SELF-ENFORCING CONTRACTS WITH PERSISTENCE∗

Martin Dumav† William Fuchs‡ Jangwoo Lee§

July 24, 2019

Abstract

The use of self-enforcing contracts in an environment with persistent shocks rationalizes: (i) that com-

pensation may appear to “reward luck;” (ii) that adverse shocks are compounded via a “low morale effect”

reducing performance even further; (iii) that countries with poorer institutions of contract enforcement

experience larger dispersions of firm productivities and higher aggregate TFP and GDP volatilities. We

relate the model’s predictions to existing empirical findings and provide new evidence that a country’s

improved legal environment decreases its aggregate volatility as well.

Keywords: Dynamic moral hazard, Productivity, Relational contracts, Persistence, Limited Commit-

ment

JEL Classification: C73, D24, D82, D86, E24, L14

∗Many thanks to Nicolas Aragon, Gregor Matvos, Andy Skrzypacz, Chad Syverson, Giorgio Zanarone, seminar participantsat Universidad Carlos III de Madrid, UT Austin, the 4th Relational Contracts Workshop at University of Chicago, the Societyfor the Advancement of Economic Theory Conference 2019 at Ischia, and the 4th FTG European Summer Meeting at Madrid,for helpful comments. Dumav gratefully acknowledges the financial support from the Ministerio Economia y Competitividad(Spain) through grants ECO2017-86261-P and MDM 2014-0431. Fuchs gratefully acknowledges support from the ERC Grant# 681575.†Universidad Carlos III de Madrid‡Universidad Carlos III de Madrid and University of Texas at Austin§University of Texas at Austin

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Conflict-of-interest Disclosure Statement

Martin Dumav

Hereby I declare that I have no conflicts of interests with regards to our submitted paper: Self-Enforcing

Contracts with Persistence. I gratefully acknowledge support from Ministerio Economia y Competitividad

(Spain) through grants ECO2017-86261-P and MDM 2014-0431.

William Fuchs

I have no conflicts of interests with regards to our submitted paper entitled Self-Enforcing Contracts with

Persistence . I gratefully acknowledge support from the European Research Council Grant 681575.

Jangwoo Lee

I have no conflicts of interests with regards to our submitted paper: Self-Enforcing Contracts with Persistence.

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1 Introduction

Many, if not most, economic interactions are carried out with very incomplete or no formal contracts at all.

In their absence, to overcome the possible hold-up problems, economic agents often rely on the repeated

nature of their interactions to establish relational or self-enforcing contracts. In this paper we show how

limited external enforceability coupled with persistent productivity shocks can generate several interesting

empirical predictions. We show these predictions are consistent with both micro and macro data, and further

establish a new link between a country’s legal environment and its aggregate-level TFP volatility.

Concretely, we build on the model of relational contracts a la Levin (2003) by allowing for persistence in

productivity and show the following three related results:

(i) agents can appear to be rewarded for luck;

(ii) in periods of distress the firm’s profitability is further worsened by “low morale” effect, the difficulty

in incentivizing workers to work hard and/or not to leave the firm;

(iii) the enforceability constraint, which captures the temptation of the principal to renege on the promised

bonus, generates a multiplier effect amplifying the exogenous productivity shocks. This has two impli-

cations for economic environments where enforceability constraints are more binding:

(a) aggregate shocks result in higher TFP and GDP volatilities

(b) idiosyncratic shocks induce a larger dispersion of measured firm productivity levels within a coun-

try or industry.

We further document multiple pieces of evidence in support of (iii)-(a) and (iii)-(b). In particular, we provide

evidence that a country’s enhanced legal environment decreases its aggregate volatility. To avoid endogeneity

problems we use early European settler’s mortality rates from the seminal work Acemoglu et al. (2001) to

instrument for the quality of legal environment. Our identifying assumption hinges on Acemoglu et al.

(2001)’s argument for exclusion restriction. Namely, that although malaria and yellow fever are accountable

for most European deaths in colonization process, they had limited effect on native populations who had

developed various forms of immunity. The merits and weaknesses of the Acemoglu et al. (2001)’s instrumental

variable have extensively been discussed in later works and refer the interested reader there. 1 For the purpose

our paper, we believe that using this instrumental variable provides a better estimate than a simple OLS

estimate.

One could potentially try to come up with different theories that might coincide with our findings for aggregate

volatility and cross-sectional dispersion where the mechanism connecting rule of law and these measures is

different from the one we propose. We do not try to argue such a thing is impossible, but what we find very

unlikely is that such an alternative mechanism could simultaneously also explain the existence of reward for

luck and the morale effect. In this paper we are not taking a single empirical fact and arguing that our theory

is the only one that can support the fact. Instead, we take what we think is a reasonable model and explore

all of its empirical implications. The value of our paper thus relies on providing a parsimonious model that

can explain many different empirical findings simultaneously.

We show that even when exogenous productivity shocks are perfectly observable and independent of the

1See for instance, Glaeser et al. (2004); Albouy (2012); Acemoglu et al. (2012)

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agent’s effort, bonuses are dependent on the realization of the shocks beyond the agent’s influence. This stands

in contrast to the “Informativeness Principle” (see Holmstrom (1979)), according to which a performance

measure should only affect compensation if it provides information about the agent’s hidden effort. Instead,

the empirical literature has documented several cases in which contracts seem to be rewarding luck. Bertrand

and Mullainathan (2001) document that in the oil industry of the US the compensation of executives in major

companies positively correlates with the price of oil, an element arguably outside the executive’s control.

Garvey and Milbourn (2006) show also that there is reward for luck and further argue it is not symmetric:

good luck is rewarded while bad luck is not punished. Finally, DeVaro et al. (2018) (which we discuss

further below) show empirically that persistence of the state is important for “reward for luck.” They also

provide evidence against capture of boards by CEOs being the main driver of the observed compensation

patterns.

The economic rationale for “rewarding luck” in our model hinges on the fact that the persistent productivity

shock implies that the continuation value of the firm is higher in the good state than in the bad state. As a

result, the principal has a stronger desire to maintain the relationship in the good state and thus can credibly

promise to pay higher bonuses in such a state. The bonus payments are thus higher in the high state than

in the low state. Note that although luck determines the size of the bonus, the bonus is paid only when the

output measure directly related to the agent’s effort is indicative of effort. More specifically, the agent is

being rewarded, conditional upon success, with a state dependent lottery. For incentives what matters is the

expected payoff of that lottery.

Our second result highlights the role of persistence in productivity during bad times. When the current

state is bad, the future value from maintaining the relationship is low and the principal cannot credibly

promise to make large payments. Thus, in bad times the firm cannot motivate the agent to exert high

effort. This “low morale effect” is consistent with the evidence that labor productivity levels are pro-cyclical

(Baily et al., 2001). We can alternatively recast the model as in Fuchs (2015) where every period the agent

receives a stochastic outside option and must decide whether or not to leave the firm and take the outside

option. In such a reformulation, when the future of the firm looks bleak, the agent would be willing to accept

lower outside options. This is consistent with the findings presented by Baghai et al. (2016) which show

that Swedish firms tend to lose their key talents when their financial health deteriorates. Similarly, using

an online search platform, Brown and Matsa (2016) show that job applicants avoid companies with poor

financial conditions.

In his review on the determinants of productivity Syverson (2011) highlights:

“... robust finding in the literature ... is that higher productivity producers are more likely to

survive than their less efficient industry competitors. Productivity is quite literally a matter of

survival for businesses.”

Our predictions are consistent with this finding and further suggest that the causality can go both ways. In

particular, if a firm is more likely to survive, then it will be able to become more productive. With higher

productivity it becomes easier for the firm to maintain the relationship, i.e., the dynamic enforceability

constraint is less likely to be binding when the productivity is higher. Empirically, the survival prospects of

the firm at time t might not be fully observable to the econometrician but influence time t+ 1 productivity.

This would give appearances of a positive correlation between productivity at t and survival to t + 1, and

yet, it is survivability that determines productivity but not the converse.

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Syverson (2011) empirically shows large productivity differences within a given industry. Particularly related

to the third implication of our model, he highlights that these differences vary dramatically across countries.

While in the US the most productive firm within a given industry can be twice as productive as the least

productive one, for India and China there can be fivefold differences.2

There are many possible reasons for why these cross country differences arise; our explanation is that it is

partly driven by the observation that the US has an effective legal system that facilitates the enforceability

of contracts while these institutions are much weaker in developing countries.3 Thus, a larger fraction of

business relationships in India and China are governed by relational contracts. As a result, when two firms

in the US and two firms in India experience a pair of low and high exogenous productivity shocks, the spread

in measured productivity in India would be larger than in the US. To see why this is the case, suppose

the productivity shock does not affect the optimal level of effort. With complete contracts there are no

differences in worker effort between the two firms. As a result, the measured productivity differences in the

US firms would be just given by the spread of the exogenous shocks. Instead, with imperfect enforceability,

the optimal effort that can be induced from the workers depends on the shock realization. When the shock

is high, the firm can credibly commit to pay high bonuses and, as a result, demand high effort and become

more productive. The opposite effect takes place when the shock is negative. This implies there is an extra

wedge in the measured productivity of Indian firms given by the fact that the equilibrium effort also varies

with the exogenous shock.

If common persistent shocks impact all firms in economies, our model predicts that the volatility of aggregate

TFP and GDP of economies with poorer legal enforcement would be greater (abstracting from general

equilibrium effects). Consistent with our predictions, for a large sample of countries we find a negative

relationship between dispersion of aggregate TFP and GDP growth rates and quality of law enforcement. We

further address potential endogeneity concerns by exploiting exogenous variations in early settler mortality

rates (Acemoglu et al., 2001) to instrument for the quality of law enforcement. This identification strategy

suggests a causal relationship between the legal environment and aggregate volatility yet to appear in the

literature.

Alternatively, if persistent idiosyncratic shocks affect firm-level productivity, our model predicts that spread

in firm-level TFP in economies with poorer legal enforcement would be wider. In line with our predictions,

we document a negative relationship between the spread in firm-level TFP within a given country and the

quality of the legal system in that country. Though similar findings are also documented in Powell (2017),

our analysis corroborates the robustness of his result with an alternative measure of firm-level productivity

dispersion based on Ackerberg et al. (2015)’s estimation techniques.

There is certainly additional heterogeneity and richness that is not captured by our simple model. Yet, we

believe our model provides a very useful lens through which we can start to uncover how seemingly unrelated

findings at the firm, industry and aggregate economy level can actually be manifestations of a natural friction

economic agents must constantly deal with: namely, how to realize the gains from trade without falling prey

to the hold-up problem.

2See references therein in particular Syverson (2004) and Hsieh and Klenow (2009).3See Syverson (2011) and references therein for a rich discussion of other possible factors influencing firm productivity.

Collard-Wexler et al. (2011), for example, argue that these differences arise due adjustment costs and persistence of shocks tofactors of production.

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Related Theoretical Literature

Our model belongs to the rich literature on relational incentive contracts that builds on Bull (1987), MacLeod

and Malcomson (1989), and Levin (2003). From a technical perspective the most closely related work to ours

within this literature are Kwon (2016) and DeVaro et al. (2018). We complement Kwon (2016) and go beyond

her analysis by characterizing the optimal contracts. Similar to ours, DeVaro et al. (2018) offers a theoretical

rationale for rewarding luck. In addition, as mentioned earlier, they complement their theoretical analysis

with some corroborating evidence on CEO compensation. Although our main argument for rewarding luck

is similar in spirit, our model uses a different timing of shocks vis-a-vis theirs. In our view, our timing, with

the shock revealed after the effort choice, is more convenient when considering non-separable production

functions. Our timing implies that positive (luck) innovations can lead to higher bonuses without affecting

the current effort choice. Instead, with the DeVaro et al. (2018) timing where the shock is revealed before

the effort choice, it would become less clear whether the increase in compensation is instead driven by the

desire to elicit higher effort in the good states. From an econometrician’s perspective this is problematic

since, like the principal, the econometrician is unlikely to observe the agent’s effort. In addition to this, our

main contribution relative to their work is on highlighting the “morale effect” and the related amplification

of shocks that is induced by the lack of enforceability4.

Three other different theoretical explanations have been put forward for the well-documented “reward for

luck” phenomenon. First, Oyer (2004) argues that the shocks identified in some of the empirical studies not

only affect one firm, but rather the entire sector or industry. As a result, in the high state the aggregate

demand for the agent’s services increases. Thus, in order to retain the agent (CEO), the principal must

increase the agent’s wage. His model does not have implications for the agent’s effort, and thus, unlike our

model, does not speak to the “morale effect” or the correlation between enforceability and volatility.

The second explanation was put forward by Hoffmann and Pfeil (2010) and is also present in DeMarzo et al.

(2012). Their models, have no enforceability constraints. Instead, the principal needs to finance up-front an

agent protected by limited liability. When productivity is high, the potential cash flows are large. Thus, the

principal has stronger incentives to avoid reducing the size of the firm or triggering termination (which is

the main way of providing incentives within their model). Hence, the principal must give more rents to the

agent, and the “reward for luck” effect arises. However the equilibrium effort in their model is independent

of the productivity shock. Thus, their models do not allow for a “morale effect” nor can they be used to

study the correlation between enforceability and output volatility.

The third explanation, put forward by DeMarzo and Kaniel (2017), is of a very different nature. They

consider agents that not only care about their own compensation but also about their compensation relative

to others. When this second component of the preferences is strong, relative performance-pay may not be

optimal. As a result, they argue, under certain parameter configurations it appears as if the agents are

rewarded for luck.

When considering the “morale effect,” Barron et al. (2018) is closely related. They study the problem of an

entrepreneur which needs both: (i) to take an initial loan to finance its venture and (ii) to hire an agent to

work on it. The contract between the entrepreneur and the worker is not enforceable. Thus the continuation

value of the entrepreneur plays an important role in determining the extent to which the enforceability

4Although in their model the morale effect is implicitly present they do not explicitly discuss it. Furthermore, while in theirsettings these two effects must always co-exist, in our model we could have reward for luck without it necessarily implying thepresence of the morale effect. For more details, see Remark 1 in section 4.2.

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constraint binds. A highly levered entrepreneur is similar to the principal in our model that experiences a

low productivity shock. In both cases, their equity value is low and thus there is a limit on how large a bonus

they can credibly promise to the agent. Thus, when the firm is in poor financial shape, it cannot induce the

worker to work hard. This is similar to the “morale effect.” Their paper also provides empirical evidence

supporting the mechanism in their model.

Our results about macroeconomic volatility in productivity are related to Ramey and Watson (1997) and

Den Haan et al. (2000). Ramey and Watson (1997) develop a theory of labor contracting in which negative

productivity shocks lead to costly job loss due to limited enforceability. Their model is used in Den Haan

et al. (2000)’s dynamic general equilibrium analysis to illustrate a propagation effect through which cyclical

fluctuations in the job-destruction rate magnify the shocks to output and make them persistent. In contrast,

in our analysis, the firm’s commitment power is driven by the endogenous value of maintaining relationship

where persistence plays an important role. Furthermore, our analysis sheds lights on output volatility as well

as its level, and we provide empirical support for our model’s implications. Our empirical analysis identifies

a negative relationship between the quality of legal environment and aggregate volatility and thus takes

established findings from Aguiar and Gopinath (2007); Koren and Tenreyro (2007) further.

Finally, the idea that imperfect enforceability of contracts can lead to dispersion in firm productivity is present

in the recent work of Board and Meyer-ter Vehn (2014) and Powell (2017). Their mechanisms provide com-

plementary rationales for why poor enforcement generates productivity dispersions. Powell (2017) studies

competitive industry equilibrium where heterogeneity in firms’ transitory shocks generate cross-sectional out-

put dispersion. In an efficiency wage model where the future promises in long-term relationships incentivize

the worker’s current effort choice, Board and Meyer-ter Vehn (2014) shows that wage and productivity dis-

persions among ex ante identical firms arise from labor-market competition and on-the-job search. Like us

Powell (2017) also provides empirical evidence of the positive and meaningful correlation between contract

enforceability and productivity. We supplement his analysis with an alternative measure of productivity

dispersion based on Ackerberg et al. (2015)’s estimation method, and document a new causal link between

aggregate volatility and contract enforceability, which is absent in his work.

We see our main contribution relative to the literature, not being the first to point out a particular effect,

but rather providing a unified model capturing multiple relevant phenomena. The existing literature so far

provides distinct explanations for these facts. Instead, we offer a parsimonious rationale for all three: time-

varying limits to contract enforceability. Furthermore, we add a new link between the legal environment and

aggregate volatility to the existing empirical literature.

2 Setup

A principal (“she”) and an agent (“he”) interact repeatedly over time. Time is discrete and indexed via

t ∈ {1, 2, ...}. At the beginning of every period, the principal makes the agent an offer consisting of an

enforceable wage payment wt, a schedule of non-enforceable bonus payments {bt} and recommends the

agent take an action at. If the agent rejects, the game ends and both receive their outside options, which

we normalize to be zero. If the agent accepts, the agent privately chooses his action at ∈ [0, 1] incurring

a cost c (·). We assume c (·) is continuously differentiable, strictly increasing and strictly convex, with

c (0) = c′ (0) = 0 and 1 < c′(1) ≤ ∞.

7

The firm’s publicly observable output is composed of two additively separable components πt = yt + θt. The

first component takes on binary values y ∈ {0, 1} and is partially under the control of the agent. Specifically,

the agent’s unobservable action at determines the probability that yt = 1. On the other hand, the distribution

over publicly observable state θt ∈ {L,H} is independent of the agent’s actions. We assume H > L > 0

and it is worth noting that one can show that the value to the firm is weakly decreasing on the spread

of the shock H − L. Thus, to the extent possible, firms would want to hedge this shock, our underlying

assumption is that hedging cannot be done perfectly5. The process for θt follows a first-order Markov chain

with a symmetric persistence parameter λ := Prob (θt = θt−1) > 1/2 for both θt−1 = H,L. The model could

be extended to richer stochastic processes. Similarly, the additive separability assumption on output can

be easily relaxed. Yet, the additive structure provides a very clear benchmark since it implies the first best

action is independent of the state6.

Figure 1: Timeline

t t+1

the principal offers a

relational contract

(wt, bt)

the agent

accepts

(/rejects)

the principal

pays wage wt

the agent

chooses at

outcome yt and

state θt realizes

the principal

pays bonus bt

See Figure 1 for the order of play. The relationship terminates the first time the parties renege on payments,

or the first-time the agent rejects a contract. Let τ denote the time when the contracting relation terminates.

After termination, both players receive the value of outside option, normalized to 0.

Both players are risk-neutral, have a common discount factor δ, and have deep pockets. The principal maxi-

mizes the present value of the expected discounted output streams minus her payments to the agent:

Πt = (1− δ)Etτ∑s=t

δs−t (πs − ws − bs)

The agent maximizes the expected discounted payments received minus his cost of effort:

vt = (1− δ)Etτ∑s=t

δs−t (ws − c(as) + bs)

Given any public history ht = {θ0, θ1, ...θt−1, y1, ...yt−1, b1, ...bt−1, w1, ...wt−1} ∈ Ht, a contract specifies the

compensation mix the principal offers; whether or not the agent accepts it; and the effort and discretionary

bonus payment decisions. The compensation is given by the functions of the following form: wt : Ht → R,

bt : Ht × (Θ× Y)→ R+.

A contract is self-enforcing if it is a perfect public equilibrium of the repeated game. We are interested

in characterizing the efficient self-enforcing arrangements that would govern this relationship. As a first

step, using an extension of the methods by Spear and Srivastava (1987) and Abreu et al. (1990) we can

5Even if θt is contractible, firms cannot fully hedge θt as long as the court (or any outside third party) cannot verify whetherone party has reneged. Assume that the principal and agent write a long-term contract based on θt. The long-term contractwould always be in effect, regardless of whether a player has reneged or not. Hence, the long-term contract does not affect anyparty’s decision to renege.

6If the output is additively separable, the first-best effort level aFB satisfy: ∂∂at

(E(πt|at, θt−1) − c(at)

)∣∣∣∣∣at=aFB

= 1 −

c′(aFB) = 0.

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conveniently surmise a given public history of the game ht, into a continuation value for the agent v and the

past state θt−1. The principal is choosing the current actions (a,w, b, v′) based on the public history (v, θt−1)

This allows us to capture the principal’s problem with the following recursive representation:

Π (v, θ−1) = maxa,w,b,v′

E [(1− δ) (π − w(θ−1, v)− b (y, θ−1, θ, v)) + δΠ (v′ (y, θ−1, θ, v) , θ) |a, θ−1]

s.t.

v = E [(1− δ) (w + b− c (a)) + δv′|a, θ−1] [PK]

a ∈ arg maxa∈[0,1]

E [(1− δ) (w + b− c (a)) + δv′|a, θ−1] [IC]

δΠ(v′, θ) ≥ (1− δ) b (y, θ−1, θ, v) ∀y × θ−1 × θ × v [DEP]

Π(v′, θ) ≥ 0 [PCP]

v′ ≥ 0 [PCA]

where [PK] is the promise keeping constraint, [IC] the incentive compatibility constraint for the agent to follow

the recommended action a, [DEP] is the dynamic enforceability constraints guaranteeing that the principal

prefers to pay the promised bonus rather than reneging and effectively terminating the relationship, [PC]’s

imply that each party needs to be at least as well off as the outside option every period.

3 Basic Properties of Optimal Markovian Contracts

We formally define Markovian contracts, and show that no generality is lost by restricting attention to this

class of contracts. This definition is similar to Kwon (2016)’s history-independent contract, but two defini-

tions are not exactly identical due to different timing of the shock.

Definition 1 (Markovian Relational Contracts). A Markovian relational contract consists of the fixed

wage component wt = w(θ−1), bonus salary component bt = b(y, θ−1, θ), and recommended action at =

a(θ−1).

The general problem can be significantly simplified by restricting our analysis to Markovian contracts. This

is analogous to the main contribution in Levin (2003) showing that it is without loss to focus on stationary

contracts when shocks are i.i.d. over time, and its extension to Markovian productivity states in Kwon

(2016). Furthermore, we give a characterization of the optimal contract.

Proposition 1 (Optimality of Markovian Relational Contracts). There exists a Markovian relational con-

tract such that

(R-1) it attains maximum surplus state-by-state and it is without loss to set v′ = v; and

(R-2) it is without loss to set b∗ (y = 0, θ−1, θ, v) = 0.

This result says that for any contract that maximizes the expected total surplus we can find a Markovian rela-

tional contract that does the job. The surplus-maximizing Markovian contract also maximizes the principal’s

profit, because the principal can pay base wages to deliver the promised utility to the agent without affecting

the total surplus. The offered base wages and bonuses are only functions of the last and current realization

θ. Importantly, since beyond participation, continuation values don’t play a role, we can simply focus on

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maximizing the current total surplus generated by the pair to characterize the optimal arrangements.

Since we have established that bonuses are only paid after success, we simplify our notation and denote

the bonus payment upon success simply as b(θ−1, θ). The only role for the base wage is to ensure that the

contract delivers the right promised value to the agent. We can thus solve the problem of figuring out the

effort and bonus choices that maximize surplus, which we denote by Π(θ−1), first and then divide it between

the principal and the agent by adjusting the base wages. Also note that we can drop the participation

constraints since a contract that implements zero effort is always available and generates strictly positive

surplus. Thus, the problem simplifies to:

Π(θ−1) = maxa(θ−1),b(θ−1,θ)

(1− δ) (a(θ−1) + E(θ|θ−1)− c(a(θ−1))) + δE(Π(θ)|θ−1) (1)

subject to:

[IC] a(θ−1) ∈ argmaxa

E(b(θ−1, θ)|θ−1)a− c(a)

[DEP] δΠ(θ) ≥ (1− δ)b(θ−1, θ)

Denote the optimal effort and bonus payment by a∗(θ−1) and b∗(θ−1, θ), respectively, and resulting profit

Π∗(θ−1). The optimal bonus scheme b∗(θ−1, θ) and the optimal effort a∗(θ−1) that solves the reduced

problem (1) are related as follows:

Lemma 1. In an optimal Markovian relational contract, we have that

(E-1) (Monotonicity in effort) a∗(L) ≤ a∗(H) ≤ (c′)−1(1) = aFB ; and

(E-2) c′(a∗(θ−1)) < 1 implies b∗(θ−1, θ) = δ1−δΠ∗(θ) for both θ = H,L.

The first part of the Lemma 1 establishes that if there is a distortion in the effort level with respect to

first best it is due only to a possible underprovision of effort. This arises due to the principal’s dynamic

enforceability constraint, i.e. the temptation of the principal to default on the promised bonus. As shown in

the second part of the Lemma 1, when the effort is distorted the enforceability constraint must bind. Hence

the principal must fully pledge the principal’s future surplus (which is the max amount the principal can

credibly commit to pay) as the bonus7. Since there is more surplus in high states than in low states the

equilibrium effort must always be weakly lower in the low state.

Denote by ΠFB(θ) the maximum payoffs that can be obtained without the dynamic enforceability for the

state θ. According to Lemma 1 the dynamic enforceability constraint is the only friction precluding the

first-best outcome, and the following monotonicity result obtains:

Lemma 2. (Monotonicity in profits) In any optimal Markovian relational contract the principal’s profits

satisfy Π∗(L) < Π∗(H), Π∗(L) ≤ ΠFB(L) and Π∗(H) ≤ ΠFB(H).

7To see this, suppose that b∗(θ−1, θ) <δ

1−δΠ∗(θ) for some θ. The principal could then increase b(θ−1, θ) and simultaneously

request that the agent exert more effort. Since there is under-provision of effort that would improve efficiency.

10

4 Main Results

We are now ready to focus on the main contributions of the paper which is to show that this simple yet

natural environment has numerous implications that match well several empirical findings.

4.1 Rewarding Luck

There are numerous papers (see for example Bertrand and Mullainathan (2001), Garvey and Milbourn

(2006) and DeVaro et al. (2018)) documenting empirically that variables outside of the agent’s control

influence compensation. Consistent with this evidence, our next result shows that when designing the optimal

structure of bonus payments an observable and contemporaneous shock would not be filtered out, even when

the realization of the shock contains no information about possible deviations by the agent in his choice of

effort. Hence, it would appear as if employees are partially rewarded for luck.

Proposition 2 (Rewarding Luck). There exists a δ ∈ (0, 1) such that for all δ < δ and θ−1 ∈ Θ, the optimal

contract features b∗ (θ−1, L) < b∗ (θ−1, H).

The principal can credibly promise to pay out larger amounts when the current shock is revealed to be favor-

able. This is the case because shocks are persistent. After a good realization the future prospects of the firm

are better, hence the enforceability constraint of the firm is relaxed and thus the firm can credibly promise

to pay more in such a state. Importantly, even though the size of the bonus varies with the persistent state

θ, an element outside the agent’s control or influence and with no information content, incentives are only

really provided by the expected bonus. Additionally, the bonus is still only paid if the agent delivers on his

part of the output. Thus, although luck determines the size of the reward these rewards are still fully driven

by incentive motives.

4.2 Morale Effect

Our first main result shows that the dynamic enforcement constraint is more binding in the low state.

Proposition 3 (Morale Effect). There exists a δ ∈ (0, 1) such that for all 0 < δ < δ, the optimal contract

satisfies

(i) E(b∗ (L, θ) |θ−1 = L) < E(b∗ (H, θ) |θ−1 = H);

(ii) a∗ (L) < a∗ (H).

There is an intuitive explanation for this result that highlights the effect of persistent productivity during

bad times. When the current state is bad, the future value of the relationship is low and the principal can

no longer credibly promise to make large payments. Thus, it can no longer motivate the agent to exert very

high effort.

Our model thus rationalizes the “low morale effect” during firm distress, once the profitability is interpreted

as a driver of financial distress. In our model, firms reduce the expected bonus payments during difficult

times, to which workers respond by exerting less effort. Thus, firms’ productivity is further depressed due to

11

de-moralization of the workforce. This mechanism resonates well with multiple labor concession episodes in

the domestic steel industry during 1980s (DeAngelo and DeAngelo, 1991).

DeAngelo and DeAngelo (1991) illustrates this effect with the labor problems experienced by the US steel

plants in the 1980s. This was a challenging time for US steel producers as they faced stiff competition

from more productive Japanese counterparts coupled with a decline in global demand. Mas (2008) uses

re-sale data for industrial machinery to show how Caterpillar’s problems with its workers had a negative and

economically significant impact on the quality of the machinery it produced.

The paper by Barron et al. (2018) has a similar mechanism at play. Although the shocks in their model are

transient, their firms have different levels of leverage which generates persistence. An increase in leverage in

the previous period in their model is similar to having experienced a negative persistent shock in our model

and leads to a lower labor productivity. They use data on a large sample of European firms to show that

indeed, an increase in leverage in year t− 1 is followed by a decrease in year t productivity. They report that

one standard deviation increase in contemporary leverage is correlated with a decrease in TFP-R equal to

9-15% of the median within firm standard deviation.

Furthermore, we can recast our model as in Fuchs (2015) to rationalize why distressed firms are more likely

to lose, or less likely to attract workers. This is consistent with the empirical findings in Baghai et al. (2016),

and Brown and Matsa (2016). When a negative productivity shock hits the firm, workers with high outside

values would leave the company, because firms cannot promise sufficiently high bonuses to retain them.

Remark 1. Importantly, unlike related papers (Kwon, 2016; DeVaro et al., 2018), the reward-for-luck in our

setup is distinct from the morale effect. In our setup, the morale effect necessarily leads to the reward-for-luck

effect, but not vice versa.

Claim 4.1. a∗ (L) < a∗ (H)→ b∗(θ−1, L) < b∗(θ−1, H) for both θ−1 ∈ Θ, but the converse is not necessarily

true8.

To understand this result it is important to recall that the enforceability constraint will always be more

binding in the when θt = L than when θt = H (see Lemma 1). When the enforceability constraint first

becomes binding it implies that the principal must promise lower bonus when θt = L is realized. Absent any

other adjustments to the compensation, the expected bonus would decrease and thus effort would inefficiently

decrease. To try to avoid this inefficiency, the principal can raise the bonus conditional on θt = H being

realized. If the enforceability constraint does not bind in the high state, that means that the principal can

fully offset the decrease in expected payments and thus keep the agent at the efficient effort level regardless

of the state θt−1. This is a situation in which we have reward for luck but no morale effect. The morale effect

only arises when the enforceability constraint also starts binding for θt = H and, as a result, the expected

bonus is no longer sufficient to induce high effort. Note, that in particular, this would imply lower effort

when last period shock was low. This follows because, due to persistence, the low state is more likely to be

realized again and thus the expected bonuses must be lower.

8Under certain parameters, all optimal bonuses b∗(·) exhibit reward-for-luck, but a∗(θ−1) = aFB for both θ−1 ∈ {L,H}.Hence, even if the agent is risk averse, the agent would still be rewarded for luck. This result is available upon request. Theproof for the first part of the claim is included in the appendix.

12

4.3 Amplification and Contract Enforcement

We have focused our analysis and discussion on a single firm but our analysis at the micro level has impli-

cations from an industry wide and macroeconomic perspective. If we assume that the productivity shocks

are correlated across firms our model generates an amplification effect of these shocks when there is poor

contractual enforceability. This is consistent with the empirical findings that show that emerging economies

have much higher output volatility than developed economies.9 Naturally there could be many explanations

for this difference and indeed there is a rich literature providing a number of different rationales for these

facts. However, our proposed mechanism has an additional advantage that it can rationalize other phenom-

ena as the reward for luck and morale effect. We are unaware of any other alternative mechanisms that are

consistent with these stylized facts.

Our paper contributes to this literature by arguing that the larger reliance on self-enforcing contracts by

firms in emerging economies might provide an amplifier effect on exogenous productivity shocks leading

to larger business cycle volatility. It is well documented that emerging economies tend to have poorer

legal enforcement and a larger underground economy (Porta et al., 1998; Allen et al., 2005). As a result,

enforceability constraints play a larger role in those countries. Therefore, the amplification effect generated

by the enforceability constraints would lead to larger volatility of labor productivity and output in emerging

economies vis a vis developed ones.

To approximate the data more closely we generalize our benchmark model and introduce a parameter α

that measures the strength of contract enforcement institutions. In particular, assume that whenever the

principal reneges on her promised payments, she can walk away with only a fraction (1−α) of the sum. Weak

contract institutions amplify the effects of productivity shocks on the volatility of output, mainly because

weak institutions constrain the value promises the principal credibly makes. With this generalization, the

principal’s contracting problem (1) now takes the following form

Π(θ−1;α) = maxa(θ−1),b(θ−1,θ)

(1− δ) (a(θ−1) + E(θ|θ−1)− c(a(θ−1))) + δE(Π(θ;α)|θ−1) (2)

subject to:

[IC] a(θ−1) = argmaxa

E(b(θ−1, θ)|θ−1)a− c(a)

[DEP]′δΠ(θ;α) ≥ (1− δ)(1− α)b(θ−1, θ)

Let the corresponding optimal output in each state be given by Y (θ;α) = a∗(θ;α) +E(θ+1|θ), where a∗(θ;α)

denotes the recommended effort under the optimal relational contract.

Assumption 1. The cost function satisfies: (1) c is log-concave and (2) c′ is convex.

Note that this assumption allows for a wide range of commonly used cost functions. For example it allows

for any cost function of the form c(x) = cxn × exp(ax), a ≥ 0, c > 0, n ≥ 2 which includes the quadratic cost

case c(x) = cx2.

Proposition 4 (Amplification). If Assumption (1) holds, the dispersion of output, as measured by Y (H;α)Y (L;α) ,

decreases in the strength of contract enforcement α.

9See for example Aguiar and Gopinath (2007)

13

When α increases, the enforceability constraint is relaxed. This allows the principal to make larger com-

mitments to the agent in both states. Hence, the principal can incentivize a higher level of effort in both

states. Given the Assumption 1, we can show that the effort is sufficiently more costly at a∗(H). Thus

proportionally, effort, and hence output, increases more in the low state.

Remark 2. An alternative way to obtain the amplification at the aggregate level is to assume that a

proportion α ∈ [0, 1] of firms rely on the formal court of law and thus can fully enforce their commitments,

while the rest rely on relational contracts as in our benchmark economy. Then, as above, the dispersion of

outputs as measured by Y (H;α)Y (L;α) decreases in α.

The Proposition 4 has implications for both the volatility of the aggregate output and the idiosyncratic

dispersion in firm-level productivity. More formally, consider instead an economy populated by a unit mass

of firms. Each firm i is subject to the firm-level persistent shock θi,t. Define the firm-level output as yi,t, and

firm-level recommended effort as a∗i,t.

Consider two stochastic structures governing the evolution of θi,t. Here, the primitive ωi,tiid∼ U [0, 1] for all

i ∈ [0, 1] and t ∈ {0, 1, 2, ...}. Under the assumption 2, there is no idiosyncratic volatility as corr(θi,t, θj,t) = 1.

Under the assumption 3, there is no aggregate volatility because∫ 1

0yi,tdi = 1

2Y (H,α) + 12Y (L,α) from the

strong law of large numbers (SLLN). :

Assumption 2 (Pure aggregate volatility without idiosyncratic volatility). Define:

1. ∀t ≥ 1, θ1,t = (H − L) ∗ [1({ω1,t ≤ λ, θ1,t−1 = H}) + 1({ω1,t ≤ 1− λ, θ1,t−1 = L})] + L

2. θ1,0 = (H − L)1({ω1,0 ≤ 12}) + L

3. ∀t, θi,t = θj,t = θ1,t.

Assumption 3 (Pure idiosyncratic volatility without aggregate volatility). Define:

1. ∀t ≥ 1, θi,t = (H − L) ∗ [1({ωi,t ≤ λ, θi,t−1 = H}) + 1({ωi,t ≤ 1− λ, θi,t−1 = L})] + L

2. θi,0 = (H − L)1({ω1,0 ≤ 12}) + L

Notice that the Proposition 4 has a different implication when coupled with a different assumption. Under the

first assumption, the aggregate volatility measureE(∫ 10yi,tdi|

∫ 10θi,tdi=H)

E(∫ 10yi,tdi|

∫ 10θi,tdi=L)

decreases in α. Under the second as-

sumption, the ratio between more productive firm’s productivity and less productive counterpartE(yi,t|θi,t=H)E(yj,t|θj,t=L)

also decreases in α. Hence, the proposition 4 amplifies the fundamental shock at both levels.

Unfortunately, there are no data available on measures of α at the individual firm level to contrast with

the prediction on productivity volatility in given Proposition 4. Fortunately there are a number of well-

established indicators for the strength of contracting institutions at the country level. Since enforcement

of contracts requires well functioning court systems, we believe these measures are likely to correlate well

with the degree of reliance on self-enforcing contracts. Specifically, we use the ‘Rule of Law’ indicators from

the Worldwide Governance Index (Kaufmann et al., 2011) in the results presented below.The Rule of Law

variable measures the agents’ confidence in contract enforceability, property rights, police, courts and the

likelihood of crime and violence. Using this measure as our proxy for α, we study empirically two implications

of Proposition 4: 1) Aggregate shocks would result in larger aggregate TFP and GDP volatility in countries

14

with poor rule of law measures; 2) Idiosyncratic shocks would result in a larger dispersion of TFP measures in

countries with worse rule of law. In our data appendix, we provide a detailed account of our data construction

procedures and summary statistics in each sample.

4.3.1 Aggregate Volatility and Rule of Law

Proposition 4 under Assumption 2 implies that aggregate shocks are further amplified under poor legal

environments. To test our model’s implication in the presence of aggregate shocks, we employ ‘Adjusted

Growth Accounting and Total Factor Productivity’ from Total Economy Database (conference-board.org)

to obtain country-level GDP and TFP growth rates in percentage points. After merging with WGI, our

sample covers 123 countries from 1996 to 2016. For each country c, we compute the variable V OLc as the

volatility of the country c’s GDP (TFP) growth rates over the sample period. We calculate the Average Rule

of Law variable AV G RLc as the average of Rule of Law of the country c over the sample period. We include

Continent Dummies to partially control for region-specific effects. For all 123 countries in our sample, we

estimate:

V OLc = α+ βOLSAV G RLc + Continent Dummies + εc (3)

Figures 2a and 2b plot the Average Rule of Law against the GDP and TFP volatility, respectively, after

partialling out regional averages (Continent Dummies). Equations above scatter plots also report estimated

coefficients, robust standard errors in parentheses, and “within R-squared” values after partialling out regional

averages. According to our “within R-squared” values, within-region variations in Average Rule of Law

explains 9.83% (15.39%) of within-region variations in GDP (TFP) growth rates.

As predicted by our model, the more a country relies on formal contracts (i.e., the better the legal environment

is) the lower the TFP and GDP volatility it has experienced over the last 20 years. After partialling out

region-specific effects, one standard deviation increase in Average Rule of Law would decrease the volatility of

GDP growth rates by 0.88 percentage points and TFP growth rates by 0.99 percentage points. Given that in

our sample the volatility of annual GDP growth rates averages around 3.41 percentage points while volatility

of TFP growth rates around 3.16 percentage points, the magnitude of these effects is quite large.

This, of course, is just a correlation. It is well documented in the existing literature (Aguiar and Gopinath,

2007; Koren and Tenreyro, 2007) that developing countries’ outputs are more volatile. Since contracts

are more difficult to enforce in poor countries, even when looking at departures from regional averages, the

approach in Figure 2a and 2b cannot distinguish the effect of the legal environment from other determinants

of TFP and GDP volatility of a country.

In order to address the possible endogeneity, we exploit exogenous variations in early European settler’s (log)

mortality rates Sc from the seminal work by Acemoglu et al. (2001). After merging our original sample

with Acemoglu et al. (2001)’s dataset, our sub-sample covers 50 countries. Except for few exceptions as the

United States, Australia, and New Zealand, most countries in the sample are developing countries. To test

the inclusion restriction condition, we use our sub-sample to estimate the first-stage regression:

AV G RLc = γSc + Asia Dummy + Africa Dummy + “Other Region” Dummy + ηc

15

Figure 2: Rule of Law and Output Volatility (OLS)

(a) Rule of Law and GDP Volatility

(b) Rule of Law and TFP Volatility

Panel (a) plots standard deviations of country-level GDP growth rates against Average Rule of Law measure, afterpartialling out regional fixed effects. Panel (b) plots standard deviations of country-level TFP growth rates againstAverage Rule of Law measure, also after partialling out regional fixed effects. Above each plot is the line of best fitin red. The equation reports estimated coefficients, robust standard errors in parentheses, and “within R-squared”values after partialling out regional fixed effects.

16

Figure 3: Rule of Law and Output Volatility (IV)

(a) Rule of Law and GDP Volatility

(b) Rule of Law and TFP Volatility

Panel (a) plots standard deviations of country-level GDP growth rates against instrumented Average Rule of Lawmeasure, after partialling out regional fixed effects. Panel (b) plots standard deviations of country-level TFP growthrates against Average Rule of Law measure, also after partialling out regional fixed effects. For both panels,Acemoglu et al. (2001)’s early European settler’s (log) mortality rates Sc instruments the Average Rule of Law.Above each plot is the line of best fit in red. The equation reports estimated coefficients, robust standard errors inparentheses, and “within R-squared” values after partialling out regional fixed effects.

17

Our specification is similar to Acemoglu et al. (2001). The t-statistics for our γ estimate is −3.72. Thus,

our Cragg-Donald Wald F-statistic 13.81 exceeds Stock and Yogo (2005)’s suggested threshold value at 10.

Hence, our maximal IV estimator bias is likely to be small. So early settler’s mortality rates Sc are likely to

satisfy the inclusion restriction as well.

For 50 countries in our sub-sample, we use mortality rates Sc as an excluded instrument and estimate:

V olc = βIVAV G RLc + Asia Dummy + Africa Dummy + “Other Region” Dummy + εc (4)

Figure 3a and 3b plot results from instrumental variable regressions, after partialling out regional dummies.

According to our “within R-squared” values, within-region variations in Average Rule of Law explains 8.15%

(21.53%) of within-region variations in GDP (TFP) growth rates.

After partialling out region-specific effects and instrumenting with mortality rates Sc, one standard devia-

tion increase in Average Rule of Law would decrease the volatility of GDP and TFP growth rates by 0.62

percentage points after rounding. Note that for this sub-sample the volatility of annual GDP growth rates

averages 2.76 percentage points while the volatility of TFP growth rates is 2.71 percentage points. Thus,

the magnitudes remain economically significant when using IV to estimate the effects. Hence, in conjunction

with the Proposition 4, our IV estimates suggest that for these countries output may be more volatile due

to their larger reliance on relational contracting.

In untabulated regressions, we also estimate βOLS with our sub-sample for both GDP and TFP growth rates

without any excluded instrument. OLS estimates are very similar to IV estimates. One standard deviation

increase in Average Rule of Law is associated with decrease in annual GDP volatility by .45 percentage

points and annual TFP volatility by .59 percentage points. Indeed, Durbin-Wu-Hausman F-tests fail to

reject the null hypotheses that the instrumental variable estimates are significantly different from the OLS

estimates. As long as mortality rates satisfy exclusion restrictions, the Durbin-Wu-Hausman F-tests suggest

that endogeneity is not as severe.

In the data appendix, we use aggregate variance instead of volatility, and cluster for standard errors at

different levels to check robustness of our analysis. These additional analyses yield qualitatively similar

conclusions.

4.3.2 Dispersion in Firm-Level Productivity and Rule of Law

In the presence of idiosyncratic shocks, the same mechanism has implications for the cross-sectional distribu-

tion for firm-level TFP. Namely, the amplification effect, implies that firms in countries with poor contract

enforcement have a larger dispersion of TFP at any given point in time as well. The comparison between

the findings by Syverson (2004) for the US and Hsieh and Klenow (2009) for India and China are clearly in

line with our predictions. While for the US there is a twofold difference in productivity between the 10th

and 90th percentile firm, in China and India there is a fivefold difference.

To obtain firm-level TFP estimates, we download firm-level financial statements from ORBIS database.

We closely follow Gopinath et al. (2017)’s data construction procedure and use Ackerberg et al. (2015)’s

methodology to estimate firm-level TFP. We refer the interested readers to our data appendix for a more

detailed account. The final sample contains 4, 786, 856 firm-years observations from 27 countries. We merge

18

this sample with WGI data. For each country c, we compute TFP9010c as the TFP difference between the

firm with the top 10% productivity and the bottom 10% firm in a country c in the logarithmic scale. Also,

AV G RLc denotes the average rule of law of the country c over the corresponding sample period. Since

all countries in the sample belong either to Asia or Europe, we include a separate dummy variable Euro

indicating whether the country belongs to Europe or not. We estimate the following regression:

TFP9010c = α+ βOLSAV G RLc + Euro + εc (5)

Figure 4 shows that the relationship implied by Proposition 4 is indeed consistent with what can be ob-

served more broadly. Within R-squared value suggests that within-region variations in Average Rule of Law

would explain 78.5% of within-region variations in country-level dispersion in firm productivity. In data

appendix, we check the robustness of our finding with alternative measures of country-level dispersion in

firm productivity and different econometric specifications. We find qualitatively similar results.

The countries with better contract enforcement have, on average, lower dispersion of TFP than those with

weaker contract enforcement. One standard deviation increase in Average Rule of Law would decrease the

productivity gap between the top 10% most productive firm and the bottom 10% firm by 0.409 in logarithm

scale. This translates into reduction in productivity gap by 33.6%.

Since sample countries in the ORBIS universe do not overlap with those in Acemoglu et al. (2001)’s dataset,

we cannot conduct an instrumental variable regression based on Acemoglu et al. (2001)’s European settler

mortality rates. We are hopeful that future research can explore this possibility with new datasets.

Figure 4: Rule of Law and Firm-Level TFP Dispersion

Figure 4 plots the TFP difference between the firm with the top 10% productivity and the bottom 10% firm in acountry c in the logarithmic scale against the Average Rule of Law measure. Above the plot is the line of best fit inred. The equation reports estimated coefficients, robust standard errors in parentheses, and “within R-squared”values after partialling out regional fixed effects.

19

5 Conclusion

We have shown that a very natural and parsimonious economic environment that features time-varying yet

persistent prospects in value creation and limits to enforceability of future promises delivers predictions that

are consistent with a wide range of empirical findings. Although many these empirical facts had mostly

been documented before and individual theories had been put forward for many of them, our paper pro-

vides a unique and simple framework that sheds light on how all these facts might be connected. Finally,

we contribute to empirical literature by identifying a new link between legal environment and aggregate

volatility.

References

Abreu, D., D. Pearce, and E. Stacchetti (1990). Toward a theory of discounted repeated games with imperfect

monitoring. Econometrica, 1041–1063.

Acemoglu, D., S. Johnson, and J. A. Robinson (2001). The colonial origins of comparative development: An

empirical investigation. American economic review 91 (5), 1369–1401.

Acemoglu, D., S. Johnson, and J. A. Robinson (2012). The colonial origins of comparative development: an

empirical investigation: comment. American Economic Review 102 (6), 3077–3110.

Ackerberg, D. A., K. Caves, and G. Frazer (2015). Identification properties of recent production function

estimators. Econometrica 83 (6), 2411–2451.

Aguiar, M. and G. Gopinath (2007). Emerging market business cycles: The cycle is the trend. Journal of

Political Economy 115 (1), 69–102.

Albouy, D. Y. (2012). The colonial origins of comparative development: an empirical investigation: comment.

American Economic Review 102 (6), 3059–76.

Allen, F., J. Qian, and M. Qian (2005). Law, finance, and economic growth in china. Journal of Financial

Economics 77 (1), 57–116.

Baghai, R., R. Silva, V. Thell, and V. Vig (2016). Talent in distressed firms: Labor fragility and capital

structure. Working Paper .

Baily, M. N., E. J. Bartelsman, and J. Haltiwanger (2001). Labor productivity: structural change and cyclical

dynamics. Review of Economics and Statistics 83 (3), 420–433.

Barron, D., J. Li, and M. Zator (2018). Managing debt in relational contracts. Working Paper .

Bertrand, M. and S. Mullainathan (2001). Are ceos rewarded for luck? the ones without principals are. The

Quarterly Journal of Economics 116 (3), 901–932.

Board, S. and M. Meyer-ter Vehn (2014). Relational contracts in competitive labour markets. The Review

of Economic Studies 82 (2), 490–534.

Brown, J. and D. A. Matsa (2016). Boarding a sinking ship? an investigation of job applications to distressed

firms. The Journal of Finance 71 (2), 507–550.

20

Bull, C. (1987). The existence of self-enforcing implicit contracts. The Quarterly Journal of Eco-

nomics 102 (1), 147–159.

Collard-Wexler, A., J. Asker, and J. De Loecker (2011). Productivity volatility and the misallocation of

resources in developing economies. Technical report.

DeAngelo, H. and L. DeAngelo (1991). Union negotiations and corporate policy: A study of labor concessions

in the domestic steel industry during the 1980s. Journal of Financial Economics 30 (1), 3–43.

DeMarzo, P. and R. Kaniel (2017). Relative pay for non-relative performance: Keeping up with the joneses

with optimal contracts. Working Paper .

DeMarzo, P. M., M. J. Fishman, Z. He, and N. Wang (2012). Dynamic agency and the q theory of investment.

The Journal of Finance 67 (6), 2295–2340.

Den Haan, W. J., G. Ramey, and J. Watson (2000). Job destruction and propagation of shocks. American

Economic Review 90 (3), 482–498.

DeVaro, J., J.-H. Kim, and N. Vikander (2018). Non-performance pay and relational contracting: Evidence

from ceo compensation. The Economic Journal 128 (613), 1923–1951.

Fuchs, W. (2015). Subjective evaluations: Discretionary bonuses and feedback credibility. American Eco-

nomic Journal: Microeconomics 7 (1), 99–108.

Garvey, G. T. and T. T. Milbourn (2006). Asymmetric benchmarking in compensation: Executives are

rewarded for good luck but not penalized for bad. Journal of Financial Economics 82 (1), 197–225.

Glaeser, E. L., R. La Porta, F. Lopez-de Silanes, and A. Shleifer (2004). Do institutions cause growth?

Journal of economic Growth 9 (3), 271–303.

Gopinath, G., S. Kalemli-Ozcan, L. Karabarbounis, and C. Villegas-Sanchez (2017). Capital allocation and

productivity in south europe. The Quarterly Journal of Economics 132 (4), 1915–1967.

Hoffmann, F. and S. Pfeil (2010). Reward for luck in a dynamic agency model. The Review of Financial

Studies 23 (9), 3329–3345.

Holmstrom, B. (1979). Moral hazard and observability. The Bell Journal of Economics, 74–91.

Hsieh, C.-T. and P. J. Klenow (2009). Misallocation and manufacturing tfp in china and india. The Quarterly

Journal of Economics 124 (4), 1403–1448.

Kaufmann, D., A. Kraay, and M. Mastruzzi (2011). The worldwide governance indicators: methodology and

analytical issues. Hague Journal on the Rule of Law 3 (2), 220–246.

Koren, M. and S. Tenreyro (2007). Volatility and development. The Quarterly Journal of Economics 122 (1),

243–287.

Kwon, S. (2016). Relational contracts in a persistent environment. Economic Theory 61 (1), 183–205.

Levin, J. (2003). Relational incentive contracts. American Economic Review 93 (3), 835–857.

MacLeod, W. B. and J. M. Malcomson (1989). Implicit contracts, incentive compatibility, and involuntary

unemployment. Econometrica, 447–480.

Oyer, P. (2004, August). Why do firms use incentives that have no incentive effects? The Journal of

Finance 59 (4), 1619–1649.

21

Porta, R. L., F. Lopez-de Silanes, A. Shleifer, and R. W. Vishny (1998). Law and finance. Journal of Political

Economy 106 (6), 1113–1155.

Powell, M. L. (2017). Productivity and credibility in industry equilibrium. Rand Journal of Economics,

forthcoming .

Ramey, G. and J. Watson (1997). Contractual fragility, job destruction, and business cycles. The Quarterly

Journal of Economics 112 (3), 873–911.

Spear, S. E. and S. Srivastava (1987). On repeated moral hazard with discounting. The Review of Economic

Studies 54 (4), 599–617.

Stock, J. and M. Yogo (2005). Asymptotic distributions of instrumental variables statistics with many in-

struments, Volume 6. Chapter.

Sundaram, R. K. (1996). A first course in optimization theory. Cambridge university press.

Syverson, C. (2004). Market structure and productivity: A concrete example. Journal of Political Econ-

omy 112 (6), 1181–1222.

Syverson, C. (2011). What determines productivity? Journal of Economic literature 49 (2), 326–65.

22

Data Appendix

Sources and Sample Construction

A. Rule of Law Measure

We obtain this measure directly from the Worldwide Governance Indicators (WGI) project website (Kauf-

mann et al., 2011). The database covers 216 countries from 1996 to 2016.

B. Country-Level GDP / TFP Growth Rate

We use country-level GDP/TFP growth rates in the Adjusted Growth Accounting and Total Factor Pro-

ductivity, 1950-2016 from the Conference Board (TCB) Total Economy Database website. After we merge

this database with WGI database, our sample covers 123 countries over 1996-2016. For each country c, the

variable V OLc denotes the volatility of the country c’s GDP (TFP) growth rates from 1996 to 2016. The

Average Rule of Law AV G RLc denotes the average of Rule of Law of the country c from 1996 to 2016.

C. Early European Settler’s Mortality Rates

We obtain early European settler’s mortality rates used in Acemoglu et al. (2001) from Daron Acemoglu’s

website. After merging with TCB Total Economy Database and WGI, our sub-sample covers 50 countries.

D. Firm-Level TFP Dispersion

We closely follow data construction procedures in Gopinath et al. (2017). We download all unconsolidated

financial statements of all manufacturing firms (2-digit NACE code from C.10-C.33) with from the entire

ORBIS universe. Then, we drop entire history of a firm if the firm has any basic accounting errors in any year

(e.g. negative total assets, employment, sales). After removing observations that indicate basic accounting

errors, we focus on ones that has non-missing data on total assets, tangible fixed assets, sales, operating

revenues, employment, and country information. We also keep a firm-year only if value added, defined as

the difference between the operating revenue and material costs, is positive. For informativeness, we drop

countries with less than one hundred observations (firm-years). After these procedures, we have 4,786,856

firm-years with 27 countries. Nominal values are deflated via Producer Price Indices (PPI) obtained from

the Datastream database.

We adopt a standard methodology for TFP estimation in the literature. For each country c, we use the

methodology from Ackerberg et al. (2015) to separately estimate parameters βl,c and βk,c:

log(yi,c,t) = log(Zi,c,t) + βl,clog(li,c,t) + βk,clog(ki,c,t) + εi,c,t

log(yi,c,t) denotes the nominal value added deflated by PPI for the domestic manufacturing firms, log(li,c,t)

the nominal wage bill deflated by the same PPI, and log(ki,c,t) the value of the sum of tangible and intangible

fixed assets deflated by PPI for the investment good.10 We use intermediate goods as a control variable.

For each country c, LN TFP9010c denotes the TFP difference between the firm with the top 10% produc-

tivity and the bottom 10% firm in a country c in the logarithmic scale. As samples above, AV G RLc denotes

10We deflate all variables by economy-wide PPI’s for following countries, because other PPI’s were are not available: Bosniaand Herzegovina, Montenegro, the Republic of Macedonia, Romania, Serbia, and Ukraine.

23

the average rule of law of the country c over the corresponding sample period. After merging the ORBIS

sample with WGI data, our final sample contains 27 countries.

Summary Statistics

Table 1: Sample After Merging WGI and TCB Total Economy Database

Variable Obs Mean Median Std. Dev.Average Rule of Law 123 .097 -.105 1.028Volatility of GDP Growth Rates 123 3.411 2.812 2.137Volatility of TFP Growth Rates 123 3.159 2.748 2.026

Table 2: Sub-Sample of Table 1 After Merging with Acemoglu et al. (2001)

Variable Obs Mean Median Std. Dev.Average Rule of Law 50 -.187 -.456 .928Volatility of GDP Growth Rates 50 2.763 2.425 1.578Volatility of TFP Growth Rates 50 2.71 2.513 1.289Log(Settler Mortality) 50 4.495 4.266 1.275

Table 3: Sample After Merging WGI and ORBIS

Variable Obs Mean Median Std. Dev.Average Rule of Law 27 .843 .944 .81490/10 Difference in Firm-Level TFP 27 1.246 1.093 .45

24

Robustness Checks for Empirical Analyses

In this section, we check robustness of our empirical analyses. In Table 4, we use aggregate variance itead of

aggregate volatility to check the robustness of equation 3. In Table 5, we check the robustness of equation

4 by clustering at different levels and alternative econometric specifications. Furthermore, we use aggregate

variance instead of aggregate volatility in Table 6. In Table 7, we again use aggregate variance instead of

aggregate volatility to check the robustness of equation 5. While most results involving TFP growth rates

are statistically significant, results for GDP growth rates are less often statistically significant.

Moreover, in untabulated regressions, to address the finite sample bias associated with robust standard errors,

we also estimate standard errors without adjusting for heteroskedasticity. Most of our results involving TFP

growth rates remain statistically significant even under non-robust standard errors, but the result involving

GDP growth rates is more mixed11. We believe that TFP results are more relevant for our theoretical results,

because other determinants of GDP growth rates (e.g. population growths) may potentially confound our

regressions results involving aggregate GDP growth rates.

Table 4: Varc = α+ βOLSAV G RLc(+Continent Dummies) + εc

Var GDP Var GDP Var GDP Var GDP Var TFP Var TFP Var TFP Var TFPAVG RL -8.036 -8.036 -11.306 -11.306 -8.539 -8.539 -11.088 -11.088

(2.341) (3.374) (3.880) (7.232) (2.201) (2.743) (3.611) (5.735)Region FE No No Yes Yes No No Yes YesClustering No Yes No Yes No Yes No Yes

(Continent)

The table reports ordinary least squares (OLS) regression results of a country’s variance of aggregate growth rateson its average rule of law. Var GDP (Var TFP) is calculated as the variance of aggregate GDP (TFP) growth ratesover 1996-2016 from TCB Total Economy Database. AVG RL is calculated as the average of rule of law measureover the from World Governance Index. Region FE denotes whether continent dummies are included in the righthand side. Parentheses reports standard errors. Clustering denotes whether standard errors are clustered atcontinent levels. All standard errors are robust to heteroskedasticity.

Table 5: V olc = α+ βIVAV G RLc(+Region Dummies) + εc

Vol GDP Vol GDP Vol GDP Vol TFP Vol TFP Vol TFPAVG RL -.517 -.676 -.676 -.664 -.669 -.669

(.250) (.157) (.403) (.225) (.148) (.296)Region FE No Yes Yes No Yes YesClustering No Yellow Fever Region No Yellow Fever Region

The table reports the second-stage regression results of a country’s volatility of aggregate growth rates on itsaverage rule of law using from the instrumental variable (IV) method. The only excluded instrumental variable isAcemoglu et al. (2001)’s logarithm of early European settler’s mortality. Vol GDP (Vol TFP) is calculated as thevolatility of aggregate GDP (TFP) growth rates over 1996-2016 from TCB Total Economy Database. AVG RL iscalculated as the average of rule of law measure over the from World Governance Index. Region FE denotes whetherregion dummies (Asia, Africa, and Others) are included in the right hand side. Yellow Fever denotes whether thevector yellow fever is present today. Parentheses reports standard errors. Clustering denotes the level at whichstandard errors are clustered. All standard errors are robust to heteroskedasticity.

11These results are available upon request.

25

Table 6: V arc = α+ βIVAV G RLc(+Region Dummies) + εc

Var GDP Var GDP Var GDP Var GDP Var TFP Var TFP Var TFP Var TFPAVG RL -3.108 -3.108 -4.771 -4.771 -3.589 -3.589 -3.680 -3.680

(1.593) (.851) (2.092) (2.202) (1.445) (1.133) (1.552) (1.683)Region FE No No Yes Yes No No Yes YesClustering No Yes No Yes No Yes No Yes(Region)

The table reports the second-stage regression results of a country’s variance of aggregate growth rates on its averagerule of law using from the instrumental variable (IV) method. The only excluded instrumental variable is Acemogluet al. (2001)’s early European settler’s (log) mortality rates Sc. Var GDP (Var TFP) is calculated as the volatilityof aggregate GDP (TFP) growth rates over 1996-2016 from TCB Total Economy Database. AVG RL is calculatedas the average of rule of law measure over the from World Governance Index. Region FE denotes whether regiondummies (Asia, Africa, and Others) are included in the right hand side. Parentheses reports standard errors.Clustering denotes the level at which standard errors are clustered. All standard errors are robust toheteroskedasticity.

Table 7: TFP9010c = α+ βAV G RLc(+Euro) + εc

SD TFPc = α+ βAV G RLc(+Euro) + εc

TFP9010 TFP9010 TFP9010 SD TFP SD TFP SD TFP SD TFPAVG RL -.4728 -.4728 -.483 -.2215 -.2215 -.2244 -.2244

(.0588) (.0376) (.018) (.0249) (.0094) (.0254) (.004)Euro Dummy No No Yes No No Yes Yes

Clustering No Yes Yes No Yes No Yes(Region)

The table reports the ordinary least squares (OLS) regression results of a country’s firm-level productivitydispersion measures on its average rule of law. TFP9010 is calculated as the difference in logged firm-level totalfactor productivity (TFP) between top 10% and bottom 10% for a country over the sample period. TFP9010 iscalculated as the volatility of logged firm-level total factor productivity (TFP) within a country over the sampleperiod. AVG RL is calculated as the average of rule of law measure over the from World Governance Index. RegionFE denotes whether region dummies (Asia, Africa, and Others) are included in the right hand side. Parenthesesreports standard errors. Clustering denotes the level at which standard errors are clustered. All standard errors arerobust to heteroskedasticity.

26

Appendix for Proofs

Proof of Proposition 1. Suppose that the optimal dynamic contract generates surplus s∗(θ) starting from

the initial state θ and that it specifies after the public history ht−1 and state θt−1 the fixed wage payments

wt(ht−1, θt−1) and contingent payments bt(h

t−1, θt−1, yt, θt) which induce from the agent effort at(ht−1, θt−1)

and yields him continuation expected value vt(ht−1, θt−1, yt, θt). Denote the value of the principal’s outside

option as π and the agent’s outside option as u.

Fix θ and let v∗(θ) ∈ [u, s∗(θ) − π] be the agent’s value from the optimal contract. We now construct from

the optimal contract a Markovian relational contract that transfers any variation in the continuation value

into the bonus payments that implements et(ht−1, θt−1) and yields the agent value v∗(θt−1).

Define the history-dependent bonus payments:

bt(ht−1, θt−1, yt, θt) = bt(h

t−1, θt−1, yt, θt) +δ

1− δ

(vt(h

t−1, θt−1, yt, θt)− v∗(θt−1))

and the fixed payments:

wt(ht−1, θt) = v∗(θ) + c

(at(h

t−1, θt−1))− Eyt,θt

[bt(h

t−1, θt−1, yt, θt)|at(ht−1, θt−1), θt−1

]

Notice that the contract(wt(h

t−1, θt), bt(ht−1, θt, yt, θt+1)

)induces the same effort level at(h

t−1, θt−1) as in

the optimal contract and yields the agent his outside value v∗(θt−1).

Now define the Markovian bonus payments b∗′(θt−1, yt, θt) as the bonus payments b that maximize the

expected profit, which is equal to the expected per-period joint surplus, for each state:

b∗′(θ−1, y, θ) = argmaxbt(ht−1,θ−1,·;·),ht−1,t

Eθ[θ|θ−1] + at(ht−1, θ−1)︸ ︷︷ ︸

=Ey[y|at(ht−1,θ−1)

]−c(at(h

t−1, θ−1))

s.t. at(ht−1, θ−1) = argmax

a,θ′Ey,θ′

[bt(h

t−1, θ, y, θ′)|a, θ−1

]− c(a)

Let a′(θ) be the solution

a′(θ−1) = argmaxa

Ey,θ[b∗′(θ−1, y, θ)|a, θ−1

]− c(a)

and define the state-dependent wages:

w∗′(θ−1) = v∗(θ−1) + c(a′(θ−1))− Ey,θ′[b∗′(θ−1, y, θ)|a′(θ−1), θ−1

](6)

By construction the Markovian contract (w∗′, b∗′) is self-enforcing, yields to the agent value v∗(θt−1) and

to the principal s∗(θt−1) − v∗(θt−1). Finally, because this Markovian contract repeats in each following

period, the continuation contract is self-enforcing. Thus the constructed contract is self-enforcing and gives

a per-period expected surplus s∗(θt−1).

Notice that taking v∗ ∈ [u,min{s∗(H)− π, s∗(L)− π}], a state-independent value for the agent, and setting

27

w∗′(θ−1) = v∗ − Ey,θ[b∗′(θ−1, y, θ)|a′(θ−1), θ−1

]+ c(a′(θ−1)) for all θ−1 redistributes the surplus from the

optimal Markovian relational contract and establishes that the property (R-1) holds in any optimal Markovian

contract.

Now, to establish the property (R-2) we start by noticing that by (R-1) the optimal Markovian relational

contract reduces to solving the following equations recursively (we illustrate for the lagged state is H and

the arguments follow analogously for the state L):

Π(H) = max{bH(·,θ),a}

(1− δ)

(Eθ[θ|θ−1 = H]− w(H)

+ a(

1− λbH(1, H)− (1− λ)bH(1, L))

− (1− a)(λbH(0, H) + (1− λ)bH(0, L)

))+ δ[λΠ(θH) + (1− λ)Π(θL)]

[IC] c′(a) = λ(bH(1, H)− bH(0, H)) + (1− λ)(bH(1, L)− bH(0, L));

[PC’s] Π(H) ≥ π, v∗ ≥ u;

[DEPH] bH(1, H) ≤ δ

1− δΠ(H);

[DEPL] bH(1, L) ≤ δ

1− δΠ(L);

[DEAH] − bH(0, H) ≤ δ

1− δv∗ − u.

[DEAL] − bH(0, L) ≤ δ

1− δv∗ − u.

Notice that by (R-1) without loss we set u∗(θ) = u and hence ensuring the participation constraints can

safely be dropped and that since u = 0 the dynamic enforcement constraints [DEPH] and [DEPL] reduce to

respectively, bH(0, H) ≥ 0 and bH(0, L) ≥ 0. Furthermore, using 6 we have wH = u+c(a)−a(λbH(1, H)+(1−λ)bH(1, L))− (1− a)(λbH(0, H) + (1− λ)bH(0, L)) and the principal’s contracting problem further simplifies

to solving:

Π(H) = max{bH(1,θ′),bH(0,θ′)}θ′ ,a

(1− δ)[Eθ′ [θ′|θH ] + a− c(a)

]+ δ[λΠ(θH) + (1− λ)Π(θL)

]s.t.

[IC] c′(a) = λ(bH(1, H)− bH(0, H)) + (1− λ)(bH(1, L)− bH(0, L));

[DEPH] bH(1, H) ≤ δ

1− δΠ(H);

[DEPL] bH(1, L) ≤ δ

1− δΠ(L);

[DEA] bH(0, H) ≥ 0 and bH(0, L) ≥ 0.

Notice that only the spread in bonus payments bH(1, θ)− bH(0, θ) matter for incentives. The constraint set

28

in contracting problem therefore further simplifies to:

[IC] c′(a) = λ(bH(1, H)− bH(0, H)) + (1− λ)(bH(1, L)− bH(0, L));

[DEH] bH(1, H)− bH(0, H) ≤ δ

1− δΠ(H);

[DEL] bH(1, L)− bH(0, L) ≤ δ

1− δΠ(L).

Inspecting [IC] and [DE]’s we see that without loss the principal uses contracts that do not involve paying

a bonus following low output: bH(0, θ) = 0, and bL(0, θ) = 0 follows from analogous arguments. This

establishes (R-2).

Proof of Lemma 1. To establish (E-1), we first show that max{a∗} ≤ aFB under the optimal contract

{b∗, a∗,Π∗}. Suppose to the contrary that there exists a state θ0 with a∗(θ0) > aFB . Now, consider an

alternative bonus system b(θ−1, θ) = b∗(θ−1, θ) − ε × 1(θ−1 = θ0) by arbitrarily small ε > 0. A new effort

level a(θ0) under b would result in an effort level higher than aFB but strictly less than a∗(θ0), from the IC

condition a(θ−1) = g[E(b(θ, θ−1)|θ−1)]. Since the function x − c(x) is concave and attains its maximum at

aFB , decrease in effort a(θ0) would increase a − c(a). From the system of Bellman equations, the following

relationship holds:[Π(H)

Π(L)

]=

[γ 1− γ

1− γ γ

][a(H)− c(a(H)) + H

a(L)− c(a(L)) + L

](7)

From (7), a strict increase in a − c(a) would strictly increase all of principal’s surpluses Π as well. Denote

the new principal’s surpluses under b as Π.

Notice that new bonus payments never exceed bonus pays under the original contract, so that:

(1− δ)b(θ−1, θ) ≤ (1− δ)b∗(θ−1, θ) ≤︸︷︷︸DEP

δΠ∗(θ) < δΠ(θ)

Yet, this new contract obtains strictly higher joint payoffs in both states compared to initial optimal contract

without violating dynamic enforceability conditions, which contradicts the optimality of the initial contract.

It therefore follows that max{a∗} ≤ aFB .

Next, we show that a∗(H) ≥ a∗(L) under the optimal contract {b∗, a∗,Π∗}. Suppose to the contrary and

a∗(H) < a∗(L). Now, as we have observed in the previous part of the current proof that max{a∗} ≤ aFB

must hold, it must be that a∗(H) < a∗(L) ≤ aFB .

From the agent’s incentive compatibility condition, expected bonus payments should be equal to the agent’s

marginal cost of effort. Thus,

E(b∗(L, θ)|θ−1 = L) = c′(a∗(L)) > c′(a∗(H)) = E(b∗(H, θ)|θ−1 = H)

where the inequality follows from the assumption that a∗(L) > a∗(H) and c′′ > 0.

29

Now, construct a new payment scheme b as follows:

b(H,H) = b∗(L,L)− 1− λλ

(δ

1− δΠ∗(L)− b∗(L,H)

)b(H,L) =

δ

1− δΠ∗(L)

b(L, θ) = b∗(L, θ) ∀θ ∈ Θ

It is routine to verify that E(b(θ, θ−1)|θ−1) = E(b∗(L, θ)|θ−1 = L) for ∀θ−1 ∈ Θ. Denote the new recom-

mended actions under b as a and the principal’s surplus from this new contract by Π. Recall that x− c(x) is

concave and attains maximum at aFB . Since the recommended effort in the high state a(H) = a∗(L) ≤ aFB ,

the new contract increases the effort level a(H) compared to the initial contract, due to the incentive com-

patibility condition. This results in a higher a(H)− c(a(H)), which would strictly increase all of principal’s

surpluses under new payment b from a similar representation as (7).

Summarizing the observations thus far, we have the following relationships hold:

Π(H) > Π(L) > Π∗(L), Π(H) > Π∗(H)

where the first inequality follows from H > L and that principal’s surpluses in all states would strictly

increase under the new payment. The other inequalities merely state that the principal’s surpluses increased.

Now, it remains to verify that dynamic enforceability constraints are satisfied under the new payment system.

It is routine to check:

δΠ(H) > δΠ∗(H) ≥ (1− δ)b∗(L,H) = (1− δ)b(L,H)

δΠ(L) > δΠ∗(L) ≥ (1− δ)b∗(L,L) = (1− δ)b(L,L)

δΠ(L) > δΠ∗(L) = (1− δ)b(H,L)

which follow from the definition of b, the system of equations at the beginning of the proof, or DEP of the

initial optimal contract. Now,

δΠ(H) > (2− 1

λ)δΠ∗(L) + (

1

λ− 1)δΠ∗(H)

≥ (2− 1

λ)δΠ∗(L) + (

1

λ− 1)(1− δ)b∗(L,H)

= δΠ∗(L) + (1− 1

λ)δΠ∗(L) + (

1

λ− 1)(1− δ)b∗(L,H)

≥ (1− δ)b∗(L,L) + (1− 1

λ)δΠ∗(L) + (

1

λ− 1)(1− δ)b∗(L,H)

= (1− δ)b(H,H)

where the first inequality follows from the observation Π(H) > max{Π∗} and 2 > 1λ , the second and the

third inequality from dynamic enforceability constraints of the original contract. Finally, the last equality

follows from the definition of b(H,H). As in the first part of the proof to establish max{a∗} ≤ aFB , the

existence of this new contract {b, a, Π} contradict the optimality of the originally proposed contract. These

two contradictions establish the part (E-1).

Denote g := (c′)−1 the inverse of the marginal cost and θ = E(θ+1|θ) for convenience. Suppose to

30

the contrary. Denote {b∗, a∗,Π∗} an optimal Markovian contract in which a∗(θ′) < g(1) = aFB with

(1 − δ)b∗(θ′, θ′′) < δΠ∗(θ′′) for some (θ′, θ′′) ∈ Θ2. Fix such θ′, θ′′ and let ε := 12 min{ δ

1−δΠ∗(θ′′) −b∗(θ′, θ′′), 1−E(b∗(θ′,θ|θ−1=θ′))

λ } > 0. Consider the alternative bonus system by b(θ−1, θ) = b∗(θ−1, θ) + ε ×1((θ−1, θ) = (θ′, θ′′)). Under this new bonus system {b}:

E(b(θ′, θ)|θ−1 = θ′) ≤ E(b∗(θ′, θ)|θ−1 = θ′) + λε < 1

E(b(θ′, θ)|θ−1 = θ′) ≥ E(b∗(θ′, θ)|θ−1 = θ′) + (1− λ)ε > E(b∗(θ′, θ)|θ−1 = θ′)

To establish (E-2), denote the principal’s surpluses induced from b as Π, and the new recommended effort

levels a. From similar representation as 7, Π(θ) > Π∗(θ) ∀θ, because a(θ′) − c(a(θ′)) > a∗(θ′) − c(a∗(θ′))from a∗(θ′) ∈ (a∗(θ′), aFB) and a(θ) ≥ a∗(θ) ∀θ. Now, we will arrive at a contradiction if the new contract

{b, a, Π} satisfies all the dynamic enforceability conditions, because the new contract strictly does better than

the optimal contract. If (θ−1, θ) 6= (θ′, θ′′),

(1− δ)b(θ−1, θ) = (1− δ)b∗(θ−1, θ) ≤ δΠ∗(θ) < δΠ(θ)

Moreover, if (θ−1, θ) = (θ′, θ′′),

(1− δ)b(θ′, θ′′) = (1− δ)[b∗(θ′, θ′′) + ε] ≤︸︷︷︸Def’n of ε

1

2[δΠ∗(θ′′) + (1− δ)b∗(θ′, θ′′)] ≤ δΠ∗(θ′′) < δΠ(θ′′)

Proof of Lemma 2. Applying the result (E-1), a∗(H) − c(a∗(H)) + H > a∗(L) − c(a∗(L)) + L. From the

representation (7), this implies that the joint payoff is strictly higher under the favorable state and hence it

establishes Lemma 2.

Proof of Proposition 2. Let δ = 11+ΠFB(H)

so that δ1−δΠFB(H) = 1. Choose any δ < δ. Using incentive

compatibility condition,

c′(a∗(H)) = E(b∗(H, θ)|θ−1 = H) ≤ δ

1− δΠFB(H) < 1

where the second inequality holds from dynamic enforceability condition, and the last strict inequality from

δ < δ.

Now, due to weak monotonicity in the effort level (E-1) and the observation above, it must be that both

states’ effort levels fall strictly below the first best level (c′)−1(1). This implies that b∗(θ−1, θ) = δ1−δΠ∗(θ)

∀ θ−1, θ ∈ Θ due to Lemma 1.

Now applying the Lemma 1 yields:

b∗(θ−1, L) =δ

1− δΠ∗(L) <

δ

1− δΠ∗(H) = b∗(θ−1, H)

31

Proof of Proposition 3. The existence of such δ = δ follows from analogous arguments as Proposition 2.

From the agent’s incentive compatibility,

c′(a∗(H)) = E(b∗(H, θ)|H) =δ

1− δE(Π∗(θ)|H)

>δ

1− δE(Π∗(θ)|L) = E(b∗(L, θ)|L) = c′(a∗(L))

where the strict inequality in the middle follows from the strict monotonicity in payoffs, establishing the part

(i). Finally, c′′ > 0 would imply that a∗(H) > a∗(L), establishing the part (ii).

Proof of Claim 4.1. We first show that b∗(L,L) < b∗(L,H). By Lemma 1 and a∗ (L) < a∗ (H), we must have

strict underprovision of effort when θ−1 = L: a∗ (L) < a∗ (H) ≤ aFB . We can apply the second part of Lemma

1 to a∗(L) < aFB , so that all dynamic enforceability conditions bind when θ−1 = L: b∗(L, θ) = δ1−δΠ∗(θ).

By Lemma 2, we have:

b∗(L,L) =δ

1− δΠ∗(L) <

δ

1− δΠ∗(H) = b∗(L,H)

Now it suffices to show b∗(H,L) < b∗(H,H). Suppose to the contrary, so that b∗(H,H) ≤ b∗(H,L).

b∗(H,L) ≤ δ1−δΠ∗(L) from its dynamic enforceability condition. Hence, b(H, θ) ≤ δ

1−δΠ∗(L), so that the

expected bonus E(b∗ (H, θ) |θ−1 = H) ≤ δ1−δΠ∗(L).

Since a∗(L) < aFB , all dynamic enforceability conditions bind when θ−1 = L: b∗(L, θ) = δ1−δΠ∗(θ). Hence,

E(b∗ (L, θ) |θ−1 = L) = δ1−δ

(λΠ∗(L) + (1 − λ)Π∗(H)

). Thus, from the previous paragraph, we must have

E(b∗ (H, θ) |θ−1 = H) < E(b∗ (L, θ) |θ−1 = L)

From the agent’s incentive compatibility conditions,

c′(a∗(H)) = E(b∗ (H, θ) |θ−1 = H) < E(b∗ (L, θ) |θ−1 = L) = c′(a∗(L))

Since the marginal cost of effort is strictly increasing, this implies that a∗(L) > a∗(H). But this contradicts

the first part of Lemma 1 that a∗(L) ≤ a∗(H) ≤ aFB .

Proof of Proposition 4. Outline of the Proof

The proof is quite long, so it is helpful to sketch the gist of proof before we start the proof. It would be useful

to revisit this outline after each step to keep track of what is going on. The figure 5 is a graphical illustration

of the proof. In the step 1, we can divide [0, 1] into two distinct segments: I1 = {α ∈ [0, 1] : a∗(H,α) < aFB},and I2 = {α ∈ [0, 1] : a∗(H,α) = aFB}. Next, we show that I1 = [0, α0) and I2 = [α0, 1] for some α0. In step

2, we show that Y (H,α)Y (L,α) is decreasing in α in the closed interval I2. According to one of our auxillary claims

presented below, a∗(θ, α) is weakly increasing in α. Hence, the logic is that Y (H,α) = a∗(H,α) +E(θ|θ−1 =

H) = aFB + E(θ|θ−1 = H) has no more room for further increase because it already reached the maximal

possible level, whereas the denominator Y (L,α) can potentially increase when α increases. The step 3 is a

preliminary step before implicit differentiation in step 4, to check whether we can indeed implicitly differenti-

32

Figure 5: Graphical Illustration

ate Y (H,α)Y (L,α) w.r.t. α’s in I1. It turns out that we can implicitly differentiate everywhere on I1, except at most

one α ∈ I1. We shall denote such point as α0, the region on the left side of α0 as IL1 = [0, α0), the region on

the right side of α0 as IR1 = (α0, α0). So even when α0 does exist, we can still use implicit differentiation to

analyze Y (H,α)Y (L,α) in IL1 ∪ IR1 . In step 4, we implicitly differentiate the ratio Y (H,α)

Y (L,α) in the region IL1 ∪ IR1 (or

I1 if α0 does not exist), and show that the ratio ∂∂α

Y (H,α)Y (L,α) ≤ 0 whenever implicit differentiation is possible.

After we are done with the step 4, we know that Y (H,α)Y (L,α) is decreasing in each of region IL1 , IR1 (or I1 if α does

not exist) and I2, but we still do not know what might happen around α0 and α due to jumps. The last step

in step 5 will analyze local behaviors around α0 and α0 with “pasting” argument. Thanks to the continuity

of Y (H,α)Y (L,α) , irregular jumps are ruled out. Hence, we conclude by showing that Y (H,α)

Y (L,α) is decreasing in all of

[0, 1] and finish the proof.

Auxiliary Claims

33

Though often referenced throughout the main proof for the amplification effect, these auxillary claims are

not directly related to the proof of the amplification effect and can be verified independently of the main

proof. Hence, their proofs will be presented after the main proof. 12

Lemma 3 (Monotonicity). In any optimal Markovian relational contract the constraint optimal effort and

the principal’s profits satisfy

(M’-1) a∗(L,α) ≤ a∗(H,α) ≤ (c′)−1(1) = aFB ; and

(M’-2) Π∗(L,α) < Π∗(H,α), Π∗(L,α) ≤ ΠFB(L) and Π∗(H) ≤ ΠFB(H).

Lemma 4 (Binding DE’s). In any optimal Markovian relational contract, we have that a∗(θ−1) < aFB

implies (1− α)b(θ−1, θ) = δ1−δΠ(θ) for both θ = H,L.

Lemma 5 (Relationship between value function and recommended effort). In any Markovian relational

contract, the principal’s surplus can be expressed according to the following system of equations:[Π(H,α)

Π(L,α)

]=

[γ 1− γ

1− γ γ

][a(H,α)− c(a(H,α)) + H

a(L,α)− c(a(L,α)) + L

]

Here, H = E(θ|θ−1 = H) = λH+(1−λ)L and L = E(θ|θ−1 = L) = (1−λ)H+λL, so that H > H > L > L.

Lemma 6 (Continuity). In any optimal Markovian relational contract, we have that

(C-1) Both optimal recommended efforts a∗(H,α), a∗(L,α) are continuous with respect to α.

(C-2) Both principal’s value functions Π∗(H,α) and Π∗(L,α) are continuous with respect to α.

Lemma 7 (Monotonicity in α). In any optimal Markovian relational contract, we have that

(I-1) a∗(H,α) and a∗(L,α) are weakly increasing with respect to α everywhere in α ∈ [0, 1].

(I-2) Π∗(H,α) and Π∗(L,α) are weakly increasing with respect to α everywhere in α ∈ [0, 1].

Lemma 8. In any optimal Markovian relational contract with a∗(H,α) < aFB , (1−δ)δ (1 − α)c′′(a∗(θ)) −

γ(1− c′(a∗(θ))) ≥ 0 for both θ = L,H.

Lemma 9. In any optimal Markovian relational contract with a∗(H,α) < aFB , c′(a∗(H,α))

a∗(H,α)+H≤ c′(a∗(L,α))

a∗(L,α)+L.

Proof of Amplification Effect

12Yet, these proofs for auxillary claims are NOT fully independent of each other; claims that appear earlier would be used inlater claims.

34

Step 1) Let I1 := {α ∈ [0, 1] : a∗(H,α) < aFB}, and I2 := {α ∈ [0, 1] : a∗(H,α) = aFB}. Then, there exists

a unique α0 ≥ 0 such that I1 := [0, α0) and I2 := [α0, 1]. (I1 can be empty if α0 = 0)

We first show that 1 ∈ I2, so that I2 is non-empty. When α = 1, DE-P is no longer binding and we are

effectively in a standard principal agent with a risk-neutral agent without any bounds on transfer amount.

Hence, we can make the agent the residual claimant and achieve the first best. Thus 1 ∈ I2.

By continuity of a∗(H,α) from Lemma (C-1), I2 = [α ∈ [0, 1] : a∗(H,α) = 1] is closed. This implies

that inf I2 ∈ I2. Let α0 := inf I2. Now, fix any α ∈ [α0, 1]. Since a∗(H,α) is weakly increasing in α,

aFB = a∗(H,α0) ≤ a∗(H,α). Now, applying Lemma (M’-1), a∗(H,α) ≤ aFB . Combining two conclusions,

we have that aFB ≤ a∗(H,α0) ≤ aFB , or a∗(H,α0) = aFB . This implies that α ∈ I2. Thus, [α0, 1] ⊂ I2. By

definition of infinimum and supremum, I2 ⊂ [inf I2, sup I2] ⊂ [α0, 1]. Hence, I2 = [α0, 1]. Moreover, by mono-

tonicity shown in Lemma (M’-1), a∗(H,α) > aFB is impossible. Hence, from the definition, I1 = [0, 1] \ I2.

Thus, I2 = [0, α0).

Step 2) Y (H,α)Y (L,α) is weakly decreasing in I2.

By definition of I2 := {α ∈ [0, 1] : a∗(H,α) = aFB}, Y (H) = aFB + H is independent of α in I2. In

contrast, by Lemma (I-1), Y (L) = a∗(L,α) + L is a weakly increasing function from Lemma (I-1). More-

over, a∗(L,α) + L > 0, so that 1a∗(L,α)+L

is always a weakly decreasing function. Hence, when α ∈ I2,

Y (H,α)Y (L,α) = aFB+H

a∗(L,α)+Lis a weakly decreasing function. This completes the step 2.

Step 3) Then, a∗(θ, α) is differentiable everywhere in (0, α0) except at most one point.

By monotonicity in Lemma (M’-1), a∗(L,α) ≤ a∗(H,α). By assumption α ∈ (0, α0), a∗(H,α) < aFB Hence,

the recommended effort level for both states H,L are strictly less than the first best effort level aFB . This

implies that dynamic enforceability conditions for all possible past and contemporaneous shocks bind by

Lemma 4:

(1− δ)δ

b∗(H,H,α)× (1− α) = Π∗(H,α),(1− δ)δ

b∗(H,L, α)× (1− α) = Π∗(L,α)

(1− δ)δ

b∗(L,H, α)× (1− α) = Π∗(H,α),(1− δ)δ

b∗(L,L, α)× (1− α) = Π∗(L,α)

Notice that E(b∗(H, θ, α)|θ−1 = H) = λb∗(H,H,α) + (1 − λ)b∗(H,L, α) and E(b∗(L, θ, α)|θ−1 = L) =

λb∗(L,L, α) + (1− λ)b∗(L,H, α). Thus, taking the convex combination of each row, we have that:

35

(1− δ)δ

(1− α)[λb∗(H,H,α) + (1− λ)b∗(H,L, α)] = [λΠ∗(H,α) + (1− λ)Π∗(L,α)]

(1− δ)δ

(1− α)[λb∗(H,H,α) + (1− λ)b∗(H,L, α)] = [λΠ∗(L,α) + (1− λ)Π∗(H,α)]

We can plug in the incentive compatibility condition c′(a∗(θ−1, α)) = E(b∗(θ−1, θ, α)|θ−1) to express the

equations above more concisely: :

[(1−δ)δ (1− α)c′(a∗(H,α))

(1−δ)δ (1− α)c′(a∗(L,α))

]=

[λ 1− λ

1− λ λ

][Π∗(H,α)

Π∗(L,α)

]

Also, by Lemma 5, for any Markovian contract,

[Π∗(H,α)

Π∗(L,α)

]=

[γ 1− γ

1− γ γ

][a∗(H,α)− c(a∗(H,α)) + H

a∗(L,α)− c(a∗(L,α)) + L

].

Let γ := γλ+ (1− γ)(1− λ) ∈ ( 12 , 1) 13 . Two systems of equations above can be combined as:

[(1−δ)δ (1− α)c′(a∗(H,α))

(1−δ)δ (1− α)c′(a∗(L,α))

]=

[γ 1− γ

1− γ γ

][a∗(H,α)− c(a∗(H,α)) + H

a∗(L,α)− c(a∗(L,α)) + L

](DE2)

.

This system of equations forms a necessary condition for optimal effort choices a∗(H,α), a∗(L,α). Now, we

shall totally differentiate the system of equations above with respect to α, treating a∗(H,α) and a∗(L,α) as

endogenous variables.

[(1−δ)δ (1− α)c′′(a∗(H,α)) ∂

∂αa∗(H,α)

(1−δ)δ (1− α)c′′(a∗(L,α)) ∂

∂αa∗(L,α)

]−

[(1−δ)δ c′(a∗(H,α))

(1−δ)δ c′(a∗(L,α))

]=

[γ 1− γ

1− γ γ

][(1− c′(a∗(H,α))) ∂

∂αa∗(H,α)

(1− c′(a∗(L,α))) ∂∂αa

∗(L,α)

]13 This can be established through following algebraic manipulations:

γ = γλ+ (1− γ)(1− λ) <︸︷︷︸λ> 1

2

γλ+ (1− γ)λ = λ < 1

γ − (1− γ) =︸︷︷︸1−γ=(1−γ)λ+λ(1−γ)

γλ+ (1− γ)(1− λ)− [(1− γ)λ+ γ(1− λ)] = (2λ− 1)(2γ − 1) >︸︷︷︸γ> 1

2,λ> 1

2

0

36

Then, moving the vector

[(1−δ)δ (1− α)c′′(a∗(H,α)) ∂

∂αa∗(H,α)

(1−δ)δ (1− α)c′′(a∗(L,α)) ∂

∂αa∗(L,α)

]to the right hand side of the equation,

−

[(1−δ)δ c′(a∗(H,α))

(1−δ)δ c′(a∗(L,α))

]=

[γ 1− γ

1− γ γ

][(1− c′(a∗(H,α))) ∂

∂αa∗(H,α)

(1− c′(a∗(L,α))) ∂∂αa

∗(L,α)

]−

[(1−δ)δ (1− α)c′′(a∗(H,α)) ∂

∂αa∗(H,α)

(1−δ)δ (1− α)c′′(a∗(L,α)) ∂

∂αa∗(L,α)

]

Now, the first row of the right hand side of the equation above can be written as:

γ(1− c′(a∗(H,α)))∂

∂αa∗(H,α) + (1− γ)(1− c′(a∗(L,α)))

∂

∂αa∗(L,α)− (1− δ)

δ(1− α)c′′(a∗(H,α))

∂

∂αa∗(H,α)

=[γ(1− c′(a∗(H,α)))− (1− δ)δ

(1− α)c′′(a∗(H,α))]∂

∂αa∗(H,α) + (1− γ)(1− c′(a∗(L,α)))

∂

∂αa∗(L,α)

=[γ(1− c′(a∗(H,α)))− (1−δ)

δ (1− α)c′′(a∗(H,α)) (1− γ)(1− c′(a∗(L,α)))] [ ∂

∂αa∗(H,α)

∂∂αa

∗(L,α)

]

An almost identical operation yields that the second row of the right hand side is equal to:

[(1− γ)(1− c′(a∗(H,α))) γ(1− c′(a∗(L,α)))− (1−δ)

δ (1− α)c′′(a∗(L,α))] [ ∂

∂αa∗(H,α)

∂∂αa

∗(L,α)

]

Thus, the system of equations is equivalent to:

−

[(1−δ)δ c′(a∗(H,α))

(1−δ)δ c′(a∗(L,α))

]

=

A(α):=︷ ︸︸ ︷[γ(1− c′(a∗(H,α)))− (1−δ)

δ (1− α)c′′(a∗(H,α)) (1− γ)(1− c′(a∗(L,α)))

(1− γ)(1− c′(a∗(H,α))) γ(1− c′(a∗(L,α)))− (1−δ)δ (1− α)c′′(a∗(L,α))

][∂∂αa

∗(H,α)∂∂αa

∗(L,α)

](IFT1)

Given this representation, our claim on differentiability of a∗(θ, α) will be established if we show that

det(A(α)) 6= 0 for at most one point in (0, α0)14. Using the formula for determinant, we can write:

14This is well known. For instance, see Sundaram (1996)

37

det(A(α)) = [γ(1− c′(a∗(H,α)))− (1− δ)δ

(1− α)c′′(a∗(H,α))][γ(1− c′(a∗(L,α)))− (1− δ)δ

(1− α)c′′(a∗(L,α))]

− (1− γ)(1− c′(a∗(L,α)))× (1− γ)(1− c′(a∗(H,α)))

= (1− γ)2(1− c′(a∗(L,α)))(1− c′(a∗(H,α)))

× [[γ

1− γ− (1− δ)

δ

1− α1− γ

c′′(a∗(H,α))

(1− c′(a∗(H,α)))][

γ

1− γ− (1− δ)

δ

1− α1− γ

c′′(a∗(L,α))

(1− c′(a∗(L,α)))]− 1]

When α ∈ (0, α0), both a∗(L,α) ≤ a∗(H,α) < aFB = (c′)−1(1). Hence, c′(a∗(L,α)) < 1, c′(a∗(H,α)) < 1 by

strict monotonicity of c′. Hence, (1− γ)2(1− c′(a∗(L,α)))(1− c′(a∗(H,α))) 6= 0. Thus, we have:

det(A(α)) = 0 ↔ [γ − (1− δ)δ

(1− α)c′′(a∗(H,α))

(1− c′(a∗(H,α)))][γ − (1− δ)

δ(1− α)

c′′(a∗(L,α))

(1− c′(a∗(L,α)))] = (1− γ)2

Now, h(x) := (1−δ)δ (1 − α) c′′(x)

1−c′(x) − γ for convenience. Choose any α1, α2 ∈ (0, α0), with detA(α1) =

detA(α2) = 0. According to the equivalence relationship above, this means that h(a∗(H,α1))h(a∗(L,α1)) =

h(a∗(H,α2))h(a∗(L,α2)) = (1−γ)2. According to our Lemma 8, (1−δ)δ (1−α)c′′(a∗(θ, α))−γ(1−c′(a∗(θ, α))) ≥

0 if α ∈ (0, α0). Since h(x) = [ (1−δ)δ (1− α)c′′(x)− γ(1− c′(x))]× 1

1−c′(x) and c′(a(θ, α)) < 1 in the region of

our interest, all four h(a∗(H,α1)), h(a∗(L,α1)), h(a∗(H,α2)), h(a∗(L,α2)) should be strictly increasing.

h(a∗(H,α1))h(a∗(L,α1))− h(a∗(H,α2))h(a∗(L,α2))

=[h(a∗(H,α1))− h(a∗(H,α2))]h(a∗(L,α1)) + h(a∗(H,α2))[h(a∗(L,α1))− h(a∗(L,α2))]

One can see that h(a∗(θ, α)) is a strictly increasing function with respect to α in (0, α0), because both

a∗(θ, ·) h(·) are strictly increasing functions in the region of interest. Since h(a∗(L,α1)), h(a∗(H,α2)) > 0,

the expression above is strictly negative or positive if α1 6= α2. Yet, since h(a∗(H,α1))h(a∗(L,α1)) =

h(a∗(H,α2))h(a∗(L,α2)) = (1− γ)2 for detA(α1) = detA(α2) = 0 to hold, it must be that α1 = α2. Thus,

there is at most one point where det(A(α)) 6= 0.

Step 4) If α ∈ {α ∈ (0, α0) : det(A(α)) 6= 0}, ∂∂α

Y (H,α)Y (L,α) ≤ 0.

Recall that the system of equations that characterize the ∂∂αa

∗(H,α), ∂∂αa

∗(L,α) in I1 is:

−

[(1−δ)δ c′(a∗(H,α))

(1−δ)δ c′(a∗(L,α))

]

=

A(α):=︷ ︸︸ ︷[γ(1− c′(a∗(H,α)))− (1−δ)

δ (1− α)c′′(a∗(H,α)) (1− γ)(1− c′(a∗(L,α)))

(1− γ)(1− c′(a∗(H,α))) γ(1− c′(a∗(L,α)))− (1−δ)δ (1− α)c′′(a∗(L,α))

][∂∂αa

∗(H,α)∂∂αa

∗(L,α)

]

38

If α ∈ {α ∈ I1 : det(A(α)) 6= 0}, we can post-multiply 1−δδ and the inverse matrix A(α)−1 on both sides:

[∂∂αa

∗(H,α)∂∂αa

∗(L,α)

]=

(1− δ)δ

1

det(A(α))[(1−δ)δ (1− α)c′′(a∗(L,α))− γ(1− c′(a∗(L,α))) (1− γ)(1− c′(a∗(L,α)))

(1− γ)(1− c′(a∗(H,α))) (1−δ)δ (1− α)c′′(a∗(H,α))− γ(1− c′(a∗(H,α)))

][c′(a∗(H,α))

c′(a∗(L,α))

]

From the matrix multiplication on the right hand side, notice that:

∂

∂αa∗(H,α) =

1

det(A(α))

(1− δ)δ︸ ︷︷ ︸>0

[(1− δ)δ

(1− α)c′′((a∗(L,α)))− γ(1− c′(a∗(L,α)))]c′(a∗(H,α))︸ ︷︷ ︸≥0 by lemma 8

+ (1− γ)(1− c′(a∗(L,α)))︸ ︷︷ ︸>0 by α∈I1

c′(a∗(L,α))︸ ︷︷ ︸>0

]

∂

∂αa∗(L,α) =

1

det(A(α))

(1− δ)δ︸ ︷︷ ︸>0

[(1− δ)δ

(1− α)c′′(a∗(H,α))− γ(1− c′(a∗(L,α)))]c′(a∗(L,α))︸ ︷︷ ︸≥0 by lemma 8

+ (1− γ)(1− c′(a∗(H,α)))︸ ︷︷ ︸>0 by α∈I1

c′(a∗(H,α))︸ ︷︷ ︸>0

]

(IFT2)

We know from Lemma (I-1) that a∗(H,α), a∗(L,α) are strictly increasing with respect to α in I1. Hence, we

must have det(A(α)) > 0.

Now, Y (θ, α) = a∗(θ, α) + θ > 0. Thus, we may log-differentiate the ratio Y (H,α)Y (L,α) and show that:

∂

∂α

Y (H,α)

Y (L,α)≤ 0 ↔

∂∂αY (H,α)

Y (H,α)≤

∂∂αY (L,α)

Y (L,α)↔ (

∂∂αa

∗(H,α)

a∗(H,α) + E(θ|θ−1 = H)) ≤ (

∂∂αa

∗(L,α)

a∗(L,α) + E(θ|θ−1 = L))

We shall plug in the expressions from IFT2 to show the last inequality above. Since 1det(A)

(1−δ)δ is a common

term for both ∂∂αa

∗(H,α), ∂∂αa

∗(L,α) in IFT2 and is strictly positive, the last inequality is equivalent to:

[( (1−δ)δ (1− α)c′′(a∗(L,α))− γ(1− c′(a∗(L,α))))c′(a∗(H,α)) + (1− γ)(1− c′(a∗(L,α)))c′(a∗(L,α))]

a∗(H,α) + H

≤[( (1−δ)

δ (1− α)c′′(a∗(H,α))− γ(1− c′(a∗(H,α))))c′(a∗(L,α)) + (1− γ)(1− c′(a∗(H,α)))c′(a∗(H,α))]

a∗(L,α) + L

(V1)

We can also obtain an equivalent inequality by collecting terms based on 1− c′(a∗(L,α)) inside the bracket

39

[ ] on the left hand side and 1− c′(a∗(H,α)) inside the bracket on the right hand side:

[( (1−δ)δ (1− α)c′′(a∗(L,α))c′(a∗(H,α))− (1− c′(a∗(L,α)))(γc′(a∗(H,α))− (1− γ)c′(a∗(L,α)))]

a∗(H,α) + H

≤[( (1−δ)

δ (1− α)c′′(a∗(H,α))c′(a∗(L,α))− (1− c′(a∗(H,α)))(γc′(a∗(L,α))− (1− γ)c′(a∗(H,α)))]

a∗(L,α) + L

(V2)

Now, we will further divide into two different cases, (A) 1−c′(a∗(L,α))a∗(H,α)+H < 1−c′(a∗(H,α))

a∗(L,α)+L , and (B) 1−c′(a∗(L,α))a∗(H,α)+H ≥

1−c′(a∗(H,α))a∗(L,α)+L . Even though two inequalities V1 and V2 are both equivalent to ∂

∂αY (H,α)Y (L,α) ≤ 0, V1 is easier to

establish under the first case, whereas V2 is easier to show under the second case.

Case A) 1−c′(a∗(L,α))

a∗(H,α)+H< 1−c′(a∗(H,α))

a∗(L,α)+L

First, note that:

0 ≤((1− δ)δ

(1− α)c′′(a∗(L,α))− γ(1− c′(a∗(L,α)))) < ((1− δ)δ

(1− α)c′′(a∗(H,α))− γ(1− c′(a∗(H,α))))

whereas the first inequality follows from the first part of the Lemma 8, and the second step from the convexity

of c′, c, and monotonicity in effort level. Now, the left hand side of V1 is:

[( (1−δ)δ (1− α)c′′(a∗(L,α))− γ(1− c′(a∗(L,α))))c′(a∗(H,α)) + (1− γ)(1− c′(a∗(L,α)))c′(a∗(L,α))]

a∗(H,α) + H

=[(1− δ)δ

(1− α)c′′(a∗(L,α))− γ(1− c′(a∗(L,α)))]c′(a∗(H,α))

a∗(H,α) + H+ (1− γ)

(1− c′(a∗(L,α)))

a∗(H,α) + Hc′(a∗(L,α))

< [((1− δ)δ

(1− α)c′′(a∗(H,α))− γ(1− c′(a∗(H,α))))]︸ ︷︷ ︸>

(1−δ)δ (1−α)c′′(a∗(L,α))−γ(1−c′(a∗(L,α)))

c′(a∗(H,α))

a∗(H,α) + H︸ ︷︷ ︸>0

+(1− γ)1− c′(a∗(H,α))

a∗(L,α) + L︸ ︷︷ ︸>

(1−c′(a∗(L,α)))

a∗(H,α)+H; Case A

c′(a∗(L,α))︸ ︷︷ ︸>0

≤ [((1− δ)δ

(1− α)c′′(a∗(H,α))− γ(1− c′(a∗(H,α))))]︸ ︷︷ ︸>0

c′(a∗(L,α))

a∗(L,α) + L︸ ︷︷ ︸by lemma 9

+(1− γ)1− c′(a∗(H,α))

a∗(L,α) + L︸ ︷︷ ︸>0

c′(a∗(H,α))︸ ︷︷ ︸>c′(a∗(L,α))

the last expression coincides with the right hand side of inequality V1, the version we wanted to establish.

Case B) 1−c′(a∗(L,α))a∗(H,α)+H ≥ 1−c′(a∗(H,α))

a∗(L,α)+L

Here, we start by showing:

40

γc′(a∗(H,α))− (1− γ)c′(a∗(L,α)) >︸︷︷︸c′(a∗(H,α))>c′(a∗(L,α))

γc′(a∗(L,α))− (1− γ)c′(a∗(H,α)) > 0

The first inequality is easy to see from the monotonicity in effort level given in Lemma (M’-1); the second

inequality is less obvious. This second inequality is equivalent to γ(1−γ) >

c′(a∗(H,α))c′(a∗(L,α)) . We can use DE2 to

rewrite c′(a∗(H,α))c′(a∗(L,α)) :

c′(a∗(H,α))

c′(a∗(L,α)))=︸︷︷︸

DE2

γΠ∗(H,α) + (1− γ)Π∗(L,α)

γΠ∗(L,α) + (1− γ)Π∗(H,α)=

Π∗(L,α) + γ(Π∗(H,α)−Π∗(L,α))

Π∗(L,α) + (1− γ)(Π∗(H,α)−Π∗(L,α))

=1 + γ( Π∗(H,α)

Π∗(L,α)) − 1)

1 + (1− γ)( Π∗(H,α)Π∗(L,α)) − 1)

Note that γ1−γ > 1 from γ > 1

2 , and Π∗(H,α)) > Π∗(L,α). This implies that1+γ(

Π∗(H,α)Π∗(L,α)

−1)

1+(1−γ)(Π∗(H,α)Π∗(L,α))

−1)is between

1 and γ1−γ , so that c′(a∗(H,α))

c′(a∗(L,α)) <γ

1−γ .

Now, the left hand side of V2 is:

[( (1−δ)δ (1− α)c′′(a∗(L,α))c′(a∗(H,α))− (1− c′(a∗(L,α)))(γc′(a∗(H,α))− (1− γ)c′(a∗(L,α)))]

a∗(H,α) + H

=((1− δ)δ

(1− α)c′′(a∗(L,α))c′(a∗(H,α))

a∗(H,α) + H− (1− c′(a∗(L,α))

a∗(H,α) + H(γc′(a∗(H,α))− (1− γ)c′(a∗(L,α)))

≤ ((1− δ)δ

(1− α)c′′(a∗(L,α))︸ ︷︷ ︸>0

c′(a∗(L,α))

a∗(L,α) + L︸ ︷︷ ︸by lemma 9

− (1− c′(a∗(H,α))

a∗(L,α) + L︸ ︷︷ ︸≤ 1−c′(a∗(L,α))

a∗(H,α)+H; Case B

(γc′(a∗(H,α))− (1− γ)c′(a∗(L,α)))︸ ︷︷ ︸>0

< ((1− δ)δ

(1− α)c′′(a∗(H,α))︸ ︷︷ ︸≥c′′(a∗(L,α)) from convexity of c′

c′(a∗(L,α))

a∗(L,α) + L︸ ︷︷ ︸>0

− (1− c′(a∗(H,α))

a∗(L,α) + L︸ ︷︷ ︸>0

(γc′(a∗(L,α))− (1− γ)c′(a∗(H,α)))︸ ︷︷ ︸<(γc′(a∗(H,α))−(1−γ)c′(a∗(L,α)))

Hence, we showed V2 under Case B. Hence, Y (H,α)Y (L,α) is locally decreasing in α when α ∈ {α ∈ I1 : det(A(α)) 6=

0}.

Step 5: Y (H,α)Y (L,α) is decreasing everywhere on [0, 1].

Y (H,α)Y (L,α) = a∗(H,α)+H

a∗(L,α)+Lis continuous in [0, 1], because a∗(θ, α) ≥ 0 is continuous by Lemma (C-1). Recall

that α0 = inf{α : a∗(H,α) = aFB} and α0 ∈ (0, α0) in which detA(α) = 0}. If α0 does not exist, step

41

2 and step 4 show that Y (H,α)Y (L,α) is piece-wise decreasing in each region (0, α0), (α0, α0), (α0, 1]. If α0 exists,

step 2 and step 4 show that Y (H,α)Y (L,α) is piece-wise decreasing in each region (0, α0), (α0, 1]. Since Y (H,α)

Y (L,α) is

everywhere continuous, the piece-wise monotonicity would make Y (H,α)Y (L,α) a decreasing function everywhere.

This completes the proof.

Proofs for Auxillary Results

Proofs for Lemma 3, and Lemma 4 are almost identical as proofs for Lemmas 2, and 1, with analogous

modifications. Lemma 5 follows from the system of Bellman equations, as in the equation (7) in the proof of

Lemma 1.

Proof of Lemma (I-1). Suppose to the contrary, so that there exists a triple (θ0, α1, α2) with α1 < α2 such

that the optimal effort a∗(θ′, α2) < a∗(θ′, α1). Fix α1, α2 and let Θ = {θ′ ∈ Θ : α1 < α2, a∗(θ′, α2) <

a∗(θ′, α1)}. Now, define a new bonus system under α2 as following: for all θ−1 ∈ Θ, b(θ−1, θ, α2) :=

b∗(θ−1, θ, α1); for all other θ−1 /∈ Θ, let b(θ−1, θ, α2) := b∗(θ−1, θ, α2).

If θ′ ∈ Θ so that a∗(θ′, α2) < a∗(θ′, α1), the new effort a∗(θ′, α2) should satisfy:

a∗(θ′, α2) =︸︷︷︸IC

(c′)−1(E(b(θ−1, θ, α2)|θ−1 = θ′)) = (c′)−1(E(b∗(θ−1, θ, α1)|θ−1 = θ′))

=︸︷︷︸IC

(c′)−1(c′(a∗(θ′, α1))) = a∗(θ′, α1) = maxα1,α2

a∗(θ′, α)

Similarly, if θ′ /∈ Θ so that a∗(θ′, α2) ≥ a∗(θ′, α1), the new effort a∗(θ′, α2) should satisfy:

a∗(θ′, α2) =︸︷︷︸IC

(c′)−1(E(b∗(θ−1, θ, α2)|θ−1 = θ′)) = (c′)−1(E(b(θ−1, θ, α2)|θ−1 = θ′))

=︸︷︷︸IC

(c′)−1(c′(a∗(θ′, α2))) = a∗(θ′, α2) = maxα1,α2

a∗(θ′, α)

In other words, this bonus system {b(·, ·, α2)} will solicit maxα1,α2

a∗(θ, α) for each θ. Furthermore, observe

that for all θ′ ∈ Θ, a∗(θ′, α1) < a∗(θ′, α2) ≤ aFB by monotonicity (M’-1), so a∗(θ′, α1) − c(a∗(θ′, α1)) <

a∗(θ′, α2)− c(a∗(θ′, α2)).

Now, by the system of equations 5, for any stationary contract,

[Π∗(H,α)

Π∗(L,α)

]=

[γ 1− γ

1− γ γ

][a∗(H,α)− c(a∗(H,α)) + H

a∗(L,α)− c(a∗(L,α)) + L

]

42

We can use a similar representation for the new continuation values under the new bonus system {b(·, ·, α2)}for Π(·, α2), as well. Since the new bonus system solicits higher effort level max

α1,α2

a∗(θ, α), the system of

equations above shows that Π(θ, α2) ≥ max{Π∗(θ, α2),Π∗(θ, α1)}. Now, for all θ−1 ∈ Θ,

(1− δ)(1− α2)b(θ−1, θ, α2) = (1− δ)(1− α2)b∗(θ−1, θ, α1)

<(1− δ)(1− α1)b∗(θ−1, θ, α1) ≤︸︷︷︸DE-P for (θ−1,θ,α1)

δΠ∗(θ−1, α1) ≤ δΠ(θ−1, α2)

Now, for all θ−1 /∈ Θ,

(1− δ)(1− α2)b(θ−1, θ, α2) = (1− δ)(1− α2)b∗(θ−1, θ, α2)

≤︸︷︷︸DE-P for (θ−1,θ,α2)

δΠ∗(θ−1, α2) ≤ δΠ(θ−1, α2)

Since Θ 6= ∅ and a∗(θ′, α2)− c(a∗(θ′, α2)) < a∗(θ′, α2)− c(a∗(θ′, α2)), Π∗(θ, α2) < Π(θ, α2). This contradicts

the optimality of the bonus system {b(·, ·, α2)}, because we obtained strictly higher continuation values

without violating the dynamic enforceability condition. This contradiction establishes (I-1).

Proof of Lemma (I-2). Note that x− c(x) is a strictly increasing function on [0, aFB ]. By monotonicity (M’-

1), a∗(L,α) ≤ a∗(H,α) ≤ aFB . Hence a∗(H,α)− c(a∗(H,α)) and a∗(L,α)− c(a∗(L,α)) are both increasing

in α ∈ [0, 1] as well. By Lemma 5, for any stationary contract,

[Π(H,α)

Π(L,α)

]=

[γ 1− γ

1− γ γ

][a∗(H,α)− c(a∗(H,α)) + H

a∗(L,α)− c(a∗(L,α)) + L

]

Hence, Π∗(H,α) and Π∗(L,α) are both strictly increasing in α.

Proof of Lemma 8. Suppose to the contrary. We will first assume so that (1−δ)δ (1− α)c′′(a∗(H,α))− γ(1−

c′(a∗(H,α))) < 0 for some α ∈ I1. Our goal is to show that we can raise continuation values without violating

dynamic enforceability conditions by raising bonus by small amounts, which contradicts the optimality of the

bonus and recommended effort policy. Denote the bonus system {b∗(·, ·, α)}, the current recommended efforts

{a∗(·, α)} and continuation values {Π∗(·, α)} under which (1−δ)δ (1−α)c′′(a∗(H,α))− γ(1− c′(a∗(H,α))) < 0

is true.

For any ε ∈ (0, 1−c′(a∗(H,α))2 ), we may define the new bonus payments {b(·, ·, α)}, and new recommended

43

effort policies {a∗(·, α)} as:

b(H,H,α) = b∗(H,H,α) +γ

γε, b(H,L, α) = b∗(H,L, α) +

1− γγ

ε, b(L,L, α) = b∗(L,L, α), b(L,H, α) = b∗(L,H, α)

a∗(H,α) = (c′)−1(λb(H,H,α) + (1− λ)b(H,L, α)) =︸︷︷︸γ=γλ+(1−λ)(1−γ)

(c′)−1(λb(H,H,α) + (1− λ)b(H,L, α)︸ ︷︷ ︸=c′(a∗(H,α))

+ε)

= (c′)−1(c′(a∗(H,α)) + ε)

a∗(L,α) = (c′)−1(λb(L,L, α) + (1− λ)b(L,H, α)︸ ︷︷ ︸=c′(a∗(L,α))

= a∗(L,α)

Given our range for ε, c′(a∗(H,α)) = c′(a∗(H,α)) + ε < 1. Since a∗(H,α) < a∗(H,α) < aFB , a∗(H,α) −c(a∗(H,α)) < a∗(H,α) − c(a∗(H,α)). From the system of equations in Lemma 5, new continuation values

{Π(·, α)} are also pinned down. Also, it must be that Π(H,α) > Π(H,α) and Π(L,α) > Π(L,α). Our next

task is show that the new bonus system {b(·, ·, α)} satisfies the dynamic enforceability condition.

For convenience, denote the increment in the new recommended effort level a∗(H,α) relative to the old effort

level a∗(H,α) by η(ε) := a∗(H,α)− a∗(H,α). Observe that:

δ

1− δΠ(H,α)− (1− α)b(H,H,α)

=︸︷︷︸Lemma 5

δ

1− δ[γ(a∗(H,α)− c(a∗(H,α) + η(ε)) + H) + (1− γ)(a∗(L,α)− c(a∗(L,α)) + L)]− (1− α)(b∗(H,H,α)

+δ

1− δγη(ε)− (1− α)

γ

γε

=δ

1− δ[γ(a∗(H,α)− c(a∗(H,α)) + H) + (1− γ)(a∗(L,α)− c(a∗(L,α)) + L)]− (1− α)(b∗(H,H,α)︸ ︷︷ ︸

=0 ∵ Π∗(H,α)=(1−δ)δ b∗(H,H,α) from a∗(H,α)<aFB

+δ

1− δγ(η(ε)− [c(a∗(H,α) + η(ε))− c(a∗(H,α))])− (1− α)

γ

γε

=δ

1− δγ(η(ε)− [c(a∗(H,α) + η(ε))− c(a∗(H,α))])− (1− α)

γ

γε =: f(ε)

Clearly, f(0) = 0. Since dη(ε)dε |ε=0 = 1

c′′(a∗(H,α)) ,

f ′(0) = γδ

1− δ1− c′(a∗(H,α))

c′′(a∗(H,α))− γ

γ(1− α)

By the initial assumption that (1−δ)δ (1−α)c′′(a∗(θ−1))− γ(1− c′(a∗(θ−1))) < 0, f ′(0) > 0. This implies that

the f(ε) > 0 for ε > 0 small. This implies that δ1−δ Π(H,α) − (1 − α)b(H,H,α) > 0 if our choice of ε > 0

was small enough. Similarly, an almost identical procedure shows that δ1−δ Π(L,α) − (1 − α)b(H,L, α) > 0

as well if our choice of ε was small enough. Hence, {b(·, ·, α), a∗(·, α), Π(·, α)} satisfies the dynamic en-

forceability conditions for small enough ε > 0, but attains higher continuation values from Lemma 5. This

contradicts the optimality of the initial contract {b∗(·, ·, α), a∗(·, α),Π∗(·, α)}. This contradiction implies that(1−δ)δ (1− α)c′′(a∗(H,α))− γ(1− c′(a∗(H,α))) ≥ 0.

44

We can derive a similar contradiction if we assume (1−δ)δ (1 − α)c′′(a∗(L,α)) − γ(1 − c′(a∗(L,α))) < 0 by

increasing the b(L, ·, α) incrementally instead of b(H, ·, α). Hence, it must be that (1−δ)δ (1−α)c′′(a∗(θ, α))−

γ(1− c′(a∗(θ, α))) ≥ 0 for both θ = H,L.

Proof of Lemma 9. We want to show:

c′(a∗(H,α))

a∗(H,α) +H≤ c′(a∗(L,α))

a∗(L,α) + L↔ c′(a∗(H,α))

c′(a∗(L,α))≤ a∗(H,α) +H

a∗(L,α) + L

From the assumption that c(·) is log-concave, ddx log[c(x)] = c′(x)

c(x) is a decreasing function. Since a∗(H,α) ≥a∗(L,α) from (M’-1), we have that:

c′(a∗(H,α))

c(a∗(H,α))≤ c′(a∗(L,α))

c(a∗(L,α))↔ c′(a∗(H,α))

c′(a∗(L,α))≤ c(a∗(H,α))

c(a∗(L,α))

Moreover,

c′(a∗(H,α))

c′(a∗(L,α))=γ(a∗(H,α)− c(a∗(H,α)) + H) + (1− γ)(a∗(L,α)− c(a∗(L,α)) + L)

γ(a∗(L,α)− c(a∗(L,α)) + L) + (1− γ)(a∗(H,α)− c(a∗(H,α)) + H)

<a∗(H,α)− c(a∗(H,α)) + H

a∗(L,α)− c(a∗(L,α)) + L

The equality above follows from Lemma 5. Since x − c(x) is a strictly increasing function on [0, (c′)−1(1)],

a∗(L,α)− c(a∗(L,α)) ≤ a∗(H,α)− c(a∗(H,α)) holds from the assumption that a∗(H,α) < aFB = (c′)−1(1)

and the monotonicity in effort a∗(L,α) ≤ a∗(H,α). Hence, it must be that a∗(H,α) − c(a∗(H,α)) + H ≥a∗(L,α)− c(a∗(L,α)) + L, which shows the inequality above.

Now, since a∗(H,α)+H

a∗(L,α)+Llies between a∗(H,α)−c(a∗(H,α))+H

a∗(L,α)−c(a∗(L,α))+Land c(a∗(H,α))

c(a∗(L,α))15, it follows that:

c′(a∗(H,α))

c′(a∗(L,α))≤ min{a

∗(H,α)− c(a∗(H,α)) + H

a∗(L,α)− c(a∗(L,α)) + L,c(a∗(H,α))

c(a∗(L,α))} ≤ a∗(H,α) +H

a∗(L,α) + L

This completes the proof.

15This follows from the following fact: assume a1, a2, b1, b2 > 0 and a1a2≤ b1

b2, then a1

a2≤ a1+b1

a2+b2≤ b1

b2. These inequalities are

strict if a1a2

< b1b2

.

45