Optimal cost of signaling through education

Joanna Franaszek, European University Institute

October 12, 2012

1 Introduction

The last 20 years have been a golden age for the Polish tertiary education institutions. The number ofstudents in post-Soviet Poland increased dramatically and the percentage of students within the age group19-24 has quadrupled in 20 years, from 9.8% in 1990/91 to 40.8% in 2010/2011 (see Fig. 1).

There is an anecdotal evidence that a growing number of alumni exceeds the capacity of the labor market,resulting in a gap between a high supply of educated workers and little demand for them. Some elds ofstudies (notably, humanities and social sciences) are believed to generate unemployment and since thehigher education in Poland is publicly funded, the question of eectiveness of the investment in studentsmay naturally arise.

In this article we verify, how serious the problem of overeducation is and we oer a way to decrease it byendogenous reduction of a number of students. We examine a simple model of signaling in the job marketand check (under very specic assumptions) how increasing the cost of education may result in a bettermatch on the job market.

The structure of this paper summarizes the procedure we used to answer the research question. First,we examine a standard model of signaling with a continuous set of agents (of measure 1) and discrete eortchoice, which we shall interpret as a choice of eld of study - with a possibility of not going to universityat all. Our assumptions mimic the standard ones, with an exception, that we assume that there is a xedcost of going to university and additional disutility from studying a certain eld with a certain productivitylevel. We prove the existence and nd the characterization of semi-separating equilibria, in which the agentsdivide themselves into subgroups, depending on their productivity level. We also prove that for a given costof education the equilibrium is uniquely dened through the subset of eorts, chosen by agents.

We shall assume that the demand side of the labour market is mildly imperfect - in particular, thereexist a maximal capacity of high-productivity job positions. However, it is not recognized or internalizedby agents, that make educational decisions. As a result, some mismatch arises in the market, which can bedecreased only by forcing agents to make other school choices through policy changes.

We follow with a simple calibration exercise. Given the data on recent alumni's eld of study andprofessional career, we assume that their choice of eort had arisen within a semi-separating equilibrium ina signaling game. We therefore try to nd a disutility function that would satisfy the equilibrium conditionswith wages and choices observed on the market. We also examine what is the range of overeducation on thelabor market suggested by the data, in terms of overeducation, i.e. the ratio of employers whose job requireless schooling than they actually obtained.

The calibration is followed by a sensitivity analysis - varying the cost of education, we verify, whetherbetter match on the labor market could be obtained. In particular, we check whether the issue of overe-ducation could be reduced through the policy changes, in particular, through an introduction of a tuitionfee.

Finally, we discuss strengths and weaknesses of our approach and suggest some extensions and amend-ments.

1

Figure 1: Net enrollment ratio to higher education (percentage of people aged 19-24, who are enrolled in auniversity) for Poland in 1990-2011

Source: Wspczynnik skolaryzacji - szkolnictwo wysze, http://www.studenckamarka.pl/serwis.php?s=73&pok=1922.

Based on the data from Central Statistical Oce: www.stat.gov.pl

2 Literature review

2.1 Signaling

The standard signaling model of education dates back to the seminal article by Spence (1973), where theagents' types are discrete (high or low). A fully continuous case is considered in Mailath (1987), whereimportant results considering dierentiability and existence of equilibria are set. Most of the models in theliterature we found either mimic the Spence's model with two types (e.g. Ma, Weiss (1993)) or considera fully continuous case, where one-to-one correspondence between signals and types could be found, as inEscriche et al. (2002). We investigate a model with a discrete set of signals (with |S| > 2) and continuumof types. This makes the analysis slightly more dicult, as the standard conditions from Mailath (1987)cannot be properly dened. However, we guess the form of an equilibrium and prove that it exists under mildconditions. Moreover, we argue that in the class of symmetric equilbria, only semi-pooling (and pooling)equilibria exist and provide characterization for them.

There is not much empirical work on signaling games, probably due to scarcity of data. The notableexception is Tyler et al. (2000), where the signaling eect of education is properly tested and shown to besignicant for high-school dropouts in the US. This observation enables us to justify (at least partially) ourassumption that education has only a signaling eect on the job market, without increasing the productivity.

2.2 Overeducation

Overeducation is dened as an excess of a worker's attained or completed level of schooling, comparing tothe level of schooling required for the job the worker holds (Leuven, Oosterbeek (2011); Sicherman (1991);Duncan, Homan (1982)). This general denition could be made more precise, by specifying, how theeducational requirements for the job are analyzed and measured. In particular, as is pointed out in Groot,Maasen van den Brink (2000), overeducation could be measured either by respondent's subjective opinion,expressed in a questionnaire, or by objective measures, referring either to requirements expressed in joboers, or to an average level od education within the occupation of the worker. The latter approach, isrepresented by e.g. Sicherman (1991), where large sample from PSID data is used to measure overeducationin a relation to the level of schooling of other workers employed in the same job. The subjective approachis used e.g. in Alba-Ramirez (1993), for an analysis of ECTV Spanish data, or Linsley (2005) for statisticsarising from Australian NLC survey. Due to the character of dataset, we follow their path, and examinepotential overeducation in worker's own evaluation of the appropiateness of their schooling, as reported inHigher Education as a Generator of Strategic Competences (HEGESCO) project.

The empirical studies that are cited in this paper (e.g. Alba-Ramirez (1993); Duncan, Homan (1982);Linsley (2005)) all use as a theoretical base some version of Mincer human capital equation. Although, on

2

the ground of theory, there were attempts to explain overeducation with a signaling model (Spence (1973)),we are not aware of any empirical exercise in this framework.

An important assumption needs to be made, when refering to overeducation in the signaling model. Noticethat in the classic signaling model that we use, overschooling understood as attaining more education thanstrictly required to perform the tasks of one's jobs (Linsley (2005)) arises always for all the workers, since,by assumption, the only role of education is to signal one's productivity. Therefore, we propose anotherinterpretation of overschooling: we assume that if the agents send a signal by completing a given levelof schooling, they would perceive themselves as overeducated if and only if, their job position does notcompensate them for their cost of signaling, i.e. they consider their choice of education as ex-post too high.Such a mismatch may arise if the demand side of the job market is not perfectly exible, i.e. only some ofthe workers are correctly matched to the job position they are suitable for.

3 Model

3.1 Setup

3.1.1 Agents

We will consider a measure one of agents with secondary education that have to decide about their futurecareer. Agents dier in productivity - i.e. their type - which is a private information of each agent. Produc-tivity could be signaled through tertiary education: getting a master diploma entail a xed cost of studying,and provides some disutility to the student, who has to exercise his brain. However, a diploma serves as asignal on the job market, and, therefore, studying can result in a higher wage in the future. We shall assumethat the tertiary education decision is a choice among a nite sets of eort levels, which we shall interpretas elds of study.

The idea behind extending the signal space to more than a binary choice is that the students signaltheir productivity not only through the mere fact of obtaining a diploma, but also through the type of thediploma, that would dierentiate between more and less productive students. The elds of study could betherefore divided into those that are more pleasant (i.e. provide less disutility), but result in a lower futurewage and elds that provide more disutility, but guarantee higher wage.

Going to the university entails a xed cost c, which may be interpreted in the general model as the tuitionfees and other costs associated with getting a higher education - cost of moving to another city, renting aat etc. This cost is borne by every student, regardless of the eld of study. In principle, c could be negativeand then it would be interpreted as a constant part of the disutility function (in our setting, we assume thatdisutility is normalized). Notice, that if some part of c are tuition fees, then c is subject to policy changes.

Moreover, we shall assume that dierent elds of study require dierent eorts and the eort levels areone-dimensional real variables - in particular, they could be put in a linear order from studies that requirelittle eort to those that require a lot of hard work. Our assumptions abstract from individual's talents andpreferences, assuming that every high-school graduate could in principle study any eld - from humanitiesto medicine and engineering - and the choice would entail a eld-specic eort that would signal the player'stype and result in eld-specic wage.

Specically, we shall consider a signal space S with eort levels 0 = e0 < e1 < . . . < eS . The agent'sproblem is

maxeS

{w(e) v(e, ) c 1{e 6=e0}

}.

Notice that the wages oered by the rms w(e) take into account the expected productivity of the agent,and therefore, are the function of his signal. Agent anticipates this process and knows the distribution oftypes in the whole population. The assumptions about the disutility function we make are quite standard:

Assumption 1. Let v(e, ) be a disutility function of studying. We will assume that v : [0,) [b, B] Rand satises:

1. v(0, ) = 0,

2. ve(e, ) > 0,

3

3. v2e(e, ) 0,

4. v(e, ) < 0,

5. ve(e, ) < 0,

6. limB v(e, ) = 0 and limb+ v(e, ) =

where f ix denotes an i-th partial derivative of f with respect to x.

Note, that we assume that the disutility is dened not only for eorts in S, but also for a whole haline this would enable us to use calculus in proofs. The assumptions reect our economic intuition about how thedisutility should look like. In particular, we assume that increasing eort entails bigger disutility (2.) andthe bigger the eort is, the more dicult it is to increase it further (3.). Moreover, more productive agentshave lower disutility from exerting a given eort (4.), and nd it easier to increase it than the less productiveagents (5.). Finally, exerting no eort gives no disutility to the agent (1.), which is just a normalizationexercise. We also assume that Inada conditions hold (6.), but most of the general results of the paper wouldhold also in the absence of the boundary limit conditions.1

We guess that in the equilibrium, lower types chose lower eort levels and receive lower wages. Therefore,we will make a correspondence between eorts and wages and therefore guess that the order of the wagesrepresents the order of eorts.2

3.1.2 Firms and the regulator

As in standard signaling exercises, we assume there are at least two rms, to make the market competitionexplicit. The competition drives the rms' prot to zero, therefore forcing the rms to oer wage contractsthat cover the expected roductivity of hired workers:

w(ei) = E (| ei) ,

where ei = { : ei = arg maxe w(e) v(, e)} is the anticipated set of types, who chose a signal ei.Moreover, we assume that the market is not completely exible. In particular, the rms cannot oer an

unlimited number of contracts for any number of productive workers they face. It might be due to the costsassociated with creating new job positions, or by the fact, that rms require a certain ratio of productive vs.unproductive workers. In particular, for any class of jobs that require at least a given level of productivity, there could potentially be created at most a() job positions available for new workers. If the number ofworkers with productivity signaled to be at least is less a(), then all the workers are employed in jobsappropiate to their signal. If, however, the number of produtive workers exceeds a(), only a() randomlychosen workers are employed, while the rest must search for jobs in a lower class, i.e. job positions thatwould also accept workers with a lower signal.

We assume that the level of a(), as an aggregate of individual rms' decisions is unknown to marketparticipants and is not internalized by the future workers, deciding about their level of education. It mightcorrespond to the assumption, that the inexibility of the market is only temporary e.g. for a periodneeded to form a new job positions and the market is generally perceived as almost exible. It seemsreasonable to assume that the dierence between the supply and demand of productive workers is small, andtherefore, might not be perceived as a real threat to the market participants. We shall assume, on the otherhand, that a() is known to3 the regulator, who decides about the cost of study. Manipulations with thecost c can generally lead to oversupply or undersupply of workers, in reference to the market satiation levela().

1In particular, Inada conditions are used to prove an existence of semi-separating equilibrium for two signals. If (6) doesnot hold, only an existence of a pooling equilibrium could be proven.

2The interpretation of such dened productivity might seem controversial, but is not. The fact that some people chose hu-manities over engineering even if engineer's wages are higher than historians' would in our model mean that humanities studentshave lower productivity than engineering students. However, productivity should not be mistaken for talent, intelligence orskills, rather some personal characteristics valued (through wages) in the market. The assumption that those characteristicsare one-dimensional is clearly a severe simplication, but it makes the model controllable.

3or might be estimated by

4

3.1.3 Distribution of types

We shall assume that the type space is continuous and bounded, with a cumulative distribution function F .We shall assume that the distribution of types has no atoms and therefore F is continuous.

Fact 2. Let F be a continuous cumulative distribution function and U be a uniform random variable on

[b, B]. Then X = F1(

1BbU

)follows a distribution F .

Fact 2 has an interesting consequence, which states, that we can essentially consider only a uniformdistribution.

Corollary 3. Let follow any continuous distribution function F and v(, e) be a disutility function satis-fying Assumption 1. Then there exist Unif [b, B] and disutility function v(, e), such that v also satisesAssumption 1.

Proof. Let v(, e) = v(F1

(/(B b)

), e). Since F is increasing, also F1is increasing and it is easy to

verify that properties (1)-(5) from Assumption hold.

Notice that in our exercise, the distribution F is only needed to estimate v. By Corollary 3 we canassume without loss of generality that F is uniform and estimate the disutility function corresponding tothe uniform distribution. To avoid excessive notation, we shall denote from this point the desired disutilityfunction as v(, e), remembering that now Unif [b, B].

3.2 Equilibrium

3.2.1 Semi-pooling equilibria

We shall look for a specic class of equilibria in the signaling model, symmetric equilibria, in which all theagents choose ex-ante (i.e. before learning their type) the same strategy, that denes their eort, given theirtype.

Denition 4. A symmetric equilibrium for a signalling game is a function e : S, such that for an agentof type , function e() is the best response to all the other players choosing according to e(). Specically,e() satises:

e() = maxeS

{w(e |other players choose according to e) v(, e) c 1{e 6=e0}

}.

Notice that since a single agent is of measure zero, by changing unilaterally her decision, she does not aectother player's payo function. Therefore, the equilibrium is simply dened through an equation:

e() = maxe()

w(e()) v(, e()) c 1{e()6=e0}.

Fact 5 (Trivial). The function e : S cannot be 1 1, therefore no separating equilibria exist.Fact 6. The equilibrium function e() is locally constant and weakly increasing in .

Fact 6 is intuitive and we leave it without proof4. Notice, that by the fact that e() is an increasingstep function, all the equilibria must be partition equilibria, in a sense that the inverse images of subsequenteort levels dene intervals (possibly degenerated) that sum up to the type space [b, B]. This intuition isformalized in the two denitions:

Denition 7. Let 3 be a space of types andS = {e0, . . . , eS} be the eort space. In this environment, a(generalized) semi-pooling equilibrium of sizeM+1 is dened by a subset of eortsM = {ei0 , . . . , eiM } Sand a sequence of thresholds 0 = 0 < 1 < . . . < M+1 = B such that all agents of type [m, m+1]choose e = eim for m = 0, . . . ,M . The sequence {i}i would be called an equilibrium partition for asubsetM.

In particular, if e() is an equilibrium in a sense od Denition 4, then M = e1(S) and [m, m+1] =e1(eim).

4The rst part of the sentence is abvious. To give the idea of the proof of the second part, notice that if both and e werecontinuous variables, the statement is an immediate consequence of an Envelope Theorem. For e discrete, some minor technicaldiculties must be considered, however, the proof is more tedious than ingenious.

5

Notice that a semi-pooling equilibrium of size 1 is called in the literature a pooling equilibrium.However, to avoid confusion in our further results, we will keep our generalized notion throughout the paper.The discussion above could be concluded with a statement:

Corollary 8. In a signaling game with a continuum of types and discrete signal, the only symmetric equilibriaare (generalized) semi-pooling equilibria.

We will now examine some properties of the equilibrium. Let us recall that, following 3, we assumed that Unif [b, B].

Lemma 9. For eort levels e0, . . . , eM , and following a uniform distribution on [b, B], the semi-poolingequilibrium wages satisfy:

w(ei) =i+1 + i

2for i = 0, 1, . . . S.

Proof. (Simple) By an assumption that rms make no prots, in a semi-pooling equilibrium with thresholds0 = 0 < 1 < . . . < S+1 = B we have w(ei) = E |(i,i+1) =

(i + i+1

)/2.

Corollary 10. Given market wages w1, . . . , wS, the semi-pooling equilibrium is a set of thresholds denedsequentially:

0 = b and S = B

i = 2wi1 i1 for i = 1, . . . , S

3.2.2 Binary signal

Let us rst consider a basic model, where there only two possible signals: getting a higher education diploma(denoted by e1 = 1) or not (denoted by e0 = 0). We argue, that in such a model there exist a threshold oftype , such that agents whose productivity is above choose to study and those with productivity below thethreshold prefer to go to the job market with a secondary education diploma.

Proposition 11. For a given distribution of types : [b, B] R and cost of education c, such that if onlyc B E, there exists a semi-pooling equilibrium in which all agents of type > choose e = 1 and allagents of type < choose e = 0 for some threshold .

Proof. Observe that a set of symmetric strategies {(es = 1 | s > ), (es = 0 | < )}s is a Nash equilibrium,if it satises:

1. Individual rationality:

w(1) v(1, ) c 0 for > (1)w(0) v(0, ) 0 for < (2)

2. Incentive compatibility:

w(1) v(1, ) c w(0) v(0, ) for > (3)w(0) v(0, ) w(1) v(1, ) c for < (4)

Notice rst that condition (2) is trivially satised by the assumption that v(0, ) = 0 and > 0, so only (1)needs to be veried.

The incentive compatibility conditions could be rewritten as:

w(1) w(0) v(1, ) v(0, ) + c for > w(1) w(0) v(1, ) v(0, ) + c for <

The wage wedge w(1)w(0) for a given and the distribution F is simply a number. Moreover, functionv(1, ) v(0, ) is dierentiable, therefore continuous in , so by taking we infer that it must be:

6

w(1) w(0) = v(1, ) + c. (5)To nish the proof of existence, we must only verify that for a given c, F and v there exists such that

(5) holds. Observe that the LHS, observe that it could be rewritten in terms of F only:

w(1) w(0) = E( | > ) E( | < ) = BxdF (x)

1 F () bxdF (x)

F ().

Since we assume that F is uniform, we infer (e.g. from Lemma 4) that

w(1) w(0) = B + 2 + b

2=B b

2.

Notice that the RHS of (5) is a decreasing function of . By Inada conditions (Assumption 1.6), we have:

limb+

v(1, ) + c = and limB

v(1, ) + c = c.

Therefore if only c < (B b)/2, the formula v(1, ) + cw(1) +w(0) attains two dierent signs at the endsof interval [b, B], therefore, there exists a solution to (5). Moreover, since the formula is decreasing in , thesolution is unique.5

3.2.3 Finite space of signals

Now, we can move to the general case of a nite number of possible signals. Assume the signal space is adiscrete set of size S + 1 with eort levels 0 = e0 < e1 < . . . < eS for S > 1. We shall interpret e0 as adecision not to study, and e1, . . . , eS as a choice to study in one of S elds.

Theorem 12. For a given distribution of types : [b, B] R and cost of education c, there exists sumenumber M such that for any 1 M M there exists a semi-pooling equilibrium of size M .

In particular, an equilibrium partition for an equilibrium of size M could be chosen for a subset of Mminimal eorts, i.e. {e0, e1, . . . , eM}.

Proof. The argument follows a truly brilliant idea introduced by Crawford, Sobel (1982) and only neededto be adapted to our setting and the presence of eorts, that do not appear in their model. For details, seeAppendix A.

Remark 13. A semi-pooling equilibrium with eorts {ei0 < . . . , < eiM } is a sequence b = 0 < 1 < . . . 1. Observe that a set of symmetric strategies {(es = em | s [m, m+1])}s is a Nash equilibrium, if it satises:

1. Individual rationality

w(em) v(em, ) c 1{m6=0} 0 for [m, m+1] (9)

2. Incentive compatibility:

w(em) v(em, ) c 1{m6=0} w(ej) v(ej , ) c 1{j 6=0} j 6=m for [m, m+1] (10)

Observe that by Fact 6 restrictions do matter (i.e. could be binding) only for neighboring subintervals, i.e.a sucient set of conditions is just:

w(em) w(em1) v(em, ) v(em1, ) c 1{m=1} for (m1, m

)(11)

w(em) w(em1) v(em, ) v(em1, ) c 1{m=1} for (m, m+1

)(12)

Notice that the LHS is in fact a function of the partition:

m+1m12 v(em, ) v(em1, ) c 1{m=1} for

(m1, m

)m+1m1

2 v(em, ) v(em1, ) c 1{m=1} for (m, m+1

)By continuity, we must have:

m+1 m12

= v(em, m) v(em1, m) c 1{m=1}for m = 1, . . . ,M, (13)

with boundary conditions:0 = b, M+1 = B (14)

Notice that (13) for given {e0, . . . , eM} is a set ofM non-linear equations inM+2 variables 0, . . . , M+1.We already know from Proposition 11 that it has a solution at least for M = 1. We want to determine whatis the maximal M that allows us to construct the equilibrium.We shall proceed as in Crawford, Sobel (1982). Let us use the following notation: let i denote a sequencei0 < . . . <

ii that satises a set of conditons (13). Let

K(x) = max{i : there exist b < x < 2 < . . . < i B, satisfying (13) with a subset {e0, . . . , ei}}.

Since we have a total of S possible eort levels, K(x) is bounded from above by S. Let M = supx(0,1]K(x).

Then M is a maximal size of an equilibrium.W have proven now the existence of M . Let us take 1 = arg maxK(x), so that K(1) = M. Now we

would like to show that for any 1 M M we could construct an equilibrium of size M .Let K(x) be a sequence of length K(x) that solves (13) with a boundary condition

K(x)1 = x. We might

notice that since all the functions in (13) are continuous, then the solutions of the system vary continuously

with boundary conditions. Therefore, if only K(x)K(x) < 1, then the function K(x) is continuous at x and since

it is a integer-valued function, it is locally constant around x. Moreover, the function can change by at mostone in a disutility point. Given that two arguments, since K(1) = M and K(B) = 1, we conclude thatK(x) must take all integer values 1 M M on an interval [1, B].

We shall also examine, what happens to the choice of eorts, when K varies with x. If K(x) is discontinu-ous at x, then K(x) satises both (13) and (14), so it is an equilibrium of a size K(x). We might also notice,

that the discontinuity may happen only if K(x)K(x) = 1, so that the biggest eort eK(x) becomes redundant, in

a sense, that it is not chosen anymore by a postive measure of agents. Therefore at discontinuity points ofK(x) = M , a set of eort choices changes from {e0, . . . , eM} to {e0, . . . , eM1}.

14

Proof of Proposition 14. Assume there exists an equilibrium b = 0 < 1 < . . . < M+1 = B, such asdescribed in the Proposition. We shall show that such equilibrium is unique.

Assume the contrary, i.e. there exists another equilibrium b = t0 < t1 < . . . < tM+1 = B, such thatall the eorst from S are chosen with a positive probability. Notice that the sequences {i}Mi=1 and {ti}Mi=1are two dierent solutions to a set of equations (6) with boundary conditions (7). Take j = maxi{i 6= ti}.Without loss of generality, we can assume that j > tj . Since 1 j M and for i > j we havei = ti, wecan use the conditions (6) to write:

[v(eij , j) v(eij1 , j)

][v(eij , tj) v(eij1 , tj)

]=tj1 j1

2,

and the LHS by Lagrange's mean value theorem (used twice), could be rewritten in terms of v's derivative:

ve(e, )

(eij eij1

) (j tj

)=tj1 j1

2,

where e [eij1 , eij ] and [tj , j ]. By Assumption 1, ve is strictly negative, and the terms in bracketsare strictly positive, therefore the LHS is strictly negative. We therefore conclude that the RHS must bestrictly negative as well, and we get j > tj j1 > tj1. By repeating the exercise for m = j 1 we get:[

v(eij1 , j1) v(eij2 , j1)][v(eij1 , tj1) v(eij2 , tj1)

]+tj j

2=tj2 j2

2,

and, proceeding iteratively we prove that j > tj i > ti i < j. But since it must also hold for i = 0, itcontradicts the fact that the sequences {i}M+1i=0 and {ti}

M+1i=0 satisfy (7). Therefore, the initial assumption

that there exist two dierent equilibria must have been false.

15

Appendix B: Partial Pareto-order of semi-pooling equilibria

This part is an attempt to generalize the results from Crawford, Sobel (1982), regarding the Pareto-dominanceof equilibria of higher order (called more informative). It must be noted, however, that our model issignicantly dierent from the one in Crawford, Sobel (1982) in particular, sending a signal is (usually)costly. Thus, their strong results do not hold in general in our settings. Although we cannot state thatequilibria of higher order are Pareto superior to all the equilibria of lower order (and we are certain that thissentence is in general not true), it can be shown that high-order equilibrium Pareto-dominates at least someclass of lower-order equilibria.

All proofs in this part are at least partly based on the ideas from Crawford, Sobel (1982).

Lemma 15. If{m}mis an equilibrium partition for some eort levels E 3 ej , then for any m j + 1 it

must be mej 0.

Proof. Let us consider an incremental change in ej , keeping all the other eorts the same. We will show that

since the equilibrium is a solution to (6) with boundary conditions (7), for any m > j+ 1 the derivative mejhas always the same sign. To see this, notice that M+1 = 1 and therefore the derivative is zero. Moreover,dierentiating (6) for m = M gives us:

12

M1ej

=Mej

(v(eiM , M ) v(eiM1 , M )

).

Since the element in brackets is nonpositive, we conlude that the two partial derivatives must have the samesign. The next step (backwards) is to consider the dierentiated equation for m = M 1:

12

M2ej

=M1ej

(v(eiM1 , M1) v(eiM2 , M1)

) M

ej,

which suggests that also M2ej has the same sign. We can repeat the exercise until m > j+ 1, showing that{kej

}kj+1

is of the same sign.

To claim that the sign is in fact negative, we shall turn to economic arguments. Notice that increasingejwould not make the choice of ej+1 more attractive for agents, who have already chosen less than ej .However, it will make ej+1 more attractive to agents, who had chosen ej . Since one eort increases while theother remain the same, the alternative option which is a bigger eort becomes relatively cheaper. Therefore,j+1 decrease and so do all other k.

Lemma 16. If there exists an equilibrium {Mi }i of size M with eorts EM and an equilibrium {M+1i }i of

size M + 1 with eorts {ei0} EM such that ei0 < min EM , then for all i M we have Mi1 M+1i Mi .

Proof. Let us denote ei1 := min EM . Consider the partition {M+1i }i and a graduate increase in ei0 , so thatei0 ei1 . Then, {M+1i }i all decrease (by Lemma (15)). However, in the end, when ej ej+1 the thresholdsjumps by one, therefore M+1k collapses into

Mk1. This proves that

Mk1

M+1k Mk at least for 1.

Proposition 17. If there exists an equilibrium {Mi }i of size M with eorts EM and an equilibrium {M+1i }i

of size M + 1 with eorts {ei0} EM such that ei0 < min EM then the more informative (higher-order)equilibrium Pareto dominates the less informative equilibrium.

Proof. Without loss of generality, we can assume that the eorts in the bigger equilibrium are just M + 1rst states, i.e. EM+1 = {e0, . . . , eM+1}. Let (M) be an equilibrium of size M . Let x = (x0 , . . . , xM+1)be an increasing sequence that satises (13) for i = 2, . . . ,M with eorts EM+1 and boundary conditionsx0 = b,

xM = x,

xM+1 = B. Notice that if x = M1(M), then the sequence must be in fact (M) (one of

the eorts became redundant). On the other hand, if x = M (M + 1), then x = (M + 1). Therefore, x

provides a continuous mapping between the two equilibria.The interval [M1(M), M (M + 1)] is nongenerate, by Lemma 16. We would like to show that agent's

utility is increasing in x for x [M1(M), M (M + 1)].

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First, let us notice, that the rms gain 0 in any equilibrium. However, it is not the case for agents.Denote by EU(x) the expected utility (ex ante) in an equlibrium partition x with eorts {e0, . . . , eM+1}.Then, we have:

EU(x) =

M+1j=0

xj+1

xj

xj+1 + xj

2(B b) v(ej , )d

=

=B + b

2 1B b

M+1j=0

xj+1

xj

v(ej , )d

The rst element of the sum is just an average wage ex ante and does not depend on x. The second elementis more troublesome. However, by Leibniz rule we have:

x

xj+1

xj

v(ej , )d =xj+1x

v(ej , xj+1)

xjx

v(ej , xj ).

Taking the sum we get a simple formula:

EU(x)

x= 1

B b

M+1j=1

[xj+1x

v(ej , xj+1)

xjx

v(ej , xj )

]= 1

B b

Mj=1

xjx

[v(ej ,

xj ) v(ej1, xj )

].

Sincexjx 0 (a non-trivial consequence of Lemma 15) and v(ej ,

xj ) v(ej1, xj ) < 0, by the assumption

on v, then EU(x)x 0. Therefore utility grows as we move from the less informative to a more informativeequilibrium.

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