Policy Making with Reputation Concerns∗

Qiang Fu†

National University of Singapore

Ming Li‡

Concordia University and CIREQ

First draft: July 2008

This version: May 2010

Abstract

We study the policy choice of an office-holding politician who is concerned with the public’s

perception of his capabilities. The politician decides whether to maintain the status quo or

to conduct a risky reform. The success of the reform depends critically upon the abilities of

the politician, which is privately known to the politician. The public observes both his policy

choice and the outcome of the reform, and forms a posterior on the true ability of the politician.

We show that politicians may engage in socially detrimental reform in order to be perceived

as more capable. Conservative institutions that thwart reform may potentially improve social

welfare.

JEL Nos: C72, D72, D82

Keywords: Reform, Reputation, Ability, Conservatism

∗We are grateful to Parimal Bag, Michael Baye, Oliver Board, Kim-Sau Chung, Rick Harbaugh, Navin

Kartik, Kyungmin Kim, Tilman Klumpp, Jingfeng Lu, Marco Ottaviani, Ivan Png, Larry Samuelson, Joel

Sobel, and Bernard Yeung for their helpful comments. We gratefully acknowledge valuable input from

the seminar and conference audiences at the Midwest Economic Theory Meeting–Fall 2008, the Canadian

Economic Association Meeting–2009, the Far Eastern Meeting of the Econometric Society–2009, the Montreal

Theory Reading Group, and the Hong Kong University of Science and Technology. Part of the research was

completed while Fu was a CIREQ visitor, and he is grateful for the hospitality.†Department of Business Policy, National University of Singapore, 1 Business Link, SINGAPORE, 117592,

Tel: (65)65163775, Email: bizfq@nus.edu.sg‡Department of Economics, Concordia University 1455 Boul.de Maisonneuve O., Montreal, QC, H3G

1M8, CANADA, Tel: +1(514)8482424 Ext. 3922, Email: mingli@alcor.concordia.ca.

1 Introduction

She (Emma) was not much deceived as to her own skill either as an artist or a

musician, but she was not unwilling to have others deceived, or sorry to know

her reputation for accomplishment often higher than it deserved.

Emma, vol. 1, ch. 6, by Jane Austen, English Author

Love of fame brings about eccentricity, and being eccentric brings danger to

oneself; therefore the sages exhorted against the love of fame.

Xing xin za yan, Li Bangxian, Chinese Poet

In making decisions and taking action, we are often concerned about inferences that

people draw about us from our choices and/or their consequences. To a large extent, our

success, professional or otherwise, is determined by these inferences, as they shape our rep-

utation. A good reputation not only generates psychological satisfaction and improves one’s

social status, but also facilitates various tangible gains such as better opportunities in career

development.

Reputation or career concerns form one important dimension of the informal incentives

that motivate economic agents to carry out various activities. They loom large, and perhaps

more conspicuously, in the public sector or in non-profit organizations, where formal contracts

based on explicit performance-based incentives are usually rare. There are many examples

that illustrate the ubiquitous and non-trivial influence of reputation. The reputation of a

technocrat’s professional competence often determines his ability to either reach a higher

rung on the hierarchical ladder, or resume an alternative career in the private sector after

his term of service is over. A bureaucrat in the Securities and Exchange Commission, for

instance, often seeks a lucrative job offer from private financial firms after he leaves office.

Career politics provide more salient examples of this. As Irving Rein, Philip Kotler, and

Martin Stoller (1987) state, politics is an “image intensive sector,” where “image building

and transformation truly dominate.” The likelihood of a politician being re-elected may

depend to a large extent on the public’s perception of his capabilities. For instance, in the

aftermath of the economic turmoil, commentators deemed Gordon Brown to have lost his

“reputation for economic competence” “through a combination of appallingly bad luck and

even worse misjudgment,”1 which would eventually cost him his premiership. Alternatively,

1Source: Fraser Neslon, “Brown’s Reputation for Economic Competence Has Gone. The Tories Should

Seize the Chance.” http://www.spectator.co.uk, January 23rd, 2008.

2

a politician in office may simply be concerned about how the public evaluates his legacy

when he steps down.

Due to the strength and prevalence of such informal incentives, it is not unusual for policy

makers to take action to enhance their reputation. For instance, Frederick Sheehan (2009)

comments that Alan Greenspan deliberately built up his own reputation of competence

in designing monetary policy, and went to great lengths to protect his reputation. In the

current paper, we identify one particular context in which such concerns affect the behaviour

of individuals – they may take on risky or innovative initiatives whose success depends on

their capabilities, so as to manipulate the public’s perception of their talents, even though

they know that they have low capabilities and hence a poor chance of success. As Tereza

Capelos (2005) states, “political actors often engage in controversial activities that challenge

their reputation.” She points out that politicians risk losing their support “after showing

inexperience, or wrong judgment.” Our analysis predicts that undertaking risky actions can

be interpreted as attempts on the part of a politician to protect his reputation.

In our model, an office-holding politician decides whether to adopt a reform proposal or

to maintain the status quo. The politician’s capability level can be either high or low, and

is privately known to only himself. If the politician chooses to maintain the status quo, then

the performance of the policy does not depend on his capabilities. If the politician chooses

reform, however, the performance depends on the inherent merit of the reform proposal,

as well as the implementation by the politician. A high-capability politician is better at

implementing reform, and successful reform is more likely in this case.2 The public observes

both the policy choice of the politician and the resultant performance. The public then

forms an assessment of the capabilities of the politician. The politician cares only about the

public’s perception of his capabilities.

Throughout this paper, the decision maker in our model is referred to as a “politician,”

and the risky option he takes is referred to as “reform.” However, our analysis may encompass

a variety of environments, such as a judge who has to decide whether or not to exercise his

power to strike down a law, a prosecutor who has to decide whether or not to file charges

against a crime suspect, a CEO who has to decide whether or not to implement an expansion

plan, or even a doctoral candidate who must decide whether or not to pursue a cutting-edge

research project.

The politician’s competence is reflected by his ability to gather information and to make

2This assumption can be related to the concept of “state capacity,” which is proposed by Theda

Skocpol (1985). She argues that ambitious reform attempts often fail because bureaucrats usually lack

the required competence to administer their reform.

3

sensible decisions in situations of uncertainty. The policy performance is independent of the

politician’s capabilities when the status quo is maintained: a continuation of the existing

policy minimizes uncertainty, and allows the politician to borrow from past experience and to

utilize existing information. In contrast, when the politician abandons the status quo, more

uncertainty arises. The reform succeeds only if the politician chooses appropriate action

in each contingency. For instance, if the US President pushes through a fiscal stimulus

plan, which may help rescue the economy out of a recession, the ultimate success of the

plan depends on how funds are allocated to optimize its effectiveness. In another example,

although the acquisition of Compaq by Hewlett-Packard has proven its merit over the years,

it is a widely held belief that the initial fiasco was due to the flawed management of the

merger by its CEO at the time, Carly Fiorina. The politician is more likely to implement

a successful reform if he has a superior ability to gather information in the presence of

uncertainty.

There is a continuum of Perfect Bayesian equilibria in this game. Each equilibrium is

characterized by a distinct cutoff, such that the high-type politician reforms if and only if

the reform proposal carries a potential value that exceeds the cutoff. We find that there

exists no fully-separating equilibrium. A high-type politician is always “eager” to reveal

more information by undertaking reform: he reforms with probability one once the value of

the available proposal exceeds the prevailing cutoff in the equilibrium. The low type mimics

his high-type counterpart with a positive probability. Although the reform undertaken by

a low type fails with a higher probability, his reputation concerns “force” him to do so,

because he would otherwise suffer from a more unfavourable assessment. Our analysis yields

a number of interesting observations.

• Pressure to prove oneself. The probability of reform by the low type decreaseswith the public’s initial assessment of the likelihood that the politician is capable. For

the low type, a greater gain in reputation can be expected from reform if the initial

assessment is less favourable. Due to this effect, reform can be predicted to occur less

often in the equilibrium when the initial assessment of the politician is more favourable

(that is, when the politician is more likely to be a high type), even without knowing

his true type. The decreased frequency of reform by the low type more than offsets the

increased frequency of reform due to the higher likelihood of the high type (the high

type reforms with probability one in equilibrium if he reforms at all).

We interpret these results as an indication of the pressure to prove oneself. This

phenomenon is present in the intellectual, political, and social aspects of our lives.

4

Young or less established individuals are usually seen as being more progressive and

opposed to the status quo, in contrast to senior or more established individuals, who

usually behave more prudently and conservatively. A famous example is the “‘Young’

Turks” reform movement, which agitated against the Ottoman Empire in the early

20th century, thus building a rich tradition of dissent and paving the foundation of

modern Turkey.3 The term “‘Young’ Turks” today represents progressive individuals

who are eager to bring about widespread change.

• Tough act to follow. The higher the capability differential between the high typeand the low type, the less likely is the low type to undertake reform. To put it simply,

the widened gap makes successful mimicry more difficult. This effect causes the low

type to reform less often in order to avoid failure.

• Thwarted good reforms. We consider the design of the optimal (welfare-maximizing)constitution or bureaucratic rule that restricts the discretion of the politician. Assume

that a legislature enforces a threshold rule – it prohibits reform unless the value of

the reform proposal exceeds a threshold. A higher, or more conservative, threshold

has two competing effects. First, it discourages a low-capability politician from un-

dertaking detrimental reform. However, it also prevents a high-capability politician

from undertaking beneficial reform. We find that the social optimum requires proper

“conservatism”: the optimal threshold rule must thwart otherwise ex ante beneficial

reform. Our analysis on the optimal institution thus lends support to the bureaucratic

rules in various organizations that restrict the discretionary power of politicians or

bureaucrats in carrying out risky activities.4 It also provides an alternative rationale

for the often observed organizational resistance to policy reform and the widely dis-

cussed bias towards the status quo. As pointed out by Raquel Fernandez and Dani

Rodrik (1991), “one of the fundamental questions in political economy” has been why,

ex ante, governments fail to carry out ex post efficiency-enhancing reform.

• Opportunities hurt and “optimism” requires more conservatism. In an envi-3The Young Turks originated from the secret societies of progressive and modernist university students

and military cadets, who advocated reformation of the Ottoman administration and promoted social and

political changes against the monarchy. The Young Turk revolution re-established the constitutional era in

1908. As a nationalist party, Young Turks dominated the domestic politics of Turkey thereafter for an entire

decade.4For instance, in the debate on judicial restraint or judicial activism, one popular argument is to encourage

judges to refrain from exercising their power to strike existing laws.

5

ronment in which good reform proposals are more likely to emerge, it is not necessarily

true that social welfare will be higher. On the one hand, society gains more from

the efficient reform undertaken by the high-type. On the other hand, it “forces” the

low type to reform more often, as the choice to forego reform will be more likely to

be attributed by the public to the politician’s lack of ability, instead of to the lack

of opportunities (i.e. low-valued reform proposals are realized). A more favourable

environment may paradoxically lead to a decrease in social welfare. This observation

compels us to study how the welfare-maximizing institutional rule should respond to

changes in the environment. Under plausible conditions, we find that a more favourable

environment can lead to a more conservative bureaucratic rule.

In the rest of this section, we discuss the link between our paper and the relevant litera-

ture. In Section 2, we set up the model. In Section 3, we characterize equilibria of the model

and present comparative statics of relevant environmental factors. In Section 4, we discuss

the welfare implications of our equilibrium results and consider the issue of institution design.

In Section 5, we conclude. All proofs are collated in the Appendix.

Relationship to the Literature

The notion of career or reputation concerns can be traced back to the pathbreaking work of

Bengt Holmström (1982, 1999). An enormous amount of scholarly effort has been devoted

to exploring the incentive effects of reputation or career concerns in a wide array of envi-

ronments, including corporate decision making (e.g. Bengt Holmström and Joan E. Ricart i

Costa 1986, Jeffery Zwiebel 1995, and Adam Brandenburger and Ben Polak 1996), economic

agents’ effort supply (e.g. Holmström 1999 and Alberto Alesina and Guido Tabellini 2007),

and experts’ strategic advising activities (e.g. Stephen Morris 2001 and Marco Ottaviani

and Peter Norman Sørensen 2006). The literature reveals in various contexts that concerns

regarding public or market perceptions distort managerial decision making, and lead man-

agers to ignore their own useful information and instead, to strategically manipulate the

belief of the public or the market.5

Our paper explores (1) the politician’s incentives to take on risky actions, which signal

his competence; and (2) the optimal (welfare-maximizing) bureaucratic rule that restricts

the politician’s discretion when he is subject to reputation concerns.

5For instance, Brandenburger and Polak (1996), David S. Scharfstein and Jeremy C. Stein (1990), and

Ottaviani and Sørensen (2006) all share this feature.

6

Hence, our paper belongs to the strand of career concern literature that focuses on agents’

incentives to undertake a risky project. The setup of our paper is a variation of the example

introduced in Section 3.2 of Holmström’s (1999) seminal paper. The common feature is that

the politician’s (decision maker’s) talent is only relevant when the reform (risky project)

is undertaken. Hence, more information can be revealed when the risky activity is carried

out.6 Two features distinguish our setup from Holmström’s (1999): first, we assume the

politician’s talent is his private information, while he assumes symmetric information and

symmetric information updating; second, in our model, the probability of success for each

type is common knowledge, while in Holmström’s (1999), it is the private information of the

agent. As a consequence, in our model, the choice to undertake reform can be used as a

signal of talent by the politician, which is not possible in Holmström’s setup.

Holmström and Ricart i Costa (1986) study managers’ incentives to make a new invest-

ment when the manager is subject to career concerns. Similar to Holmström’s (1999) setup,

the type of the manager is unknown to all players. Benjamin E. Hermalin (1993) shows that a

risk-averse agent with career concerns may have the incentive to choose a more risky project,

when the risk levels of projects are commonly known to both the principal and the agent.

However, in his model, a risky project is a worse indicator of the agent’s talent, while in our

model, reform reveals more information. Gary Biglaiser and Claudio Mezzetti (1997) study

in a symmetric-information model politicians’ incentives to implement major new projects.

Voters evaluate the incumbent’s ability based on his performance in the project. They show

that the incumbents’ project choices can be either too radical or too conservative, depending

on the bias of the median voters. They also briefly discuss an extension where the incum-

bent possesses private information about his type and show that full separation is impossible.

However, they do not fully characterize all the equilibria in that case. Zwiebel (1995) also

explores how reputation concerns moderate a manager’s incentive to undertake risky innova-

tion. He shows that managers with intermediate capabilities may resist innovation that can

be beneficial to the firm. In the setting of Zwiebel (1995), the manager’s innovative action is

unobservable. Hence, the setting of Zwiebel (1995) does not involve costly signalling action

on the part of the decision maker. We also arrive at the opposite conclusion that there can

be too much reform when the politician cares about his reputation.7

6The inclusion of a “status quo” option that does not reveal the right action to take for the risky

option is also present in Amal Sanyal and Kunal Sengupta’s (2006) paper. They study a game of strategic

communication in which the expert is career-concerned in the sense of Ottaviani and Sørensen (2006).7Robert A. J. Dur (2001) and Peter Howitt and Ronald Wintrobe (1995) also explore scenarios in which

there is too little change in policy.

7

Our study includes flavours from both the literature of signalling and that of career

concerns, which places it in the company of a handful of other studies, including the no-

table examples of Canice Prendergast and Lars Stole (1996), Gilat Levy (2007), and Wei Li

(2007).8 In a recent paper, Kim-Sau Chung and Péter Esö (2008) build a model in which

a worker chooses a task to both signal his capabilities to potential employers and learn

about his capabilities himself, as only he has imperfect knowledge of it. They assume that

the more difficult task is a worse (less informative) device for assessing the capability of a

worker; meanwhile in our setting, undertaking the more difficult task (reform) allows for

more information to be revealed.

Our work is related to that of Sumon Majumdar and Sharun W. Mukand (2004) and

Guido Suurmond, Otto H. Swank, and Bauke Visser (2004), as both sets of authors consider

the incentives of agents in the public sector to undertake risky projects, which signal their

types. Majumdar and Mukand (2004) study the dynamic incentives of a government within

an election cycle and its policy persistence. The government can be either too radical or

too conservative in equilibrium. Suurmond, Swank, and Visser (2004) contend that the

presence of career concerns can be socially beneficial, as it can encourage a smart agent

to expend more effort in gathering information. In both studies, the performance of the

risky project can be depicted by a binary indicator (success or failure). The likelihood of

its success is pre-determined while a more capable agent can discover the pre-determined

“suitability” of the project more precisely. In contrast, we assume that (1) the performance

of reform is a continuous measure, depending on the quality of both the project per se and

the implementation by the politician; (2) the politician’s ability determines the quality of ex

post implementation; and (3) the quality of the project is random. This setting allows us to

study the issue of institutional design concerning the proper level of tolerance for reform.

Our paper is also closely related to a simultaneous and independent paper by Ying

Chen (2010). She analyzes a model in which an agent chooses between a risky project,

whose payoff to a principal is uncertain, and a safe project, whose payoff to the principal

is constant. Chen (2010) also assumes that the likelihood of success of the risky project

depends on the agent’s ability. Our paper and Chen’s (2010) are different from each other

in a few respects. First, in the setup of Chen (2010), the likelihood of success also depends

on a random variable, whose realization is observable only to the agent, while the principal’s

8Prendergast and Stole (1996) argue that career concerns induce young investors to overreact to new

information they receive, so as to signal that they are fast learners. Wei Li (2007) makes a similar point in

the case of experts providing advice to decision makers. We discuss Levy’s work at the end of the literature

discussion.

8

payoffs from success or failure of the risky project are prefixed. In contrast, in our model,

the public’s payoffs from a successful or failed reform are dependent on a random variable,

whereas the success and failure probabilities are affected by the politician’s ability alone.

Second, in Chen’s (2010) model, even if the risky project fails, the agent’s reputation is still

higher than that from choosing the safe project.9 In our model, the politician’s reputation

from a failed reform is lower than that from choosing the status quo. Third and the most

important, Chen (2010) has a different focus than we do. She shows that the agent takes

excessive/inadequate risks when he does/does not know his own type. In contrast, we focus

on a setting in which the politician knows his type. In addition to identifying the problem

of a politician undertaking excess reform, we also study the impact of various environmental

factors on the politician’s risk taking behaviour and the design of optimal institution that

remedies this problem.

Our analysis of optimal institution design in the presence of reputation concerns is con-

ceptually related to that in a small number of other papers, which study the ramifications of

various institutions in career concerns models. Andrea Prat (2005) argues that transparency

in an organization may hurt as the agent may take revealed action to influence the princi-

pal’s posterior instead of seeking the best interests of the organization. Gilat Levy (2007)

shows that in a committee of voters with career concerns, radical actions are more likely

to be accepted when the voting process is transparent. To our knowledge, our paper might

be one of the first to explicitly investigate an institutional remedy for inefficient risk taking

when the decision maker has reputation concerns. Our result that restrictions on changes to

the status quo could be welfare-improving complements other justifications of institutional

conservatism, for example, those offered by Li, Hao (2001) and Young K. Kwon (2005). Our

analysis suggests that institutional barriers (bureaucracy) that limit the amount of discre-

tion that the decision maker can exercise may be welfare-enhancing. Our paper echoes the

conclusion of Jean Tirole (1986) in this respect.

2 Setup

A risk-neutral politician makes a policy choice between two alternatives: maintaining the

status quo or implementing a reform. If the politician retains the status quo, the outcome of

this polity, y, is deterministic, which we normalize to 0. In contrast, if the politician chooses

to undertake the reform, uncertainty will arise and the politician must take an action to

9In her model, the agent is prevented from choosing the risky project only by monetary incentives.

9

address it. The outcome of a reform is given by

y = θ − (a− ω)2.. (1)

where θ measures the intrinsic value of reform, ω is the true state of the world, and a is the

action taken by the politician in response to his assessment of ω. The politician observes

the value of the reform, θ, before choosing whether or not to implement it. It is common

knowledge that θ is continuously distributed on [−θ1, θ2] with a distribution function F anddensity function f , where −θ1 < 0 < θ2, and θ1, θ2 ∈ (1, 2). The state of the world, ω, maytake two values from Ω = {−1, 1}, each with a probability 1/2. Neither the politician nor thepublic observes the true state. The action a can be chosen from A = {−1, 1}. Thus, whenthe politician implements a reform, he achieves the best outcome when his action matches

the state of the world.

The likelihood of success for the reform is affected by the politician’s talent, t, which

can be high (t = H) or low (t = L). The talent of the politician is his private information.

A high-talent politician receives an informative signal σ ∈ {−1, 1}, which matches the truestate with a probability

q = Pr(σ = ω) >3

4.

In contrast, a low-talent politician’s signal is completely uninformative.10 It should be noted

that the assumption q > 3/4 does not affect the strategic analysis. However, without this

assumption, no reform can be ex ante socially beneficial. Further, let α be the probability

of t = H, which is commonly known. It is the public’s prior on the politician, which can

also be viewed as the proportion of high-capability politicians in the population.11

The setup of our model differs from the existing literature in a number of ways. Here,

we stress two essential features of our model. First, the distinction between policies (status

quo or reform) and actions is important in our model. Policies are macro-level or “strategic”

10Although we do not model how the politician obtains his signal, the politician’s talent in our model

may be interpreted as the ability to gather information from various sources. The US presidential historian,

Erwin C. Hargrove (1966, pp 70-73 and pp 114-116), paints two completely different pictures of Franklin D.

Roosevelt and Herbert Hoover with respect to information gathering. Roosevelt brought together experts

who held a great variety of views and balanced them off against each other, while Hoover did not enjoy

obtaining critical advice from anyone.11There is literature that analyzes the composition of politicians as a group, which is complementary to

our research, in that it offers an explanation for why politicians may consist of a significant proportion of

low-ability individuals. Francesco Caselli and Massimo Morelli (2004), Matthias Messner and Mattias K.

Polborn (2004), and Andrea Mattozzi and Antonio Merlo (2007, 2008) have offered various explanations for

why political processes tend to select low-ability individuals to be politicians.

10

decisions such as whether to reform financial regulations or whether to start a war. In

contrast, actions are micro-level or “tactical” decisions such as which instrument of regulation

to introduce in overhauling the financial system or how many troops to deploy in the war.

Although there may be general agreement about how desirable a reform is (θ), there may

well be disagreement over the optimal way to implement the reform (a). The true nature

of the problem (ω) determines which action is ex post suitable for implementing the reform.

Second, in contrast to many existing career-concern models with risky experimentation (e.g.

Majumdar and Mukand 2004 and Suurmond, Swank, and Visser 2004), the outcome of a

reform is measured by a continuous variable, instead of a binary indicator (e.g., success or

failure) alone. It depends on both the quality of the reform proposal (θ) and the quality

of implementation (|a− ω|), while both are subject to random perturbation. This setupenriches our analysis in two aspects. First, it enables an analysis of institution design.

A more sophisticated trade-off is involved in determining the proper level of institutional

conservatism. Second, a comparative static analysis may be performed on the probability

distribution of the value of reform, which may provide the answer to questions like “should

the institution become more or less conservative when the ex ante prospects of the reform

improve?”.

We assume that the proportion of “good” politicians in the population is small:12

α <1

2.

Upon receiving σ (either informative or uninformative), the politician takes an action.

The public observes the politician’s policy choice (status quo or reform) and the final

outcome.13 Their updated belief, or the reputation of the politician, can be written as

µi(y) ≡ Pr(t = H| y, i)

by Baye’s rule, where i = 0 indicates the status quo and i = 1 indicates reform. Borrowing

from much of the career concern literature, we assume that the politician’s payoff depends

purely on his reputation. The politician therefore chooses the action that maximizes his

reputation.

12This regularity assumption is only required so that in the extreme case where the high type’s signal is

perfectly informative, the low type still has an incentive to undertake reform and mimic the high type (see

the proof of Lemma 1).13In our setup, whether or not the public observe the action is inconsequential. Once the politician chooses

reform, the belief of the public is determined only by whether the outcome is a “failure” or “success.”

11

3 The Analysis

We adopt the solution concept of Perfect Bayesian Equilibrium for our analysis. This requires

that (1) the politician and the public form Bayesian beliefs, (2) the politician chooses the

action that maximizes his expected reputation if he undertakes reform, and (3) the politician

chooses reform or the status quo to maximize his expected reputation.

When the politician chooses to reform and takes an action a, the expected outcome of

the reform is given by

E(y) = θ − Eω∈{−1,1}(a− ω)2 ≥ 0. (2)

Since a low-type politician’s signal is uninformative and the two states are equally likely,

his choice of a is ex ante irrelevant. A high-type politician maximizes his probability of

success by following his signal, i.e., choosing a = σ.

In the first-best situation, a politician would undertake reform if and only if the expected

outcome E(y) is non-negative. A low-type politician should never reform regardless of θ as

the expected loss from wrong actions always exceeds the benefit of reform, that is,

E(y) =1

2θ +

1

2(θ − 4) = θ − 2 < 0,

because θ2 < 2. In contrast, the expected outcome for a high-type politician is given by

E(y) = θ − 4(1− q).

Thus, the high type should undertake reform if and only if the value of reform is sufficiently

high, i.e., θ ≥ 4(1− q).We now formally analyze the politician’s policy choice. Let ρt(θ) be the probability

with which a type-t politician chooses reform when its value is θ. We focus on Monotone

Equilibria, where the politician’s probability of undertaking reform is non-decreasing in θ,

the potential value of reform. Let

θt ≡ inf{θ| ρt(θ) > 0}, t = H,L.

Thus, a type-t politician undertakes reform with a positive probability only if the value of

reform exceeds the cutoff θt.

In monotone equilibria, when the politician maintains the status quo, his reputation

among the public will be

µ0 =αF (θH) + α

∫ θ2θH

[1− ρH(θ)]f(θ)dθ[αF (θH) + α

∫ θ2θH

[1− ρH(θ)]f(θ)dθ+(1− α)F (θL) + (1− α)

∫ θ2θL

[1− ρL(θ)]f(θ)dθ

] . (3)

12

When the politician implements a reform of value θ, his reputation will become

µs =αqρH(θ)f(θ)

αqρH(θ)f(θ) + (1− α)12ρL(θ)f(θ)

when the reform succeeds, and

µf =α(1− q)ρH(θ)f(θ)

α(1− q)ρH(θ)f(θ) + (1− α)12ρL(θ)f(θ)

when the reform fails. Define

qt =

{q for t = H;12

for t = L.

If a type-t politician implements a reform with value θ, he receives an expected payoff

µt = qtµs + (1− qt)µf .

We first establish the following lemma.

Lemma 1. (No full separation.) In any equilibrium that involves a positive probability of

reform, (1) the cutoffs for reform must be the same for the low type and the high type, i.e.,

θL = θH = θ; and (2) the high-type politician plays a pure strategy ρH(θ) = 1 for any

θ ∈ [θ, θ2].

The above lemma states that there is no full separation of the two types. The same cutoff

level θ = θL = θH applies to both types of politician. When θ exceeds this threshold θ, the

high type always undertakes reform and the low type undertakes reform with a positive

probability. We now turn to the characterization of the equilibria of the game. For the

moment, we assume that the politician is maximally empowered. He is allowed to initiate a

reform for any θ ∈ [−θ1, θ2]. We will explore the optimal institutional rule that restricts hisstrategy space later in this paper.

Proposition 1. There exists a continuum of Perfect Bayesian Equilibria with cutoffs θ̄ ∈[−θ1, θ2]. For any θ, there exists a unique equilibrium probability ρ∗ ∈ (0, 1), such that thelow-type politician undertakes reform with the probability ρ∗ whenever he receives a signal

θ ≥ θ. The equilibrium probability ρ∗ solves1

1 + λ(α)A=

1

2· 1

1 + λ(α)B+

1

2· 1

1 + λ(α)C, (4)

where

λ(α) =1− αα

, A = 1 + (1− ρ)κ(θ̄), κ(θ̄) = 1− F (θ̄)F (θ̄)

, B =12ρ

q, C =

12ρ

1− q.

13

A continuum of perfect Bayesian equilibria exists in this game. Each is characterized by

a distinct cutoff θ. When θ exceeds the cutoff, the low type mimics his high-type counterpart

and conducts the reform with a probability ρ∗. Even though the probability of success is

only 1/2, it is optimal for the low type to reform because the choice of reform can be a signal

of high talent, which manipulates the posterior of the public.

Comparative Statics

We now examine how the politician’s equilibrium behaviour varies with environment param-

eters. First, we focus on the comparative statics in a given equilibrium with a fixed θ. In

the equilibrium, the high-type politician reforms with probability one whenever θ exceeds θ,

while the low type reforms with a probability ρ∗. Hence, in this equilibrium, reform occurs

with a probability

ρ̄ = [1− F (θ)][α + (1− α)ρ∗]. (5)

The main results are summarized in the following proposition.

Proposition 2. Fix any equilibrium with a cutoff θ.

1. The probability of reform by the low type, ρ∗, is strictly decreasing in α, the public’s

prior. The overall likelihood of reform, ρ̄, also strictly decreases with α.

2. The probability of reform by the low type, ρ∗, is strictly decreasing in q. The overall

likelihood of reform, ρ̄, also strictly decreases with q.

3. Let ρ and ρ′ denote, respectively, the equilibrium probabilities of the low type undertak-

ing reform associated with distributions F (·) and G(·). Let ρ̄ and ρ̄′ be their counter-parts for the overall likelihood of reform. For a given θ, then, ρ > ρ′ and ρ̄ > ρ̄′ if F (·)first order stochastically dominates G(·).

Now, we discuss the intuition and implications of these results. Part 1 of Proposition 2

states that the low type conducts more reforms, when the public holds a less favourable

prior assessment, or the proportion of capable politicians in the population is smaller. A

more favourable prior assessment increases a politician’s loss from a failed reform, which

consequently weakens his incentive to reform. By contrast, a less favourable prior assessment

would strengthen his incentive to take risk, because it implies a smaller loss from a failed

reform but a larger gain from an accidental success. This is then interpreted as the pressure

to prove oneself phenomenon.

14

Part 1 of Proposition 2 further shows that less reform would take place overall when

the public has a more favourable assessment of the politician’s talent (or there is a higher

proportion of capable politicians). Note that

∂ρ̄

∂α= [1− F (θ)][1− ρ∗ + (1− α)∂ρ

∗

∂α]. (6)

Two competing forces come into play when α is higher. On the one hand, since the low type

reforms less often than the high type, more reform would be expected when there is a higher

proportion of the high type. This effect is depicted by the term (1−ρ∗). On the other hand,a larger α leads the low type to reform less, which tends to reduce the frequency of reform.

The latter effect is witnessed in the term (1 − α)∂ρ∗∂α

. Our analysis shows that the second

effect always dominates.

This result yields an empirically testable hypothesis: when there is a smaller proportion of

capable politicians in the population or when the public holds a more pessimistic prior view,

more reform is expected. Conversely, the public observes less reform when the politician has

a better reputation. This conclusion is drawn without knowledge of the true type of the

politician, which is his private information and is unverifiable.

Part 2 of Proposition 2 states that a low-type politician would mimic his high-type

counterpart less often when the latter becomes more capable. The logic of this result is as

follows. When the high type has a more accurate signal, the public is more likely to attribute

an unsuccessful reform to a low-type politician, which unambiguously increases his cost of

conducting reform. To put it simply, a higher q makes it more difficult for a low type to

mimic his high-type counterpart, and therefore leads to a lower probability of reform by the

low type. This result is interpreted as the tough act to follow phenomenon.

The distribution of θ does not qualitatively alter the main prediction of our analysis,

but it quantitatively affects the equilibrium behaviour. Part 3 of Proposition 2 describes

its effect on ρ∗. A stochastically dominant distribution implies that the probability mass is

shifted upward. Hence, favourable reform proposals are more likely to be realized. Given

the better prospect of reform, the public would then believe that a no-reform outcome is

more likely to be caused by the politician’s lack of talent, instead of a lack of opportunities

(a lower θ is realized). The public’s assessment of the politician’s ability is therefore lowered

when they observe no reform, and this “forces” the low type to reform more often. This

result yields interesting welfare implications, which are discussed later in this paper.

15

Comparison across Equilibria

Analogous to standard signalling game, our analysis yields multiple equilibria. Each is

characterized by a distinct cutoff θ. We now explore how the equilibrium behaviour of a

low-type politician (ρ∗) would differ across equilibria.

Lemma 2. The equilibrium probability of reform by the low type, ρ∗, strictly decreases with

the cutoff θ.

The intuition is consistent with that of Part 3 of Proposition 2. A higher cutoff θ implies

that it is less likely for the value of available reform proposals to exceed the equilibrium

cutoff. This increases the payoff of the low type when he does not reform, as the public is

more likely to interpret it as the result of there being no opportunity.

A higher cutoff θ in the prevailing equilibrium reflects escalated conservatism or stronger

resistance to reform. Lemma 2 allows us to further explore a politician’s preference for “con-

servatism”. We are interested in the following question: Does a politician prefer equilibria

with more reform or less reform?

A less conservative equilibrium exerts mixed effects on the payoff of a high-type politician.

On the one hand, it allows the public to infer his true type more often. On the other hand,

an equilibrium with a lower cutoff θ encourages his low-type counterpart to conduct more

reform, which makes his reform less informative and tends to offset the gains he may obtain

from a successful reform. With a slight abuse of notation, let us denote by Eµt(θ) the ex

ante expected payoff of a type-t politician in an equilibrium with a cutoff θ. Our analysis

leads to the following proposition.

Proposition 3. The low-type politician always prefers an equilibrium with a higher cutoff

θ; while the high-type politician always benefits from an equilibrium with a lower θ. That is,dEµH(θ)

dθ< 0, and dEµL(θ)

dθ> 0.

Proposition 3 states that the high type always prefers an equilibrium where he can reform

more, while the low type suffers from this. Two forces contribute to the low type’s aversion

to reform. First, it embodies the logic of Part 3 of Proposition 2: less reform allows the

low type to pool with his high-type counterpart more often and to reveal less information.

Second, when the low type reforms less often, the public would believe that a reform is

increasingly likely to be implemented by the high type, which mitigates the damage to the

low type when he fails in his reform.

16

4 Institutional Design

In the preceding analysis, we have assumed that the politician in office is maximally em-

powered, and is subject to no institutional constraints in making a choice between policy

alternatives. Based on the equilibrium analysis above, we may now turn to the investigation

of the optimal institution that governs the politician’s scope of discretion.

Consider a context in which a legislature exists, with the goal of maximizing social wel-

fare. The legislature can be a parliament, an advisory committee, a board of directors, etc.

It enforces a limit of authority by restricting the action space of the politician. The rule set

by the legislature can be understood as constitution, or the widely observed organizational

bureaucracies (see Tirole 1986), which restrict the discretion of the decision maker. Such

institutional restrictions on a decision maker’s discretion are prevalent in various organi-

zations. For instance, the US President must obtain congressional approval for his policy

choices. Military commanders have to honor “rules of engagement” in the use of force. An

administrator of the Environmental Protection Agency has to rely on limited authority and

resources to regulate polluting industries. In addition, judges are often encouraged to refrain

from exercising their power to strike down existing laws.

In particular, the institution of focus in this setting resembles a “rule of engagement.”

The legislature cannot observe the true type of the politician. However, the behaviour of the

politician is subject to its regulation. The legislature sets a threshold θ̂ and a politician is

authorized to undertake a reform only if the potential value of the reform proposal exceeds

θ̂. The implementation of the rule may be interpreted in two ways. First, it may be assumed

that the politician is subject to ex post auditing or monitoring, and would be held accountable

and be subject to severe non-pecuniary punishment, e.g., termination of his career, if the rule

was breached and a reform with θ < θ̂ was carried out. As mentioned earlier, the assumption

of θ2 < 2 guarantees that the true value of θ can be correctly inferred once the outcome y

of a reform is realized. Second, analogous to Tirole (1986), it may be assumed that the

politician can provide a partially verifiable report on the realization of θ to the legislature

when he advocates a reform, but such information is neither verifiable nor accessible ex ante

to the general public.14

The authority granting rule aims at maximizing social welfare. A higher θ̂ represents a

more conservative rule that grants less authority to the politician; while a lower θ̂ represents

14In addition, it should be noted that the legislature does not use contingent monetary transfer to elicit

desirable action. The performance of a decision maker can be non-contractible in a wide variety of settings.

Consider the examples of career politicians, supreme court justices, and district attorneys, for instance.

17

a more liberal rule that is more tolerating of reform. As aforementioned, the society may

expect a positive gain from the reform that is undertaken by the high-type politician (when

θ is sufficiently high), while always expects a net loss from the reform that is undertaken

by the low type. A trade-off can be triggered when a more conservative rule is adopted.

By restricting reform, it could reduce the damage from the latter on the one hand, while it

could also lead to lesser gain from the former on the other.

As seen in the previous section, without further restrictions on out-of-equilibrium beliefs,

there exist multiple equilibria. It remains the case, when a threshold rule θ̂ is enforced. Each

equilibrium is characterized by a distinct cutoff θ ∈ [θ̂, θ2]. The multiplicity of equilibria thusprevents us from drawing conclusive predictions on the behaviour of the politician when he

is subject to a particular institutional rule. Our subsequent analysis proceeds in two steps.

First, we employ a popularly adopted refinement technique to select the most plausible

equilibrium when an institutional rule θ̂ is in place. We follow Jeffrey S. Banks and Joel

Sobel (1987) and apply the “Divinity Criterion” to select the most plausible equilibrium,

which turns out to be unique. Second, we characterize the optimal institutional rule based

on the unique equilibrium prediction through our refinement.

4.1 Equilibrium Refinement

Analogous to other conventional refinement techniques for signalling games, the Divinity Cri-

terion seeks to impose additional restrictions on out-of-equilibrium beliefs. Fix an arbitrary

equilibrium. When an unexpected signal is received, the receiver has to form a conjecture

about the type of sender who deviates from the equilibrium path. The criterion is built

upon the notion that a sender is willing to deviate by sending an unexpected signal only if

he hopes for a payoff that is higher than that obtained in the equilibrium. Consider two

different types of senders. If one type is more likely to benefit from a given deviation, then

the receiver should believe that the former type deviates at least no less often than the latter.

The receiver must assign in his posterior more weight to the type that is more likely to gain

from the given deviation. Banks and Sobel (1987) contend that an equilibrium is sensible

only if it can be supported by a belief system that satisfies such requirement. A formal

definition of this refinement criterion is provided in the Appendix. Our analysis leads to the

following.

Proposition 4. When a threshold rule θ̂ is implemented, there is a unique equilibrium that

satisfies the Divinity Criterion. In this equilibrium, θ = θ̂.

18

Among all equilibria that are permitted by the threshold rule, only the most aggressive

equilibrium (with θ = θ̂) satisfies the Divinity Criterion. This result is underpinned by the

fact that a capable politician always prefers to reform as much as possible, as evidenced

by Proposition 3. When the high type reforms more, the low type is in turn “forced”

to follow suit. The refinement criterion simply requires the belief system to reflect this

fact. This refinement exercise paves a foundation for our subsequent analysis of the optimal

institutional design, i.e. the welfare-maximizing θ̂∗.

4.2 Institutional Design: Optimal “Rule of Engagement”

In this subsection, we explore the welfare maximizing threshold rule. Based on Theorem 4,

under a threshold rule θ̂, the most plausible equilibrium in the subsequent game is the one

with the cutoff θ = θ̂. Consider an arbitrary threshold rule θ̂. The social welfare in this

equilibrium can be written as a function

W = α

∫ θ2θ̂

[θ − 4(1− q)]f(θ)dθ︸ ︷︷ ︸W1

+ (1− α) ρ∗|θ=θ̂∫ θ2θ̂

(θ − 2)f(θ)dθ︸ ︷︷ ︸W2

. (7)

By implementing a proposal of value θ ≥ θ̂, the high-type politician contributes an expectedoutcome θ−4(1− q), while the low-type generates a loss θ−2. The term W1 thus representsthe overall net gain from the reform that is undertaken by the high type; while the term

W2 depicts the overall loss from the inefficient reform that is undertaken by the low type.

Apparently, the optimal rule θ̂∗

must exceed 4(1− q).Consider an arbitrary reform proposal with a value θ ∈ [θ̂, θ2]. The ex ante expected

outcome of this proposal under the threshold rule θ̂ is given by

E(y| θ, θ̂) = α[θ − 4(1− q)] + (1− α) ρ∗|θ=θ̂

(θ − 2),

which, for a given θ̂, strictly increases with θ. Define ρ ≡ limθ↑θ2 ρ∗. We have the following.

Lemma 3. Whenever(1− α)ρ

α<θ2 − 4(1− q)

2− θ2, (8)

there exists a unique θ̂0∈ (4(1− q), θ2) that solves

E(y| θ̂, θ̂) = α[θ̂ − 4(1− q)] + (1− α) ρ∗|θ=θ̂ (θ̂ − 2) = 0.

Further, θ̂0

exhibits the following property: for any θ̂ ∈ [−θ1, θ2],

E(y|θ̂, θ̂) T 0 if and only if θ̂ T θ̂0. (9)

19

Consider an arbitrary threshold rule θ̂. The expression in (9) depicts the expected out-

come from a “marginal” reform proposal, i.e. the proposal with a value of exactly θ̂. The

property of θ̂0

demonstrated by (9) yields interesting implications. Specifically, the threshold

rule θ̂0

can be used as a natural benchmark. If the prevailing rule θ̂ is less conservative than

θ̂0, it must admit “bad” reform: reform with a value in [θ̂, θ̂

0) would be allowed, which yields

negative expected outcome.15 In contrast, if the prevailing rule θ̂ imposes more restrictions

than θ̂0, it must thwart otherwise “good” reform: reform with a value in [θ̂

0, θ̂) would be

prohibited, which would otherwise yield a positive expected outcome. However, a threshold

rule θ̂0, by its very definition, completely rules out “bad” reform, while it does not thwart

otherwise beneficial reform. Hence, is θ̂0

the optimal cutoff θ̂∗

that maximizes social welfare?

If not, then would the optimal institution be more conservative or less conservative, i.e., does

the optimum require θ̂∗< θ̂

0or θ̂

∗> θ̂

0?

Our analysis yields the following result.

Proposition 5. A unique socially optimal cutoff θ̂∗∈ (θ̂

0, θ2) exists if and only if (8) is

satisfied; otherwise, the public prefers no reform at all, i.e., θ̂∗

= θ2.

This proposition states that a unique optimal threshold exists, and the optimum θ̂∗

must exceed θ̂0

whenever θ̂0

exists. The welfare maximizing institutional rule requires more

conservatism than θ̂0. In order to understand its logic, let us now analyze the marginal

impact of an increase in θ̂ on social welfare. Taking first order derivative of (7) with respect

to θ̂ yields

dW

dθ̂= f(θ̂)

−α[θ̂ − 4(1− q)]︸ ︷︷ ︸a

− (1− α) ρ∗|θ=θ̂ (θ̂ − 2)︸ ︷︷ ︸b

+(1− α)d ρ∗|θ=θ̂ /dθ̂

f(θ̂)

∫ θ2θ̂

(θ − 2)f(θ)dθ︸ ︷︷ ︸c

. (10)

An increase in θ̂ affects W through three venues. First, it reduces the beneficial reform

that is undertaken by the high type, and therefore decreases the gains from reform by

the high-type politician. This loss is shown by the term a, which is negative whenever

θ̂ > 4(1−q). Second, a higher cutoff θ̂ (directly) reduces the expected loss from the inefficientreform that is undertaken by the low type. This (direct) effect is embodied by the term b.

Third, it leads the low-type politician to refrain from undertaking reform for any given θ ≥ θ̂

15Based on the definition of θ̂0, under the threshold θ̂ < θ̂

0, even a reform with a value that is higher than

θ̂0

may still incur an expected loss.

20

(because d ρ∗|θ=θ̂ /dθ̂ < 0 by Lemma 2), which further reduces the loss from the inefficientreform that is undertaken by the low type. This positive (indirect) effect is depicted by the

term c.

The decomposition of dW/dθ̂ demonstrates that θ̂0

is never the optimal threshold. When

θ̂ = θ̂0

is enforced, social welfare can be increased by raising θ̂: the sum of the first two terms

simply boils down to E(y| θ̂0, θ̂

0), and is equal to zero based on the definition of θ̂

0, but the

last term, c, remains positive. It implies that W can be further increased by increasing θ̂

from θ̂0: although a more conservative threshold would deter otherwise productive reform,

it further deters the detrimental reform that is undertaken by the low type by decreasing ρ∗.

The reduced loss could more than compensate for the sacrificed gain from those otherwise

efficient reforms with value in (θ̂0, θ̂∗). Therefore, the social optimum must require a cautious

attitude about potential reform, despite it inhibiting seemingly beneficial reform.

Reform can be permitted, i.e., θ̂∗< θ2, if and only if condition (8) is met. Because ρ

decreases with α (by Proposition 1), the left hand side of (8) strictly decreases with α. Hence,

this condition is more likely to be met with a larger α, i.e., the presence of a higher proportion

of high-talent politicians in the population. When the talent required for successful reform is

very scarce, the public would not expect sufficient gain from reform. The public then prefer

no reform at all.

Similarly, the condition is more likely to be met with a larger q. In other words, reform

is socially beneficial only when the success of reform is sufficiently likely.

These arguments further lead to more general conclusions on the impact of α and q on

the properties of θ̂∗∈ (θ̂

0, θ2).

Proposition 6. The socially optimal cutoff θ̂∗

decreases with α and q.

A greater α or q always allows for less restriction on the politician’s activities.

Example: Uniform Distribution

Proposition 3 demonstrates that the equilibrium behaviour depends on the properties of the

distribution of θ. We now discuss its impact on the optimal threshold rule θ̂∗. To allow for

a handy and informative analysis, consider an example in which the value of reform follows

a uniform distribution

F (θ) =θ + θ1θ2 + θ1

and the high-talent politician receives a perfect signal with q = 1.

21

An increase in θ2 implies that the probability mass of the distribution is shifted upward,

high-valued reform proposals are more likely to occur, and more beneficial opportunities can

be expected. The environment thus seems to favor more reform. Before we examine its

impact on the socially optimal institutional rule, let us examine its welfare implications in

an arbitrary equilibrium with a fixed θ. Figure 1 testifies to a non-monotonic relationship

between social welfare and θ2 when θ is given. The society may not be better off when more

opportunities are available. The logic can be seen in Proposition 3: for a given cutoff θ, a

stochastically dominant distribution of θ forces the low type to reform more, which increases

the loss from his inefficient reform.

The ambiguous welfare implication compels us to further look into its implications on the

socially optimal institution θ̂∗. The implications of a higher θ2 on the social optimum θ̂

∗are

also ambiguous. On the one hand, low-valued reform proposals would emerge less often, and

cause less damage, which encourages a more liberal rule to reap more benefits from reform.

On the other hand, it could demand a more conservative rule in order to discipline the low

type. Our analysis leads to the following proposition.

Proposition 7. The socially optimal cutoff point for reform, θ̂∗, strictly increases with θ2.

We find that when the probability mass of the uniform distribution is shifted upward,

i.e., when more opportunities for reform can be expected, it unambiguously lifts the optimal

cutoff θ̂∗. That is, a more optimistic environment requires additional caution and a more

conservative socially optimal rule.

5 Concluding Remarks

In this paper, we study a politician’s incentive to implement reform when his true ability

is privately known but he is concerned about the public’s perception of his abilities. The

politician thus chooses his policy and action to maximize his reputation payoff. We find

that a high-talent politician always attempts to reform as much as possible, which “forces”

his low-talent counterpart to mimic with a positive probability. Socially inefficient reform

therefore results. Further, we explore the socially optimal level of empowerment, and find

that the social optimum can be achieved only if the prevailing institutional rule implements

proper conservatism and deters some otherwise efficient reform.

22

Figure 1: An example that demonstrates the non-monotonic effect of θ2 on social welfare

(θ1 = 1.1, q = 1.0, α = 0.2, and θ̄ = 1.2).

Our paper sets forth a simple theoretical framework to investigate a politician’s incentive

to undertake innovative but risky actions when he has reputation concerns. It leaves open

many possibilities for extensions and variations. For instance, the model may be extended

to allow for a larger strategy space, or to allow the payoff of the politician to depend on

the realized outcome of his policy choice. Although we believe extensions in these directions

would not yield predictions that fundamentally depart from those that come out of the

current setting, these more comprehensive settings may yet spawn richer comparative statics

that further add to our understanding of this issue. The analysis may also be extended to

a dynamic setting. For example, the “pressure to prove oneself” result points toward the

following conjecture: a politician who has failed in the past is more likely to take radical

action in the future. Past failure lowers his rating among the public, which therefore makes

it more lucrative for him to pursue accidental success in the future.

23

Appendix: Proofs

Proof of Lemma 1

Proof. First, in any equilibrium, the low type does not reform when the realized value of

reform satisfies θ < θH . If he did in equilibrium, since the high type does not reform for

θ < θH , the public must assign probability one to him being the low type regardless of

success or failure, thereby leaving him worse off than if he does not reform. Hence, we must

have θL ≥ θH in any equilibrium. In the following argument, we show that their strategiesfollow the same cutoff θ = θL = θH .

Second, observe that q > 1/2 implies that µs > µf as long as ρH(θ) > 0 and ρL(θ) > 0.

But, this implies that µH(θ) > µL(θ). Thus, whenever both types choose reform with positive

probabilities, the high type must choose it with probability one.

Third, we claim that whenever the high type chooses reform with a positive probability,

the low type must do so as well. We have shown that whenever both types choose reform

with positive probability, the high type’s probability of reform is one and therefore at least

as high as the low type’s. Therefore, the overall probability for the low type to choose the

status quo, P0L, is weakly higher than that for the high type, P0H . Thus, if the low type

chooses the status quo, his reputation is µ0 = αP0HαP0H+(1−α)P0L

≤ α.However, if he deviates and undertakes reform, he is believed to be a high type with

probability one if q < 1. If q = 1, his payoff depends on the public’s off-equilibrium belief

when reform fails. However, he succeeds with probability 12, and the resulting expected

payoff still exceeds α. Therefore, it cannot be that the low type always chooses the status

quo when the high type chooses reform. This completes our proof.

Proof of Proposition 1

Proof. We now determine the low-type politician’s probability of reform for a proposal with

value θ, which we denote by ρ(θ) to economize on notation. By (3), if the politician maintains

the status quo, his payoff is

µ0 =αF (θ)

αF (θ) + (1− α)F (θ) + (1− α)∫ θ2θ

[1− ρ(θ)]f(θ)dθ̂

=α

α + (1− α)F (θ)+∫ θ2θ

[1−ρ(θ)]f(θ)dθF (θ)

. (11)

24

Note that it does not depend on θ. On the other hand, if the low-type politician undertakes

the reform, his payoff is given by

µL(θ) =1

2· qαf(θ)qαf(θ) + 1

2(1− α)ρ(θ)f(θ)

+1

2· (1− q)αf(θ)

(1− q)αf(θ) + 12(1− α)ρ(θ)f(θ)

=1

2· α

α +12(1−α)ρ(θ)

q

+1

2· α

α +12(1−α)ρ(θ)

1−q

. (12)

If the low-type plays a completely mixed strategy, ρ(θ) ∈ (0, 1), we need to equate (11)and (12), which implies that ρ(θ) must be a constant ρ regardless of the value θ. Conse-

quently, in equilibrium,

α

α + (1− α)F (θ)+(1−ρ∗)[1−F (θ)]F (θ)

=1

2· α

α + (1− α)12ρ∗

q

+1

2· α

α + (1− α)12ρ∗

1−q

, (13)

which we may rewrite as

1

1 + λ(α)A=

1

2· 1

1 + λ(α)B+

1

2· 1

1 + λ(α)C, (14)

where

λ(α) =1− αα

, A = 1 + (1− ρ)κ(θ̄), κ(θ̄) = 1− F (θ̄)F (θ̄)

, B =12ρ

q, C =

12ρ

1− q.

This is the same equation as (4). The expression λ(α) is the likelihood ratio of the low type

versus the high type, κ(θ̄) is the likelihood ratio of reform having good prospects versus

bad prospects, and A, B, and C are respectively the likelyhood ratios of the low type

not reforming, having a successful reform, and having a failed reform versus the high type

obtaining each outcome. Consider the equilibrium condition (14). Note that its LHS is µ0

and its RHS is µL. When ρ = 0, µ0 ≤ α, while µL = 1 as B = C = 0. Therefore, µ0 < µL.

By contrast, when ρ = 1, µ0 = α as A = 1, and µ1L < α, which can be seen from the fact

that when ρ = 1

αµH + (1− α)µL = α,

while µL < µH . Therefore, µ0 > µL.

Both the RHS and LHS of (14) are continuous in ρ. Furthermore, it is straightforward

to show that the LHS strictly increases with ρ, while the RHS strictly decreases with ρ.

Hence, we conclude that there must exist a unique ρ∗ ∈ (0, 1) that solves (14).

25

Proof of Proposition 2

Proof. Part 1 Consider the equilibrium condition (14). We have shown above that the left

hand side of (14) is increasing in ρ∗ and the right hand side decreasing in ρ∗. Note that A,

B, and C do not contain α in their expressions. Thus, we may write

∂(LHS −RHS) of (14)∂α

= − 1α2

[− A

(1 + λ(α)A)2+

1

2· B

(1 + λ(α)B)2+

1

2· C

(1 + λ(α)C)2

].

We want to evaluate the above derivative at the value of ρ that satisfies (14). Observe that

0 < B < C as q ≥ 3/4 > 1/2, we may conclude then B < A < C based on (14). From (14),we obtain

A

1 + λ(α)A=

1

2· B

1 + λ(α)B+

1

2· C

1 + λ(α)C.

Therefore,

1

2· B

(1 + λ(α)B)2+

1

2· C

(1 + λ(α)C)2

=A

1 + λ(α)A

[B

1+λ(α)B

B1+λ(α)B

+ C1+λ(α)C

· 11 + λ(α)B

+

C1+λ(α)C

B1+λ(α)B

+ C1+λ(α)C

· 11 + λ(α)C

].

The expression in the brackets is a convex combination of 11+λ(α)B

and 11+λ(α)C

. Since 0 <

B < C, the former is larger, but the coefficient on the former is smaller than 12. Using (14),

we have1

2· B

(1 + λ(α)B)2+

1

2· C

(1 + λ(α)C)2<

A

(1 + λ(α)A)2.

Hence, at the value of ρ that satisfies (14),

∂(LHS −RHS) of (14)∂α

> 0.

Thus, by the implicit function theorem, the probability of reform by the low type, ρ∗, is

decreasing in α, the probability of high type.

Next, we verify the comparative statics of ρ̄. Because ∂ρ∗

∂α< 0, we only need to show∣∣(1− α)∂ρ∗

∂α

∣∣+ ρ∗ > 1. We have∣∣∣∣(1− α)∂ρ∗∂α∣∣∣∣+ ρ∗

=(1− α)α2

ρ∗2[1−[4q(1−q)][4q(1−q)+λ(α)ρ∗]2

[κ(θ̄) + 1 + 4q(1−q)[1−[4q(1−q)][4q(1−q)+λ(α)ρ∗]2 ]

+ ρ∗

26

Rearranging the equilibrium condition leads to

(1− ρ∗)κ(θ̄) = ρ∗(λ(α)ρ∗ + 1)

4q(1− q) + λ(α)ρ∗− 1

=ρ∗(λ(α)ρ∗ + 1)− 4q(1− q)− λ(α)ρ∗

4q(1− q) + λ(α)ρ∗

=λ(α)ρ∗2 + ρ∗ − λ(α)ρ∗ − 4q(1− q)

4q(1− q) + λ(α)ρ∗,

which yields

κ(θ̄) =λ(α)ρ∗2 + ρ∗ − λ(α)ρ∗ − 4q(1− q)

[4q(1− q) + λ(α)ρ∗](1− ρ∗),

and therefore

κ(θ̄) + 1 =λ(α)ρ∗2 + ρ∗ − λ(α)ρ∗ − 4q(1− q) + [4q(1− q) + λ(α)ρ∗](1− ρ∗)

[4q(1− q) + λ(α)ρ∗](1− ρ∗)

=ρ∗[1− 4q(1− q)]

[4q(1− q) + λ(α)ρ∗](1− ρ∗).

Hence,

[κ(θ̄) + 1 +4q(1− q)[1− 4q(1− q)]

[4q(1− q) + λ(α)ρ∗]2]

=ρ[1− 4q(1− q)]

[4q(1− q) + λ(α)ρ∗](1− ρ∗)+

4q(1− q)[1− 4q(1− q)][4q(1− q) + λ(α)ρ∗]2

=1− 4q(1− q)

[4q(1− q) + λ(α)ρ∗]2(1− ρ∗)[4q(1− q) + λ(α)ρ∗2].

We then obtain ∣∣∣∣(1− α)∂ρ∗∂α∣∣∣∣+ ρ∗

=(1− α)α2

·ρ∗2[1−4q(1−q)]

[4q(1−q)+λ(α)ρ∗]21−4q(1−q)

[4q(1−q)+λ(α)ρ∗]2(1−ρ∗) [4q(1− q) + λ(α)ρ∗2]+ ρ∗

=(1− α)α2

· (1− ρ∗)ρ∗2

[4q(1− q) + λ(α)ρ∗2]+ ρ∗.

For our purpose, we only need to show (1−α)α2· ρ∗2[4q(1−q)+λ(α)ρ∗2] > 1. Rewrite it as

(1−α)α2·

ρ∗2

[4q(1−q)+λ(α)ρ∗2] =1α· λ(α)ρ

∗2

[4q(1−q)+λ(α)ρ∗2] =1α· 1[4q(1−q)λ(α)ρ∗2

+1]. Hence, it suffices to show 1

[4q(1−q)λ(α)ρ∗2

+1]> α.

We claim 1[4q(1−q)λ(α)ρ∗2

+1]> 1

2> α, i.e., 4q(1− q) > λ(α)ρ∗2. To show that, recall the equilibrium

condition 1+(1−ρ∗)m] = ρ∗(λ(α)ρ∗+1)

4q(1−q)+λ(α)ρ∗ , which impliesρ∗(λ(α)ρ∗+1)

4q(1−q)+λ(α)ρ∗ > 1⇔ ρ∗(λ(α)ρ∗+1) >

4q(1−q)+λ(α)ρ∗ ⇔ λ(α)ρ∗2+ρ∗ > 4q(1−q)+λ(α)ρ∗. Because λ(α) > 1, λ(α)ρ∗2 > 4q(1−q)must hold.

27

Part 2 The equilibrium condition can be rewritten as

g(ρ∗, α, q, θ) ≡ [1 + (1− ρ∗)κ(θ̄)]− ρ∗[λ(α)ρ∗ + 1]

4q(1− q) + λ(α)ρ= 0. (15)

Since q ≥ 34, G(ρ∗, α, q) is decreasing with q. Further,

∂g(ρ∗, α, q, θ)

∂ρ∗= −

[κ(θ̄) + 1 +

4q(1− q)[1− [4q(1− q)]2][4q(1− q) + λρ∗]2

]< 0.

Recall that κ(θ̄) = [1− F (θ)]/F (θ). We then obtain

dρ∗

dq= −

∂g(ρ∗,q)∂q

∂g(ρ∗,q)∂ρ∗

< 0.

That ρ̄ is decreasing in q is an immediate consequence by its definition in (5).

Part 3 Consider the equilibrium condition (14). Since F first order stochastically domi-

nates G, we have F (θ) < G(θ). This implies that for any given ρ, LHS of (14) for F is lower

than that for G, since κ(θ̄) is larger for F than for G.

As we have shown above, LHS of (14) strictly increases with ρ, while RHS strictly de-

creases. Thus, only if ρ > ρ′ can make (14) hold for both distributions. From this, the

definition of ρ̄ in (5), and the assumption that F first order stochastically dominates G, we

can immediately see that ρ̄ > ρ̄′.

Proof of Lemma 2

Proof. Recall the equilibrium condition (15). When θ increases, κ(θ̄) ≡ 1−F (θ)F (θ)

must decrease,

which causes g(ρ∗, α, q, θ) to decrease. Further, as we have shown in the proof for previous

results, g(ρ∗, α, q, θ) strictly decreases with ρ∗. By the implicit function theorem, we establish

that when θ increases, ρ∗ must decrease.

Proof of Proposition 3

Proof. Recall that the equilibrium is defined by the equation

α

1 + (1−α)(1−ρ∗)[1−F (θ)]

F (θ)︸ ︷︷ ︸µ0

=1

2[

α

α + (1− α)12ρ∗

q︸ ︷︷ ︸µs

+α

α + (1− α)12ρ∗

1−q︸ ︷︷ ︸µf

].

The politician in office receives a payoff µ0 when he maintains the status quo. He receives a

payoff µs when he successfully implements a reform and µf when he fails. In any equilibrium

28

with a given θ, the type-t politician receives a payoff

µt =

{qtµ

s + (1− qt)µf , for θ ≥ θ;µ0, for θ < θ

.

Hence, in this equilibrium, the expected payoff of a type-t politician is given by

E(µt) = µ0F (θ̄) + [qtµ

s + (1− qt)µf ][1− F (θ̄)].

First, we claim that when θ̄ increases, E(µH) and E(µL) change in opposite directions.

Therefore, the first part of the proposition implies the second part. This claim is an impli-

cation of the fact αE(µH) + (1− α)E(µL) = α, or

E(µH) = 1− λ(α)E(µL).

Now, we prove the first part of the proposition. For a low-type politician, E(µL) = µ0

because µ0 = 12µs + 1

2µf . Hence, we need only verify dµ

0

dθ> 0. Define

H(ρ∗, θ) =α

1 + (1−α)(1−ρ∗)[1−F (θ)]

F (θ)︸ ︷︷ ︸µ0

− 12

[α

α + (1− α)12ρ∗

q︸ ︷︷ ︸µs

+α

α + (1− α)12ρ∗

1−q︸ ︷︷ ︸µf

].

We havedµ0

dθ=∂µ0

∂θ+∂µ0

∂ρ∗· ∂ρ

∗

∂θ=∂µ0

∂θ+∂µ0

∂ρ∗· [−∂H(ρ

∗, θ)

∂θ�∂H(ρ∗, θ)

∂ρ∗].

Because ∂H(ρ∗,θ)

∂θ= ∂µ

0

∂θ, we then have dµ

0

dθ= ∂µ

0

∂θ[1 − ∂µ0

∂ρ∗�∂H(θ,ρ

∗)∂ρ∗

]. We must have 1 −∂µ0

∂ρ∗�∂H(θ,ρ

∗)∂ρ∗

> 0 because ∂H(ρ∗,θ)

∂ρ∗= ∂µ

0

∂ρ∗− 1

2(∂µ

s

∂ρ∗+ ∂µ

f

∂ρ∗), while ∂µ

0

∂ρ∗> 0, ∂µ

s

∂ρ∗, ∂µ

f

∂ρ∗< 0.

Divinity Criterion and Proof of Proposition 4

We first formally translate the notion of the Divinity Criterion into our context. Fix an

equilibrium with a cutoff θ > θ̂. Suppose that an unexpected reform with a value θ ∈ [θ̂, θ)takes place. The public infers from its outcome the value of θ. The public forms a set

of beliefs φθ ≡ {ρ̃H(θ), ρ̃L(θ)}, where ρ̃t(θ) specifies the probability of a type-t politicianto undertake this reform. Given this conjecture, a type-t politician, when deviating, has a

payoff

µt(θ;φθ) = qt ×αρ̃H(θ)q

αρ̃H(θ)q +12(1− α)ρ̃L(θ)

+(1− qt)×αρ̃H(θ)(1− q)

αρ̃H(θ)(1− q) + 12(1− α)ρ̃L(θ).

Let µ∗t denote the payoff of a type-t politician in the equilibrium. Further define Φtθ ≡

{φθ|µt(θ;φθ) > µ∗t}. We then have the following.

29

Definition 1. Under Divinity Criterion, the out-of-equilibrium belief φθ satisfies:

ρ̃t(θ) ≥ ρ̃t′(θ) if Φt′

θ ⊂ Φtθ, with t ∈ {H,L} and t 6= t′.

In what follows, we formally establish Proposition 4.

Proof. Consider an arbitrary equilibrium with a cutoff θ > θ̂. Suppose that an unexpected

reform is undertaken, and the public observes that the reform has a potential value θ ∈ [θ̂, θ).Define α̃ ≡ αρ̃H(θ)

αρ̃H(θ)+(1−α)ρ̃L(θ). By taking this reform, the high type has an ex ante expected

payoff

µH(θ; α̃) = q ×α̃q

α̃q + 12(1− α̃)

+ (1− q)× α̃(1− q)α̃(1− q) + 1

2(1− α̃)

= q × 11 + 1

2α̃q(1− α̃)

+ (1− q)× 11 + 1

2α̃(1−q)(1− α̃).

She has an incentive to deviate if and only if πH(θ)−µ0 ≥ 0. The low type, by contrast, hasan ex ante expected payoff

µL(θ; α̃) =1

2× 1

1 + 12α̃q

(1− α̃)+

1

2× 1

1 + 12α̃(1−q)(1− α̃)

.

She has an incentive to deviate if and only if πL(θ) − µ0 ≥ 0. Because 11+ 12α̃q

(1−α̃) >

11+ 1

2α̃(1−q) (1−α̃), we see that µH(θ) − µ0 > 0 whenever µL(θ) − µ0 ≥ 0. It implies that the

high type is always more likely to deviate by undertaking an expected reform than the low

type. The out-of-equilibrium belief must require α̃ ≥ α to reflect this fact.We now prove µH(θ) > α. To see this, observe that

µH(θ; α̃) = q ×α̃q

α̃q + 12(1− α̃)

+ (1− q)× α̃(1− q)α̃(1− q) + 1

2(1− α̃)

> [α̃q +1

2(1− α̃)]× α̃q

α̃q + 12(1− α̃)

+ [α̃(1− q) + 12

(1− α̃)]× α̃(1− q)α̃(1− q) + 1

2(1− α̃)

= α̃ > α,

where we have used the fact that q > 1/2.

Given such a belief, the high type must deviate when θ is realized, because his expected

payoff µH(θ) > α > µ0. The original equilibrium cannot be sustained by a belief system

that satisfies Divinity Criterion.

30

Proof of Lemma 3

Proof. Consider the value of

E(y| θ̂, θ̂) = α[θ̂ − 4(1− q)] + (1− α) ρ∗|θ=θ̂ (θ̂ − 2).

When θ̂ = 4(1− q), it must be negative. When θ̂ approaches θ2, we have its value approachα[θ2− 4(1− q)] + (1−α)ρ(θ2− 2), which is positive if and only if

(1−α)ρα

< θ2−4(1−q)2−θ2 . Further

recall that E(y| θ, θ̂) strictly increases with both θ and θ̂. There must exist a unique θ̂0

that

solves the equation.

Lemma 4 and Its Proof

Because f(θ̂) > 0 for all θ̂ ∈ [−θ1, θ2], the sign of (10) is the same as that of dWdθ̂ �f(θ̂). Forour purpose, it suffices to explore dW

dθ̂�f(θ̂). We then establish the following lemma.

Lemma 4. The expression dWdθ̂

�f(θ̂) strictly decreases with θ̂.

Proof. Recall that the equilibrium condition (15) for θ = θ̂ can be written as

g(ρ∗, κ) = [1 + (1− ρ∗)κ(θ̂)]− ρ∗(λρ∗ + 1)

4q(1− q) + λρ∗= 0,

where κ(θ̂) = [1 − F (θ̂)]/F (θ̂), as defined in (14). Hence, we have ∂g(ρ∗,κ(θ̂))∂κ

= λ(1 − ρ∗).Because ∂g(ρ

∗,κ(θ̂))∂ρ∗

= −[κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ]2 ] < 0, we must have

dρ∗

dκ=

1− ρ∗

κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2,

and thereforedρ∗

dθ̂�f(θ̂) = − 1− ρ

∗

[κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2 ][F (θ̂)]2.

We now claim −dρ∗dθ̂

�f(θ̂) strictly decreases with θ̂. We have

d[−dρ∗dθ̂

�f(θ̂)]

dθ̂=

−dρ∗dθ̂ [κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2][4q(1−q)+λρ∗]2 ][F (θ̂)]2−(1− ρ∗)

d{[κ(θ̂)+1+ 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2][F (θ̂)]2}

dθ̂

{[κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2 ][F (θ̂)]2}2

.

Note that −dρ∗dθ̂

[κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2 ][F (θ̂)]2 = (1 − ρ∗)f(θ̂). We then only need to

proved{[κ(θ̂)+1+ 4q(1−q)[1−[4q(1−q)]

2]

[4q(1−q)+λρ∗]2][F (θ̂)]2}

dθ̂> f(θ̂). Rewrite [κ(θ̂)+1+ 4q(1−q)[1−[4q(1−q)]

2][4q(1−q)+λρ∗]2 ][F (θ̂)]

2 as

F (θ̂) + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2 [F (θ̂)]2. When θ̂ increases, both 4q(1−q)[1−[4q(1−q)]

2][4q(1−q)+λρ∗]2 and F (θ̂) strictly

increases. Hence,d{ 4q(1−q)[1−[4q(1−q)]

2]

[4q(1−q)+λρ∗]2][F (θ̂)]2}

dθ̂> 0. Furthermore, dF (θ̂)

dθ̂= f(θ̂). We then estab-

lish our claim.

31

Proof of Proposition 5

Proof. If(1−α)ρα≥ θ2−4(1−q)

2−θ2 , then θ̂0

does not exist. Any reform with a value θ < θ2 must lead

to negative expected outcome. Hence, no reform is ex ante beneficial, which implies θ̂∗

= θ2.

If(1−α)ρα

< θ2−4(1−q)2−θ̂ , then θ̂

0exists. dW

dθ̂�f(θ̂)

∣∣∣θ̂=θ̂

0> 0, but dW

dθ̂�f(θ̂)

∣∣∣θ̂=θ2

< 0 (because

(1−α)ρα

< θ2−4(1−q)2−θ̂ ), then there must exist a unique θ̂

∗∈ (θ̂

0, θ2) that solves

dW

dθ̂�f(θ̂) = 0.

Proof of Proposition 6

Proof. Suppose that an interior optimum with θ̂∗∈ (0, θ2) exists. Define k ≡ [−dρ

∗

dθ̂�f(θ̂)].

Then the optimal condition is

υ ≡ α[θ̂ − 4(1− q)] + (1− α)ρ∗(θ̂ − 2)− (1− α)k∫ θ2θ̂

(2− θ)f(θ)dθ = 0. (16)

Apparently, dυdθ̂

= −ddWdθ̂f(θ̂)

dθ̂> 0. We now claim dυ

dα> 0. Taking first order derivative of υ

yields

dυ

dα= [θ̂ − 4(1− q)]− ρ∗(θ̂ − 2) + (1− α)dρ

∗

dα(θ̂ − 2)

+k

∫ θ2θ̂

(2− θ)f(θ)dθ − (1− α)∂k∂α

∫ θ2θ̂

(2− θ)f(θ)dθ.

It suffices to show k strictly decreases with α and q. Recall by the proofs of previous results:

−dρ∗

dα=

ρ∗2[1−[4q(1−q)][4q(1−q)+λρ∗]2

[κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2 ]·∣∣∣∣dλ(α)dα

∣∣∣∣ .Note −d

dρ∗

dθ̂

dα= −d

dρ∗dα

dθ̂. Hence, we now evaluate −dρ∗

dαwith respect to θ̂. We first rearrange it

as

−dρ∗

dα=

ρ∗2[1−[4q(1−q)][4q(1−q)+λρ∗]2

[κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2 ]·∣∣∣∣dλ(α)dα

∣∣∣∣=

(1− ρ∗)[κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]

2][4q(1−q)+λρ∗]2 ]

· [1− [4q(1− q)]

· ρ∗2

1− ρ∗· 1

[4q(1− q) + λρ∗]2.

By the proof of Lemma 2, (1−ρ∗)

[κ(θ̂)+1+4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2]

decreases with θ̂. We claim ρ∗2

1−ρ∗ ·

32

1[4q(1−q)+λρ∗]2 also decreases with θ̂. Evaluate it with respect to θ̂ yields

ρ∗(2− ρ∗)dρ∗dθ̂

(1− ρ∗)2· 1

[4q(1− q) + λρ∗]2

+ρ∗2

1− ρ∗·

−2λdρ∗dθ̂

[4q(1− q) + λρ∗]3.

Because dρ∗

dθ̂< 0, we need to show (2−ρ∗)[4q(1−q)+λρ∗]−2λρ∗(1−ρ∗) > 0, which is obvious

because (2− ρ∗)[4q(1− q) + λρ∗]− 2λρ∗(1− ρ∗) = (2− ρ∗)[4q(1− q) + λρ∗]− λρ∗(2− 2ρ∗),and 2− ρ∗ > 2− 2ρ∗.

We further claim θ̂∗

decreases with q. To show that, we have to prove dυdq> 0. We have

dυ

dq= 4α + (1− α)dρ

∗

dq(θ̂ − 2)− (1− α)dk

dq

∫ θ2θ̂

(2− θ)f(θ)dθ.

It would suffice to show dkdq< 0. We use the same technique as above. We have

−dρ∗

dq=

4(2q−1)ρ∗(λρ∗+1)[4q(1−q)+λρ∗]2

[κ(θ̄) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2 ].

We then claim − ∂2ρ∗∂q∂θ̂

< 0. Rewrite −dρ∗dq

as

−dρ∗

dq=

1− ρ∗

[κ(θ̂) + 1 + 4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2 ]· 1

1− ρ∗· 4(2q − 1)ρ

∗(λρ∗ + 1)

[4q(1− q) + λρ∗]2.

Because 1−ρ∗

[κ(θ̂)+1+4q(1−q)[1−[4q(1−q)]2]

[4q(1−q)+λρ∗]2]

and 11−ρ∗ decreases with θ̂, we only need to show

ρ∗(λρ∗+1)[4q(1−q)+λρ∗]2

decreases with θ̂. Taking first order derivative of it with respect to θ̂ yields

d ρ∗(λρ∗+1)

[4q(1−q)+λρ∗]2

dθ̂=

(2λρ∗ + 1)dρ

∗

dθ̂[4q(1− q) + λρ∗]2(1− ρ∗)

−2ρ∗(λρ∗ + 1)(1− ρ∗)[4q(1− q) + λρ∗]λdρ∗dθ̂

+ρ∗(λρ∗ + 1)[4q(1− q) + λρ∗]2 dρ∗dθ̂

(1− ρ∗)2[4q(1− q) + λρ∗]4

=

dρ∗

dθ̂

[4q(1− q) + λρ∗]3

×

[(2λρ∗ + 1)[4q(1− q) + λρ∗](1− ρ∗)− 2λρ∗(λρ∗ + 1)(1− ρ∗)

+ρ∗(λρ∗ + 1)[4q(1− q) + λρ∗]

].

By Lemma 2, dρ∗

dθ̂< 0. Hence, it remains to verify that the item in bracket is positive. This

33

is obvious because[(2λρ∗ + 1)[4q(1− q) + λρ∗](1− ρ∗)− 2λρ∗(λρ∗ + 1)(1− ρ∗)

+ρ∗(λρ∗ + 1)[4q(1− q) + λρ∗]

]> λρ∗ [(2λρ∗ + 1)(1− ρ∗)− 2(λρ∗ + 1)(1− ρ∗) + ρ∗(λρ∗ + 1)]

= λρ∗[−ρ∗ + ρ∗(λρ∗ + 1)] > 0.

Proof of Proposition 7

Proof. We examine how a higher upper support θ2 could affect dW/dθ̂ for any given θ̂. When

the high-type politician is perfectly informed, a closed form for ρ∗ is obtained as

ρ∗ = 1− α1− α

F (θ̂).

The first-order derivative of the welfare function is derived as follows

dW

dθ̂=

1

θ2 + θ1

−αθ̂ − (1− α)ρ∗(θ̂ − 2)+(1− α)dρ∗/dθ̂f(θ̂)

∫ θ2θ̂

(θ − 2)f(θ)dθ

=

1

θ2 + θ1

{−αθ̂ + (1− α)[1− α(θ̂+θ1)

(1−α)(θ2+θ1) ](2− θ̂)+α(θ2−θ̂)

2[4− (θ2 + θ̂)]

}.

The optimal cutoff θ̂∗

is determined by the equation

υ = −αθ̂ + (1− α)[1− α(θ̂ + θ1)(1− α)(θ2 + θ1)

](2− θ̂) + α(θ2 − θ̂)2

[4− (θ2 + θ̂)] = 0.

By Lemma 4, υ strictly decreases with θ̂. We only need to show ∂υ∂θ2

> 0. Apparently,

(1−α)[1− α(θ̂+θ1)(1−α)(θ2+θ1) ](2− θ̂) increases with θ2. We claim

α(θ2−θ̂)2

[4− (θ2 + θ̂)] increases withit as well. Taking first order derivative of (θ2− θ̂)[4− (θ2 + θ̂)] yields 4− (θ2 + θ̂)− (θ2− θ̂) =4− 2θ2 > 0, which completes the proof.

References

Alesina, Alberto and Guido Tabellini, “Bureaucrats or Politicians? Part I: A Single

Policy Task,” American Economic Review, March 2007, 97 (1), 169–179.

Banks, Jeffrey S. and Joel Sobel, “Equilibrium Selection in Signaling Games,”

Econometrica, May 1987, 55 (3), 647–61.

Biglaiser, Gary and Claudio Mezzetti, “Politicians’ decision making with re-election

concerns,” Journal of Public Economics, December 1997, 66 (3), 425–447.

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Brandenburger, Adam and Ben Polak, “W