The risk-incentive trade-off in competitive search

GETULIO VARGAS FOUNDATION SÃO PAULO SCHOOL OF ECONOMICS

PAULA FERREIRA ONUCHIC

THE RISK-INCENTIVE TRADE-OFF IN COMPETITIVE

SEARCH

PAULA FERREIRA ONUCHIC

THE RISK-INCENTIVE TRADE-OFF IN COMPETITIVE

SEARCH

Dissertação apresentada à Escola de Economia de São Paulo da Fundação Getulio Vargas como requisito para obtenção do título de Mestre em Economia

Campo de Conhecimento: Microeconomia

Orientador: Prof. Dr. Braz Ministério de Camargo

Onuchic, Paula Ferreira

The Risk-Incentive Trade-Off in Competitive Search / Paula Ferreira Onuchic – 2015

30 f.

Orientador: Braz Ministério de Camargo

Dissertação (mestrado) - Escola de Economia de São Paulo.

1. Risco (Economia). 2. Incerteza (Economia). 3. Relações trabalhistas. I. Camargo, Braz Ministério de. II. Dissertação (mestrado) - Escola de

Economia de São Paulo. III. Título.

PAULA FERREIRA ONUCHIC

THE RISK-INCENTIVE TRADE-OFF IN COMPETITIVE

SEARCH

Dissertação apresentada à Escola de Economia de São Paulo da Fundação Getulio Vargas como requisito para obtenção do título de Mestre em Economia

Campo de Conhecimento: Microeconomia

Orientador: Prof. Dr. Braz Ministério de Camargo

Data de Aprovação: 23/03/2015 Banca examinadora:

Prof. Dr. Braz Camargo (Orientador) FGV-EESP

Prof. Dr. Klênio Barbosa FGV-EESP

ABSTRACT

I use the competitive search framework to model a job market with heterogeneous workers in which there is a moral hazard problem in the employer-worker relation. In this setting, I can predict how contracts react to changes in underlying parameters of the market (in particular, the production risk), as well as how the probability of each type of worker being hired responds. My main finding is that while at the individual level there is a negative risk-incentive trade-off, general equilibrium effects imply that the effect can be positive at the aggregate level depending on the market search frictions and the distribution of types. My re-sults help shed some light on some puzzling empirical findings on the risk-incentives trade-off.

RESUMO

Usando a abordagem de competitive search, modelo um mercado de trabalho com

tra-balhadores heterogêneos no qual há um problema de risco moral na relação entre firmas e trabalhadores. Nesse contexto, consigo prever como contratos reagem a mudanças nos parâmetros do mercado (em particular, o risco de produção), assim como a variação da prob-abilidade dos trabalhadores serem contratados. Minha contribuição principal é ver que, no nível individual, existe uma relação negativa entre risco e incentivos, mas efeitos de equilíbrio geral implicam que essa relação pode ser positiva no nível agregado. Esse resultado ajuda a esclarecer resultados empíricos contraditórios sobre a relação entre risco e incentivos.

Contents

1 INTRODUCTION . . . 8

2 LITERATURE . . . 10

3 THE MODEL . . . 12

3.1 Environment . . . 12

3.2 Equilibrium . . . 13

4 CHARACTERIZATION . . . 16

5 AVERAGE BONUS . . . 19

5.1 Continuous Distribution of Types . . . 19

5.2 Two Types . . . 21

6 CONCLUSION . . . 24

Bibliography . . . 25

1 Introduction

The trade-off between risk and incentives has been a matter addressed by much theoretical and empirical work on contract theory. Traditionally, the agency framework considers a risk-neutral principal proposing take-it-or-leave-it contracts to a risk-averse agent.1 _The choice this employer makes is between providing incentives to the worker by tying pay to performance and rewarding him for bearing risk. In this context, the theory predicts that higher idiosyncratic variance in the worker’s output will make principals propose less incentive pay. With higher risk, performance becomes a noisier estimator of the worker’s effort, making bonus pay riskier to the agent, who will demand a better compensation for it.

Empirically, this relation has been tested in different settings, such as CEO compensa-tion, agricultural contracts and franchising. However, the results have been inconclusive, often pointing towards a positive rather than negative relation between output risk and in-centive pay.2 _{These puzzling observations have motivated researchers to look for alternatives} to risk-sharing to explain the contract choice by principals, such as transaction costs or task delegation as in Prendergast (2002). In the present work, I attempt to reconcile the tradi-tional moral-hazard theory with the empirical findings by modeling the market in a general equilibrium setting, rather than in partial equilibrium as usually done.

I use a competitive search equilibrium framework, as in Moen (1997) and Shimer (1996), with risk averse workers that are heterogeneous in their risk aversion and homogeneous risk neutral firms. In this setting, I can predict how contracts react to changes in the production risk, as well as how the probability of each type of worker being hired responds. The change in probability of employment may affect the distribution of risk aversion among the workers that are hired and, as the less risk averse workers are associated with higher bonus levels than the more risk averse ones, the distribution of bonus levels in the contracts that are actually celebrated in this economy may also shift. In particular, the average level of bonus among the workers that are hired can change.

The main finding of my work is that while at the individual level the negative risk-incentive trade-off traditionally predicted holds, general equilibrium effects that change the distribution of types among the workers that are hired imply that the effect over the average bonus can be positive. I find sufficient conditions on the market search frictions and on the distribution of types that guarantee that the overall effect of increasing production risk on the average bonus in this economy either positive or negative.

1

Holmstom and Milgrom (1987)

2

2 Literature

This work is related to two strands of the literature. First, it relates to research trying to explain the inconsistency between theoretical results and empirical observation regarding the risk-incentive trade-off. Serfes (2005, 2008) considers that (i) the risk aversion of the workers is not observable or only partially observable to the empirical researcher and (ii) firms and workers are matched assortativelly according to the levels of risk aversion of each worker and production risk of each firm. As riskier firms are matched with lower risk aversion agents, the overall risk-incentive trade-off may point in the positive direction when empirical researchers do not control for workers’ risk aversion.

Serfes (2005) does not propose an equilibrium model, rather assumes that firms and work-ers match assortativelly to reach the conclusion. As for Serfes (2008), it uses the Shapley and Shubik assignment game to motivate the assortative matching. The drive of the inconsistency between empirical and theoretical results proposed by Serfes (2005, 2008) is closely related to this work. Alternatively to Serfes, I consider a model set in competitive search equilibrium with only worker heterogeneity. An important difference between my work and Serfes (2005, 2008) is that I can relate my results to characteristics of the market and workers, such as market search frictions and the distribution of risk aversion.

Empirical literature on the matter does acknowledge that it is important to control for heterogeneity in unobservables such as risk aversion. However, most works only use proxies for risk aversion of workers, such as wealth and property. Ackerberg and Botticini (2002) show that such approach is not enough to solve the endogenous distribution of risk aversion problem. In particular, the CEO compensation literature (Aggarwal and Samwick, 1999, for example) attempts to control for the risk aversion heterogeneity by using individual fixed effects. In fact, in the CEO compensation case, it appears that the risk-incentive trade off is negative, as predicted by theory.

3 The Model

3.1

Environment

I consider an economy set in a competitive search framework. There exists a measure 1 of workers who are heterogeneous in their risk aversion searching for job vacancies and free entry, at cost k > 0 of homogeneous firms searching for employees. One firm has to match with exactly one worker to produce output. Once there is a match, the production faces a standard moral hazard problem (e.g. Holmstrom and Milgrom, 1987). Firms can join the market by paying entry cost and posting a vacancy, specifying and committing to the contract it is offering to workers.

Each unemployed worker observes all the contracts posted in the market and directs his search to any of the vacancies he likes. Each agent directs its search to a single vacancy, but could use a mixed strategy to decide which one. If a match is formed, the posted contract determines the payoff to the worker and to the firm involved in the match. All workers and firms left unmatched receive zero payoff.

Preferences: Firms and workers face two types of decisions. The first type relates to effort and production once they are engaged in a match. The second type regards the searching behavior, choosing between higher probabilities of matching and better matches. I consider these to be decisions of different natures, as the job search decision is a more long term choice, while the effort decision is a more short term on-the-job choice.

To stylize this fact, I allow workers to account differently for each type of decision, by allowing u(w, a)to be the utility function with which the workers account for the first type

of decisions, while V( ¯w) is the one used to account for the second type. Hence, the overall

utility of a worker that receives a contractψ is given byV( ¯w(ψ)), wherew(ψ)¯ is the certainty

equivalent wage yielded by contract ψ, given by u( ¯w) = _Eψ[u(w, a)] with w being income

and a costly effort.

I impose u(w, a) = 1−e−η(w−h(a))_{, where h(a) =} a2

2 , to follow the standard moral hazard model, meaning workers are risk averse with coefficient of absolute risk-aversion equal to η,

and letV(0) = 0, V′

(·)>0, V′′

(·)≤0. Workers are heterogeneous in terms of η, while they

do not differ in terms of V. Risk aversionη∈(0,η]¯ is distributed according toG(η)and types

are observable by all.

in the literature, I assume that firms are risk neutral.

Production and Contracts: Output of a match is given by y = a+ξ, where ξ ∼ N(0, σ2_{) is a shock that affects production. Since effort is costly and unobservable, firms} post contracts that motivate workers to exhort effort. As usual in the literature, I restrict these to linear contracts, making workers income equal to w = c+by, where c is the fixed

part and b the bonus level (power of the contract).

Search Technology: Firms post vacancies indexed by (c, b, η), specifying the contract

it is committing to and the type it searches for. Because the types are observable, the market is segmented into markets for each type of worker. Inside a market, workers compete with each other for the job vacancies and firms compete with each other to haver their vacancies filled. The degree of competition is captured by the ratio of workers to firms, denoted by

λ∈[0,∞] and referred to as queue length. There are search frictions in the sense that even

if the number of firms and workers in a market is the same (λ = 1), there is a positive

probability that buyers and sellers are left unmatched. When a firm faces queue length λ,

it matches with probability m(λ). Pairwise matching requires that p(λ) = m(λ)/λ is the

probability that a worker finds a match.

I assume that the number of matches in a market with u unemployed workers and v

vacancies is given by a CES matching functionM(u, v) = uv(ur_+vr₎−1

r, withr >0, implying

m(λ) =M(λ,1) =λ(λr_{+ 1)}−1

r and p(λ) =M(λ,1)/λ= (λr+ 1)− 1

r. The idea that relatively

more workers make it easier for a firm to fill a vacancy and harder for a worker to find a job is captured bym(λ) strictly increasing and p(λ)strictly decreasing in λ. The parameterr >0

measures how frictional the search process is; the lowerr, the more frictional the process.

3.2

Equilibrium

I follow the concept of Competitive Search Equilibrium proposed in Moen (1997) and Shimer (1996). Each contract posted in the market for each type of worker forms a submarket. The payoff of each individual firm or worker depends on which submarket they choose to join and on the queue length that is formed in this submarket, which in turn arises from the decisions of other individual firms and workers.

lengthλ, he gets the following expected payoff:3

Wη(c, b, λ) =

m(λ)

λ V

c+ (1−ησ2)b

2

2

+

1− m(λ)

λ

V(0) (3.1)

For firms, the expected payoff of posting contract (c, b) for type η workers under queue lengthλ is:

Jη(c, b, λ) = m(λ)

(1−b)b−c−k (3.2)

Firms post profit maximizing contracts and earn zero profit because of the free entry condition, which also makes firms indifferent between serving each type of worker. Workers direct their search to the submarkets that maximize their expected payoff conditional on the contracts posted in each of the markets, and on the search behavior of other workers. In equilibrium, a single contract will be posted in the market for each type of worker. However, beliefs about the queue lengths are defined for all possible contracts that are actually not offered in equilibrium by themarket utility condition that determines that all contracts that

are offered must yield the same market utility to the workers of each type.

A competitive search equilibrium is composed of the following equilibrium objects: Ψη ⊂

R2 are the contracts offered in equilibrium for typeη workers; the equilibrium expected type η worker payoff, denoted byW¯η ∈R+; the functions that give the queue length expected for

each contract in the market for each type of worker, λη :R2 →R+.

Definition 1. Ψη ⊂R2, W¯η ∈R+ and λη :R2 →R+, η ∈(0, ηmax], is a competitive search

equilibrium if it satisfies:

1. Profit Maximization: For allψη = (c, b)∈Ψη and η ∈(0, ηmax], (c, b, λη(c, b)) solve

the problem:

max

ˆ

c,ˆb,λˆ

Jη(ˆc,ˆb,λ)ˆ

subject to Wη(ˆc,ˆb,λ)ˆ >W¯η

2. Optimal Search: For allη ∈(0,η]¯,

¯

Wη =max

0, max

(c,b)∈Ψη

Wη(c, b, λη(c, b))

, if Ψη 6=∅

¯

Wη = 0, if Ψη =∅

3

¯

w(c, b) =c+b2 2 −

η

2σ2 is the certainty equivalent wage yielded to the typeηworker under contract (c,b).

The worker chooses to exhort effort a∗_{=b as it maximizes the expected payoff:}

a∗_{=argmaxac+ba−}b2

2ησ

2₋a2

3. Free Entry:

Jη(c, b, λη(c, b)) = 0, ∀(c, b)∈Ψη, η∈(0, ηmax]

4 Characterization

In this section, I characterize the contracts proposed to each type of worker, as well as the queue lengths formed in the market for them. From the definition of equilibrium, the free entry condition implies that firms are indifferent between serving the market for each type of worker among the types that are served. A firm that decides to serve the market for type

η workers, solves the following problem(Pη):

max

c,b,λ m(λ)

(1−b)b−c−k

subject to m(λ)

λ V

c+ (1−ησ2)b

2

2

>W¯η

(Pη)

Lemma 1 shows that, for each type of worker, there is a single level of bη that may solve

(Pη). This bonus level is the same equilibrium bonus found in the partial equilibrium models

(Holmstrom and Milgrom (1987)), and is decreasing both in the risk aversion coefficient and on the market riskσ2_.

Lemma 1. When a type η worker is hired, the bonus level celebrated in the contract is

bη = ₁₊1_ησ2, for all η ∈(0,η]¯.

Proof. Holding cand λ constant, b = ₁₊1_ησ2 solves the constrained maximization. Hence, in any solution(c∗

, b∗

, λ∗

) to(Pη), it must be true that b∗ = ₁₊1_ησ2.

The choice of bonus level completely determines the amount of effort the worker will exhort and hence the expected level of production. In turn, this level of production, jointly withηandσ2_{determine the "surplus" to be split between firm and worker, equal to} 1

2(1+ησ2₎.4 I can now rewrite problem (Pη) as (Pη′), where the firm is choosing which level of certainty

equivalent w¯ to offer to workers. It can either give more surplus to the worker (in terms of the certainty equivalent yielded), which attracts more workers to the vacancy, or keep a bigger portion of it and get a lower probability of matching.

4

Lemma 2. For each η ∈ (0,η]¯, finding (c, b, λ) that solves problem (Pη) is equivalent to

finding ( ¯w, λ) that solve the following problem (P′

η).

max

¯

w,λ m(λ)

1

2(1 +ησ2₎−w¯

−k

subject to m(λ)

λ V( ¯w)>W¯η

(P′

η)

Proof. Proof is in the appendix.

Free entry condition also determines that firms have an outside option payoff of 0, achieved if they do not post any vacancies. Lemma 3 will show that this implies that workers with too high risk aversion are not served by any firms.

Lemma 3. No firms direct search to workers of types η > 1−2k

2kσ2 :=ηmax.

Proof. The probability that a firm matches in a market,m is bounded above by 1; while no workers direct search to vacancies that provide w <¯ 0. Hence, the payoff a firm achieves by

posting a vacancy in the market for type η workers is bounded above by 1

2(1+ησ2₎ −k. This implies that firms can only achieve negative payoffs by serving markets of workers of types

η > ηmax, which determines that no firms will serve such markets.

The solution to (P′

η), along with free entry, define the equilibrium levels of W¯, w¯ and λ

for the workers that do get served. Lemma 4 gives the condition for the equilibrium queue lengthλ in the market for each type.

Lemma 4. For each worker type that is served by firms, η∈(0, ηmax), the equilibrium queue

length λ is unique and satisfies the following condition:

H(λ) = V

′ 1

2(1+ησ2₎− _mk_(λ)

V ₂₍₁₊1_ησ2₎−

k m(λ)

− m(λ)

r+1

k(1−m(λ)r₎ = 0 (4.1)

Proof. Proof is in the appendix.

The following proposition shows that individuals with higher η will be associated with

markets with higher equilibrium queue length, hence lower probability of matching to workers. These more risk-averse workers will also receive a lower equilibrium level ofw¯. The intuition

here is that more risk-averse workers produce a lower surplus in a match, making them less attractive to firms, that are now willing to search for them only if there is a higher probability of matching.

Proof. Proof is in the appendix.

Proposition 2 considers how queue length depends on how risk averse workers are regard-ing their searchregard-ing behavior. It shows that when workers are more risk averse, they direct their search towards vacancies that guarantee a higher probability of matching, even when this means a lower certainty equivalent level once matched, implying that equilibrium queue length and certainty equivalent are lower for all types.

Proposition 2. LetV be a CARA utility function, that isV( ¯w) = 1−e−ϕw¯_{, withϕ being the}

coefficient of risk aversion of workers relative to their searching decisions. For every worker

type η ∈(0, ηmax), λ(η, ϕ) and w(η, ϕ)¯ are both decreasing in ϕ.

Proof. Proof is in the appendix.

Proposition 3 shows that, when the searching process is highly frictional, workers and firms cannot find each other, implying that for a firm to have even a very small probability of matching, the queue length must go to infinity, meaning that the measure of firms entering the market goes to zero. The opposite extreme case happens when the market becomes frictionless. In this case, there is perfect coordination in the market, in the sense that firms with vacancies and unemployed workers are always able to find each other. Hence, there is one firm joining the market for each of the unemployed workers existent and both workers and firms match with certainty. In both the extreme scenarios considered, the level of risk aversionη and market risk σ2 _{play no role in determining the probability of matching of the} workers.

Proposition 3. Let the workers be risk-neutral in terms of their searching behavior,V( ¯w) = ¯

w. Then:

1. When the market is very frictional, r → 0, λ(η) → +∞ and p(λ(η)) = m_λ(λ_(η(η₎)) → 0, ∀η ∈(0, ηmax).

2. When the market becomes frictionless, r→+∞, λ(η)→1 and p(λ(η)) = m_λ(λ₍(_ηη₎)) →1, ∀η ∈(0, ηmax).

5 Average Bonus

Knowing the distribution of typesGin the economy, and the equilibrium values ofλfor each

of these types, I can write the distribution of types among the hired workers as:

F(η) =

ˆ η

0

p(λ(ˆη)) dG(ˆη)

ˆ η¯

0

p(λ(ˆη)) dG(ˆη)

Knowing this distribution and that the equilibrium bonus level of each type is given by

b(η) = 1

1+ησ2, I see that the average bonus level across the economy is:

bavg =

ˆ η¯

0

1

1 +ησ2dF(η)

If there is an increase in the market risk level, σ2_{, there are two effects on b}

avg. The first

effect, that I call Partial Equilibrium Effect, is the decrease in the equilibrium bonus level for each of the worker types. This effect is the one that was found on the traditional partial equilibrium moral hazard models and will always have a negative impact onbavg. The second

effect (General Equilibrium Effect) relates to the change in the distribution of types among the hired workers (F) resulting from an increase in σ2 _{and may be a positive or a negative} impact onbavg.

The equilibrium condition in Lemma 4 implies that, whenσ2_{increases,λ(η)also increases,} for all types, hence all types of workers now get a lower probability of matching. These probability changes for each of the types have an impact on the distribution F that may in

turn have a positive or a negative effect on bavg. For instance, suppose the probability of

match has a much stronger decrease for higher types than for lower types, implying a decrease inF in first order stochastic. In this case, as b(η) is decreasing in η, a the decrease in F in

first order stochastic (General Equilibrium Effect) would have a positive impact onbavg. It is

possible, in certain circumstances, that this effect outweighs the Partial Equilibrium Effect, making the overall impact on bavg of an increase in σ2 a positive one.

5.1

Continuous Distribution of Types

indexing the distribution of types, as well as the equilibrium levels of bonus and queue length by the risk aversion coefficient η of the workers. However, I can also write η(b) =˜ 1−b

bσ2 being the type of worker that will get bonus b if hired; and use my model to find λ(b)˜ , the queue

length that forms in the market for workers that gets bonusb. Writing in this form is useful

for my purpose as _λ(b)˜ _{does not vary with σ}2_{, as can be seen below in the equation that} determines_λ(b)˜ _.

H(˜λ) = V

′ b

2 −

k m(˜λ)

V b

2 −

k m(˜λ)

− m(˜λ)

r+1

k(1−m(˜λ)r₎ = 0 (5.1)

FromG, I can write the distribution of workers in the economy that potentially get bonus

equal to or less thanb ∈[bmin,1)5 as 6:

˜

G(b) = G

1−2k

2kσ2

−G 1−b bσ2

G 1−2k

2kσ2

, with density g(b) =˜

g 1−b bσ2

b2_σ2_G 1−2k

2kσ2

The distribution of bonus levels among the hired workers in this economy, in turn, can be written as:

˜ F(b) =

ˆ b

bmin

p(λ(ˆb)) d ˜G(ˆb)

ˆ 1

bmin

p(λ(ˆb)) dG(ˆb) =

ˆ b

bmin

p(λ(ˆb))˜g(ˆb) dˆb

ˆ 1

bmin

p(λ(ˆb))˜g(ˆb) dˆb

Now the average bonus level among the hired workers is the expected value of this random variable b with distribution _F˜_.

bavg =

ˆ 1

bmin

ˆ_{b d ˜F(ˆb)}

When there is an increase in the production riskσ2_{, there is no change in}˜_{λ(b), as discussed} above. Hence, the workers that are potentially hired under bonus levelb are still hired with the same probabilityp(˜λ(b)). However, what changes is which type of worker that is now to

earn this bonus level, as when σ2 _{increases each type is now associated with a lower bonus.} This effect changes the distribution_F˜ _{of bonus among the hired workers by changing the}

distribution_G˜_{. On the one hand, each type of worker is now associated with a lower bonus,}

causing an increase in the measure of workers with lower bonus levels. On the other hand, 5

Workers with bonus lower than bmin := 2k are those with η > 1−2k

2kσ2 that are not served by any firms. On the other hand, the bonus level goes to 1 as η goes to 0.

6

G 1−2k

2kσ2

is the total measure of workers that potentially get served in this economy, as the free entry condition guarantees that workers with η > 1−2k

2kσ2 cannot be searched for. G 1−

b bσ2

the workers that were served but had really high risk aversion and were associated with really low bonus levels now cease to be served and are out of the market, causing a decrease in the measure of workers with lower bonus levels. As Proposition 1 will show, the overall direction of this effect can be determined by the characteristics of the distribution of types G in this

economy.

Proposition 4. Let ǫg(η) := g

′_(η)η

g(η) be the elasticity of the density g.

1. If ǫg(η) is strictly increasing in η, then an increase in σ2 causes F˜ to increase in first

order stochastic and bavg to increase.

2. If ǫg(η) is constant in η, then an increase in σ2 causes has no effect on F˜ or bavg.

3. If ǫg(η) is strictly decreasing in η, then an increase in σ2 causes F˜ to decrease in first

order stochastic and bavg to decrease.

Proof. Proof is in the appendix.

5.2

Two Types

Proposition 4 shows that, in many cases when G is a continuous distribution of types, the

search frictions in the economy are not important in determining the effect of an increase in production risk on the average bonus level among the hired workers. However, turning to a the case where there are only two types of workers, search frictions now become important. There is a measureαof workers with lower risk-aversion,η1, and a measure(1−α)of workers

with higher risk aversion, η2. The expression for bavg in this case is:

bavg(σ2) =

αp(λη1(σ

2₎₎ 1

1+η1σ2

+ (1−α)p(λη2(σ

2₎₎ 1

1+η2σ2

αp(λη1(σ2)) + (1−α)p(λη2(σ2))

As the riskσ2 _{increases, the bonus level for the two types decrease, pressuring the average} bonus downwards. As for the probabilities of matching, they also decrease for the two types, as the equilibrium queue length in both markets, λη1(σ

2_{) and λ}

η2(σ

puts an upward pressure onbavg. This is shown below.

dbavg(σ2)

dσ2 ∝

Partial Equilibrium Effect

z }| {

−αp(λη1(σ

2_{)) + (1−α)p(λ}

η2(σ

2₎₎

αp(λη1(σ

2_))η

1

(1 +η1σ2)2

+(1−α)p(λη2(σ

2_))η

2

(1 +η2σ2)2

+α(1−α)

1 1 +η1σ2

− 1

1 +η2σ2

p(λη2(σ

2₎₎dp(λη1(σ

2₎₎

dσ2 −p(λη1(σ

2₎₎dp(λη2(σ

2₎₎

dσ2

| {z }

General Equilibrium Effect

As discussed in the characterization session, when the market is completely frictionless (r → +∞), the probability of matching of the workers is not affected by either the risk

aversion levelη or the production riskσ2 7_{. This explains the next proposition.}

Proposition 5. When the market is frictionless, r → +∞, the average bonus among the hired workers is decreasing in the production risk level, dbavg(σ2)

dσ2 <0.

Proof. Proof is in the appendix.

When the market is perfectly frictionless, the proportion of each type of worker among the hired workers does not change when there is an increase in the riskσ2_{. This implies that} the General Equilibrium Effect does not take place and the overall impact of the increase in risk amounts to the Partial Equilibrium Effect, which always points in the negative direction. In case where the market becomes too frictional,r →0, the probability of matching for either

type of worker goes to 0, implying that jobs are not formed and contracts are not established and that it makes little sense to consider the average bonus.

When there are search frictions, but in a low enough level, General Equilibrium Effects become important. Proposition 6 shows that, if type 2 workers are sufficiently risk averse, the General Equilibrium Effect is positive and outweighs the Partial Equilibrium Effect. Proposition 6. Let r >1 and hold η1 constant at η1 = ˜η1. Then there exists η¯2 ∈( ˜η1,1₂−_kσ2k2)

high enough such that dbavg(σ2)

dσ2 >0 and η2

¯

∈( ˜η1,1₂−_kσ2k2) low enough such that

dbavg(σ2)

dσ2 <0.

Proof. Proof is in the appendix.

Similarly to the results with a continuous distribution of types, in the case with two types and sufficiently low search frictions, the direction of the effect of increasingσ2 _{on the average} bonus among the hired workers depends on how the types are distributed, particularly in this case on how high is the risk averseness of the type 2 agent.

7

As η2 becomes high, the probability that type 2 workers are hired becomes more sus-ceptible to changes in σ2_{. This makes the General Equilibrium effect positive and strong,} possibly outweighing the Partial Equilibrium Effect and hence implying that the average bonus increases withσ2_{. On the other hand, as η}

2 decreases, the difference between the two types becomes lower, making the General Equilibrium effect weaker. In the limit, as the two types become equal, the impact ofσ2 _onb

avg is only the Partial Equilibrium Effect, which is

6 Conclusion

Bibliography

1 Ackerberg, Daniel A., and Maristella Botticini. "Endogenous matching and the empirical determinants of contract form." Journal of Political Economy 110.3 (2002): 564-591.

2 Aggarwal, Rajesh K., and Andrew A. Samwick. "The Other Side of the Trade-off: The Impact of Risk on Executive Compensation." Journal of Political Economy 107.1 (1999): 65-105.

3 Chiappori, Pierre André, and Bernard Salanié. "Testing contract theory: A survey of some recent work." No. 738. CESifo Working Paper, 2002.

4 Grund, Christian, and Dirk Sliwka. "Evidence on performance pay and risk aversion." Economics Letters 106.1 (2010): 8-11.

5 Guerrieri, Veronica, Robert Shimer, and Randall Wright. "Adverse selection in competitive search equilibrium." Econometrica 78.6 (2010): 1823-1862.

6 Holmstrom, Bengt, and Paul Milgrom. "Aggregation and linearity in the provision of intertemporal incentives." Econometrica (1987): 303-328.

7 Jin, Li. "CEO compensation, risk sharing and incentives: Theory and empirical results." Manuscript. Cambridge: Massachusetts Inst. Tech (2000).

8 Moen, Espen R. "Competitive search equilibrium." Journal of Political Economy 105.2 (1997): 385-411.

9 Moen, Espen R., and Åsa Rosén. "Incentives in competitive search equilibrium." The Review of Economic Studies (2011).

10 Pandey, Priyanka. "Effects of technology on incentive design of share contracts." American Economic Review (2004): 1152-1168.

11 Petrongolo, Barbara, and Christopher A. Pissarides. "Looking into the black box: A survey of the matching function." Journal of Economic literature (2001): 390-431.

12 Prendergast, Canice. "The Tenuous Trade-off between Risk and Incentives." Journal of Political Economy 110.5 (2002): 1071-1102.

14 Serfes, Konstantinos. "Endogenous matching in a market with heterogeneous principals and agents." International Journal of Game Theory 36.3-4 (2008): 587-619.

15 Shimer, Robert. "Essays in search theory." Diss. Massachusetts Institute of Technology, 1996.

A Appendix

Proofs of Results

Proof of Lemma 2. From Lemma 1, I know that b that solves (Pη) is bη = ₁₊1_ησ2. Using this, (Pη) becomes:

max

c,λ m(λ)

ησ2

(1 +ησ2₎2 −c

−k

subject to m(λ)

λ V

c+ 1−ησ

2

2(1 +ησ2₎2

>W¯η

(A.1)

Now using w¯ =c+ 1−ησ2

2(1+ησ2₎2, I rewrite the problem again as a problem of choosing w¯ rather than cand get(P′

η).

Proof of Lemma 4. As m(λ) is a strictly increasing function of λ and m_λ(λ) a strictly

de-creasing one, the constraint in (P′

η) must be binding. I can then write w¯ in terms of λ and

solve the problem for λ.

λ ∈ (λmin,+∞), where λmin is such that ₂₍₁₊1_ησ2₎ − _m(λk

min) = 0. There is an unique

solution for λ as lim

λ→λmin

H(λ)>0,H′_(λ)<0 and lim

λ→+∞H(λ)<0.

The First Order Condition and Free Entry give, after simple algebra, the conditionH(λ) = 0in the lemma.

Proof of Proposition 1. Increasingη makes H(λ) higher for every level of λ. In turn, as H′

(λ) < 0, I conclude that λ(η) increases in η. As λ(η) incrases, it must be the case that

V′ 1 2(1+ησ2)−

k m(λ)

V 1

2(1+ησ2)

also increases, which will happen when w(η) =¯ 1

2(1+ησ2₎ −

k

m(λ) decreases. Hence, w(η)¯ is decreasing in η.

Proof of Proposition 2. Using the CARA utility specification forV, the equilibrium

con-ditionH(η) = 0 becomes:

˜

H(λ) = ϕ exp

−ϕ ₂₍₁₊1_ησ2₎ − _mk_(λ)

1−exp−ϕ 1

2(1+ησ2₎ − _mk_(λ)

− m(λ)

r+1

k(1−m(λ)r₎ = 0 (A.2)

(7) gives that the higher ϕ, the lower H(λ)˜ is, for any given level of λ(η) and any η, and

Proof of Proposition 3. Using V( ¯w) = ¯w, the equilibrium condition now becomes: λr+1

(λr_{+ 1)}r+1r

=m(λ)r+1 _{= 2k(1 +ησ}2_{) (A.3)}

From this condition, as r → 0, in equilibrium m(λ) → 1 and λ → 1, implying also that p(λ)→1.

In the second case, asr→+∞ the probability that a firm finds a match,m(λ), tends to 2k(1 +ησ2_{)and w¯ →0, while λ →+∞, making the probability that a worker finds a match}

p(λ) = m_λ(λ) →0.

Proof of Proposition 4. I am going to prove the first statement of Proposition 1. The

second ad third statements can be proved in a similar fashion.

Supposeǫg(η)is strictly increasing inη. I want to show thatF˜ is increasing (in first order

stochastic) inσ2_{, implying thatb}

avg is also increasing in σ2.

˜ F(b) =

ˆ b

bmin

p(λ(ˆb))˜g(ˆb) dˆb

ˆ 1

bmin

p(λ(ˆb))˜g(ˆb) dˆb

, where g(b) =˜ g

1−b bσ2

b2_σ2_G 1−2k

2kσ2

⇒ d ˜F(b)

dσ2 ∝

ˆ b

bmin

p(λ(ˆb))d˜g(ˆb) dσ2 dˆb

ˆ 1

bmin

p(λ(ˆb))˜g(ˆb)dˆb

−

ˆ 1

bmin

p(λ(ˆb))d˜g(ˆb) dσ2 dˆb

ˆ b

bmin

p(λ(ˆb))˜g(ˆb)dˆb

I write d˜g(b)

dσ2 asκ(b)˜g(b), whereκ(b) = d˜g(b)

dσ2 _g˜(1_b). I claim (and show at the end of this proof) that κ(b) is strictly increasing in b. Using this, I conclude that, for all b∈(bmin,1):

d ˜F(b) dσ2 ∝

ˆ b

bmin

p(λ(ˆb))κ(ˆb)˜g(ˆb)dˆb

ˆ 1

bmin

p(λ(ˆb))˜g(ˆb)dˆb

−

ˆ 1

bmin

p(λ(ˆb))κ(ˆb)˜g(ˆb)dˆb

ˆ b

bmin

p(λ(ˆb))˜g(ˆb)dˆb

< κ(b)

ˆ b

bmin

p(λ(ˆb))˜g(ˆb)dˆb

ˆ 1

bmin

p(λ(ˆb))˜g(ˆb)dˆb

−κ(b)

ˆ 1

bmin

p(λ(ˆb))˜g(ˆb)dˆb

ˆ b

bmin

p(λ(ˆb))˜g(ˆb)dˆb = 0

⇒ d ˜F(b)

It is easy to see that for b = bmin and b = 1, d ˜ F(b)

dσ2 = 0. Hence, F˜ is increasing (in first order stochastic) in σ2_{, implying trivially thatb}

avg =

´1

bmin

ˆ_{b d ˜F(ˆb) is also increasing in σ}2_. Now I will show that κ(b) is strictly increasing in b.

κ(b) = d˜g(b) dσ2

1 ˜

g(b) =−

1 σ2 −

1 σ2

1−b bσ2

g′ 1−b bσ2

g 1−b bσ2

+

1 σ2

1 G 1−2k

2kσ2

1−2k 2kσ2

=− 1

σ2

1 +ǫg

1−b bσ2

− 1

G 1−2k

2kσ2

1−2k 2kσ2

Asǫg(b) is increasing in b, and 1_bσ−2b decreasing in b, κ(b) is strictly increasing in b.

Proof of Proposition 5. As seen above:

dbavg(σ2)

dσ2 ∝ −

p(λη1(σ

2_{)) +p(λ}

η2(σ 2₎₎

p(λη1(σ

2_))η

1

(1 +η1σ2)2

+p(λη2(σ

2_))η

2

(1 +η2σ2)2

+

1 1 +η1σ2

− 1

1 +η2σ2

p(λη2(σ

2₎₎dp(λη1(σ

2₎₎

dσ2 −p(λη1(σ

2₎₎dp(λη2)

dσ2

where p(λ) = m(λ)

λ ⇒p( ˆ˜ m) = ˆ m

m−1_{( ˆm)} and m(λη(σ

2_{)) =2k(1 +ησ}2₎1+1r

dm(λη(σ2))

dσ2 =

2kη 1 +r

2k(1 +ησ2)1+rr

⇒ lim

r→+∞

dm(λη(σ2))

dσ2 = 0⇒_r→lim+∞

dp(λη(σ2))

dσ2 = 0 for η∈ {η1, η2}

Also, lim

r→+∞p(λη(σ

2_{)) = 1, for η∈ {η} 1, η2}

⇒ lim

r→+∞

dbavg

dσ2 ∝_r→lim+∞−

p(λ(η1)) +p(λ(η2))

p(λ(η1))η1

(1 +η1σ2)2

+ p(λ(η2))η2 (1 +η2σ2)2

=−2

η1

(1 +η2σ2)2

+ η2

(1 +η2σ2)

<0

Proof of Proposition 6. Letr >1 and hold η1 constant at η1 = ˜η1.

p(λη(σ2)) =

1− 2k(1 +ησ2)r+1r 1

r

⇒ dp(λη(σ

2₎₎

dσ2 =

−2kη r+ 1

1− 2k(1 +ησ2)r+1r 1−r

r

2k(1 +ησ2)r−+11

First, i will show that there existsη¯2 ∈( ˜η1,1₂−_kσ2k2) high enough such that

dbavg(σ2)

(i) p(λη2(σ

2_{))is decreasing in η}

2 and lim

η2→1−2k 2kσ2

p(λη2(σ

2_{)) = 0}

(ii) dp(λη2(σ 2₎₎

dσ2 is decreasing in η2 and lim

η2→1−2k 2kσ2

dp(λη2(σ 2₎₎

dσ2 =−∞

(i) and (ii) imply that dbavg(σ2)

dσ2 is increasing in η2 and lim

η2→1−2k 2kσ2

dbavg(σ2)

dσ2 = +∞

⇒∋η¯2 ∈( ˜η1,

1−2k

2kσ2 ) such that

dbavg(σ2)

dσ2 >0

Now it is easy to see that, as η2 →η˜1, the general equilibrium effect is lowered and dbavg(σ 2₎ dσ2 decreases.

lim

η2→η˜1

dbavg

dσ2 ∝_ηlim₂→η˜1

−p(λ(˜η1)) +p(λ(η2))

p(λ(˜η1))˜η1

(1 + ˜η1σ2)2

+ p(λ(η2))η2 (1 +η2σ2)2

=−p(λ(˜η1)) +p(λη˜1))

p(λ(˜η1))˜η1

(1 +η1σ2)2

+ p(λ(˜η1))˜η1 (1 + ˜η1σ2)2

<0

⇒∋η2 ¯

∈( ˜η1,

1−2k

2kσ2 ) such that

dbavg(σ2)

dσ2 <0

Proof of Theorem 1 - Existence and Uniqueness of Equilibrium. Let(Pη)be the

fol-lowing constrained optimization problem, withWη(c, b, λ) as defined in (1) and Jη(c, b, λ)as

defined in (2):

max

c,b,λ Wη(c, b, λ)

subject to Jη(c, b, λ)>0

(Pη)

Now consider the larger problem(P) of solving (Pη)for all η. More precisely, the set I∗

In step 1, I establish that any competitive search equilibrium solves the contained opti-mization problem (P) and, in step 2, i show that any allocation that solves this program is

a part of an equilibrium. Then I follow to step 3 and show that (P) has an unique solution.

STEP 1

Let{(Ψη)η∈(0,η¯],( ¯Wη)η∈(0,η¯],(λη)η∈(0,η¯]}be an equilibrium allocation with(c∗η, b

∗

η)∈Ψη and

λ∗

η =λη(c∗η, b

∗

η). We must prove that, for every η ∈ (0,η]¯, (c

∗

η, b

∗

η, λ

∗

η) solves the constrained

optimization problem(Pη). First, the free entry condition in the equilibrium definition implies

that (c∗

η, b

∗

η, λ

∗

Wη(cη, bη, λη)> Wη(c∗η, b

∗

η, λ

∗

η)must not satisfy the constraint and hence that(c

∗

η, b

∗

η, λ

∗

η)solves

the constrained maximization.

Assume by way of contradiction that there exists (c′

η, b

′

η, λ

′

η) such that Wη(c′η, b

′

η, λ

′

η) >

Wη(c∗η, b

∗

η, λ

∗

η) and Jη(c′η, b

′

η, λ

′

η)) > 0. Then, by continuity of Wη, there exists λ′′η > λ

′

η such

that Wη(c′η, b

′

η, λ

′′

η) > Wη(c∗η, b

∗

η, λ

∗

η) and Jη(c′η, b

′

η, λ

′′

η) > 0, because Jη is strictly increasing

in λη. Hence, Jη(c′η, b

′

η, λ

′′

η) > Jη(c∗η, b

∗

η, λ

∗

η) and (c

∗

η, b

∗

η, λ

∗

η) does not satisfy Maximization of

Vacancy Value, the first equilibrium condition. (CONTRADICTION)

STEP 2

Take(cη, bη, λη)η∈(0,η¯] that solves the program(P).

Define Γ := {(Ψη)η∈(0,η¯],( ¯Wη)η∈(0,η¯],(˜λη)η∈(0,η¯]} such that Ψη = {(cη, bη)} and W¯η =

Wη(cη, bη, λη) for each η ∈ (0,η]¯. Let λ˜η(c′η, b

′

η) be such that Wη(c′η, b

′

η,λ˜η(c′η, b

′

η)) = ¯Wη

for any (c′

η, b

′

η)∈R2.

Condition (2 - Optimal Search) of equilibrium is satisfied trivially by Γ. Assume by way

of contradiction that condition (3 - Free Entry) is not satisfied, that is, thatJη˜(cη˜, b˜η, λη˜)6= 0 for some η˜∈ (0,η]¯. Then either Jη˜(cη˜, bη˜, λη˜)<0 which contradicts the fact that (cη˜, bη˜, λη˜) satisfies the constraint, or Jη˜(cη˜, bη˜, λη˜) > 0. In this case, continuity of Jη˜ implies that there exists λ′

˜

η < λ˜η such that Jη˜(cη˜, bη˜, λ′η˜) > 0 and Wη˜(cη˜, bη˜, λ′η˜) > Wη˜(c˜η, bη˜, λη˜), which contradicts the fact that (cη˜, bη˜, pη˜)solves problem (Pη˜).

Now I still need to show that condition (1- Profit Maximization) is satisfied. Assume by contradiction that, for someη ∈ (0,η]¯, there exists (c′

η, b

′

η, λ

′

η) such that Wη(c′η, b

′

η, λ

′

η)>

¯

Wη and Jη(c′η, b

′

η, λ

′

η) > Jη(cη, bη, λη). Wη being strictly decreasing in λη and continuity

of Jη imply that there exists λ′′η < λ

′

η such that Wη(c′η, b

′

η, λ

′′

η) > W¯η = Wη(cη, bη, pη) and

Jη(c′η, b

′

η, λ

′′

η)>Jη(cη, bη, λη)>0. This contradicts the fact that(cη, bη, λη)solves the problem

(Pη).

STEP 3