Essays in Applied Game Theory and Political Economy

Essays in Applied Game Theory and

Political Economy

Christopher Kai Steverson

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Economics

Adviser: Professor Faruk Gul

June 2015

 Copyright by Christopher Kai Steverson, 2015.

Abstract

This dissertation provides a game theoretic study of the incentives and dynamics of govern-

ment action. In chapter 1, I study how the political incentives for providing legal protections

to minority workers is inﬂuenced by the mobility of workers across cities. I build a model that

shows how cities that provide insuﬃcient protection to minorities can do better at attracting

workers and end up being larger overall. To the extent that policy makers care about city size

and city growth, my result identiﬁes an incentive that reinforces the inadequate protection

of minorities in the workplace.

In chapter 2, I build a model to explain electoral cycling, which is the regularity observed

in the data that the longer a party is in power the more likely it loses the next election. I study

an inﬁnite horizon model of political competition where parties choose between nominating

moderate candidates, who are better at winning elections, and partisan candidates who

energize and grow the base which helps to win future elections. Hence, parties face a trade-

oﬀ between winning current and future elections. In the presence of such a trade-oﬀ, parties

win elections by sacriﬁcing their future prospects which naturally leads to electoral cycling.

In chapter 3, I explore novel consequences of the learning model of economic voting.

Economic voting is the empirical phenomenon that a strong economy makes incumbent re-

election more likely while a weak economy makes incumbent defeat more likely. Previous

work in the literature explains this phenomenon with voter learning; a strong economy is

taken as a positive signal about the incumbent’s policy or competence. I introduce the idea

that voters who learn should also experiment, that is they should care about how much

learning a policy allows when deciding whom to elect. In particular voters will, at times,

prefer a policy that creates worse outcomes but that allows for more learning.

iii

Acknowledgements

I owe an enormous debt to Faruk Gul and Wolfgang Pesendorfer for their constant support

and guidance. They taught me everything I know about economic research, and without

their help none of this would have been possible. I am grateful to Stephen Morris for many

helpful and insightful conversations on the various chapters of this work. Over the years

Roland Benabou, Dilip Abreu, Sylvain Chassang and Matias Iaryczower have all provided

invaluable comments and advice.

Thank you to my friends and peers for their help and support: Doron Ravid, Nemanja

Antic, Brian So, Alex Jakobson, Giri Parameswaran, Nikhil Gupta, Ishita Rajani, Justin

Weidner, and Jay Lu. I learned a great deal from each of them, and their companionship

made the journey fun as well as fruitful.

I am grateful to Kathleen DeGennaro, Ellen Graf, Laura Hedden, Geralyn McDermott,

Matthew Parker, and Barbara Radvanady for always providing assistance when I have been

in need.

I am always very grateful to my loving family who support me in all my endeavors: my

parents Misako and Randy, and my sister Erica.

And ﬁnally, I owe a special thanks to Caitlin McGugan. In the time that I have known

her she has provided an endless source of support and wisdom.

To my parents.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Discrimination and Mobility in a Search Model 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Steady State Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.1 Pure Strategy Steady States . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.6 Lump Sum Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.7 General Enforcement Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.7.1 Existence with general enforcement regimesI . . . . . . . . . . . . . . 35

1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Losing to Win: How Partisan Candidates Help Parties Win Future Elec-

tions 38

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Motivating the Model: Evidence and Theory . . . . . . . . . . . . . . . . . . 42

2.2.1 Abstention due to Alienation . . . . . . . . . . . . . . . . . . . . . . 42

2.2.2 Habitual Voting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.3 The Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.2 The Decision to Vote . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4.2 Characterization and Discussion . . . . . . . . . . . . . . . . . . . . . 49

2.4.3 Sketch of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5 Oﬃce Motivated Parties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Economic Voting and Experimentation 57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Value Functions and Incentives . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5 Electoral outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A Appendix to Chapter 1 78

A.1 Proof of proposition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.1.1 Proof of Lemma A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

A.2 Proof of Proposition 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.3 Proof of Proposition 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.4 Proofs from section 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.4.1 Proof of Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.4.2 Proof of Proposition 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . 91

vii

B Appendix to Chapter 2 96

B.1 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.1.1 Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.1.2 Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

B.2 Proof of proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

B.2.1 Showing σ

is an equilibrium . . . . . . . . . . . . . . . . . . . . . . . 98

B.2.2 Proof that σ

is unique . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.3 Proof of Proposition 2.2 (Oﬃce Motivated Candidates) . . . . . . . . . . . . 108

B.3.1 Patient Oﬃce Motivated Parties . . . . . . . . . . . . . . . . . . . . . 109

B.3.2 Impatient Oﬃce Motivated Parties . . . . . . . . . . . . . . . . . . . 109

C Appendix to Chapter 3 110

C.1 Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

C.2 Two Useful Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

C.3 Proofs of Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

C.3.1 Proof of proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 114

C.3.2 Proof of proposition 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . 119

C.3.3 Proof of proposition 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . 120

C.3.4 Proof of proposition 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . 120

Bibliography 122

viii

Chapter 1

Discrimination and Mobility in a

Search Model

1.1 Introduction

Tiebout (1956) developed the idea that individuals can escape undesirable local policies by

relocating. This dynamic of “voting with your feet” leads, in principle, to well-managed

cities attracting more workers and growing, while poorly managed cities shrink. To the

extent that policy makers care about city growth, mobility creates pressure for reform.

Even in the absence of reform, poorly-managed cities becomes less relevant as they become

smaller and their policies impact fewer people. In democratic societies, mobility is especially

important in the protection of minorities who cannot, by themselves, muster the votes for

change. Moving may be the only recourse when faced with a discriminating majority.

In this chapter, I analyze the extent to which Tiebout-style competition aﬀects the legal

protection of minorities in the labor market. To this end, I consider two cities that diﬀer in

the degree to which they enforce the equal protection of minorities in the the workplace. In

one city, enforcement is high and minorities are treated equally. In the other city, enforcement

is low and ﬁrms are tacitly allowed to violate the labor rights of minorities for monetary gain.

My main result is that, for certain parameters, the city with low enforcement will be larger,

and, in extreme cases, will contain all of the non-minority workers and half of the minority

workers. In other words, I ﬁnd that mobility has the potential to reward rather than punish

inadequate legal protections for minorities. The result requires a tacitly allowed form of

discrimination in that ﬁrms cannot openly advertise a minority speciﬁc wage. This tacit

discrimination creates a transfer to non-minority workers that would not occur under more

explicit wage discrimination.

Labor law violations are a widespread problem in the United States that dispropor-

tionately impacts minorities. In a 2001 study, the U.S. Department of Labor randomly

investigated ﬁrms within certain industries and found wage violations in as many as 78 per-

cent of garment factories in Los Angeles and 60 percent of nursing homes nationwide (U.S.

Department of Labor (2001)). These violations took many forms including under counting

hours, failing to pay overtime rates, and paying less than the minimum wage. Moreover, mi-

norities are disproportionately likely to experience these violations. For example, Bernhardt

et al. (2013) ﬁnds that within low wage occupations, African Americans and immigrants

have roughly twice the odds to experience minimum wage violations as white workers

. In-

adequate enforcement of labor laws is often cited as one of the prime causes of labor law

violations

. By examining the interaction of worker mobility with legal protections of minori-

ties, this chapter contributes to the understanding of why inadequate protection of minority

groups may arise and why it persists.

Consider a mobile workforce living in two cities. Workers are split into two groups, a

disadvantaged minority and an advantaged majority. Each worker draws an idiosyncratic

city preference that helps ensure no city is ever empty. The distribution from which this

preference is drawn treats the cities symmetrically. Each city has its own labor market

captured using a search model with frictions, where ﬁrms post wages and randomly match

with unemployed workers. A free entry condition ensures all ﬁrms earn a proﬁt of zero.

See also Brotherton et al. (2013).

For example see Bobo (2011)

Firms must post the same wage for both types of workers, but can choose whether or not to

hire any worker they meet. Upon matching, workers draw a random labor cost speciﬁc to

that ﬁrm, which determines whether they accept the job oﬀer. Without the random labor

costs, in equilibrium ﬁrms would always oﬀer a wage of zero.

Firms have occasional opportunities to violate the labor rights of the workers they employ.

Speciﬁcally, ﬁrms may fail to give workers their full wage, which will be referred to as wage

theft. In both cities wage theft is illegal, but enforcement is imperfect and varies across cities

and across types of workers. In what will be called the high enforcement city, all workers are

protected equally, and ﬁrms are caught often enough to deter them. In the low enforcement

city, protection is unequal, and ﬁrms are less likely to be caught while underpaying minority

workers. This leads to systematic wage theft against the minority, while majority workers

remain well protected. In both cities, ﬁrms that are caught are forced to pay a penalty,

and the match between worker and ﬁrm is ended. This enforcement policy will mean that,

in the low enforcement city, the minority workers are not only paid less but also have a

higher unemployment rate. That minority workers are disadvantaged in both compensation

and unemployment is necessary for the results. Capturing both of these dimensions in one

mechanism is done for the sake of simplicity.

My main result establishes that, when the average length of employment is suﬃciently

long, there will exist a low enforcement policy that ensures the low enforcement city is always

larger than the high enforcement city. In other words, when workers can vote with their

feet, they vote in favor of inadequate protection for the minority. For extreme parameters,

the low enforcement city will contain all of the majority workers and half of the minority

workers. This case also identiﬁes a somewhat counter intuitive mechanism for segregation.

One might think that inadequate protection would drive away minority workers from the

low enforcement city. Instead it is the lack of wage theft against minorities that drives away

all the majority workers from the high enforcement city. The net result is that a city that

protects its workers equally ends up being segregated, while the city where a minority is

inadequately protected attracts a broader cross section of workers.

The mechanism whereby the low enforcement city grows larger works through the equi-

librium wage. In the low enforcement city, the possibility of underpaying minority workers

grants a monetary gain to the ﬁrms. But the free entry condition ensures ﬁrms earn zero

proﬁts, and hence this monetary gain translates into a higher equilibrium wage. Majority

workers, who are safe from wage theft, beneﬁt from this higher wage. However, the minority

workers will still have a lower eﬀective wage, accounting for the wage theft they experience.

In this way, low enforcement acts as a transfer from minority workers to majority work-

ers. This transfer attracts more majority workers to the low enforcement city, while driving

away minority workers. The city preference ensures that the number of workers who are

attracted or driven away depends continuously on the diﬀerential in eﬀective wages across

cities. Workers with a strong enough city preference will accept lower compensation to live

in their preferred city. Therefore, which city is larger depends on how the beneﬁt to majority

workers from low enforcement compares to the cost imposed on minority workers.

The key mechanism is that the wage increases with a diﬀerential in the unemployment

rate across types. When minority workers have a higher unemployment rate, they make up a

larger portion of the unemployment pool, and hence ﬁrms are more likely to match with them.

And when hiring a minority workers is more likely, ﬁrms have a higher expected monetary

gain from wage theft, which pushes up the wage in order to satisfy the zero proﬁt condition.

A higher wage raises the eﬀective compensation received by minority workers, partially

compensating them for the wage theft they experience. Hence, with an unemployment

diﬀerential, the impacts of wage theft can be more beneﬁcial to majority workers than they

are detrimental to minority workers, and the low enforcement city will attract more workers

overall.

To further illustrate the mechanism, consider the extreme case where the unemployment

pool in the low enforcement city is comprised entirely of minority workers. Firms entering

the market know they will surely hire a minority worker and will always pay them below the

posted rate. In equilibrium, this underpayment becomes priced in and acts only to relabel

the meaning of the wages. Therefore, the wage rises until it fully compensates the minority

workers for the wage theft they experience. Majority workers, even those not currently

unemployed, make their location decisions based on the prevailing wage in each city. Hence,

in steady state, majority workers will strictly prefer the low enforcement city, while minority

workers will be close to indiﬀerent, so the low enforcement city will attract more workers

overall.

One consideration not accounted for in the previous argument is that the diﬀerential in

unemployment rates is supported by minority workers losing their jobs more often, which

makes the low enforcement city less attractive to both ﬁrms and d workers. However this

eﬀect becomes insigniﬁcant when employment length is very long, which is the key parameter

in the results. When losing a job is rare, an increase in the job losing rate for minority workers

can be small in absolute terms while being large in relative terms. In other words, a minority

worker can lose their job at rate that is simultaneously low in absolute terms and many times

the rate of the majority workers. And due to discounting, a small job losing rate has little

impact on where workers choose to live.

An important ingredient in the above argument is a tacit form of discrimination. Firms

are de facto permitted to underpay minority workers but cannot openly advertise a minority

speciﬁc wage. If ﬁrms could oﬀer diﬀerent wages, the labor market in the low enforcement

city would functionally segment into two portions, one for each type of worker. Under such

segmentation, wages of the two types of workers would be independent, and the mechanism

by which the low enforcement city gains more workers would disappear. Tacit discrimination

is needed for majority workers to gain from wage theft against minority workers, which drives

my result.

While ﬁrms cannot post explicitly diﬀerent wages, they can decide who to hire. In

particular, ﬁrms in the low enforcement city could pursue the strategy of only hiring the

more proﬁtable minority workers. This strategy would create an informal segmentation of

the market. But when meeting a majority worker, the ﬁrm must either hire the current

prospect at the posted wage or continue searching and incur further search costs. As long

as the beneﬁt to wage theft is not too large, the ﬁrm will prefer to hire the majority worker

over searching, and this informal segmentation will not occur.

One limitation of the model is that cities are ﬁxed as either low or high enforcement

instead of being allowed to choose their policies. Incorporating an endogenous policy decision

would not allow for use steady state analysis. Given any initial state, a city that changes its

enforcement policy would not jump instantaneously to a new steady state but instead would

converge there over time. To study endogenous policy choice properly, it would be necessary

to evaluate the dynamic path out of steady state, which is diﬃcult. My model oﬀers an

analysis of the impact of low enforcement, and provides an argument for its persistence

regardless of how it arises. And, under the reasonable assumption that cities value long term

growth, my analysis identiﬁes an incentive for cities to provide inadequate legal protection

for minorities.

This chapter brings together to two distinct literatures: the literature on worker mobility

and the literature on the economics of discrimination. Another paper that combines these

literatures is Margo (1990), who looks at whether mobility constrained discrimination in the

southern United States at the beginning of the twentieth century. However his model of the

economy is frictionless and has no unemployment, and so cannot capture the key mechanism

of this chapter. Moreover, he ﬁnds that mobility did constrain discrimination, coming to the

opposite conclusion I do. This chapter also connects to the labor search literature through

the modeling techniques employed. I will now discuss the connection of this chapter to the

literatures on discrimination, mobility and labor search.

The economics of discrimination literature dates back at least as far as Becker (1971),

who argued that discriminating ﬁrms will be less proﬁtable and will be bought out in the

long run. Goldberg (1982) adds nuance to this observation by pointing out that if ﬁrms gain

utility from hiring majority workers rather than dis-utility from hiring minority workers,

then it is possible discriminatory ﬁrms will not want to sell. They may accept lower proﬁts

in order to retain the utility gained from hiring their favorite type of worker.

Black (1995) and Bowlus and Eckstein (2002) embed Becker’s argument into a labor

search model with ﬁrm entry. Black argues that discriminating ﬁrms will only be driven out

of the market if the pool of potential ﬁrms is suﬃciently large. In particular, Black assumes

that ﬁrms diﬀer in their productivity and more productive discriminatory ﬁrms remain in

business even with entry. However, the productivity cutoﬀ above which ﬁrms enter will

be lower for non-discriminatory ﬁrms. Black use techniques similar to Borjas and Bronars

(1989), who study the impact of consumers who dislike buying from self-employed minorities.

Another strain of the economics of discrimination literature starts with Arrow (1973), who

showed how discrimination can arise without any taste for discrimination, but instead using

incomplete information. In his model, ﬁrms infer a worker’s private education investment

based on whether the worker is a minority. This creates discriminatory equilibria in which

all minorities make low investments and are assigned to low paying jobs that have small

complementarities with education. In a similar vein, Aigner and Cain (1977) and Lundberg

and Startz (1983) show how discrimination will arise if employers are more uncertain about

the skill of minority workers than the skill of majority workers.

Coate and Loury (1993) extend Arrow’s model in order to consider the impact of aﬃr-

mative action. They ﬁnd that, under aﬃrmative action policies, discriminatory equilibria

still exist and may even become worse. Ros´en (1997) builds a model similar to Arrow’s but

with labor market search. Ros´en (1997) shows that equilibria without discrimination are

unstable to small perturbations. Lang et al. (2005) builds a model of discrimination using

a labor search market without incomplete information. In their model, when ﬁrms choose

between hiring two diﬀerent workers, they break indiﬀerence by hiring majority workers over

minority workers. Eeckhout (2006) as well as Peski and Szentes (2013) echo Ros´en (1997) in

arguing that equilibria with discrimination tend to be more stable than equilibria without

discrimination.

This chapter contributes to the literature on the economics of discrimination by consid-

ering the impact of worker mobility. All the papers mentioned above take place in a single

labor market and rule out the possibility that minority workers could choose where to live

and work.

This chapter also connects to the literature on mobility started by Tiebout (1956). Much

of the theoretical literature following Tiebout has focused on the impact of mobility on

ﬁscal policy. Oates (1999) surveys the discussion on whether, given mobility, public good

provision should be done at the federal or local level. Ellickson (1971) as well as Epple et al.

(1984) and Epple and Romer (1991) have looked at the degree to which mobility constrains

redistribution. A contribution this chapter makes to this literature is incorporating a labor

search model into a Tiebout style mobility model. However, this chapter does not allow for

endogenous policy, which is a feature common to this literature.

There is also a literature that studies how workers locate within cities. Of particular

relevance is the spatial mis-match hypothesis, ﬁrst introduced by Kain (1968) and of which

Holzer (1991) provides a good survey. Spatial mis-match studies whether increased suburban-

ization has worsened unemployment for inner city workers, and in particular for minorities,

by moving jobs further away into the suburbs. Gobillon et al. (2007) surveys the theoretical

models of the spatial mis-match hypothesis. Unlike in this chapter, none of these models

look at diﬀerences in the legal protection of minorities across locations. Moreover, typically

these models ﬁx wages exogenously, which leaves no scope for the mechanism driving my

main result. One exception to this is Brueckner and Zenou (2003) who allow for endogenous

wages, but only for the minority group, the majority group wage is still ﬁxed exogenously.

Lastly, this chapter connects to the labor search literature through the modeling tech-

niques employed. My model combines elements of the random search model of Mortensen

and Pissarides (1994) as well as models where ﬁrms post wages such as Moen (1997). How-

ever, in Moen (1997) workers costlessly and immediately see every distinct posted wage.

Each worker decides the wage level at which he wants to apply and engages in costly search

among ﬁrms at that particular wage. This makes diﬀerent wages act as eﬀectively diﬀerent

markets. A key departure in my model is that workers instead come across posted wages

randomly, similar to Mortensen and Pissarides (1994). Once a worker sees a wage, he decides

whether to apply or to keep looking. My modeling of wage theft uses techniques similar to

those of Albrecht and Axell (1984) and Burdett et al. (2003). Those papers are interested in

workers stealing from ﬁrms within the labor market. In my model, however, it is the ﬁrms

that steals from the workers.

The rest of the chapter is organized as follows. Section 1.2 analyzes a simple example

that illustrates how the main mechanism functions. Section 1.3 gives the formal description

of the model. Section 1.4 characterizes the optimal behavior of agents and provides a formal

deﬁnition of steady state equilibrium. Section 1.5 presents the results. Sections 1.6 and 1.7

consider extensions. And section 1.8 concludes.

1.2 A Simple Example

This section analyzes a simple example that illustrates the mechanism underlying the main

result of this chapter. There are two cities, labeled 1 and 2, each with a large population

of ﬁrms and workers. Time is discrete and all agents are risk neutral and discount at rate

β ∈ (0, 1). Workers live forever and come in two types, an advantaged majority (a) and a

disadvantaged minority (d). Both cities are split evenly between the two types of workers.

City 1 is the low enforcement city where ﬁrms are tacitly allowed to discriminate against d

workers, leading to those workers being both paid less and losing their job more often. City

2 is the high enforcement city where a and d workers are treated equally.

Unlike in the full model, workers cannot move. Instead I will consider two new workers,

one of each type, who don’t yet live in a city and are considering where to locate. These new

workers start their life unemployed after deciding where to live. Each of these new workers

draws a idiosyncratic city preference shock η from distribution Γ, which is symmetric around

zero and full support. Each η is a payoﬀ that a worker gets for living in city 2. Since η

is symmetrically distributed around 0, the baseline expectation is for each city to get one

worker. When employment length is suﬃciently long, city 1 will have an advantage in

attracting workers and will do better than gaining one worker on average. In the full model

where workers can move, this will translate into city 1 being larger.

One worker matches with one ﬁrm to produce a per period surplus of 1. In city j, workers

earn a per period wage of w

, which will be pinned down by a zero proﬁt condition. In city

1, d workers lose a ﬁxed amount y of their wages due to wage theft by ﬁrms. Hence, the

eﬀective wage of a d worker in city 1 is w

− y. Employment length is T periods for all

workers, except for d types in city 1 who lose their job more often and have employment

lasting only

√

T periods. For simplicity I restrict attention to values of T that are perfect

squares. When employment ends, the worker becomes unemployed and the ﬁrm dies. Each

period, ﬁrms enter each city and match randomly with a worker. Enough ﬁrms enter to

match all the unemployed workers, meaning that unemployment lasts for only one period.

Firms pay entry cost k > 0, and a free entry condition pins their expected discounted proﬁt

to zero. Because matching is random, the probability that an entering ﬁrm matches with an

i worker equals the proportion of the unemployed pool that is made up of i types.

In city 2, let U

denote the lifetime discounted value of an unemployed worker net of the

city preference η. The same value in city 1 depends on the worker’s type and will be denoted

and U

. A new worker of type i moves to city 1 if and only if

η ≤ U

− U

Hence the expected number of new workers city 1 will attract is



− U



+ Γ



− U



. (1.1)

I now analyze and compute the steady state. Steady state requires the number of workers

exiting and entering employment to be equal, which pins down the city 2 unemployment rate

T +1

. Given the diﬀerent employment lengths, the unemployment rate of d and a type

workers will be

√

T +1

and

T +1

respectively. Let m

be the proportion of the unemployment

pool made up of d type workers, then m

is deﬁned by:

T + 1

T + 1 +

√

T + 1

Moreover, m

equals the probability an entering ﬁrm matches with a d type worker. Notice

that m

increases in T and goes to one as T goes to inﬁnity.

Now consider the zero proﬁt condition in city 2, which can be written as



1 − w





1 − β



1 − β

= k.

The term 1 −w

is the ﬁrm’s per period proﬁt once a match is made. And the term

(

1−β

)

1−β

captures both that employment only lasts T periods and that searching takes one period. In

city 1, the zero proﬁt condition can be written as



1 − m



1 − w





1 − β



1 − β

+ m



1 − w

+ y





1 − β

√



1 − β

= k. (1.2)

The left-hand side of the above equation has two terms, one for each type of worker.

Each of those terms gives the discounted proﬁt the ﬁrm earns from employing that type

of worker, weighted by the probability of matching with that type. The ﬁrst term on the

left-hand side is the term for the a workers, and it reﬂects that ﬁrms earn a per period proﬁt

of 1 − w

for T periods from matching an a worker. The second term on the left-hand side

is the term for d workers, and it reﬂects that ﬁrms earn a per period proﬁt of 1 − w

+ y

for

√

T periods from matching a d worker. The a worker term pushes w

towards w

, while

the d worker term pushes w

towards w

+ y when T is large. To see why, notice that the

a worker term exactly matches the zero proﬁt condition in city 2, and hence this term by

itself would set w

= w

. When T is large, the diﬀerence between 1 − β

and 1 − β

√

very small, and ignoring this diﬀerence the d worker term by itself would set w

= w

+ y.

The value of w

depends on the entire left-hand side of equation 1.2, and hence, for large T

the following inequalities hold:

< w

+ y.

The above inequality shows us that wage theft acts as a transfer from d workers to a workers.

The value of w

−w

is the wage gap for a workers between city 1 and city 2. While w

−y−w

is the gap in the eﬀective wage for d workers across cities. Hence, the above inequality implies

that a workers earn more in city 1 and are better oﬀ, while d workers earn less in city 1 and

are worse oﬀ.

Whether w

is closer to w

or w

+ y depends on m

. As m

becomes closer to one, the

d worker term of equation 1.2 dominates, and that pushes w

towards w

+ y. Hence, for

larger m

, the wage loss experienced by d workers in city 1 is mitigated, and the wage gain a

workers experience increases. In this way, the composition of the unemployment pool plays

a crucial role in determining the impacts of wage theft. And since m

increases in T , for

suﬃciently large T , a workers gain more than d workers lose, which means city 1 will do

better at attracting new workers.

To see this argument more formally, consider the limit as T goes to inﬁnity. The value

of m

goes to one which implies that wages become:

lim

T →∞

= 1 −

1 − β

k + y, and lim

T →∞

= 1 −

1 − β

In this limit the d workers are earning the same eﬀective wage in both cities, while a workers

earn strictly more in city 1. The lifetime value of an unemployed worker in city 2 can be

written as

∞

t=0

−

∞

t=0

t(T +1)

= w

β − β

T +1

(1 − β

T +1

) (1 − β)

The above expression captures that city 2 workers are employed for T periods at a time

earning w

, interspaced with single periods of unemployment where they earn nothing. And

the lifetime values of unemployed d and a workers in city 1 can be written as



− y



β − β

√

T +1

(1 − β)



1 − β

√

T +1



and

= w

β − β

T +1

(1 − β

T +1

) (1 − β)

And as T goes to inﬁnity, U

− U

goes to zero, and U

− U

goes to

yβ

1−β

. And since Γ is

symmetric around zero it must be that

Γ (0) + Γ



yβ

1 − β



> 1.

And the left-hand side of this inequality is how many new workers city 1 will attract in the

limit as T goes to inﬁnity. Hence, when T is suﬃciently large, city 1 will expect more than

half of the new workers; that is city 1 will have an edge in attracting workers.

To highlight the importance of the search features of the model, consider what would

happen if, in lieu of discrimination by ﬁrms, city 1 made lump sum transfers from d workers

to a workers. In this case, city 1 gains no advantage in attracting new workers. So now

suppose a and d workers earn the same wage and have the same employment length in both

cities. However, city 1 imposes a per-period lump sum tax of t

> 0 on d workers and splits

the proceeds evenly among all a type workers. So a workers receive a negative lump sum

tax of t

< 0. Budget balance requires that

= 0.

Let U

include the lump sum transfer, while still not including the city preference η. Then

it must be that for a worker of type i:

1 − β

= U

And since the budget balance condition requires t

= −t

, it must be that



− U



= −



− U



Using the symmetry of Γ around zero and equation 1.1, it turns out that city 1 expects exactly

one of the new workers. Under lump sum transfers, the probability a workers move to city 1

goes up by the same amount that the probability that d workers move to city 1 goes down.

Hence, on net, lump sum transfers do not help city 1 attract more workers. It is only with

the search mechanism included that mobility rewards inadequate protections for minorities.

In the full model, this same analysis carries through provided that the distribution of city

preferences has an increasing hazard rate and ﬁnite expectation. In fact, lump transfers

makes city 1 strictly smaller than city 2. This analysis is done in section 1.6 below.

1.3 Model

There are two cities, labeled 1 and 2, each of which contains a large number of workers

and ﬁrms. Time moves continuously and all agents discount the future at rate r > 0. One

worker matches with one ﬁrm to produce a ﬂow of proﬁt π > 0. However, workers and ﬁrms

must search for each other, which involves frictions and takes time. There is a unit mass of

workers who can choose to reside in either city and live forever. The mass of ﬁrms in each

city is endogenously determined by a free entry condition that sets proﬁts to zero. I will

analyze the steady states of this model.

Workers come in two types, an advantaged majority (a) and a disadvantaged minority

(d). The size of these populations are ﬁxed at s

and s

respectively, with s

+ s

= 1.

Throughout subscripts will refer to a worker’s type and superscripts will refer to cities. City

1 will be the low enforcement city where d workers are inadequately protected. City 2 will

be the high enforcement city that protects all workers equally at a high level.

Workers can move between cities costlessly at any time. However, when an employed

worker moves he give up his current job and becomes unemployed. Workers have a ﬁxed

preference regarding where to live. This is modeled by receiving ﬂow payoﬀ ηγ ∈ R while

living in city 2. The parameter γ > 0 controls the importance of the city preference. The

random variable η is drawn from cdf Γ, which is symmetrically distributed around 0, has full

support and is diﬀerentiable. Each worker’s η is drawn once before the market opens and

does not change.

Unemployed workers search costlessly for a job. Vacant ﬁrms search for workers at a

ﬂow cost of k > 0. If there are a measure of υ searching ﬁrms and µ searching workers in

a given city, then a ﬂow of m (υ, µ) matches are created. As is standard, assume m (·, ·) is

continuous, non-negative, increasing in both its arguments and constant returns to scale

Constant returns to scale means the ﬂow rate of matches depends only on q =

. Firms

meets workers at rate

(q) =

m (υ, µ)

= m



1, q

−1



and workers meet ﬁrms at rate

(q) =

m (υ, µ)

= m (q, 1) .

An example of m () is Cobb-Douglas where the exponents add to one.

Notably, α

is increasing in q and α

is decreasing in q. I also impose the standard assumption

that

lim

q→0

(q) = ∞ and lim

q→∞

(q) = 0.

Every worker and ﬁrm have the same probability of matching. Hence, when a ﬁrm meets a

worker, the probability that the worker is of type i is equal to the proportion of unemployed

workers who are i types in that city.

A free entry condition determines the number of searching ﬁrms in each city by setting

expected discounted proﬁt of an entering ﬁrm to zero. Upon entering, ﬁrms post a wage

w which they cannot change. When a match is created the worker observes this wage as

well as a random match speciﬁc labor cost c distributed according to cdf G. Assume G has

support [0, ∞), is strictly concave and is twice continuously diﬀerentiable

. The ﬁrm sees

the worker’s type and can choose whether to extend a job oﬀer to the worker at the posted

wage. Notably, ﬁrms can condition their hiring decision, but not the level of the wage, on

a worker’s type. The worker can also choose whether to accept the oﬀer not. The random

labor cost ensures the worker’s probability of accepting a job moves continuously with the

wage, without which the ﬁrms would always have an incentive to slightly lower the wage,

leading to an equilibrium wage of zero. Once employment begins, the worker gets ﬂow utility

of w − c and the ﬁrm gets ﬂow proﬁt of π − w. Matches break exogenously and randomly

with arrival rate δ > 0. When the match ends, workers go back into the unemployment pool

of their preferred city and ﬁrms die

Firms will have occasional opportunities to engage in wage theft by underpaying the

workers they employ. These opportunities arrive at rate λ > 0 and allow the ﬁrm to take

a lump sum amount of x > 0 from the worker. In both cities, wage theft is illegal, and if

caught the ﬁrm must return x and pay penalty P

> 0 to the worker. Where the penalty

Examples of G include the exponential or a normal distribution truncated at 0

Because a searching ﬁrm has proﬁt zero, having ﬁrms die at the end of the match is equivalent to allowing

them to search for a new worker.

is speciﬁc to city j. Moreover, the match between the worker and ﬁrm is broken by the

enforcement agency.

Enforcement, however, is imperfect and varies across cities and across the types of work-

ers. In city 2, both types of workers are protected equally, while in city 1, low enforcement

of the equal right of minorities leads to a type workers being more carefully protected than d

type workers. Let ρ

denote the probability that wage theft against a type i worker is caught

in city j. As a baseline model, I consider a very stark enforcement regime. In particular set

= ρ

= 1.

Hence, all workers, except d types in city 1, are perfectly protected and therefore ﬁrms will

not attempt to underpay them. Moreover we’ll assume

x (r + δ)

π + (r + δ) (x + P

)

which ensures that ﬁrms in city 1 always take advantage of opportunities to underpay d

workers. This stark enforcement regime greatly eases the exposition and is without loss of

generality. Section 1.7 allows for more general enforcement regimes and shows that the main

results still hold.

Notice that d workers in city 1 are both eﬀectively paid less and lose their job more often.

Disadvantage in both these dimension is necessary for the main result. A more expansive

model could model these two dimensions separately, but I do it within one mechanism for

the sake of simplicity.

Let y

be the expected monetary gain from wage theft against a worker of type i in city

j. That is

= x



1 − ρ



− P

One can think of wage theft as reducing the wage by λy

, where λ is the arrival rate of

wage theft opportunities. In particular, at an oﬃcial wage of w, the eﬀective wage paid by a

ﬁrm engaging in wage theft is w −λy

. Additionally, wage theft increases the rate at which

matches end from δ to δ + λρ

1.4 Steady State Equilibrium

This section analyzes the worker’s and ﬁrm’s behavior in more detail as well as provide a

formal deﬁnition of steady state equilibrium. Section 1.4.1 specializes the analysis to pure

strategy equilibria. Sharper conclusions hold for such equilibria and they play a central role

in the proof of the main result. Throughout, superscripts will to refer to cities and subscripts

will refer to worker’s type. When it is clear a variable is referring to city 2, subscripts will

be dropped since city 2 treats both types of workers equally.

Each worker makes two choices: whether to accept a job and which city to live in. These

decisions depend on the random labor cost (c) and city preferences (η) respectively, both of

which are drawn from a continuous distribution. Hence, indiﬀerence occurs with probability

zero, and without loss of generality I treat the workers as employing pure strategies that

depend on c and η.

Firms choose what wage to post and whether to hire workers that they meet. Since ﬁrms

learns nothing between posting the wage and deciding whether to hire a worker, I treat them

as making both decisions simultaneously. Hence, upon entering the market, each ﬁrm chooses

a triplet (w, h

), where w denotes the wage and h

is the probability they hire a worker of

type i. All ﬁrms in the same city employ the same mixed strategy over (w, h

), denoted

by F

. In both cities I can restrict the ﬁrm’s choice of wage to the interval [0, π + λy

], since

at any wage above π + λy

the ﬁrm will always make a negative proﬁt. Let F be the set of

cumulative distribution functions on [0, π + λy

] ×[0, 1] × [0, 1], which implies that F

∈ F.

The ﬁrm also chooses whether to engage in wage theft. Denote by e

∈ [0, 1] the prob-

ability a ﬁrm underpays a worker of type i in city j when the opportunity arises. The

assumptions on ρ

ensure that

= e

= 0 and e

= 1.

There are also three general equilibrium variables to characterize: the ratio of searching

ﬁrms to searching workers in city j denoted by q

, the unemployment rate for a worker of

type i in city j denoted by u

, and lastly the proportion of city j’s unemployment pool made

up of type i workers denoted by m

I now examine the worker’s problem in more detail. Let U

be the lifetime discounted

expected payoﬀ of an unemployed worker of type i in city j. Similarly let W

(w, c) be the

discounted lifetime value of an employed worker of a type i worker in city j who is being paid

wage w and has realized labor cost c. Both U

and W

(w, c) are net of the city preference

η. The value of W

(w, c) is characterized by the following continuous time value function.

(w, c) = w −λe

− c +



δ + λe



− W

(w, c)



(1.3)

The worker’s payoﬀ ﬂow while employed is w − λe

− c, which is the eﬀective wage

minus the cost of working. The worker becomes unemployed at rate δ + λe

, either due to

an exogenous break or because the ﬁrm is caught underpaying. And U

−W

(w, c) captures

the utility loss when the worker loses his job.

Now imagine a worker deciding whether to accept a wage oﬀer w, and who draws labor

cost c for the job in question. The worker accepts the oﬀer if and only if W

(w, c) ≥ U

Hence, the equation 1.3 above generates a cutoﬀ c

(w) such that a worker of type i in city

j will accept the oﬀer if and only if c is below c

(w). And c

(w) obeys

(w) = w −λe

− rU

. (1.4)

Using c

(w), we characterize U

as follows:

= α







(w)





(w, c) |c ≤ c

(w)



− U



(w, h

, h

) . (1.5)

Since searching is free to workers, there is no ﬂow utility term in the above equation. Workers

encounters ﬁrms at rate α

), and that ﬁrm will have posted wage w and will use hiring

rates h

, h

according to the mixed strategy F

. For a ﬁxed realization of (w, h

, h

), the

joint probability that the ﬁrm hires the worker and the worker accepts is h



(w)



. Once

hired the term E



(w, c) |c ≤



(w)



− U

is the expected value of employment to the

worker over being unemployed. The value of employment is conditional on the labor cost

being low enough so that the worker will accept the job.

When accounting for the preference shock η, an unemployed worker in city 2 has a lifetime

value of rU

+ γη. Recall that γ is the parameter that measures the importance of the city

preference shock. The preference shock only applies while living in city 2, hence the lifetime

utility of an unemployed worker in city 1 is still just rU

. So an unemployed worker will

choose to reside in city 1 if and only if η is below η

where

γη

= rU

− rU

. (1.6)

And, in steady state, employed workers do not change cities because they lose their job when

moving. Since all workers become unemployed eventually, where workers live in steady state

is determined by the cutoﬀ rule η

. The fraction of i workers living in city 1 is Γ (η

), and

the mass of such workers is s

Γ (η

). Hence the outcome of the competition between cities

for workers depends entirely on η

, which depends entirely on how (U

, U

) compares with

Now consider the ﬁrm’s problem. Let V

be the discounted lifetime value of a searching

ﬁrm in city j. Let J

(w) be the discounted lifetime value of a city j ﬁrm matched with

worker of type i at wage w. J

(w) is characterized by the following continuous time value

function:

(w) = π − w + λe



δ + λe



− J

(w)



The ﬁrm receives a ﬂow proﬁt of π minus the eﬀective wage w − λe

. The ﬁrm loses its

employee with probability δ+λe

, which causes the ﬁrm to experience a loss of V

−J

(w).

Since free entry requires that searching ﬁrms make zero proﬁt, set V

= 0 and derive:

(w) =

π − w + λe

r + δ + λe

I now analyze the ﬁrm’s choice of the wage w and when to hire a worker. The ﬁrm’s

optimal decision depends on the composition of the unemployment pool given by m

, and

solves the following continuous time bellman equation:

= max

w,h







−k +

i∈{a,d}

G(c

(w))(J

(w) − V

)







. (1.7)

This equation characterizes the discounted lifetime value of a searching ﬁrm, which is

the left-hand side of the equation. The searching ﬁrm pays ﬂow cost k and meets a worker

of type i at rate m

). For a given (w, h

, h

), h

G(c

(w)) is the joint probability that

the ﬁrm hires the worker and the worker accepts. And the value to the ﬁrm of employing a

worker over searching is J

(w) −V

. Since V

= 0, and since the ﬁrm takes q

as given, any

w, h

, h

that solves the above maximization problem also solves:

= max

w,h

i∈{a,d}



(w)



(w) . (1.8)

From this maximization problem one can see that for any ﬁxed wage w, ﬁrms set h

= 1

whenever J

(w) > 0, and sets h

= 0 when J

(w) < 0. When J

(w) = 0 the ﬁrm is

indiﬀerent and can hire at any probability between 0 and 1. The expression maximized in

the above equation is diﬀerentiable except when c

(w) = 0, since the cdf G has support

[0, ∞] creating a kink at zero. However, the pdf G

jumps up at zero which makes the proﬁt

function locally convex. Hence it is impossible for a wage at one of the kinks to be optimal

and it follows that, for any ﬁxed choice of h

, h

, the ﬁrm’s choice of wage w must obey the

following ﬁrst order condition.

i∈{a,d}



(w)



(w) −



(w)



r + δ + λe

= 0 (1.9)

Lastly I verify that the restriction on ρ

is suﬃcient to ensure that d types workers in

city 1 are always underpaid. A ﬁrm paying wage w beneﬁts from underpaying if and only if

π − w + λy

r + δ + λρ

≥

π − w

r + δ

The left-hand side of the inequality is J

(w) when setting e

= 1, while the right-hand side

is J

(w) when setting e

= 0. Rearranging the inequality and substituting out y

yields

≤

x (r + δ)

π − w + (r + δ) (x + P

)

And no worker will ever accept a negative wage so I only consider weakly positive wages.

And the right-hand side of the inequality is increasing in w, and at w = 0 is equal to the

upper bound on ρ

. Hence the assumption on ρ

ensures that the ﬁrms in city 1 set e

= 1.

Now I characterize the three market variables, q

, m

and u

. I can pin down q

using

the free entry condition by setting V

to zero in each city and rewriting equation 1.7 as:

0 = rV

= −k + α





, (1.10)

where M

is deﬁned in equation 1.8. Since α

(·) is strictly decreasing, equation 1.10 uniquely

pins down q

. To derive the unemployment rate u

, note that in steady state the number of

workers leaving employment must equal the number entering which gives that



1 − u



δ + λρ



= u







(w)



(w, h

, h

) .

The left-hand side of this equation is the rate at which workers of type i lose their job

multiplied by the employment rate, while the right-hand side of the equation gives the rate

at which workers gain a job multiplied by the unemployment rate. Rearranging the above

equation gives that:

δ + λe

+ α

)



(w)



(w, h

, h

)

. (1.11)

To derive the fraction of the unemployment pool made up of d types in city 1, m

, weight

the unemployment rates u

by how many workers of each type live in city 1, which is given

by s

Γ (η

) u

Γ (η

) u

+ s

Γ (η

) u

(1.12)

And of course m

is simply 1 −m

. The values of m

and m

can be similarly derived, but

since city 2 treats all workers equally it is unimportant. The city 2 steady state equations

can all be written without reference to m

or m

Notice that, given q

, U

and F

, one can uniquely derive all the other steady state

variables. The values for c

(w) , η

are pinned respectively by equations 1.4, 1.6, and

1.12. Hence, I can deﬁne a steady state equilibrium in the following way.

Deﬁnition 1.1. A steady state equilibrium consists of the variables, (q

, q

, U

), and

a pair of ﬁrm’s actions (F

, F

) ∈ F × F such that, having derived c

(w) , η

, m

from the

above equations:

1. Every (w, h

, h

) in the support of F

solves condition 1.8.

2. U

satisﬁes equation 1.5.

3. q

satisﬁes equation 1.10.

Proposition 1.1. For all parameters a steady state equilibrium exists.

Proof. See Appendix.

With the restrictions on ρ

, the equations that deﬁne steady state are continuous, and

since ﬁrms can employ mixed strategies their best response functions are convex valued.

Hence, existence can be proved with a standard ﬁxed point argument. In appendix section

A.4, I establish existence in the more general case without the assumptions on ρ

. In this

more general case, continuity is no longer assured. In particular, the worker’s labor cost

cutoﬀ c

(w) jumps discontinuously at the wage where the ﬁrm is indiﬀerent about engaging

in wage theft. I am able to restore continuity by giving ﬁrms the ability to cheap talk with

the workers. The cheap talk plays a role similar to public randomization. And then I show

that an equilibria in which the cheap talk is never used always exists, and so I can remove

the cheap talk while maintaining existence.

1.4.1 Pure Strategy Steady States

This section studies steady state equilibria where ﬁrms in the same city employ the same

pure strategy. In the proof of the main result, we are able to focus on a portion of the

parameter space where only these type of steady states exist. So in each city, all ﬁrms post

the same wage w

and make the same hiring decisions h

, h

. Therefore, a worker of type i

in city j will always use the same labor cost cutoﬀ c

, when choosing whether to accept a

job oﬀer. Combining equations 1.3 and 1.5 pins down U

while solving out W

) G





− λy

− E



c|c ≤ c



r + δ + λe

+ α

) G





(1.13)

Using the characterization of c

from equation 1.4 U

can be substituted out to get:

− λe

= c

)

r + δ + λρ



− c



dG (c) . (1.14)

The right-hand side of the above equation strictly increases in c

, which means c

can be

uniquely determined in terms of



, q

, h



. The ﬁrm’s ﬁrst order condition then becomes:

i∈{a,d}

r + δ + λe





π − w

+ λe



− G





= 0. (1.15)

The free entry condition can also be rewritten as:





i∈{a,d}





π − w

+ λe

r + δ + λe

− k = 0 (1.16)

The expression for the unemployment rate becomes simpler as well.

δ + e

λρ

) G





+ δ + e

λρ

. (1.17)

An important fact that will be used later is that for any city j, ﬁxing m

, h

, there ex-

ists a unique solution



, c

, w



to equations 1.14 through 1.16. And given



, c

, w



for both cities, one can uniquely determine the rest of the equilibrium variables including

city size. Uniqueness of the solution can then be proven in the following way. First ﬁx any

, h

, q

, w

, and notice that there is a unique c

that satisﬁes equation 1.14, since the

right-hand of that equation is strictly increasing in c

. Then ﬁx any m

, h

, q

, and let

be deﬁned as a function of w

, and it turns out that the left-hand side of the ﬁrm’s ﬁrst

order condition, equation 1.15, is decreasing in w

even taking into account how c

moves

with w

. This uniquely pins down w

and c

for any ﬁxed values of m

, h

, q

. Lastly, ﬁx

only m

, h

and make a similar argument on the free entry condition, equation 1.16.

1.5 Results

Now that I have formally deﬁned steady state equilibria, I turn to establishing the main claim

of this chapter, that the city that inadequately protects a disadvantaged minority gains an

advantage in attracting workers. The following proposition provides a formal statement of

this advantage.

Proposition 1.2. There exists a δ

> 0, such that for all δ < δ

there will be an open set

of enforcement regimes (ρ

, P

) such that in every steady state more workers live in city 1

than in city 2.

Proof. See Appendix.

Proposition 1.2 says that, when employment length is suﬃciently long, a city can gain

an advantage in attracting workers by inadequately protecting minorities. The process of

voting with one’s feet leads to workers voting in favor of inadequate protection of minorities.

Far from protecting disadvantaged minorities, mobility has the perverse eﬀect of promot-

ing discrimination against them. Notably, this result does not involve selecting particular

equilibrium since it applies to all steady states at the speciﬁed parameters. Also, while not

explicitly stated, the open set of (ρ

, P

) found in the proposition respects the inequality on

that ensured ﬁrms always engage in wage theft against d workers in city 1.

Proposition 1.2 chooses (ρ

, P

) so that the degree of wage theft is not too severe in city

1. In particular, y

, which measures the ﬁrm’s net gain from wage theft including penalties

paid when caught, cannot be too large. Consider that, when λy

is larger than π, only wages

higher than π gives d workers positive eﬀective wages. And at any wage above π, ﬁrms earn

negative proﬁt from a type workers, and hence never hire them. Therefore, at any wage,

either d workers will not accept it or a workers will never be hired. Hence the market will

segment, with some ﬁrms posting wages appropriate for a types, and some ﬁrms posting

wages appropriate for d types. And this segmentation prevents the mechanism underlying

the main result from working. However, for the correct choice of (ρ

, P

), the possibility of

being penalized for wage theft will ensure y

is small enough to prevent this segmentation

from occurring. It should be noted that segmentation can occur when λy

is less than π,

and bounds on y

suﬃcient to rule it out are found in the appendix.

To gain an intuition on how city 1 attracts more workers, consider what happens as δ goes

to 0. In this limit, the a workers never lose their job and have a negligible unemployment

rate. The d workers, on the other hand, still lose their job whenever the ﬁrm is caught

underpaying. Hence, the unemployment pool in city 1 is made up entirely of d type workers.

Entering ﬁrms in city 1 only match with d type workers, and they know that any posted wage

w will result in an eﬀective wage of w −λy

. So one can think of the ﬁrms as relabeling the

wages and acting as if there is no wage theft. In other words, if w

would have been the wage

without wage theft, then the ﬁrms will post wage w

+ λy

instead. Hence, d type workers

earn the same eﬀective wage in both cities, which induces roughly half of them to move to

each city. On the other hand, a type workers in city 1 will earn the full w

+ λy

, inducing

a majority of them to live there. Therefore, city 1 will have more workers overall. The one

point the above argument glosses over is that enforcement of wage theft causes matches to

to end at rate δ + λρ

instead of rate δ for d type workers. However, the proposition allows

us to choose ρ

, and ρ

is chosen to be small enough to not interfere with the above analysis.

To complement the intuition given above I now sketch some of the more technical details.

The proof proceeds in three main steps. First, focus on pure strategy equilibria where ﬁrms

hire all willing workers and consider the limit as δ goes to zero. I will show that, in that

limit, city 1 is always larger. I show next that, for the correct choice of enforcement regime

(ρ

, P

), as δ goes to zero all equilibria will indeed be in pure strategy equilibria where

ﬁrms hire all willing workers. I then ﬁnish the proof by applying an upper hemicontinuity

argument to prove that city 1 must be larger for small, but strictly positive, values of δ.

To study the limit case as δ goes to 0 focus on equations 1.14 through 1.16, those equations

apply to pure strategy steady states and are well deﬁned at δ = 0. Recall that these equations

uniquely characterize



, q

, w



for any ﬁxed values of h

and m

. Now set h

= 1, since

we are focusing on steady states where ﬁrms hire all willing workers. Set m

= 1, since, at

the δ = 0 limit, in city 1 a workers never lose their job while d workers still do. And the

solution to equations 1.14 through 1.16 does not depend on m

, so it can be set to anything.

With h

= m

= 1 there is a unique solution for



, q

, w



, which in turn determines

and city sizes. It turns out there exists a ρ

> 0 such that city 1 will have more workers

as long as ρ

< ρ

. And from here on I will only consider ρ

< ρ

To ensure that all steady states are pure strategy where ﬁrms hire all workers, note

that these properties always hold in city 2. In city 2 all workers are equally proﬁtable to

ﬁrms, which implies all workers must give strictly positive proﬁts so that ﬁrms can recover

their search costs. Hence ﬁrms must hire all willing workers. Similarly, since all workers

act identically in city 2, ﬁrms will only oﬀer wages that every workers accepts with positive

probability, which avoids the kink that occurs when c

(w) = 0. Without that kink the

ﬁrm’s proﬁt function is strictly concave and all ﬁrms must oﬀer the same wage. And these

arguments can be extended to city 1 as long as y

is not too large and δ is small enough.

And for any ρ

, I can choose P

to achieve any y

. And since y

is decreasing in P

, the fact

that we are making y

small mean the condition that P

≥ 0 will not be violated.

The last step is to relate the δ = 0 limit to what happens at small positive values of

δ. The equations 1.14 through 1.16 used to study the δ = 0 limit are continuous in both

the equilibrium variables and the parameter δ. Hence, the set of solutions to that system

will change upper hemicontinuously as one moves δ, and city sizes must converge upper

hemicontinuously to the δ = 0 limit. And it must follow that city 1 has more workers when

δ is suﬃciently small, which ﬁnishes the proof.

I now examine how much larger city 1 can be compared to city 2. The intensity of the

city preference, captured by parameter γ, determines how large a size diﬀerential is possible.

For example, as γ goes to inﬁnity workers care only about their city preference, and each

city must have exactly half of each type of workers. I instead study the case as γ goes to

zero, and see how much larger city 1 can be compared to city 2, for the correct choice of

(ρ

, P

) and δ.

Proposition 1.3. For any ε > 0, there exists a γ

> 0 such that for all γ < γ

there will

exist an open set of the parameters (P

, ρ

, δ) such that in every steady state Γ (η

) ≥

− ε

and Γ (η

) ≥ 1 −ε.

Proof. See Appendix.

Recall that Γ (η

) is the fraction of type i workers that live in city 1. This result says

that, in the most extreme case, city 1 will have all of the a workers and half of the d workers.

To gain an intuition for this result, recall that I argued above that, in the δ = 0 limit, d type

workers earn close to the same wage in both cities while a workers earn strictly more in city

1. And as γ goes to zero, workers care only about their earnings, and so d workers will split

themselves evenly between the cities, while all a workers will choose to live in city 1.

Besides demonstrating how much better city 1 can do at attracting workers, proposition

1.3 identiﬁes a somewhat counter-intuitive mechanism for segregation. One might reasonably

expect that low enforcement would drive away the d workers, and hence with low γ, city 1

could only attract one type of worker while city 2 could attract both types. Proposition 1.3

turns this on its head and says that only city 1 can attract both types of workers. Instead

of inadequate protection of minorities chasing away d workers, it is strong protection of

minorities that chases away a workers. The net result is that a city that protects its workers

equally ends up being segregated, while the city where a minority is inadequately protected

attracts a broader cross section of workers.

One concern raised by proposition 1.2 is it may involve unrealistic unemployment rates.

In particular, the unemployment rates may be very low due to the requirement of small δ.

Another objection is that, given the intuition of the δ = 0 case where the unemployment

pool in city 1 is entirely d workers, the ratio of the unemployment rates may need to be

unrealistically skewed. However, the main result is robust enough to address both of these

concerns. In particular one can match any single unemployment rate, or any ratio of unem-

ployment rates in city 1, provided I choose the functional forms and parameters correctly.

This is captured by the following result.

Proposition 1.4. The following two things hold:

1. Fix any ε > 0 and u

∈ (0, 1) and ﬁx i, j with associated unemployment rate u

. Then

there exists functional forms for m (·, ·) , Γ, G and parameters, such that in every steady

state city 1 has more workers than city 2, and u

∈ (u

− ε, u

+ ε).

2. For any ε > 0 and z > 1 there exists functional forms for m (·, ·) , Γ, G and pa-

rameters, such that in every steady state city 1 has more workers than city 2, and

∈ (z −ε, z + ε).

Proof. See Appendix

To prove this result I employ the simplest functional forms. In particular set the matching

function to be

m (µ, υ) = mµ,

which implies workers always meet ﬁrms at the constant rate m. And I set both Γ and

G to be uniform distributions. Technically the uniform distribution does not satisfy the

requirements that Γ has full support and G has support on [0, ∞]. However I can modify

the tails of Γ and G to have the correct supports while delivering the same steady states

as the uniform distribution. Consider that rU

will never be below zero or above π, which

allows us to impose the following bound:

|η

| ≤

And this bound implies that, while the mass that Γ puts below

−π

or above

matters, the

shape of Γ in those regions does not matter. Hence one can modify the uniform distribution

in order to have an inﬁnite tail, without changing any of the features of steady state. And

a similar argument can be made for G.

Given the functional forms I am able to solve the model more explicitly and can prove

proposition 1.4 using similar methods to the proof of proposition 1.2, except now I allow δ

to be arbitrary and vary r instead. As I take r to be large, workers and ﬁrms become very

impatient and the eventual possibility of losing a job seems increasingly distant. So from the

perspective of the agents, a large r and a small δ serve a similar function. However it is δ that

matters for the unemployment rate and not r. This allows us to match any unemployment

rate will still maintaining city 1 as larger.

1.6 Lump Sum Transfers

In order to highlight the necessity of the search elements of the model to the main result, I

consider a more direct form of discrimination. Suppose that, in lieu of allowing discrimination

by ﬁrms, city 1 directly transfers money from the minority workers to the majority workers

via a lump sum tax. I show that, under these transfers, city 1 will always be strictly

smaller than city 2, provided that Γ has an increasing hazard rate and a ﬁnite expectation.

Hence, I get the opposite eﬀect as proposition 1.2: mobility punishes discrimination against

a disadvantaged minority. Moreover, the conditions imposed on Γ are satisﬁed by many of

the usual distributions, such as the Normal and Logistic distributions.

Start by setting ρ

= 1 for all i, j, which ensures all workers are treated equally in both

cities. For simplicity set γ = 1, but any positive γ can be used while obtaining similar

results. Suppose that city 1 imposes a lump sum tax of t

> 0 on d workers and splits the

proceeds evenly among the a workers. The a workers receive a negative lump sum tax of

< 0, determined by the following budget constraint.

Γ (η

) t

+ s

Γ (η

) t

= 0 (1.18)

Since the transfers are lump sum, they only impact the worker’s decision of where to live.

The wages oﬀered by ﬁrms and the speed at which workers ﬁnd jobs will be the same in

both cities for all workers. Hence, in every steady state it must be that

+ t

= rU

+ t

= rU

Recall that U

is the determining factor in how workers make their location decision. And

using equation 1.6, one can see that a worker of type i chooses to live in city 1 if and only

if η ≤ η

where

= −t

and η

= −t

And the size of city 1 will be s

Γ (−t

) + s

Γ (−t

), which demonstrates that the search

elements are now irrelevant to city size. Workers make their location decision based solely

on the transfers they receive and their city preference shock.

Recall that Γ has the monotone hazard ratio when

(η)

1−Γ(η)

is weakly increasing. By

symmetry of Γ around zero, that is equivalent to saying that

Γ(η)

(η)

is weakly increasing.

This latter form of the monotone hazard ratio will be used in the proof of the following

proposition.

Proposition 1.5. If Γ has a monotone hazard ratio and a ﬁnite expectation, then for every

> 0 city 2 will have more workers than city 1.

Proof. It will suﬃce to prove that t

= t

= 0 is the transfer scheme that makes city 1 as

large as possible. Consider the following maximization problem:

max

)∈R

Γ (−t

) + s

Γ (−t

)

such that

Γ (−t

) t

+ s

Γ (−t

) t

= 0.

First I will rule out the possibility that this maximization problem tries to push t

to inﬁnity.

Since Γ is full support, the ﬁnite expectation assumption requires that t

(−t

) goes to zero

as t

goes to inﬁnity. Using the monotone hazard ratio we can conclude that t

Γ (−t

) also

goes to zero as t

goes to inﬁnity. Hence, sending t

to inﬁnity yields zero revenue and can’t

be optimal. So I can restrict t

to a compact set with the property that there is an interior

solution. Using the ﬁrst order conditions of the above maximization problem one can derive

that

− t

Γ (−t

)

(−t

)

−

Γ (−t

)

(−t

)

Now suppose that the optimal solution involves t

> 0. From the budget constraint it must

be that t

< 0, which makes the left-hand side of the above equation strictly negative.

However, −t

< 0 < −t

implies the right-hand side of the above equation is weakly positive

by the monotone hazard property. Hence t

> 0 is not a solution to the maximization

problem. And a similar argument rules out t

< 0. And so it must follow that t

= t

= 0 is

the unique transfer scheme that maximizes the size of city 1, which concludes the proof.

1.7 General Enforcement Regimes

In this section, I relax the restrictions we imposed on the enforcement regimes and show

the main result still holds. City 2 still treats all workers equally, and catches ﬁrms engaging

in wage theft at rate ρ

, and levies penalty P

. City 1 catches ﬁrms at rates ρ

and ρ

depending on whether the ﬁrm is underpaying an a worker or a d worker respectively. Firms

that are caught in city 1 pay a penalty P

that is independent of the worker’s type. I show

that, as long δ is suﬃciently small, there is an enforcement regime (ρ

, ρ

, P

) that ensures

city 1 is always larger, regardless of the enforcement regime (ρ

, P

) used by city 2. The

intuition for this result is that allowing wage theft by ﬁrms only beneﬁts the city if workers

are treated unequally. Hence, city 2 attracts the most workers under perfect enforcement,

where ρ

equals 1, which is precisely the case considered in proposition 1.2.

Existence of steady state is more diﬃcult to establish with general enforcement regimes.

In particular discontinuities will appear in the labor cost cutoﬀ used by the workers. In

section 1.7.1, I will show how to restore continuity by granting the ﬁrms the ability to cheap

talk with the workers, and then showing the cheap talk is actually unnecessary for the steady

state. I now state the main result of this section.

Proposition 1.6. There exists δ

> 0, such that for all δ < δ

there exists an open set of

city 1 enforcement regimes (ρ

, ρ

, P

) such that, for any enforcement regime (ρ

, P

) used

in city 2, in every steady state city 1 is larger than city 2.

Proof. See Appendix.

Proposition 1.7 replicates ther main result using general enforcement regimes. The result

says that, with the right enforcement regime and with δ small enough, city 1 is larger than

city 2 no matter what enforcement regimes city 2 uses. The key step, a proof of which is

found in the appendix, is showing that ρ

= 1 maximizes U

. Since city size is directly tied

to U

, ρ

= 1 makes city 2 as large as possible. So simply choose (ρ

, ρ

, P

) in the same way

as in the proof of proposition 1.2, and since city 1 must be larger than city 2 when ρ

= 1,

it must be larger under all other enforcement regimes.

The intuition for why ρ

= 1 maximizes U

is as follows: when all workers are treated

equally, wage theft serves only to shorten employment length. Firms will underpay all

workers equally, hence they perfectly price this into the original wage. So the only net eﬀect

of wage theft is that occasionally the enforcement agency catches the ﬁrm and breaks the

match.

1.7.1 Existence with general enforcement regimesI

I now establish existence of a steady state for arbitrary enforcement regimes. First notice

that whether a ﬁrm will want to engage in wage theft will depend on the wage they are

paying. There will be a cutoﬀ wage ω

, above which a city j ﬁrm employing a type i worker

will always engage in wage theft, and below which they will never engage in wage theft. At

wage ω

they will be indiﬀerent. The value of ω

is given by

= π −

(r + δ) y

The value of ω

depends only on exogenous parameters because free entry ensures that the

ﬁrm’s payoﬀ from losing the match is zero. Let e

(w) ∈ [0, 1] be the probability that a

city j ﬁrm engages in wage theft against a type i worker each time the opportunity arises.

Optimality requires that e

(w) equals 1 for w > ω

, and e

(w) equals 0 for w < ω

. Hence,

at the wage of ω

, the labor cost cutoﬀ will experience a discontinuity for worker of type i

in city j. To see this recall that the labor cutoﬀ is

(w) = w −λe

(w) y

− U

No matter what value e





takes, there will be a discontinuity in c (·) at ω

. Hence, one

can no longer straightforwardly apply a ﬁxed point theorem to prove existence. This is an

example of the type of existence problem that arises in dynamic games with continuous

action spaces, as discussed in Harris (1985); Harris et al. (1995). Harris et al. (1995) shows

that in multi-stage games public randomization ensures existence. However the model is not

a multi-stage game so their results cannot be directly applied. Instead I allow the ﬁrms to

engage in cheap talk with the workers, which plays a similar role to public randomization

and is very much in the spirit of Harris et al. (1995). I will also show that the equilibrium

that arises with cheap can be interpreted as equilibrium without cheap talk. Hence cheap

talk only serves as a proof technique and an equilibrium exists in the original model without

cheap talk.

To add cheap talk, I expand the model so that when a ﬁrm meets a worker, they are able

to engage to say how often they will underpay the worker, i.e. how they will set e

(w). At

any wage other than ω

, this talk is purely cheap and the worker will ignore it. However,

when oﬀering wage ω

, the ﬁrm will be indiﬀerent about wage theft, which allows cheap talk

to be credible. Hence I can restrict attention to equilibria in which ﬁrms commit to e

when

posting wage ω

, which will restore continuity. I will show how this works in city 2, and a

similar argument for city 1 can be found in the appendix.

In city 2, change the action space of ﬁrms so that, instead of a choosing a wage, they

choose z ∈ [0, ¯z

], where ¯z

= π + λy

+ 1. One should interpret z ∈ [0, ω

) as oﬀering wage

z, and never engaging in wage theft since w < ω

. In the range z ∈ [ω

, ω

+ 1] the ﬁrm

oﬀers wage ω

, and credibly commits to engage in wage theft with probability z − ω

each

time the opportunity arises. Finally, in the range z ∈ (ω

+ 1, ¯z

] the ﬁrm is oﬀering wage

z −1 and will always engage in wage theft. I can then deﬁne continuous functions ˜w

(z) and

˜e

(z), which give the implied wage and wage theft probability as functions of z. Writing out

˜w

(z) explicitly gives that:

˜w

(z) =

(

z if z ∈ [0, ω

]

if z ∈ [ω

, ω

+ 1]

z −1 if z ∈ [ω

+ 1, ¯z

] .

And deﬁning ˜e

(z) explicitly gives:

˜e

(z) =

(

0 if z ∈ [0, ω

]

z −ω

if z ∈ [ω

, ω

+ 1]

1 if z ∈ [ω

+ 1, ¯z

] .

The key observation is that ˜e

and ˜w

are both continuous in z. The worker’s labor cost

cutoﬀ can be written as

(z) = ˜w

(z) − λy

˜e

(z) − rU

which is a continuous function of z. In a similar way I can rewrite all of the relevant steady

state conditions in terms of ˜w

(z) and ˜e

(z) and they will all be continuous. This allows

the use of a ﬁxed point theorem to prove that a steady state with cheap talk exists.I

I now show that the cheap talk is not needed. At wage ω

, ﬁrms are only indiﬀerent about

engaging in wage theft once the worker has accepted the job. Before the worker accepts,

ﬁrms will always want to promise not to underpay in order to increase the probability of

acceptance. Therefore, optimality will require that any ﬁrm that posts wage ω

will always

use the cheap talk to set e

= 0. Hence, cheap talk can taken away and it will be simply a

feature of equilibrium that ﬁrms never engage in wage theft at wage ω

1.8 Conclusion

In this chapter I analyzed the interaction of mobility and discrimination in the labor market.

My main result was that inadequate protection of a disadvantaged minority has the potential

to give a city an advantage in attracting workers. When policy makers care about long term

city growth, this creates an incentive for low enforcement of the equal protection of minorities.

Even when policy makers do not respond to these incentives, the growth of a city leads to a

larger number of the disadvantaged minority being adversely impacted. The result required

a tacit form of discrimination of the minority workers, where the ﬁrms could deprive minority

workers of their wages, but are not allowed to explicitly advertise a minority speciﬁc wage.

This chapter contributes to the discussion on why economically exploited minorities arise

and why they persist.

Chapter 2

Losing to Win: How Partisan

Candidates Help Parties Win Future

Elections

2.1 Introduction

In the 1964 U.S. presidential election, Republican nominee Barry Goldwater was widely

considered too ideologically extreme to be elected. The New York Times said Goldwater

threatened to turn the GOP into a ”right-wing splinter group” and had ”minimal” chance of

winning

. As predicted, Goldwater lost in a landslide with less than 40% of the vote, winning

only six states. However, conservatives now argue that Goldwater’s nomination energized

the base and gave the party a momentum that led to future victories (Middendorf (2006)

p.xii; Will (1994)). Goldwater himself writes in his autobiography that he never believed he

had a chance of winning, yet decided his candidacy could rally the base and give voice to

the conservative cause (Goldwater (1988), p. 154).

”The Goldwater Nomination”, The New York Times, July 16, 1964, p. 30.

Goldwater’s case exempliﬁes the thesis of this work: partisan candidates are less elec-

torally viable but energize the base, helping to win future elections. This creates a trade-oﬀ

between winning today and winning tomorrow. The goal of this chapter is to formalize this

trade-oﬀ and explore its consequence. I build an inﬁnite horizon model of two party com-

petition, where in each period both parties nominate a citizen candidate to represent them

in an election. In the unique subgame perfect equilibrium moderate candidates always win,

and yet partisan candidates are nominated to help the party win future elections.

The model has two key ingredients: that partisan candidates energize the base and that

this lasts until the next election. Energizing the base potentially has many dimensions, but

for the purposes of this work I speciﬁcally mean increasing turnout. Both of these ingredients

ﬁnd strong support in the empirical literature. Partisan candidates increasing turnout among

their base ﬁnds support in the literature on ”abstention due to alienation”. The idea that

increasing turnout today impacts turnout tomorrow is found in the literature on ”habitual

voting”. Section 2.2 further discusses the evidence underlying these assumptions.

The most striking feature of the equilibrium is that parties cycle in and out of power in a

regular and predictable fashion, a pattern which has strong empirical support. This cycling

is a natural consequence of the core trade-oﬀ of the model. In order to win, a party has

to nominate a moderate, but in doing so it neglects its base and sets itself up for a weaker

future. Thus the longer a party is in power the weaker and more likely to lose it becomes.

This prediction has been strongly empirically veriﬁed and given various names: the ”cost

of ruling” (Paldam (1986); Lewis-Beck and Nadeau (2004)), ”the time for a change eﬀect”

(Abramowitz (1988, 1996); Abramowitz and Saunders (2008)), the ”electoral pendulum”

(Norpoth (1996, 2004)), and ”realignment theory” (Key (1959); Merrill et al. (2008)).

This chapter provides a new explanation for the observed cycling and provides an equilib-

rium model that produces this behavior. The most common explanation given is the natural

variation in the business cycle. However, the papers cited above include economic controls

and still conclude that there is a separate penalty for being in power. Other explanations

given for the cycling are over-reach by the party in power and an innate desire for novelty

by voters. In this chapter I give a novel reason: the tension between nominating a viable

candidate or one that will energize the base. Moreover, the previous literature has focused

on estimating this phenomenon and providing econometric models. Since parties will manip-

ulate and respond to any electoral force of importance, a strategic model is needed, which I

provide in this chapter.

The model also predicts cycles in two other areas: ideology of nominees and turnout. In

equilibrium, the parties oscillate between partisan and moderate candidates. In particular, a

party that just lost power will be more ideologically extreme and gradually move to the center

over time. This oscillation will create a companion cycle in the turnout of the ideological

base of that party. The longer a party remains in power the less energized its base becomes,

and turnout falls. Once the party leaves power it focuses on energizing its base and turnout

rises again. These novel and testable predictions highlight potentially interesting empirical

relationships produced by the model. As far as I know, little to no empirical work has been

done on these predictions, and they present an interesting direction for future work.

As remarked above, the inﬁnite horizon model presented here will have a unique subgame

perfect equilibrium. This is important, since it allows for the distinct testable predictions

that I emphasized as the key results. Uniqueness holds even as the discount rate goes to one,

which is striking considering the model belongs to a class games for which a folk theorem

has been proved (Dutta (1995)). This can occur because there is only a single set of payoﬀs

that are both feasible and above the appropriately deﬁned min-max. The intuition behind

this and a sketch of the proof will be discussed in more detail in section 2.4.

The rest of the chapter is organized as follows. Section 2.1.1 brieﬂy reviews the formal

literature on candidate selection. Section 2.2 motivates and discusses the key mechanism of

the model. Section 2.3 formally describes the model. Section 2.4 characterizes the equilib-

rium and discusses its key features and predictions. Section 2.5 checks robustness by looking

at purely oﬃce motivated parties with varying levels of patience. Section 2.5 concludes.

2.1.1 Related Literature

The formal literature on candidate selection began with the insights of the median voter

theorem, which goes back at least as far as Downs (1957). The median voter theorem

delivers two distinct testable implications: moderate candidates perform better in election

and both parties should converge to the same position. The empirical evidence strongly

supports the ﬁrst of these claims and strongly disagrees with the second (see Ansolabehere

et al. (2001) and Erikson and Wright (2005)). The subsequent literature has worked on

modifying the model to explain divergence. To do so it must address the following question:

if moderates perform better in elections why are partisans candidates ever nominated?

The most direct way to explain divergence is assuming that parties have an intrinsic

preference for partisan candidates, i.e., a policy motive. This tack was ﬁrst taken by Wittman

(1983) and later developed by Calvert (1985) and Roemer (1994). This works if the basic

Downsian model is modiﬁed to include uncertainty. Another tack, taken by Palfrey (1984),

argues that the possibility of entry by a third party cause divergence. Adams and Merrill

III (2003) argue that partisan candidates will beat moderates by increasing turnout in the

base which compensates for votes lost from the center. However, this explains divergence at

the cost of contradicting the evidence that moderate candidates do better in elections. And

while the focus on turnout resembles the argument in this chapter, they conclude partisan

candidates win in the present while I conclude they help parties win in the future.

The key innovation found in this chapter is the use of a multi period setting to address

candidate selection. All of the models discussed above are one period, and in a one period

model parties can only care about two things: winning today and policy today. And each of

the above models gives one of those two reason as to why partisan candidates are nominated.

Because my model is multi-period parties can care about something new: winning and policy

in the future. This allows for the central story of this chapter: partisans energize the base

which helps parties win in the future. A multi-period model also has the advantage that it can

explain inter-temporal patterns in elections. This allows my model to explore phenomenon

such as the electoral cycling highlighted in the introduction.

2.2 Motivating the Model: Evidence and Theory

In this section I discuss and motivate the two key ingredients of the model: that partisan

candidates increase the turnout of the base and that this eﬀect lasts until the next election.

The ﬁrst ingredient links to the literature on ”abstention due to alienation” and the second

to ”habitual voting”. These are discussed in sections 2.2.1 and 2.2.2 respectively. Crucially

these literatures connect in that people who did not vote previously are more likely to have

their turnout decision aﬀected by the ideology of the candidate. In other words ”habitual

voters” are less likely to ”abstain due to alienation”. This connection leads to the key

mechanism of the model: that partisan candidates can inﬂuence whether voters acquire the

habit of voting. This mechanism is fully discussed in 2.2.3.

2.2.1 Abstention due to Alienation

Zipp (1985) examined how the ideological position of the candidates relative to the voter

inﬂuences turnout. Zipp scored the ideological position of each individual and each candidate

on a range of policy issues. He found that individuals who were far from either candidate

voted less, an eﬀect referred to as ”abstention due to alienation”. Of course the ﬂip side

of this eﬀect is that people who are ideologically close to at least one candidate vote more.

This captures the idea that a partisan candidate is capable of energizing the base.

More recently Plane and Gershtenson (2004) and Adams et al. (2006) have conducted

similar studies and also found a signiﬁcant ”abstention due to alienation” eﬀect. Adams et

al. assumes the eﬀect works through a threshold, that is an individual votes only if the closest

candidate is close enough. Using maximum likelihood they estimated the determinants of

this threshold. They ﬁnd that having voted in the past election signiﬁcantly reduces how

close the candidate has to be. In other words non-voters are more likely to require a candidate

with similar ideology to decide to vote.

2.2.2 Habitual Voting

Plutzer (2002) notes: ”[v]irtually all major works on turnout have concluded that voting be-

havior is, in part, a gradually acquired habit”. The observation that underlies this conclusion

is that voting is a highly stable behavior. This can be observed from both cross-sectional and

panel data. Miller and Shanks (1996) look at cross-sectional cohort data and ﬁnds cohort

turnout rises for the ﬁrst two or three election before hitting a long ”turnout plateau” and

then tailing oﬀ at old age. Similarly Plutzer (2002) looks at panel data from U.S. presi-

dential elections and notes that of the 516 respondents who voted in both 1968 and 1972

97% voted again in 1976. More generally numerous studies show previous voting is a robust

predicator of current voting in multivariate analysis (e.g. Franklin (2004); Brody (1977)).

The interpretation given to these results is that the population can be divided into two

groups: voters and non-voters. During young adulthood individual citizens sorts themselves

into these groups and, once settled, are high unlikely to shift from one to the other.

For the purposes of this work what matters is that there is a causal link between vot-

ing today and voting tomorrow. That such a link exists has been conﬁrmed by both the

instrumental variable and experimental approach. Green and Shachar (2000) instrumented

for past voting using aggregate variables from the past elections known to inﬂuence voting,

such as perceived closeness of the election or the ideological gap between candidates. They

concluded that past voting causes current voting. Gerber et al. (2003) conducted an ex-

periment where the treatment groups was urged to vote through direct mail or face to face

canvassing. The treatment was more likely to vote in the current election and this diﬀerence

persisted in subsequent elections. In fact those who voted in the treatment group became

indistinguishable from those who voted in the control group. These two studies point to

something intrinsic to voting that makes future voting more likely.

2.2.3 The Mechanism

The most common explanation given for habitual voting is that voting involves a one time

ﬁxed cost. Plutzer (2002) puts it as follows:

As young citizens confront their ﬁrst election, all of the costs of voting are magni-

ﬁed: they have never gone through the process of registration, may not know the

location of their polling place, and may not have yet developed an understanding

of party diﬀerences and key issues. Moreover, their peer group consists almost

entirely of other nonvoters: their friends cannot assure them that voting has been

easy, enjoyable, or satisfying.

Of course there is variation, some people enjoy engaging in the political process, even in

their ﬁrst election. The crucial point is, all else equal, having experience with voting makes

it easier, and hence more likely.

This explanation of habitual voting connects in a crucial way with the evidence mentioned

above that non-voters have a much higher ”abstention due to alienation” threshold. In other

words those who still face the ﬁxed cost to voting are more likely to require a candidate with

a similar ideology to be convinced to vote. This evidence isn’t perfect in that non-voters

are deﬁned as someone who didn’t vote last period, not someone who has never voted.

Nevertheless this highly suggests that a partisan candidate brings ﬁrst time voters from the

political base to the polls, overcoming the initial cost to voting, and pushing them onto the

path of habitual voting. More habitual voters in the base leads to higher turnout in the

future, enhancing the party’s future election prospects. This is the key mechanism by which

partisan candidates help parties win in the future.

2.3 The Model

2.3.1 Basics

I start with an overlapping generations model with citizens who live for 2 periods and discount

the future at β ∈ (0, 1). Each generation contains a continuum of citizens of measure .5. Each

period there is an election in which each citizen has the right, but not the obligation, to vote.

Each generation of citizens is split into three equally sized groups based on ideology {l, m, r}

The l and r groups are called partisan citizens, and form the base of two inﬁnitely lived

parties L and R. Type m citizens don’t have a party and are referred to as moderate citizens.

Each period the parties nominate a citizen candidate from one of the three groups to represent

it in the election. For simplicity I suppose the parties will not choose a candidate from the

other party’s base. Thus parties make a binary choice between a moderate candidate or

a partisan candidate from its own base. In order to avoid mixed strategy equilibrium, the

party that won last period has to choose its candidate ﬁrst.

The candidate with a larger measure of votes wins the election. If the party that won last

period decides to keep the same candidate, then that candidate is known as an incumbent.

In the event of a tie, incumbent candidates win with probability p ∈ (.5, 1). If there is no

incumbent then ties are decided by a 50-50 coin ﬂip.

The parties are inﬁnitely lived with discount rate β and are both policy and oﬃce mo-

tivated. A party receives payoﬀ γ > 0 for winning an election and 0 for losing. The policy

payoﬀ of a party is 0 if someone from its base wins, −.25 if a moderate wins and −1 if some-

one from the opposite base wins. This is consistent with quadratic loss in a spatial model.

To write this out formally, in any period t let w

, y

be the party and ideology, respectively,

The assumption that all three groups are the same size is not essential. It is important that the l and r

groups are the same size.

of the winning candidate. Thus w

∈ {L, R} and y

∈ {l, m, r}. Party L maximizes:

∞

t=0

) + u

))

where v

(L) = γ > 0, v

(R) = 0 and u

(l) = 0, u

(m) = −.25, u

(r) = −1. R has

symmetric preferences.

2.3.2 The Decision to Vote

I follow Riker and Ordeshook (1968) and assume people vote because they gain consumption

value from voting. This could take many forms: voters could enjoy expressing their opinion,

fulﬁlling their moral or civic duty (as in Feddersen and Sandroni (2006)) or feel social pressure

to vote. The alternative approach would be to assume people vote because they might change

the outcome. This view is implausible in a large election and becomes even more implausible

if there is a cost to voting. And since the motivation of this model, discussed in section 2.2,

involves a ﬁxed cost of voting, the pivotal voter view is an unpalatable choice.

Citizens choose whether to vote and who to vote for after observing which candidates

have been nominated. In each period, voting yields utility of D > 0 which is net of costs

such as time spent driving to the polling place, waiting in line etc. A citizen i, when voting

for the ﬁrst time, face an additional cost of α

. Candidates energize the ideological group

they come from and reduce the ﬁrst time cost by δ > 0

I restrict α

to two values α

< α

. These two types are evenly distributed throughout

the three ideological groups and there is strictly positive mass of both. The a

types ﬁnd

engaging with the political process relatively painless (or even enjoyable) and always vote in

both periods of their life. For simplicity set α

= 0. The α

types need to be energized or

else they will ”abstain due to alienation” and fail to gain the habit of voting. Speciﬁcally,

if a candidate from their own ideological group is nominated in their ﬁrst period of life,

Alternatively the citizen utility’s from voting could increase by δ when energized.

then they become habitual voters and will always vote. In all other cases they never vote.

Formally, this is given by (1 + β)D < α

< δ + (1 + β)D.

Conditional on voting, all citizens vote their sincere preference. Every citizen’s ﬁrst choice

is a candidate with the same ideology. Partisan citizens (l and r types) prefer a moderate

to a candidate with the opposite ideology. And moderate citizens are indiﬀerent between l

and r candidates. When faced with two moderate candidates, partisan citizens vote with

their own party. Moderates break indiﬀerence on ideology using 50-50 randomization. These

preferences lead to the following result.

Lemma 2.1. In every election the following statements hold:

1. A moderate candidate will always defeat a partisan candidate.

2. Suppose both parties choose moderate candidates. Then if party i chose a partisan

candidate last period and party j chose a moderate last period than party i’s candidate

will win. If both parties chose a partisan candidate last period, or neither did, then the

election is a tie.

Proof. In appendix.

Lemma 2.1 formalizes the central logic of this work: that partisan candidates aren’t

as electorally viable as moderate candidates but help parties win in the future. Part 1

says a partisan can never defeat a moderate, which makes speciﬁc how they are electorally

unviable. Part 2 explains what happens if both parties pick a moderate. In that case,

moderate citizens will split their vote evenly and partisan citizen will vote with their own

party. Thus the election is won by the party with higher turnout from their base, which in

turn is determined by which party energized their base last period. In this way, partisan

candidates help parties win future election by drawing in new voters and raising future

turnout. In the next section, I study how these forces play out in the strategic interaction

of the two parties.

2.4 Equilibrium

In this section I analyze the game described above using the solution concept of subgame

perfect equilibrium. The model above will have a unique subgame perfect equilibrium, which

allows for sharp predictions. This section is split into three parts. Section 2.4.1 discusses

two key assumptions. Section 2.4.2 presents the unique equilibrium in detail and gives a

discussion of its important features. Section 2.4.3 provides a sketch of the proof.

2.4.1 Assumptions

Throughout this section the following two assumptions are maintained

Assumption 1 : γ <

1 + β

2β(1 −β)

; Assumption 2 : β >

1 − p

The ﬁrst assumption says the parties can’t be too oﬃce motivated and the second says

that they can’t be too impatient. If the parties are patient enough (β → 1), both conditions

are always satisﬁed. Assumption 1 gives the allowed level of oﬃce motivation as a function

of the patience of the parties and always holds for β is close enough to 1 or close enough to

0. Moreover any γ ≤ 2.4 satisﬁes assumption 1 no matter the level of β. Assumption 2 says

the parties have to be patient compared to the advantage incumbents get in the case of ties.

As p → 1 any level of β works, as p → .5 then β needs to approach 1.

Assumption 1 is not essential; in section 2.5 I show purely oﬃce motivated parties still

retain the central characteristics of the equilibrium. The main diﬀerence with purely oﬃce

motivated parties is partisan candidates manage to win in equilibrium. Counterintuitively

this implies a policy motive makes it less likely for partisan policies to get enacted. On the

other hand, assumption 2 is essential. In section 2.5 I show that impatient parties always pick

moderate candidates in every period. It is intuitive that short sighted parties won’t invest

in future elections. And without this forward looking behavior my model largely resembles

a standard one shot spatial model and the logic of the median voter theorem applies.

2.4.2 Characterization and Discussion

Here I characterize and discuss the unique subgame perfect equilibrium, with strategies

denoted by σ

= (σ

, σ

). The equilibrium ends up being Markovian, so it can be described

using a state that includes the party that won and the positions taken in the previous period.

To denote these states I will use the notation (I, x, y) where I is the party that won last

period, x is that party’s last period position, and y is the losing party’s previous position.

For example (R, m, l) means the R party won last period with a moderate candidate, and

the L party lost with a partisan candidate.

In the equilibrium the parties will use symmetric strategies, so only the states where the

L party is the incumbent need to be described. This is done in the following table:

State L’s Position R’s Position R’s Position if L deviates

(L, m, m) m r m

(L, m, r) m m r

(L, l, r) m m m

Note the table doesn’t include state (L, l, m) because it is impossible since the moderate

candidate picked by party R can’t lose to the partisan candidates picked by L.

Proposition 2.1. Both parties employing strategy σ

is the unique subgame perfect equilib-

rium.

Proof. In appendix.

By examining the table it can be seen that this equilibrium has the property that after

any history the cyclical pattern displayed in ﬁgure 1 below will result. Each box in the ﬁgure

represents a period. The ﬁrst line in the box is the state, and the second line describes what

happens.

The ﬁrst thing to notice is a moderate candidate wins in every period, and yet partisan

candidates are regularly nominated. To see why consider the lower left hand box with

state (R, m, m) which means L is out of power and neither party has the advantage of an

energized base. R goes ﬁrst and picks a moderate candidate. L could pick a moderate

Figure 2.1: Cycle in Equilibrium

(R,m,m)

R-m Beats L-l

(R,m,l)

L-m Beats R-m

(L,m,m)

L-m Beats R-r

(L,m,r)

R-m Beats L-m



candidate which would result in a tie and give L probability 1 −p <

of winning. Instead L

decides to concede the current election and pick a partisan. This leads to the upper left hand

box (state (R, m, l)) where both parties pick moderates, but L wins for certain because they

energized their base last period. Hence L trades oﬀ a chance of victory (at state (R, m, m))

for certain victory in the next period (at state (R, m, l)). Notice that 1 − p represents how

much L is giving up by picking a partisan. This makes clear the role of assumption 2, the

parties have to be suﬃciently patient relative to what they are giving up by conceding the

election. In the remaining two boxes, (L, m, m) and (L, m, r), the same story occurs with

the role of the two parties switched.

One feature that might seem puzzling is that R chooses a moderate candidate in the

upper left box in state (R, m, l). R knows L’s base is energized, and can foresee defeat, so

why doesn’t he choose a partisan candidate to build a stronger future? Well if R chooses a

partisan then party L can also choose a partisan and win for sure (recall R has to go ﬁrst).

And since R has a policy motive he prefers to lose to a moderate than lose to a partisan.

That R acts in this way is the content of assumption 1. Thus the policy motive of the

parties acts to prevent partisan candidates from winning, which is a counterintuitive result.

This point is exactly how the equilibrium with purely oﬃce motivated diﬀers from the above

cycle, and will be discussed in more detail in section 2.5.

The most distinctive result that comes out of the equilibrium is that parties cycle in

power in a predictable fashion. This is a result of the fundamental tension between winning

today and tomorrow. In equilibrium, only moderates ever win, but they fail to energize the

base leaving the winning party with a weakened future. As discussed in the introduction, this

result ﬁnds strong support in data. This chapter contributes to the literature by providing

a novel reason why this cycling occurs as well providing a model with strategic interaction

that produces this result.

The equilibrium cycle also creates patterns in the ideology of the candidates nominated

and turnout among the parties’ bases. The logic of the model requires parties neglect their

base while in power. So once out of oﬃce they need to reenergize their base before being

able to win elections. This implies that newly out of power parties should nominate more

partisan candidates and see a surge in turnout among young voter in their base. As far as I

know, little empirical work has been done on these question and represents strong testable

predictions of the model. Cohen et al. (2008) (pg. 92) provides some suggestive evidence,

but this is a direction that warrants further investigation.

A third feature of the equilibrium is that parties only manage to elect moderates despite

nominating partisan candidates. The median convergence theorem holds more strongly in

policies implemented than in candidates chosen. This can occur because parties nominate

partisan candidates to invest in the future, not to win today. There is limited evidence

to suggest parties converge in policy outcomes (see Imbeau (2001) for a survey) and quite

strong evidence they diverge in candidates nominated (see Ansolabehere et al. (2001) and

Erikson and Wright (2005)). But this is another testable prediction that requires further

study.

2.4.3 Sketch of Proof

Verifying that σ

is an equilibrium is simply an application of the one shot deviation prin-

ciple: see appendix for calculations. The calculations are made manageable by the fact the

equilibrium is Markovian. Additionally the strict inequalities in the assumptions implies the

one shot deviation principle holds strictly at every history, which will be used to prove the

equilibrium is unique.

The intuition for why the equilibrium is unique is the model is almost a zero sum game.

To see this note that if a moderate always wins then the only variation in payoﬀs comes

from who get the oﬃce motivated payoﬀ, which is zero sum. And as seen in Lemma 2.1

a moderate always defeats a partisan, so a partisan can only win if the other party allows

it. And the parties won’t allow it as long as they are suﬃciently policy motivated, which is

guaranteed by assumption 1.

The key step of the proof of uniqueness is showing that σ

is both strongly eﬃcient and

pushes both players to their min-max payoﬀ. Thus the equilibrium is the only arrangement

of strategies that is individually rational for both players. Consider any history h that begins

a period, so that it is the party in power’s turn to move. Let V

(h) be the continuation

payoﬀ party i receives at history h if both parties are employing strategy σ

. The proof

establishes the following three statements:

1. At history h, each party can achieve at least V

(h), no matter the strategy of the other

party.

2. σ

is strongly eﬃcient in the sense that the sum of continuation payoﬀs at history h

can never exceed V

(h) + V

(h)

3. In any equilibrium the continuation payoﬀ of both parties at history h must be V

(h).

That statement 3 follows from 1 and 2 is obvious. Statement 2 can be seen by noting

that if a partisan candidate wins the parties combined payoﬀ is γ − 1 and if a moderate

candidate wins it is γ −.5 and these are the only two possibilities. Thus combined payoﬀ is

maximized when a moderate always win, which happens in equilibrium σ

starting from any

history that begins a period. To see statement 1 suppose that party L commits to playing

strategy σ

, then notice that, no matter what R does, σ

ensures an r type will never win.

Thus it is intuitive the worst case for L will have a moderate winning in every period

. But

when a moderate always wins the game is zero sum, and so the strategy that minimizes L’s

payoﬀ is the strategy that maximizes R’s payoﬀ, which is σ

. Thus L’s worst case is V

(h)

and hence he can always achieve this payoﬀ by playing σ

Given statement 3 the rest of the proof proceed as follows. Take any history h that

doesn’t start a period, i.e. where it is the out of power party’s turn to move. Without loss of

generality let it be L’s turn to move. Once L moves the next history will be one that starts

a period. Using statement 3, for any action a, L

s continuation payoﬀ must be V

((h, a)).

This, plus the fact that the one shot deviation principle holds strictly for σ

, means that

in any equilibrium L must take the same action as σ

at history h. And this can be used

to show the continuation payoﬀs for both parties at h must be V

(h). And then this same

argument can be applied again on histories that start a period, which completes the proof.

2.5 Oﬃce Motivated Parties

In this section I consider purely oﬃce motivated parties with varying level of patience. The

key feature, that parties concede elections to invest in the future, is preserved with patient

oﬃce motivated parties but not when they are short sighted. This shows that assumption 2

was essential while assumption 1 was not. The only change to the model is to the parties’

payoﬀ. Each party gets a payoﬀ of 1 for winning and 0 for losing and discount the future at

β. The needed level of patience is deﬁned as:

(p) =

p −



− 4 (1 − p)



p −



1/2



p −



It can be easily veriﬁed that for any p ∈ (.5, 1) we have that β

(p) ∈ (0, 1) and

(p) is decreasing in p. When the parties are patient they follow strategy pair σ

oﬃce

Showing this step is somewhat involved and requires assumption 1. See Lemma B.1 in the appendix for

full details.



L,oﬃce

, σ

R,oﬃce



, which is described in the following matrix. Just as in σ

the equilibrium

is symmetric so I only describe the states where L is in power.

State L’s Action R’s Action R’s Action if L deviates

(L, m, m) m r m

(L, m, r) l r m

(L, l, r) m m m

Proposition 2.2. If β > β (p)

then σ

oﬃce

is the unique sub game perfect equilibrium. If

β < β

(p) then in the unique sub game perfect equilibrium, after any history, within two

period both parties will nominate moderate candidate forever.

Proof: In Appendix.

By examining σ

oﬃce

it can be seen that, after any history and within 2 periods, the cycle

displayed in Figure 2 below results.

Figure 2.2: Cycle in Equilibrium

(L,l,r)

L-m ties R-m

(R,m,m)

R-m Beats L-l

(R,m,l)

L-l Beats R-r

(L,m,r)

R-r Beats L-l

(L,m,m)

L-m Beats R-r

(R,r,l)

R-m ties L-m



*











*

Unlike the previous cycle this one involve ties. Where the ties occur the party in power

did not keep their incumbent candidate, so the election is decided by a fair coin ﬂip. The

boxes where the ties occur have two arrows coming out to indicate the two possibilities, each

of which occur with probability

. Conceding the current election to invest in the future

still occurs at states (R, m, m) and (L, m, m) (the two center boxes), which is the same

states where it happened in the original equilibrium. Thus the central feature of the original

equilibrium is maintained with purely oﬃce motivated parties.

The main diﬀerence from the original equilibrium occurs at states (R, m, l) and (L, m, r),

which are the lower left and upper right boxes respectively. Under σ

, in these states both

parties move to the middle and the side with the energized base wins with certainty. In σ

oﬃce

the side with the energized base still wins, but both parties choose partisan candidates. The

reason for this is the party without the energized base knows they are going to lose no

matter what. Their only motive for taking a moderate position is to keep a partisan from

the other side out of power. Since there is no policy motive they don’t care about that and

choose instead to pick a partisan candidate to build a stronger future. This provides the

counterintuitive result that a policy motive prevents partisan candidates from winning.

Proposition 2.2 also says that impatient parties converge to the center in every period.

Myopic parties do not care about investing in the future and thus have no incentive to pick

partisan candidates. If parties ignore the future my model reduces to a Downsian model

where a median voter theorem type result holds. This result could be extended to impatient

parties with a policy motive.

2.6 Conclusion

In this work I formalized the idea that partisan candidates are less electorally viable, but

energize the base which is valuable for future elections. I motivated this mechanism with

empirical evidence on habitual voting and abstention due to alienation. The key result of

the model is it provides a novel reason for why parties cycle in power in a regular fashion.

That the party in power has trouble energizing their base and so their electoral strength

erodes over time.

The core innovation of this work is that it builds a strategic model of candidate selection

where parties care about future as well as the present. That is parties are patient enough

to consider the ramiﬁcation of their actions on future elections. This work has focused

speciﬁcally on the mechanism of partisan candidates energizing the base. But this is just

one example of a larger principle; exploring other ways forward looking behavior can impact

political strategies presents an important direction for future work.

Chapter 3

Economic Voting and

Experimentation

3.1 Introduction

Economic events impact elections, a phenomenon referred to as economic voting (Kramer,

1971; Tufte, 1978; Lewis-Beck and Stegmaier, 2000). Speciﬁcally, a strong economy makes

incumbent re-election more likely, while a weak economy makes it less likely. Previous work

such as Rogoﬀ (1990), Persson and Tabellini (1990) and Alesina et al. (1993) explains this

behavior using a simple model of voter learning, where a strong economy is taken as a

positive signal about the incumbent’s policies and/or competence, leading to re-election. In

this chapter, I introduce the idea that voters who have the opportunity to learn should care

about learning when deciding whom to elect. In particular they may elect a party with

a policy that creates worse economic outcomes but that allows for more learning. I refer

to this as experimentation, and it is the focus of this chapter. I build a stylized model of

experimentation in the context of economic voting and explore its consequences.

The papers cited above do not allow for experimentation, but rule it out using a series of

assumptions that make their setup analogous to a two-period model with only one election.

The voter can only value learning if there is a future election at which to use the information

learned. Hence in these models, the voter does not care about learning at the time of the

election and will never experiment.

I introduce experimentation within an inﬁnite horizon model that captures the following

story. There are two parties, Left and Right, each of which advocates for a distinct ideological

policy. Each period, a median voter elects a party, which then chooses what policy to

implement. The parties cannot commit to policies before the election, which limits the voter’s

control. The voter is indiﬀerent about ideology and cares only about growing the economy.

There is an underlying state of the world that determines which ideological policy is actually

best for the economy. None of the players can observe the state directly, and it changes over

time. All the players attempt to infer the state by observing how the economy performs

under diﬀerent policies. However, economic performance has an i.i.d. noise component that

makes this inference diﬃcult. Parties also have the ability to implement a compromise policy

that includes some of the ideas of each party.

The process whereby the players learn about the state has two key features: extreme

economic outcomes are more informative, and ideological policies allow for more learning than

the compromise policy. Spectacular policy failures or successes allow for strong inferences.

In contrast, it is unclear what to infer from average economic outcomes. Moreover, the

compromise policy blocks inference, since it contains ideas from both parties. As a result,

both parties can take credit for good economic outcomes or pass blame for poor ones. In

order to generate sharp results and for tractability, I employ assumptions that deliver a very

stark learning process. In particular I assume the i.i.d. noise component of the economic

outcome is distributed uniformly. This has the implication that learning is all or nothing:

either the state is revealed or nothing is learned.

The ﬁrst insight the model delivers is that a voter who cares only about the economy will,

at times, vote for inferior economic policy in order to experiment. For example suppose the

voter believes the underlying state likely favors the Left party, but is faced with a Left party

who wants to implement the compromise policy and a Right party who wants to implement

its ideological policy. The Right party’s policy is inferior for immediate economics outcomes,

but it sets up a clean test between the ideologies. Whether it succeeds or fails, the voter

will have learned something. Hence a voter who values learning suﬃciently will elect the

Right party, even though doing so is likely to harm the economy. In other words the voter

experiments with the Right party. Proposition 3.1 provides a natural deﬁnition of the voter’s

value for learning that fully characterizes whether experimentation occurs in equilibrium. I

then show that experimentation will occur if the underlying state is suﬃciently persistent

but will not occur if the state is suﬃciently impersistent.

The second insight the model produces is that when experimentation is possible, economic

voting incentives matter greatly even during calm economic times. In the classic formulation

economic voting matters more with larger economic ﬂuctuations. The model still captures

this channel but also introduces a new one. During times of moderate economic outcomes, the

voter has diﬃculty making inferences about the underlying state. Hence, after a suﬃciently

long period of calm economic times, the voter becomes highly uncertain about the underlying

state. This raises the voter’s desire for learning and makes experimenting with the disfavored

ideological policy less costly. So it is precisely during calm economic times that the voter’s

desire for learning is highest. In proposition 3.1, I show that if experimentation occurs at

all then it will occur during calm economic times. In this way the incentives underlying

economic voting can be important even in the absence of a proximate economic event.

I also explore the electoral outcomes and party behavior that occur in equilibrium. Classic

economic voting behavior still takes place in every equilibrium in the model: after any

economic event that is large enough to reveal the underlying state, the voter will always elect

the party the state favors, and that party will implement their ideological policy. When the

state has just been revealed, the favored party and the voter both want the same thing: the

favored party’s ideological policy.

To further explore electoral outcomes I study equilibria with and without experimentation

separately. Proposition 3.3 examines equilibria with experimentation. It says that, in any

equilibrium with experimentation, after a suﬃciently long period of calm economic times the

party in power will switch every period and only ideological policies will be implemented.

This continues until the next economic outcome large enough to reveal the state. The

implementation of ideological policies is an expression of the voter’s desire to learn, which

is high during calm economic times because the voter becomes very uncertain about the

underlying state. Switching the party in power every period is required to give the parties

the proper incentives. Since the party in power knows it will lose the next election unless

learning occurs, it will desire to implement ideological policies. Hence, the voter’s desire for

experimentation causes high political turnover during calm economic times. This reinforces

the earlier point that economic voting incentives matter even without a proximate economic

event. In equilibria without experimentation the exact opposite outcome occurs: proposition

3.4 says that after a suﬃciently long period of calm economic times, the same party will

remain in power every period and will always implement the moderate policy.

Recall that experimentation is more likely when the underlying state of the world is

persistent. The interpretation of a persistent state is that the policy needs of the economy

are stable. Hence, the model has the counter-intuitive implication that, during moderate

economic times, stable economic fundamentals lead to higher political turnover and unstable

fundamentals lead to lower political turnover.

I solve the model by looking for mixed strategy Markov perfect equilibria that treat

the parties symmetrically. The focus on symmetric equilibria is based on the idea that

the median voter is not invested in either party. Asymmetric equilibria would allow the

voter to coordinate better with one of the parties, and hence prefer that party to be in

power. Moreover, the parties in the model are entirely symmetric, so focusing on symmetric

equilibria is natural. The Markov assumption, as usual, helps with tractability and to cut

through the multiplicity problem prevalent in inﬁnite horizon models. This is especially

important given that the model belongs to a class of games for which Dutta (1995) has

proved a folk theorem. Technical considerations aside, the Markov assumption is appropriate

in these large election games due to the diﬃculty in coordinating many voters. Coordinating

on complex history-dependent equilibria seems implausible with a large electorate. Of course,

the model only has a single median voter, but this is a useful abstraction, and one can think

of the voter as representing a large electorate in the background.

The model assumes a high level of voter sophistication, which can be viewed as unrealistic.

However, the behavior I examine is interesting independent of the realism of the voter.

Exploring optimal voter behavior is important even if just to see how far away it is from

actual voter behavior. Additionally this chapter is undertaking the important theoretical task

of taking the learning rationale for economic voting seriously and exploring it to its natural

end. Moreover, the sophistication required by the model is not altogether implausible. It

is true that the calculations required of the voter are fairly complex, but the strategies are

quite straightforward. The strategies of the voter depend on something quite simple and

plausible: the voter’s belief about which party’s policy is better for the economy. This

can be represented by a single value and is something that does factor into actual voting

behavior.

This chapter connects to two distinct literatures. It is most directly linked to the literature

cited above relating economic voting to learning (Rogoﬀ, 1990; Persson and Tabellini, 1990;

Alesina et al., 1993). However, as previously discussed, these models employ assumptions

to ensure the equilibrium is the same as a two period model, with the two periods just

continually repeated. They also only allow elections every other period. Hence these models

are functionally two period models with only one election. In such a model, the voter learns

before the election, but at the time of the election he no longer cares about additional

learning since there is no future election. Callander (2011) explores voter learning in a truly

dynamic model with a much richer learning environment than employed this chapter, but

he employs myopic players who do not value the future, which rules out any possibility of

experimentation.

A second literature studies learning about polices in the absence of parties. Piketty (1995)

and Strulovici (2010) look at models where individuals vote directly on policies. Removing

the parties removes the agency problem, which is one of the main focuses of this chapter.

Also, many of the key results of this chapter are framed in terms of the parties and whom the

voter chooses to elect. In both papers mentioned above, agents learn based on their private

information and make no inference from aggregate variables. In contrast I focus precisely on

learning from aggregate outcomes and how that inﬂuences the interaction of the players.

The rest of this chapter is organized as follows. Section 3.2 formally describes the model

and discusses how the learning process works. Section 3.3 lays out the value functions of the

players and demonstrates a few useful facts about player behavior. Section 3.4 discusses when

and why the voter experiments in equilibrium. Section 3.5 studies the electoral outcomes in

equilibria both with and without experimentation. Section 3.6 concludes.

3.2 Model

This section presents the formal model. I build a stark model both for tractability and to

allow for sharp results. This section starts with a terse, but complete, description of the

model and then expands on and discuss the most important element: learning process.

Model Description: There are three inﬁnitely lived players: two parties, L and R, and

a voter v. Time is discrete and denoted t = 1, 2.... Each period the voter elects a party who

then publicly selects a policy x

∈ {−1, 0, 1}. The parties cannot commit to a policy before

the election. The policies −1 and 1 represent extreme or ideological policies associated with

one of the parties. Party L prefers policy −1, and party R prefers policy 1. The 0 policy

is a compromise, which both parties are lukewarm about. Full preferences will be described

below.

After the policy is selected the economic outcome y

is revealed which represents the

growth rate of the economy. The value of y

follows the process

= −(x

− s

)

+ ε

The random variable ε

is i.i.d. and distributed uniformly on the interval



−2

1−r



, with

r ∈ (0, 1). The variable s

is an underlying state of the world determined at the start of

every period. The variable s

can be either −1 or 1 and follows a Markov switching process

where s

= s

t−1

with probability γ and s

6= s

t−1

with probability 1 − γ. As a non-triviality

condition restrict γ ∈



, 1



. None of the players observe s

or ε

, and all players form beliefs

on s

using the public history of x

and y

. Denote by p

the belief that the state equals 1,

and p

can be deﬁned as:

= Pr



= 1|{x

, y

}

j=1

, s



There is no asymmetric information so all players always share the belief p

By inspecting the process for y

it becomes clear that s

determines which policy is best

for the economy. If s

were known, then setting x

= s

would maximize y

. This captures

the idea that each party’s ideological policy is right for the economy in some state of the

world. In practice the voter is uncertain about s

, and p

captures his belief about which

policy is more likely to be correct. In other words the L party is belief-favored whenever

and the R party is belief-favored whenever p

All players discount at β ∈ (0, 1). The voter cares only about the economy and maximizes

∞

s=1

s−1

The parties are both oﬃce and policy motivated while caring nothing about the economy.

The L party’s policy payoﬀ is −x

, and their oﬃce payoﬀ gives them λ > 0 in each period

they are elected and −λ in each period they are not elected. Let I

∈ {−1, 1} be an indicator

function that takes value 1 if L is elected at t and −1 otherwise. Then the L party maximizes

∞

s=1

s−1

λ − x

)

The R party’s payoﬀs are similar: they get policy payoﬀ x

, and λ for winning and −λ for

losing. Hence the R party maximizes

∞

s=1

s−1

(−I

λ + x

)

Notice that the sum of the two parties’ payoﬀs is 0. However, the game is not zero-sum

because of the existence of the voter.

I analyze this model using mixed strategy symmetric Markov perfect equilibrium where

the Markov state is the belief p

. Symmetry imposes that at beliefs p and 1 − p, L and R

act symmetrically and the voter treats them symmetrically. Formally, the strategies of the

parties are functions σ

: [0, 1] → ∆ {−1, 0, 1}, and the strategy of the voter is a function

: [0, 1] → ∆ {L, R}. I abuse notation and let σ

(a; p) be the probability that player i

takes action a at belief p. The symmetry assumption imposes that

(−1; p) = σ

(1; 1 − p)

(1; p) = σ

(−1; 1 − p)

(0; p) = σ

(0; 1 − p)

(L; p) = σ

(R; 1 −p) .

Since the parties can use mixed strategies, standard ﬁxed point arguments ensure equilibrium

existence. A formal existence proof is given in the appendix. Throughout, the following

assumptions are maintained:

2 − r

< β and λ >

1 − β + 2βr

β (1 − r)

The ﬁrst assumption ensures the players are forward looking enough to care about learning.

The second assumption ensures the parties are suﬃciently oﬃce motivated.

Learning Process: At the start of each period t, all the players have a shared prior

about s

denoted p

. After observing x

, y

, they use this information to update to a new

belief denoted by v

. Note that v

will not equal p

t+1

, since the players will update again

using their knowledge of the Markov process that s

follows. The next period belief p

t+1

set by

t+1

= v

+ (1 − γ) v

The process for updating from p

to v

is very stark due to the assumption that ε

distributed uniformly. For extreme y

, the underlying state s

is completely revealed, and

for moderate y

nothing is learned. Hence v

can only take one of three values: 0,1 or p

. To

see this, suppose the −1 policy is implemented, then there are two possible distributions for

. If s

= −1 then y

∼ U



−2

1−r



, or if s

= 1 then y

∼ U



−4 +

−2

1−r

, −4 +

1−r



. Any

−2

1−r

is outside the support when s

= −1, and hence for any such y

it can be inferred

that s

= 1. Similarly if y

> −4+

1−r

then it must be that s

= −1. For intermediate values

of y

nothing can be learned since both of the possible distributions have the same density

in this region, and hence v

= p

. With the compromise policy, y

has the same distribution

regardless of s

, and hence no matter the outcome nothing can be learned and v

= p

For clarity, I will that say learning occurs if and only if s

is revealed. Note that learning

is distinct from p

changing. For any ideological policy (x ∈ {−1, 1}) learning occurs with

probability 1−r and no learning with probability r. Conditional on learning, with probability

the state is revealed to be 1 which gives p

t+1

= γ, and with probability 1 − p

the

state is revealed to be −1 which gives p

t+1

= 1 − γ. If there is no learning then p

t+1

γp

+ (1 − γ) (1 − p

). If the policy is moderate (x = 0) there is never any learning and

t+1

= γp

+ (1 − γ) (1 − p

) with probability 1.

The learning process used here is very simple and yet captures the essence of the story

told in the introduction. In particular, it captures the idea that large shocks create learning,

and that moderate policies make learning more diﬃcult. Both of these features are fairly

robust and would remain in a wide variety of speciﬁcations.

I conclude this section with a technical remark on the belief process. As discussed above,

the belief moves either by jumping to the extremes, (γ, 1 − γ) or moving in discrete jumps

towards

. Hence, only a countable number of beliefs are ever possible. This feature simpliﬁes

the technical analysis and is the main tractability advantage of the all or nothing learning

process. The possible beliefs can be calculated by forward solving the equation p

t+1

γp

+ (1 − γ) p

starting from γ and 1 − γ.

3.3 Value Functions and Incentives

In this section I discuss the incentives of the players and highlight some useful facts about

their behavior. I do this primarily by examining the value functions of the voter and the

parties. Using the Markov assumption, the value functions depend only on the current belief

about s

, denoted by p. Recall that

p = Pr



s = 1|{x

, y

}

j=0

, s



and I say that L is belief-favored when p <

and R is belief-favored when p >

First consider the voter’s decision of which party to elect. The Markov assumption

ensures that the voter only cares about the implemented policy. In equilibrium the voter

forecasts what each party will do, and then elects the party whose policy he prefers. This

allows the voter’s decision problem to reduced simply comparing diﬀerent policies. I will

abuse notation and let W (p) represent the voter’s equilibrium payoﬀ at belief p, and I let

W (x; p) be the equilibrium payoﬀ at belief p if policy x is implemented. I use Pr (q|x, p) to

denote the probability that next period’s belief is q given current belief p and policy x. I

can characterize W (x; p) as follows:

W (x; p) = E [y|x, p] + β

Pr (q|x, p) W (q) .

The right-hand side of the above equation has two terms: E [y|x, p] is the expected

economic outcome, and

Pr (q|x, p) W (q) is the voter’s continuation payoﬀ. Notice the

policy only impacts the voter’s continuation payoﬀ by changing the Pr (q|x, p) term. In other

words, the policy only impacts the voter’s future payoﬀ through the learning process. Hence

the voter’s choice of policy depends on two things: the current economy and learning. The

voter’s concern about the economy is captured by the following expression:

E [y|x, p] =

(

−4p if x = −1

−4 (1 − p) if x = 1

−1 if x = 0

If p >

or p <

, then the voter is quite certain of s

and the belief favored ideological

policy is best for the economy today. If p ∈





then the voter is uncertain of the state and

the compromise policy is best for today’s economy

. Moreover, the disfavored ideological

policy is always the worst policy for today’s economy.

Now consider the voter’s second concern: the learning consequences of policies captured

by Pr (q|x, p). The process governing Pr (q|x, p) was laid out in section 3.2. One important

point to recall is that the two ideological policies have the same learning consequences.

Hence Pr (q| −1, p) = Pr (q|1, p) for all q, p. This implies the voter always prefers the favored

This analysis clariﬁes the need for γ > .75. If γ < .75 then p ∈





will always be true and hence the

voter will never be sure enough about the s

to be willing to gamble on ideological policy. This means they

can never value learning and the model becomes quite trivial.

ideological policy over the unfavored one. In other words W (−1; p) ≥ W (1; p) if and only

if p ≤

Whenever learning occurs, s

is fully revealed, and hence, the voter gets continuation

payoﬀ of either W (γ) or W (1 −γ). It turns out that the voter’s continuation payoﬀ is the

same regardless of what was learned, as is captured by the following lemma.

Lemma. In any symmetric MPE W (γ) = W (1 − γ).

Proof. Follows directly from the symmetry of the model and assumption of a symmetric

MPE. 

This result says that the voter has the same continuation value from learning regardless of

what s

is revealed. Intuitively, this represents that the voter is not invested in the ideological

debate between the parties. Moreover, this result allows reference to a single unambiguous

value of learning for the voter. This will be crucial in the statement and interpretation of

my results. Additionally, this simpliﬁes the learning process from the voter’s perspective:

the compromise policy allows no learning and the ideological policies create learning with

probability 1 − r and no learning with probability r.

I now turn to the parties’ incentives. The only decision the parties make is what policy to

implement once elected. So only the party’s preferences over policies need to be considered.

The value functions of the parties are:

(x; p) = −x + −I

λ + β

Pr (q|x, p) V

(q) (3.1)

(x; p) = x − I

λ + β

Pr (q|x, p) V

(q) (3.2)

For concreteness I will discuss L’s value function (equation 3.1) but everything said

applies equally to R. The term −x represents L’s current policy payoﬀ. The term I

captures the oﬃce motivated payoﬀ. Recall that I

takes value 1 if the L party wins and −1

if the R party wins. At the time the party chooses the policy the election has already occurred

and the oﬃce payoﬀ is already sunk. However, the parties will account for getting the oﬃce

payoﬀ in future periods in making their decision. The value β

Pr (q|x, p) V

(q)) represents

L’s continuation payoﬀ and, as with the voter, the policy only impacts the continuation

payoﬀ through the learning process. The ﬁrst thing to notice is that the parties will never

implement the opposite ideological policy. This follows from the fact that the ideological

policies have the same learning consequences. Hence, without loss of generality suppose the

parties make a binary choice between the compromise policy and their own ideological policy.

An important implication of the zero sum property is that both parties never want to

implement the compromise policy in the same period. Intuitively, blocking learning is the

parties’ only motivation for the compromise policy, and if blocking learning helps one party

it must hurt the other party. Hence both parties can’t want to block learning at the same

time, which can be seen formally from the following expression:

(p, 0) − V

(p, −1) = −V

(p, 0) + V

(p, −1) = V

(p, 1) − V

(p, 0) − 2. (3.3)

The ﬁrst equality is a consequence of the zero sum property. The second equality occurs

since policies −1 and 1 have the same learning consequences and diﬀer only in today’s policy

payoﬀ. Hence V



−1



= V

(p, 1) − 2 for all p. Party L desiring to implement policy 0 is

the same as the far left of equation 3.3 being positive, which means the far right must be

positive. And that can only happen if V

(p, 1)−V

(p, 0) > 0, which implies party R strictly

prefers to implement their ideological policy. And this shows it is impossible for both parties

to desire to implement the compromise policy at the same p.

3.4 Experimentation

This section deﬁnes what it means for the voter to experiment, and discusses why and when

it occurs.

Deﬁnition. Experimentation occurs if the voter elects the belief unfavored party who im-

plements their ideological policy with probability 1. Experimentation occurs in a equilibrium

if there exists a belief p where experimentation occurs that is reached with probability 1 as

t → ∞.

To understand this deﬁnition recall that the belief unfavored party’s ideological policy

is the worst possible policy for today’s economy (y

). The economy would have done better

with the other party no matter what policy they would have implemented. Hence, the only

justiﬁcation for the voter’s choice is a desire to increase learning, i.e. to experiment. An

alternative deﬁnition would have been to deﬁne experimentation as occurring when the voter

makes a choice that is worse for today’s economy than what the other party would have

implemented. However, this alternative deﬁnition would require knowing counter-factual

information: what the unelected party would have done and whether it would have been

better for the economy. On the other hand, the deﬁnition given above can be identiﬁed from

what actually occurred along with beliefs about which party the public trusts more with the

economy, a subject of constant polling. Additionally, this deﬁnition makes experimentation

harder to ﬁnd, biasing the results against ﬁnding experimentation.

I now state my ﬁrst result about experimentation.

Proposition 3.1. In any symmetric MPE:

1. Experimentation will occur in equilibrium if and only if the voter’s valuation of learning

(W (1 − γ)) is strictly greater than

−1

1−β

β(1−r)

2. There exists γ

, γ

∈



, 1



such that for all γ > γ

experimentation always occurs

and for all γ < γ

experimentation never occurs.

3. If W (1 − γ) >

−1

1−β

β(1−r)

then experimentation will occur regularly after a suﬃciently

long period of moderate economic shocks (no learning), and will continue until the next

large economic shock. In particular there is a p

such that as long as the belief

remains in (p

, 1 − p

) experimentation will occur every other period.

Proof. In appendix. 

Experimentation requires that the voter, who cares only about the economy, to vote for

the worse economic policy. Proposition 3.1 illuminates why and when this would occur. Part

1 of the proposition says that a voter’s willingness to experiment depends on his continuation

value after learning occurs, which is captured by W (1 −γ). With a high enough value of

W (1 − γ), the voter is willing to accept worse economic outcomes today in order to learn

more and get better outcomes in the future. The second part of the proposition relates exper-

imentation to the primitives of the model. The value of learning depends on the persistence

of the underlying state captured by γ. A high value for γ makes the underlying state very

persistent, which makes learning valuable and leads to experimentation. Conversely, a low

value for γ prevents experimentation. The interpretation of γ is the stability of the policy

needs of the economy. In an economy with rapidly changing policy needs, experimentation

serves little purpose. Conversely, if a policy that works today is likely to work far in the

future, then learning about the policies is worthwhile.

The third part of proposition 3.1 says that the incentives behind economic voting are

important even without a proximate economic event. In the classic formulation, economic

voting is a response to immediate economic events and has more salience with larger ﬂuc-

tuations. This part of the proposition broadens the view of when economic voting incentive

matter by showing that it is precisely during calm economic times that the voter is most

likely to experiment. Moderate economic outcomes block learning which causes the belief

p to drift towards

. After a suﬃciently long period of such times the voter becomes very

uncertain about the state captured by p ∈ (p

, 1 − p

), which drives the voter’s desire to

learn. It also makes experimenting with the disfavored ideological policy less costly since the

voter only weakly believes the disfavored policy is wrong for the economy. In this way, eco-

nomic voting incentives remain salient even during calm economic times. This point will be

strengthened in section 3.5 where I show this motive to experiment will also impact election

outcomes.

This proposition also yields testable predictions of the model: the party which voters

disagree with on the economy will occasionally win with an extreme platform. Moreover,

this will occur during calm economic times and to continue regularly until the next large

economic shock. This prediction is testable in part because experimentation was deﬁned

in an observable way. Part 2 of the proposition can also be tested using the interpretation

of γ as the stability of the policy needs of the economy. The prediction would be that

experimentation is more common in places or times with more stable policy needs.

The full proof of proposition 3.1 can be found in the appendix, but I provide a brief

sketch here. I start with part 1 of the proposition. First, suppose experimentation occurs

and we want to show that

W (1 − γ) >

−1

1 − β

β (1 − r)

Without loss of generality consider p <

, which means the L party is belief favored. Ex-

perimentation when p <

requires the R party to be elected and to implement policy 1.

Since the voter prefers −1 to 1 this can only happen if the L oﬀered policy 0. Set p

to be

the next period belief when no learning occurs, so that p

= (1 −p) (1 −γ) + pγ. Using the

value functions from above, the voter prefers policy 1 over 0 if and only if:

0 ≤ −2 (1 −2p) − 1 + β (1 −r) (W (γ) − W (p

)) .

This can be rewritten as

W (p

) +

1 + 2 (1 − 2p)

β (1 − r)

≤ W (1 − γ) .

Since p <

, it must be that

1+2(1−2p)

β(1−r)

. So it would suﬃce to show that W (p

) ≥

−1

1−β

. The formal proof of this point can be found in the appendix but the intuition is as

follows: the voter likes learning enough that he prefers even the wrong ideological policy

over the compromise policy. Hence it is intuitive his worst case scenario is always getting

the compromise policy and that scenario gives payoﬀ

−1

1−β

. So it follows that W (p

) ≥

−1

1−β

For the other direction suppose that

W (1 − γ) >

−1

1 − β

β (1 − r)

and we want to show experimentation occurs. For concreteness suppose L is belief advan-

taged (p <

). Well, high W (1 − γ) makes the voter value learning enough that always

getting policy −1 is their best case. For p close to

the voter is close to indiﬀerent between

policies −1 and 1. Recall, at least one of those will always be oﬀered. So the voter can

achieve close to their best case by electing whoever oﬀers an ideological policy. To do this,

equilibrium incentives will require that the R will be elected for some p close to

. To see

this suppose the L party wins for all p close to

. Then L has a strong incentive to imple-

ment policy 0, which will keep p close to

by blocking learning. But L cannot be elected

with policy 0, which creates a contradiction. Hence R must win for some p close to

and

implement policy 1.

Part 2 follows quite naturally from part 1. If γ is very close to 1 then learning lasts for a

long time which is valuable to the voter. Also, with high γ the belief advantage party is not

worried about learning and is happy to implement their ideological policy which reduces the

agency problem. So we can show as γ → 1 that

W (1 − γ) → 0 >

−1

1 − β

β (1 − r)

and the result follows from part 1 of the proposition. Note that the inequality 0 >

−1

1−β

β(1−r)

is guaranteed by the assumptions on β. On the other side, if γ is too low that learning isn’t

useful to the voter. Recall that for any belief p ∈





the compromise policy maximizes

E [y

]. And if γ = .75 the lowest (highest) the belief can ever be is





. So there must

exists an upper-bound for W (1 −γ) that converges to

−1

1−β

−1

1−β

β(1−r)

as γ → .75.

3.5 Electoral outcomes

In this section I discuss the electoral outcomes that occur in equilibrium. This includes who

the voter elects and what policy gets implemented. I start by replicating the basic economic

voting result.

Proposition 3.2. In any symmetric MPE, after any economic shock large enough to cause

learning the incumbent party is re-elected if the economic outcome was higher than E [y

and the out of power party is elected if the outcome was lower than E [y

Proof. See Appendix

Proposition 3.2 replicates the basic economic voting result and with the same reasoning as

found in previous works. Large economic shocks allow the voter to make an inference about

the underlying state. The voter infers the state favors the incumbent following a positive

shock, and infers the state favors the challenger following a negative shock. And when the

party is highly conﬁdent s

favors them they have little incentive to block learning which

reduces the agency problem between the voter and the favored party. Following learning the

voter both strongly believes in the favored party and has a reduced agency problem with

them. Both of these forces push the voter to re-elect the favored party leading to the result.

For other electoral outcomes consider equilibria with and without experimentation sep-

arately. First I examine equilibria with experimentation.

Proposition 3.3. In any symmetric MPE with experimentation, there exists a p

such

that for all p ∈ (p

, 1 − p

) incumbents always lose and the party in power always implements

their own ideological policy.

Proof In appendix.

To interpret this result recall that the belief drifts towards

during moderate economic

times and only moves away from

with extreme economic outcomes. Hence, proposition 3.3

says that, in any equilibrium with experimentation, after a suﬃciently long period of calm

economic times the party in power will switch every period and only ideological policies will

be implemented. This continues until the next economic outcome large enough to reveal

. Hence, during a calm economy I can expect to see highly ideological policies and high

political turnover. This reinforces the earlier point that economic voting incentives matter

even without a proximate economic event.

The fact that only ideological policies are implemented is an expression of the voter’s high

desire to learn during calm economic times. Recall from the previous discussion of proposition

3.1 that p close to

makes the voter desire experimentation and learning generally. Moreover,

p close to

means the voter views the two ideological policies similarly. Hence, the voter will

elect whichever party oﬀers an ideological policy with a slight preference for the belief-favored

policy. And recall from the discussion in section 3 that at least one party will implement

an ideological policy at all times. And so only parties who intend to implement ideological

policies will ever be elected.

The switching of who is in power every period is required by equilibrium in order to give

parties the proper incentives. If no learning leads to an incumbent being re-elected, then the

party in power has a strong incentive to block learning. And when p is close to

, the parties

are naturally wary of learning since they are unsure of s

. These two incentives will push

the party to implement a compromise policy. But the voter would never have elected them

in that case. Hence equilibrium requires that the incumbent party always loses. In this way

the voter’s desire for experimentation causes high political turnover during calm economic

times. In the next result I consider equilibria without experimentation.

Proposition 3.4. In any equilibrium without experimentation there exists a p

such

that for all p ∈ (p

, 1 − p

) the incumbent always win and implements a moderate policy.

Proof. In Appendix.

The intuition behind this result is that when the value of learning is low and p is close to

the voter desires to play it safe with a compromise policy. In equilibrium this takes the form

of electing the same party for as long as learning doesn’t occur. This gives that party who

is being elected a strong incentive to block learning and so they implement the compromise

policy. This results directly contrasts to the previous results where calm economic times lead

to parties only implementing ideological policies with a high degree of political turnover.

One consequence of the previous two results together is an inverse relationship between

the persistence in the economic fundamentals and the stability of political outcomes. Recall

that one interpretation of γ is the stability of the policy needs of an economy. And recall from

proposition 3.1 that high γ leads to experimentation which leads to high political turnover

during calm economic times. Conversely low γ leads to no experimentation and low political

turnover during calm economic times. This result is counter-intuitive and is driven by the

equilibrium incentives created by the voter’s desire to learn.

3.6 Summary

In this chapter I laid out a theory of experimentation in the context of economic voting. The

starting point is the idea that economic voting can be explained by voters making inferences

from economic outcomes. I extended this idea by allowing voters to care about the learning

consequences of policies which was not allowed in previous models. This leads to the idea

that a voter who cares only about the economy may vote against their immediate economic

interest in order to learn; in the other words the voter may experiment with inferior policies.

My model delivers insights about when and why experimentation will occur as well the

accompanying electoral patterns. Whether experimentation occur depends on how much the

voter values learning given by the expression W (1 − γ). This can be linked to the primitives

in term of the persistence of the underlying state. With a highly persistence state, learning

is naturally valuable and experimentation occurs. Conversely with an impersistent state,

learning is not valuable and experimentation won’t occur.

I also showed that experimentation is most likely to occur during a calm economy. During

a calm economy the voter has a hard time making inferences and becomes uncertain about

the state. This make learning valuable to the voter and makes experimentation with the

unfavored policy less costly. Experimentation also occurs during a calm economy and is

accompanied by a high degree of political turnover. This is required by equilibrium in order

to give the correct incentives to the parties. In equilibria without experimentation calm

economic times are associated with low political turnover. Since experimentation occurs

when the policy needs of the economy are stable this sets up an inverse relationship between

that kind of stability and political stability. In other words experimentation requires stability

in the fundamentals and creates instability in political outcomes.

Appendix A

Appendix to Chapter 1

A.1 Proof of proposition 1.2

We start by studying equations 1.14 through 1.16 and setting δ = 0, m

= 1 and h

= 1.

With m

and h

pinned down, those equations give a unique solution for the variables

, q

, c

, w

, q

, from which all other steady state variables can be derived. We will

refer to this solution as the δ = 0 steady state. Now consider what happens to the δ = 0

steady state if we set ρ

= 0. It is clear the equations for city 1 are the same as for city 2

when we replace w

−λy

, q

, c

with w

, q

, c

. Hence it must be that, in the δ = 0 solution

when ρ

= 0, we have w

− λy

= w

as well as q

= q

and c

= c

. Examining equation

1.13 which deﬁnes U

, we can show that U

= U

and U

> U

. To see why U

> U

, set

= c

and compare the equations. And since the labor cost cutoﬀs c

are optimally chosen,

the a type workers must do better while using cutoﬀ c

compared to c

. And hence we get

that city 1 has a number of workers equal to:



− rU



+ s



− rU



+ s



− U



So at the δ = 0 steady state, when we set ρ

= 0 city 1 has more workers. And the

equations used to deﬁne the δ = 0 solution move continuously in ρ

and give a unique

solution. Hence the δ = 0 solution must also move continuously in ρ

. It follows that we

can ﬁnd a ρ

> 0 such that for all ρ

∈ (0, ρ

), city 1 has more workers at the δ = 0 steady

state. Now consider the following lemma.

Lemma A.1. There exists a y

> 0 and a m

∈ (0, 1) such that for all y

< y

, in every

steady state h

= 1 for all i, j. If, in addition, m

> m

, every steady state is in pure

strategies. Moreover y

and m

are independent of ρ

and γ.

The proof of this lemma is given in section A.1.1 below. From now on we ﬁx any y

, ρ

such that y

< min





and ρ

< min



(r+δ)

(

)

π−w



. We let P

move as necessary to

gain our desired y

. And these bounds also ensure for us that P

≥ 0 and ρ

is small enough

so that ﬁrms all underpay d workers in city 1.

From now on we will only vary δ and consider all other parameters ﬁxed. Fix any

δ > 0

and we will only consider δ ∈





. The rest of the proof will be concerned with proving

that, when δ is small enough all steady states must be close enough to the δ = 0 solution so

that city 1 has more people, given how we chose (ρ

, P

I ﬁrst claim that as δ goes to 0, m

goes to 1 in every steady state. Since we are setting

< y

we know that h

= 1. So in any steady state we can write that:

δ + λρ

+ α

)

G (c

(w)) dF

(w)



δ + α

)

G (c

(w)) dF

(w)



Deﬁne ¯q such that α

(¯q)

= k. Since J

(w) <

, ¯q is an upper bound on q for all values of

δ. Deﬁne

U as:

U =

(¯q) π

r + α

(¯q)

< π.

And

U is an upper bound on U

for all values of δ. We can also ﬁnd a L > 0 such that

)

G (c

(w)) dF

(w) ≥ L, for every δ ∈





, which will show how to derive below.

It then follows that:

δ + λρ

+ α

(¯q)



δ + L



As δ goes to 0 the right-hand side of the above inequality goes to inﬁnity. From which it

follows m

goes to 1 since:

Γ (η

) u

Γ (η

) u

+ s

Γ (η

) u

Γ(η

)

Γ(η

)

(

−

)

The inequality follows from the fact that Γ (η

) ≤ 1 and that rU

∈ [0, π] which implies

γη

≥ π. And the far right-hand side term converges to 1 as

converges to inﬁnity, so it

must be that m

converges to 1 as δ goes to zero.

So from now we only consider δ low enough so that m

> m

and hence all steady states

have to be pure in strategies with h

= 1. Now deﬁne Φ (m

, δ) as the unique solution to

equations 1.14 through 1.16 at m

, δ while setting h

= 1. And we don’t worry about m

since the solution to the equations 1.14 through 1.16 does not depend on it in any way. And

since the equations deﬁning Φ are continuous in m

, δ, and Φ is single valued, it must follow

that Φ is continuous as well.

Now suppose for contradiction there is no δ

such that, in all steady states, city 1 is

bigger. Then we could build a sequence δ

such that δ

→ 0 as n → ∞, and at each δ

there

exists steady state with equilibrium values



, q

, c

a,n

, c

d,n

, w

, q

, c

, m

d,n



where city 2 is

weakly larger than city 1. These variables must satisfy equations 1.14 through 1.16 which

mean that



, q

, c

a,n

, c

d,n

, w

, q

, c



= Φ



d,n



Hence, we are looking at a sequence



, q

, c

a,n

, c

d,n

, w

, q

, c

, m

d,n

, δ



∞

n=1

where

→ 0 and m

→ 1. And we can put bounds on all the variables meaning we are in

compact space. So that sequence must have a convergent sub-sequence. Without loss of

generality let the convergent sub-sequence be the original sequence. And by continuity of

Φ, it must follow that



, q

, c

a,n

, c

d,n

, w

, q

, c

, m

d,n

, δ



converges to Φ (0, 1) . But Φ (0, 1)

is exactly the δ = 0 steady state solution where we already showed that city 1 was strictly

larger. And city size is a continuous function of those variables, hence there must be some

point along that sequence where city 1 is larger, which gives us our contradiction. .

Deriving L: We now show to derive the lower bound L we used above. First we use the

upper bound

U to ﬁnd a lower bound 0 < q < q

by setting

J = max



w − r



π − w

r +

δ + λρ

and setting

(q) J = k

This works because in any steady state the ﬁrm can take whichever worker type makes up

at least half the unemployment pool and plan to only hire those workers. And

) J is

the worst case proﬁt from this plan. Similarly, using the fact that U

≥ 0, we can derive

a w > 0 such all ﬁrms will oﬀer wages above w. Using that we can derive a lower bound

0 < U ≤ U

, where

rU = α

(q) G





w − E



c|c ≤



r +

And in any steady state with equilibrium values of q

, c

we must have that:

≤







(w)



(w)

And combining this with our the fact that rU ≤ rU

, we get a lower bound L > 0 on

)

G (c

(w)) dF

(w) of:

L =

(q) G





w − E



c|c ≤



r +

So we can ﬁnd our desired lower bound L, which holds for all δ ∈





A.1.1 Proof of Lemma A.1

Showing that ﬁrms in city 2 all post the same wage and hire all willing workers is immediate

and discussed in the main body of the text. We only need to prove the lemma for city 1.

Deﬁne

(rU) = arg max

G (w − rU) (π −w)

Note that w

(rU) does not directly depend on ρ

, δ, P

or γ, which becomes crucial later.

Also note that the above maximization problem is strictly concave and hence w

is pinned

down by the following ﬁrst order condition:

− rU ) (π − w

) − G (w

− rU ) = 0.

By the implicit function theorem we can show that

∂w

∂rU

∈ (0, 1) .

Moreover it is clear that whenever rU < π we must have w

(rU) ∈ (rU, π). In any equilib-

rium all wages w oﬀered must obey

min







, w





+ λy



≤ w ≤ max







, w





+ λy



. (A.1)

This follows from the ﬁrm’s ﬁrst order condition from equation 1.9, since it can be veriﬁed

that, for any w < min {w

(rU

) , w

(rU

) + λy

}, the ﬁrst order condition is strictly pos-

itive, and at any w > max {w

(rU

) , w

(rU

) + λy

}, the ﬁrst order condition is strictly

negative.

Recall we derived

U < π which is an upper bound on U

for all values of δ, ρ

, y

and γ.

Now deﬁne y

such that





+ λy

= π.

Choose any y

< y

. And combining this deﬁnition with our condition from equation A.1,

we get that at any wage w that a ﬁrm may post in steady state:

w ≤ max







, w





+ λy



< w





+ λy

< π

And at any wage below π all workers gives the ﬁrm positive proﬁts which is suﬃcient to

show that ﬁrms hire all willing workers, which is the ﬁrst thing we wanted to show.

Now we want to ﬁnd the appropriate m

such that for all m

> m

all steady states are

in pure strategy. Given that we are ﬁxing y

> y

, we know that h

= 1 in all steady states.

The ﬁrm’s choice of wage obeys the following maximization problem:

max



w − rU

− λy



(w) + h



w, m



where



w, m





1 − m





w − rU



(w) .

And for any ε > 0 we can ﬁnd a m

close enough to 1 such that |h (w) | < ε for all wages

w ∈ [0, π + λy

]. And the solution to the above maximization problem when h (w, m

) = 0

is simply w

(rU

) + λy

. By a continuity argument, for any ε

> 0 we can ﬁnd a m

to 1 such that for any wage w posted in steady state must obey:

w ∈





+ λy

− ε

, w





+ λy

+ ε



From this we can derive that:

(w) = w −rU

> w





+ λy

− ε

− rU

And using our upper bound ¯q we get:

(¯q)

(¯q) + r





+ λy

+ ε



Combining the last two inequalities gives us that:

(w) >





+ λy





(α

(¯q) + r)



− 2ε

> w

(0)



(¯q) + r



− 2ε

Since w

(0) > 0, we can choose ε

small enough to get that c

(w) > 0 is assured. And

by a similar proof we can show that c

(w) > 0 if ε

is small enough. And that means we

can avoid the kinks that occur when c

(w) = 0. And away from those kinks the ﬁrm’s proﬁt

function is strictly concave. Hence when m

is close enough to 1, all ﬁrms must post the

same wage.

A.2 Proof of Proposition 1.3

The proof of this proposition is a corollary to the proof of proposition 1.2. Nowhere in the

proof of lemma A.1 was γ referenced, so we can ﬁnd the y

and m

independently of γ.

Moreover, the equations that characterize the δ = 0 solution do not rely on γ in any way.

Hence we can derive U

in the δ = 0 steady state independently of γ. And the size of city 1

that we derived at the δ = 0 solution was:



− rU



+ s



− rU



+ s



− rU



And since rU

−rU

> 0, if we choose γ small enough we can get Γ



−rU



to be arbitrarily

close to 1. And, then we extend this from the δ = 0 steady state to small positive δ using

the same upper hemicontinuity argument as in the proof of proposition 1.2.

A.3 Proof of Proposition 1.4

We will show how to match any ratio of unemployment rates. The proof for matching levels

of unemployment rate is similar. So we want to match unemployment rate ratio z =

> 1,

choosing the functional forms and the parameters. Set the matching function as

m (µ, υ) = mµ,

where µ and υ are the measures of searching workers and ﬁrms respectively. This functional

form implies that α

(q) = m for all q. And we set G to be uniform on the interval [0, N] and

Γ to be uniform on [−N, N]. Where N > π + λy

. Fix any ρ

, P

> 0 such that 2y

∈





We set γ = 1 and we will vary r while choosing k so that rk = 1 always holds. The other

parameters we take as ﬁxed. Consider the following system of equations.

δ + λρ

+ m

(

−(1−m

)λy

)

δ + m

(

)

+ s

This system of equations is simply the deﬁnitions of u

, m

in a pure strategy steady

state, putting in our functional forms and setting w

and w

+ m

+ λy

. We solve

this system for (m

, u

) while considering the other variables ﬁxed. It can be shown that

this reduces to a quadratic equation where (m

, u

) has only one solution where m

positive. Moreover, as we vary δ from 0 to ∞, the positive solution moves continuously and

the ratio

goes from ∞ to 1. By the intermediate value theorem we can xix δ so that the

solution, which we denote by ( ˆm

, ˆu

), has the property

ˆu

= z. And since z > 1 it must

follow that ˆm

> s

Now I claim there exists an r

such that for all r ≥ r

, every steady state is in pure

strategies where ﬁrms always underpay d workers in city 1 and all ﬁrms set h

= 1. Recall

that our restriction on ρ

that ensured ﬁrms engage in wage theft was

x (r + δ)

π + (r + δ) (x + P )

And as r goes to inﬁnity the right-hand side goes to 1, which makes this always hold.

Additionally, note that as r goes to inﬁnity rU

goes to zero for all i, j. And using condition

A.1 from above, and the fact that 2y

∈





as well, we can make a similar argument as

we did in the proof of lemma A.1 to show that all steady states are pure strategy.

From now we will only consider r ≥ r

. Putting in our functional forms into the ﬁrm’s

maximization problem we get that the ﬁrm solves:

max

− rU

− λe

π − w + λe

r + δ + λe

Taking the ﬁrst order condition gives us:



1 +

λe





π − 2w + 2λe

+ rU



= 0

As we take r → ∞ we get that rU

→ 0 from which it follows that

→

+ λm

and w

→

And these converges are exactly what deliver the system of equations we considered

earlier. Hence, as r goes to inﬁnity, u

, u

, m

converge to the ˆu

, ˆu

, ˆm

that we derived

above. Hence for r large enough we must close to our desired ratio z. All that remains to

be shown is that city 1 is larger than city when r is large. Since Γ is uniform, we can write

the size of city 1 as:

(rU

− rU

)

(rU

− rU

)

Hence city 1 will be larger if and only if

+ s

− r

> 0,

Our deﬁnition of U

can be written as:





− λy

− E



c|c ≤ c



1 +

λe

(

)

And for any worker of type i in city j, the term c

is chosen to maximize U

. Hence in every

steady state we get that

≥

mG (c

)



− λy

− E [c|c ≤ c

]



1 +

λe

)G(c

)

Deﬁne ˆc

, which is the limit of c

as we take r → ∞. Then we get that limit of

+ s

− r

is bounded below by



ˆc



− E



c|c ≤ ˆc



+ s



ˆc



− λy

− E



c|c ≤ ˆc



−mG



ˆc



− E



c|c ≤ ˆc



Using the limit we derived on w

, w

we get that converges to



ˆc



λ ˆm

− s

λy



And we already showed that ˆm

> s

so that term is strictly greater than 0. Hence it must

follow that for r large enough we have city 1 is larger than city 2. And we are done.

A.4 Proofs from section 1.7

A.4.1 Proof of Existence

We have two enforcement regimes (ρ

, ρ

, P

) in city 1 and (ρ

, P

) in city 2. As usual deﬁne

= x (1 −ρ

) − ρ

. And similarly y

= x (1 −ρ

) − ρ

. Also deﬁne ω

= π −

(r+δ)y

and ω

= π −

(r+δ)y

. These are the wage cutoﬀ above which the ﬁrm will always engage in

wage theft and below which it will never engage in wage theft. For simplicity we will treat

the case where ω

6= ω

in city 1. If ω

= ω

it must be that ρ

= ρ

and we simply treat

city 1 like city 2. Without loss of generality, let ω

< ω

and deﬁne ¯y = max {y

, y

}. Set

¯z

= π + 2 + λ¯y and ¯z

= π + 1 + λ¯y.

The main technique in this proof is we will reinterpret the ﬁrm’s action. We already

outline how to do this in city 2. For completeness we do the analogous thing for city. In city

1, instead of a wage, the ﬁrm will choose z ∈ [0, ¯z

]. We will deﬁne ˜w

(z) , ˜e

(z) that

tells us, for any z, what wage the ﬁrm is oﬀering and with what probability he will engage

in wage theft. We deﬁne ˜w

˜w

(z) =

(

z if z ∈ [0, ω

]

if z ∈ [ω

, ω

+ 1]

z −1 if z ∈ [ω

+ 1, ω

+ 1]

if z ∈ [ω

+ 1, ω

+ 2]

z −2 if z ∈ [ω

+ 2, ¯z

]

and deﬁne ˜e

(z) as

˜e

(z) =

(

0 if z ∈ [0, ω

]

z −ω

if z ∈ [ω

, ω

+ 1]

1 if z ∈ [ω

+ 1, ¯z

]

and ﬁnally

˜e

(z) =

(

0 if z ∈ [0, ω

+ 1]

z −ω

− 1 if z ∈ [ω

+ 1, ω

+ 2]

1 if z ∈ [ω

+ 2, ¯z

] .

And it is clear that ˜w

, ˜e

are all continuous function. We will also deﬁne functions to

capture the worker’s actions. Set

U =

, ¯c = π. Deﬁne ˜c

: [0, ¯z

] ×





→ [0, ¯c] as:

˜c

(z, U) = −rU + ˜w

(z) − ˜e

(z) λy.

Deﬁne ˜c

similarly. Clearly these functions are continuous in z. And these functions represent

the cutoﬀ on the labor cost below which the worker will accept an oﬀer of z. Now we can

redeﬁne the value functions of the agents.The ﬁrm’s value from a match at action z becomes:

(z) =

π − ˜w

(z) + ˜e

(z) λy

r + δ + λ˜e

(z) ρ

The worker’s value from a match given ﬁrm action z and labor cost cutoﬀ of c is simply

(z, c) =

˜w

(z) − c − ˜e

(z) λy

r + δ + λρ

˜e

(z)

The value of a searching ﬁrm becomes:

= max

z,h







−k +

i∈{a,d}

G(˜c

(z))(

(z) −

)







And we redeﬁne the ﬁrm’s mixed strategy F

as measures on the borel sets on of [0, ¯z

] ×

[0, 1] ×[0, 1] endowed with the weak topology. Then we can give the value of an unemployed

worker as:

= α







˜c

(z)





(z, c) |c ≤



˜c

(w)



−



(z, h

, h

) .

And all these value functions are now continuous in their arguments. There is a slight

diﬃculty in deﬁning u

which we deal with below. But otherwise our redeﬁnition of the ﬁrm’s

action has made everything continuous, and the existence proof can proceed using standard

Kakutani ﬁxed point arguments, which we omit. And the equilibrium we ﬁnd in this way

will be an equilibrium of the original game, for the same reasoning we gave in section 1.7.1

above. In particular ﬁrms in city 1 will never ﬁnd it optimal to choose z ∈ (ω

, ω

+ 1] or

z ∈ (ω

+ 1, ω

+ 2]. Hence at every wage oﬀered in equilibrium, there will only be one level

of wage theft that will be expected to occur. And we can reinterpret the equilibrium with

the expanded action space as in an equilibrium of the original model.

We now deal with the diﬃculty in deﬁning the unemployment rate. The reason for this

diﬃcultly is that workers of the same type may now lose their job at a diﬀerent rates. In

fact, since the rate of job loss depend on z, it is possible that workers of the same type

lose their job at a inﬁnitely many diﬀerent rates. So we now have to keep track of the

distributions of z in the employed population in order to derive the unemployment rate.

Let B([0, ¯z

]) be the borel measures on [0, ¯z

]. Then in each city we will have a probability

measure L

: B([0, ¯z

]) → [0, 1] where the value L

(A) is the percentage of employed workers

of type i in j working for a ﬁrm who posted any z ∈ A. Using L

, the number of workers of

type i in city j who lose their job at any z ∈ A can be written as:



1 − u





δ + λρ

˜e

(z)



(z) .

And the number of workers of type i in city j gaining a job at any z ∈ A is







˜c

(z)



(z, h

, h) .

And the steady state requirement is that exit and entries equal for all A ∈ B([0, ¯z

]). So we

get that for all A ⊂ B([0, ¯z

])







˜c

(z)



(z, h

, h) =



1 − u





δ + λρ

˜e

(z)



(A) . (A.2)

This deﬁnes u

in terms of L

(A), but we would like to derive u

without reference to such

an object. Extend L

to have the same domain as F

and it must be that L

and F

are

absolutely continuous with respect to each other. Hence we can apply the Radon-Nikoyn

theorem to tell us that:



1 − u



(A) = u







˜c

(z)





δ + λρ

˜e

(z)



(z, h

, h

) .

And then we simply set A = [0, ¯z

] and we get that:



1 − u



= u





[0,¯z

]



˜c

(z)





δ + λρ

˜e

(z)



(z, h

, h

) ,

which leads to



1 + α

)

[0,¯z

]

(

˜c

(z)

)

(

δ+λρ

˜e

(z)

)

(z, h

, h

)



which is our desired deﬁnition.

A.4.2 Proof of Proposition 1.6

Given the arguments made in the main body of the text we only need to prove the following

lemma.

Lemma. Fix all the parameters except (ρ

, P

). Then U

is maximized whenever ρ

= 1.

Proof: We ﬁrst deﬁne a few useful variables. For any ﬁxed q, let ˜w (q) , ˜c (q) be the

unique solution to the following system of two equations:

˜w = ˜c +

(q)

r + δ

˜c

(˜c − c) dc (A.3)

(˜c) (π − ˜w) − G (˜c) = 0 (A.4)

It can be shown that ˜w (q) is increasing in q, a fact we will use later.

Let (U

, w

, q

, c

) be the city 2 steady state variables when ρ

= 1. It can be shown

that w

= ˜w (q

) and c

= ˜c (q

). Now ﬁx any ρ

, P

with associated equilibrium variables

, F

, q

, e

(w) , c

(w). Recall that e

(w) ∈ [0, 1] is the probability that a city 2 ﬁrms

engages in wage theft given the opportunity. Optimality of the ﬁrm will ensure there is some

cutoﬀ wage ω

such that w > (<) ω

⇒ e

(w) = 1 (0).

The rest of the proof works as follows. We suppose for contradiction that U

> U

. We

then show that q < q

, and then we use this inequality show that w − e

(w) λy

≤ w

for

every w in the support of F

. Together these inequalities imply that U

< U

, giving us our

contradiction and ﬁnishing the proof.

Proof that q

< q

Firm’s optimality requires that any wage w posted in city 2 has to solve:

max



w − rU

− e

(w) λy



π − w + e

(w) λy

r + δ + λe

(w) ρ

(A.5)

And it is clear that the ﬁrm’s proﬁt is weakly higher if they could commit to never engaging

in wage theft by setting e

(w) = 0. Hence we can write that that

max



w − rU

− e

(w) λy



π − w + e

(w) λy

r + δ + λe

(w) ρ

≤ max



w − rU



π − w

r + δ

And the right-hand side of that inequality is strictly decreasing in U

and using that U

> U

we get

max



w − rU

− e

(w) λy



π − w + e

(w) λy

r + δ + λe

(w) ρ

< max

G (w − rU

)

π − w

r + δ

And the right hand side is the maximization problem the ﬁrm’s face in the ρ

= 1 case.

Hence, using the free entry condition it follows that q

< q

as desired.

Proof that w − e

) λy

≤ w

Note that equation A.5 can have at most one discontinuity at wage ω

, and on either side

of this discontinuity the problem is strictly concave. Hence F

can have at most ﬁnite wages

it mixes over. Let f

(w) be the probability of oﬀering any gave wage. Now we can deﬁne

using equation equation 1.13 as

)

(w) G (c

(w)) E [W

(w, c) |c ≤ c

(w)]

r +

(q) f (w) G (c

(w))

Let ¯w be the wage in the support of F

that gives the highest value for w − λy

(w).

Suppose the worker always gets wage oﬀer ¯w, but still employs cutoﬀ c

( ¯w). This makes the

worker weakly better oﬀ, giving us

≤

) G (c

( ¯w)) E [W

( ¯w, c) |c ≤ c

( ¯w)]

r + α

(q) G (c

( ¯w))

Which can be further re-arranged into

≤

(q) G (c

( ¯w)) (w − λye

( ¯w) −E [c|c ≤ c

( ¯w)])

r + δ + λe

( ¯w) ρ

+ α

) G (c

( ¯w))

And using the deﬁnition of c

(w) we know that

¯w − λy

( ¯w) −c

( ¯w) = rU

≤

(q) G (c

( ¯w)) (w − λye

( ¯w) −E [c|c ≤ c

( ¯w)])

r + δ + λe

( ¯w) ρ

+ α

) G (c

( ¯w))

Which we can rearrange into the following inequality

¯w − λy

( ¯w) ≤ c

( ¯w) +

)

r + δ

( ¯w)



( ¯w) −c



dG (c)

I now claim that the ﬁrm’s ﬁrst order condition must always hold in one direction for ¯w that

is:



( ¯w)



π − ¯w − λe

( ¯w)



− G



( ¯w)



≥ 0. (A.6)

First notice that c

( ¯w) > 0 because otherwise the wage ¯w would give zero proﬁt and hence

cannot be optimal for the ﬁrm. So the kink that occurs there is avoided. The only other

problem that might occur is if c

(w) is discontinuous at ¯w. This can only happen at the

wage where the ﬁrm is indiﬀerent between underpaying and not underpaying, and it must be

that for all w < ¯w we have e

(w) = 0 and for all w > ¯w we have e

(w) = 1. Then optimality

for the ﬁrm would require that e

( ¯w) = 0, otherwise the ﬁrm could oﬀer wage ¯w − ε and

have a discontinuous jump in the acceptance rate while giving up almost no proﬁt. Hence

we have the ﬁrm’s proﬁt function is diﬀerentiable from the left at ¯w and so if inequality A.6

fails to hold the ﬁrm could raise their proﬁt by oﬀering a wage slightly below ¯w.

Now suppose for contradiction that w

< ¯w −λy

( ¯w). Well we know that w

= ˜w (q

)

and we can use the above inequality in equation A.6 to show that



( ¯w)



(π − ˜w (q

)) − G



( ¯w)



> 0.

And this inequality is decreasing in the c

term, so using the equation A.3 it must be that

˜c (q

) > c

( ¯w). Then, recalling that q

< q

we get

¯w − λy

( ¯w)

≤ c

( ¯w) +

)

r+δ

( ¯w)

( ¯w) −c) dG (c) < ˜c (q

) +

)

r+δ

˜c(q

)

(˜c (q

) − c) dG (c)

= ˜w (q

) .

Which is a contradiction. Hence, we can infer that w

≥ ¯w − λye ( ¯w), which is what we

wanted to show. 

Appendix B

Appendix to Chapter 2

B.1 Proof of Lemma 2.1

B.1.1 Part 1

WLOG suppose party L chooses a partisan candidate (from l) and party R choose a moderate

candidate (from m). We want to show that the m candidates gets a strictly high measure

of votes. Well, all of the m and r types citizens will vote for the moderate candidate, all of

the l types citizens will vote for the L candidate. So the most votes the partisan candidate

can get is

Now to ﬁgure out how many votes the moderate candidate gets consider two cases. Let

x ∈ (0, 1) be the fraction of citizens who are α

type.

Case 1) Last period R choose a moderate. That means both the old and young generation

of the m types are energized so they all vote. So R gets measure

votes from the m types.

Among the r types at least all the citizens with cost α

will vote, and since x > 0 there is

a strictly positive measure of them. So R get’s strictly more than

measure of votes and

wins for sure.

Case 2) Last period R choose a partisan. That means the old generation of the r types

are energized and young generation of the m types are energized. The α

types always vote,

hence R gets at least votes of



x +

(1 − x)





x +

(1 − x)







And since x > 0 that value is strictly greater than

and hence R wins for sure.

And these two cases exhaust all the possibilities. 

B.1.2 Part 2

If both parties choose a moderate then they split the votes from m evenly. The partisan

citizens each vote for their own party, so the outcome is determined by which party has a

higher turnout from their base. The young citizens in both bases are not energized, so only

the α

types vote, and each base has the same number of those. So all that matters is the

turnout from the old citizen in the base. Of those the α

always vote, and the α

types

vote iﬀ they were energized last period. If both or neither bases were energized last period

then there is a tie. If one base was energized and the other wasn’t, the side who energized

the base gets a higher turnout and wins. 

B.2 Proof of proposition 2.1

To make this simpler we will introduce some notation. In addition to the states that begin

periods such as (L, m, m) we will introduce states in the middle of periods. For example

(L, m, m, l) means last period L won last and both parties played m. This period L has

already gone and chosen l, and now it is R’s turn to play. Also I will let V

(σ

, σ

|h)

denote the payoﬀ for player i at history h given strategies σ

, σ

. As shorthand I will write

(h) = V

(σ

, σ

|h), where σ

is the unique equilibrium discussed in the text. And since

is Markovian I can abuse notation and write V

(s) for any state s.

B.2.1 Showing σ

is an equilibrium

This can be veriﬁed using the one shot deviation principle, which will hold strictly at every

point. I will only consider states where L won last period, all the other states are entirely

symmetric.

Calculating Continuation Payoﬀs

I will do the calculations state by state. I will go through the ﬁrst two calculations carefully,

and then more quickly. Using the cycle depicted in ﬁgure 1 we can determine that a m

candidate wins in every period. Also L will cycle between winning two periods in a row and

losing two periods in a row. So his payoﬀ stream for this state looks like:

(L, m, m) = γ − .25 − .25β −.25β

+ β

(γ + −.25) + β

(−.25 + γ)...

= γ +

−.25

1 − β

+ γβ

1 + β

1 − β

0.25β

− γ + βγ − β

γ + 0.25

− β

+ β − 1

Now consider state (L, l, r). Here both parties will choose an m candidate, and the

outcome will be decided by a coin ﬂip sending us to state (L, m, m) or (R, m, m). So we get

(L, l, r) = .5(−.25 + γ + βV

(L, m, m)) + .5(−.25 + βV

(R, m, m))

We need to calculate V

(R, m, m). Well by symmetry we know V

(R, m, m) =

(L, m, m). And since in the cycle a me a moderate candidate wins in every period we

know that:

(L, m, m) + V

(L, m, m). =

−.5 + γ

1 − β

So we can calculate that

(L, l, r) = .5(−.25 + γ + βV

(L, m, m)) + .5(−.25 + β(

−.5 + γ

1 − β

− V

(L, m, m))

4(1 − β)

(2γ − 1)

Now we will go more quickly.

(L, m, r) =

−.25

1 − β

+ γβ

1 + β

1 − β

0.25β

− β

γ + 0.25

− β

+ β − 1

Now we calculate R’s payoﬀ in these states.

(L, m, m, m) =

−.5 + γ

1 − β

− V

(L, m, m) =

4γβ − β

− 1

4 (1 − β) (β

+ 1)

(L, m, r, m) =

−.5 + γ

1 − β

− V

(L, m, r) =

4γ − β

− 1

4 (1 − β) (β

+ 1)

(L, l, r, m) =

−.5 + γ

1 − β

− V

(L, l, r) =

2γ − 1

4(1 − β)

(L, m, m, l) = −.25 + γ + βV

(R, m, l)

= −.25 + γ + βV

(L, m, r) = −.25 + γ + βV

(L, m, r) =

− β

+ β − 1



0.25β

− γ + βγ − β

γ + 0.25



(L, m, r, l) = γ + βV

(R, r, l) = γ + βV

(L, l, r)

4γ − β − 2βγ

4 (1 − β)

(L, l, r, l) = −.25 + γ + βV

(R, m, l) = −.25 + γ + βV

(L, m, r)

0.25β

− γ + βγ − β

γ + 0.25

− β

+ β − 1

And that covers all the states. Next we check to see if a deviation in each of these cases

is proﬁtable.

Checking (one shot) Deviations for L

Consider the states (L, m, m). If L deviates to l, R will choose m and L gets payoﬀ stream

−.25 + βV

(R, m, l) = −.25 + βV

(L, m, r, m)

= −.25 + βV

(L, m, r, m) =

0.25β

− γβ + 0.25

− β

+ β − 1

So we have the equilibrium action is strictly optimal iﬀ

0.25β

− γ + βγ − β

γ + 0.25

− β

+ β − 1

−

0.25β

− γβ + 0.25

− β

+ β − 1

> 0 ⇐⇒

+ 1

(1 − β) > 0

100

Which always holds given our assumptions on β, γ. Now consider state (L, l, r). If L deviates

to pick l he gets payoﬀ

−.25 + βV

(R, m, l) =

0.25β

− γβ + 0.25

− β

+ β − 1

So the equilibrium action is strictly optimal if and only if

2γ − 1

4(1 − β)

−

0.25β

− γβ + 0.25

− β

+ β − 1

> 0 ⇐⇒ 0.5

+ 1

(1 − β) > 0

Which always holds. Now consider state (L, m, r). If L deviates R will pick r and win

for sure. By symmetry it must be that V

(R, r, l) = V

(L, l, r) = V

(L, l, r, m) this yields

payoﬀ:

−1 + βV

(R, r, l) = −1 + βV

(L, l, r, m) =

3β + 2βγ − 4

4 (1 − β)

So the equilibrium action is strictly optimal when

0.25β

− β

γ + 0.25

− β

+ β − 1

−

3β + 2βγ − 4

4 (1 − β)

> 0 ⇐⇒

0.25 (3β

− 2βγ + 2β

γ + 3)

+ 1

> 0 ⇐⇒

γ(−2β + 2β

) > −3 −3β

⇐⇒ γ <

−3 − 3β

−2β + 2β

3(1 + β

)

2β(1 −β)

And assumption 1 is a suﬃcient condition to guarantees this holds.

Checking (one shot) Deviations for R

Consider state (L, m, r, m). If R deviates to r he gets payoﬀ

−.25 + βV

(L, m, r) = −.25 + βV

(L, m, r, m)

101

And since in equilibrium a moderate candidate always win, and R gets to win at least once,

we know that V

(L, m, r, m) >

−.25

1−β

. Thus it must hold that

(L, m, r, m) > βV

(L, m, r, m) −.25

Which is precisely what we need for this one shot deviation to be strictly unproﬁtable. Now

consider state (L, m, m, m). If R deviates to m he gets payoﬀ

(1 − p)(−.25 + γ + βV

(R, m, m)) + p(−.25 + βV

(L, m, m))

And hence the deviation is strictly unproﬁtable iﬀ

(L, m, m)(1 −pβ) > −.25 + (1 − p)(γ + βV

(R, m, m))

And using that by symmetry V

(R, m, m) = V

(L, m, m) we get the deviation in unproﬁtable

when:

+ 1

(p − 1 + pβ) > 0 ⇐⇒ β >

1 − p

Which is precisely assumption 2. Now consider state (L, l, r, l). If R deviates to r he gets

payoﬀ

p(−1 + βV

(L, l, r)) + (1 −p)(γ + βV

(R, r, l)) =

p(−1 + β

2γ − 1

4(1 − β)

) + (1 − p)(γ + β

4(1 − β)

(2γ − 1)) =

4 (β − 1)

(4p + β − 4γ − 4pβ + 4pγ + 2βγ − 4pβγ)

So this deviation is strictly unproﬁtable when;

102

0.25β

− γ + βγ − β

γ + 0.25

− β

+ β − 1

−

4 (β − 1)

(4p + β − 4γ − 4pβ + 4pγ + 2βγ − 4pβγ) > 0

⇐⇒ −

0.25

+ 1



− 4pγ − 4p + 2βγ − 4pβ

+ 2β

γ − 4pβ

γ + 1



> 0

⇐⇒ β

− 4pγ − 4p + 2βγ − 4pβ

+ 2β

γ − 4pβ

γ + 1 < 0

⇐⇒ β

+ 2βγ + 2β

γ + 1 − 4p



γ + 1 + β

+ β



< 0

We know p ∈ (.5, 1) so that 4p > 2. And clearly (γ + 1 + β

+ β

γ) > 0. Hence a suﬃcient

condition for the above inequality is

+ 2βγ + 2β

γ + 1 − 2



γ + 1 + β

+ β



< 0 ⇐⇒ −β

+ 2γβ − 2γ − 1 < 0

Which holds because β < 1 ⇒ 2γβ − 2γ < 0. Now consider state (L, m, m, l). If R deviates

to r he gets payoﬀ

.5(−.25 + γ + βV

(R, r, l)) + .5(−.25 + βV

(L, l, r))

= −.25 + .5γ + .5β(V

(R, r, l) + V

(R, r, l))

= −.25 + .5γ + .5β

−.5 + γ

1 − β

2γ − 1

4(1 − β)

So the equilibrium action is strictly proﬁtable when

− β

+ β − 1



0.25β

− γ + βγ − β

γ + 0.25



−

2γ − 1

4(1 − β)

> 0 ⇐⇒ 0.5

γ (1 − β)

+ 1

> 0

Which always holds. Now consider state (L, l, r, m). If R deviates he gets payoﬀ

−.25 + βV

(L, m, r) =

0.25β

− γβ + 0.25

− β

+ β − 1

So the equilibrium action is strictly optimal when:

103

4(1 − β)

(2γ − 1) −

0.25β

− γβ + 0.25

− β

+ β − 1

> 0 ⇐⇒ 0.5

γ(1 −β)

+ 1

> 0

Which always holds.

Now the ﬁnal state (L, m, r, l). If R deviates to m he gets payoﬀ

−.25 + γ + βV

(R, m, l) = −.25 + γ + βV

(L, m, r) =

0.25β

− γ + βγ − β

γ + 0.25

− β

+ β − 1

So the equilibrium action is strictly optimal when

−β + 4γ − 2βγ

4 (1 − β)

−

0.25β

− γ + βγ − β

γ + 0.25

− β

+ β − 1

> 0

⇐⇒

0.25

+ 1



+ 2βγ + 2β

γ + 1



> 0

Which always holds.

And so we’ve proved there is no proﬁtable one shot deviation and it follows that σ

a subgame perfect equilibrium. Moreover we’ve shown all one shot deviations are strictly

unproﬁtable, a fact we will use later. 

B.2.2 Proof that σ

is unique

Two Lemmas

I will start by establishing some lemmas.

Lemma B.1. Take any history h

that starts a period and suppose party L pursues strategy

from here on. Then following σ

is the unique strategy of R that minimizes the payoﬀ of

L. The same statement holds switching the role of L and R.

Proof.

104

Fix any history h

that starts a period t. We assume L follows σ

from here on. Thus R

in minimizing L’s payoﬀ is trying to solve

min

(σ

, σ

)

In this problem we can consider L actions ﬁxed and known, this is identical so solving a

standard dynamic programming problem. I claim that to prove Lemma B.1 it will suﬃce to

prove the following statement:

Statement A: Take any history h, when it is R’s turn to act, and that follows h

,and in

which L has never deviated from σ

after h

, then L is strictly worse oﬀ if R plays σ

opposed to any one shot deviation from σ

That this statement implies Lemma B.1 can be shown through standard dynamic pro-

gramming arguments. In particular there is a constant discount rate β < 1, and the per

period payoﬀs are bounded, so the usual theorems apply. The uniqueness of σ

follows from

the strictness assumed in the statement.

Proving Statement A

Here we show Statement A holds. Fix the appropriate histories h

and h. Let S be the

state at history h. We assumed that h was such that it is R’s turn to act, and L never

deviates, hence it can only be at one of 6 states:

S ∈ {(R, m, m), (R, m, l), (R, r, l), (L, m, m, m), (L, m, l, m), (L, l, r, m)}

Notice that for every state except (R, m, l), no matter what one shot deviation R pursues

a moderate candidate will win every period from here on (this depends on the fact that L

is using σ

.) And whenever a moderate candidate always win the policy payoﬀ for both

parties is always −.25 and the only variability is which party gets the γ in each period, which

means the parties payoﬀ will sum to −.5 + γ in every period. Thus for every state except

105

(R, m, l) a one shot deviation makes L strictly better oﬀ iﬀ it makes R strictly worse oﬀ.

And we already showed above that every one shot deviation from σ

makes R strictly worse

oﬀ and hence would make L strictly better oﬀ.

So now we have just one state left: (R, m, l). If R doesn’t deviate L gets payoﬀ:

4γ − β

− 1

4 (1 − β) (β

+ 1)

If R deviates then L gets the σ

equilibrium payoﬀ from (R, m, l, r) in game G. Which is

the same payoﬀ R receives in states (L, m, r, l). Checking from above this is just

4γ − β − 2βγ

4 (1 − β)

So not deviating makes L strictly worse oﬀ when

4γ − β

− 1

4 (1 − β) (β

+ 1)

−

4γ − β − 2βγ

4 (1 − β)

< 0

⇐⇒ −

4 (β

+ 1)



− 2βγ + 2β

γ + 1



< 0

⇐⇒ γ(−2β + 2β

) > −1 −β

⇐⇒ γ <

1 + β

2β(1 −β)

And this is guaranteed to hold by assumption 1 

Lemma B.2. Fix any equilibrium σ

, σ

and any history h that starts a period. Then:

(σ

, σ

|h) = V

(h)

Proof.

Fix an equilibrium (σ

, σ

), and a history h that starts a period. Consider party L. Note

that σ

is a feasible strategy, which L could pursue. And by Lemma B.1 the lowest payoﬀ

106

he can get is the case where R pursues σ

, which would yield V

). Thus by optimality

must do at least as well. We can apply the symmetric argument to R. and we get the

following inequalities:

(σ

, σ

) ≥ V

) and V

(σ

, σ

) ≥ V

) (*)

Notice if a moderate candidates wins the combined period payoﬀ of the parties must be

γ −.5, if an extreme candidates win it must be γ − 1. These are the only two possibility so

we have the combined period payoﬀ must range between these numbers, which implies that

γ − 1

1 − β

≤ V

(σ

, σ

) + V

(σ

, σ

) ≤

γ − .5

1 − β

(**)

But notice that in the equilibrium σ

for any history h that starts a period the moderate

candidates will always wins in every period, thus V

) + V

) =

γ−.5

1−β

. So if either

inequality in equation * was strict the upper bound on equation ** would be violated. And

thus the inequalities in * must hold with equality, which is what we wanted to show.

Proof of Uniqueness Given Lemmas B.1 and B.2

Take any equilibrium σ

, σ

. First take any history h and. WLOG suppose it is R’s turn

to act. 3 cases.

Case 1: h starts a period.

If σ

(h) recommends a diﬀerent action than σ

(h) with positive probability then by

Lemma B.1 we must have V

(σ

, σ

|h) > V

(h). But by optimality of an equilibrium we

must have V

(σ

, σ

|h) ≥ V

(σ

, σ

|h) > V

(h). And this violates Lemma B.2.

Case 2: h is in the middle of a period. By optimality we must have for any action a

(σ

(h)) + V

(σ

, σ

|(h, σ

(h))) ≥ g

(a) + V

(σ

, σ

|(h, a))

107

And in particular that must hold for the action a = σ

(h). And for any action c

that R takes, the next history will start a period, and so we can apply Lemma A2 to say:

(σ

, σ

|(h, c)) = V

(h, c) for any action c. So we can write that:

(σ

(h)) + V

(h, σ

(h)) ≥ g

(σ

(h)) + V

(h, σ

(h))

But since the one shot deviation principle holds strictly for σ

this can only happen if

(h) = σ

(h) as desired. 

B.3 Proof of Proposition 2.2 (Oﬃce Motivated Candi-

dates)

First set

(p) =

p −



− 4 (1 − p)



p −



1/2



p −



And note for any p ∈ [.5, 1] we have that β

(p) ∈ (0, 1) and β ∈ (0, β

(p)) implies

p (1 + β − β

) − 1 +

< 0 and β ∈ (β

(p) , 1) implies p (1 + β − β

) − 1 +

> 0.

Deﬁne σ

oﬃce

State L’s Action R’s Action R’s Action if L deviates

(L, m, m) m r m

(L, m, r) l r m

(L, l, r) m m m

And σ

short

as:

State L’s Action R’s Action R’s Action if L deviates

(L, m, m) m m m

(L, m, r) l r m

(L, l, r) m m m

108

An examination of σ

short

will quickly verify that any state will lead back to (L, m, m) or

(R, m, m) within two periods, at which point it will alternate stochastically between those

two states forever.

B.3.1 Patient Oﬃce Motivated Parties

Here we assume β > β

(p). We want to show σ

is the unique equilibrium. This is just an

application of the one shot deviation principle. I won’t go through all the calculations here,

it is similar to what was done above. The only diﬃcult calculation is when party R is faced

with state (L, m, m, m). In this case it turns out it is proﬁtable to not deviate if and only if

p (1 + β − β

) − 1 +

> 0 which is guaranteed by the fact that β > β

(p) .

To show σ

oﬃce

is unique as again similar to the proof of proposition 2.1, except now it

is even easier. The equivalent of Lemma B.1 is now immediate because the game is zero

sum, and the rest of proof proceeds similarly. The proof also relies on the one shot deviation

principle holding strictly at every point, which it does.

B.3.2 Impatient Oﬃce Motivated Parties

Here we assume β < β

(p) and we want to show that σ

short

is the unique equilibrium. Again

this is very similar to what we just did in the proof of proposition 2.1. At state (L, m, m, m)

party ﬁnds playing m to be optimal exactly when β < β

(p). Optimality of R playing m at

state (L, l, r, l) holds when β < 1/2p, but it can be easily veriﬁed this is guaranteed by the

fact that β < β

(p). And uniqueness is proven just as above.

109

Appendix C

Appendix to Chapter 3

C.1 Notation and Preliminaries

In this section we discuss the notation used in the appendix and show a few useful inequalities

that describe the behavior of the players. We use a diﬀerent notation in the appendix than

was employed in the paper. In particular we restrict the beliefs to the countable set of

achievable beliefs. Just after learning occurs we know that the belief p ∈ {1 −γ, γ}. And

when no learning occurs we the belief evolves as p

= (1 −p) (1 − γ) + pγ. And these are the

only two ways the belief can evolve. Hence we can ﬁnd all possible beliefs by forward solving

this equation starting from 1 −γ and γ. In particular we let p

−1

= 1 − γ and p

= γ. Then

we recursively deﬁne p

n+1

= (1 −γ) p

+ γp

. So the set {p

−1

}

∞

n=1

is the set of all reachable

beliefs less than

and {p

}

∞

n=∞

is the set of all reachable beliefs greater than

We also show a few inequalities that characterize the players’ behavior. The inequalities

are derived from the value functions and equalities we discussed in section 3.3. Without

loss of generality, we still display these inequalities for beliefs less than

but analogous

statements would hold for beliefs greater than

. First at any belief p

−1

the voter prefers

110

policy −1 over policy 0 if and only if:



−1

n+1



1 − 2 (2γ − 1)

β (1 − r)

≤ W



−1



And the voter prefers policy 1 over policy 0 if and only if:



−1

n+1



1 + 2 (2γ − 1)

β (1 − r)

≤ W



−1



And the voter always prefers policy the favored ideological policy over the unfavored one

so we don’t need to rank −1 and 1. Now for the parties. At belief p

−1

party L prefers policy

−1 over if and only if

(2γ − 1)



−1



− V



−1

n+1



≥

−1

β (1 − r)

The L party never implements policy 1 so we don’t have state inequalities about 1. Using

the zero sum property we can show the R party prefers policy 1 over 0 if and only if

(2γ − 1)



−1



− V



−1

n+1



≤

β (1 − r)

And these four inequalities characterize what policies the players prefer. They will be

used extensively in the proofs below.

C.2 Two Useful Lemmas

In this section we prove two very useful lemmas that we will makes use of in the proof of the

propositions.

Lemma C.1. If at state p the voter ever prefers policy 0 over policy −1 then it must be that

W (p) ≤

−1

1−β

111

Proof.

WLOG take p

−1

and suppose the R party is chosen and implements 0 with positive

probability at this belief. We want to show that W (p

−1

) ≤

−1

1−β

. First take the case that

for all m > n either x

−1

= 0 OR 0 is the voter’s most preferred policy. This would give

us that: W (p

−1

) = −1 + βW



−1

m+1



OR W (p

−1

) ≤ −1 + βW



−1

m+1



. Either way we can

solve forward and show thatW (p

−1

) ≤

−1

1−β

.as desired.

Now take the other case where there exists some m > n such that x

6= 0 and 0 is not the

voter’s most preferred policy. WLOG let this be the smallest such m. Since 0 is not the most

preferred policy, −1 must be. By choice of m as the smallest we can use the same logic as

above to derive that: W



−1

n+1



≤ −1−β +..−β

m−n−2

+β

m−n−1

W (p

−1

). That −1 is optimal

for the voter at m requires that W (p

−1

) ≤ −2+2 (2γ − 1)

+β (1 − r) W



−1



+βrW (p

m+1

Combining this with the above equation we get that:



−1

n+1



≤

−1 − β − ... −β

m−n−2

+ β

m−n−1



−2 + 2 (2γ − 1)

+ β (1 − r) W



−1



+ βrW (p

m+1

)



Voter’s optimality at m also requires that W



−1

m+1



1−2(2γ−1)

β(1−r)

≤ W



−1



. Plugging in

and simplifying yields:



−1

n+1



≤

−1 − β − .. −β

m−n−2

+β

m−n−1



−2 + 2 (2γ − 1)

+ βW



−1



+ r



−1+2(2γ−1)

1−r



112

By assumption, at n the voter prefers 0 over −1 which has optimality condition of W



−1

n+1



1−2(2γ−1)

β(1−r)

≥ W



−1



. Plugging in and rearranging gives and using the fact that m > n

implies −2 (2γ − 1)

< −2 (2γ − 1)

gives:



−1

n+1



(1 − β

m−n

) <

−1 − β + .. −β

m−n−2

+ β

m−n−1



−1 + 2 (2γ − 1)

1−r

(−2 (2γ − 1)

(1 − r))



Which becomes:



−1

n+1



−1−β−..−β

m−n−2

−β

m−n−1

1−β

m−n

= −

1−β

m−n

1−β



1−β

m−n



−1

1−β

As desired 

Lemma C.2. In any symmetric MPE:

1. W (p

−1

) ≥

−2

1−β

for all n

2. If W (1 − γ) ≥ (>)

−1

1−β

β(1−r)

then W (p

−1

) ≥ (>)

−1

1−β

for all n,

3. If W (1 − γ) ≤

−1

1−β

β(1−r)

then W (p

−1

) ≥

−2

1−βr

β(1−r)W

(

−1

)

1−βr

for all n.

Proof of part 1: If the voter only elect the belief-favored party, then the worst case

value for E [y

] is −2. This is a feasible strategy so it must be that the voter gets payoﬀ

greater than

−2

1−β

at all times.

Proof of part 2: Suppose W (1 −γ) ≥

−1

1−β

β(1−r)

. Fix any n. One possible strategy

the voter can pursue is always electing the L party at all p

−1

with m ≥ n until the next

time the state is revealed then revert to the equilibrium strategy. The L party will only

113

implement −1 or 0 (or some mixture) and I claim the worst possible outcome for the voter

in this case can face is x

= 0 for all m ≥ n. In this case the payoﬀ would be:W (p

−1

) =

−1

1−β

for all m ≥ n. The principle of optimality tells us that a suﬃcient condition for this to be

the worst possible path overall is that, for each m ≥ n, deviating to −1 would improve the

voter’s payoﬀ. Well we know this holds exactly when:

−1

1−β

1−2(2γ−1)

β(1−r)

≤ W



−1



. And a

suﬃcient condition for that to be true for all m ≥ n is that W



−1



≥

−1

1−β

β(1−r)

, which

we assumed to be true. So the voter has a strategy available that has a worst case payoﬀ of

−1

1−β

, and it follows that W (p

−1

) ≥

−1

1−β

as desired.

Proof of part 3: Now assume W (1 − γ) ≤

−1

1−β

β(1−r)

. WLOG we consider p

−1

Again consider the strategy where the voter only elects the belief favored party until the next

learning event then returns to his equilibrium strategy. Consider a modiﬁed problem where

the voter always gets period payoﬀ of −2 for the favored ideological policy until the next

learning event. At the next learning event the voter gets continuation payoﬀ W (1 − γ). This

modiﬁed problem is (weakly) worse under every possible path. Hence if we consider the worst

possible path in this modiﬁed game until the next learning event that path yield payoﬀ weakly

below W (p

−1

). I claim the worst case in this modiﬁed problem using the favored party only

strategy is when the voter gets the favored ideological policy in every period. In this case the

voter’s payoﬀ is:

−2

1−βr

β(1−r)

1−βr

W (1 − γ). And to verify this is the worst case we can apply

the principle of optimality. All we need to show is that

−2

1−βr

β(1−r)

1−βr

W (1 − γ) +

β(1−r)

≥

W (1 − γ). Which we can rewrite as:

−1

1−β

β(1−r)

≥ W



−1



. Which we assumed holds.

Hence the worst case payoﬀ in the modiﬁed game is

−2

1−βr

β(1−r)

1−βr



−1



, which is a lower

bound on the worst case payoﬀ in the original game. 

C.3 Proofs of Propositions

C.3.1 Proof of proposition 3.1

Proof of part 1.

114

First suppose that W



−1



−1

1−β

β(1−r)

. I will show that experimentation occurs at

some belief p

−1

. Of course by symmetry experimentation occurs at p

. And at least one

of these beliefs will occur in any equilibrium with probability 1 as t → ∞.

First I will show that for n large enough, at any belief p

−1

the voter prefers both −1 and

1 to 0. Well W



−1



−1

1−β

β(1−r)

immediately ensures ensure the voter’s best option is

always −1 whenever p <

. This is just a consequence of lemma C.2 combined with lemma

C.1. From that it follows that we can put an upper bound on the voter’s payoﬀ for any n:

W (p

−1

) ≤

−2+βr(1−r)W

(

−1

)

1−βr

2(2γ−1)

1−βr(2γ−1)

. Now we want to show that for n large enough the

voter strictly prefers 1 over 0 at p

−1

. Suppose for contradiction the voter prefers 0 over 1.

Optimality requires that:



−1



≤ W



−1

n+1



1 + 2 (2γ − 1)

β (1 − r)

≤

−2 + βr (1 − r) W



−1



1 − βr

2 (2γ − 1)

n+1

1 − βr (2γ − 1)

1 + 2 (2γ − 1)

β (1 − r)

Rearranging we get:



−1



≤

−1

1 − β

β (1 − r)

1 − βr

1 − β

2 (2γ − 1)

n+1

1 − βr (2γ − 1)

2 (2γ − 1)

β (1 − r)

And as we take n to ∞ the righthand side converges to

−1

1−β

β(1−r)

. And so W



−1



−1

1−β

β(1−r)

implies there is an n

large enough such that for all n > n

the above inequality

fails and the voter strictly prefers −1 and 1 to 0 at p

−1

. Notice this implies that at any

such belief R can only get elected if it implements policy 1 with probability 1. For if R

implemented policy 0 with positive probability that means L would strictly prefers policy

−1. And hence the voter would have elected L instead. And by a similar argument L can

only get elected if they implement −1 with positive probability.

115

All that needs to be show now is the the R party wins at p

−1

for some n > n

. Well

suppose not for contradiction. Then the L party has the option of playing x = 0 at all such

beliefs. This will yield a payoﬀ of

1−β

. Hence optimality requires that V

−1

) ≥

1−β

for

all n > n

. Recall that we assumed λ >

1−β+2βr

β(1−r)

1−β

β(1−r)

. Hence it follows that V



−1

n+1



β(1−r)

. But for n suﬃciently large that means V



−1

n+1



> (2γ − 1)



−1



β(1−r)

. And

this is exactly the optimality condition that says party L strictly prefers policy 0 at p

−1

And we know if party L prefers 0 then party R must strictly prefer 1. But since n > n

the

voter would elect party R over L in this case. And that gives us our contradiction. Hence

R must be elected for some n > n

. And this ﬁnishes the ﬁrst direction.

Now suppose that experimentation occurs. WLOG there exists an n where R is elected

at p

−1

and implements policy 1 with probability 1. This requires that the voter weakly

prefers policy 1 to policy 0, otherwise policy 1 is the voter’s worst outcome and they would

have elected L instead.

. Hence the voter’s optimality condition requires that W



−1

n+1



1+2(2γ−1)

β(1−r)

≤ W



−1



. Suppose for contradiction that W



−1



≤

−1

1−β

β(1−r)

. Then by

Lemma C.2 we’d have that W



−1

n+1



≥

−2

1−βr

β(1−r)W

(

−1

)

1−βr

. Combining this with the voter’s

optimality condition yields:

−2

1−βr

β(1−r)W

(

−1

)

1−βr

1+2(2γ−1)

β(1−r)

≤ W



−1



. And that can

rearranged into:

−1

1 − β

β (1 − r)



1 − βr

1 − β



2 (2γ − 1)

β (1 − r)

≤ W



−1



Which is the desired contradiction. Hence

−1

1−β

β(1−r)

< W



−1



must hold.

Proof of Part 2:

Proof for γ

. Here we prove that γ close to 1 implies W



−1



−1

1−β

β(1−r)

. Fix

any equilibrium and let n

be the lowest n such that L party would not implement policy

−1 if elected at p

−1

. If such a period does not exist we set n

= ∞. I claim that n

→ 1

as γ → 1. This can be proved using methods similar to proof of proposition 3.2. Well one

Recall we know the L party will never give positive probability to policy 1 in equilibrium

116

partial strategy the voter can pursue is to always elect the ideological dominant party at any

−1

or p

where n < n

. This would produce payoﬀ of:



−1



k=1



−2 + 2 (2γ − 1)

+ β (1 − r) W



−1





+ (βr)



−1



Which we can rearrange into:



−1





1 −

β(1−r)



1−(βr)



1−βr



−2



1−(βr)



1−βr

2(2γ−1)



1−(βr(2γ−1))



1−βr(2γ−1)

+ (βr)

W (p

)

Notice that (βr (2γ − 1))

≤ 0 and from lemma C.2 we know that W (p

) ≥

−2

1−β

. Hence

we can write that:



−1



≥



1 −

β(1−r)



1−(βr)



1−βr



−1



−2



1−(βr)



1−βr

2(2γ−1)



1−(βr)



1−βr(2γ−1)

+ (βr)

−2

1−β



And as γ → 1 and n

→ ∞ the right hand side of that inequality converges to 0. And

recall we assumed β >

2−r

which is suﬃcient to ensure that

−1

1−β

β(1−r)

< 0. And so for γ

close enough to 1 it follows that W



−1



−1

1−β

β(1−r)

as desired.

Proof for γ

: Now we prove that γ close to .75 implies W



−1



≤

−1

1−β

β(1−r)

. Well

notice that the largest value of E [y] the voter can ever get is policy −1 at p

−1

, which yields

value −2 + 2 (2γ − 1), and this represents the largest possible period the voter can ever get.

And hence in any equilibrium W (p

) ≤

−2+2(2γ−1)

1−β

→

−1

1−β

as γ → .75 as desired.

Proof of part 3. In the proof for part 1 we proved there existed a n

such that for all

n > n

at p

−1

the voter prefers policies −1 and 1 to policy 0. Furthermore we showed that

R can only win if they implement 1 with probability 1 and L can only win if they implement

117

−1 with positive probability. Here we will show there exists a m

> n

such that the same

party is not elected with positive probability at p

−1

and p

−1

n+1

for all n > m

. The other side

will be implied by symmetry.

Suppose for contradiction that R is elected with positive probability at beliefs p

−1

, p

−1

n+1

where we can think of n as arbitrarily high. We already showed that L can’t be in power at

−1

for all n > n

and a similar proof works for R. Hence WLOG we can suppose R is not

in power at p

−1

n+2

. For ease of exposition we will set (2γ − 1)

= 0. The exact same proof

can be down without making this simpliﬁcation and just carrying around extra terms.

At p

−1

n+2

we know the L party is elected and must play −1 with positive probability.

Which means that either L plays −1 with probability 1 or is indiﬀerent between playing

−1 and 0. Either way we can write that V



−1

n+2



= 1 + λ + βrV



−1

n+3



. Combining

that with the fact that we know that at p

n+1

the R party is elected and plays −1 we get:



−1

n+1



= −(1 + λ) (1 −βr) + (βr)



−1

n+3



The optimality condition for L at n + 2 is

β(1−r)

≥ V



−1

n+3



. We can use that to get:



−1

n+1



≤ −(1 + λ) (1 −βr) +

(βr)

β(1−r)

. And the optimality condition for R at n + 1is

β(1−r)

≤ V



−1

n+1



. This leads to the following inequality:

β (1 − r)

≤ V



−1

n+1



≤ −(1 + λ) (1 − βr) +

(βr)

β (1 − r)

We can re-arrange this to get:

1−(βr)

β(1−r)

≤ −(1 + λ) (1 − βr). Which gives us our con-

tradiction since the left hand side is negative and the right hand side is positive. And our

simpliﬁcation that (2γ − 1)

= 0 is irrelevant because both sides are strictly bounded away

from 0 and don’t depend on n.

Now suppose that L is in power at p

−1

and p

−1

n+1

with n arbitrarily high. Again we make

the simpliﬁcation that (2γ − 1)

= 0. Using similar logic to what we just did we can derive

the condition that: 1 + λ + βrV



−1

n+2



≤ V



−1

n+1



≤

β(1−r)

. And we know R can never

be in power at p

−1

and p

−1

m+1

for m > n. And we can use that to calculate a lower bound

of V



−1

n+2



≥ −

1+λ

1+βr

. Using that our inequalities become: 1 + λ −βr



1+λ

1+βr



≤

β(1−r)

. We

118

can rearrange this to get: λ ≤

1+2βr−β

β(1−r)

. But we assumed that λ >

1+2βr−β

β(1−r)

giving us our

contradiction. And again because our assumption involves a strictly inequality which does

not depend on n, our assumption that (2γ − 1)

= 0 becomes irrelevant.

And this completely the proof 

C.3.2 Proof of proposition 3.2.

We will show that R cannot win at belief p

−1

= 1 −γ. The other case is entirely symmetric.

First suppose for contradiction that R wins and implements 0 with positive probability. By

Lemma C.1 it must follow that W



−1



≤

−1

1−β

. Additionally we know the L party would

have implemented −1. Since −1 is always preferred to 1, electing R can only be optimal for

the voter only if the voter prefers 0 to −1. This requires that W



−1



1−2(2γ−1)

β(1−r)

≥ W



−1



Optimality also requires W



−1



≥ −2 + 2 (2γ − 1) + β (1 − r) W



−1



+ βrW



−1



since

the voter could have elected L. Combining these together and simplifying we get:



−1



(1 − β) ≥ −2 + 2 (2γ − 1) + r



−1 + 2 (2γ − 1)

1 − r



Using that 2γ − 1 >

we get that:



−1



(1 − β) > −2 + 1 −r



1 − 1

1 − r



= −1 ⇒ W



−1



−1

1 − β

And that is a contradiction, and the R party can’t be elected and implement 0 with

positive probability.

Now assume for contradiction the R party is elected and implements 1. Well the voter

always prefers −1 to 1 so it must be that the L party was oﬀering 0 with positive probability

and the voter prefers 1 to 0. Optimality in this case requires that: W



−1



1+2(2γ−1)

β(1−r)

≤



−1



. And we know that W



−1



= −2 − 2 (2γ − 1) + β (1 −r) W



−1



+ βrW



−1



which implies that W



−1



−2−2(2γ−1)+βrW

(

−1

)

1−β(1−r)

. Combining that with our optimality

119

condition yields:



−1



1 + 2 (2γ − 1)

β (1 − r)

≤

−2 − 2 (2γ − 1) + βrW



−1



1 − β (1 −r)

Rearranging:



−1





1 − β

1 − β (1 −r)



≤

−1

1 − β (1 −r)

−

1 + 2 (2γ − 1)

β (1 − r) (1 − β (1 − r))

Using again that (2γ − 1) >



−1



≤

−1

1 − β

−

1 + 2 (2γ − 1)

β (1 − r) (1 − β)

−1

1 − β

−

β (1 − r) (1 − β)

−3

1 − β

And we have showed in lemma C.2 that W (p

−1

) ≥

−2

1−β

for all n in any equilibrium so

that isn’t possible. Hence it follows that the L party must win at p

−1



C.3.3 Proof of proposition 3.3

The proof of this is implied by the proof of part 3 of proposition 3.1.

C.3.4 Proof of proposition 3.4

It suﬃces to show that, in any equilibrium without experimentation, for p

−1

close enough

the R party is every elected then they are elected at p

−1

for all m > n. If this were

true then there are two possible cases, the belief advantage party is always in power for p

near

or the belief dis-advantaged party is always in power. And in either case it is easy to

check the party in power would only play policy 0 since doing so ensures they stay in power

forever.

Now we just need to show that statement holds. Well suppose R is in power at p

−1

but

L takes power at p

−1

n+1

where we can think of n as being arbitrarily large. Since there is no

120

experimentation R must implement policy 0 at p

−1

which implies L oﬀered policy −1. A

stronger version of lemma C.1 (proof available from the author) can be shown where this

implies that for all m > n the voter must prefer 0 over −1 at p

−1

. Hence for L to be

elected at p

−1

n+1

it must be that party R weakly prefers policy 1 and L must implement 0

with positive probability. Hence we can write R’s optimality conditions at n and n + 1 as

(2γ − 1)



−1



− V



−1

n+1



≥

β(1−r)

and (2γ − 1)

n+1



−1



− V



−1

n+2



≤

β(1−r)

. We

also know that V



−1

n+1



= λ + βV



−1

n+2



because either L implements policy 0 or they

mix which means they are indiﬀerent to doing that and what they actually do. Using that

we can rewrite the ﬁrst inequality as: (2γ − 1)



−1



−λ + −βV



−1

n+2



≥

β(1−r)

. Since

n is arbitrary large we can again use the approximation (2γ − 1)

≈ 0 giving us the two

inequalities of:

−λ

−

(1−r)

≥ +V



−1

n+2



and V



−1

n+2



≥

−1

β(1−r)

. And for that to hold

requires that

−λ

−

(1−r)

≥

−1

β(1−r)

which can be reduced to : βλ ≤ −

1−β

1−r

which is obviously

impossible. Hence we have our contradiction and this completes the proof. 

121

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