Selective Incentives∗

Giuseppe Dari-Mattiacci†and Alex Raskolnikov‡

September 26, 2018

Abstract

Many public and private regulators do not rely on mandates and cannot tailor their regula-tory instruments to the characteristics of individual agents. Rather, they use selective incentives–amix of carrots and sticks intended to induce some agents both to participate in the regulatoryscheme and to comply with the regulator’s conditions for participation. We show that whenthe participation and the compliance decisions are considered jointly, we must revise long-heldbeliefs about the effects of rewards and punishments, the benefits of monitoring, and the interac-tion between the design of substantive rules and their enforcement. We conclude that regulatorsshould invest more in improving the accuracy of enforcement rather than relying on the prob-ability and magnitude of punishments and rewards. We illustrate our findings with examplesranging from the U.S. auto industry to tax incentives to international climate change negotia-tions.

Keywords: carrots, sticks, incentives, participation, accuracy.JEL codes: K10, K20, K42.

1 Introduction

When governments forgo command-and-control regulation, they rely on incentives to shape be-havior. Private parties often lack the power to command and control, so they rely on incentiveseven more. With limited exceptions, taxes and subsidies, rewards and sanctions, carrots and sticksare the all-important regulatory and contractual tools. This article investigates how a regula-tor—public or private—should design these tools in order to achieve two of the most commonreal-world objectives: participation in an incentive-based regime and compliance with its require-ments.

Consider a firm that underestimated the demand for its product. To increase productionquickly, the owner of the firm (the principal) would like to extend the plant’s working hours.The owner offers the workers (agents) a carrot—higher wages for overtime work. Some of thefirm’s workers decide to participate, others decline. Moreover, while some participants work hardduring the overtime shift, others slack off, frustrating the owner’s objective.

∗We received valuable feedback from Gerrit De Geest, Ezra Friedman, and workshop participants at Columbia Univer-sity, University of Chicago, and the annual meetings of ComplianceNet and the American Law and Economics Association.All mistakes are ours.

†Professor of Law and Professor of Economics, University of Amsterdam and Joseph P. Cunningham Visiting Professorof Commercial and Insurance Law at Columbia University School of Law (Fall 2018).

‡Wilbur H. Friedman Professor of Tax Law, Columbia University School of Law.

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Manuscript

It would seem that the owner should simply deny overtime pay to the slackers, or perhapsfire them altogether. Unfortunately for the owner, it cannot observe each worker’s effort; it canonly observe each worker’s output. Poor effort often leads to low output, but sometimes evengreat effort yields poor results due to factors outside of a worker’s control. Punishing low outputwith a stick will occasionally penalize high-effort workers. This would reduce their incentive towork hard (comply with the owner’s requirements) and to sign up for overtime work altogether(participate in the incentive regime).

Thus, the challenge facing the principal is not just to design an incentive scheme that wouldinduce a particular number of workers to volunteer for overtime. The challenge is to induce onlythose who would work hard to sign up for extra work. That is, the principal must not simplyincentivize the agents, it must select the “right” agents to incentivize. So the principal’s challengeis to design a set of selective incentives.1

Government regulators often face this challenge just as private parties do. The incentive schememay aim to stimulate job creation, research and development, a particular geographic area, or aspecific activity. The incentive may even be unintended by the principal—a quasi-incentive thatwe explain below. The unobservable feature may be the agent’s mental state, level of care, or,more generally, compliance effort. The incentive regime may be part of financial regulation, taxlaw, agricultural policy, or any other government scheme. Even international agreements may giverise to the dual challenge of inducing participation and compliance by using selective incentives.So the application of our analysis is truly broad.

The dual challenge that we study here is familiar in law and economics. In mechanism designtheory, the individual rationality constraint underlies the decision to participate and the incentivecompatibility constraint underlies the decision to comply (Emons 1994). The economic analysis oflaw studies both the level of care (or, more generally, compliance) and the level of activity (that is,participation) (Polinsky and Shavell 2007). The tradeoff between achieving deterrence (inhibitingnon-compliance) and reducing chilling (limiting non-participation) (Kaplow 2011) is also closelyrelated to our inquiry.

Our focus, however, is different from all these analyses. We do not assume—as virtually all ofthe literature just mentioned does—that the principal’s objective is to maximize efficiency or wel-fare. Nor do we posit that the regulatory environment, whether public or private, is in equilibrium.Instead, we make two very basic assumptions about the principal’s goals. First, the principal hasa preference, however formed, regarding the desirable extent of the agents’ participation. Second,if an agent would not comply if he participated, the principal prefers for that agent to abstain fromparticipation altogether. In terms of our initial example, the owner wants only certain number ofworkers to work overtime. And if a worker is exhausted and has no energy to work hard duringextra hours, the owner wants this worker to stay home rather than come in and slack off.

Thus, we investigate how a principal should set selective incentives to achieve her own objec-tives rather than ideal theoretical goals. Our principal is neither a benevolent social planner nor anindividual utility maximizer. Rather, our principal is a manager aiming to achieve a particular par-ticipation/compliance combination that the principal prefers for the reasons only she knows. Ofcourse, if the principal prefers to maximize efficiency and concludes that particular levels of par-ticipation and compliance would do so, our prescriptions would tell the principal how to achieve

1Political science literature follows Mancur Olson’s lead in using the term selective incentives to mean incentives that“punish[ ] or reward[ ] individuals according as they do or do not bear the costs of collective action” (Olson 1992). We usethe term more broadly to mean incentives “relating to or involving the selection of the most suitable or best qualified.”(Oxford Dictionary of English Language, https://en.oxforddictionaries.com/definition/selective).

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(or at least approach) the desired outcome given the tools at her disposal.We posit that the principal may influence agents’ behavior by using five of the most common

instruments familiar to public and private regulators alike. The principal may vary the size ofthe carrot, the magnitude of the stick, the agent’s effort required to participate in the regime, theaccuracy in identifying compliers and violators, and the rate of monitoring (auditing) participantsfor potential violations. Thus, the factory owner in our example may increase or decrease theovertime pay, impose varying sanctions on slackers, and vary the minimum number of overtimehours. The principal may also try to better distinguish diligent workers from slackers (that is,reduce false positives and/or false negatives), and to monitor each worker’s output more or lessclosely.

One further assumption adds to the realism of our model and drives its results. We posit thatagents are heterogeneous with respect to at least one significant characteristic other than their costof compliance (which, of course, also differs among agents). In the main version of the model weassume that agents vary in the (unobservable) private benefit from participation. In an extensionwe show that our results hold when heterogeneity relates to the principal’s accuracy in identifyingviolations.

The meaning of benefit heterogeneity is obvious. As for the accuracy heterogeneity, it reflectstwo plausible real-world features. First, while the principal has a view about her own ability torecognize compliers and violators, agents need not share the same view. Some agents perceivethe principal to be highly accurate; others believe that the principal is rather error-prone. Thus,agents vary in their perceptions. It is also possible that, perceptions aside, the principal’s accuracyactually varies from one agent to the next. Some workers in our example may perform simpleindividual tasks. Output is a reliable indicator of effort for these workers. Other workers maywork in teams and tackle multi-faceted problems. Output is a less reliable indicator of effort forthese individuals.

Agents’ heterogeneity with respect to private benefits and (their views of) the principal’s accu-racy is not only plausible; a model that ignores either heterogeneity would have a rather distantrelationship to reality. In contrast, carrots, sticks, and required effort are much more likely to beuniform. A principal may simply announce an overtime wage, or a cash grant, or a tax credit,or a fine, or the minimum hours, investments, or actions required to qualify for (or avoid) theprincipal’s carrots and sticks.2

1.1 Summary of the Results

Two insights drive our results. First, given a particular combination of incentives, heterogeneousagents fall in one of two groups. In one group, all agents choose to participate. We call this thefull participation group. Some of the participating agents choose to comply while others choose toviolate the principal’s rule. In another group, only some agents choose to participate. We call thisthe partial participation group. In that group, we show that all agents who choose to participatealso choose to comply with the principal’s requirement. Overall, we obtain a realistic picturewhere some agents participate and comply, some participate and violate, and some abstain fromparticipation altogether.

Second, the agents in the full participation group must decide whether to comply or violate.This is the standard setting examined in the carrots-and-sticks literature. The basic result in that lit-

2We discuss the realism of this assumption in Section 6.

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erature is that carrots and sticks are substitute.3 The same holds for agents in the full participationgroup in our model. But agents in the partial participation group face a different decision: theychoose whether to comply or abstain from participation. We show that for these agents carrotsand sticks are complements rather than substitutes. Overall, carrots and sticks acts as both com-plements and substitutes in the entire population of agents. This combination makes the analysismore complex and the findings more nuanced than received wisdom would have them.

The payoffs from our model confirm some well-known results and seemingly contradict others.The contradictions are apparent rather than real; they arise because we analyze a different prob-lem than those studied in the literature. To start, we follow many others in finding that greateraccuracy increases compliance with the law. The new and more nuanced result is that increasingaccuracy in identifying violators (reducing false negatives) is reduces the number of participantswhile increasing accuracy in identifying compliers (reducing false positives) increases that num-ber. If a principal is able to adjust either type of accuracy, our findings tell the principal how toimprove compliance while also increasing participation or reducing it.

Our next result is that raising the required effort has a hidden cost. Not surprisingly, increasingthe required effort reduces the number of participants. We show that the reduction is of exactlythe wrong kind from the principal’s perspective. Greater required effort leads to fewer compliersand more violators. This finding is a stark formal illustration of the general view that design oflegal rules and their enforcement are inextricably linked (Raskolnikov 2013, Easterbrook 1985).

Turning to carrots and sticks, we show that while more generous carrots have the expectedeffect of inducing greater participation by compliers, their effect on compliance rate (the share ofcompliers among all participants) is ambiguous. This conclusion expands the literature on car-rots and sticks that generally does not focus on possible non-compliance of carrot recipients. Wefind that sticks are the least useful selective incentive. Although higher sticks depress partici-pation, their effects on the number of compliers and on the rate of compliance are ambiguous.This drawback of sanctions is new to the law enforcement literature. The novelty is not surpris-ing given that literature’s focus on mandatory command-and-control settings rather than optionalincentive-based regimes studied here.

We find more novel and somewhat unexpected results when we consider imperfect monitor-ing. The well-developed enforcement literature posits that greater monitoring (and higher sanc-tions) improve deterrence (Becker 1968). Yet we find that in optional regimes this is not the case.Greater monitoring has an ambiguous effect on both the number of compliers and the rate ofcompliance. The result mimics our conclusions about the effect of higher sanctions. Both highersanctions and stricter monitoring increase the expected cost for violators. So changes in both in-struments give rise to the same ambiguities.

Finally, we show that adding positive or negative inducements that are not conditional on com-pliance adds nothing of value. One may induce participation not only by a carrot for complyingparticipants, but also by a tax on all non-participants. The inverse of that tax is a subsidy to allparticipants—compliers and violators alike. Any such tax (or subsidy), we show, amounts to si-multaneously increasing the carrot and reducing the stick by a fixed amount, without expandingthe set of outcomes that a regulator may achieve.

3For example, one can tax pollution or subsidize pollution abatement—the resulting incentive to pollute is the same.As is long-known, however, this result does not quite hold even in the typical settings considered by the carrots-and-sticksliterature (Polinsky 1979).

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1.2 Related Literature

Our results relate to several large literatures, but build on a small number of key contributions.Starting with the latter, Png (1986) introduced the idea of adding carrots to offset the regulator’s in-accurate determination of liability. He also identified the tradeoff between participation and com-pliance. He concluded that to induce individuals to make socially optimal choices the regulatorshould award carrots to all participants (drivers involved in accidents in his example)—whetheror not they followed the principal’s rule (exercised due care). Png (1986) did not model hetero-geneity of the agents’ benefits or the principal’s accuracy—heterogeneity that is essential to ourfindings.

Levmore (1986) observed that a regulator trying to incentivize behavior with carrots may needto introduce sticks to prevent false claiming of carrots. That is precisely the problem we studyhere. Levmore’s (1986) observation was brief, however. He did not offer a model, did not considervariations in accuracy or required effort, and did not analyze any heterogeneity among the agents.

Kaplow (1994) was first to formally study the interaction between accuracy and sanctions in alegal system. He also extended Png’s (1986) analysis of the chilling effects in the optimal deter-rence framework (Kaplow 2011). In contrast with our model, Kaplow did not consider variablecarrots as policy instruments, did not model heterogeneous error rates, and did not allow agentsto choose between the three alternative courses of action that we study. Moreover, as most deter-rence literature does, Kaplow and Png studied how varying the model parameters affects socialwelfare, not participation and compliance that are the focus here. So while Png (1986) and Kaplow(1994, 2011) provide important foundations for our analysis, their contributions differ from ours.

Kleven and Kopczuk (2011) were first to analyze government benefits (carrots) jointly withvariations in complexity, which they modeled as costly increases in enforcement accuracy. Theydid not include sticks or heterogeneous perceptions of error rates as we do. Moreover, the cost ofgreater accuracy born by agents played a key role in their model while it is absent from ours.

Shavell (2017) suggests that subsidies may be appropriate to account for positive externalitiesof activities regulated by strict liability or negligence regimes. He does not consider the effects ofinaccurate enforcement that are key to our model.

Broadening the lens, our contribution relates to several large literatures. Economic analysisof accident law and the theory of optimal deterrence both consider the two margins of behaviorthat we study: the activity level (participation) and the level of care (compliance). However, forthe most part, the models in these literatures assume perfect accuracy in determination of liability.Analysis of inaccurate enforcement—the key feature of our model—is limited (Shavell 1987: 79,Polinsky and Shavell 2007: 427), and the most relevant contributions are those discussed above.

Our work is also related to the substantial literature on the economic analysis of carrots andsticks. Much of that literature considers positive and negative incentives as substitutes (Ben Sha-har and Bradford 2012). A few contributions stress the differences between carrots and stickswhile remaining faithful to the idea that policymakers use either carrots or sticks (Polinsky 1979,Wittman 1984, De Geest and Dari-Mattiacci 2013).

Even authors favoring joint use of carrots and sticks believe that the resulting incentives act ona single margin (Andreoni 2003; Ben Shahar and Bradford 2012; Gilpatric 2009; Moldovanu 2012).In contrast, we study the effect of combining rewards and punishments to affect two separate de-cisions: whether or not to participate and, conditional on participation, whether or not to comply.Dari-Mattiacci (2009) investigates how agents make decisions along both of these margins but hestudies the use of a single incentive.

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A large literature on mechanism design bears only slight resemblance to our inquiry. Thatliterature studies a setting similar to ours—incentives designed under imperfect information toovercome both the incentive compatibility constraint (assure compliance) and the individual ra-tionality constraint (induce participation) (Emons 1994: 487). But the critical aspects of that liter-ature are absent from our analysis. We do not assume the existence of a revelation mechanism orany other equilibrium strategy. Moreover, our objective is not to find individually rational andincentive-compatible mechanisms that assure both budget neutrality and efficiency—the main ob-jective of the mechanism design theory (Emons 1994).

Overall, although the selective incentives schemes are pervasive in public and private law,there appears to be no formal analysis that evaluates the effects of various selective incentives onparticipation and compliance in a realistic setting with heterogeneous agents. This is our task here.

We develop the model in Section 2, consider extensions in Section 3, and discuss our resultsin Section 4. Section 5 illustrates our model’s relevance in a variety of public and private lawsettings ranging from U.S. auto industry to law firm hiring, tax incentives, financial regulation, andinternational climate change negotiations. A brief conclusion follows in Section 6. The Appendixcontains the technical details of the proofs.

2 The model

2.1 Model setup

Consider a principal aiming to regulate a population of agents. The principal requires participat-ing agents to comply with the principal’s rule of conduct by exerting effort. Each agent choosesindependently whether to participate in the activity and, if so, whether to comply with the prin-cipal’s rule of conduct. (Those who do not participate in the activity are not subject to the regu-lation.)4 Agents derive a benefit from participation b > 0 and exert a costly effort e > 0, both ofwhich vary across agents. Private benefits may vary for many reasons. In our overtime work ex-ample, some workers may want to work overtime just to get extra pay. Others may also hope for apromotion. And a few may derive a non-pecuniary benefit from helping the company succeed. Ofcourse, workers also vary in their costs of working overtime. Thus the agent’s two-dimensionaltype is (b, e), and the population of agents is described by the density function f (b, e).5

If an agent abstains from the regulated activity, he earns a payoff normalized to zero. If theagent participates, he earns a direct benefit b6 and decides whether to comply with the principal’sregulation or to violate it. Complying agents exert the required effort r > 0 at a cost re > 0; violatorsincur no costs.

To induce agents to participate and exert effort, the principal rewards compliance with a carrotc > 0 and punishes violations with a stick s > 0.7 (We start by assuming that monitoring takesplace with probability p = 1. We consider the general case of 0 < p ≤ 1 in Section 3.1.)

4In Section 3.2, we show that our results also apply when the principal directly regulates participation through taxesand subsidies that are not conditional on agents’ compliance.

5Note that we do not make any assumption as to whether b and e are independently distributed or not.6Some activities are costly to the agent; that is, we could allow b to be negative. Setting a zero lower bound for b,

however, is without loss of generality. Allowing for negative benefits for participants is equivalent to setting a positivebenefit for non-participants, which would not affect our results.

7We define carrots and sticks in an unambiguous way: a carrot is a monetary payment from the principal to the agentconditional on (detected) compliance, while a stick is a monetary payment from the agent to the principal conditional on(detected) violation.

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The principal’s qualification of an agent’s conduct as compliance or violation does not per-fectly overlap with the agent’s actual behavior. The principal may erroneously classify an agent’sconduct as violation while in fact the agent exerted effort (a false positive, akin to convicting theinnocent) or, vice versa, the principal could erroneously detect compliance while the agent in factdid not exert effort (a false negative, akin to acquitting the guilty). To capture this aspect of the

problem we introduce the probabilities qk >12 of correctly identifying compliance (hence a false

positive occurs with probability 1 − qk) and qv >12 of correctly identifying violation (hence a false

negative occurs with probability 1 − qv).8 Given this setup, an agent of type (b, e) anticipates thepayoffs reported in Table 1.

Action Payoff

The agent participates and complies Πk (b, e) = b + qkc − (1 − qk) s − reThe agent participates but violates Πv (b) = b + (1 − qv) c − qvs

The agent does not participate 0

Table 1: Agent’s payoffs

Starting with the bottom row, if the agent does not participate, his (normalized) payoff is zero.If he participates but violates the principal’s rule of conduct, the agent will earn the benefit b, willface a stick s with probability qv (that is, if the principal correctly qualifies the agent’s conductas a violation of the rule), and will face a carrot with probability 1 − qv (that is, if the principalerroneously qualifies the agent’s conduct as compliance). The agent will incur zero cost of effort.We denote the payoff of an agent who participates but violates the rule as Πv (b). Note that thispayoff is independent of the agent’s effort cost. Finally, if the agent participates and complies, heearns the benefit b, faces a carrot with probability qk (that is, if the principal correctly qualifies hisconduct as compliance), faces a stick with probability 1 − qk (that is, if the principal erroneouslyqualifies his conduct as a violation), and incurs costly effort equal to re. We denote the payoff ofan agent who participates and complies as Πk (b, e), which depends both on b and on e.

2.2 Key insight illustrated

To illustrate the basic insight that drives our results, consider a simple example (here we markvariables with an over-line to avoid confusion with the same variables used in the general model).The principal sets the carrot c = 80 and the stick s = 20. The principal’s accuracy is q = 3

4 bothfor compliers and violators. All agents have the same null benefit from participation, and the

regulation requires unitary effort (that is, b = 0 and r = 1).

If an agent exerts effort, he expects to receive qc − (1 − q) s − e = 34 80 − 1

4 20 − e = 55 − e. If,

instead, the agent does not exert effort, he expects to receive (1 − q) c − qs = 14 80 − 3

4 20 = 5. Notethat the payoff from compliance equals the payoff from violation for agents with effort cost equal

8By imposing that both probabilities are greater than 12 we simply restrict attention to the plausible scenarios where

the principal’s enforcement technology fares better than a coin toss. If that were not the case, the principal could improvematters by doing either of two simple things. She could apply carrots and sticks randomly, thereby improving the accuracy

of her enforcement technology from qk <12 to q

′

k =12 . Or she could apply a carrot upon detecting a violation and a stick

upon detecting compliance. By doing so, the principal would improve the accuracy of her enforcement technology from

qk <12 to q

′′

k = 1− qk >12 . The same would apply to qv. Therefore, it is not realistic to expect that enforcement technologies

characterized by qk or qv lower than 12 would be implemented on a relevant scale.

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to the compliance threshold ek ≡ (c + s) (2q − 1) = 50. An agent complies if e < 50 and violatesotherwise.

The next question is whether agents participate in the regulated activity. Since violators earna positive payoff (equal to 5), compliers must also earn a positive payoff, because the conditionfor choosing compliance over violation is that the payoff from compliance be the higher of thetwo. As a result, all agents participate in the regulated activity. We will refer to this case as “fullparticipation.”

So far, our analysis reflects the common wisdom: compliance increases if the principal is moreaccurate, if the carrot is higher, or if the stick is higher.9 In particular, the carrot and the stick aresubstitutes: as long as their sum remains constant, at the margin the principal can increase thecarrot and reduce the stick (or vice versa) while keeping the compliance threshold ek unchanged.

Now consider a modified example with c = 60, s = 40, and the same q as above. Clearly ek

remains unchanged because the sum of the carrot and the stick has not changed. The payoff for

violators is now equal to 14 60 − 3

4 40 = −15, which is negative. So violators will not participate inthe regulated activity. Moreover, at the margin, some potential compliers will also find it moreadvantageous to abstain from participation. More precisely, a (potentially) complying agent willparticipate if his payoff from compliance is positive, that is qc − (1 − q) s − e > 0. This will be thecase if the agent’s effort cost e is less than the participation threshold ep ≡ qc − (1 − q) s = 35. Thelower threshold reveals that fewer agents participate and comply compared to the previous case.We will refer to this case as “partial participation.”

Note that in the partial participation case compliance still increases in q and c, but it now de-creases in s. In particular, the carrot and the stick are complements: to keep compliance constant

their ratio must be kept equal to cs =

1−qq . The reason for this change is that in the partial partic-

ipation case the participation threshold dominates the compliance threshold. Some agents (thatis, those with ep ≤ e < ek) would comply if they participated, but they do not participate becauseparticipation yields a negative payoff.

In short, in the full participation case agents choose between compliance and violation—forthem carrots and sticks are substitutes. In the partial participation case agents choose betweencompliance and non-participation—for them carrots and sticks are complements. Varying thecarrot or the stick has different effects on these two groups of agents.

In real-world settings agents vary along several dimensions. (We consider variation in benefitsin the main model of Section 2.1 analyzed below, and variation in the (perceptions of)principal’saccuracy in Section 3.3.) As we show below, the full participation and the partial participationcases may well take place at the same time. When this happens, the effects of carrots and sticks onparticipation and compliance decisions may become ambiguous. The same is true of the effects ofsome other common regulatory instruments. These ambiguities bear on policy choices in impor-tant and so far unexplored ways. In the following section we drop the simplifying assumptionsmade in this example and return to the general setup introduced at the outset.

2.3 Analysis

It is useful to begin the analysis of the model introduced in Section 2.1 by defining the followingthreshold level for b

bv ≡ qvs − (1 − qv) c

9That is, ek increases in q, c, and s.

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which is the level of b such that Πv (b) = 0. Specifying this benefit threshold bv allows us topartition the population of agents in two groups:

• High-benefit agents (with b > bv) participate in the activity irrespective of whether they com-ply or violate. To see why, note that if b > bv, then Πv (b)> 0 and hence the payoff of violatorsis positive. If the agent complies, he must also earn a positive payoff as well because he willcomply only if Πk (b, e) > Πv (b), and we know that the latter is positive. Overall, high-benefit agents form the full participation group—they all take part in the activity. Whetherthe agent complies or violates depends on his cost of effort e. More precisely, the agent willcomply if e < ek where the compliance threshold ek is

ek ≡c + s

r(qk + qv − 1)

which is the level of e such that Πk (b, e) = Πv (b). The agent will violate if e ≥ ek. When thecompliance threshold controls, carrots and sticks are substitutes.

• Low-benefit agents (with b ≤ bv) participate in the activity only if they comply. To see why,note that if b ≤ bv, then Πv (b)≤ 0 and hence the payoff of violators is (weakly) negative. Thepayoff for compliers may be positive or negative. Overall, low-benefit agents form the partialparticipation group where all participants comply, all violators abstain from participation,and some would-be compliers choose not to participate as well. A potential complier willparticipate only if Πk (b, e)> 0 at his level of e. More precisely, the agent will comply if e < ep

where the participation threshold ep is

ep (b) ≡1

r(b + qkc − (1 − qk) s)

which is the level of e such that Πk (b, e) = 0. The agent will participate and comply ife < ep (b) and will not participate if e ≥ ep (b). Note that ep (b) controls both complianceand participation because an agent who does not have incentives to comply would earn anegative payoff by participating and violating. When the participation threshold controls,carrots and sticks are complements.

In sum, ek is the compliance threshold for participants, which determines the choice between com-plying and violating, on the assumption that the agent participates. This threshold controls theagent’s compliance decision if b > bv, that is, in cases where both complying and violating agentsparticipate. In contrast, ep (b) is the participation threshold for complying agents. It controls be-havior when b ≤ bv, that is, when violating agents do not find it advantageous to participate. Inthis case, the alternative to compliance is abstention rather than violation.

We impose no constraints on the possible benefits of heterogeneous agents in our model. There-fore, some agents in our model belong to the full participation group while others belong to thepartial participation group. Figure 1 characterizes the agents’ behavior. Quite intuitively, agentswith low costs of effort participate and comply. Those with large benefits and high effort costs par-ticipate but violate. The remaining agents (those with low benefits and high effort costs) abstainfrom participation.

The next two propositions show how changes in selective incentives affect the outcomes of in-terest to most real-world principals. These outcomes are the absolute number of complying agents

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Figure 1: Agent’s behavior (simulation parameters: qk = qv =34 , c = s = 1

2 , r = 1).

(“compliance”), the absolute number of violating agents (“violations”), overall participation (com-pliance plus violations), and the ratio of the number of compliers over the number of participants(“compliance rate”). The results of both propositions are summarized in Table 2, proofs are in theAppendix.

Proposition 1. An increase in the accuracy in identifying violations, qv, results in an increase in com-pliance, a decrease in violations and participation, and an increase in the compliance rate. An increase inthe accuracy in identifying compliance, qk, results in an increase in compliance, a decrease in violations, anincrease in participation, and an increase in the compliance rate. An increase in the required effort resultsin a decrease in compliance, an increase in violations, and a decrease in participation and in the compliancerate.

An increase in accuracy has the plausible effect of increasing compliance and reducing vio-lations. This is a well-known result (Becker 1968, Png 1986). However, we also show that theeffect on participation depends on whether accuracy increases with respect to identifying viola-tors or compliers. Greater accuracy in identifying compliers, qk, increases the participation andcompliance thresholds but has no effect on the benefit threshold. As a result, some former vio-lators start complying, and some former non-participants do the same, but the decision betweennon-participating and violating is unaffected. So higher qk leads to greater participation becausesome former non-participants start complying and that is the only change made by former non-participants.

In contrast, greater accuracy in identifying violators, qv, increases both the compliance thresh-old and the benefit threshold, while keeping the participation threshold unchanged. This meansthat some former violators start complying, and some former violators stop participating, but thedecision between non-participating and complying is unaffected. So higher qv leads to lower par-ticipation because the only change in participation comes from some violators deciding to abstainfrom participating.

Making the rule more demanding (increasing r) reduces compliance and participation in a pre-

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dictable way but also increases violations, negatively affecting the compliance rate. The reason isthat effort requirements have no bearing on the choice between violating the rule and not partici-pating at all (that is, bv is unaffected). Hence, at the margin, greater required effort induces thosewho participate anyway to violate the rule rather than comply with it.

Proposition 2. An increase in the carrot, c, results in an increase in compliance and participation, whilethe effect on violations and the compliance rate is ambiguous. An increase in the stick, s, results in a decreasein violations and participation, while the effect on compliance and the compliance rate is ambiguous.

Carrots and sticks have predictable effects on participation, which increases with carrots anddecreases with sticks. However, while carrots unambiguously increase compliance, they may ormay not reduce the occurrence of violations. The reason is that, by drawing more agents to theactivity, carrots also increase the payoff for those who choose to participate and violate the rule.Similarly, sticks unambiguously reduce violations but may or may not increase the number ofcomplying agents. This is because while sticks induce some violators to comply, sticks also pushsome compliers away from the regulated activity. The balance of these opposing effects dependson values of the various policy parameters (and in the Appendix we show that these results donot depend on a particular distribution of the agents’ types and do not disappear if each point onthe graph is given equal weight, which is the case, for instance, if e and b are independently anduniformly distributed on the unit interval.)

(a) The ambiguous effect of carrots (from c = 12 to c

′

= 34 ) (b) The ambiguous effect of sticks (from s = 1

2 to s′

= 34 )

Figure 2: Effects of an increase in the carrot or the stick (simulation parameters: qk = qv =34 , r = 1).

Figure 2 visualizes these results. An increase in the carrot pushes up both the compliance andthe participation thresholds but reduces the benefit threshold bv. As a result, the compliance areaexpands, the no-participation area contracts (resulting in more participation), but the effect on theviolation area is ambiguous. The increase in the compliance threshold causes some violators tocomply. The decrease in bv, however, enlarges the full participation group, causing some non-participants to become violators.

Similarly, an increase in the stick increases ek and bv, but reduces ep (b). As a result, the violationarea contracts, the no-participation area expands (resulting in less participation), while the effecton the compliance area is ambiguous. The increase in ek induces some violators to comply, whilethe reduction in ep (b) leads some compliers to abstain from participation.

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Policy variable Compliance Violations Participation Compliance rate

qv Increase Decrease Decrease Increaseqk Increase Decrease Increase Increaser Decrease Increase Decrease Decreasec Increase Ambiguous Increase Ambiguouss Ambiguous Decrease Decrease Ambiguous

Table 2: Effects of an increase in the policy variables on compliance, violations and overall partici-pation

3 Extensions

3.1 Imperfect monitoring

In the basic model, we assumed that the principal monitors each agent with certainty. Here, weintroduce imperfect monitoring (or auditing) in the model. We posit that the principal monitorseach agent with probability p ∈ (0,1). As a result of imperfect monitoring, agents who are notmonitored may–in the general case–receive a carrot with probability φc ≥ 0, a stick with probabilityφs ≥ 0, or neither of the two with the residual probability 1 − φc − φs ≥ 0.10

To illustrate, if φc = 1, then non-monitored agents receive a carrot with certainty. This casecorresponds to a subsidy given to agents who engage in the activity and are not found in violationof the rule. Similarly, if φs = 1, then non-monitored agents are subject to a stick with certainty.This amounts to taxing the activity unless the agent is found to comply. Finally, if φc = φs = 0, thencarrots and sticks are applied only to monitored agents. In the general case, non-monitored agentsmay be taxed or subsidized with some probability.11

Action Payoff

The agent participates and complies Πk (b, e) = b + pqkc − p (1 − qk) s − re + (1 − p) (φcc − φss)The agent participates but violates Πv (b) = b + p (1 − qv) c − pqvs + (1 − p) (φcc − φss)

The agent does not participate 0

Table 3: Agent’s payoffs with imperfect monitoring

From the payoffs reported in Table 3, we can calculate the relevant thresholds with imperfectmonitoring:

biv = p (qvs − (1 − qv) c)− (1 − p) (φcc − φss)

eik = p c+s

r (qk + qv − 1)ei

p (b) = 1r [b + p (qkc − (1 − qk) s) + (1 − p) (φcc − φss)]

Imperfect monitoring has two effects on the thresholds that we consider in the analysis. First,for all thresholds, imperfect monitoring dilutes the effect of carrots and sticks proportionally tothe probability of monitoring p. Second, while the compliance threshold ei

k does not depend onthe treatment of non-monitored agents, the other two thresholds do. The factor (1 − p) (φcc − φss)

10See Dari-Mattiacci et al. 2009 examining the incentive problems that arise from rewarding or punishing non-monitoredagents.

11In Section 3.2 we examine the classical case of taxes and subsidies that are not conditional to the agent’s behavior.

12

is the net expected payment that non-monitored agents receive, which is positive if φcc > φss, thatis, if non-monitored agents are rewarded in expectation and negative otherwise.

It is easy to verify (see Appendix) that the expanded model including imperfect monitoringgenerates exactly the same results as the basic model with respect to the selective incentives con-sidered there. In addition, however, when monitoring is imperfect, there is an additional policyvariable: the probability of monitoring.

Proposition 3. Propositions 1 and 2 hold also with imperfect monitoring, p < 1. In addition, an increasein the probability of monitoring, p, has ambiguous effects on compliance, violations, participation and thecompliance rate.

As it is easy to verify by inspection, an increase in p results in an increase of eik while having an

ambiguous effect on biv and ei

p (b). In particular, biv increases in p if (qv − φs) s > (1 − qv − φc) c, that

is, if the expected stick for violators exceeds their expected carrot, net of all payments to or by non-monitored agents, and decreases in p otherwise. The intuition is that if the expected carrot is largeenough, increasing the probability of monitoring induces violators to participate in the activity(due to lower bi

v). Similarly, the threshold eip (b) increases in p only if (qk − φc) c > (1 − qk − φs) s,

that is, if compliers face an expected carrot larger than the expected stick. In this case, increas-ing monitoring induces some violators to start complying. These ambiguities make it impossibleto predict the effect of increased monitoring levels on compliance, violations, participation, andrelative compliance in the general case.

Let us now turn to a realistic sub-case in which non-monitored agents receive a carrot withcertainty, that is, where φc = 1 and φs = 0. In this case, we have

biv = p (qvs + qvc)− c

eik = p c+s

r (qk + qv − 1)ei

p (b) = 1r [b + p (qkc − (1 − qk) s) + (1 − p) c]

It is easy to see that biv now increases in p while ei

p (b) decreases in p. This resolves some but notall ambiguities. In particular, the effect of an increase in p on absolute and relative complianceremains ambiguous as illustrated in Figure 3.

Corollary 1. When non-monitored agents receive a carrot then an increase in the probability of monitoringresults in a decrease in violations and in participation and has ambiguous effects on compliance and on thecompliance rate.

An increase in monitoring when φc = 1 leads to an increase in ek and bv, but reduces ep (b).These are the same shifts that result from an increase in sticks. This similarity is quite intuitive.Greater monitoring amounts to a greater expected cost for violators and mistakenly punishedcompliers. The resulting consequences and ambiguities are the same as when an increase in thestick raises the same expected cost.

3.2 Taxes and subsidies

Let us now examine whether adding a tax to the model expands the set of outcomes that theprincipal can reach. Assume that the principal taxes non-participation in the activity by a fixedamount τ. Obviously, and in contrast with the carrot and the stick in our model, this tax does notdepend on the compliance decision of participating agents. Alternatively, the principal subsidizes

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Figure 3: Effects of a change in apprehension rate (from p = 23 to p

′

= 1; other simulation parame-

ters: qk = qv =35 , c = 1

4 , s = 34 , r = 1, φc = 1, φs = 0).

the regulated activity (again, regardless of the principal’s findings of compliance or violation) bythe same amount τ. (If τ is negative we have a tax for participants or, equivalently, a subsidy fornon-participants.) The new payoff matrix is as follows:

Action Payoff

The agent participates and complies Πk (b, e) = b + qkc − (1 − qk) s − re + τThe agent participates but violates Πv (b) = b + (1 − qv) c − qvs + τ

The agent does not participate 0

Table 4: Agent’s payoffs

We have now the following thresholds:

bτv = qvs − (1 − qv) c − τ

eτk = c+s

r (qk + qv − 1)eτ

p (b) = 1r (b + qkc − (1 − qk) s + τ)

Clearly the compliance threshold is unaffected by τ, because every participant receives τ whetherhe complies or not. In contrast, participation will be incentivized by τ, both for compliers and forviolators, since τ reduces bτ

v and increases eτp (b). The addition of τ, however, does not expand the

set of outcomes that the principal can reach. To see why, consider that the principal could replicatethe same compliance and participation thresholds as above by simply adjusting the carrot andthe stick instead of implementing a tax or a subsidy. More precisely, the principal could increasethe carrot by an amount equal to τ and reduce the stick by the same amount; that is, the princi-

pal could set c′

= c + τ and s′

= s − τ to mimic the thresholds reported above. The compliance

threshold is clearly met, because c′

+ s′

= c + s. The participation threshold is also met because we

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have

ep (b) = 1r

[

b + qkc′

− (1 − qk) s′]

= 1r [b + qk (c + τ)− (1 − qk) (s − τ)]

= 1r [b + qkc − (1 − qk) s + τ]

= eτp (b)

Likewise, for the benefit threshold we have

bv = qvs′

− (1 − qv) c′

= qv (s − τ)− (1 − qv) (c + τ)= qvs − (1 − qv) c − τ= bτ

p

In essence, the principal can replicate the effects of taxes and subsidies by keeping the sum ofthe carrot and the stick constant (so as to keep the compliance threshold constant) while alteringtheir ratio c

s . That ratio should increase if the activity is subsidized (that is, if τ is positive) anddecrease if the activity is taxed (that is, if τ is negative), leading to the following proposition:

Proposition 4. Complementing carrots and sticks with a tax or a subsidy on the regulated activity does notaffect any of the results; the principal can replicate the effect of any tax or subsidy, τ, by setting c′ = c + T

and s′

= s − T where:

T =τ

p + (1 − p) (φc + φs)

If monitoring is perfect (p = 1) then T = τ as illustrated above. Even if monitoring is imper-fect (p < 1) we still have T = τ if φc + φs = 1, that is, if non-monitored agents are always eitherrewarded or punished. The common case where all unmonitored participants receive a carrot isone particular example when this condition is met. In the general case, the adjustment to the ratioof carrots and sticks needs to account for the treatment of non-monitored agents. The intuitionis that, if there is a positive probability that a non-monitored agent is subject neither to a carrotnor to a stick, that agent fails to be compensated for the effect of the tax or subsidy and hence thecorrection on the carrot and the stick needs to take that eventuality into account.

3.3 Agents’ perceptions of accuracy

In the model considered thus far, agents vary with respect to their costs of effort and the benefitthat they derive from participation. Here we report the results on an alternative model in whichthe benefit from participation is uniform across the population and, for simplicity, is normalized tozero, while the principal’s accuracy (qk and qv) varies among the agents. (The full formal analysisis in the Appendix.)

This variation is so likely that assuming it away would considerably limit the realism of themodel. This is so for two reasons. First, whatever principal believes about her own accuracy, itis difficult for the principal to convey this belief to the agents, and all but impossible to do socredibly. Second, it is implausible to assume that the principal is equally accurate in evaluatingcompliance of every agent.

Following the same logic as we used in analyzing heterogeneous benefits, we define a thresholdlevel of the (perceived) accuracy in detecting violations, q̂v, which separates agents into the fulland partial participation groups. Then in each group we identify a compliance threshold—which

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is now a function of the perceived accuracy of detection both violations and compliance, and aparticipation threshold—which is a function of the perceived accuracy in detecting compliance.The analysis then proceeds as in the main model and we derive the same comparative staticsresults.

4 Discussion

Our results shed new, and sometimes unexpected, light on the generally familiar economic analy-sis of incentives. To start, we find that accuracy is the most effective regulatory instrument. Greateraccuracy leads to more compliers and fewer violators. Notably, an increase in accuracy is the onlymeasure that has these salutary effects.

Not all accuracy improvements are the same, however. The traditional view supporting thepreponderance of the evidence standard in civil litigation is that it “results in a roughly equalallocation of the risk of error between litigants” (Grogan v. Garner, 498 U.S. 279, 286 (1991)). Falseacquittals, that is, are as likely as false convictions. It is not hard to imagine that even in non-litigation settings many principals may want to roughly balance the two types of errors.

Kaplow (2012) criticizes this balancing on welfarist grounds. We offer a more practical reasonto differentiate between false negatives and false positives. Reducing both types of errors increasesboth the number of compliers and the compliance rate, but the effects on participation differ. Whileincreasing the accuracy in identifying violations reduces participation, increasing the accuracy inidentifying compliance increases participation. For a principal who aims to achieve a particularlevel of both participation and compliance, increases in the two types of accuracy are clearly notinterchangeable.

While our general conclusion that a more accurate enforcement leads to better results is hardlysurprising, our overall conclusion regarding carrots and sticks is less obvious and, as far as weknow, is new in the literature. Quite simply, neither carrots nor sticks are effective selective incen-tives. Greater carrots yield the expected benefit of increasing compliance. But we show that theireffect on compliance rate is ambiguous. A regulator may well decide that an increase in the num-ber of compliers combined with an even greater increase in the number of violators is a negativerather than a positive development. Expending resources on greater carrots while facing a possi-bility of such outcome may give pause to a regulator. Sticks are even less attractive. It is unclearwhether higher sticks improve compliance under any definition of the term. Both the number ofcompliers and the rate of compliance may increase or decline when the principal increases thestick.

Turning to the consequences of making a rule more demanding reveals more results that areboth new and concerning for regulators. Generally, the required effort is not viewed as germaneto the enforcement analysis. That effort has to do with the design of the rule rather than its en-forcement. This logic may explain the lack of attention to the effects of the required effort onparticipation and compliance in the literature.

We show that the design of a rule cannot be separated from its enforcement. In fact, raising therequired effort is especially detrimental to the principal’s objectives. Greater required effort lowersthe number of compliers and increases the number of violators—both effects are the opposite ofwhat a principal would want.

Some implications of this finding are startlingly obvious. Instead of hiring one hundred work-ers to work eight extra hours each the employer should hire two hundred workers to work four

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extra hours each in two shifts. This change will improve the production results without any changein traditional enforcement parameters and, most likely, any meaningful increase in the employer’scosts. In fact, the employer is likely to save costs because it will need to reduce the carrot to preventparticipation from increasing as a result of the decrease in the extra-hour requirement.

Our analysis of imperfect monitoring (detection) confirms the well-known result that uncertaindetection dilutes incentives. Moreover, if p is low, the benefit threshold is small or even negative,so every agent is in the full participation group. A low p also makes the compliance thresholdsmall, meaning that most participants violate the rule. Such widespread noncompliance, however,is not observed in practice. This is a well-known puzzle (Graetz and Wilde 1985, Weisbach 2002).

If low level of monitoring leads to widespread noncompliance, and perfect monitoring is likelyto produce meaningful compliance (as we conclude above), one is lead to the long-dominant viewthat increasing the rate of monitoring increases compliance (Becker 1968). We show, however, thatthis view is wrong when the principal deploys selective incentives rather than command-and-control regulation. Greater monitoring has an ambiguous effect both on the number of compliersand on the rate of compliance. This finding further highlights the unique complexities of regulat-ing optional selective-incentives schemes.

In sum, our analysis confirms some well-known results but also offers a number of novel in-sights. The high payoff from improving accuracy in identifying violators, the ineffectiveness ofhigher sanctions as a selective incentive, and the ambiguous consequences of greater monitoringare just some examples of the latter. We consider the practical importance of our results next.

5 Applications

Our model aims to inform real-world decisionmakers. In this section we consider whether ouranalysis reflects existing private and public regulatory regimes and whether our conclusions shedlight on actual regulatory reforms. We answer both questions in the affirmative.

5.1 Private Law

Our leading vignette—a firm searching for productive overtime workers—is quite realistic, and itsuggests that our model applies to private contracting. We now offer two examples demonstratingthat our analysis bears directly on the efforts to improve selective incentives undertaken overseveral decades by multiple players in two important industries.

The first industry is the U.S. auto and, more broadly, heavy machinery equipment manufac-turing. The so-called original equipment manufacturers (OEMs) like General Motors, Caterpillar,and Cummins must choose among tens of thousands of potential suppliers for thousands of partsthat go into producing an automobile, a tractor, or a locomotive (Bernstein 2015). Supplier failuresmay have major negative effects on OEMs, as the tragic debacles related to Firestone tires (Wald2000) and Takata airbags (Spector and Fujikawa 2016) aptly demonstrate. So it is critical for OEMsto induce the best suppliers to compete for OEMs’ business. It is also critical to make sure that thesuppliers chosen by OEMs actually are—and continue to be—the best.

The multi-decade history of OEM-supplier relationships reflects a dramatic shift in OEMs’supplier-selection approach. More and more, OEMs are de-emphasizing carrots and sticks andare focusing on increasing accuracy in identifying and keeping the best suppliers.

17

In the middle of the twentieth century, the Big Three automakers purchased most of their partsfrom multiple suppliers, imposing harsh competition on them (Kenworthy, Macaulay, and Rogers1996). By the 1980s, OEMs switched to single-sourcing, selecting one supplier for any given part.Securing the status of a sole supplier was a huge carrot compared to the previous inter-suppliercompetition regime. Yet the stick increased as well. Suppliers had little security. Most supplierexecutives believed that OEMs would readily switch to a new single-source supplier (thus denyingthe initial supplier all of its business with the OEM) if the competing supplier offered a lowerprice for the same quality product (or, presumably, higher quality for the same price) (Kenworthy,Macaulay, and Rogers 1996 p. 649-50).

In the past decade or so, however, OEM’s focus shifted decisively to selecting the best suppliersand continuously evaluating whether the selected suppliers remain the best. The emphasis isno longer on rewards and punishments, but on jointly working through problems when theyarise. The first critical goal of this process is to assure that the suppliers truly comply with theirformal and informal commitments to OEMs. The second goal is to avoid penalizing suppliersfor occasional uncharacteristic failures (Bernstein 2015 p. 570-71; 576-85; Gilson, Sabel, and Scott2018). In other words, OEMs are now very focused on the accuracy of their evaluations of supplierquality.

Overall, the U.S. auto industry has undergone a multi-decade shift reflecting some of the keypayoffs from our model. It learned that improving accuracy has a higher payoff than an increasingemphasis on carrots and sticks.

It appears that another important industry has gone in the opposite direction. Over the pastseveral decades, major American law firms have moved decisively towards higher rewards, costliersanctions, and surprising inattention to the accuracy in making their most important business de-cisions—admitting new partners.

Over the past several decades, the world of large law firms has undergone a dramatic transfor-mation. Young associates starting their careers at America’s largest two hundred or so law firmsin the 1980s had rather settled expectations. They would aspire to become partners at their firms,see their partnership earnings grow gradually in lock-step with other partners in their cohort,and eventually retire after a multi-decade career with the firm that they joined as wide-eyed lawschool graduates (Galanter and Henderson 2008, Henderson 2006). Today, almost no part of thispicture remains true. The lock-step compensation model is all but gone (Henderson and Bierman2009). The number of single-tier partnerships is shrinking (Henderson and Bierman 2009, Hen-derson 2006). Lateral partner moves are omnipresent (Henderson and Bierman 2009, Press andCho 2014). And the spread in profits between the highest- and lowest-paid partners at the samefirm has skyrocketed (Gluckman 2017, Hildebrandt Consulting 2018). Law firms compete fiercelyfor high-performing partners, and partners are not shy about choosing the highest bidder for theirservices, possibly again and again.

This picture of law firms relying on increasingly large carrots would be incomplete withoutmentioning increasingly large sticks. In the past, law firm partnership was not too different fromacademic tenure (Galanter and Henderson 2008, Press and Cho 2014). Partners were rarely fired.And because there was no such thing as non-equity partnership, partners were not de-equitized.Today, the brass ring is not nearly as appealing as it used to be. Partners work harder than everbefore (Henderson 2006). De-equalizations are far from uncommon (Cecchi-Dimeglio and Simons2018, Galanter and Henderson 2008, Henderson 2006, Press and Cho 2014). Without lock-stepcompensation, partners may see their shares and profits reduced. And even outright partner dis-missals are not unheard of (Galanter and Henderson 2008, Press and Cho 2014). Clearly, the mod-

18

ern world of large law firms features much higher carrots and sticks compared to several decadesago.

Given the explosion of lateral partner moves (Galanter and Henderson 2008, Henderson andZorn 2015), and the substantial cost of hiring a lateral partner (Bruch, Dewey, and Kovalan 2015),one would think that law firms have developed sophisticated ways of screening the potentiallateral hires. It turns out, however, that quite the contrary is true.

Major American law firms—the ultimate experts in carrying out due diligence on behalf oftheir clients—do shockingly little due diligence for themselves (Simons and Ellenhorn 2018). Theresults, not surprisingly, are disappointing (Bruch, Dewey, and Kovalan 2015). For example, “only28 percent of managing partners believe that lateral hiring has been effective at their firm” (Hen-derson and Zorn 2015). Recent estimates show that thirty to forty percent of lateral partnersdepart within five years of joining the firm (Cecchi-Dimeglio and Simons 2018, Simons and El-lenhorn 2018). This state of affairs is precisely what our model would predict. No matter howhigh-powered the incentives become, they will remain ineffective if the accuracy is low.

5.2 Government Regulation

Selective incentive schemes are pervasive in public law as well. And the regulatory challenges thatwe investigate are very familiar to public regulators. The U.S. government spends multiple billionseach year on cash grants (Goodwin and Smith 2018; Theodos, Stacy and Ho 2017), loan assistance(Goodwin and Smith 2018), price and revenue protection programs (Glauber 2018; Sumner 2018),insurance subsidies (Glauber 2018; Sumner 2018), and other carrots designed to induce participa-tion in a wide variety of government schemes aimed at anything from peanut farming (Goodwinand Smith 2018) to community development (Theodos, Stacy and Ho 2017) to workplace diversity(Sturm 2006). Small businesses alone benefit from fifty two different federal assistance programsadministered by the Departments of Agriculture, Commerce, Housing and Urban Development,and the Small Business Administration (U.S. Government Accountability Office 2018). U.S. statesspend billions more on similar inducements (Patrick 2014).

Many of these programs present regulators with the challenges of managing participation andcompliance, and regulators respond to these challenges in the ways that we model. The naïve viewthat any carrot would induce carrot-claiming clearly fails to reflect reality even when the regulatedparties are informed and sophisticated. U.S. cotton subsidies enacted in 2015, for example, areclaimed by less than 25 percent of eligible recipients (Glauber 2018). The results are similarlydisappointing for a recently enacted dairy subsidy (Sumner 2018).

Some grant programs challenge the recipients (and the regulators) with “a complicated labyrinthof criteria”, making enforcement errors all but certain (Howells 19970, pp. 393-94, 412). Somecarrot-claimers skirt the rules, failing to comply either with their letter or their spirit (Goodwinand Smith 2018; Howells 1997; Theodos, Stacy and Ho 2017). In response, regulators use penal-ties to deter improper carrot-claiming (Howells 1997). And although the regulators often react toan unsuccessful selective incentive scheme by reducing the carrot or eliminating the program alto-gether (Sumner 2018; Theodos, Stacy and Ho 2017), commentators emphasize the need to improvethe program’s accuracy (Howells 1997; Theodos, Stacy and Ho 2017).

There are also examples of more recent, innovative grant programs that, like OEMs describedabove, abandon the traditional compliance emphasis on higher sticks. Instead, regulators focuson working jointly with regulated parties to identify and evaluate problems as they arise, aimingto make sure that compliers are not mistaken for violators and vice versa (Sturm 2006). These

19

programs—such as the workplace diversity ADVANCE program—emphasize the accuracy of se-lective incentives just as our analysis does.

The U.S. government spends billions more in selective incentives outside of the federal bud-get. The so-called tax expenditures—special tax breaks in the Internal Revenue Code designed toinduce participation in a particular activity—offer potential participants a carrot just like outrightcash grants do. And for the largest eight hundred or so companies under a continuous IRS audit(Elliott 2012), the only uncertainty surrounding the carrot arises from inaccurate enforcement bythe government auditors that is key to our model.

In fact, all three of the largest corporate tax expenditures of the recent past—the deductionfor accelerated depreciation, the domestic production activities deduction (repealed at the end of2017), and the research and development (R&D) tax credit—illustrate every feature of our model(Raskolnikov and Dari-Mattiacci 2018). The R&D credit, to take one example, is a carrot. Its falseclaiming is punished with a tax penalty stick. Over the years Congress has varied both on multipleoccasions. The same is true of the required effort and the accuracy of the government’s eligibilitydeterminations. Given that Congress tends to tinker with tax incentives particularly often, ouranalysis should be especially valuable in that context.

Two further insights highlight the wide application of our results. First, the carrot need not takethe form of a payment or a reduction in a price or tax liability. Any government-created benefit isa carrot that may be a part of a selective incentive regime.

The U.S. patent system is a case in point. The government incentivizes innovation with acarrot of exclusive use of the patented invention during the patent term. The invention’s noveltyis an imperfectly observable characteristic that occasionally leads to inaccurate patent-grantingdecisions by the U.S. Patent and Trademark Office. And Congress has tried to deter bad-faithclaims on novelty by penalizing them with criminal fines—a stick if there ever was one (Yelderman2017).

The second insight is that the carrot need not be intended as such by the regulator. Considerthe Federal Reserve’s so-called discount window that offers eligible banks access to short-termliquidity under certain conditions. The discount window was put in place in 1913 to establishthe Fed as the lender of the last resort, and possibly to incentivize small-business lending by localbanks (Hunt 2014). It was certainly not created to incentivize trading firms to become banks.Yet this was exactly how it worked when Morgan Stanley and Goldman Sachs chose to becomebanks in the midst of the financial crisis (Sorkin and Bajaj 2008). The discount window acted as anunintended quasi-carrot.

The Volcker Rule enacted in the aftermath of the crisis in order to stop proprietary tradingby deposit-taking institutions completed the quasi-incentive regime that we model. The discountwindow is a quasi-carrot. The broad enforcement powers of the Fed are a stick (Hamilton 2017).And the purpose-based definition of proprietary trading, as well as the hedging and market-making exceptions, in the Volcker Rule inevitably give rise to false positives and false negatives.Other examples of quasi-incentive schemes are easy to find (Raskolnikov and Dari-Mattiacci 2018).

5.3 International Relations

Finally, we turn to international relations. Short of going to war, sovereign states cannot imposetheir will on other states. Every international regulatory regime is an optional selective-incentivemechanism. So our findings are particularly relevant to international relations.

20

Consider the example that Ben-Shahar and Bradford offer to illustrate their proposal for a si-multaneous use of carrots and sticks to create “reversible rewards” (Ben-Shahar and Bradford2012). Environmentally-conscious countries (let us call them the “EU states”) want to persuadeother nations—most notably China and the United States—to join a global climate change treaty.12

Ben-Shahar and Bradford focus on designing a stronger incentive to participate (which theyrefer to as not “violating” the EU wishes). But the authors also note in passing that a participatingstate may decide not to “fully comply” with the treaty.

This non-compliance problem is central to our inquiry. And some of our main results—theimportance of accuracy and the ineffectiveness of sanctions—are particularly applicable in theinternational setting where neither command-and-control regulation nor armed conflict are viableoptions for regulating multi-party arrangements.

In sum, the examples discussed in this Section demonstrate that selective incentives are perva-sive in public and private law. Public and private regulators routinely adjust these incentives withthe aim of improving participation with the incentive regimes and compliance with the regimes’requirements. Yet there appears to be no prevalent view about which incentive should be adjustedand how. U.S. automakers and the National Science Foundation implementing the ADVANCEprogram focus on accuracy, and so does the U.S. Treasury and the Internal Revenue Service whendealing with large corporate tax incentives. But law firms prefer raising carrots and sticks whileCongress lowers them for both farm- and non-farm subsidies. Most regulators tinker with therequired effort in one way or the other. Our analysis of selective incentives should aid all theseregulators in making more informed judgements.

6 Conclusion

This paper analyzes an old problem using a novel approach. We aim to inform any principal—anemployer, a contractor, a government agency, or an international body—about the most effectiveuse of incentives that the principal has at her disposal. We use the term “effective” rather than“efficient” deliberately. Our analysis is not aimed at maximizing efficiency, though it would informan efficiency-maximizing principal how to reach the principal’s goals that she identified on herown. Rather, we investigate how a principal can achieve two common real-world objectives ofparticipation and compliance using widespread regulatory instruments.

Our main result is to arrange the tools in the regulator’s toolbox in order of their effectiveness.We show that increasing the enforcement accuracy is the most effective policy tool, followed bycarrots. The use of sticks and reliance on the probability of monitoring (auditing) are problematicbecause both have ambiguous effects on absolute and relative compliance. Finally, increasing theeffort requirements of a regulation or a contract is the most ineffective of the policies consideredbecause it leads to fewer compliers and more violators.

Our analysis has limitations. We discuss four of them here. The first limitation reflects a well-known and persistent challenge to the economic analysis of law. Agents in our model cannot affectthe principal’s accuracy by adjusting their behavior. Agents either comply or violate; they cannotbarely cross the line or engage in egregious noncompliance. This assumption is more realistic insome settings than in others.

12Presumably, the desired degree of participation here approaches one hundred percent. Yet one can imagine specialcases of small, poor countries for whom the amount of pollution is relatively small and the cost of pollution abatementwould be so high (relative to their GDP) that it may make sense not to impose strict pollution requirements on thesecountries at least for some time.

21

The assumption of agents’ binary behavior is standard in law and economics. The reason forit is well-known: eliminating this assumption leads to indeterminate results (Craswell and Calfee1986; Shavell 1987). Several recent contributions show that in certain settings this indeterminacydisappears (Baker and Raskolnikov 2018; Dari-Mattiacci 2005). Our analysis does not contributeto further resolving this long-standing challenge.

Another limitation is our principal’s inability to tailor its instruments to individual agents. Thisinability is one of the key distinctions between a constrained model like ours and the mechanismdesign literature (Kaplow 2017). In reality, regulators usually deploy crude, uniform instrumentsrather than variable incentives (Kaplow 2017, Montero 2005, Stavins 2003). So our assumption ofuniform carrots, sticks, and required effort is quite realistic in many settings.

We do not model costs of various selective incentives because we are not concerned withwelfare-maximizing policy interventions. These costs are likely to vary. Carrots are usually morecostly for the principal than sticks. Increasing the rate of monitoring is likely to consume moreof the principal’s resources than adjusting the required effort. Our contribution is to clarify thebenefits of various selective incentives, leaving it to the principals (and to future work) to balancethese benefits against costs.

Finally, and at the risk of stating the obvious, our model is a simplified version of reality. Notonly agents’ benefits and principal’s accuracy vary simultaneously in real-world settings, otherfactors such as risk aversion, uncertainty aversion, and various heuristics and biases likely comeinto play. It is unlikely that introducing further complexity into the model will resolve the ambigu-ities that we identify. If anything, further complexity may negate some of our clear results. So, asalways, any real-world regulator following our suggestions would do well by closely monitoringthe actual responses to any change in selective incentives (Duflo 2017).

References

[1] Andreoni, James, William Harbaugh, and Lise Vesterlund. 2003. The Carrot or the Stick: Re-wards, Punishments, and Cooperation, 93 American Economic Review 93:893-902.

[2] Baker, Scott, and Alex Raskolnikov. 2017. Harmful, Harmless, and Beneficial Uncertainty inLaw. Journal of Legal Studies 46: 281-307.

[3] Ben-Shahar, Omri, and Anu Bradford. 2012. Reversible Rewards. American Law and EconomicsReview 15:156-80.

[4] Bernstein, Lisa. 2015. Beyond Relational Contracts: Social Capital and Network Governancein Procurement Contracts. Journal of Legal Analysis 7:561-621.

[5] Bruch, Nicholas, Eric Dewey, and Steven Kovalan. 2015. Surmounting the Lateral Partner Hir-ing Challenge: Lessons Learned, Best Practices, and Tools for Success. ALM Legal Intelligence.

[6] Cecchi-Dimeglio, Paola, and Hugh A. Simons. 2018. Lateral Moves: An Empirical Investiga-tion of Cyclicality, Directional Mobility, and 5-Year Retention Rate by Gender and Age Cohort.Journal of Legal Profession 42:171-195.

[7] Dari-Mattiacci, Giuseppe. 2005. Errors and the Functioning of Tort Liability. Supreme CourtEconomic Review 13:165–87.

22

[8] Dari-Mattiacci, Giuseppe. 2009. Negative Liability, Journal of Legal Studies 38:21-60.

[9] Dari-Mattiacci, Giuseppe, Gerrit De Geest and Jacques J. Siegers. 2009. Annullable Bonusesand Penalties. International Review of Law and Economics 29:349-359.

[10] De Geest, Gerrit and Giuseppe Dari-Mattiacci. 2013. The Rise of Carrots and the Decline ofSticks. University of Chicago Law Review 80:341-93.

[11] Duflo, Esther. 2017. The Economist as Plumber. American Economic Review: Papers & Proceed-ings 107: 1-26.

[12] Easterbrook, Frank H. 1985. Detrebling Antitrust Damages. Journal of Law and Economics 28:445-468

[13] Elliott, Amy. 2012. Audit Proof? How Hedge Funds, PE Funds, and PTPs Escape the IRS. TaxNotes 136: 351-355.

[14] Emons, Winand. 1994. The Provision of Environmental Protection Measures Under Incom-plete Information: An Introduction to the Theory of Mechanism Design. International Reviewof Law and Economics 14:479-91.

[15] Galanter Marc, and William Henderson. 2008. The Elastic Tournament: A Second Transfor-mation of the Big Law Firm. Stanford Law Review 60:1867-1929.

[16] Galle, Brian. 2012. The Tragedy of the Carrots: Economics and Politics in the Choice of PriceInstruments, Stanford Law Review 64:797—850.

[17] Gilpatric, Scott M. 2009. Risk Taking in Contests and the Role of Carrots and Sticks, EconomicInquiry 47:266-77.

[18] Gilson, Ronald J., Charles F. Sabel, and Robert E. Scott. 2018 The Adaptive Contract: Col-laboration and Innovation In an Uncertain World. Unpublished manuscript. Columbia LawSchool.

[19] Glauber, Joseph A. 2018. Unraveling Reforms? Cotton in the 2018 Farm Bill. American Enter-prise Institute.

[20] Gluckman, Nell. 2017. Changing Compensation Strategies Put Partners Under Pressure.American Lawyer, February 2017.

[21] Goodwin, Barry K., and Vincent H. Smith 2018. Reflections on the US Peanut Program. Amer-ican Enterprise Institute.

[22] Graetz, Michael J., and Luois L. Wilde. 1985. The Economics of Tax Compliance: Fact andFantasy. National Tax Journal 38:345-63.

[23] Hamilton, Jesse. 2017. Deutsche Bank Is First Bank Busted for Breaking Volcker Rule.Bloomberg, April 20.

[24] Henderson, William D. 2006. An Empirical Study of Single-Tier versus Two-Tier Partnershipsin the AmLaw 200. North Carolina Law Review 84:1691-1750.

23

[25] Henderson, William D., and Leonard Bierman. 2009. An Empirical Analysis of Lateral LawyerTrends from 2000 to 2007: The Emerging Equilibrium for Corporate Law Firms. GeorgetownJournal of Legal Ethics 22:1395-1420.

[26] Henderson, William D., and Christopher Zorn. 2015. An Rx for Lateral Heartache. AmericanLawyer, February 9.

[27] Hildebrandt Consulting. 2018. Citi 2018 Client Advisory.

[28] Howells, Louise A. 1997. Looking for The Butterfly Effect: An Analysis of Urban EconomicDevelopment Under the Community Development Block Grant Program. Saint Louis Univer-sity Public Law Review 16:383-417.

[29] Hunt, Daniel J. 2014. Just Grin and Bear It: Why Consistent Use of Individual Bailouts Un-der Section 13(3) of the Federal Reserve Act Is a Necessary Evil to Combat Economic “MassDestruction.” George Mason Journal of International Law 6:59-105.

[30] Kaplow, Louis. 1994. Accuracy in the Determination of Liability. Journal of Law and Economics37:1-15.

[31] ____________. 2011. Optimal Proof Burdens, Deterrence, and the Chilling of Desirable Behav-ior. American Economic Review: Papers & Proceedings 101:277-80.

[32] ____________. 2012. Burden of Proof. Yale Law Journal 121:738-859

[33] ____________. 2017. Optimal Regulation with Exemptions. Working Paper No. 23887. Na-tional Bureau of Economic Research, Cambridge, Mass.

[34] Kenworthy, Lane, Stewart Macaulay, and Joel Rogers. 1996. “The More Things Change . . . ”:Business Litigation and Governance in the American Automobile Industry. Law and SocialInquiry 21:631-78.

[35] Kleven, Henrik Jacobsen, and Wojciech Kopczuk. 2011. Transfer Program Complexity and theTake-Up of Social Benefits, American Economic Journal: Economic Policy 3:54-90.

[36] Lazear, Edward P. 1991. Labor Economics and the Psychology of Organization, Journal of Eco-nomic Perspectives 5:89-110.

[37] Levmore, Saul. 1986. Waiting for Rescue, An Essay on the Evolution and Incentive Structureof the Law of Affirmative Obligations. Virginia Law Review 72:879-941.

[38] Moldovanu, Benny, Aner Sela, and Xianwen Shi. 2012. Carrots and Sticks: Prizes and Punish-ments in Contests, Economic Inquiry 50:453-62.

[39] Montero, Juan-Pablo. 2005. Pollution Markets with Imperfectly Observed Emissions. RANDJournal of Economics 36: 645-660.

[40] Olson, Mancur. 1992. Do Lawyers Impair Economic Growth. Law and Social Inquiry 17:625-633.

[41] Patrick, Carlieanne. 2014. Does Increasing Available Non-tax Economic Development Incen-tives Result in More Jobs? National Tax Journal 67:351-86.

24

[42] Png, I.P.L. 1986. Optimal Subsidies and Damages in the Presence of Judicial Error, InternationalReview of Law and Economics 6:101-105.

[43] Polinsky, Mitchell A. 1979. Notes on the Symmetry of Taxes and Subsidies in Pollution Con-trol. Canadian Journal of Economics 12:75-83.

[44] Polinsky, Mitchell A., and Steven Shavell. 2007. The Theory of Public Enforcement of Law. Pp403-454 in vol. 1 of The Handbook of Law and Economics, edited by Mitchell Polinsky and StevenShavell. Amsterdam: North Holland.

[45] Press, Aric, and Mary Cho. 2014. The State of the Big Law Market. ALM Legal IntelligenceReport.

[46] Raskolnikov, Alex. 2014. Irredeemably Inefficient Acts: A Threat to Markets, Firms, and theFisc. Georgetown Law Journal 102: 1133-1189.

[47] Shavell, Steven. 2017. *******. Working Paper. Harvard University Law School. Cambridge,MA

[48] ____________. 1987. Economic Analysis of Accident Law. Cambridge, MA: Harvard UniversityPress.

[49] Spector, Mike and Megumi Fujikawa. 2016. Millions More Cars With Takata Air Bags Re-called; Honda, Fiat Chrysler, Toyota, Nissan, Subaru and More Kick Off Latest U.S. Recalls.Wall Street Journal, May 27.

[50] Simons, Hugh A., and Michael A. Ellenhorn. 2018. Recent Survey Shows Firms Aren’t DoingRight Due Diligence on Laterals. American Lawyer, January 28.

[51] Sorkin, Andrew Ross, and Vikas Bajaj. 2008. Shift for Goldman and Morgan Marks the Endof an Era. New York Times, September 21.

[52] Stavins, Robert N. 2003. Experience with Market-Based Environmental Policy Instruments.Pp 355-435 in Vol. 3 of Handbook of Environmental Economics, edited by Karl-Göran Mäler andJeffrey Vincent. Amsterdam: Elsevier Science.

[53] Sumner, Daniel A. 2018. Dairy Policy Progress. American Enterprise Institute.

[54] Theodos, Brett, Christina Plerhoples Stacy, and Helen Ho. 2017. Taking Stock of the Commu-nity Development Block Grant. Urban Institute. Washington D.C.

[55] U.S. Government Accountability Office. 2018. Economic Development and Small Business.

[56] Wald, Matthew. 2000. Tread Failures Lead to Recall of 6.5 Million Firestone Tires. New YorkTimes, August 10.

[57] Weisbach, David. 2002. Ten Truths About Tax Shelters. Tax Law Review 55: 215-253.

[58] Wittman, Donald. 1984. Liability for Harm or Restitution for Benefit?, Journal of Legal Studies13:57-80.

[59] Yelderman, Stephen. 2017. The Value of Accuracy in the Patent System. The University ofChicago Law Review 84:1217-1284.

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A Appendix

A.1 Proof of Proposition 1

Proof. Proposition 1 can be easily proved by the comparative statics for the three relevant thresh-olds thresholds bv, ek and ep reported in Table 5, which is trivial to verify.

Policy variable bv ek ep (b)

qv Increase Increase Constantqk Constant Increase Increaser Constant Decrease Decreasec Decrease Increase Increases Increase Increase Decrease

Table 5: Comparative statics

A.2 Proof of Proposition 2

Proof. The results on compliance and participation follow trivially from the fact that ek and ep (b)increase in c, while bv decreases in c. The effect on violations is ambiguous because the violationregion shifts over different types as c increases, which in turn is due to the fact that ek moves upand bv moves to the left. Hence, the result depends on the relative frequency of those types in thepopulation of agents, that is, on f (b, e).

Let us now examine carrots. The absolute number of violating agents is given by

V ≡

∫

∞

bv

∫

∞

ek

f (b, e)dedb

Therefore, we have:∂V∂c = −

∂bv∂c

∫

∞

ekf (bv, e)de

−∂ek∂c

∫

∞

bvf (b, ek)db

= (1 − qv)∫

∞

ekf (bv, e)de

−qk+qv−1

r

∫

∞

bvf (b, ek)db

If b and e are independently and uniformly distributed on the unit interval, the ambiguity

persists and we have that ∂V∂c > 0 iff

1 − ek

1 − bv>

1

r

qk + qv − 1

1 − qv

Concerning sticks, the results on violations and participation follow trivially from the fact thatek and bv increase in v, while ep (b) decreases in s. The effect on compliance is ambiguous becausethe compliance region shifts over different types as s increases because ek moves up and bv movesto the right; hence the result depends on f (b, e). More formally, the absolute number of complyingagents is given by

K ≡

∫ bv

0

∫ ep(b)

0f (b, e)dedb +

∫

∞

bv

∫ ek

0f (b, e)dedb

26

Therefore, we have:∂K∂s = ∂bv

∂s

∫ ep(bv)0 f (bv, e)de

+∫ bv

0∂ep(b)

∂s f(

b, ep (b))

db

−∂bv∂s

∫ ek0 f (bv, e)de

+ ∂ek∂s

∫

∞

bvf (b, ek)db

= −1−qk

r

∫ bv

0 f(

b, ep (b))

db

+qk+qv−1

r

∫

∞

bvf (b, ek)db

where note that the first and the third line cancel each other out because ep (bv) = ek. If b and e are

independently and uniformly distributed on the unit interval, we have that ∂K∂s > 0 iff

1 − bv

bv>

1 − qk

qk + qv − 1

which again shows that the ambiguity persists even if the agents’ types are uniformly and inde-pendently distributed.

A.3 Proof of Proposition 4

Proof. The results follow from the comparative statics reported in Table 6, to be compared withTable 5. Accordingly, we can easily derive Table 7: which proves the proposition.

Policy variable biv ei

k eip (b)

qv Increase Increase Constantqk Constant Increase Increaser Constant Decrease Decreasec Decrease Increase Increases Increase Increase Decreasep Ambiguous Increase Ambiguous

Table 6: Comparative statics

Policy variable Compliance Violations Participation Compliance rate

qv Increase Decrease Decrease Increaseqk Increase Decrease Increase Increaser Decrease Increase Decrease Decreasec Increase Ambiguous Increase Ambiguouss Ambiguous Decrease Decrease Ambiguousp Ambiguous Ambiguous Ambiguous Ambiguous

Table 7: Effects of an increase in the policy variables on compliance, violations and overall partici-pation with imperfect monitoring

27

A.4 Proof of Proposition 4

Proof. Let c′

= c + T and s′

= s − T. It is easy to see that c′

+ s′

= c + s and hence eik = eτ

k . Note also

that the value of T that guarantees that biv = bτ

v also necessarily yields eip (b) = eτ

p (b). To derive thevalue of T note that

biv = bτ

v⇐⇒

p(

qvs′

− (1 − qv) c′)

− (1 − p)(

φcc′

− φss′)

= p (qvs − (1 − qv) c)− (1 − p) (φcc − φss)−

⇐⇒

p (qvs − (1 − qv) c)− (1 − p) (φcc − φss)− pT − (1 − p) (φc + φs)T = p (qvs − (1 − qv) c)− (1 − p) (φcc − φss)−⇐⇒

T [p + (1 − p) (φc + φs)] = τ

which yields the result.

A.5 Varying perceptions of accuracy

Here we consider an alternative model where all agents have a benefit of participation b = 0.Agent’s, however, vary in their perception of the accuracy of enforcement by the principal so thatthe three-dimensional agent type is (e,qk,qv), where we assume that e is distributed according tof (e) and is independent of (qk,qv) which are distributed according to g (qk,qv) and are correct

on average, that is, the mean of the distribution is equal to the true levels of accuracy(

qgk ,q

gv

)

.

Additionally, we assume that a change in qgk or q

gv shifts the distribution according to the single-

crossing property. We will show that all the results obtained in the basic setup are confirmed inthis alternative setup.

The agents’ payoffs are:

Action Payoff

The agent participates and complies Πk (e,qk) = qkc − (1 − qk) s − reThe agent participates but violates Πv (qv) = (1 − qv) c − qvs

The agent does not participate 0

Table 8: Agent’s payoffs

As in the basic model, we can define three thresholds:

q̂v ≡ cc+s

ek (qk,qv) ≡ c+sr (qk + qv − 1)

ep (qk) ≡ 1r (qkc − (1 − qk) s)

The threshold q̂v is the level of an agent’s perception of the enforcement accuracy, qv, thatmakes the agent indifferent between violation and abstention, that is, such that Πv (qv) = 0. Thisthreshold is analogous to the threshold bv in the basic model and partitions the agent’s populationinto two groups: the willing participants, with qv < q̂v—whose compliance decision is controlled

28

by the compliance threshold ek (qk,qv)—and the reluctant participants, with qv ≥ q̂v—whose com-pliance decision is controlled by the participation threshold ep (qk).

If c > s, we have 12 < q̂v < 1. In this case, the threshold q̂v lies in the interval of admissible values

for qv. This implies that, stochastically, some agents are in the full participation group and others

are in the partial participation group. If instead c ≤ s, then q̂v ≤12 and hence all agents are in the

group with qv > q̂v, that is, all agents are in the partial participation group and either participateand comply or stay away from the regulated activity. Figure 4 illustrates how agents of differenttypes behave when c > s.

Figure 4: Agent’s behavior (simulation parameters: qk =34 , c = 3

4 , s = 14 , r = 1).

We are interested in establishing the expected levels of compliance and participation in thepopulation. We start by summarizing the comparative statics of the thresholds of interest in Table9. Recall that here, while the first three variable are policy variables, qk is an agent’s characteristicand varies across agents.

Policy variables q̂v ek (qk,qv) ep (qk)

r Constant Decrease Decreasec Increase Increase Increases Decrease Increase Decrease

qk Constant Increase Increase

Table 9: Comparative statics

Accordingly, we can derive the results in Table 10, which are in accordance with those obtainedin the basic model.

While the last three lines in Table 10 can be trivially derived from Table 9, the first two lines

require a more involved derivation. The result on the effect of qgk follows from the comparative

statics in Table 9 and the single-crossing property: an increase in qgk results in a larger probability

29

Policy variable Compliance Violations Participation Compliance rate

qgv Increase Decrease Decrease Increase

qgk Increase Decrease Increase Increaser Decrease Increase Decrease Decreasec Increase Ambiguous Increase Ambiguouss Ambiguous Decrease Decrease Ambiguous

Table 10: Effects of an increase in the policy variables on compliance, violations and overall par-ticipation with imperfect monitoring

mass on higher values on qk, which in turn correspond to higher values of the compliance and

participation thresholds, yielding the result. Finally, the result on the effect of qgv follows from the

single-crossing property: an increase in qgv results in a larger probability mass on higher values on

qv; by inspecting Figure 4 it is easy to see that as we move to higher levels of qv compliance weaklyincreases, violations decrease (initially in a continuous fashion and then discontinuously to zero)and participation increases (in a discontinuous fashion), which yield the result.

30