Unemployment Benefits and Social
Differentiation
Johannes Rincke∗
March 28, 2003
Abstract
The paper shows that unemployment benefits can be used as a selection device. It is argued that status generating social interactions are
related to productivity differences. Therefore, identification of less productive individuals may be valuable for the middle class. For a given
benefit, the unskilled are the first to switch from employment to unemployment since both their wage and their status as employed is the
lowest among all productivity groups. With labor market status being
observable, unemployment benefits can therefore serve to identify the
unskilled. If the status effect is sufficiently strong, the middle class will
politically support a benefit scheme that leads to a self-selection of unskilled individuals into unemployment and thereby to increased social
differentiation.
Keywords: Unemployment benefits, voting behavior
JEL Classification: D72; D31; H24
∗
Centre for European Economic Research, L7, 1, D-68161 Mannheim, Germany, phone:
+49/621/1235-217, email: rincke@zew.de. I would like to thank Robert Schwager, Ingolf
Schwarz and Hans Peter Grüner for valuable comments and suggestions. Financial support
by the Fritz-Thyssen-Stiftung is gratefully acknowledged.
1
1
Introduction
Unemployment benefits figure prominently in the social security systems of
many countries. Roughly summarized, the underlying reason for this is traditionally seen as being twofold [see, e.g., Baily (1977), Mortensen (1977), SaintPaul (2000)]: Firstly, unemployment benefits insure the individual against
the risk of job loss. For adverse selection reasons, this kind of insurance is
provided as a public good and not via markets. Secondly, the existence of unemployment benefits tends to rise the outside option of employed individuals.
Therefore unemployment benefits are seen, in a politico-economic perspective,
as an instrument of insiders at the labor market to extract rents.
Although convincing and more or less standard in the literature, both arguments are limited in that they concentrate solely on economic motivations of
individuals. Especially in connection with social security transfers and other
institutions that redistribute income this view might well be judged restrictive. Corneo and Grüner (2000, 2002) present empirical evidence that individual preferences for redistribution are shaped by status considerations.1 In
sociological terms, social status is defined as a ranking of individuals based on
their traits, assets or actions. As the literature on social status and economic
behavior has shown, social motivations play a prominent role in shaping the
behavior of individuals whenever this ranking is affected.2 Furthermore, social
status often depends on the reference group individuals belong to. Given that
the unskilled and less educated dominate the group of unemployed, individuals receiving unemployment benefits are supposed to be negatively affected by
this group association in terms of their social status. But this also means that
not belonging to the group of unemployed has strong positive status effects.
This status effect is at the focus of the model presented in this paper. The
key aspects of the model can be summarized briefly. I follow Cole et al. (1992)
and Corneo and Grüner (2000) in modelling social status as utility derived
from matching with a social partner.3 Admittedly, this is a somehow crude
way of describing an extremely complex phenomenon that is, perhaps, not well
understood by economists. Nevertheless, this simple way of modelling captures
the basic fact that social status is generated primarily by social interactions
1
Boeri et al. (2001) provide a detailed survey of individual attitudes on the welfare state
in France, Germany, Italy and Spain.
2
For a survey, see Weiss and Fershtman (1998).
3
The basic framework of the model is borrowed from Corneo and Grüner (2000). However,
they do not model a labor supply decision. Instead, they discuss a lump-sum redistribution
scheme with a proportional tax on wealth endowments.
2
and not by market transactions or, in a broad sense, economic decisions. Furthermore, it allows to model social decisions that determine status by using
comparatively simple tools.
Individuals are assumed to be only imperfectly informed about potential
social partners. Technically, an individual’s desirability as a social partner,
i.e. its matching value, and its productivity are private information. At the
same time, productivity and matching value are correlated. This correlation is
crucial because it introduces a link between economic and social decisions. The
assumption may be justified in that a large fraction of what is conventionally
seen as productivity enhancing skills and abilities might also be valuable in
a social context. Similarly, it could capture a positive correlation between
productivity (and therefore wages) on the one hand and the probability of
acquiring a certain degree of education on the other. Conventionally, better
education is seen to expand the opportunities for social interaction.
Individuals can learn about potential social partners by observing their economic decisions. First of all, in a stylized labor market individuals decide
whether to be employed or not. It is natural to assume that this decision is
perfectly observable. Secondly, an individual’s level of consumption can be imperfectly observed. Based on a broad measure of consumption, say the type of
car one is driving or the house one is living in, individuals estimate the income
level of potential partners. Due to the correlation between wage and matching
value consumption signals carry information about the value of individuals as
social partners.
In this framework, the majority outcome of a voting over a tax-financed
flat-rate unemployment benefit scheme is analyzed.4 For a given benefit, the
unskilled are the first to switch from employment to unemployment since both
their wage and their status as employed is the lowest among all productivity
groups. With labor market status being observable, unemployment benefits
can therefore serve to identify the unskilled. In equilibrium, individuals can,
by their decision to be employed, signal a certain minimum ability of providing
status to a social partner. A benefit scheme that leads unskilled individuals
to self-select into unemployment therefore improves the status position of the
political decisive middle class. If the concern for social status is sufficiently
strong, a political equilibrium with self-selection of the unskilled is supported
4
Most OECD countries have earnings-related schemes, but after a certain period of unemployment the payments from unemployment insurance are replaced by unemployment
assistance, a flat-rate scheme in almost all cases. See OECD (1997) for an overview of
existing schemes and Atkinson and Micklewright (1991) for a discussion of institutional
differences between unemployment insurance and unemployment assistance.
3
by the middle class.
The paper relates to the literature in several ways. The effect of unemployment insurance on individual labor supply decisions has been investigated
using search models [see, e.g, Mortensen (1977)]. In an empirical study using
Swedish data, Carling et al. (2001) find strong evidence for the prediction that
a reduction of benefits leads to an increase in job finding rates. Nickell (1998)
uses a panel of 20 OECD countries and concludes that high replacement rates
contribute to high unemployment. The empirical evidence for sorting of the
less productive into unemployment is also persuasive. Dreze and Sneessens
(1997) and Nickell and Bell (1997) provide descriptive statistics about unemployment in OECD countries, showing a marked decline in unemployment as
education rises. Nickell and Bell (1997) also point out that only in countries
having an unemployment benefit scheme the unskilled have much higher unemployment rates than the skilled.
In a vast body of literature surveyed by Karni (1999), the optimal design of
unemployment insurance schemes has been analyzed. There is also a number
of studies modelling voting over unemployment benefits, among them Wright
(1986) and Pallage and Zimmermann (2001). In both strands of the literature,
individuals face some risk of becoming unemployed, and unemployment benefits serve as insurance. In contrast to this, I abstain from modelling market
imperfections leading to job losses or persistent unemployment. I even abstain from modelling a fully-fledged labor market with endogenous wages and
assume that individuals can run a firm generating income depending only on
individual productivity.
It is often criticized that the institutional features of unemployment benefit
schemes in many countries have strong adverse incentive effects for less productive unemployed. A main point of criticism is the huge implicit marginal
tax rate that results from reducing the transfer for individuals who earn additional income such that the net income is more or less independent of the
individual labor supply decision. It could be argued that discouraging less
productive individuals from supplying labor is an instrument used by insiders
to prevent competition from unemployed. The alternative explanation offered
by the present paper refers to the observability of the labor supply decision.
The crucial point is that a benefit scheme effectively promoting zero-one decisions of less productive individuals at the labor market is suitable to maintain
the quality of what can be seen as a signal for a certain minimum productivity. By introducing a participation barrier, a distinction between employed
and unemployed is introduced that otherwise would not be available. Since
4
in equilibrium only the least productive are unemployed, the benefit scheme
helps to separate productivity types.
The paper is organized as follows. The model is introduced in the following
section. In section 3, the matching technology is described. Section 4 identifies
environments in which a complete self-selection of less productive individuals
into unemployment occurs in equilibrium. Section 5 concludes.
2
The model
There is a unit mass continuum of individuals. Individuals are characterized
by their productivity type. There are three types of individuals: L, M and H.
For simplicity, let all subsets of individuals of a specific type have the same
Lebesgue measure. That is, the distribution of types is symmetric. Individuals
derive utility from consumption and from matching with another individual.
With respect to the formation of pairs, what matters for each individual is the
matching value of its partner. Let the utility function be
U = c + v(µ),
(1)
where c is consumption and v(µ) is the utility from matching with a partner
µ. The matching value of an individual is the utility the partner derives when
matching with the respective individual. Therefore, the matching value measures the ability of individuals of generating matching utility for a social partner, i.e. the desirability of individuals in social interactions. An individual’s
productivity translates into matching value in a deterministic way: M -types
have matching value m, L-types m − δ and H-types m + δ, where m > δ > 0.
Each individual is endowed with an amount s of the consumption good and
one unit of labor that can be used in individual firms to produce additional
units of the consumption good. Let us refer to the individually produced consumption good as income. Opening a firm and producing income is modelled
as a zero-one-decision. In the following, I call this the employment decision.
Conditioned on opening a firm, let M -types have income e, L-types e − ∆ and
H-types e+∆, where e > ∆ > 0. If the endowment is not used, income is zero.
The income of employed individuals is subject to the tax t. Thus an employed
M -type, for example, has consumption cM = s + e − t. Call individuals with
zero income unemployed, and assume that it is public information whether
an individual is unemployed or not. Both the income of employed individuals
and the matching values are private information, but consumption serves as a
5
signal: it is imperfectly observed by all other individuals. To be specific, let
individual i be of type K = L, M, H. Then its consumption signal is

 c +² >0
if i is employed
K
i
γi =
(2)
 0
otherwise ,
where the errors ²i are identically
and
£
¤ independently distributed according
to a uniform distribution on − f2 , f2 . Note that γ is a signal for employed
individuals and an indicator that is zero if i is unemployed and takes on strictly
positive values if i is employed.5 In order to have a meaningful signalling device
and to keep things as simple as possible, assume ∆ < f < 2∆. We therefore
have, for employed individuals, overlapping supports of the signal between
neighboring types, but no overlapping for the signals of L- and H-types.
In what follows, I will analyze a four stage game: At the first stage, there is
pairwise majority voting over an unemployment benefit T ≥ 0. At the second
stage, employment decisions are made and the budget-balancing income tax t
is determined. The political decision is revised and the transfer is set to its
default value zero if T > 0 has been chosen as public policy and more than
one third of the population applies for the benefit, or if the lowest possible
consumption signal of employed individuals would not be strictly positive any
more, i.e. if cL (t) − f2 ≤ 0.6 After a revision, all individuals who applied for
the transfer once again have to make their employment decision. The effective
T then is payed out to the unemployed, be it the one fixed in the political
process or the default value zero. At the third stage consumption takes place.
Consumption signals are sent and used to learn about the matching value of all
other individuals. At the last stage, individuals voluntarily match into pairs.7
The feature of policy revision built into the game reduces its complexity a
lot. It simplifies the reasoning when solving the game by ruling out equilibria
that would be supported only by extreme realizations for parameters anyway.
Furthermore, it greatly reduces the space we have to devote to the description
of individual behavior off the equilibrium path. Building a policy revision
into the game which effectively rules out certain ‘extreme’ outcomes may be
justified by the following argument. Suppose your government promises to pay
a benefit in case you should become unemployed. Furthermore, suppose that,
5
An alternative model could have stochastic type-specific firm profits and perfectly observable consumption levels.
6
If cL (t) − f2 = 0 were a valid signal, one could not distinguish between an employed
L-type having this signal and an unemployed individual.
7
The details of the matching technology are described in the next section.
6
for whatever reason, everyone stops working and applies for the benefit. Do
you believe you would get anything?8
3
The matching technology
In order to be able to model social interactions as a two-sided matching game,
two proper sub-populations are needed. In the following, the construction of
these sub-populations is described.
Once consumption has taken place the population can be partitioned into
subsets according to the signals. Firstly, there is a set of unemployed. Secondly,
there are employed individuals. The employed can be sorted according to
their signal.9 Define γ K (t) ≡ cK − f2 and γ K (t) ≡ cK + f2 . Now partition
the whole interval [γ L (t), γ H (t)], which is the domain of possible signals, into
five intervals Γ1 = [γ L (t), γ M (t)), Γ2 = [γ M (t), γ L (t)), Γ3 = [γ L (t), γ H (t)),
Γ4 = [γ H (t), γ M (t)), Γ5 = [γ M (t), γ H (t)]. Let us define sets Sh , h = 1, 2, . . . , 5
where S1 is the set of all individuals having a signal in Γ1 , S2 is the set of
all individuals having a signal in Γ2 and so on. Note that by evaluating the
Lebesgue measure of S1 , S3 and S5 all individuals can compute the shares of
employed individuals in all types. Furthermore, if the set a given signal belongs
to has positive measure, individuals can compute the probability PK (γ, t, p)
of the event that the sender of a signal γ is of type K, given the tax t and
the vector p = (pL , pM , pH )0 , where pK indicates the share of employed type
K-individuals. This probability is determined according to

αγ,K (t) pK

:
γ>0
αγ,L (t) pL +αγ,M (t) pM +αγ,H (t) pH
PK (γ, t, p) =
(3)
1−pK

:
γ=0 ,
3−(pL +pM +pH )
where

 1
αγ,K (t) =
 0
if γ ∈ [γ K (t), γ K (t)]
otherwise
.
The upper line on the right hand side of (3) refers to signals of employed
individuals and the lower to those of unemployed.
8
Interestingly, the political debate in Sweden in the 1990s culminating in a reduction of
the replacement rate from 90 to 75 percent focused on the fiscal crisis induced by increased
unemployment, not on adverse incentive effects. See Carling et al. (2001).
9
Note that t is a tax collected from each employed individual, thereby shifting the whole
distribution of the signals.
7
Note that I did not say anything about conditional probabilities for signals
which are elements of zero-measure sets. For completeness, I could specify
beliefs, in some cases even arbitrary ones, but as we will see shortly, I actually
do not need to do so in order to describe behavior in the matching game.10
Using the conditional probabilities derived above, each individual can compute the expected utility that is associated with choosing a partner having
signal γ, where
Ev(·|γ, t, p) = PH (γ, t, p)(m + δ) + PM (γ, t, p)m + PL (γ, t, p)(m − δ). (4)
In order to be able to describe stable matchings as the outcome of a twosided matching game I divide the population into two sub-populations of equal
measure. Call the set of unemployed individuals S0 . Then, by definition, we
5
S
have [0, 1] =
Sk . Now, divide each of the sets Sk , k = 0, 1, . . . , 5 into two
k=0
disjoint sets S1k and S2k such that λ(S1k ) = λ(S2k ) ∀k, where λ(X) is the
Lebesgue measure of X. Finally, construct two sub-populations defined as
S1 ≡
5
[
k=0
S1k
and
S2 ≡
5
[
S2k .
(5)
k=0
Note that λ(S1 ) = λ(S2 ) and that both sub-populations are identical up to
subsets of individuals with measure zero.
4
An equilibrium with self-selection into unemployment
The game is analyzed by backward induction. As a first step, I will describe
individual behavior in the matching process. Then I go on to the second stage
and analyze employment decisions. Finally, voting behavior at the first stage
of the game is analyzed.
10
Note that, if γ > 0 and the relevant set has measure zero, then the sender is identified
if T > 0. This is because only S2 and S4 can possibly have as elements individuals of more
than one type, and due to the restriction concerning the maximum share of unemployed in
the population these sets must have positive measure if T > 0.
8
4.1
Matching into pairs
At the matching stage of the game, individuals from both sub-populations S1
and S2 match voluntarily into pairs, using the information about potential
partners made available by the signals. For simplicity, assume that individuals
receive utility zero from staying unmatched. Following Cole et al. (1992)
and Corneo and Grüner (2000), I abstain from describing the details of the
two-sided matching process11 . Stable matchings are characterized as follows:
Let µ : S1 → S2 be a voluntary matching if the following two conditions
hold: (i) There does not exist i, j ∈ S1 , i 6= j such that Ev(i|γi , t, p) >
Ev(j|γj , t, p) and Ev(µ(j)|γµ(j) , t, p) > Ev(µ(i)|γµ(i) , t, p), where µ(i) ∈ S2 is
i’s matching partner. (ii) There does not exist a pair of individuals i ∈ S1
and j ∈ S2 where both individuals are unmatched. (iii) For any set B ⊂
µ−1 (S2 ), λ(µ(B)) = λ(B). If the first condition is violated, there exists a
pair of individuals that blocks the actual matching because they both prefer
to leave their assigned partners and match with one another. The meaning
of the second conditions is that individuals who can match with one another
but are unmatched form a blocking pair. The third conditions means that if
all individuals from two sets are to be matched with one another, both sets
must have the same Lebesgue measure. Note that condition (ii) does not
preclude the existence of a set of unmatched individuals. But, if it exists, it
must necessarily have measure zero. Too see this, suppose a subset of S1 with
positive measure stays unmatched. Call this subset S10 . Since λ(S1 ) = λ(S2 )
and the measure of matched individuals must be equal in both S1 and S2 , there
also exists a set S20 ⊂ S2 of unmatched individuals. This violates condition (ii).
A symmetric reasoning with respect to subsets of S2 completes the argument.
With the description of the matching technology and the conditions for a
voluntary matching at hand I am now in a position to characterize the set of
stable outcomes of the two-sided matching process.12
Proposition 1
Consider all i who are elements of some Sk , k = 0, 1, . . . , 5 having positive
measure. In every stable matching, Ev(µ(i)|γµ(i) , t, p) = Ev(i|γi , t, p) for all
such i, except possibly for a zero-measure set.
Proof. The proof is by contradiction. Consider the subset S1m with the
11
For an introduction into two-sided matching problems, see Roth and Sotomayor (1990).
The proof of proposition 1 is inspired by Corneo and Grüner (2000) p. 1494, but more
general. Due to the endogenous distribution of income a large variety of possible distributions
of signals has to be covered.
12
9
highest expected matching value of all S1k , k = 0, 1, . . . , 5 having positive
measure. By definition and because only zero-measure sets can stay unmatched, in every stable matching we can partition S1m into disjoint subsets
S1mk , k = 0, 1, . . . , 5 such that, for all k, all i ∈ S1mk are matched with some
µ(i) ∈ S2k . Now suppose it exists S1mk for some k with λ(S1mk ) > 0 and for
all i ∈ S1mk we have Ev(µ(i)|γµ(i) , t, p) < Ev(i|γi , t, p). Then, since λ(S1k ) =
λ(S2k ) ∀k, there exists a S̃2 ⊂ S2 with λ(S̃2 ) = λ(S1mk ) and all µ(j) ∈ S̃2
being matched with some j ∈ S1 such that Ev(j|γj , t, p) < Ev(µ(j)|γµ(j) , t, p)
while, by definition, Ev(µ(j)|γµ(j) , t, p) = Ev(i|γi , t, p). This cannot be a stable
matching since all i ∈ S1mk would prefer to be matched with some µ(j) ∈ S̃2
and vice versa. Hence ∀i ∈ S1m (except possibly for a zero-measure set),
Ev(µ(i)|γµ(i) , t, p) = Ev(i|γi , t, p) holds. Now let S1n be the subset with the
highest expected matching value of all S1k having positive measure once S1m
is removed from S1 . Repeating the same kind of argument as before for S1n
and all remaining S1k with λ(S1k ) > 0 shows that in a stable matching only
zero-measure sets of individuals can possibly be matched with partners from
sets which differ with respect to the expected matching value of their elements.
Intuitively, because both sub-populations are identical up to (possibly) a
zero-measure set of individuals and all individuals have identical preferences
over potential social partners, we must have that in a voluntary matching
the typical pair (i, µ(i)) has signals (γi , γµ(i) ) carrying the same informational
value in the sense that based on γi and γµ(i) , a third individual would estimate
the matching values of both i and µ(i) as being equal. All individuals would
like to have one of the most attractive partners, but competition for highquality matches leads to a sorting of individuals into symmetric matches. Thus,
the symmetry of the distribution of signals in the sub-populations translates
into symmetry of matching for the typical pair. Proposition 1 is somewhat
weak in that it does not say anything about expected matching utilities for
a possibly existing zero-measure set of individuals. Covering zero-measure
sets is not possible because in almost all cases we could, starting from any
stable matching, reassign some individuals to partners with different expected
matching values such that the resulting matching would also be stable. For
example, suppose all individuals are employed and all matches are symmetric.
Starting from this stable matching, take a zero-measure set S̃1h ⊂ S1h for
some h and reassign the elements to partners from some S2l , l 6= h. Call the
0
set of elements of S2l who are rematched S̃2l . Define S1h
≡ S1h \ S̃1h and
0
0
0
S2l ≡ S2l \ S̃2l . Since λ(S1h ) = λ(S1h ) and λ(S2l ) = λ(S2l ), one can now
0
0
rematch all i ∈ S1h
with elements from S2h and all j ∈ S2l
with elements
10
from S2l . After all these reassignments, there does not exist a blocking pair of
individuals. Conditions (ii) and (iii) are also fulfilled. Therefore, the resulting
matching is stable. On the other hand, proposition 1 is rather strong. It shows
that if a set Sk , k = 0, 1, . . . , 5 has positive measure, then all individuals i ∈ Sk
associate probability zero to the event that in a stable matching they end up
with an asymmetric match, i.e. that they are matched with a partner who
differs in terms of expected matching value.
Because proposition 1 does not say anything about what kind of match
individuals should expect if they are element of some possibly existing zeromeasure set Sk , I specify the following beliefs: (i) All individuals believe that
they will be self-matched if they are element of some zero-measure set Sh ,
h = 1, 2, . . . , 5. To be self-matched means to be matched with a partner
who has the same matching value as oneself. To illustrate, consider the case
pL = 0, pM = pH = 1. A single L-type considering the alternative of being
employed then believes to be matched with an L-type if γ ∈ Γ1 should be
drawn as his signal. This is what we should expect, given that he is identified
as an L-type, a randomly drawn partner from the set of unemployed is of type
L with probability one and all other individuals have an expected matching
value strictly higher than m − δ.13 (ii) All individuals believe that they will be
matched with an L-type if they are element of S0 and λ(S0 ) = 0. If the rate of
unemployment is zero, choosing to be unemployed is believed to be associated
with the worst match available.
As mentioned above, one could specify additional beliefs saying how individuals assign probabilities to the event that the sender of signal γi is of type K
given that i ∈ Sk and λ(Sk ) = 0. But, since for the determination of expected
matching utilities it suffices to know the beliefs about matching partners, I
abstain from doing so.
4.2
Employment decisions
At the second stage, all individuals simultaneously face their employment decision. Note that for given T , the tax that equalizes the government’s budget
depends on the share of unemployed, i.e. t = t(T, p).
Each individual takes the actions of others as given and optimizes against
them. Let individual i be of type K. Suppose i chooses to be employed. Her
13
The same argument holds for the cases pM = 0, pL = pH = 1 and pH = 0, pL = pM = 1
(see proposition 3). All other cases are of minor importance since they become relevant only
if T = 0 results from the first stage (see lemma 1).
11
expected matching utility, given employment decisions of all other individuals
resulting in λ(Sk ) > 0 for all k, then equals
Z∞
EvK (t, p) =
Ev(i|γ, t, p)
αγ,K (t)
dγ,
f
(6)
−∞
α
(t)
where γ,K
is the density of the signal for employed K-types. If i chooses to
f
be unemployed, her expected matching utility is
EvK (γ = 0, t, p) = Ev(i|γ = 0, t, p).
(7)
If the actions of all other individuals at stage two give rise to zero-measure
sets in which i could possibly end up with her γ, her expected matching utility
must be formulated according to the specified beliefs.
An equilibrium profile of strategies must, for any T , prescribe employment
decisions for all individuals that are simultaneously optimal. Let me start with
some results stating properties that an equilibrium profile of strategies must
or cannot have.
Proposition 2 Suppose T > 0. There cannot exist an equilibrium where sets
of different types having positive measure are unemployed such that λ(S0 ) ≤
1/3 and γ L (t) = s + e − ∆ − t(T, p) − f2 > 0.
Proof. Suppose 0 < pL < 1, 0 < pM < 1 and pH = 1, where pL + pM ≥ 1,
i.e. λ(S0 ) ≤ 1/3, and γ L (t) > 0. Employment decisions do not trigger the
policy revision. Hence T is the effective transfer. Employment decisions can
only be optimal for all individuals if, given pL , pM and pH , all L-types and all
M -types are exactly indifferent between working and applying for the transfer.
If L-types and M -types pool in applying for the transfer, they must have the
same expected matching utility from this option. Therefore, both types also
must have the same expected utility from working. This cannot be the case,
because both the net income and the expected matching utility of working M types are higher than those of working L-types. Similar arguments show that
there cannot exist an equilibrium where sets of L-types and H-types, M -types
and H-types or all types having positive measure are unemployed.
Proposition 3 Neither an equilibrium with 0 ≤ pM < 1, pL = pH = 1 nor
one with 0 ≤ pH < 1, pL = pM = 1 can exist as long as γ L (t) > 0.
12
Proof. Suppose 0 < pM < 1, pL = pH = 1 in equilibrium, and suppose
γ L (t) > 0. M -types must be indifferent between working and applying for the
transfer. The indifference condition is
e − t(T, p) + EvM (t, p) = T + m.
(8)
Using (3), (4) and (6) yields
γZL (t)
e − t(T, p) +
γ M (t)
m(1 + pM ) − δ
dγ +
(1 + pM )f
γZ
M (t)
γ H (t)
γ H (t)
Z
m
dγ +
f
γ L (t)
m(1 + pM ) + δ
dγ = T + m.
(1 + pM )f
(9)
This reduces to t(T, p) = e − T . At the same time, L-types’ behavior must be
optimal:
γ M (t)
Z
e − ∆ − t(T, p) +
m−δ
dγ +
f
γ L (t)
This reduces to
δ
t(T, p) ≤ e − ∆ −
1 + pM
γZL (t)
γ M (t)
µ
m(1 + pM ) − δ
dγ ≥ T + m (10)
(1 + pM )f
¶
∆pM
+ 1 − T,
f
(11)
which contradicts t = e − T . If pM = 0, an M -type considering the alternative
of being employed beliefs to be self-matched if his signal should lie in Γ3 .
Therefore, one gets t(T, p) ≥ e − T and t(T, p) ≤ e − ∆ − δ − T from both
types’ optimality conditions, once again a contradiction. A similar argument
shows that 0 ≤ pH < 1, pL = pM = 1 cannot be an equilibrium.
The only transfer which could, in general, lead to an effective unemployment
rate higher than one third is T = 0. This is because, if T > 0 is fixed
by the political process, an unemployment rate higher than one third will
trigger the default transfer, which again is zero. The following lemma serves
to characterize equilibrium employment decisions for T = 0.
Lemma 1 For T = 0 (determined either by the political process or as the default transfer), equilibrium strategies for all individuals must specify to choose
employment.
13
Proof. Suppose T = 0 has been implemented and all individuals choose
to work. Then for all types deviating and going for a zero transfer leads
to strictly less consumption and strictly decreased expected matching utility.
This is because a single individual believes to be matched with an L-type and
therefore has expected matching utility m − δ, which is for all types less than
what a working individual could expect.
By very similar arguments it can be shown that a strategy profile leading to
a set of unemployed individuals with measure zero cannot be an equilibrium.
All strategy profiles leading to a set of unemployed individuals with positive
measure can be assigned to one of the following cases:
0 ≤ pL < 1, pM = pH = 1;
0 ≤ pM < 1, pL = pH = 1;
0 ≤ pH < 1, pL = pM = 1;
pL = 1, 0 ≤ pM < 1, 0 ≤ pH < 1;
0 ≤ pL < 1, pM = 1, 0 ≤ pH < 1;
0 ≤ pL < 1, 0 ≤ pM < 1, pH = 1;
0 ≤ pK < 1 ∀ K = L, M, H.
For all cases, one can easily find individuals for whom a deviation would be
profitable (details omitted).
Lemma 1 shows that employment is optimal for all individuals if T = 0.
The next lemma states that for all T > 0 that might be implemented in the
first stage we find at least one profile of simultaneously optimal employment
decisions.
Lemma 2 Suppose T > 0. Then all strategy profiles leading to λ(S0 ) > 1/3
are Nash equilibria in the subgame starting after T has been determined.
Proof. If strategies are such that λ(S0 ) > 1/3 if T > 0 has been fixed in the
political process, the default T = 0 is implemented anyway. Thus no individual
has an incentive to deviate from its strategy.
I am now going to characterize transfers which might be followed by employment decisions leading to a strictly positive effective rate of unemployment (a
strictly positive tax rate) in equilibrium. From the observations made so far
we already know that while doing so only employment decisions leading to
0 ≤ pL < 1, pM = pH = 1 need to be considered. In the following, assume
for simplicity that the endowment s is sufficiently high to guarantee γ L (t) > 0
as long as T is such that an equilibrium with a strictly positive effective tax
exists in the subgame starting after T has been chosen.
Note that L-types must be indifferent between being employed and being
14
unemployed if individual decisions leading to 0 < pL < 1 are required to be
simultaneously optimal. Given pM = pH = 1 the indifference of L-types can
be stated as
γ M (t)
Z
e−∆−t+
m−δ
dγ +
f
γ L (t)
γZL (t)
γ M (t)
m + pL (m − δ)
dγ = T + m − δ,
(1 + pL )f
(12)
where t indicates any tax rate that might be determined irrespective of the
budget constraint. Basic manipulations yield
µ
¶
δ
∆
t=e−∆+
1−
− T.
(13)
1 + pL
f
In figure 1, the indifference conditions for pL = 1 and pL = 0 are displayed.
To the left of L1 are points at which, given pM = pH = 1, it is simultaneously
optimal to work for all type-L individuals. Here, because no one goes for
the transfer, we must have t(T, p) = 0 in equilibrium anyway. To the right
of L0 there are points at which, given pM = pH = 1, it is simultaneously
optimal for all L-types to go for the transfer (pL = 0). Here, as long as the
self-selection of types into employment and unemployment does not change,
budget equalization requires t(T, p) = T /2. Let T1 be the transfer at which L1
and the T -axis intersect. T0 denotes the intersection of L0 and t = T /2. The
corresponding taxes are t1 = 0 and t0 .
The next task is to identify the set of budget balancing pairs (t, T ) for which
0 ≤ pL ≤ 1, pM = pH = 1 reflect optimal employment decisions of all types.
Let us first look at the L-types. The government’s budget equation can be
written as
(2 + pL )t = (1 − pL )T.
(14)
Solving for the share of working L-types gives pL = (T − 2t)/(T + t). Using
the indifference condition (13), substituting for pL and solving for T gives two
separate functions
¤1
2v + z − t 1 £
± (3t − 2v)2 + z(4v + z + 6t) 2 ,
(15)
4
4
³
´
where I substituted v ≡ e − ∆ and z ≡ δ 1 − ∆
. It is easy to verify that
f
T (+,−) =
T (−) ≤ 0 for all positive t, and that T (+) is real valued, continuous and convex
15
δ
2
t
6
L
@0
@
@
t0
L
@1
@
0
@
³
1−
∆
f
´
δ
2
>e−∆
t
@
@ M
@ 1
@
@
@
@
@
@
@
@
@
@
@
@
@
@
Tmax
T1 T0
1−
∆
f
´
<e−∆
6@
@
@
@ M1
L0
@
@
@
@
L
@
@1 @
1
@ @ t = 2 T@
@
@ @
t0
@
@ @
@
@ @
@ @
@
0
T0 T1
Tmax
@
t = 12 T @
@
³
@
T
T
Figure 1: Equilibrium employment decisions for varying transfers.
in t. Since the budget is balanced and the indifference condition for L-types
is met at points (T1 , 0) and (T0 , t0 ), these points must lie on T (+) . As shown
in figure 1, T (+) might have its minimum on [0, t0 ], but this is not necessarily
(+)
the
³ case.´ Evaluating the derivative at t = 0, one³ finds ´dT (0)/dt < 0 if
δ 1− ∆
< 4(e − ∆). Furthermore, T0 > T1 if δ 1 − ∆
> 2(e − ∆). One
f
f
can also use (13) and the budget to eliminate the transfer. This gives
µ
¶
∆
3t
δ
1−
−
= 0.
(16)
e−∆+
1 + pL
f
1 − pL
Using the implicit function rule shows that d pL /d t < 0. An increase in the tax
is always associated with a lower share of type-L individuals being employed.
Now let us turn to the M -types. Consistency of their employment decisions
in an equilibrium with 0 ≤ pL < 1, pM = pH = 1 requires
γZL (t)
T +m−δ ≤ e−t+
γ M (t)
m + pL (m − δ)
dγ +
(1 + pL )f
γ H (t)
Z
γ L (t)
m
dγ +
f
γZ
M (t)
m + 2δ
dγ. (17)
f
γ H (t)
Note that if this condition is exactly binding, the equivalent condition for
H-types is not, because H-types have higher income and a higher expected
16
matching utility if they choose to be employed.14 Furthermore, any (t, T )
satisfying (13) also satisfies (17) for any pL ∈ [0, 1). To see this, write (17) as
µ
¶
µ
¶
δ
∆
pL δ
∆
t≤e+δ+
1−
−
1−
− T,
(18)
2
f
1 + pL
f
and compare with (13).
In order to determine Tmax , the highest transfer we have to consider explicitly, substitute pL = 0 as the polar case. The set of points (t, T ) at which
M -types are exactly indifferent between being employed and being unemployed
is displayed as M1 in figure 1. Let T be the transfer defined by the intersection
of M1 and the budget line. Tmax is then given by max{T1 , T }.
From the characterizing properties of equilibrium given so far it is immediately obvious that for any T > Tmax that might be determined in the first
stage, equilibrium strategy profiles necessarily must trigger the default transfer
T = 0.
Lemma 3 Equilibrium strategy profiles must specify employment decisions
leading to an effective transfer of zero if T > Tmax was determined in the
political process.
In the preceding paragraphs we found that multiple equilibria exist in almost all subgames starting after the transfer has been fixed. In particular,
if dT (+) (0)/dt < 0, then there exist transfers to the left of T1 which might in
equilibrium be followed by a zero rate of unemployment as well as different
strictly positive rates of unemployment and corresponding taxes. I conclude
this section with
Lemma 4 The following strategy profile constitutes a Nash equilibrium in every subgame starting after the transfer has been chosen:
For all types, strategies specify to work in case the political process should be
revised. Furthermore, for each M -type and each H-type, strategies specify to
work if T ≤ Tmax and to choose not to work if T > Tmax .
For L-types, strategies condition on two possible cases that might occur:
³
´
i) 2δ 1 − ∆
> e − ∆: Each L-type’s strategy specifies to work if T ≤ T1 ,
f
to choose employment with probability pL if T ∈ (T1 , T0 ], where pL is jointly
determined by (13) and the budget, and to choose not to work if T > T0 .
14
Thus I do not have to deal with the employment decisions of H-types explicitly.
17
δ
2
³
∆
f
´
1−
≤ e − ∆: Each L-type’s strategy specifies to work if T ≤ T1 and
ii)
not to work if T > T1 .
For the moment, consider only the set D of equilibria where employment decisions following the equilibrium strategies do not induce a reversion of the
political process for all T ≤ Tmax . It is easy to see that the strategy profile
described by Lemma 4 has the following property: For all transfers that might
be chosen by the electorate, it specifies employment decisions such that there
does not exist another equilibrium from D leading to a strictly lower rate of
unemployment.
4.3
Voting over unemployment benefits
We are now in a position to find the Condorcet winner of the voting stage,
i.e. the transfer which beats all alternatives in a pairwise majority voting.
In a given pairwise comparison, all individuals vote for the transfer which
maximizes their expected utility. Because subgames starting after a transfer
has been fixed generally have multiple Nash equilibria, we need to specify
which equilibrium is expected to occur. It seems reasonable to assume that
for all transfers that might be determined in the political process individuals
expect the Nash equilibrium described in lemma 4. This assumption could
be weakened in the light of lemma 3. The crucial point, however, is that the
electorate must be assumed to ‘coordinate’ expectations with regard to Nash
equilibria in subgames following after a transfer T ≤ Tmax has been chosen.
Proposition 4 Suppose all individuals expect equilibrium employment decisions as described in lemma 4 and have beliefs about matching partners as
specified above.³Then ´
the following holds:
δ
∆
(i) Suppose 2 1 − f > e − ∆. Then the unique Condorcet winner is T0 , the
smallest transfer that triggers self-selection of all L-types into unemployment
in equilibrium. ³
´
≤ e − ∆. Then the set of Condorcet winners is
(ii) Suppose 2δ 1 − ∆
f
{T |T ∈ R+ \ (T1 , Tmax ]} if Tmax = T and {T |T ∈ R+ } if Tmax = T1 .
³
´
δ
∆
Proof. Part (i). Suppose 2 1 − f > e − ∆ holds and recall that T0 > T1
in this case. Consider first transfers satisfying 0 ≤ T ≤ T0 . To derive the
expected utility of L-types as a function of pL (taking pM = pH = 1 as given)
18
in this range, evaluate the left-hand side of (12), substituting for t using (13)
and the budget to get
·
µ
¶¸
2 + pL
δ
∆
EUL (pL ) =
e−∆+
1−
+ s + m − δ.
(19)
3
1 + pL
f
Note that EUL (pL ) is convex in pL and that EUL (pL )|pL =0 > EUL (pL )|pL =1 .
Thus L-types must prefer T0 compared to any strictly smaller T .
Setting up the expected utility of M -types for all 0 ≤ T ≤ T0 as a function
of pL , one finds
µ
¶
∆
δ
1+
.
(20)
EUM (pL ) = EUL (pL ) + ∆ +
2
f
Equation (20) shows that in a pairwise comparison between transfers 0 ≤ T ≤
T0 , M -types behave exactly as L-types.
Now consider transfers satisfying T0 < T ≤ Tmax . For all T in this range, selfselection of L-types into unemployment is expected as the Nash equilibrium in
the subgame starting after the transfer has been fixed. Because t = T /2, the
expected utility of M -types is a strictly decreasing function of T in this range.
Therefore, M -types strictly prefer T0 over all T0 < T ≤ Tmax .
The expected utility of H-types is constant up to T1 and strictly decreasing
for all T ∈ (T1 , Tmax ] because their expected matching utility is unchanged
while the tax is strictly positive and increasing. All Nash equilibria in subgames
starting after a transfer higher than Tmax has been fixed have a zero effective
rate of unemployment. Thus all types are indifferent between all T in {T |T ∈
R+ \ (T1 , Tmax ]}. Therefore, there does not exist a T ∈ R+ which beats T0 in
a pairwise comparison.
³
´
δ
∆
Part (ii): Suppose 2 1 − f ≤ e − ∆ and note that T0 ≤ T1 in this case. If
Tmax = T , all types are indifferent between all T in {T |T³ ∈ R+ ´\ (T1 , Tmax ]}.
From (13), t = 0 and pL = 1 we have T1 = e − ∆ + 2δ 1 − ∆
. Using the
f
³
´
for all T ∈ (T1 , Tmax ]. Let ŨK = ŨK (T )
condition, this implies T > δ 1 − ∆
f
be the expected utility of a type-K individual
as³a function
h
´i of the transfer.
1
∆
Then E ŨM (T1 ) − E ŨM (T )|T ∈(T1 ,Tmax ] = 2 T − δ 1 − f
> 0. For the Htypes, the same argument as in part (i) holds. If Tmax = T1 , all types are
indifferent between all T ∈ R+ .
Depending on the parameters, either a transfer is chosen which is followed by
a rate of unemployment and a tax of zero, or the transfer which is the smallest
19
one to trigger complete self-selection of L-types into unemployment. Note that
δ is restricted by an upper bound m. Thus, by simultaneously increasing both
parameters
³
´ while holding the difference fixed, we can always find a δ such that
δ
∆
1 − f > e − ∆ holds.
2
The main result of the paper can thus be summarized as follows. If productivity differences between types coincide with sufficiently large differences in
social values attached to individuals, a political equilibrium with tax financed
unemployment benefits leading to a self-selection of the least productive types
into unemployment exists. In this equilibrium, the fact that employment decisions are observable and only unemployed are entitled to receive the benefit is
used by the middle class to induce self-sorting of less desirable social partners.
Hence, public policy serves as an instrument to relegate less productive (and
socially less valuable) individuals into a separated lower class league before it
comes to choosing social partners. In this equilibrium, members of the middle
class have to pay taxes to finance the transfers to the unemployed. However,
they gain in terms of social status, and these gains more than outweigh the
tax cost. At the same time, members of the class of less productive individuals
lose in terms of social status by self-selecting into unemployment. They are
compensated for this loss by a benefit which exceeds the income they would
earn if they supplied labor. To see this, note that an L-type’s loss in expected
matching utility (or social status) from choosing unemployment instead of employment given that all other L-types are unemployed is δ (1 − ∆/f ). The
equilibrium benefit T0 = (2/3)[e − ∆ + δ (1 − ∆/f )] exceeds the net income
L-types receive by running a firm and paying the tax by the same amount and
thus exactly balances this loss. Given that an effective benefit higher than T0
will never gain support by another group, what matters for the L-types’s voting behavior is the comparison between a benefit leading to full employment
and T0 . The crucial parameter is δ, measuring the social distance between
different types. The socially more distant different types are, the higher is the
L-types’ loss in social status that comes along with self-selection into unemployment compared to the laissez-faire. But the transfer depends positively on
δ, and the net gain for L-types associated with switching from the laissez-faire
to a benefit T0 is
· µ
¶
¸
1 δ
∆
E ŨL (T0 ) − E ŨL (T = 0) =
1−
− (e − ∆) .
(21)
3 2
f
Hence, if the social distance is sufficiently large, L-types politically support
T0 . M -types gain what L-types lose by switching from a laissez-faire to T0 :
the gain in terms of expected matching utility of the former is exactly what is
20
lost by the latter. The cost of inducing self-selection of the less desirable social
partners is, however, partly shifted to the H-types, who have to pay half of the
tax burden. This is why both L-types and M -types prefer the self-selection
outcome compared to a laissez-faire if δ is high enough.
5
Concluding remarks
It is widely criticized that certain features of unemployment benefit schemes
especially in many European countries have discouraging effects for the permanently unemployed to engage in job search activities. One of the main issues
in the ongoing discussion on political reform of unemployment benefit schemes
is that minimum benefits are close to or even above the wage level for less productive workers. At the same time, lower income groups face a huge implicit
tax rate through the withdrawal of benefits.
In this paper, I offer a model in which two essential characteristics of individuals, namely productivity and social value, i.e. their desirability as social
partners, are correlated. As usual, labor supply decisions are affected by the
outside option provided by an unemployment benefit scheme. The political
attitudes of voters towards these benefits are then partly driven by social motivations: Anticipating the self-selection outcome, members of the middle class
might vote for a level of benefits that is just sufficient to induce the least productive individuals to opt for the benefit. The gain of middle class voters comes
as an improved social status fueled by an increase in social differentiation.
The insight generated is that one of the institutions figuring prominently in
most social welfare systems, in addition to supplying insurance and to prevent
poverty, may serve as a social selection device. This may help to understand
why political reform of discouraging benefit schemes is often slow and far from
unanimously supported by the middle class. The paper thus points out that
differences in institutional arrangements may be quite important in shaping
individual political attitudes towards redistribution. If specific arrangements
used to identify entitled individuals serve as a selection device, competition for
social status can foster redistribution in a way that might be in conflict with
traditional objectives of social security systems.
21
References
Atkinson, Anthony B. and John Micklewright (1991), Unemployment
Compensation and Labor Market Transitions: A Critical Review, Journal of Economic Literature 29 (4), 1679-1727.
Baily, Martin N. (1977), Unemployment Insurance as Insurance for Workers,
Industrial and Labor Relations Review 30 (4), 495-504.
Boeri, Tito, Axel Börsch-Supan and Guido Tabellini (2001), Would
You Like to Shrink the Welfare State? A Survey of European Citizens,
Economic Policy: A European Forum, vol. 0, no. 32, 7-44.
Cole, Harold L., George J. Mailath and Andrew Postlewaite (1992), Social Norms, Savings Behavior, and Growth, Journal of Political Economy
100 (6), 1092-1125.
Carling, Kenneth, Bertil Holmlund and Altin Vejsiu (2001), Do Benefit
Cuts Boost Job Finding? Swedish Evidence from the 1990s, Economic
Journal 111 (474), 766-790.
Corneo, Giacomo and Hans Peter Grüner (2000), Social Limits to Redistribution, American Ecnomic Review 90 (5), 1491-1507.
Corneo, Giacomo and Hans Peter Grüner (2002), Individual Preferences
for Political Redistribution, Journal of Public Economics 83, 83-107.
Drèze, Jacques H. and Henri Sneessens (1997), Technological Development, Competition from Low-Wage Economies and Low-Skilled Unemployment, in: Snower, Dennis J. and Guillermo de la Dehesa, Unemployment Policy. Government Options for the Labour Market, Cambridge
University Press, 250-277.
Karni, Edi (1999), Optimal Unemployment Insurance. A Survey, Southern
Economic Journal 66 (2), 442-465.
Mortensen, Dale T. (1977), Unemployment Insurance and Job Search Decisions, Industrial and Labor Relations Review 30 (4), 505-517.
Nickell, Stephen and Brian Bell (1997), Would Cutting Payroll Taxes on
the Unskilled Have a Significant Impact on Unemployment?, in: Snower,
Dennis J. and Guillermo de la Dehesa (eds), Unemployment Policy. Government Options for the Labour Market, Cambridge University Press,
296-328.
22
Nickell, Stephen (1998), Unemployment: Questions and Some Answers,
Economic Journal 108 (448), 802-816.
OECD (1997), The OECD Job Strategy – Making Work Pay (Taxation,
Benefits, Employment and Unemployment), Paris.
Pallage, Stéphane and Christian Zimmermann (2001), Voting on Unemployment Insurance, International Economic Review 42 (4), 903-923.
Roth, Alvin E. and Marilda A. Sotomayor (1990), Two-Sided Matching. A
Study in Game-Theoretic Modelling and Analysis, Cambridge University
Press.
Saint-Paul, Gilles (2000), The Political Economy of Labour Market Institutions, Oxford University Press.
Weiss, Yoram and Chaim Fershtman (1998), Social Status and Economic
Performance: A Survey, European Economic Review 42, 801-820.
Wright, Randall (1986), The Redistributive Roles of Unemployment Insurance and the Dynamics of Voting, Journal of Public Economics 31,
377-399.
23