Bonus Payments versus Efficiency Wages in the Repeated Principal-Agent Model with Subjective...

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American Economic Journal: Microeconomics 2012, 4(3): 34–56 http://dx.doi.org/10.1257/mic.4.3.34

Bonus Payments versus Efficiency Wages in the Repeated Principal-Agent Model with Subjective Evaluations†

By Lucas Maestri*

We study an infinitely repeated principal-agent model with subjec-tive evaluations. We compare the surplus in efficiency-wage equilib-ria and in bonus-payments equilibria. The agent receives a constant wage and is motivated by the threat of dismissal in efficiency-wage equilibria. The agent receives a bonus and quits the relationship after disagreements between his self-evaluation and the principal’s performance appraisal in bonus-payments equilibria. We construct a class of equilibria with bonus payments that approach efficiency as patience increases. In contrast, payoffs from efficiency-wage equi-libria are bounded away from the Pareto-payoff frontier for any dis-count factor. (JEL D82, J33, J41)

Most of the agency literature focuses on jobs for which there are objective measures of output. However, the principal and the agent may have different

assessments about the agent’s work. This asymmetry of information may tempt the principal to minimize her expenditures by underreporting the agent’s performance and consequently paying him a lower bonus. Aware of that, the agent may be less motivated to work. Hence, principal-agent models with subjective evaluations give rise to interesting strategic behavior.

We study a repeated principal-agent model with subjective evaluations. The agent exerts an effort which leads to the output distribution in each period. The principal privately observes the output, while the agent observes a correlated signal. We compare the efficiency of two important classes of equilibria: bonus payments and efficiency wages. The principal motivates the agent by the promise of a bonus in the case of a good evaluation in bonus-payments equilibria. The agent receives a constant wage and is entirely motivated by the threat of being fired in efficiency-wage equilibria.

Our first main contribution is to show that (asymptotic) efficiency can be obtained under bonus payments. We construct equilibria in which the agent receives a bonus if and only if the principal observes good performance. The bonus payments in this class of equilibria resemble the bonuses used in environments with objective evalu-ations. However, our construction entails an important difference: the agent uses his own private information to review the principal’s evaluation every T periods.

* Toulouse School of Economics, Manufacture des Tabacs, 21 Allee de Brienne, 31000 Toulouse, France (e-mail: [email protected]). I am very grateful to Larry Samuelson for his encouragement and advice. I also thank Dino Gerardi, Johannes Hörner, and Priscila Souza. Errors are my own.

† To comment on this article in the online discussion forum, or to view additional materials, visit the article page at http://dx.doi.org/10.1257/mic.4.3.34.

ContentsBonus Payments versus Efficiency Wages in the Repeated Principal-Agent Model with Subjective Evaluations 34I. Model 36II. Bonus Payments 38A. Strategies 38B. Equilibrium 39III. Efficiency Wages 42A. Preliminaries 42B. The Game with Communication 42C. The Bound on Efficiency 44IV. Discussion 45Appendix 46References 55

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VoL. 4 No. 3 35MAEsTrI: BoNus PAyMENTs

If the agent’s signals and the principal’s rewards disagree in every period of this review phase, then the agent disciplines the principal by quitting the relationship. Otherwise, the agent remains in the job and waits T more periods to assess the prin-cipal’s evaluation again. We show that payoffs in this class of equilibria approach the Pareto-payoff frontier as patience increases. This demonstrates that bonus pay-ments can perform well under subjective evaluations when the parties are patient.

Our second main contribution consists in showing that the surplus of efficiency-wage equilibria is always bounded away from the Pareto-payoff frontier. We show that the incentive provision implies that the agent must expect being fired with a higher probability after observing a bad signal. Accordingly, a certain amount of surplus must be destroyed every time that the agent learns negative information about his performance. This implies a minimal amount of surplus destruction per period. As a consequence, we obtain a lower bound on the surplus of efficiency-wage equilibria which is independent of the discount factor.

It is not hard to show that the use of bonus payments are optimal under objective evaluations and that efficiency wages are always suboptimal. Our paper shows that this result is asymptotically robust to subjective evaluations. That is, bonus pay-ments always dominate efficiency wages when the parties are patient.

Agency with subjective evaluation has been recently studied by Levin (2003) and MacLeod (2003). Levin (2003) studies a repeated agency framework in which the principal privately observes the output. He restricted attention to equilibria with the “Full Review Property.” That is, all private information is revealed after every period in equilibrium. The agent is motivated by the threat of dismissal in the optimal equilibrium with this property. MacLeod (2003) examines a static environment in which the principal and the agent observe signals of the outcome of the agent’s effort. He discusses how equilibria in the static game can be used to construct equilibria in an infinite-horizon model in which all private information is revealed after every period. Under this restriction, optimal equilibria entail the use of performance pay and the agent uses a quitting device to compel the principal to truthfully reward him. Both papers restrict attention to equilibria in which all relevant private information is revealed after every period. Because of that, these papers find that the expected surplus is bounded away from the first-best surplus for any discount factor. Our paper departs from these studies by looking at equilibria in which the parties may aggregate private information over many periods before revealing it.

Fuchs (2007) studies a repeated principal-agent model in which the principal privately observes the output, while the agent does not observe any signal of the outcome of his effort. Fuchs analyzes equilibria in which the principal aggregates the information acquired in many periods before evaluating the agent. He finds that the surplus-maximizing equilibria are achieved by efficiency wages. The principal should pay the agent a constant wage, give him no feedback about his performance and discipline him by the threat of dismissal. Furthermore, Fuchs constructs a class of efficiency-wage equilibria in which payoffs approach the Pareto-payoff frontier as patience increases.

In the class of equilibria analyzed by Fuchs, the principal aggregates the infor-mation of many periods (review phase) before deciding whether to fire the agent. In his construction, the agent is fired only if the principal observed a low output

36 AMErIcAN EcoNoMIc JourNAL: MIcroEcoNoMIcs AugusT 2012

in each period of the review phase. Since the agent observes no signal about his performance, he is kept ignorant about the likelihood of punishment over the entire review phase. The agent believes that each period may be pivotal and trigger a high punishment if it leads to a low output. It is possible to make each review phase longer and to destroy a lower amount of surplus from earlier termination as the discount factor increases. This equilibrium construction hinges on the agent learning no information along the review phase. Suppose that the agent observes a signal which is correlated to the output. After observing many good signals within a block, the agent attributes very small probability to being fired in the end of the block. Hence, the amount of surplus destruction due to earlier termination must be very large to keep the agent motivated to exert effort for the remainder periods of the block. This limits the rate at which one can increase the length of the block as the discount factor increases. As a consequence, the equilibrium surplus is bounded away from the first-best one.

Fuchs’ paper is complimentary to ours. For a given discount factor, efficiency wages perform well if the principal and the agent’s information are not very corre-lated. On the other hand, if the principal and the agent’s signals are highly correlated and the discount factor is high, then the use of bonus payments brings large gains in efficiency relatively to efficiency-wages.

The model is presented in Section I. Bonus-payments equilibria are analyzed in Section II. Efficiency-wage equilibria are studied in Section III. Section IV dis-cusses the results. Omitted proofs are in the Appendix.

I. Model

A principal (henceforth she) and an agent (henceforth he) are involved in a repeated game with option of exit. Time evolves discretely: t = {0, 1, …}. Both par-ties discount payoffs by δ ∈ (0, 1).

If the principal and the agent remain in the relationship in period t, the agent chooses an effort level e t ∈ [ 0,

_ e ], _ e < 1. Effort entails a strictly increasing, strictly

convex and differentiable cost, c, with c (0) = c′ (0) = 0 and lim e↑ _ e c′ (e) = ∞. The relationship generates an output every period y t , which can be low ( y L ) or high ( y H ). The level e t of effort leads to high output with probability e t .

The principal privately observes the output. The agent observes a signal in period t, x t A , which can be low ( x L A ) or high ( x H A

).Assumption 1 below characterizes the signal distribution.

ASSUMPTIOn 1: conditional on low (high) output the agent observes a high sig-nal with probability α L ( α H ) satisfying

0 < α L < α H < 1.

Assumption 1 states that the agent observes a correlated signal of his perfor-mance. The case in which α L = α H is in Fuchs (2007).

Each party makes a nonnegative transfer to the other at the end of each period. We write τ t P for the transfer made by the principal in period t and τ t A for the transfer


made by the agent in period t. We assume that both parties are risk-neutral, expected-utility maximizers and have outside option equal to 0.

The principal’s payoff is given by

(1) u P ( { y t , τ t A , τ t P , e t } t=0 ∞ ) ≜ (1 − δ ) ∑

t=0

∞

δ t [ y t + τ t A − τ t P ],

while the agent’s payoff is given by

(2) u A ( { y t , τ t A , τ t P , e t } t=0 ∞ ) ≜ (1 − δ ) ∑

t=0

∞

δ t [ τ t P − τ t A − c ( e t )].

The surplus of the relationship, denoted by s * , is given by

(3) s * ≜ max e∈[0,

_ e ] E [ y | e ] − c (e) > 0.

Furthermore, we write e * for the most efficient effort level: e * = arg ma x e∈[0,

_ e ]

E [ y | e ] − c (e). Since c′ (0) > 0 and y H > y L we have e * > 0.The game unfolds as follows within each period. (i) The agent chooses an effort

e t ∈ [0, _ e ]. (ii) The agent observes his private signal and the principal observes the

product. (iii) Each party makes a voluntary transfer. (iv) Both parties simultane-ously decide whether to quit the relationship. If one of the parties leaves the rela-tionship in the end of a period, the game ends and each party receives a continuation payoff equal to his/her outside option. Otherwise a new period starts.

A private history of length t for the principal, h t P , consists of the outputs realized in previous periods { y 0 , … , y t−1 } and the transfers made in previous periods { τ 0 P , τ 0 A , … , τ t−1 P

, τ t−1 A }. We set h 0 P = ~. A private history of length t for the agent, h t A , consists

of the effort levels chosen in all prior periods { e 0 , … , e t−1 }, the signals received in all prior periods { x 0 A , … , x t−1 A

} and the transfers made in all prior periods { τ 0 P , τ 0 A , … , τ t−1 P

, τ t−1 A }. We set h 0 A = ~.

A pure strategy for the principal, σ P ∈ Σ P consists of a transfer function for each period τ t P : h t P × { y L , y H } → ℜ + and a quitting function for each period Q t P : h t P × { y L , y H } × ℜ + × ℜ + → {Stay, Quit}.

A pure strategy for the agent, σ A ∈ Σ A , consists of an effort function for each period e t : h t A → [ 0,

_ e ], a transfer function for each period: τ t A : h t A × [ 0, _ e ]

× { x L A , x H A } → ℜ + and a quitting function for each period Q t A : h t A × [ 0,

_ e ] × { x L A , x H A }

× ℜ + × ℜ + → {Stay, Quit}1.With a little abuse of notation we use ( σ P , σ A ) to represent mixed strategies.

Strategies ( σ P , σ A ) and private histories ( h t P , h t A ) define a continuation payoff for the principal at the beginning of period t:

(4) V P (( σ P , σ A ) | h t P ) ≜ E [(1 − δ ) ∑ s=t

∞

δ s−t [ y s + τ s A − τ s P ] | σ P , σ A , h t P ];1 We do not need to define strategies if any player has chosen to quit the relationship prior to period t, because

in this case the game is over.


and a continuation payoff for the agent at the beginning of period t:

(5) V A (( σ P , σ A ) | h t A ) ≜ E [(1 − δ ) ∑ s=t

∞

δ s−t [ τ s P − τ s A − c ( e s )] | σ P , σ A , h t A ].The solution concept adopted is perfect Bayesian equilibrium.

II. Bonus Payments

In this section, we analyze equilibria in which the principal pays the agent a bonus after good performance and no bonus after bad performance. The agent is entirely motivated by the perspective of receiving a bonus in these equilibria. Moreover, the agent disciplines the principal by the threat of quitting the relationship. We call these equilibria bonus-payments equilibria.

A. strategies

We construct equilibria in stationary review strategies.2 A T-period stationary review strategy is defined as follows: the agent works for the principal for a pre-determined number of periods T < ∞. In each of these periods, we say that the principal pays a bonus if τ t P > 0. After the agent has worked for the principal for T periods, the agent uses the private information acquired in the last T periods to choose whether to quit the relationship or not. If the agent does not quit, the prin-cipal hires the agent for T more periods and the relationship restarts with a clear record. We call each subset of T periods a block. We work with equilibria in which both players play the same strategy in each block. Thus, with a little abuse of nota-tion, we write h t P ( h t A ) for the private history acquired by the principal (agent) in the first t periods of each block.

The agent chooses effort e = 0 in period 0 and effort e * in all periods t ∈ {1, … , T − 1}.3 The agent will be required to make a time invariant transfer to the principal τ t A = λ if t ∈ {0, … , T − 2} and τ t A = 0 if t = T − 1. The principal makes no transfer to the agent in period 0. In every period that the agent is supposed to exert effort (t > 0), the principal chooses whether to pay the agent a time invariant bonus B > 0 or not. That is, we have τ 0 P = 0 and τ t P ∈ {0, B} for all t ∈ {1, … , T − 1}.

In bonus-payments equilibria, the agent motivates the principal to pay a bonus by the threat of quitting the relationship. Therefore, the decision rule Q T A is essential to the analysis. It is convenient to define a scoring rule in our construction of Q T A . A scoring rule is a device used by the agent to assess the principal’s evaluation. Formally:

DEFInITIOn 1: A scoring rule is Ξ = { θ t } t=1 T−1 , with θ t : { x L A , x H A } × {0, B} →

Δ ({0, 1}).

2 This follows several papers which analyze repeated games with private monitoring such as Matsushima (2004).3 The induced inefficiency will vanish as the length of the blocks increases (which will happen as the parties

become more patient.


According to Definition 1, in period t the agent uses the signal observed and the transfer made by the principal to generate an element of {0, 1}. This corresponds to a score for that period. The number 0 corresponds to a bad score and 1 to a good one. Thus, the scoring rule Ξ conveys information regarding the principal’s evaluation of each one of the periods.

In our construction, the agent uses only the information in the scoring rule to decide between quitting or not the relationship in the end of the block. More pre-cisely, we have Q T A : {0, 1} T−1 → {0, 1}. We will require Q T A ((0, … , 0)) = 0 and Q T A (Ξ) = 1 if Ξ ≠ (0, … , 0). That is, the equilibrium strategy will request the agent to quit the relationship if and only if his scoring rule attributes the lowest score to the principal in all periods of the block (except for period 0, when no score is given).

The scoring rule depends on whether α L ≤ 1 − α H or α L > 1 − α H . We consider the case in which α L ≤ 1 − α H here. The other case can be handled analogously.

The equilibrium scoring rule Ξ * is determined according to the following strategy:

Pr ( θ t * = 0) =

⎧⎪⎨⎪⎩

0

1

( α L

_ 1 − α H )

if ( x t A , τ t P ) ∈ {( x L A , 0), ( x H A , B)}

if ( x t A , τ t P ) = ( x H A , 0)

if ( x t A , τ t P ) = ( x L A , B).

The scoring rule defined above prescribes that in period t the agent gives a score 1 to the principal when he receives a bonus and observes a high signal or when he receives no bonus and observes a low signal. In the other two cases, that is, when the principal paid the agent a bonus and he observed a low signal, and when the principal paid him no bonus and he observed a high signal, the agent gives a score of 0 to the principal with positive probability. The idea behind this construction is to let the agent use his private information to punish the principal after events indicating higher like-lihood of deviation (for example, not paying a bonus after observing a high signal).

We should explain why the agent may punish the principal even when the latter paid him a bonus in a certain period t. The reason is that we would like to leave the principal unaware about the realization of Ξ * = {0, … , 0} after any sequence of outputs, including a sequence of good ones. Consider a hypothetical scoring rule which prescribes that the principal is never punished if she pays a bonus. Hence, she will be sure that the event Ξ = (0, … , 0) will not happen after paying a bonus in the first period. In this case, she will have no incentive to pay a bonus to the agent in the future periods of the block. Our construction rules out this possibility.

Finally, notice that the agent does not give a score to the principal in period 0. This is due to the fact that the agent is not supposed to choose high effort in the first period of the block and the principal is not supposed to give him a transfer in that period.

B. Equilibrium

Consider a strategy σ P * that induces the payment of a bonus to the agent in period t > 0 if and only if y H is observed in that period. Lemma 1 shows that if the principal


follows σ P * she will remain ignorant about the realization of Ξ * = {0, … , 0} after any output realization along the block.

LEMMA 1: Assume that the agent chooses e t = e * for all t ∈ {1, … , T − 1} and fol-lows the scoring rule Ξ * to quit the relationship. If the principal follows the strategy σ P * then Pr ( Ξ * = {0, … , 0} | h t

P ) = Pr ( Ξ * = {0, … , 0} |

_ h _ t P ) for any pair of (on-path)

histories h t P , _ h _ t P .

The principal has two possible deviations in each period: pay a bonus when she observes y L and pay no bonus after observing y H . Lemma 2 shows that both devia-tions increase the probability of Ξ * = {0, … , 0}. Therefore, the principal will never choose the first deviation. We show that the marginal increase in the probability of Ξ * = {0, … , 0} is increasing in the number of deviations in Lemma 2. Hence, if it is never optimal for the principal to deviate once during a block, then there is no profitable deviation.

LEMMA 2: Assume that the agent chooses e t = e * for all t ∈ {1, … , T − 1} and follows the scoring rule Ξ * to quit the relationship. consider any pair of histories h T

P− , _ h T P− and assume that h T

P− differs from _ h T P− in only one of the following: (i) In

some t ′ ∈ {1, … , T − 1} the principal observed y L and paid a bonus in h T P− but not in

_ h T P− ; (ii) In some t ′ the principal observed y H and paid no bonus in h T

P− , but she paid a bonus in

_ h T P− . Then

(6) Pr ( Ξ * = {0, … , 0} | h T P− ) − Pr ( Ξ * = {0, … , 0} |

_ h T P− )

≥ (min {( α H _ α L ), ( 1 − α L

_ 1 − α H )} − 1) α L T−1 > 0.

Before checking other incentive constraints, we construct the payments B and λ. They are set such that: (i) the promise of being rewarded after good performance motivates the agent to choose high effort in every period; (ii) the agent obtains zero expected surplus in each period. In order to accomplish (i), we need

(7) e * = arg max ̃ e ̃ e B − c ( ̃ e ).

Therefore, we set B = c′ ( e * ). notice that the agent makes a transfer λ to the prin-cipal in periods { 0, … , T − 2}, while he exerts effort in periods {1, … , T − 1}. Thus, in order to accomplish (ii), we need

(8) − λ + δ [ e * c′ ( e * ) − c ( e * )] = 0.

Hence, we set λ = δ [ e * c′ ( e * ) − c ( e * )].In equilibrium, if the agent does not pay the transfer to the principal in any period

t ≠ T − 1, then the latter quits the relationship in the end of the same period. Hence,


incentives to pay the transfer are (weakly) satisfied. From (7) and (8), the agent is indifferent between quitting or not in the beginning of each T-periods block. Therefore, we can assume that the agent follows the scoring rule Ξ * to quit the rela-tionship. We write σ A * for the agent’s strategy according to which the agent chooses effort e * after every (on-path) history and quits the relationship in the end of each block according to the scoring rule Ξ * .

We continue with the analysis of the principal’s incentives to follow σ P * . Let s T denote the expected surplus of each block: s T = (1 − δ ) ∑ t=1

T−1 δ t s * . notice that under the equilibrium strategies ( σ P * , σ A * ) the relationship is dissolved at the end of each block with probability α L T−1 . Hence, the value of the relationship for the princi-pal in the beginning of each block under ( σ P * , σ A * ), denoted V T P , satisfies

(9) V T P = s T + δ T (1 − α L T−1 ) V T P .

We say that a strategy ̃ σ P is an “one-period” deviating strategy if its on-path behavior prescribes that the principal deviates at most once during a block. Consider the “one-period” deviating strategy ̃ σ P according to which the principal never pays a bonus in period 1. Also, ̃ σ P prescribes that if t > 1 the principal pays a bonus if and only if a high signal is observed. It is straightforward to verify that since δ < 1 this is the most profitable “one-period” deviation. not paying the bonus in period 1 increases the likelihood of Ξ * = {0, … , 0} by [( α H

_ α L ) − 1] α L T−1 . Therefore, there is no “one-period” deviation if

(10) [( α H _ α L ) − 1] α L T−1 δ T−1 V T P ≥ B (1 − δ ).

Moreover, from Lemma 2 the marginal increment in the probability of Ξ * = { 0, … , 0} is increasing in the number of past deviations. It follows that (10) is necessary and sufficient for the optimality of σ P * . In Proposition 1, we check that the conditions above are satisfied for high discount factors, which guarantees the existence of bonus-payments equilibria.

The relationship becomes more profitable for the principal relative to the cost of paying a bonus (1 − δ )B as the parties become more patient. One can then increase the length of the block T and consequently the efficiency of the equilibrium. As a result, we can always find δ * ∈ (0, 1) such that for all δ > δ * the principal’s expected payoff V T P is close to s * . This result is formally stated in Proposition 1 below.4

PROPOSITIOn 1: For every ε > 0, there exists δ * ∈ (0, 1) such that, for all δ > δ * there exists a PBE ( σ P * , σ A * ) such that V A (( σ P * , σ A * ) | h 0 A ) + V P (( σ P * , σ A * ) | h 0 P ) ≥ s * − ε.

4 This construction borrows a lot from previous results. It is inspired by the works of Compte (1998), Matsushima (2004), and Obara (2009).


III. Efficiency Wages

A. Preliminaries

The purpose of this section is to show that correlated private information imposes a bound on the expected surplus of efficiency-wage equilibria. We start by defining efficiency-wage equilibria.

DEFInITIOn 2: A PBE ( σ P , σ A ) is an efficiency-wage equilibrium if on the equi-librium path:

(i) τ t A = 0 for all t ≥ 0;

(ii) For every pair ( ̂ h t P , y t ) and ( ̃

h t P , ̃ y t ) such that the worker is not fired in the end

of period t we have τ t P ( ̂ h t P , y t ) = τ t P ( ̃

h t P , ̃ y t );

(iii) The agent chooses e t = e * with probability 1 after every history in which he is hired;

(iv) The agent never quits the relationship.

The class of equilibria specified in Definition 2 contains the class of equilibria shown to be optimal in Fuchs (2007) in a model in which the principal privately observes the output, but the agent does not observe any signal. According to (i) and (ii), the agent receives a transfer that depends only on the duration of the relation-ship. Requirements (iii) and (iv) impose that it is optimal for the agent to choose the efficient effort after every history until he is fired by the principal.

Lemma 3 shows that given an efficiency-wage equilibrium we can always find another efficient-wage equilibrium which yields the same expected surplus and in which the agent receives a constant wage

_ τ P = δE [ y | e * ] until he is fired. A similar result is contained in Fuchs (2007), Theorem 1.

LEMMA 3: For every efficiency-wage equilibrium in which τ t P is not constant for all t, there exists another efficiency-wage equilibrium yielding the same expected surplus in which τ t P = δE [ y | e * ] =

_ τ P for all t such that the relationship persists in t + 1.

For the rest of this section, we restrict attention to equilibria satisfying (i)−(iv) in which τ t P =

_ τ P in every period t.

B. The game with communication

In order to obtain an upper bound to the surplus of efficiency-wage equilibria, we assume the existence of a benevolent mediator who is capable of enforcing transfers


and with whom the parties can communicate.5 This enables us to invoke the revela-tion principle for dynamic games (Myerson 1986).

We modify our original model as follows. The parties communicate simultaneously with a mediator. In any period t, the principal sends a message m t P ∈ { y L , y H } to the mediator immediately after observing the product. The agent sends a message m t A ∈ { x L A , x H A

} to the mediator immediately after observing his signal.6 According to the revelation principle, we can confine attention to equilibria in which both parties inform the mediator about the true realization of their private information after every history.

In efficiency-wage equilibria, the agent pays τ t A = 0 in every period and chooses e t = e * until he is fired. Thus, we can assume that the mediator sends no recommen-dation to the agent. Also, since the principal’s transfers are history independent, we assume that the mediator only makes a recommendation to the principal in period t when the latter should fire the agent in that period.

Since the purpose of this section is to obtain an upper bound on payoffs of effi-ciency wage equilibria, we relax the problem by assuming that the principal always informs the mediator about the true realization of her signal and that she always fol-lows the mediator’s recommendations. This implies that we have two active players in this game: the mediator and the agent.

The game unfolds as follows within each period. (i) The agent chooses an effort e t . The effort stochastically determines the output and the agents’ private signal. (ii) The principal observes the output and the agent observes his private signal. (iii) Each party sends a message to the mediator. (iv) The mediator sends a message to the principal recommending to fire the agent, or the mediator sends no message to the principal. (v) The principal pays no transfer to the agent and fires him if she receives a message from the mediator. The game ends and each party receives a con-tinuation payoff equal to his/her outside option if the agent is fired. Otherwise, the principal pays

_ τ P to the agent and a new period starts.A private history of length t for the mediator, h t M , is an element of { y L , y H } t+1

× { x L A , x H A } t+1 .

The mediator commits to a strategy σ M ∈ Σ M which consists of a firing rule: t : h t M → Δ ({0, 1}), where t = 1 is a firing recommendation.

In this section, a pure strategy for the agent, σ A , consists of an effort function e t : h t A → [0,

_ e ] and a message function m t A : h t A × [0, _ e ] × { x L A , x H A

} → { x L A , x H A }.7

We write σ A ** for a strategy under which the agent chooses e t = e * for all t and always announces his private information truthfully on the equilibrium path. We write Σ A for the set of mixed-strategies for the agent.

Given a pair of strategies ( σ M , σ A ) and h t A , the agent continuation payoff is defined by

(11) V A (( σ M , σ A ) | h t A ) ≜ E [(1 − δ ) ∑ s=t

∞

δ s−t [ τ s P − c ( e s )] | σ M , σ A , h t A ], 5 Lemma 3 guarantees that in the absence of mediator it is without loss to assume that wages are constant in

efficiency-wage equilibria. We conjecture that this result remains true once a mediator is introduced.6 Since we study equilibria in which e t = e * as long as the agent is hired, it is not necessary to assume that the

agent reports his effort level.7 Since the agent never pays the principal in efficiency-wage equilibria, he does not have any incentive to quit

the relationship. Therefore, we assume (without loss) that he never quits.


and the principal continuation payoff is defined by8

V P (( σ M , σ A ) | h t A ) ≜ E [(1 − δ ) ∑ s=t

∞

δ s−t [ y s − τ s P ] | σ M , σ A , h t A ].An equilibrium is defined below:

DEFInITIOn 3: An efficiency-wage equilibrium is a strategy profile ( σ M ** , σ A ** ) such that

V A (( σ M ** , σ A ** ) | h 0 A ) ≥ V A (( σ M ** , σ A ) | h 0 A ) for every σ A ∈ Σ A .

C. The Bound on Efficiency

The main result of this section is Proposition 2 below. Proposition 2 states that the surplus of every efficiency-wage equilibrium is bounded away from the Pareto-payoff frontier. We show that under correlated private information the agent learns about the likelihood of punishment in each period. Accordingly, aggregating infor-mation of many periods before firing the agent becomes less effective for the princi-pal. As a result, we have excessive surplus destruction.

PROPOSITIOn 2: For every δ ∈ (0, 1) for every efficiency-wage equilibrium ( σ M ** , σ A ** ) we have

V A (( σ M ** , σ A ** ) | h 0 A ) + V P (( σ M ** , σ A ** ) | h 0 A ) ≤ s * − Λ,

where

Λ = c ( e * )( e * (1 − α H ) + (1 − e * )(1 − α L ) ___ 1 + e * α H + (1 − e * ) α L ) × ( e * (1 − α H ) + (1 − e * )(1 − α L )

___ e * (1 − α H ) )

× [( e * α H __ e * α H + (1 − e * ) α L )

− ( e * (1 − α H ) ___ e * (1 − α H ) + (1 − e * )(1 − α L ) )]

> 0.

8 We define the principal’s continuation payoff given the agent’s information because we did not define the principal’s information set. This is done only to simplify the notation.


For some intuition, consider the case in which the agent does not have access to communication. Hence, the firing rule t depends only on the realized output y t = { y 0 , … , y t−1 }. Write V A (( σ M ** , σ A ** ) | y t , y i ) for the expected payoff of the agent in period t given the mediator information { y t , y i }. We have

(12) V A (( σ M ** , σ A ** ) | h t A )

≜ − (1 − δ ) c ( e * )

+ ∑ y t ∈ { y L , y H } t

Pr ( y t | h t A )⎡⎢⎣

e * V A ( σ M ** , σ A ** ) | y t , y H )

+ (1 − e * ) V A (( σ M ** , σ A ** ) | y t , y L ) ⎤

⎥⎦

.

Using (12) we obtain the following necessary condition for incentive compatibility

(13) ∑ y t ∈ { y L , y H } t

Pr ( y t | h t A )⎡⎢⎣

V A (( σ M ** , σ A ** ) | y t , y H )

− V A (( σ M ** , σ A ** ) | y t , y L ) ⎤⎥⎦

> (1 − δ ) c′ ( e * ).

The agent becomes more confident on y L than y H when he observes x L . One can use (13) to show that the agent expects a higher amount of surplus destruction after this bad signal. This fact can be used to obtain an upper bound to the surplus of efficiency-wage equilibria which is independent of the discount factor. In the Appendix, we prove the result for equilibria in which the firing rule may depend on the agent’s announcements.

For an intuition, notice that the principal should fire the agent with higher prob-ability after events indicating that a lower level of effort was exerted. However, the agent learns about the likelihood of punishment since he observes a signal which is correlated with his performance. This leads to a lower bound on the difference between the agent continuation payoff after a good signal and his continuation payoff after a bad signal. We also show that this lower bound is proportional to the period’s weight: (1 − δ ). Since the principal is always indifferent between firing the agent or not, we obtain a lower bound on the difference between the partner-ship continuation payoff after a good signal and after a bad signal which is pro-portional to (1 − δ ). This immediately implies that, for any discount factor, the set of efficiency-wage equilibrium payoffs is bounded away from the feasible Pareto payoff frontier.

IV. Discussion

Every time that the agent observes a bad signal he expects to lose his job with a higher probability under efficiency wages. This leads to a lower bound on the amount of surplus destruction per period. As a result, efficiency-wage payoffs are bounded away from the Pareto-payoff frontier for any discount factor.


The agent is entirely motivated by short-run incentives under bonus payments. He uses his private information to discipline the principal to be fair in her evaluation by threatening to quit the relationship. Since quitting becomes more costly for the principal, one can increase the length of the review phase as the discount increases. Therefore, the agent aggregates more information before his quitting decision as the discount factor increases, which leads to a more powerful statistical test and conse-quently to less surplus destruction. Furthermore, the principal is kept ignorant about the likelihood of punishment. As a result, bonus payments approach efficiency as the discount factor increases.

It is known that performance pay dominates efficiency wages under objective evaluations. The model shows that this prediction remains true under subjective evaluations when the parties are sufficiently patient.

Our bound on the surplus of efficiency wages depends on the correlation between signals. Indeed, if x L and x H are sufficiently close then the bound is close to zero. Therefore, efficiency wages may perform well if the agent receives very noisy infor-mation. On the other hand, Propositions 1 and 2 imply that if the agent’s information is not very noisy and the discount factor is high then the introduction of bonus pay-ments brings large gains in efficiency.

Appendix

PROOF OF LEMMA 1: Let h t−1 P

be any private (on-path) history. Assume that the principal observed y H in period t and paid the agent a bonus in t. In this case Pr ( θ t * = 0 | h t−1 P

, y H , τ t P = B) is equal to (1 − α H )( α L

_ 1 − α H ) = α L . On the other hand, if the principal had

observed y L in period t and had not paid a bonus in that period, we would have Pr ( θ t * = 0 | h t−1 P

, y L , τ t P = 0) = α L , which concludes the Lemma.

PROOF OF LEMMA 2:Consider a history h T

P− in which the principal deviated s ∈ {0, … , T − 2} times. Let s = v + z, where v is the number of periods in which she was supposed to pay a bonus and did not, and z is the number of periods in which she was not supposed to pay a bonus but she paid. We have

Pr ( Ξ * = (0, … , 0) | h T P− ) = α H v

[( 1 − α L _ 1 − α H ) α L ] z α L T−s .

now, consider another history _ h T P− in which the principal deviates one additional

period. In this case, Pr ( Ξ * = (0, … , 0) | h T P− ) is at least

α H v [( 1 − α L

_ 1 − α H ) α L ] z α L T−s−1 min { α H , α L ( 1 − α L _ 1 − α H )}.


Since min { α H , α L ( 1 − α L

_ 1 − α H )} > α L , the Lemma follows.

PROOF OF PROPOSITIOn 1: Remember from Lemma 2 that the marginal increment in the probability of

Ξ * = {0, … , 0} is increasing in the number of past deviations. Hence, the con-straint (10) guarantees that the principal pays the bonus if and only if she observed a low output level.

Fix the discount factor δ. Using equation (9) we have

(14) V T P = [ (1 − δ ) ∑ t=1 T−1 δ t s *

__ 1 − δ T (1 − α L T−1 ) ].

There exists a largest integer such that (14) and (10) hold. name it T (δ ). It is straightforward to see that there exists δ′ ∈ (0, 1) such that δ > δ′ implies T (δ ) > 0 and that we have lim δ↑1 T (δ ) = ∞. For simplicity, assume that (10) holds with equality. Then using (10) and (14) we can rewrite V T (δ ) as

V T (δ ) P =

⎡⎢⎢⎣

(1 − δ )( ∑ t=1 T−1 δ t s * − δ B __

[(( α H _ α L ) − 1) − 1] ) ___

1 − δ T

⎤⎥⎥⎦

.

Thus, we have li m δ↑1 V T (δ ) P

= s * , which concludes the proof.

PROOF OF LEMMA 3: Consider an efficiency-wage strategy profile ( σ A , σ P ). Assume that there is

a history h t P ∪ { y } on the equilibrium path such that τ t P ≠ _ τ P and, following this

payment, the relationship persists in the next period. We will construct another effi-ciency-wage profile ( ̃ σ A , ̃ σ P ) such that he principal pays9

_ τ P whenever the agent is not fired in the end of the period.

Preliminaries.—Let t 1 ∈ 핅 be the first period such that the agent is fired with positive probability. notice that since the game is discounted we must have t 1 < ∞.

Consider any (on-path) private history of the principal h t 1 P and y ∈ { y L , y H }.

We claim that the expected continuation profit of the principal at h t 1 P ∪ { y},

V P ( h t 1 P ∪ { y}), should be zero.

CLAIM 1: V P ( h t 1 P ∪ { y}) = 0.

9 notice that if the principal will fire the agent at t then she must set τ t P = 0.


PROOF:Clearly, V P ( h t 1 P

∪ { y}) ≥ 0. We will show that we cannot have V P ( h t 1 P ∪ { y}) > 0.

Remember that the output is privately observed by the principal while the agent observes only her own effort, transfers and quitting decisions. Moreover, since the distribution over { y L , y H } × { x L A , x H A

} has full support under e * the requirement (iii) in Definition 2 prescribes that if the agent is not fired at t 1 she will keep on choosing e * in every future period (until she is fired). Hence, the principal’s continuation value at h t 1

P ∪ { y} does not depend on the history (on the equilibrium path). Therefore, if for some h t 1

P ∪ { y} we have V P ( h t 1 P ∪ { y}) > 0 then V P ( ̃

h t 1 P ∪ { ̃ y }) > 0 for every

(on-path) history ̃ h t 1 P and ̃ y ∈ { y L , y H } which contradicts the assumption that the agent

is fired with positive probability at t 1 and proves the claim.next, we claim that there is no period t such that the agent is fired with probabil-

ity 1 before t + 1.

CLAIM 2: There is no time period t such that the agent is fired with probability 1 before t + 1.

PROOF:Assume the claim is not true and let ̃ t be the smallest integer such that the agent is

fired with probability 1 before ̃ t + 1. Hence, if the relationship has lasted until ̃ t the principal should fire the agent with probability 1 at the end of ̃ t . As a consequence, the agent expects to receive no transfer with probability 1 in the end of ̃ t . Thus, he chooses 0 effort at ̃ t , contradicting requirement (iii) in Definition 2 and proving the claim.

next, consider the (on-path) sequence of payments in the case that the agent is not fired: { τ s P } s=0

∞ . We claim that for each t ′ > t 1 we have

(1 − δ ) ∑ s= t 1

t ′ −1

δ s− t 1 (δE [ y | e * ] − τ s P ) ≤ 0.

Suppose that there exists t ″ > t 1 such that (1 − δ ) ∑ s= t 1 t ″ −1 δ s− t 1 (δE [ y | e * ] − τ s P ) > 0.

Hence, the continuation strategy at any (on-path) history h t 1 −1 P ∪ { y} of waiting until

period t ″ to fire the agent yields a positive profit, which contradicts Claim 1.next, let t 2 be the smallest time period in { t 1 + 1, t 1 + 2, …} such that the agent

is fired with positive probability. Since at any (on-path) history h t 1 −1 P ∪ { y} the strat-

egy of waiting until the end of period t 2 to fire the agent10 is a weak best-response, we have

(1 − δ ) ∑ s= t 1

t 2 −1

δ s− t 1 (δE [ y | e * ] − τ s P ) = 0.

10 In this case, the principal makes no transfer before firing the agent at t 2 .


By the same reason, letting { t i * } i=1 ∞ be the sequence of time periods such that the agent is fired with positive probability, for every i ∈ 핅 we have: t i+1 * − 1

(15) (1 − δ ) ∑ s= t i *

t i+1 * −1

δ s− t i * (δE [ y | e * ] − τ s P ) = 0.

New Equilibrium: We construct a new equilibrium strategy profile ( ̃ σ A , ̃ σ P ).

Principal Strategy ̃ σ P : Payments: For every on path history h t P , the principal pays τ t P =

_ τ P whenever the strategy prescribes that the agent is not fired in the end of the period. Firing Decisions: Under the original strategy σ P , the probability that the principal fires the agent at each (on-path) history h t P ∪ { y t } (with t ∈ { t i * } i=1 ∞ ) depends only on the outputs { y 0 , … , y t } ⊂ h t P ∪ { y t } and we write it as Q t P ({ y 0 , … , y t } | σ P ). We define ̃ σ P such that for every t ∈ { t i * } i=1 ∞ and for every on-path history ̃ h t

P ∪ { y t } containing { y 0 , … , y t } the principal fires the agent with prob-ability Q t P ({ y 0 , … , y t } | ̃ σ P ) = Q t P ({ y 0 , … , y t } | σ P ). The principal does not fire the agent on any on-path history h t ′ −1 P

∪ { y t ′ } if t ′ ∉ { t i * } i=1 ∞ . off-path: After any own deviation or detectable deviation of the agent,11 ̃ σ P prescribes that the principal sets τ t P = 0 and fires the agent in the end of the period.

Agent’s Strategy ̃ σ A : Payments: Set τ t A = 0 for all t. Effort: Choose e t = e * in every period as long as no own deviation has happened. Quitting: never quit the relationship. off-path: If a detectable deviation from the principal or an own detectable deviation has happened, choose e t = 0 in every future period t. Finally, consider a history ̃ h t

A under which the agent has chosen { e s } s=0 t−1 with e s ∈ [0,

_ e ] for

every period s < t and no detectable deviation has happened. This history can be uniquely mapped into a history h t A such that the agent had chosen e s ∈ [0,

_ e ] for

every period s < t and no detected deviation had been detectable under ( σ P , σ A ). The strategy profile σ A prescribes a randomization over effort levels at h t A : e t ( h t A ) ∈ Δ [0,

_ e ]. We define ̃ σ A such that ̃ e t ( ̃

h t A ) = e t ( h t A ).

Verifying that ( ̃ σ P , ̃ σ A ) is a PBE: It is straightforward to check that the princi-pal has no profitable deviation. next, consider a period t and assume that a detect-able deviation from any player has occurred. Since the principal fires the agent in the end of the period, choosing effort 0 and making no transfer is a best-response for the agent. Hence, to complete the verification that ( ̃ σ P , ̃ σ A ) is a PBE, it suf-fices to show that ̃ σ A is a best-response in any history ̃ h t

A containing the efforts { e s } s=0 t−1

∈ [0, _ e ] t and no detectable deviation. Let t j * be the minimum ele-

ment of { t i * } i=1 ∞ weakly greater than t and let ̃ σ A ′ another continuation strat-egy in which (w.l.o.g.) τ s A = 0 for all s and the agent never quits. Write 1A for

11 An action is a detectable deviation if: (i) it is not prescribed by the equilibrium strategy; (ii) it is observable by the opponent player.


the indicator function of the event A. We wish to show that V A (( ̃ σ A , ̃ σ P ) | ̃ h t A )

≥ V A (( ̃ σ A ′ , ̃ σ P ) | ̃ h t A ) or12

(16) V A (( ̃ σ A , ̃ σ P ) | ̃ h t A )

=

⎧

⎪⎪⎨⎪⎪⎩

(1 − δ )E [ ∑ s=t t j * −1 δ s−t ( _ τ P − c ( e s )) − δ t j * −t c ( e s ) | ̃ σ A , ̃ σ P , ̃ h t

A ] + (1 − δ )E [1 { Q t j * P

= 1} ∑ s= t j * t j+1 * −1 δ s−t ( _ τ P − δc ( e s+1 )) | ̃ σ A , ̃ σ P , ̃ h t

A ] + (1 − δ )E [1 { Q t j * P

= 1, Q t j+1 * P = 1} ∑ s= t j+1 *

t j+2 * −1 δ s−t ( _ τ P − δc ( e s+1 )) | ̃ σ A , ̃ σ P , ̃ h t A ]

+ ...

⎫

⎪⎪⎬⎪⎪⎭

≥

⎧

⎪⎪⎨⎪⎪⎩

(1 − δ )E [ ∑ s=t t j * −1 δ s−t ( _ τ P − c ( e s )) − δ t j * −t c ( e s ) | ̃ σ A ′ , ̃ σ P , ̃ h t

A ] + (1 − δ )E [1 { Q t j * P

= 1} ∑ s= t j * t j+1 * −1 δ s−t ( _ τ P − δc ( e s+1 )) | ̃ σ A ′ , ̃ σ P , ̃ h t

A ] + (1 − δ )E [1 { Q t j * P

= 1, Q t j+1 * P = 1} ∑ s= t j+1 *

t j+2 * −1 δ s−t ( _ τ P − δc ( e s+1 )) | ̃ σ A ′ , ̃ σ P , ̃ h t A ]

+ …

⎫

⎪⎪⎬⎪⎪⎭

= V A (( ̃ σ A ′ , ̃ σ P ) | ̃ h t A ).

notice that we can uniquely map the history ̃ h t A to a history h t A composed of

efforts { e s } s=0 t−1 ∈ [0,

_ e ] t and no detectable deviation under ( σ P , σ A ). Remember that by construction ̃ σ A leads to the same effort distribution at ̃ h t

A as σ A at h t A . We can also select a different strategy σ A ′ which leads to the same effort distribution over [0,

_ e ] ∞ at h t A (under ( σ P , σ A )) as ̃ σ A ′ at ̃ h t

A . Since V A (( σ A , σ P ) | h t A ) ≥ V A (( σ A ′ , σ P ) | h t A ) we have

(17)

⎧

⎪⎪⎨⎪⎪⎩

(1 − δ )E [ ∑ s=t t j * −1 δ s−t ( τ s P − c ( e s )) − δ t j * −t c ( e s ) | σ A , σ P , h t A ]

+ (1 − δ )E [1 { Q t j * P = 1} ∑ s= t j *

t j+1 * −1 δ s−t ( τ s P − δc ( e s+1 )) | σ A , σ P , h t A ]

+ (1 − δ )E⎡⎢⎣

1 { Q t j * P = 1, Q t j+1 * P

= 1} ∑ s= t j+1 *

t j+2 * −1 δ s−t ( τ s P − c ( e s+1 )) | σ A , σ P , h t A

⎤⎥⎦ + …

⎫

⎪⎪⎬⎪⎪⎭

≥

⎧

⎪⎪⎨⎪⎪⎩

(1 − δ )E [ ∑ s=t t j * −1 δ s−t ( τ s P − c ( e s )) − δ t j * −t c ( e s ) | σ A ′ , σ P , h t A ]

+ (1 − δ )E [1 { Q t j * P = 1} ∑ s= t j *

t j+1 * −1 δ s−t ( τ s P − δc ( e s+1 )) | σ A ′ , σ P , h t A ]

+ (1 − δ )E

⎡⎢⎣

1 { Q t j * P = 1, Q t j+1 * P

= 1} ∑ s= t j+1 *

t j+2 * −1 δ s−t ( τ s P − δc ( e s+1 )) | σ A ′ , σ P , h t A

⎤⎥⎦ + …

⎫

⎪⎪⎬ .⎪⎪⎭

12 We define ∑ s=k k−1 g s = 0.


Using (15), it is straightforward to see that for all l ≥ 0 we have

E⎡⎢⎣1 { Q t j * P

= 1, … , Q t j+l * P = 1} ∑

s= t j+l *

t j+l+1 * −1

δ s− t j+l * ( _ τ P − δc ( e s+1 )) | ̃ σ A , ̃ σ P , ̃ h t A ⎤⎥⎦

= E ⎡⎢⎣1 { Q t j * P

= 1, … , Q t j+l * P = 1} ∑

s= t j+l *

t j+l+1 * −1

δ s− t j+l * ( τ s P − δc ( e s+1 )) | σ A , σ P , h t A ⎤⎥⎦

and

E⎡⎢⎣1 { Q t j * P

= 1, … , Q t j+l * P = 1} ∑

s= t j+l *

t j+l+1 * −1

δ s− t j+l * ( _ τ P − δc ( e s+1 )) | ̃ σ A ′ , ̃ σ P , ̃ h t A

⎤⎥⎦

= E ⎡⎢⎣1 { Q t j * P

= 1, … , Q t j+l * P = 1} ∑

s= t j+l *

t j+l+1 * −1

δ s− t j+l * ( τ s P − δc ( e s+1 )) | σ A ′ , σ P , h t A ⎤⎥ .⎦

Hence (17) implies (16), which completes the proof.

PROOF OF PROPOSITIOn 2:Take any history h t A . For ( y t , x t A ) ∈ { y L , y H } × { x L A , x H A

} we write W ( h t A , y t , x t A ) for the agent’s continuation payoff conditional on the agent being truthful and obedient and the joint realization ( y t , x t A ) in period t (at the moment that the mediator receives the messages ( y t , x t A )).

For this proof, we will consider three continuation strategies when the agent starts period t with a history h t A .

Strategy 1: The obedient and Truthful (continuation) strategy. Consists of choosing effort e * in period t (and every future period) and always reporting his signal truthfully. We write V A ( e * , h t A , m t * ) for the expected continuation payoff asso-ciated to this strategy. We have

(18) V A ( e * , h t A , m t * )

= e * [ α H W ( h t A , y H , x H A ) + (1 − α H ) W ( h t A , y H , x L A )]

+ (1 − e * )[ α L W ( h t A , y L , x H A ) + (1 − α L ) W ( h t A , y L , x L A )]

− (1 − δ ) c ( e * ).

Strategy 2: Effort 0 Followed by Announcing Low signal (continuation) strategy. Consists of choosing effort e t = 0 and announcing x L A in period t after h t A . In every period s > t, the agent chooses effort e * and reports the signal truthfully. We write


V A (0, h t A , m t ) for the expected continuation payoff associated to this deviation. We have

(19) V A (0, h t A , m t ) = W ( h t A , y L , x L A ).

Strategy 3: The optimal Effort Followed by Lying (continuation) strategy. Consists of choosing effort e * and always announcing x L A in period t after h t A . In every period s > t, the agent chooses effort e * and reports the signal truthfully. We write V A ( e * , h t A , ˜ m t ) for the expected continuation payoff associated to this deviation. We have

V A ( e * , h t A , ˜ m t ) = e * [ α H W ( h t A , y H , x L A ) + (1 − α H )W ( h t A , y H , x L A )]

+ (1 − e * )[ α L W ( h t A , y L , x L A ) + (1 − α L )W ( h t A , y L , x L A )]

− (1 − δ )c ( e * ).

We write W ( h t A , x A ) for the expected continuation payoff of the agent conditional on the truthful and obedient strategy and on observing the signal x A ∈ { x L A , x H A

} :

(20) W ( h t A , x L A ) = ( (1 − e * )(1 − α L ) ___ e * (1 − α H ) + (1 − e * )(1 − α L ) )W ( h t A , y L , x L A )

+ ( e * (1 − α H ) ___ e * (1 − α H ) + (1 − e * )(1 − α L ) )W ( h t A , y H , x L A ),

W ( h t A , x H A ) = ( (1 − e * ) α L

__ e * α H + (1 − e * ) α L )W ( h t A , y L , x H A

)

+ ( e * α H __ e * α H + (1 − e * ) α L

)W ( h t A , y H , x H A ).

LEMMA 4: If strategy 1 is a best-response then

(21) [ e * α H + (1 − e * ) α L ] W ( h t A , x H A )

+ [ e * (1 − α H ) + (1 − e * )(1 − α L )]W ( h t A , x L A )

− (1 − δ )c ( e * ) ≥ W ( h t A , y L , x L A ).


PROOF:Follows directly from the comparison of the payoffs from strategies 1 and 2.

LEMMA 5: For any γ > 0 if W ( h t A , x L A ) ≥ W ( h t A , x H A ) − γ then

W ( h t A , y H , x L A ) − W ( h t A , y L , x L A ) ≥ ( e * (1 − α H ) + (1 − e * )(1 − α L ) __ e * (1 − α H ) )

× [(1 − δ )c ( e * ) − γ ( e * α H + (1 − e * ) α L )].

PROOF:If W ( h t A , x L A ) ≥ W ( h t A , x H A

) − γ, then

( e * α H + (1 − e * ) α L )W ( h t A , x H A )

+ ( e * (1 − α H ) + (1 − e * )(1 − α L ))W ( h t A , x L A ) − (1 − δ )c ( e * )

≤ W ( h t A , x L A ) + γ ( e * α H + (1 − e * ) α L ) − (1 − δ )c ( e * ).

Since the LHS of the previous inequality majorizes the RHS of (21), using (20) one obtains

( e * (1 − α H ) ___ e * (1 − α H ) + (1 − e * )(1 − α L ) )[W ( h t A , y H , x L A ) − W ( h t A , y L , x L A )]

≥ (1 − δ )c ( e * ) − γ ( e * α H + (1 − e * ) α L ).

Rearranging the last inequality we obtain the desired result.

LEMMA 6: For any γ > 0 if W ( h t A , x L A ) ≥ W ( h t A , x H A ) − γ, then

W ( h t A , x H A ) ≥ W ( h t A , x L A ) + ϑ [c ( e * )(1 − δ ) − γ ( e * α H + (1 − e * ) α L )],

where

ϑ = [( e * α H __ e * α H + (1 − e * ) α L

) − ( e * (1 − α H ) ___ e * (1 − α H ) + (1 − e * )(1 − α L ) )]

× ( e * (1 − α H ) + (1 − e * )(1 − α L ) ___ e * (1 − α H ) ).


PROOF:If strategy 3 is not a profitable deviation, then the agent prefers announcing truth-

fully a high signal:

(22) ( (1 − e * ) α L __ e * α H + (1 − e * ) α L ) W ( h t A , y L , x H A

) + ( e * α H __ e * α H + (1 − e * ) α L ) W ( h t A , y H , x H A

)

≥ ( (1 − e * ) α L __ e * α H + (1 − e * ) α L

)W ( h t A , y L , x L A ) + ( e * α H __ e * α H + (1 − e * ) α L

)W ( h t A , y H , x L A )

= W ( h t A , x L A )

+ [( e * α H __ e * α H + (1 − e * ) α L ) − ( e * (1 − α H )

___ e * (1 − α H ) + (1 − e * )(1 − α L ) )] × [W ( h t A , y H , x L A ) − W ( h t A , y L , x L A )].

Applying Lemma 5 into (22) we obtain the desired inequality.

LEMMA 7: W ( h t A , x L A ) ≤ W ( h t A , x H A ) − ζ (1 − δ ), where

ζ = ( ϑc ( e * )(1 − δ ) __

1 + e * α H + (1 − e * ) α L ).

PROOF:Assume that W ( h t A , x L A ) − W ( h t A , x H A

) ≥ − γ. From Lemma 6 we have W ( h t A , x H A ) −

W ( h t A , x L A ) ≥ ϑ [c ( e * )(1 − δ ) − γ ( e * α H + (1 − e * ) α L )]. Hence,

γ ≥ ϑ [c ( e * )(1 − δ ) − γ ( e * α H + (1 − e * ) α L )] or

(23) γ ≥ (1 − δ )( ϑc ( e * ) __

1 + e * α H + (1 − e * ) α L ).

We claim that W ( h t A , x L A ) − W ( h t A , x H A ) ≤ − ζ (1 − δ ). Otherwise W ( h t A , x L A ) −

W ( h t A , x H A ) ≥ − ζ (1 − δ ) + ε for some ε ∈ (0,

ζ (1 − δ ) _ 2 ). Letting γ′ = (ζ (1 − δ ) − ε)

the inequality (23) implies ζ (1 − δ ) − ε ≥ ζ (1 − δ ), which is an absurd.

Conclusion of the Proposition: Take an efficiency-wage equilibrium ( σ M ** , σ A ** ). Consider a period t on the equilibrium path. If the agent is not fired in the end of t, the principal pays

_ τ P = δE [ y | e * ] and the agent chooses effort e * in the next period. This leads to an expected profit of δE [ y | e * ] for the principal. Hence, the principal’s continuation profit is always (1 − δ )E [ y | e * ] in the beginning of each (on-path) his-tory h t A in which the agent is hired.


Let ε > 0 and let _ V A be the supremum of the agent’s continuation payoff over

all on-path private histories h t A . Take an on-path history ̃ h t A such that V A (( σ M ** ,

σ A ** ) | ̃ h t A ) ≥

_ V A − ε. Since the agent exerts effort and reports his information

truthfully,

(24) V A (( σ M ** , σ A ** ) | ̃ h t A )

= [ e * α H + (1 − e * ) α L ]W ( h t A , x H A )

+ [ e * (1 − α H ) + (1 − e * )(1 − α L )]W ( h t A , x L A ) − (1 − δ )c ( e * ).

Clearly, W ( h t A , x i A ) ≤ (1 − δ ) _ τ P + δ _ V A for x i A ∈ { x L A , x H A

}. Using Lemma 7, we obtain

V A (( σ M ** , σ A ** ) | ̃ h t A )

= W ( h t A , x H A ) − (1 − δ )c ( e * )

+ [ e * (1 − α H ) + (1 − e * )(1 − α L )](W ( h t A , x L A ) − W ( h t A , x H A ))

≤ (1 − δ )( _ τ P − c ( e * )) + δ _ V A

− (1 − δ )[ e * (1 − α H ) + (1 − e * )(1 − α L )]ζ.

Since V A (( σ M ** , σ A ** ) | ̃ h t A ) ≥

_ V A − ε and

_ τ P = δE [ y | e * ], we have

_ V A ≤ (δE [ y | e * ] − c ( e * )) + ( ε _

1 − δ )

− [ e * (1 − α H ) + (1 − e * )(1 − α L )]ζ.

Since ε is arbitrary and the continuation profit for the principal is always (1 − δ )E [ y | e * ] in the beginning of each (on-path) history h t A in which the agent is hired, an upper bound to the surplus is

_ V A + (1 − δ )E [ y | e * ], which is bounded

above by

s * − [ e * (1 − α H ) + (1 − e * )(1 − α L )]ζ.

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