
r(JUl _ JUO)= z+
)s
[Ro- R1 +
F(s)dF(s)].
Using this in (2.39) we get the implicit solutions for the entry productivities (2.26) and (2.27)
in the text. Replacing (2.40) in (2.36) we get the creation equation(2.25). This completes the
derivation.
We now prove the main comparative statics result. Note that in the firm's creation equation
(2.25) firms choose market tightness (their vacancy opening) such that the expected cost of a
vacancy is equal to the expected productivity. Firms take the expected entry decisions (Ro,
R1 and uo) as given, so that this curve does not shift with the level of benefits, z. Increasing
unemployment insurance is represented by a shift along this curve to a new equilibrium, due to
the shift of the worker's entry curve.
To find the find how the entry curve shifts with benefits, z, we need to see how all its
components shift with z. We begin with these partial derivatives, and continue to show the
total equilibrium effect of increased z is to reduce market tightness, to increase R1, to reduce
Ro and to increase the wage/entry gap.
Claim 4 partial equilibrium effects on entry productivity (holding 9 fixed)
i) dR1 >
ii)
0
R < o0
iii) dAR >
Proof. define:
F1 :
F2
JRo
(r + s + Oq(O)/)Ro+ Oq(O)3 F(s)ds - Oq()
+ AR1 = 0
(r + A+ q(0))R1 + q(0) / F(s)ds - (r + A+ s)z- q())]y + sRo= O
R
63
so that, holding 0 fixed,
i)
dRo
(r+ (r + A+ s)A)
dz
[r+ s + q(O)/(1- F(R))][r+ A+ 0q()/(1 - F(R1))]- As
=
-A
<0
(r + A+ s)
A
[r+ s + Oq()/(1 - F(Ro))] [r + A+ Oq(O)/(1- F(R 1))]- As
>0
ii)
dR 1
dz
- [r + s + Oq(9)(1 - F(Ro))](r + A+ s) - 0
[r + s + Oq(0)/(1- F(Ro))] [r + s + Oq(0)3(1- F(R1))] - As
= [r + s + Oq()/(1 - F(Ro))] A > 0
iii)
AR _ R1 - Ro =(r + + s)
Oq(O)
[/'
r + Oq(0)/3 r + Oq(9)/3
dAR
_OAR
. OAR ORo
dz
9z
ORo Oz
OARo
F(s)ds]
OR 1
OR 1
(r+A+s) _ Oq(),F(Ro)(-A)
r + 0q(9)P r + 0q( 9)
+
(
Oq(O)/ F(R1) [r + s + Oq(0)1(1- F(Ro))] A
r + Oq(O)/3
+ F(R1) [r + s + Oq()1 (1 - F(Ro))]}
(r + A+ s) + Oq(0)PA(F(Ro)A
r + q()/
> 0
Claim 5 partial equilibrium effect on fraction of uninsured and other derivatives of dReO(holding
0
fixed):
i) -o <
ii) d(1-io)
iii)
do>
dz
-- dzd[(1-iao)s-A)io](Ri-Ro) > 0
dz
64
Proof. ii) can see, but formally,
A(1- F(Ro))
1 - uo
d (1 -
s(l - F(R1 )) + A(1- F(Ro))
(1 - 0o)ORo o(1
o)
dz
ORo
Oz +
-
o)OR 1
Oz
R1
-Af(Ro) [s(1 - F(R 1 )) + A(1 - F(Ro))] - A2 f(Ro)(1 - F(Ro)) aRo
[s(1- F(R1 )) + A(1- F(Ro))]2
Oz
A(1 - F(Ro))sf(R
1)
OR 1
[s(1 - F(R 1 )) + A(1 - F(Ro))12
z
-Af(Ro) [s(1- F(R1 )) + A(1- F(Ro)) - A(1- F(Ro))]-Q + Asf(R1)(1- F(Ro))_
[s(1- F(R1 )) + A(1- F(Ro))]2
oR 1 ]
f(R )(1 - F(Ro))OR
2
[s(l
(1-- F(Ro))
Is( -- F(R
F(R1))
+ A(1
F(Ro))] [
1)) +
- F(R))a
9
2
- YX>O sinceORozAs<0
+
1
R >
and
Oz1
As
X
2
[s(1- F(Ri)) + A(1- F(Ro))]
°Ro
aR 1 ]
=- -f(Ro)(1 - F(R 1))-Oz + f(R1)(1 - F(Ro)) Oz jZ
i)
duzo
dz
d (l-z o)
dz
iii)
d[(1- io)s - Ailo](R
1 - Ro)
dz
d[( - o)s - Aiio](
R
dz
(R1 - Ro)s
dz
- Ro)
dz
because all terms are positive. The only term we need to show is positive is
d[(1 - uo)s - Ao]
dz
d[s - lo(s + A)]
dz
=
-(s + ) dz =-(s + )(-YX) >
.
Claim 6 Market tightness declines with unemployment benefits ,
65
< O0
-
o] > O
Proof. define the two equilibrium system for ER and 0 (equations ERfirm and ERW°rk),
G1
oRo+ (1 - o)R - y +
(1 - +)
2
G2
u:
ioRo+ (1- o)R - (1 - iio)z-1
q(O + o
F(s)ds + (1-
o)
07 -[(1 - uo)s- o](R+ -ARo)°) =
-implicitly
derivatives from
and
using
the
above
partial
we have+s
rdifferentiating
implicitly differentiating and using the partial derivatives from above we have,
(0- (-)1 + )
dO*
dz
(-- 1-
F(s)ds = 0
(q'(0)l- (-_Yo)l+ )
U
66
=-- <0
+
Chapter 3
Optimal Unemployment Insurance
with Search
Summary 3 Adding risk aversion to the standard search and matching model allows for an
analytic discussion of the optimal provision of unemployment insurance. The government's
capacity to insure workers is limited by the market wage setting, which gives workers a share
in the employment surplus. When the government provides higher unemployment benefits, the
bargainedwages increase, and unemployment rises. These equilibrium responses have a negative
effect on workers' welfare if workers' bargainingpower is above a certain point, which is lower
than the matching elasticity. As risk aversion increases, workers share in the wage bargain
is smaller, and thus the equilibrium effects are attenuated. The constrained optimal provision
of unemployment benefits is a modification of the Hosios condition for efficient unemployment
insurance and highlightsthe roles of bargainingand risk aversion. The optimal level of insurance
increases with risk aversion, with the costs of creating a vacancy and with workers' higher
bargainingpower.
While the standard Search Model remains the most important framework for discussing
unemployment and analyzing labor market policies, it remains a challenge to derive analytic
solutions when workers' risk aversion is taken into account. Risk aversion, nevertheless, is the
essential reason unemployment insurance is needed. This note attempts to make some advances in understanding
the role of risk aversion in the analysis of equilibrium unemployment,
67
search frictions, wage determination, and the optimal provision of unemployment insurance.
It provides conditions under which the equilibrium is unique, shows how risk aversion reduces
workers' share in the wage bargain, and derives the optimal allocations and constrained optimal unemployment policy. It discusses whether the optimal allocation can be implemented
as a market equilibrium, and compares the efficient, unconstrained optimal, and constrained
optimal allocations and policy choices. A modification of the Hosios condition is derived for
the constrained optimal provision of unemployment insurance, which illuminates the separate
roles of bargaining and risk aversion.
When workers are risk averse, unemployment insurance is needed for consumption smoothing. An unconstrained social planner will want to maximize output and then allocate consumption equally between employment and unemployment states. However, in the matching
framework of the labor markets, once a worker and a firm match they have a surplus to split.
As long as the worker enjoys some of the bilateral surplus, the income of the employed will be
higher than the income of the unemployed. For this reason, a social planner that is constrained
by the market interactions cannot fully provide insurance, since his transfers to the unemployed
will result in higher wages for the employed. This increase in wages has also a negative effect
on workers welfare, through firms entry decision. When wages are higher, firm's profits decline,
and there are less firms entering the market. The lower number of vacant firms implies a slower
matching process for workers and an increase in unemployment.
The overall effect of unemployment insurance on workers' welfare depends on the wage response relative to the vacancy creation response. When wages are determined by bargaining,
the worker's bargaining power and his risk aversion determines the size of the wage response.
The vacancy creation response is determined by the elasticity of the matching function with
respect to vacancy creation. These opposing effects of unemployment benefits on workers' welfare exactly cancel each other for bargaining power which is lower than the matching elasticity.
These results hold regardless of the risk aversion parameter.
Higher levels of risk aversion at-
tenuates the combined equilibrium effect on wages and market tightness. The reason is that
risk averse agents settle for lower wages, and hence the smaller effect through wages.
The condition for the constrained optimal provision of unemployment insurance is thus a
modification of the Hosios (1990) efficiencyresult. When the bargaining power is higher than a
68
certain cutoff level, for instance, when it exactly equals the market tightness, the overall negative
equilibrium effect through wages and market tightness suggest the provision of benefits should
be limited. The amount of insurance should increase with workers' risk aversion, with the costs
of opening up a vacancy and with lower workers' bargaining power.
This work builds on the search model of Diamond (1982), Mortensen (1982) and Pissarides
(1990) and extends the Hosios (1990) condition for optimality to a setting with risk averse
agents. It highlights the role of bargaining power and risk aversion in determining the optimal
intervention. It follows others who analyzed the role of risk aversion in this search context.
Fredriksson and Holmlund (2001) use a search model with risk averse agents to answer different questions concerning the optimal schedule of benefits. Other authors explore aspects of
the optimal provision of unemployment benefits in different contexts. Most closely related is
Acemoglu and Shimer (1999) who ask and answer similar questions using an alternative wage
posting mechanism. Shimer and Werning (2006) find an optimality criterion for unemployment
insurance in the tradition of a partial equilibrium McCall search model. This work, in contrast,
considers the equilibrium wage adjustments. Its main contribution is in providing a better un
derstanding of the equilibrium reactions to unemployment provision through the wage bargain
and entry decision, analyzing how risk aversion affects the wage bargain and the optimality
condition, and providing a modification of the Hosios (1990) condition for the constrained optimal provision of unemployment benefits. Its shortcoming is that the solution for an internally
financed system still need to be worked out.
Section I introduces the economy and section II solves for the equilibrium. Section III derives
the unconstrained optimal allocation and unemployment insurance, and section IV solves for
the constrained optimum, taking wage negotiations and entry of firms as given. Section V
concludes.
3.1
The Economy
We consider the steady state of an infinite horizon economy. There is a measure 1 of infinitely
lived risk averse workers with a concave utility over wealth U(-) who discount the future at
rate r. A fraction u of workers are unemployed and receiving unemployment benefits. There is
69
a larger measure of firm. A measure v of firms decide to open a vacancy and incur a cost 0Y.
Denote by 0 =
the market tightness. Workers and firms match via a constant returns to scale
matching technology, where the rate of job matches per unit of time is given by m(u, v). The
rate at which vacant firms meet workers is then q(9)
m(u, v)/v , and the rate at which workers
find jobs is Oq(O).When a worker and a firm match, they negotiate a wage w(y) according to
the Nash bargaining solution and then engage in producing output y. Productivity is subject to
bad shocks which arrive at a Poisson rate of A and are followed by separation. The government
provides unemployment insurance, z, to unemployed workers, which are financed with lumpsum
taxes, r, on workers.1 Imposing a balanced budget condition requires r = (1 - Oq(8))z. Note
there are no savings in the model. We therefore repress in the notation the level of asset holdings
that a worker has, and denote his utility over wealth when unemployed and employed as U(z)
and U(w) respectively.
The economy is characterized by a series of Bellman equations which describes the flow
value of a firm or a worker being at a particular state. Denote the values of being in state s as
Js, and let the states be {Vacant, Full} for firms and {Unemployed, Emloyeoyed for workers,
so that we have s E S = {V, F, U,E}.
When opening a vacancy, a firm incurs a cost of yo, and meets a worker at a rate q(O), at
which time it moves to a filled state, and gains the differential jF
-
Jv. The value for a vacant
firm is thus given by
rJV = -- o + q(O)[JF_ jV].
(3.1)
A filled firm gains y from production, pays out wages w, and is at risk of getting hit by a bad
shock at the rate A, at which time it becomes vacant again,
rJF = y - w + A[Jv - JF].
(3.2)
Unemployedworkers enjoy their unemployment benefits net of taxes, U(z- r), and might match
with a firm at a rate Oq(O)at which time they become employed, and enjoy the extra value
'We could alternatively explore a tax t on firms.
70
fromemploymentjE - jU,
rJU = U(z - r) + Oq(8)(JE - JU).
(3.3)
When employed, a worker enjoys the net wages, but stands to loose his job at a rate A,
rJE = U(w - 7) + A[JU_ jE].
(3.4)
Wages are determined by Nash bargaining where a worker's bargaining power is . The
wage, w, maximizes the following product of worker and firm surpluses from joint production,
w = argmax(JE- jU)6(jF _ jV)l-6.
(3.5)
Free entry by firms implies the value of a vacant firm must be zero,
JV = o.
(3.6)
Finally, the evolution of unemployment is given by the net flows into and out of unemployment,
du
ddt = (1 - u)A - uq(O).
3.2
(3.7)
Equilibrium
We begin by defining, solving and characterizing the market equilibrium given the unemployment policy.
Definition 6 A steady state equilibrium is a solution to the wage, the flow variables and the
state values w, u, v, 9,JS) for s E S, such that the following hold,
1. the flow equation (3.7) holds with d = 0.
2. the Bellman equations (3.1) to (3.4).
3. the free entry condition (3.6), and
71
4. the wage equation (3.5).
The solution to the model follows. A Firm's value is given by (3.6) into (3.2)
JF= y-W
r+
A
Adding (3.1) we have the Creation condition
y-w
y
r+ A
(3.8)
q(O)
A worker values are given by (3.3 and 3.4)
U(w - r) - u(z - r)
r + A+ q(O)
jE _ JU
j
=
jE
r+A U(- ) + Oq(O) U(wr + A + Oq()
r + A + Oq(O)
A
U(z- 7)?+r +r +A +q(O)
r + + Oq(O)
Oq(O)U(w
r)
)
The solution for wages is given by the first order conditions to the Nash problem,
8ijFd(JEdwt
_ JU) +((1-p ) (jE
fiJd
J
dJF
0
JU dw
Substituting in for the values, and taking into account that wage negotiations take as given the
wages in future employment relationships, we get the Nash equation
y - w U'(w- )
Pr A
2Denote
r- AI
= (1- )
2
U(w- T)- U(z- ) 1
l \
r +Ai +tiqu)
rt . A
by JE the value from working with future employers, then the workers surplus is given by
oq(JE
_ JU)
U(-r)-U(z-r)
r+X -
r+x
72
jE _
jU =
Rearranging and using the creation condition to substitute r+A
-w =- q(O)
(1-d)
(U(t)
U'(wU(z
- r)
) = pyr3 (r + A+ q(O))
r+A
= (y - w)+
=
(y -
+
+ Oq(O)
)
Which we write as the wage equation,
(1 -)
U(
r) - U(z -)
U'(w
- r)
+
=
(y +y)
(3.9)
This fully solves for the equilibrium. The equilibrium can be characterized by the Creation
(3.8) and Wage (3.9) equations. A solution of these two equations is a market tightness, O*,and
a bargained wage, w*. Given these we can uniquely back out all the values JS above. Finally,
the steady state unemployment rate is given by setting
d
=
to get,
A
U=
Oq(O) + A
Proposition 14 The equilibrium{O*,w*} is unique if the utility exhibits decreasing or constant
absolute risk aversion in wealth.
Proof.
The creation curve (3.8) is decreasing in {0, w} space since q'(0) < 0. We next
show that the wage curve (3.9) is increasing in {, w} if the utility exhibits decreasing or
constant absolute risk aversion. Note that d < 0 (taxes decrease with the relative number of
vacancies since the pool of the unemployed decreases when it is easier for unemployed workers to
match). Furthermore, U(w,)-U(;-T)
Ul(--)
is increasing
with taxes
d [U(W-TQ-U()]/d
=
IIUCJ
ll~jW1~1
~~C rI since
111CU
I 'l(w--r)
[U(z-T)-U'(w-T)]U(w--)-U"(w-r) [U(w-r)-U(z-)] > 0 if
[U'(Z--)-U'(w-r)]> [U(w-r)-U(z-)]
U(w-
)2
U,(w--T)
U(w-)
Multiply both sides by w-Z and note that we can find J0 and 1 both in [z - r, w - r] such
U'(z-r)-U(w-r)
1
U-"(r) and [U(w-r)-U(-)]
_
(1 . Now let E(
UI(w--r)
w-z - U"(w-r) an
U'(W-T) w-Z - Ul(w-r)
be the one which minimizes U"(x)_ Then U"(-) > U"() > U"(w--r) where the last inequality is
strong
absolute
if havedecreasing
risk aversion,
- and hold(with
-) equalityfor CARA.
strong if we have decreasing absolute risk aversion, and hold with equality for CARA.
73
We also have the followingresult,
Proposition 15 Risk Aversion reduces the wages workers get in the Nash bargain.
Proof. The equation determining wages for risk neutral workers is (1 - /) (w - z) + /w =
P(y + yO). For a given market tightness this equation assigns a higher wage w to workers if
> w
U(U)-T
U(W-T)-U(z-T)
W-Z
-
z.
This is true since there exists a
E (z - r, w -
)
such that
= U'(E) > U'(w - r), where the inequality is true for any concave U. ·
Proposition 16 When the budget is balanced(r = (1- Oq())z), an increase in unemployment
benefits increases wages and reduces market tightness.
Proof. An increase in unemployment benefits, z, does not affect the Creation curve (3.8)
but shifts the wage curve (3.9) up, resulting in lower market tightness and higher wage. ·
Note the last proposition can more easily shown to be true when the budget is not balanced,
and that it does not hold (unqualified) if we finance unemployment benefits by taxing firms.
3.3
Unconstrained Optimum
In this section we find the optimal level of market tightness, wages and unemployment benefits
that maximizes workers' welfare. The government is only constrained by the matching technol-
ogy and a balanced budget requirement. To minimize distortions we assume that workers pay
for the costs of opening vacancies and their unemployment benefits through a lumpsum tax.
The government's program is,
max
max
[07r[+
e-rt[uU(z- r) + (1 - u)U(w - )]dt
subject to a per period balanced budget, and the flow of the unemployed
(z-r)u+(w
-T)(1 -u)
< -vy + vq(O)y
d-- du-- (1
v - u) - uq(O)
0
74
U
We can equivalently state the problem in terms of z - r and w - r. We then have that the social
planner would want to smooth consumption across states and provide insurance to the risk
averse agents. The optimal solution will therefore maximize total output, and then distribute
these equally between the employed and the unemployed.
The output maximizing solution is given by the program
max
e-rt((l - u)y - vy)dt
du
s.t. T = (1 - u)A - uq()
0 = -V .
U
Note that benefits, z, and and taxes, r, have fallen out of the program, as they are simply
transfers satisfying a balanced budget constraint.
Setting up the Hamiltonian and solving for a19V
H=
H
and
aH
aul =
= rT
dt
e-rt[(l - u)y - Vyo+ p(A(1- u - Oq(O)u)]
OH
OH
u
-
-
y
+ v(O)v v = rp dp
d
Rewriting these and solving for the steady state ( dt = 0)
-y
=
+
+ Oq(0)7(0)
Yo
q(O)(1- r1(0))
-
Where q7(O)denotes the negative of the elasticity of firm matching with respect to vacancies,
7(0 ) = -q'(0)
- The matching elasticity plays an important role in the incremental value of
vacancies or unemployment. Consider first the first equation, which describes the shadow value
of an increase in unemployment, /u. Increasing unemployment results in the opportunity cost
y of not producing.
This cost is discounted not only by A,r and
discount rates, but rather, by A,r and
q(O) which are the private
(O)9)q(O).
The social planner takes into account that
an increase in unemployment also lowers the matching probability for workers, and evaluates
75
this loss to be larger due to prolonged unemployment. In the second equation, the same wedge
between the private and the social returns to an increase in vacancies is present. The cost of
opening a vacancy, 7yo,is discounted not only by the expected duration, q--I, but the planner
accounts for the lower matching rate for firms and prolonged life of a vacancy, which is a larger
1
q(8)(1-7(0)) '
Putting these together we get the condition
Y
Y
Y= - ~ ~ ly
A+T+ q(O)r,()
q(s)(1
- r())'.10)
This prescribes the efficient level of market tightness
Se .
(3.10)
To maximize the utility of risk averse
agents, the government would then distribute equally consumption between the two states,
x U = xE = x, so that the resource constraint holds,
(1 - u)xU + uxE < -y 0 v + vq(O)y.
3.3.1
Decentralizing the Unconstrained Optimum
If agents are risk neutral, then the efficient level of output Oe can be achieved if workers'
bargaining power is equal to the matching elasticity. This is the well known Hosios condition
which is reproduced here. Set unemployment benefits to zero and take U(x) = x, then the wage
equation (3.9) becomes
w = (y + O)
Use this in conjunction with the creation equation (3.8) to get
Y
Y
A+ r + q(O) q(9)(1- )'
Comparing this with the condition for the unconstrained optimum (3.10) we see that the efficient
level of output can be achieved when workers' bargaining power is exactly equal to the matching
elasticity. By letting firms be accountable only for a share (1 workers for a share , with
=
) of costs and benefits, and
this exactly replicates the optimal vacancy creation rate
and steady state unemployment. Firms are internalizing the extra social benefit of creating a
76
vacancy, and discounting the creation cost appropriately.
As we move away from a risk neutral world, the efficient level of market tightness,
0e
can
still be achieved in the market. However, the bargaining power which decentralizes the efficient
level is different. In particular, define we = y - (r + A)
as the efficient wage, and then back
out the unique share ,3e which gives this wage as the outcome of the Nash bargaining solution
in (3.9). This /e is larger than the bargaining power needed in the risk neutral case, since firms
get a higher share when workers are risk averse.
Proposition 17 /3e > 7(0).(To replicate the efficient market tightness in a decentralizedequilibrium, the bargainingpower of risk averse workers must be larger than the matching elasticity).
Proof. The efficient level of market tightness e, and hence the decentralized efficient wages,
we = y - (r + A)q-,
are invariant to risk aversion. By proposition 1 wages decrease with risk
aversion. Workers with higher risk aversion must therefore have higher bargaining power to
achieve the same wage level we. ·
The second, more important implication of adding risk aversion, is that the optimal allocation prescribes equal consumption across states. This can never be achieved by the market,
since the bargaining solution always gives workers part of the surplus above their unemployment
value. In other words, the market will never fully insure workers. We highlight this point in
the following claim.
Claim 7 The optimal allocation can never be achieved as an equilibrium in a decentralized
economy.
For this reason, it is more interesting to check what would the optimal government policy
be if it is restricted by the market behavior of workers and firms. That is, if wages and entry are
determined by the market and the government can only make transfers through unemployment
insurance financed by taxes. The constrained optimum would give the market tightness which
maximizes workers' welfare when the consumption across states are not equal, and determined
by the wage bargaining outcome between workers and firms.
77
3.4
Government Constrained Optimal Policy
In this section we solve for the constrained optimal level of unemployment insurance given wages
and entry are set in the market equilibrium, and subject to a balanced budget requirement.
From our previous discussion of claim 1, we expect the role of consumption smoothing to be
limited when wage determination is governed by the Nash bargaining solution.
The government maximizes the expected lifetime utility of workers and solves the following
program:
max
e-ri[uU(z - r) + (1- u)U(w- )]dt
subject to,
y-w
7a
r+
q(9)
U'(w- r)
= /(y + y)
=
du
dt
0
(1 - Oq(9))z
= (1 - u)A- uq(O)
u
Denote by
d,
have z -
= Oq(0)z and w - 7 = w - (1 - Oq(O))z.We can form the Hamiltonian and solve
dw
the derivatives of the equilibrium (, w) with respect to z, and replace r to
H = e-rt[uU(z- 7r)+ (1- u)U(w- 7r)+ p(A(1- u) - uq(9))].
We have the FOCs dH = 0,
uU'(z - 7r)q(O)+ (1 - u)U'(w- 7)(-)(1 - Oq(O))
dO*
dO*
+uU'(z- )[Oq'(0)
+ q(0)]Z
d- + (1- u)U'(w- r)[q'(0)+ q(0)]ddz*
dz
+ q(d)]ud
+±,(-)[Oq'(9)
dw*
+(1 - u)U'(w - 7) dz =
,
78
and, dd'u= r-
dt
d
U(z- r) - U(w- r) + p(-A - Oq(O)
- (Oq'(0) q())(-)2 v) = r-
dt
At the steady state, ~d = 0, and we have
dw*
uU'(z - r)Oq(O)- (1 - u)U'(w - r)(1 - Oq(O))+ (1 - u)U'(w - 7) d
dO*
d
+[Oq'(0)
+ q(O)][uU'(z
- r)zz dz + (1- u)U'(w- r)z-dz
U(z- )-U(w-r)
dO*
dz
=0
r + A + (O)Oq(O)
Note that even a lumpsum tax, which is otherwise irrelevant in a risk neutral world, actually has a distorting effect when risk aversion moves with wealth. When the government
increases the lumpsum taxation (to finance higher unemployment benefits) workers' risk aversion increases and their bargained wages are lower (see the first proposition). The effects of an
increase in unemployment benefits have thus a complex impact. Holding risk aversion constant,
workers' bargaining position rises, increasing their bargained wages and reducing market tight..
ness. However, taking the financing into account, a lower market tightness implies workers are
unemployed longer, and a larger amount of benefits needs to be paid out and financed. The
tax burden on workers increases twice: once from the costs of increased benefits, and twice
from the cost of the endogenous increase in unemployment duration. This decrease in workers'
wealth level increases their risk aversion, which in turn means that they will settle for lower
wages. This tends to offsets the initial increase in wages due to the larger bargaining position.
3.4.1
Constrained Optimum with Outside Financing
Finding the optimal level of insurance under a balanced budget constraint complicates the
analysis since taxes depend on the endogenous market tightness. This becomes quite cumbersome even if the demand for insurance is independent of wealth levels. It is still interesting to
find out if we would want to limit the level of insurance admitting insurance is free (and worry-
ing about raising the funds elsewhere). We can alternatively add a balanced budget constraint,
Oq(O))z< B, and investigate whether there are cases where it does not bind.
79
The government maximizes workers' lifetime expected utility subject to the free entry condition, the Nash bargain outcome and the equation governing the flow of unemployment,
max
Z'VIO2W'u
e-rt[uU(z) + (1 - u)U(w)]dt
subject to,
y_ - w
r(1 -
A
) U(w) - U(z) + PW =
U'(w)
y
q(9)
P(Y + -YO)
du
dt = (1-u)A-uOq(O)
0 =
Denote by d,
u
dw' the derivative of the equilibrium (, w) with respect to z. We can form the
Hamiltonian and solve
H = e-rt[uU(z)+ (1- u)U(w)+ p(A(1- u) - uOq(8))].
We have the FOCs dH = 0:
uU'(z) + (1 - u)U'(w) d
and, d du = r-
+ A(-)[9q'(0) + q(O)]u- = 0
ddt'
U(z) - U(w) + ~/(-A - Oq(O)- (Oq'(8)+ q(O))(-) 2v) = r
u
At the steady state,
-
dt
= 0, and we have
uU'(z)+ (1- u)U'(w)
U(w) - U(z)
dz
-
dz)-(z)
dO*
r + A+ 1(0)0q(O)'
In the first equation we have unemployment
benefits affect workers' welfare through three
80
terms. The first, uU'(z), is the direct effect of benefits on the utility of the unemployed. The
- dd , is the indirect
second term, (1 - u)U'(w)
increase of the wages of the employed. Finally,
the last term, -,uq(9)(1-
(O)),dzcorresponds to the change in unemployment through the
adjustment in the equilibrium market tightness. Recall that market tightness decreases with
benefits, so that the total contribution of the last term is negative. The shadow value of an
increase in unemployment, u, is given by the last equation, and corresponds to the lost wages
U(w)- U(z), where discounting takes into account the increase in unemployment duration, and
appropriately puts a larger weight on the lost earnings
Ir+A+~i(G)eq(G)
Plugging the second equation into the first and using q(0)(1 - r(0)) = [Oq'(O)+ q(0)] we
have the equivalent expression
'(Z) + (1 -
)U'()
uU() (w(1u)U'
)
dz
--
- U(z)
d*
(0) q(0) (1 - r/(0))
r + A + 77 ()q()q
dz
We can see where the potential for limiting unemployment benefits comes from. On the LHS
we have the welfare gains from the increase in the income level of workers in both states.
The unemployed have higher income because of the higher benefits and the employed have
bargained to higher wages because of their higher outside option. If these were the only effects
of increasing benefits, and if benefits were 'free' (as they are here), the government would have
wanted to provide workers with the maximum level of benefits simply to increase their income
level. However, there is cost to providing benefits. Because firms' gains are smaller, firms
respond by a lower creation level, and the steady state unemployment level increases. The
RHS describes the loss from marginally moving to a higher unemployment level, which is equal
to the fraction of unemployed, u, times the discounted loss of increasing unemployment. Again
the loss is discounted by the social discount rate qr(O)Oq(O).
To get more of this optimality
condition, we need to substitute the equilibrium derivatives.
Equilibrium Derivatives
To simplify the optimality condition further, we want to replace
the equilibrium derivatives of (0, w) with respect to z. This also turns out to be much simpler
81
without the financing constraint.
F : q(O)(y-w)-(r+A)y=O
N : (1 -/) (U(w)-U(z)) -(y
Fo =
+ yO-w)U'(w) = O
w)
q'(0)(y-
Fw = -q(0)
F = 0O
No
-7yU'(w)
=
Nw = (1 - /3)U'(w)+ 3U'(w) -/3(y + -yO- w)U"(w)
- w)U"(w)
= U'(w) - P(y + -yO
= -(1-/) U'(z)
N
F.
Fw
dO
N
Nw
dz
Fo
Fw
No
Nw
(O - - (1 -
/)
U'(z)(-)q(O)
q'(0)(y - w) (U'(w) - 3(y + -y - w)U"(w)) - (-/3yU'(w)) (-q(O))
q(O) (1 - 3) U'(z)
q'(0)(y - w) (U'(w) - 3(y + yO- w)U"(w)) - /3yU'(w)q(O)
Fo
Fz
dw
No Nz
dz
F
Fw
No
Nw
q'(O)(y - w) (- (1 - 3) U'(z))
same
(1 -
3) U'(z)/U'(w)
1 + /3q( (r + q(O)
A+ Oq(O))RA(w)+ ;7(-0)
- Oq(O)
~~(r+A)
82
=
1
Where we have used
=1+ y-w =1+ (r +-yr\)
(y + y - w)
(y-w)
Oq(O)
+
(r + A)
I
r + A + Oq(O)
r+A
and the creation equation.
Note that
dO
0
dz
dw
7(8)(y - w) dz
Back to optimum condition We thus have the optimality condition with the equilibrium
derivatives
dO
U(w) - U(z)
uU'(z) + (1 - )U'(w) dw
r + A+ j(O)-q(O)
dw
dz
q'(0)(y - w) (1 - ) U'(z)
q'(O)(y - w) (U'(w) - (y + -y - w)U"(w)) - /-yU'(w)q(9)
dz
(1 - /3)U'(z)/U'(w)
1 + Bq (r + A+ Oq(O))RA(w)
+ ,()r+)fi
dO
0
dw
7(0)(y - w) dz
dz
Putting these together we have
dw
uU'(z) + (1 - u)U'(w)dw
=
dz
U(wr)- U(z)
r+A
-+
Use the Nash solution to replace U(w) - U(z) =
q()(1 -
6
_ )
1---(Y
+ -y - w)U'(w):
uU'(Z)+ (1-U)U(W)dw
(yA++ (O)Oq(O)
- w) 1 - (0o) ItI Oq(O) U(w
dz = 1 - u r +(y
(y - w)
n()
Note that (using the creation to substitute for y-w):
dw
u
Oq(O) r+ A ± Oq(O)
/
1 - u r + r +A +
r(O)Oq(O)1 -3
83
)
dw
dz
(y+vyB-w)
r+ ' We can manipulate
(y-w) _ r+A+q(9)
the condition to read
uU'(z)+ (1 - u)(w)zdz
dw
d
-(O)q(O)
q()(y - w) dz:'
7(0))U(
1-77(0)1
77(0)
Finally, using the steady state solution to the unemployment rate we have u
that
u+
U'(Z)
(-)U'(W)
1_
z
1-1
dw
A r + A + Oq(O)
U'(~z)
(1 - \U'(Adz + r A +A+ (0)0q(O)
1-
=
-eOq
so
- q7(8
77(0) ]
This condition for the constrained optimal level of unemployment insurance resembles the
Hosios (1990) efficiency condition but differs from it in important ways. It is useful to make one
comment even before substituting
for the equilibrium derivatives, since we can disentangle the
modifications due to the constrained nature of the problem from those due to the addition of risk
aversion. Consider first the modification due to the constrained nature of the problem. This is
easiest to see if we assume that workers are risk neutral. Note that when
/3= 77the
second term
is negative, indicating that the accumulated effect of benefits on the welfare of the employed is
= qrmaximizes the expected returns of the unemployed workers
negative. Recall that setting
in equilibrium.3 It is thus not surprising that this level of bargaining power is not optimal when
we directly care also about the employed. In fact, since income when employed is always larger
than the income when unemployed, the optimizing social planner will want to have
which
creates higher employment than / = 77.This will be achieved via a lower bargaining power since
it will give firms higher incentives to open vacancies, and increase the matching for workers.
Employment and total welfare will increase.
We can equivalently write this optimality condition as
1-/3 (uU'(z)+(1-u)U'(W)) - 1/3 0l
L ((
U dz
)
(
w
dz r +A r + + q(O) )
which might suggest that the Hosios condition is 'modified' to account for the allocation of
income between the two states.
Finally, a closed form solution which highlights the role of risk aversion can be derived.
Denote by RA(w)
3
-
the absolute risk aversion coefficientand substitute in (??) for
Pissarides (2000) Pg 187.
84
and for the steady state u = X+,
then
uU'(z) + (1 - u)U'(w)
q((r$+
(1 -
1+
r + A + q(8)
) '(z)/U'(w)
+ A + Oq(0))RA(w)+
1 - (7)1
)
O
* [L r±Ar±A+j(0)Oq(0)1-/3
r + Xr + A+ 71()Oq(O)
I - / 77(0)]=0
r(O
)
(1 )
1 + q( ( + A+ Oq())RA(w) +
u+(1-u)
L
/3
A
r±A+Oq
()
A r+
+ q()
l+r +
Ar
+ + ()q()
(3.11)
(311)
1-7()] = 0
1 -77(0)
- 0 rl
J
+(1-3)Oq(0)
(3.12)
1+
+/3-i(r + A+ Oq(O))RA(w)
+
[
3.4.2
r
A r+Ar+A+
r+A+Oq(0)
(0)q()
Ar
--.
12
/31- 1-7(0)10
71(0) ]
A 4- r(O)Oq(O) 1 -3
7710-
Interpretation
We now interpret the optimality condition (3.11) or equivalently (3.12). Increasing unemployment insurance has two consequences. The first is an increase in the utility of the unemployed
through the direct transfer. This benefit is weighted by the fraction of people enjoying it, u, or
a relative share of A,the layoff rate. The second large term corresponds to the effect of benefits
on the employed through the wage bargain outcome. An increase in benefits increases the wages
of the employed, but decreases the vacancy creation and market tightness. A variant of the
Hosios condition holds. Let /30 denote the bargaining power which makes welfare neutral to the
that is,
wage bargain,
r+\+x
q(oq)
1-j-
15_
since the elasticity of the matching function is smaller than one.4 If workers' bargaining power
is at this neutral level, 30, then as r
) 0 the gains from higher wages are washed out exactly
by the decline in market tightness. In this case, the gain from unemployment benefits is simply
A and benefits should be paid out to the limit of the budget constraint. The same prescription
is true if workers' bargaining power is low relative to the matching elasticity ( < /30 <
).
These results are summed in the following proposition,
Proposition
18 If
< /3 < 77and as r
-
O, it is optimal to provide maximal unemployment
4This elasticity is estimated to be around 0.6 See Blanchard and Diamond (1989).
85
insurance.
However, when workers' bargaining power is high (
>
O0),the government might want
to limit the provision of insurance. While directly benefiting the unemployed, there is a net
loss from the effects on wages and creation. An increase in unemployment benefits increases
workers' bargained wages due to the increase in their outside option. This increase in wages
decreases firms profits and depresses creation. As creation is depressed ( is low) there are
more unemployed workers receiving the lower unemployment benefits and less employed workers
receiving high wages. The loss due to an increase in unemployment is larger than the gain due
to higher wages when P > B0.
, the size of the negative effect of benefits depends on the sen-
- >
For a given
sitivity of wages with respect to benefits. This negative effect on welfare is smaller when
is small. That is, the need to limit unemployment insurance
(1-)
1+:
(a
(r+A+Oq())RA(w)+-
8
()
is small when risk aversion is high, when the costs of opening up a vacancy are high and when
the worker's bargaining power is high, or the elasticity of the matching function is low (for a
constant
). These results are stated in the followingproposition,
Proposition 19 The solution for optimal unemployment benefits is interior only if
> /o.
In this case, the amount of insurance increases with workers' risk aversion, with the costs of
opening up a vacancy and with workers' lower bargainingpower.
77(6)
(+A)
_
Finally note we can write a modified Hosios condition, which describes the level of 0 needed
to make some ratio of 1-/
+/)
(09)(r
770)-hold.
The government chooses z to get at that ratio,
+
l+-(r+ A+ Oq(O))RA(w)
±1
1± /(+q(O)
1-7(0)
3.5
)
r +A + q()
q()
(1- ) -q()
+
+ q(
11 -, r + A+r()Oq(9)
Conclusion
Introducing
risk averse workers into a standard search model permits solving for the optimal
unemployment insurance and comparing it with the efficient unemployment insurance program.
It was argued that the interesting problem is one which is constrained by the market behavior
of firms and workers. In this case, the consumption smoothing role of unemployment benefits
86
is limited by surplus sharing between firms and workers. In addition, unemployment benefits
have a negative effect on vacancy creation, suggesting it is beneficial to reduce insurance when
workers' bargaining power is high relative to the matching elasticity.
Some interesting questions remain. Different financing methods are bound to result in
different welfare properties, since even lumpsum taxation affects the relative marginal utilities
of workers whose risk aversion varies with wealth. A more thorough comparison of the welfare
properties of different financing methods is needed. More interesting, perhaps, are the properties
of the optimal solution in a dynamic stochastic setting. How should the social planner account
for the wage adjustments between risk averse workers and risk neutral firms when deciding on
the optimal program? These questions are for future research.
87
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