SECTlON'3.C: SIGNALING 45'
450 c HAP T E R 1 3: A D V E R S ESE L E C T ION. S I G N A LIN G , AND S C R E E N I N G
---------------------------------- -----------------------------------
Figure 13.C.l
The extensive form of
the educationsignaling
game.
Conditional onseeing
a level ofe, say.,.
firms make wage offers
simuhaneously (really a
continuous choice),
}
Worker decides
which offer 10
accept, ifany.
Firm I
Firm 2
Accept
Firm
2
~Accept Neither
,;:~--------------"":Il ....I ' - - - - Worker chooses
education level
contingent on her
type(really a
continuous choice).
Random move of
..Ao.;:"'E~---------- nature determines
worker type.
In the analysis that follows, we shall see that this otherwise useless education
may serve as a signal of unobservable worker productivity. In particular, equilibria
emerge in which high-productivity workers choose to get more education than lowproductivity
workers and firms correctly take differences in education levels as a
signal of ability. The welfare effects of signaling activities are generally ambiguous.
By revealing information about worker types, signaling can lead to a more efficient
allocation of workers' labor, and in some instances to a Pareto improvement. At the
same time, because signaling activity is costly, workers' welfare may be reduced if
they are compelled to engage in a high level of signaling activity to distinguish
t hcmsclves.
To keep things simple, throughout most of this section we concentrate on the
special case in whieh r(O,,) = r(Od = O. Note that under this assumption the unique
equilibrium that arises in the absence of the ability to signal (analyzed in Section
13.B) has all workers employed by firms at a wage of IV· =£[0] and is Pareto
efficient. Hence, our study of this case emphasizes the potential inefficiencies created
by signaling. After studying this case in detail, we briefly illustrate (in small type)
how, with alternative assumptions about the function r('), signaling may instead
generate a Pareto improvement.
A portion of the game tree for this model is shown in Figure 13.C.1. Initially, a
random move of nature determines whether a worker is of high or low ability. Then,
conditional on her type, the worker chooses how much education to obtain. After
obtaining her chosen education level, the worker enters the job market. Conditional
on the observed education level of the worker, two firms simultaneously make wage
offers to her. Finally, the worker decides whether to work for a firm and, if so,
which onc.
_1__
Moreover, if [w(-), 1(·)] satisfies (I3.B.16) and (13.B.17), then there exists a t(·) such
that [w(·), r(.),J(.)] satisfies (l3.B.14)-(13.B.16). Condition (I3.B.17), however, is exactly the
budgetconstraintfaced bya centralauthority whoruns the firms herself. Hence,wecan restrict
attention to schemes in which the authority runs the firms herself and usesa direct revelation
mechanism [w('), 1(')] satisfying (13.B.16) and (13.B.17).
Now consider any two types 0' and O· for which 1(0') = 1(0·). Setting 0 = 0' and
6= O· in condition (13.B.16), we see that we must have w(O')~ w(O·). Likewise, letting
0= o· and 6=0', wemust havew(O")~ w(O').Together,this implies that w(0') = w(0"). Since
1(0)E {O, I}, wesee that any feasible mechanism [w(· ),/(')] can be viewed as a scheme that
gives each workera choicebetween twooutcomes,(w.. 1 = I) and (w., I = 0) and satisfiesthe
budget balancecondition (13.B.17). This is exactlythe classof mechanisms studied above.
Given the problems observed in Section 13.B, one might expect mechanisms to
develop in the marketplace to help firms distinguish among workers. This seems
plausible because both the firms and the high-ability workers have incentives to try
to accomplish this objective. The mechanism that we examine in this section is that
of siqnalinq, which was first investigated by Spence (1973, 1974). The basic idea is
that high-ability workers may have actions they can take to distinguish themselves
from their low-ability counterparts.
The simplest example of such a signal occurs when workers can submit to some
costless test that reliably reveals their type. It is relatively straightforward to show
that in any subgame perfect Nash equilibrium all workers with ability greater than
owill submit to the test and the market will achieve the full information outcome
(see Exercise 13.C.I). Any worker who chooses not to take the test will be correctly
treated as being no better than the worst type of worker.
However, in many instances, no procedure exists that directly reveals a worker's
type. Nevertheless, as the analysis in this section reveals, the potential for signaling
may still exist.
Consider the following adaptation of the model discussed in Section 13.8. For
simplicity, we restrict attention to the case of two types of workers with productivities
Oil and 0L' where IJII > IJL > °and ;. = Prob (IJ =IJH ) E (0,1). The important extension
of our previous model is that before entering the job market a worker can get some
education, and the amount of education that a worker receives is observable. To
make matters particularly stark, we assume that education does nothing for a worker's
productivity (see Exercise 13.C.2 for the case of productive signaling). The cost of
obtaining education level e for a type 0 worker (the cost may be of either monetary
or psychic origin) is given by the twice continuously differentiable function c(e, 0),
with c(O, 0) = 0, c,(e, 0) > 0, c..(e, 0) > 0, c.(e, IJ) < °for all e > 0, and c..(e, 0) < °(subscripts denote partial derivatives). Thus, both the cost and the marginal cost of
education are assumed to be lower for high-ability workers; for example, the work
required to obtain a degree might be easier for a high-ability individual. Letting
U(IV, e10) denote the utility of a type IJworker who chooses education level e and
receives wage IV, we take U(IV, eI0) to equal her wage less any educational costs
incurred: U(IV, eIIJ)= IV - c(e, 0). As in Section 13.B,a worker of type IJcan earn r(O)
by working at home.
13,C Signaling
452 CHAPTER 13: ADVERSE SELECTION, SIGNALING, AND SCREENING SECTION 13.C: SIGNALING 453
---------------------------------- ~--------------------------------
Note that, in contrast with the model of Section 13.B, here we explicitly model
only a single worker of unknown type; the model with many workers can be thought
of as simply having many or these single-worker games going on simultaneously,
with the fraction of high-ability workers in the market being ,t In discussing the
equilibria or this game, we often speak of the "high-ability workers" and "low-ability
workers," having the many-workers case in mind.
The equilibrium concept we employ is that or a weak perfect Bayesian equilibrium
(see Definition 9.C.3), but with an added condition. Put formally, we require that, in
the game tree depicted in Figure 13.C.I, the firms' beliefs have the property that, for
each possible choice of e, there exists a number /lie) E [0, I] such that: (i) firm I's
belief that the worker is or type 0" after seeing her choose e is /lie) and (ii) after the
worker has chosen e, firm 2's belief that the worker is or type 0" and that firm I
has chosen wage offer w is precisely /l(e)ur(w Ie), where ur(w Ie) is firm I's equilibrium
probability of choosing wage offer w after observing education level e. This extra
condition adds an element of commonality to the firms' beliefs about the type of
worker who has chosen e, and requires that the firms' beliefs about each others' wage
offers following e are consistent with the equilibrium strategies both on and off the
equilibrium path.
We refer to a weak perfect Bayesian equilibrium satisfying this extra condition
on beliefs as a perfect Bayesian equilibrium (PBE). Fortunately, this PBE notion can
more easily, and equivalently, be stated as follows: A set of strategies and a belief
function I,(e) E [0, I] giving the firms' common probability assessment that the worker
is of high ability after observing education level e is a PBE if
(i) The worker's strategy is optimal given the firm's strategies.
(ii) The belief function /lie) is derived from the worker's strategy using Bayes'
rule where possible.
(iii) The firms' wage offers following each choice e constitute a Nash equilibrium
of the simultaneous-move wage offer game in which the probability that the
worker is or high ability is /l(e).20
In the context or the model studied here, this notion of a PBE is equivalent to the
sequential equilibrium concept discussed in Section 9.C. We also restrict our attention
throughout to pure strategy equilibria.
We begin our analysis at the end or the game. Suppose that after seeing some
education level e, the firms attach a probability of I,(e) that the worker is type 0".
lf so, the expected productivity of the worker is /l(e)OI/ + (I - /l(e»OL' In a
simultaneous-move wage offer game, the firms' (pure strategy) Nash equilibrium wage
offers equal the worker's expected productivity (this game is very much like the
Bertrand pricing game discussed in Section 12.C). Thus, in any (pure strategy) PBE,
we must have both firms offering a wage exactly equal to the worker's expected
productivity, /l(e)O" + (I - 1,(e»OL'
20. Thus, the extra condition we add imposes equilibrium-like play in parts of the tree off the
equilibrium path. See Section 9.C for a discussion of the need to augment the weak perfect Bayesian
equilibrium concept to achieve this end.
Oil ------------------------
° -----------------------,.
o
Knowing this fact, we turn to the issue or the worker's equilibrium strategy, her
choice or an education level contingent on her type. As a first step in this analysis,
it is useful to examine the worker's preferences over (wage rate, education level) pairs.
Figure IlC.2 depicts an indifference curve for each of the two types of workers (with
wages measured on the vertical axis and education levels measured on the horizontal
axis). Note that these indilTerenee curves cross only once and that, where they do,
the indifference curve of the high-ability worker has a smaller slope. This property or
preferences, known as the single-crossing property, plays an important role in the
analysis of signaling models and in models or asymmetric information more generally.
It arises here because the worker's marginal rate or substitution between wages and
education at any given (w, e) pair is (dwlde)" = c,(e, 0), which is decreasing in 0
because "...(e, II) < 0.
We can also graph a function giving the equilibrium wage offer that results for
each education level, which we denote by w(e). Note that since in any PBE
w(e) = 1,(e)O" + (I - 1,(e»IIL for the equilibrium belief function /l(e), the equilibrium
wage offer resulting from any choice or e must lie in the interval [11,.,111/].
A possible wage offer function w(e) is shown in Figure 13.C.3.
We are now ready to determine the equilibrium education choices for the two
types or workers. It is useful to consider separately two different types of equilibria
that might arise: separating equilibria, in which the two types of workers choose
different education levels, and pooling equilibria, in which the two types choose the
same education level.
Separating Equilibria
To analyze separating equilibria, let e"(II) be the worker's equilibrium education
choice as a function or her type, and let \V"(e) be the firms' equilibrium wage offer
as a function or the worker's education level. We first establish two useful lemmas.
Lemma 13.C.1: In any separating perfect Bayesian equilibrium, w"(e"(OH)) = 0H and
w"(e"(lJLl) =0L: that is, each worker type receives a wage equal to her productivity
levei.
Proof: In any PBE, beliefs on the equilibrium path must be correctly derived from
the equilibrium strategies using Bayes' rule. Here this implies that upon seeing
education level e"(OL)' firms must assign probability one to the worker being type
I)/.. Likewise, upon seeing education level e"(I)II), firms must assign probability one
l ~
Figure 13.C.2 (left)
Indifference curves for
high- and low-ability
workers: the
single-crossing
property.
Figure 13.C.3 (right)
A wage schedule.
454 CHAPTER 13: ADVERSE SELECTION, SIGNALING, AND SCREENING SECTION 13.C: SIGNALING 455
Figure 13.C.7 (right)
A separating
equilibrium with an
education choice
e"(OH) > eby
high-ability workers.
FIgure 13.C.6 (Iell)
A separating
equilibrium with the
same education
choices as in Figure
13.C.S but different
off-equilibriumpath
beliefs.
e,
II
,'(/i"l
o
II
,"(II,J
/ill
/~r'\"
./ : .'(e)
L/ :
._·-·-r :
0,. -~:.~------i--------t-----
I I
I I
I I
I I
I I
/i,. .-.-.-.-.-.--;-----------
I
I
I
I
II
"'(111.) ,'(/i"l
e
II
e'(O,,)
Type0,
Type0"
~
. l·'~---------
. I .........·*'
/ I '-.
// i "'(e)
.-- I
----------.-----------I
I
I
I
I
I
o
II
e'(/i,J
II'
"'(e'(O,J) = 01.
I
I
I
I
I
I
I
I
I
I
------------~----------
I
I
I
I
I
0" ------------ ----------
w'(e'(Od) = /iL
--- --
to the worker being type 0". The resulting wages are then exactly Ih and 0",
respectively. _
Lemma 13.C.2: In any separating perlect Bayesian equilibrium, e"(OLl = 0; that is,
a low-ability worker chooses to get no education.
Proof: Suppose not, that is, that when the worker is type 0L' she chooses some strictly
positive education level e::. O. According to Lemma IlC.l, by doing so, the worker
receives a wage equal to 0L' However, she would receive a wage of at least 0,. if she
instead chose, = O. Since choosing e =0 would have save her the cost of education,
she would be strictly better off by doing so, which is a contradiction to the assumption
that e> 0 is her equilibrium education level. _
Lemma IlC.2 implies that, in any separating equilibrium, type O,.'s indifference
curve through her equilibrium level of education and wage must look as depicted in
Figure IlCA.
Using Figure IlCA, we can construct a separating equilibrium as follows:
Let ,"(0,,) =ii, let ,"(Od = 0, and let the schedule w"(e) be as drawn in Figure
13.C.5. The firms' equilibrium beliefs following education choice e are 1I"(e) =
(w"(,) - 0d/(O" - Od. Note that they satisfy I,"(e) E [0, I] for all e;::O: 0, since
1I'"(e) E [OL' 0,,].
To verify that this is indeed a PBE, note that we are completely free to let firms
have any beliefs when e is neither 0 nor ii. On the other hand, we must have 11(0) = 0
and 1,(iI)= I. The wage offers drawn, which have w'(O) =0,- and w'(iI) = 0", reflect
exactly these beliefs.
What about the worker's strategy? It is not hard to sec that, given the wage
function w"(e), the worker is maximizing her utility by choosing e = 0 when she is
type 0,- and by choosing e = ii when she is type 0". This can be seen in Figure 13.C.S
by noting that, for each type that she may be, the worker's indifference curve is at
its highest-possible level along the schedule w"(e). Thus, strategies [e"(O), w"(e)] and
the associated beliefs lI(e) of the firms do in fact constitute a PBE.
Note that this is not the only PBE involving these education choices by the two
types of workers. Because we have so much freedom to choose the firms' beliefs off
the equilibrium path, many wage schedules can arise that support these education
Figure 13.C.4 (left)
Low-ability worker's
outcome in a
separating equilibrium.
Figure 13.C.5 (rlghtl
A separating
equilibrium:Typeis
inferred from
education level.
choices. Figure 13.C.6 depicts another one; in this PBE, firms believe that the worker
is certain to be of high quality if e ;::0: eand is certain to be of low quality if e < ii.
The resulting wage schedule has w"(e) = 0" if e ;::0: ii and w"(e) = 0L if e < e.
In thcsc separating equilibria, high-ability workers arc willing to get otherwise
useless education simply because it allows them to distinguish themselves from
low-ability workers and receive higher wages. The fundamental reason that education
can serve as a signal here is that the marginal cost of education depends on a worker's
type. Because the marginal cost of education is higher for a low-ability worker [since
(',.•(e, 0) < 0], a type 0" worker may find it worthwhile to get some positive level of
education e' > 0 to raise her wage by some amount t.w > 0, whereas a type 0Lworker
may be unwilling to get this same level of education in return for the same wage
increase. As a result, firms can reasonably come to regard education level as a signal
of worker quality.
The education level for the high-ability type observed above is not the only one
that can arise in a separating equilibrium in this model. Indeed, many education
levels for the high-ability type arc possible. In particular, any education level between
eand e, in Figure IlC.7 can be the equilibrium education level of the high-ability
workers. A wage schedule that supports education level e"(O,,) = e, is depicted in
the figure. Note that the education level of the high-ability worker cannot be below
ii in a separating equilibrium because, if it were, the low-ability worker would deviate
and pretend to be of high ability by choosing the high-ability education level. On the
other hand, the education level of the high-ability worker cannot be above e, because,
ifit were, the high-ability worker would prefer to get no education, even if this resulted
in her being thought to be of low ability.
Note that thesc various separating equilibria can be Pareto ranked. In all of them,
firms carn zero profits. and a low-ability worker's utility is 0,.:However, a high-ability
worker docs strictly better in equilibria in which she gets a lower level of education.
Thus, separating equilibria in which the high-ability worker gets education level ii
(e.g., the equilibria depicted in Figures 13.C.S and 13.C.6) Pareto dominate all the
others. The Pareto-dominated equilibria are sustained because of the high-ability
worker's fear that if she chooses a lower level of education than that prescribed in
the equilibrium firms will believe that she is not a 'high-ability worker. These beliefs
can be maintained because in equilibrium they are never disconfirmed.
456 CHAPTER 13: ADVERSE SELECTION, SIGNALING. AND SCREENING
SEC T ION 1 3 • C: S I G N A LIN G 457
--- -------------------------------------
w
Figure 13.C.l0 (right)
A pooling equilibrium.
Figure 13.C.9 (Ie")
The highest-possible
education level in a
pooling equilibrium.
Type 0.
e'
II
,'(Oil; j = L, H
w
------ ------~~~~=-~~­
,,--.~
/ : \I' '(e)
.-.,,/ \
0, ------r---------------
I
I
I
\I'
t"
0"
I
I
I
I
V~ -----~---------------­
I
I
I
I
I
E[OJ
Figure 13.C.8
Separating equilibria
may be Pareto
dominated by the
no-signaling OUlcom
(a) A.separating e
equilibrium that isnOI
Pareto domInatedby
the no,s,gnaling
outcome.
(b) A separating
equilibrium that is
Pareto dominatedby
the no-signaling
outcome.
~TypeO.
o
(b)
0, -------------------------
o
(a)
0, ------------------------
E[OJ --- --------------------
It is of interest to compare welfare in these equilibria with that arising when
worker types are unobservable but no opportunity for signaling is available. When
education is not available as a signal (so workers also incur no education eosts), we
are back in the situation studied in Section 118. In both cases, firms earn expected
profits of zero. However, low-ability workers are strictly worse off when signaling is
possible. In both cases they incur no education costs, but when signaling is possible
they receive a wage of 0, rather than £(0).
What about high-ability workers? The somewhat surprising answer is that
high-ability workers may be either better or worse off when signaling is possible. In
Figure 11C.8(a), the high-ability workers are better off because of the increase in
their wages arising through signaling. However, in Figure 13.C.8(b), even though
high-ability workers seek to take advantage of the signaling mechanism to distinguish
themselves, they are worse off than when signaling is impossible! Although this may
seem paradoxical (if high-ability workers choose to signal, how can they be worse
off"), its cause lies in the fact that in a separating signaling equilibrium firms'
expectations are such that the wage-education outcome from the no-signaling
situation, [w, e) = (£[11],0), is no longer available to the high-ability workers; if they
get no education in the separating signaling equilibrium, they are thought to be of
low ability and offered a wage of 0,. Thus, they can be worse off when signaling is
possible, even though they are choosing to signal.
Note that because the set of separating equilibria is completely unaffected by the
fraetion i. of high-ability workers, as this fraction grows it becomes more likely that
the high-ability workers are made worse off by the possibility of signaling [compare
Figures 13.C.8(a) and l3.C.8(b)]. In fact, as this fraction gets close to I, nearly every
workcr is getting costly education just to avoid being thought to be one of the handful
of bad workers!
Poolinq Equilibria
Consider now pooling equilibria, in which the two types of workers choose the same
level of education, e*(O,) = e*(III1 ) = e". Since the firms' beliefs must be correctly
derived from the equilibrium strategies and Bayes' rule when possible, their beliefs
when they see education level e* must assign probability i, to the worker being type 11/1'
Thus, in any pooling equilibrium, we must have ",*(e*) = i,OIl + (1 - i.)II, = £[0].
The only remaining issue therefore eoncerns what levels of education can arise
in a pooling equilibrium, It turns out that any education level between 0 and the
Icvcl e' depicted in Figure 13.C.9 can be sustained.
Figure 11C.lO shows an equilibrium supporting education level e'. Given the wage
schedule depicted, each type of worker maximizes her payoff by choosing education
level e', This wage schedule is consistent with Bayesian updating on the equilibrium
path because it gives a wage offer of £[0] when education level e' is observed,
Education levels between 0 and e' can be supported in a similar manner.
Education levels greater than e' cannot be sustained because a low-ability worker
would rather set e = 0 than e > e' even if this results in a wage payment of 0,. Note
that a pooling equilibrium in which both types of worker get no education Pareto
dominates any pooling equilibrium with a positive education level. Once again, the
Pareto-dominated pooling equilibria are sustained by the worker's fear that a deviation
will lead firms to have an unfavorable impression of her ability. Note also that a
pooling equilibrium in which both types of worker obtain no education results in
exactly the same outcome as that which arises in the absence of an ability to signal.
Thus, pooling equilibria are (weakly) Pareto dominated by the no-signaling outcome,
Multiple Equilibria alld Equilibrium Refinement
The multiplicity of equilibria observed here is somewhat disconcerting, As we have
seen, we can have separating equilibria in which firms learn the worker's type, but we
can also have pooling equilibria where they do not; and within each type of
equilibrium, many different equilibrium levels of education can arise, ln large part,
this multiplicity stems from the great freedom that we have to choose beliefs off the
equilibrium path, Recently, a great deal of research has investigated the implications
of putting "reasonable" restrictions on such beliefs along the lines we discussed in
Section 9.D.
To sec a simple example of this kind of reasoning, consider the separating
equilibrium depicted in Figure I3.C.7. To sustain el as the equilibrium education
level of high-ability workers, firms must believe that any worker with an education
level below eI has a positive probability of being of type ilL' But consider any
education level eE (ii, ell. A type II, worker could never be made better off choosing
such an education level than she is getting education level e = 0 regardless of what
I
___________________1 _
SECTION 13.C: SIGNALING 459
-------------------------------
---
458 CHAPTER 13: ADVERSE SELECTION, SIGNALING, AND SCREENING
Exercise l3.C.3: In the signaling model discussed in Section l3.C with r(O,,) =
r(O,,) = 0, construct an example in which a central authority who does not observe
worker types can achieve a Pareto improvement over the best separating equilibrium
through a policy that involves cross-subsidization, but cannot achieve a Pareto
improvement by simply banning the signaling activity. [Hint: Consider first a case
with linear indifference curvcs.]
wage of "'". If so, low-ability workers would choose e = 0 and high-ability workers
would choose e = ell' This alternative outcome involves firms incurring losses on
low-ability workers and making profits on high-ability workers. However, as long
as the firms break even on average, they are no worse off than before and a Pareto
improvement has been achieved. The key to this Pareto improvement is that the
central authority introduces cross-subsidization, where high-ability workers are paid
less than thcir productivity level while low-ability workers are paid more than theirs,
an outcome that cannot occur in a separating signaling equilibrium. (Note that the
outcome when signaling is banned is an extreme case of cross-subsidization.)
The case with r(O,,) = r(O,J = 0 studied above, in which the market outcome in the absence
of signaling is Pareto optimal. illustrates how the use of costly signaling can reduce welfare.
Yet, when the market outcome in the absence of signaling is not efficient, signaling's ability
to reveal information about worker types may instead create a Pareto improvement hy
leading to a more efficient allocation of labor. To see this point, suppose that we have
r =r(OL) = r(O,,), with 0L < r < 0" and £[0] < r. In this case, the equilibrium outcome
without signaling has no workers employed. In contrast, any Pareto efficient outcome must
have the high-ability workers employed by firms.
We now study the equilibrium outcome when signaling is possible. Consider, first. the
wage and employment outcome that results after educational choice e by the worker.
Following the workers choice of educational levele, equilibrium behavior involves a wage of
w'(e) = II(e)Oll + (I -1,(ellOL.1f ",'(e);;' r, then both types of workers would accept employ.
rnent: if w'(e) < r, then neither type would do so.
We now determine the equilibrium education choices of the two types of workers. Note
first that any pooling equilibrium must have both types choosing e = 0 and neither type
accepting employment. To see this, suppose that both types are choosing education level e.
Then 11(") = i. and ",'(e) = £[0] < r, and so neither type accepts employment. Hence, ife> 0,
both types would be better ofT choosing e = 0 instead. Thus, only an education level of zero
is possible in a pooling equilibrium. In this zero education pooling equilibrium, the outcome
is identical to the equilibrium outcome arising in the absence of the opportunity to signal.
The set of separating equilibria. on the other hand, is illustrated in Figure 13.C.12.In any
separating equilibrium, a low-ability worker sets e = 0, is ofTered a wage of 0,., and chooses
to work at home, thereby achieving a utility of r. High-ability workers, on the other hand,
select an education level in the interval [e. e,] depicted in the figure, are ofTered a wage of Oil'
and accept employment. Note thai no separating equilibrium can have e*(OIl) < e. since then
low-ability workers would deviate and set e = e*(O,,); also, no separating equilibrium can have
e'(Oll) > vi- since high-ability workers would then be better off setting e = 0 and working at
home.
Note that in all these equilibria. both pooling and separating. the high-ability workers are
weakly better ofT compared with the equilibrium arising without signaling opportunities and
are strictly better ofTin separating equilibria with e'(O,,) < e,. Moreover, both the low-ability
workers and the firms are equally well ofT. Thus, in the case with 0L < r < 0" and £[0] < r,
any pooling or separatingsignalingequilibrium weakly Pareto dominates the outcome arising
Figure13.C.11
Achieving a Pareto
improvement through
cross-subsidization.
I
_______l~ _
I
I
II
I
II
----------t--------------
I
I
I
I
I
Second-Best Market Intervention
In contrast with the market outcome predicted by the game-theoretic model studied
in Section I3.B (the highest-wage competitive equilibrium), in the presence of
signaling a central authority who cannot observe worker types may be able to achieve
a Pareto improvement relative to the market outcome, To see this in the simplest
manner, suppose that the Cho and Kreps (1987) argument predicting the best
separating equilibrium outcome is correct. We have already seen that the best
separating equilibrium can be Pareto dominated by the outcome that arises when
signaling is impossible. When it is, a Pareto improvement can be achieved simply by
banning the signaling activity,
In fact, it may be possible to achieve a Pareto improvement even when the
no-signaling outcome does not Pareto dominate the best separating equilibrium. To
see how, consider Figure l3.C.11. In the figure, the best separating equilibrium has
low-ability workers at point (OL'0) and high-ability workers at point (0", e). Note
that the high-ability workers would be worse off if signaling were banned, since the
point (E[OJ, O) gives them less than their equilibrium level of utility. Nevertheless,
note that if we gave the low- and high-ability workers outcomes of (IVL,O) and
(w", e,,), respectively, both types would be better off. The central authority can
achieve this outcome by mandating that workers with education levels below ell
receive a wage of "'L and that workers with education levels of at least ell receive a
firms believeabout her as a result. Hence, any belief by firms upon seeing education
level e > e other than Il(e) = I seems unreasonable. But if this is so, then we must
have w(e) = 9", and so the high-ability worker would deviate to e. In fact, by this
logic, the only education level that can be chosen by type 0" workers in a separating
equilibrium involving reasonable beliefs is e.
In Appendix A we discuss in greater detail the use of these types of reasonablebeliefs
refinements. One refinement proposed by Cho and Kreps (1987), known as
the i/lluirive criterion,extends the idea discussed in the previous paragraph to rule
out not only the dominated separating equilibria but also all pooling equilibria. Thus,
If we accept the Cho and Kreps (1987) argument, we predict a unique outcome to
this two-type signaling game: the best separating equilibrium outcome, which is
shown in Figures 13.C.S and 13.C.6.
460 CHAPTER 13: ADVERSE SELECTION, SIGNALING, AND SCREENING
- Type 0H
I
I
I
I
I
1
I
I
I
I
I
UL -----~----------t------, 1
I 1
I 1
: I
f t'2
'"----y--'
r'(OL) Possible Values of ,'(OH)
in the absence of signaling. and this Pareto dominance is strictfor (essentially) all separating
equilibria.
13.0 Screening
---
Figure 13.C.12
Separating equilibria
when r(Od = r(OH)~
re(OL.OI/)'
SECTION 13.0, SCREENtNG 461
We assume that the utility of a type °worker who receives wage wand faces task
level I 2: 0 is
u(w. 110) = w - c(l. 0).
where C(I. 0) has all the properties assumed of the function c(e.0) in Section 13.e. In
particular. c(O.0) = O. c,(I. 0) > O. c,,(I. 0) > O. C.(I. 0) < 0 for all r > O. and C'6(1. 0) < o.
As will be clear shortly. the task level I serves to distinguish among types here in
a manner that parallels the role of education in the signaling model discussed in
Section 13.e.
Here we study the pure strategy subgame perfect Nash equilibria (SPNEs) of the
following two-stage garner'?
Staqe I: Two firms simultaneously announce sets of offered contracts. A contract
is a pair (w, I). Each firm may announce any finite number of
contracts.
Staqe 2: Given the offers made by the firms. workers of each type choose whether
to accept a contract and. if so. which one. For simplicity. we assume
that if a worker is indifferent between two contracts. she always chooses
the one with the lower task level and that she accepts employment if she
is indifferent about doing so. If a worker's most preferred contract is
offered by both firms. she accepts each firm's offer with probability ].
23. For this game, the set of subgameperfect Nash equilibria is identical to thesets of strategy
profiles in weak perfect Bayesianequilibria or sequential equilibria.
24. The models in the original Rothschild and Stiglitz (1976) and Wilson (1977) analyses differ
from our model in Iwo respects. First,firms in those paperswererestricted 10offering only a single
contract.This could make sense in the productionlineinterpretation, forexample.if each firm had
only a single production line. Second, those authors allowed for "free entry," so that an additional
firm could always enter if a profitable conlracting opportunityexisted. In fact, makingthese two
changes has little effecton our conclusions.The only difference is in the preciseconditions under
whichan equilibriumexists, (For moreon Ihis,see Exercise 13.D.4.)
Thus. a firm can offer a variety of contracts; for example. it might have several
production lines. each running at a different speed. Different types of workers may
then end up choosing different contracts.i"
It is helpful to start by considering what the outcome of this game would be if
worker types were observable.To address this case. we allow firms to condition their
offer on a worker's type (so that a firm can offer a contract (WL. lel solely to type
0L workers and another contract (wli • Ill) solely to type 01/ workers).
Proposition 13.0.1: In any SPNE of the screening game with observable worker types,
a type 0, worker accepts contract (wi, til =(0" 0). and firms earn zero profits.
Proof: We first argue that any contract (wi. til that workers of type Oi accept in
equilibrium must produce exactly zero profits; that is. it must involve a wage wi = 0,.
To sec this. note that if wi > 0i' then some firm is making a loss offering this contract
and it would do better by not offering any contract to type 0, workers. Suppose. on
the other hand. that wi < 0i. and let n > 0 be the aggregate profits earned by the
two firms on type Oi workers. One of the two firms must be earning no more than
n/2 from these workers. If it deviates by offering a contract (wi + c.Ii) for any
21. The setting analyzed here is one of comperilire screenuu; of workers,since we assume that
there are several competing firms. See Section t4.C for a discussion of the monopolistic screeninu
case, wherea single firmscreens workers.
22. As was true in the case of educational signaling, the assumption that highertask levelsdo
not raise productivity is made purelyfor expositionalpurposes. Exercise 13.D.I considersthe case
in which the firms' profitsare increasing in the task level.
___1__In
Section 13.C. we considered how signaling may develop in the marketplace as a
response to the problem of asymmetric information about a good to be traded. There.
individuals on the more informedside of the market (workers) chose their level of
education in an attempt to signal information about their abilities to uninformed
parties (the firms). In this section. we consider an alternative market response to the
problem of unobservable worker productivity in which the uninformed parties take
steps to try to distinguish. or screen. the various types of individuals on the other
side of the marker.!' This possibility was first studied by Rothschild and Stiglitz
(1976) and Wilson (1977) in the context of insurance markets (see Exercise 13.0.2).
As in Section 13.C. we focus on the case in which there are two types of workers.
0L and On. with 01/ > 0L > 0 and where the fraction of workers who are of type 01/
is ;. e (0. I). In addition. workers earn nothing if they do not accept employment in
a firm [in the notation used in Section 13.B. r(0el = r(OI/) = 0). However. we now
suppose that jobs may differ in the "task level" required of the worker. For example.
jobs could differ in the number of hours per week that the worker is required to
work. Or the task level might represent the speed at which a production line is run
in a factory.
To make matters particularly simple. and to make the model parallel that in
Section IlC. we suppose that higher task levels add nothinq to the output of the
worker; rather. their only effect is to lower the utility of the worker.i! The output of
a type °worker is therefore °regardless of the worker's task level.