2015 by Iván G. López Cruz. All rights reserved. Short ... · Iván G. López Cruz . Indiana...

CAEPR Working Paper #2015-024

Policing, Schooling and Human Capital Accumulation

Iván G. López Cruz Indiana University

December, 2015 This paper can be downloaded without charge from the Social Science Research Networkelectronic library at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2714365 The Center for Applied Economics and Policy Research resides in the Department of Economics at Indiana University Bloomington. CAEPR can be found on the Internet at: http://www.indiana.edu/~caepr. CAEPR can be reached via email at [email protected] or via phone at 812-855-4050.

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http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2714365

http://www.indiana.edu/%7Ecaepr

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Iván G. López Cruz 1

Policing, Schooling and Human Capital Accumulation1

Iván G. López Cruz2

Version: First: July, 2014. Current: December, 2015

A substantial body of empirical and policy literature argues that schooling can be a powerful tool

against criminality and violence. On the other hand, recent work has demonstrated that low levels

of public safety can have serious detrimental e¤ects on educational outcomes. This paper develops

a model to analyze the roles that investments in education and in public safety have for student�s

educational attainment. The model captures the main stylized facts of the literature and explores

the optimal balance between investment in policing and schooling. The model analyses individual

decisions to accumulate violence related skills ("street capital") at the expense of human capital

formation in a setting where property rights require private e¤orts to be enforced. The model assumes

that inhabitants of a region decide, during childhood, to allocate e¤orts to schooling and/or learning

"street skills" that, as adults, will serve them in resolving violent con�icts in their favor. Hence,

if the level of public safety, which is the only mean to prevent violent confrontations, is low, the

incentives to study will also be lower. Moreover, one of the results establishes that those agents who

accumulate more human capital, and hence are more productive, su¤er a comparative disadvantage

in exerting violence because their opportunity cost of doing so is higher. Therefore, if investments in

public education increase the productivity spread between adult agents, the incentives to study might

decrease and lead to a lower output, showing that the bene�ts of schooling can only be seized if they

are complemented with enough public safety.

Key Words: Street Capital, Human Capital, Public Education, Policing, Property Rights.

Subject Classi�cation: [JEL] D74, D78, E24, I26, K42.

1 I am very grateful to Michael Kaganovich, Michael Alexeev, Gustavo Torrens, Alexander Monge-Naranjo,Dionissi Aliprantis, Eric Rasmussen, Daniel Cole, Michael Mcginnis, Bulent Guler, Amanda Michaud, JodyLipford and the seminar participants at the Jordan River Conference, the Washington University in St. Louis10th Economics Graduate Student Conference and the 52st Annual Meeting of the Missouri Valley EconomicsAssociation for useful comments and suggestions. I also gratefully acknowledge �nancial support from theVincent and Elinor Ostrom Workshop in Political Theory and Policy Analysis.

2Department of Economics, Indiana University Wylie Hall, Bloomington, IN 47405 E-mail:[email protected]

1. INTRODUCTION

Criminality and violence are important problems since they largely a¤ect the individuals�

welfare. Therefore, it is important to examine the tools that the government can use to

minimize their harmful e¤ects. Two public policies that have been explored in this context are

schooling and policing. Hence, some important questions emerge: Under what circumstances

the government should prioritize one policy over the other? Is there an optimal combination

of both? Unfortunately, there is no consensus on how to answer them.

Indeed, one can �nd evidence favoring the idea that schooling is an e¤ective tool against

criminal behaviors, sometimes superior to policing because it raises the opportunity cost of

crime (Lochner and Moretti, 2004). But on the other hand, important detrimental e¤ects that

violence has on educational outcomes have been measured in a number of papers (Gerardino,

2013). Therefore, the very existence of these results impedes to conclude that one policy is

strictly superior to the other. Accordingly, a theoretical framework able to analyze the optimal

balance between them is needed. In this paper I undertake this task by introducing a model

that explores the individual decision to accumulate human or street capital in a context with

imperfect public safety. In particular, I analyze the impact that policing, which prevents

agents from resorting to violence, and investments in public education, that increases agent�s

productivity, have on their incentives to study. Importantly, the setting introduces a novel

interpretation of public safety as a key complement to policies directed to boost individual

productivity. In short, the incentives to accumulate human capital are proved to critically

depend on the interaction of both, policing and schooling.

A central idea behind the model is that in contexts of insecurity, acquiring more education

might be a double-edged sword. In fact, taking the government�s spending in education con-

stant, an increment in any agent�s productivity comes to the expense of acquiring street skills.

Furthermore, it leads to an increment in his opportunity cost of exerting violence. As a result,

the most productive persons are the less e¤ective �ghters3 .

The strategic component present in this analysis implies that public investments in ed-

ucation might have surprising consequences. Importantly, the model establishes that under

some circumstances, more public funds assigned to education will lead to higher human capi-

tal investments (and less street capital accumulation) only if the level of public safety is high

enough. Speci�cally, if such investments widen the productivity breach between agents that

di¤er on innate learning skills, the consequent strategic weakening of the most talented induces

an overall higher investment in street capital during childhood, when types are no yet revealed.

Of course, how consequential this e¤ect is in shaping the individuals�behavior depends on how

they asses the available level of public safety.

The analysis in this paper characterizes the individuals�decision to study in terms of the

policy parameters. Hence, the model can also tackle the problem of what policy should carry a

higher weight when the goal is to maximize total welfare. Assuming that both policies compete

for the same budget, it is possible to solve for the optimal breakdown of it. Now, when does

education should be prioritized over policing depends on the rest of the parameters. For

example, if the e¤ectiveness of violence in determining the distribution of income between two

3This result is a version of the Paradox of Power, a concept introduced by Hirshleifer (1991)

2 Policing, Schooling and Human Capital Accumulation

contestants becomes higher, policing gains relative importance. If public spending in education

reduces the productivity gap between adults, the optimal fraction of the budget devoted to

education becomes larger.

This paper has a close relation with the literature known as Economics of Con�ict. In

particular the contest speci�cation that I use was �rst introduced by Hirshleifer (1988) and

further developed by Skaperdas (1992) and Skaperdas and Syropoulos (1997). In spirit, my

approach runs parallel with Gonzalez (2005) who delineates the strategic reasons for which

an agent would rationally avoid costless productivity improvements in places where property

needs to be privately defended. In Gonzalez�s words, the price of peace is poverty. The

logic underlying this conclusion is clearly described in Gar�nkel and Skaperdas (2006) in

terms of the comparative advantage concept: unilateral increments in productivity raise the

opportunity cost of �ghting over a common pool of resources. However, these papers do not

address the trade-o¤ between becoming into a better �ghter or a better producer by means of

capital investments. Most importantly, none of them incorporate the policy interventions that

I examine.

The study of how violence capacity accumulates is not new. Mocan et al. (2005) dis-

cuss a dynamic model of criminal activity. In their approach, individuals accumulate human

and criminal capital to increase the income coming from legal and illegal activities, respec-

tively. Munyo (2014) calibrates a similar model that reproduces 91% of the recent variation

in Uruguay�s juvenile crime rates. The author claims that the introduction of more indulgent

juvenile crime legislation along with an important economic crisis changed the incentives of

youths in favor of criminal activities.

My setting di¤ers from the mentioned papers. In both cases, the returns to legal and

illegal activities are exogenously determined and thus unrelated. Speci�cally, the pool of

resources from which criminals can prey upon is not produced by their victims. Hence, the

accumulation of violent skills of other agents does not interfere with the individual ability to

enjoy self-produced output. Consequently, any strategic consideration such as the comparative

advantage analysis is absent. Furthermore, the concept of criminal capital they employ di¤ers

from that one of street capital that I study. Criminal capital can only be used for predation.

Instead, in my model the street capital stock is used to privately enforce property rights over

the output, making it indispensable for consumption purposes. Ignoring that street related

skills are essential to survive in places with power vacuums leads to understate the incentives

to acquire them.

The rest of the paper is organized as follows. Section 2 describes the empirical evidence

on the e¤ectiveness of schooling in reducing crime and the harmful e¤ects of violence on

educational outcomes. Section 3 introduces the baseline model and discusses the equilibrium

concept. In Section 4 I describe the main comparative statics results. Section 5 discusses the

existence of an optimal breakdown of the government�s budget when policing and schooling

compete for it, and shows that even in the absence of budgetary competition, both policies can�t

be implemented independently from each another, they must remain coordinated. Section 6

concludes and provides some ideas for future research.


2. POLICING VS. SCHOOLING DEBATE

The impact that education has on crime rates has been widely explored in the literature.

For example, Lochner and Moretti (2004) employ US data at the state level to provide exten-

sive empirical evidence pointing out that education has itself the remedy against insecurity.

By exploiting the exogenous variation in education implied by increments in the number of

compulsory years of schooling, the authors derive a number of conclusions. First, they �nd

that an extra year of high school education signi�cantly reduces the probability of being ar-

rested. In pecuniary terms, they �nd that an increment of 1% in the high school graduation

rate for males induces a cut in the social costs of crime of 1.4 billion dollars. They argue

that the underlying intuition is that improvements in future wages through human capital

increase the opportunity cost of committing crimes. Hjalmarsson and Lochner (2012) discuss

a number of similar contributions for highly developed countries. Remarkably, all these papers

converge to the same point: increasing the years of compulsory education decreases crime rates

signi�cantly. Furthermore, they show that education is more e¤ective in reducing crime than

policing.

The role of education in reducing crime has also been explored from a purely theoretical

perspective. Hirshleifer and Kim (2003) conceive educational policy as a way through which

educated agents can transfer part of their human capital to a fraction of their uneducated

counterparts. Such donations are not altruistic; in fact, they are intended to diminish the

predatory activities against the most productive agents. In their model, public safety can

be low or high depending on the agents�ability to solve a collective action problem. When

guarding is low, education will be more e¤ective in �ghting predation if o¤ered to a targeted

number of agents. On the contrary, when guarding is high, education is o¤ered to the entire

uneducated population4 .

It is also important to recognize that the evidence favoring education over policing as a more

e¤ective policy against crime comes from the developed world (Galiani, 2014). One could say

that having a sample of developed countries, can lead to incorrect conclusions. For instance,

wealthy societies enjoy institutions that guarantee a basic level of private property protection.

Therefore, increments in wages via human capital investments are regarded as secure and

indeed, disincentives crime rates. If public safety is perceived as too low, enjoying the fruits

of human capital investments will be regarded as an unlikely outcome. Such expectations

could play an important role in shaping the individuals�decision to remain in school. Hence,

education could be ine¤ective in reducing crime rates when public safety is too low.

Indeed, the impact that crime and violence have on educational outcomes has also been

addressed. Aliprantis (2013) presents an empirical analysis to measure how exposure to vio-

lence during childhood is a strong predictor of future criminal behavior. In that paper, crime is

conceived as the ultimate consequence of a distinct educational process, that is, accumulation

of street capital which, following Anderson (1999) is de�ned as the skills and knowledge useful

for providing personal security in neighborhoods where it is not provided by state institutions.

One important conclusion is that in general, black individuals have higher probability of get-

4The authors de�ne an elitist educational policy as donations of human capital to a selected number ofagents, whereas an egalitarian educational policy consists in equal donations to all the uneducated agents.


ting engaged in criminal behaviors, but it is also true that they are exposed to a lot of violence

during childhood. Importantly, after controlling for variables such as witnessing a murder,

hearing a shooting and so on, black males tend to obtain outcomes similar to those obtained

by their white counterparts. The reported estimations reveal that exposure to violence reduces

high school graduation by around 10 percentage points and the hours worked by up to 4 hours

per week.

Along the same lines, Damm and Dustmann (2014) also measure the e¤ect of early expo-

sure to violence on later youth�s criminal behavior. Their paper exploits a natural experiment

consisting on an in�ux of refugee immigrants into Denmark and their subsequent random

assignation to neighborhoods. Interestingly, they show that the key feature that pushes young

individuals to pursue a criminal career is not the amount of crimes committed in their neigh-

borhood, but the number of criminals residing on it. Hence, is through social interactions that

crime a¤ects youths�occupational choice between legal and illegal activities. Quantitatively,

they �nd that an increment of one standard deviation in the share of criminals living in a given

neighborhood increases the probability of conviction for male youths by 9 percent. In terms of

the e¤ect of violence on educational outcomes, the authors report that individuals who lived

in areas with a high share of criminals, are less likely to be enrolled in education or working.

There are also other studies measuring the e¤ects that generalized violence, provoked by

events like civil wars, has on education. For example, Gerardino (2013) uses data from Colom-

bia to provide evidence on the existence of a gender bias of violence. The paper provides data

revealing that in Latin America the girls have been obtaining better education outcomes (en-

rolment rates and years of school completed) than the boys because the latter engage in crime

activities more often. She explains that the violence levels that Colombia experienced induced

an educational gender gap in favor of the girls. Speci�cally, the paper states that the rise in

violence against males opened job opportunities for the boys in the violence sector. Further-

more, the violence damaged the young males�perception on overall safety and life expectancy,

reducing the perceived returns to school. Other studies as Justino (2011), Leon (2012) and

Swee (2009) con�rm that violence has an overall negative e¤ect on educational attainment.

3. A MODEL OF HUMAN AND STREET CAPITAL ACCUMULATION

I begin the analysis by presenting a setting consisting in a two period model of human and

street capital accumulation. As will be explained shortly, the agents make their investment

decisions anticipating that property rights are not fully protected.

3.1. Basic assumptions

Agent�s endowments, preferences and neighborhoods. Consider two risk neutralindividuals, i = 1; 2; that reside in a particular neighborhood within an area such as the "inner

city", where many con�icts are potentially resolved out of the scope of law enforcement by

the government. Both agents live for two time periods that will be referred to as youth and

adulthood. During youth, each agent allocates a fraction gi (i 2 f1; 2g) of his time (one unitat each period) to accumulating street capital, with the rest devoted to schooling for human

accumulation purposes. As adults, the agents allocate a fraction zi of their time to exert


violence against each other and a fraction 1 � zi to produce output. Furthermore, attending

to school allows any agent to become into a more e¤ective producer, whereas street capital

enhances �ghting ability which is required to ensure some consumption out of the total output

jointly produced by both agents.

Each agent i has an innate learning ability Bi that can take on two values�Bl; Bh

("low"

and "high"), where Bl < Bh: Every agent�s ability; Bi; becomes common knowledge at the

beginning of adulthood, but during youth it remains unknown to everyone. However, the

distribution of abilities is always known: any agent will have ability Bh with probability �

and ability Bl with probability 1� �:5

Policy interventions and mechanisms of interaction. I consider two public policies:policing and schooling. Regarding policing, the government can put under police surveillance

a fraction p of the neighborhoods, which will merely prevent violence in them. Additionally,

any agent regards the safety status of his neighborhood as a random outcome: with probability

p (1 � p) a neighborhood is protected (unprotected). As for schooling, the government can

invest funds to ensure quality of public education, which increases the human capital obtained

by a student per unit of time spent in schools.

Our objective is to understand how policing and schooling interact with each other to

determine overall educational outcomes. The two channels of interaction the model comprises

are:

1. Budget competition. If both policies compete for the same budget, improving the quality

of public education will come at the cost of lowering public safety (and vice versa).

Lower levels of police surveillance translate into higher chances of having to resort to

violence to ensure consumption during adulthood which in turn will raise the incentives

to accumulate street capital.

2. Comparative advantage e¤ect. If public spending on education changes inequality be-

tween adult agents, then it also changes youth�s investment decisions. For example, if

educational spending increases more the return to education of a high ability agent than

that of a low ability agent, the high (low) ability agent will bear a higher (lower) op-

portunity cost of �ghting. As a result, the low ability agent will increase his violence

e¤ort while the one with high ability will be more specialized in production. Since the

high ability agent cannot lower his guard too much for it would entail a drastic loss in

consumption, total violence will be higher. Anticipating a more hostile environment,

young agents will invest more in street skills.

Adult neighbors. Both adult agents, after becoming aware of educational outcomes,

will either �ght over a common pool of resources formed by their individual outputs, or will

produce and consume under no con�ict. Indeed, given the distribution of abilities and the

fraction of protected neighborhoods, the pair i = 1; 2 will be neighbors in 5 possible scenarios.

If they happen to reside in a protected neighborhood, then each one of them will consume his

individual output. If they end up in an unprotected neighborhood, four possible outcomes can

5The proportion of types can be identi�ed with the presence of di¤erent kinds of families. Indeed, Schneideret al. (2005) present evidence relating family structure and children�s educational outcomes.


arise: two symmetric mano a manos and two asymmetric mano a manos, where the symmetry

refers to the vector of learning abilities involved: (Bl; Bl), (Bh; Bh), (Bh; Bl), (Bl; Bh). Figure

1 depicts all these possibilities.

Figure 1: Adult neighbors, general case

Production and con�ict technologies. The formation of street and human capital (Sand H) that takes place during the �rst period, is generated according to the functions:

Si = gi and Hi = f (E;Bi) (1� gi) (3.1)

where Si and Hi are the stocks of street and human capital, respectively, that agent i accu-

mulates when young. The return to e¤ort devoted to education, f (E;Bi) ; is assumed to be

an increasing function in the quality of public education, E, and a student�s innate learning

ability, Bi:

In the second period, the agents can produce a consumption good employing the following

production function:

Yi = Hi (1� zi) ; (3.2)

which combines the human capital of individual i (Hi) with the fraction of time that he

allocates to production (1� zi). However, the security of agent�s output might is not certain.It depends on whether the neighborhood is under police surveillance or not. For any agent

the status of a neighborhood is uncertain. Accordingly, from any agent�s perspective, with

probability p the output of each agent is totally secure while with probability 1� p it will be

contested through a violent con�ict governed by a con�ict technology � (Gi; Gj). Speci�cally,

the contestants exert violence to divide the aggregate output Yi + Yj : The con�ict technology

determines agent i�s share in the total as

� (Gi; Gj) =Gmi

Gmi +Gmj

=(Sizi)

m

(Sizi)m+ (Sjzj)

m ; (3.3)

where Gi = Sizi stands for the amount of violence that agent i exerts against his opponent

which is determined by his earlier acquired street capital Si and the fraction of time zi he

allocates to violence in adulthood. The parameter m > 0 characterizes the e¤ectiveness of

violence in reshaping the contestants�property rights over the output under dispute.

Equilibrium. The proper notion of equilibrium for this two-stage game is that of Subgame


Perfect Equilibrium. In the second period the allocation of time between production and violent

activities (z1; z2) must be a Nash Equilibrium for any realization of learning abilities (Bh; Bl),

any human and street capital accumulation decisions (H1;H2) made in the �rst period, and

any realization of police protection (p). In the �rst period forward looking agents anticipate the

equilibrium outcomes of the second period and optimally divide their childhood unit of time to

build their capital stocks. This time allocations (g1; g2) must also be a Nash Equilibrium. As

both contestants, agents 1 and 2; are ex-ante identical, it is natural to only consider symmetric

solutions for the model, that is, equilibrium outcomes where g1 = g2. More speci�cally, an

equilibrium is de�ned as follows:

Definition 1. The equilibrium strategy pro�le of any agent i = 1; 2 is composed by a

fraction of time devoted to violent activities during adulthood, zi; and a fraction of time

devoted to accumulating street capital during childhood, gi; such that:

1. zi maximizes agent i0s consumption out of total output during adulthood given agent

j0s choice zj ; where i 6= j: That is:

zi = ArgMax

8<:� (Gi; Gj) Xk2fi;jg

Hk (1� zk)

9=; ; zi 2 [0; 1]

Denote agent i0s adulthood equilibrium payo¤ as Vi(gi; gj) which depends on his street

capital and that of agent j; both of which are given by Hk = gk; k = i; j6 :

2. gi maximizes agent i0s adulthood payo¤ given agent j choice gj ;where i 6= j:

gi = ArgMax fVi(gi; gj)g ; gi 2 [0; 1]

A simpli�ed version. As mentioned above, one important channel of interaction betweenpolicing and schooling is related to the concept of comparative advantage. But this concept

only applies to situations in which the agents involved are not identical. Hence, government�s

spending on education can only a¤ect the comparative advantage of any agent in one activity,

violence or production, in the context of asymmetric confrontations. Therefore it is useful to

focus on a version of the model where only asymmetric mano a manos are allowed. To proceed

in that direction we add a couple of assumptions: one agent will be endowed with low skill

Bl while the other with high skill Bh and for each agent, his individual outcomes, Bl or Bh

come with probabilities .5.7 Notice that introducing these simplifying assumptions only a¤ect

the young agents�decisions. The adulthood problems are not a¤ected. Figure 2 depicts the

6 In the following section I present a concrete formula for Vi (gi; gj) :7 It is helpful to rationalize the story linking the complete version of the model and the simpli�ed version.

To this end, we can think of the inner city as a place populated by a continuum of agents where the fractionsof high types and low types are (.5,.5) in the simpli�ed version and (�; 1� �) in the complete version. Duringadulthood, in both versions, a random sample of agents ends up residing in unsafe locations. Now, if allthese agents, regardless their type, simultaneously select the exact location to live within the unsafe area, theresult would be the formation of random matches as in the complete version. Alternatively, if we let the hightypes choose �rst, then the low types will try to select low types as neighbors since that increase the resourcesavailable for predation. Such scenario would correspond to the simpli�ed version of the model.


possible mano a manos in the simpli�ed version.

Figure 2: Adult neighbors, simpli�ed case

Besides isolating the comparative advantage e¤ect, the simpli�ed version reduces the alge-

braic complexity of the model which in turn simpli�es the calculations required to undertake

comparative statics results. In the rest of the paper, when performing comparative statics

analysis, we provide analytical results for the simpli�ed version and numerical experiments for

the complete version. As will be clear soon, the simpli�ed version encompasses a lot of what

happens in the complete version. On the other hand, the complete version is interesting in its

own right because it enables us to understand how the model�s results change when we vary

the proportion of agents with high an low ability.

4. EQUILIBRIUM ANALYSIS

In this section, I characterize the equilibrium by means of backward induction. First, I

deduce the equilibrium behavior of adults taking the childhood decisions as given. As noted

before, the adulthood results are valid for both, the simpli�ed model and the general case.

Second, I compute the young individuals�equilibrium decisions. For the simpli�ed version I

present analytical solutions while for the general case I present numerical experiments.

4.1. Output allocation between adults through violence

Once both inner city youths grow into adulthood, with probability p; the two end up

residing in a protected area. In which case each agent chooses the time to be spent on violent

activities zi 2 [0; 1] to maximize his consumption given by CPi = Hi (1� zi) : Trivially, theoptimal choice of both contestants is to exert no violence; that is to say zPi = 0. With

probability ( 1 � p) the couple ends up living in an unprotected neighborhood without police

protection. Hence, each agent selects how to allocate a unit of time between violent activities

(zi) to maximize his share out of the common pool of resources:

CUi = � (Gi; Gj)X

k2fi;jg

Hk (1� zk) =(Sizi)

m

(Sizi)m+ (Sjzj)

m [Hi (1� zi) +Hj (1� zj)] ; (4.4)

where Sk and Hk are the street and human capitals, respectively, that were decided during

childhood.

The following lemma characterizes the second stage Nash equilibrium in the unsafe neigh-


borhood8 :

Lemma 1. Assume that agents accumulated positive quantities of both sorts of capital (for-

mally, Hi;Hj > 0 and Si; Sj > 0): Then agent i0s equilibrium time devoted to violence and

consumption are

zUi = (1� �i)�

m

1 +m

��1 +

Hj

Hi

�(4.5)

CUi = �i

�1

1 +m

�(Hi +Hj) (4.6)

respectively, where his equilibrium share �i is:

�i =1

1 +�Hi=SiHj=Sj

� mm+1

(4.7)

Thus, individuals who have accumulated relatively more human capital su¤er from a com-

parative disadvantage in the violent division of the output, result that evokes Skaperdas (1992)

who obtains the same result in a similar context. Indeed, Lemma 1 implies that agent i0s share

over the common pool of resources �i depends on i0s ratio of human to street capital (Hi=Si)

relative to that one of his opponent (Hj=Sj). Indeed, from (4.7), �i is a decreasing function of

(Hi=Si) = (Hj=Sj). Intuitively, for any agent having relatively more units of human capital per

unit of street capital the opportunity cost of predation will be relatively higher. Speci�cally,

using equations (4.5) and (4.7) we can derive the equilibrium time that agent i devotes to

violence as

zUi =

�m1+m

��1 +

Hj

Hi

�1 +

�HjSiHiSj

� mm+1

(4.8)

Hence, if i0s opponent, agent j; possesses a signi�cantly larger stock of street capital than him,

an increment in i0s human capital will decrease the value of the numerator in (4.8) relatively

more than it will decrease the denominator, yielding an overall positive e¤ect on zUi . In

other words, anyone who has a comparative disadvantage in �ghting will specialize more in

production. I will come back to this problem in the context of a symmetric subgame perfect

Nash equilibrium.

4.2. Results for the simpli�ed model

4.2.1. Schools vs. Streets.

Now lets take one step back in order to analyze the �rst stage of the game. In the simpli�ed

version where only asymmetric con�icts are possible, young agents anticipate four di¤erent

scenarios where adulthood may pass o¤. First, with probability p they live in a protected

neighborhood whereas with probability 1� p they live in an unprotected one. Second, in anyof these situations the agents need to take into account the realization of their abilities. With

8As it is proved in Skaperdas and Syropoulos (1997), the contest game between adults has a uniquepure strategy Nash Equilibrium. To ensure an interior equilibrium with 0 < zi < 1; the innequalityh(1 +Hj=Hi) =(1 + (HjSi=HiSj)

mm+1 )

i< 1

m+ 1 must hold.


probability .5 agent i knows that he will have low learning ability Bl and agent j will have high

learning ability Bh;with probability .5 the opposite will happen. Let CPi (Bi) (CUi (Bi)) be the

consumption of agent i as an adult, when residing in a protected (unprotected) neighborhood

and having a learning ability equal to Bi 2�Bl; Bh

. Therefore, the expected consumption

of a young agent i is given by9

Vi = p

�1

2CPi

�Bli�+1

2CPi

�Bhi��+ (1� p)

�1

2CUi

�Bli�+1

2CUi

�Bhi��; (4.9)

Young agents choose their time allocation between schooling and street activities to maximize

Vi: Since both agents are ex-ante identical, it is natural to explore a symmetric equilibrium

which is characterized in the following proposition.

Proposition 1. Let ! =f(E;Bh)f(E;Bl)

> 1 be the return to educational e¤ort of a high ability

agent relative to the return of the low ability agent. Then there exists !� such that if ! < !�;

the equilibrium fraction of time that the youths of either type devote to accumulating street

capital, g�; is given by:

g� =(1� p) 2zlzh

phm(!+1)2

!

i+ (1� p) (zl + zh)

(4.10)

where zl and zh represent, respectively, the time that in equilibrium a high ability agent and

a low ability agent will devote to exert violence during adulthood:

zl =mm+1 (1 + !)

1 + !m

m+1(4.11)

zh =mm+1 (1 + !)

! + !1

m+1

(4.12)

Proposition 1 conveys two important messages. First, it says that the government can

a¤ect the childhood time allocation (g�) using both available policies, policing (p) and public

education quality (E), where the latter only plays a role through the relative return to edu-

cation (!)10 . That is to say, the government can use spending on education to a¤ect young

agents�choices only if the quality of public education a¤ects the productivity spread between

adult agents, which is ultimately characterized by !. Second, it underscores the equilibrium

dependency of the decision to accumulate street capital (g�) on the extent up to which adults

resort to violence (zl and zh)11 .

Before performing any comparative statics analysis, it is convenient to list some mechanisms

through which public education spending can a¤ect income inequality, represented by ! in the

model:

1. Wage competition. As Thurow (1972) explains, if the labor market is competitive and

9Agent i0s equilibrium consumption can be written as follows: CPi (Bi) = f(E;Bi)(1 � gi), CUi (Bi) =�1

1+m

�(f(E;Bi)(1�gi)+f(E;Bj)(1�gj))"

1+

�f(E;Bj)(1�gj)=gjf(E;Bi)(1�gi)=gi

� mm+1

#10 Indeed, if (d!=dE) = 0; it follows that (dg�=dE) = 0:11 In the Appendix I discuss the conditions under which the adults�choices in the symmetric equilibrium are

interior. Indeed, both zl and zh belong to the open interval (0; 1) as long as ! does not exceed a thresholddenoted by !� (m) that depends on the parameter m:


therefore each worker is paid his marginal productivity, improving the quality of public

education increases the supply of skilled workers which in turn pushes down their wages.

Additionally, the reduction in the supply of low skill workers increases their wages. Over-

all, a better funded public education system raises output and reduces income disparities.

2. Early childhood education. Heckman (2011) argues that after the second grade, schooling

has little e¤ect on closing gaps in the capabilities that determine economic and other

adulthood outcomes. For example, class size and teacher salaries are ine¤ective in elim-

inating disparities. Indeed, the gaps emerge in earlier stages, before formal schooling

begins. Hence, government�s spending on education can only have important equalizing

e¤ects if it targets early childhood education.

3. Education and social security. Glomm and Kaganovich (2003) present a model in which

raising expenditure levels in public education increases inequality when those funds are

taken out from a PAYG social security system.

4.2.2. Comparative Statics

Violence and productivity di¤erences between adults. In view of Proposition 1,

the young agents�strategy is ultimately determined by the relative return to educational e¤ort

! through its e¤ect on the time allocation by adult agents. So far, our discussion on the

comparative advantage concept allowed us to predict that after increasing !; a high ability

agent would reduce his usage of violence (zh) and the opposite would be observed for the low

ability agent. Indeed, it is possible to show that:

@zl@!

> 0;@zh@!

< 0;

and@ (zh + zl)

@!> 0

The last inequality states that total time devoted to violence during adulthood increases

in the relative productivity spread between adults. This phenomenon occurs because the low

ability agent, when comparing the marginal cost with the marginal bene�t of attacking his

progressively relatively more productive rival, will always �nd it more rewarding to cut down

his own production and grab a larger share of the aggregate output. At the same time, for the

high ability agent increasing his production is the best way for reducing the loss that he will

experience as a result of a con�ict with his more violent neighbor. However, he can�t a¤ord to

lower his guard too much for it would entail an excessive reduction of his share, implying an

increment in the total usage of violence zh + zh12 :

Educational e¤ort and policing. The second comparative statics result tackles thee¤ect that the level of policing, p; has on the incentives to accumulate street-related skills.

The higher the chances of having to use violence (lower p), the higher the agents�incentives

12 Indeed, as the relative return to education e¤ort (!) grows, the low ability agent will tend to get fullyspecialized in violence, while the low ability agent will never be fully specialized in production. To verify thisstatement, notice that lim

!!1zl =1 but lim

!!1zh = m=(1 +m):


to become into better �ghters, which means an increment in g�: In particular, one can verify

that:@g�

@p< 0

Educational e¤ort and schooling. Next, I examine the e¤ect that public educationhas on the young agents�behavior. As Proposition 1 revealed, the quality of public education

(E) shapes young agent�s choices by a single channel, namely the relative return to education

!: Hence I �rst analyze the sign of the dependency that g�; the time that any young agent

allocates to street activities; keeps with respect to !. To that end, we have to dissect all the

e¤ects that an increment in the relative return to education (!) triggers. In particular raising

! brings about two e¤ects:

1. Preparation e¤ect. As noted above, a raise in ! implies that the total utilization of

violence during adulthood increases when the couple (i; j) ends up living in the insecure

neighborhood. As a result, young agents have more incentives to accumulate street

capital. Of course, a higher level of public safety dilutes this e¤ect for it reduces the

likelihood of being forced to use violence.

2. Rigidity e¤ect. As the relative return to education (!) increases, the shares over the total

output that any agent, say i, appropriates in each one of his possible roles (high or low

skill agent), �hi and �li, become less responsive to changes in the time that young agents

spend in the streets (gi).13 Indeed, if both agents spend the same amount of time in the

streets, that is to say gi = gj , as is the case in the symmetric equilibrium described in

Proposition 1, the marginal increments in the fractions �hi and �li with respect to gi are

decreasing in !: Formally:

@2��i@gi@!

��gi = gj< 0, � = h; l

This reduction in the responsiveness of agent i�s shares with respect to his investments

in street capital, gi, can be read as a reduction in the marginal cost of increasing the

time spent in schools, which carries the bene�t of enlarging the pie under dispute. In

an extreme case where the shares captured by the agents become constant with respect

to their street capital investments, both individuals would regard as optimal to devote

all of their childhood�s time to accumulate human capital. Altogether, less responsive

shares induce the agents to allocate less time to street activities (gi) when young. This

e¤ect is clearly reinforced by the level of public safety p; for it increases the likelihood of

consuming in a free of con�ict environment.

So far we have seen that when ! increases, two opposing forces, the preparation e¤ect and

the rigidity e¤ect, determine the direction of change of the time that young agents allocate

to street activities. The higher utilization of violence during adulthood promotes higher in-

vestments in street capital. While less responsive equilibrium shares induce a reduction in the13 Indeed, the shares that each young agent considers when solving his maximization problem can be presented

as �li =1

1+

264 1!

�1gi

�1�

�1gj

�1�375

mm+1

; �hi =1

1+

264!�1gi

�1�

�1gj

�1�375

mm+1


time spent in the streets. As argued above, the level of public safety p balances this two forces.

Hence, for a su¢ ciently high level of policing, p; the incentives to reduce g� that emerge after

an increment in the relative return to education (!), prevail over those that operate in the

opposite direction and vice versa. Therefore, one can �nd a critical level of public safety �p

such that:

if p < �p; then dg�=d! > 0 and if p > �p then dg�=d! < 0

This �nding is important since it establishes that investing in public education might

have surprising consequences in the agent�s educational e¤ort depending on the level of police

protection. In section 5 I analyze the challenges that this result poses for the government�s

problem when computing the optimal breakdown of the budget between policing and schooling.

Proposition 2 summarizes the results discussed in this section.

Proposition 2. For any interior solution to the adult agents�maximization problem with

m > 0 and ! > 1 we have the following results.

1. The time that a low (high) ability agent devotes to violence is increasing (decreasing) in

the relative return to education e¤ort !. Moreover the total time that a pair of adult

agents, one of high and another of low ability, devotes to violence is increasing in the

relative return to education e¤ort !: In other words, @zl=@! > 0; @zh=@! < 0 and

@ (zh + zl) =@! > 0:

2. The time that the youths spend acquiring street skills, g�; depends negatively on the level

of police protection p: That is (dg�=dp) < 0.

3. When the level of police protection p, is small enough, the time that the agents devote to

accumulating street capital, g�, is positively related to the type-relative return to education

!. The relation between g� and ! becomes negative when p is large enough. Formally,

there is a police protection level p 2 (0; 1), such that dg�=d! > 0 if p < p and dg�=d!

< 0 if p > p:

Government spending on education and income inequality in the data. It is usefulat this point to brie�y discuss the empirical relation between public investments in education

and income inequality. Unfortunately, the available empirical evidence is not conclusive in

this regard. For example, Martins and Pereira (2004) use data from 16 countries in the 1990s

to study the interaction between schooling and wage dispersion. They show evidence that

the returns to education are higher for the more skilled individuals, suggesting that schooling

might enlarge wage inequality. Also, there are other studies pointing to the opposite direction,

for instance, Sylwester (2002) claims that public education seems to induce a subsequent

decrease in the level of income inequality. Accordingly, in this section I didn�t impose a

speci�c dependency between the quality of public education, E, and the type-relative income

spread !: Instead, I addressed the relation between ! and g� without specifying any functional

form for f (E;B), which gives the returns to education for an agent with innate learning ability

B 2�Bl; Bk

14

14Notice that the relation between E and ! would ultimately depend on the functional form of f (E;B)


4.3. Results for the general case

4.3.1. Schools vs. Streets

Now we allow for both asymmetric and symmetric confrontations. When any agent i

considers the outcome of having agent j as an opponent, he will take into account �ve di¤erent

possibilities: peaceful production if the location has police protection, two possible asymmetric

matches and two symmetric matches whenever no police surveillance is available. Hence, agent

i0s expected payo¤ can be presented as follows15 :

Vi = p[�CPi�Bli�+ (1� �)CPi

�Bhi�] (4.13)

+(1� p) [�2CUi�Bhi jBhj

�+ (1� �)2 CUi

�BlijBlj

�+� (1� �)CUi

�Bhi jBlj

�+ (1� �)�CUi

�BlijBhj

�]

Again, the young agents divide their unit of time between schooling and street activities to

maximize Vi: Since both agents are still ex-ante identical, I focus on a symmetric equilibrium

which yields the following expression for the time that young agents spend in the streets16 :

g�=

(1� p)�12

�mm+1

�2 ��2 + (1��)2

!

�(1 + !) + � (1� �) 2zlzh

�p��+ 1��

!

�[m (1 + !)] + (1� p)

hm(1+!)2(1+m)

��2 + (1��)2

!

�+ � (1� �) (zl + zh)

i (4.14)

While the base line model enabled us to isolate the e¤ects that productivity inequality has

on the early decision to study, the extended model addresses the impact of types�heterogeneity

on the results presented in previous sections. This is important because a lower probability of

asymmetric confrontations a¤ects the expected exposure to violence during adulthood.

4.3.2. Comparative Statics

One of the advantages of the complete model is that one can study how the proportion of

high ability agents versus low ability agents � in�uences the results of the model. In this section

I present some interesting results on the behavior of g� in regards of di¤erent proportions of

high and low skill agents. The discussion on how the results for g� obtained in the simpli�ed

version continue to hold when including symmetric con�icts can be found in the appendix.

The e¤ect of a change in the proportions of types. Here I analyze the e¤ect thatdi¤erent proportions of types, determined by the value of �; have on g�; and whether the sign

of the relation between g� and � depends on the values of the type-relative return to education

! and public safety p.

To begin with, suppose that � = 0; then every agent exhibits low learning ability Bl.

Therefore, any confrontation will be symmetric and in case that agents i and j end up residing

in an unsafe location, none of them will enjoy a comparative advantage in �ghting. Next, as

� begins to grow departing from zero, two e¤ects emerge:

15The functions CPi and CUi remain the same as in section 4.1. The notation CUi (BijBj) indicates that thematch between agent i and j involves the innate learning abilities Bi and Bj :16Similar calculations as those ones involved in the proof of Proposition 3 can be replicated to obtain g� in

the current extension of the baseline model


1. Productivity e¤ect: The expected return to education for any agent increases since the

expected value of Bi approaches Bh instead of Bl which in turn increases the incentives

for accumulating human capital (g� grows): A greater level of public safety reinforces

this e¤ect, since the returns to education becomes more secure.

2. Heterogeneity e¤ect: The probability of asymmetric confrontations, given by � (1� �) ;increases as � approaches .5. Indeed, while � approaches .5 from the left, the probability

of being a low type in an asymmetric match and hence enjoying a comparative advantage

in violence is always higher than that of being on the week side in an asymmetric mano

a mano. Hence, as � approaches .5 from the left, the expected comparative advantage

in �ghting of any agent becomes greater. Once � surpasses .5, the probability of being

a high type becomes greater than that of being a low type and the heterogeneity e¤ect

eventually vanishes away. Importantly, if the level of policing is high (low), the agents

will decide to use the greater expected advantage in �ghting to reallocate time from

streets (school) to school (streets).

Panel A of Figure 3, shows a U-shape relation between g� and � for a high level of public

safety p. Indeed, for an initial increment in � the heterogeneity e¤ect and the productivity

e¤ect work together in lowering the incentives to invest in street skills. Speci�cally, if p is high

the agents will seize the greater expected comparative advantage in �ghting by decreasing the

value of g because it is unlikely to reside in an insecure neighborhood. As � keeps growing,

the improvement in the expected comparative advantage fades out, pushing g� upwards.

Clearly, panel B of Figure 3 shows the opposite scenario. When the level of police protection

is low, the relation between g� and � follows an inverted U-shape, where the underlying

intuition can be obtained by reversing the logic described in the above paragraph.

Figure 3: Accumulation of street skills vs. fraction of high types

A) High level of police protection (p = :20) B) Low level of police protection (p = :19)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2499

0.25

0.25

0.25

0.25

0.25

0.25

α

g*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.258

0.258

0.258

0.258

0.258

0.258

0.258

0.258

0.258

α

g*

In the appendix I show how the value of !; the return to education of a high skill agent

relative to that of a low skill agent a¤ects the results depicted in Figure 3.


5. POLICY IMPLICATIONS

In this section I discuss the problem of how government�s resources should be optimally

allocated to boost the returns of public education and to expand public security. Towards that

end, I breakdown the analysis in two parts. I begin by describing the problem consisting in

allocating limited resources to both budget lines, that is to say, I assume that both policing

and schooling compete for the same bulk of resources. Next, I relax the budget competition

assumption and show that even when increasing the spending on one policy without having to

reduce the budget assigned to the other, both policies need to remain coordinated. In other

words, overinvesting in schooling or policing is still possible even in the absence of budget

competition. As in section 4 I present analytical results for the simpli�ed model and use

numerical experiments to discuss the implications of the distribution of types.

5.1. Results for the simpli�ed model

5.1.1. Budget Competition

Government�s Problem. To model the problem that a policy maker faces when decidinghow many resources should be allocated to policing and schooling we need to specify two

elements: the government�s objective function and the rate at which units of policing can be

substituted with units of public education. First, I take the total expected output produced

by both agents as the government�s optimization target. Second, I assume that units of both,

education quality and policing infrastructure, can be purchased at market prices �E and �P :

The tax revenue is taken as an exogenous value denoted by I. Hence, the government�s budget

constraint is given by:

�EE + �P p = I; where 0 � p � 1 and E � 0 (5.15)

Furthermore, I assume that the tax revenue, when entirely invested in policing, will be just

enough to secure all the neighborhoods, that is to say �P = I: Additionally, a maximum

units of education quality E > 0 can be a¤orded, that is to say �EE = I. Hence, when the

government chooses to protect p% of the neighborhoods, it has to assign p% of the tax revenue

to expand police surveillance17 . Similarly, when p% of the neighborhoods are protected, the

maximum units of education quality amounts to E = (1� p)E:Functional Forms. The returns to education are given by f (E;Bi) = EBi: This func-

tional form for f; implies the independence of the relative return to education, !; from the

amount of public investments in education, E. Thus, in equilibrium, the value of ! is equal to

the ratio Bh=Bl and consequently independent of E: This simple setting is taken as a starting

point for the policy analysis. At the end of this subsection I comment on how the results would

change in a more general case.

Given the assumptions made above, the government�s problem consists in selecting a break-

down of the budget that obtains the maximum expected aggregate consumption, which in this

model is equal to the expected aggregate output and can be computed using equations (3.2)

and (C.20) in the Appendix. Now, assuming that the model is in the symmetric equilibrium17One can write the budget constraint as p = (1� �EE) =I:


described in Proposition 1, and letting � represent the fraction of the tax revenue devoted to

policing, total expected output which is the government�s objective function can be presented

as :

W (�) = [Bh +Bl]

��m+ 1

1 +m

� �(1� �)E

�[1� g� (�)] (5.16)

Interior Solution. It is easy to check that a solution containing no spending in education(� = 1) can never be optimal since it yields no output while the opposite corner leads to a

strictly positive one. Thus, the conditions under which an interior solution exists are the same

as those ones that rule out the corner that carries no spending in policing (� = 0). The

following proposition summarizes our �ndings with respect to the optimal breakdown of the

budget.

Proposition 3. Suppose that the returns to education e¤ort are given by f (E;Bi) = EBi.

Then, if the e¤ectiveness of violence, m, is not too low, the government will �nd it optimal

to spend a positive amount of resources on both policing and schooling. Formally, there exists

0 < m� < 1 such that if m > m�, the problem of maximizing expression (5.16) has an interior

solution �� 2 (0; 1) :

The existence of an interior solution to the government�s problem comes from the fact that

policing and schooling compete for the same budget. That is to say, an extra dollar assigned to

public security implies that the spending in schooling must decrease by the same amount and

vice versa. Hence, the quality of public education cannot be improved without harming public

safety and thus, without lowering the incentives to study via Proposition 2. Furthermore, this

trade-o¤ is more severe when the marginal return of violence, governed by the parameter m,

becomes higher. Consequently, as m increases, any fall in the policing capacity, p, will provoke

a more pronounced increment in g�; or in other words, a sharper fall in the time that the inner

city youths devote to school. That is why the optimal spending in policing is strictly positive

when m exceeds a threshold value m�:

In addition to its technical importance, Proposition 3 reveals that generalizing the ap-

plication of policies concerned with police surveillance and schooling can be dangerous. For

instance, suppose that an empirical analysis is based on observations collected in a location

where the e¤ectiveness of violence (m) is low18 . Hence, the natural conclusion would be that

the bene�ts of improving the quality of public education outweigh those of investing in policing

at all. Now, my setting implies that such conclusion can�t be extrapolated without running the

risk of spending the budget ine¢ ciently. Likewise, pooling observations from di¤erent regions

can also be dangerous since the average e¤ectiveness of violence might hide the heterogeneity

of the regions in that particular dimension and lead to wrong policy recommendations.

To illustrate the previous point, Figure 4 plots the expected total output as a function of

the budget�s fraction spent on policing. Panel A shows the results when the e¤ectiveness of

violence (m) is low, whereas Panel B uses a higher value of it. As can be observed, when m is

low, spending in policing will only reduce the expected output. With a higher m, an interior

18The parameter m, can be interpreted as an indicator of how severe are the violent methods used in asociety. For example, simple robbers are unable to deprive their victims from the ownership of their housesor other major assets. But kidnappers can demand a ransom that may force the victims to transfer them animportant fraction of their wealth.


solution emerges19 .

Figure 4: Optimal Policy in Di¤erent Scenarios.

Budget Competition

A) Low violence e¤ectiveness (m = :2) B) High violence e¤ectiveness (m = 5)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

µ

Exp

ecte

d O

utpu

t

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

µ

Exp

ecte

d O

utpu

tFixed policing level

C) Low policing protection (p = 0) D) High policing protection (p = :5)

1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.70.132

0.134

0.136

0.138

0.14

0.142

0.144

0.146

E

Exp

ecte

d O

utpu

t

1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.78

8.5

9

9.5

10

10.5

11

11.5

12

12.5

E

Exp

ecte

d O

utpu

t

Next, it is worth mentioning how Proposition 3 would change if a speci�c assumption linking

the quality of public education, E, and the type-relative return to education, ! , had been

made: For example, suppose a positive relation between both; that is d!=dE > 0: The question

would be whether under such circumstances a corner solution with no police protection is more

likely to arise as compared to the case where d!=dE = 0. The answer would be no, and the

reason is that a very high value of E (close to �E) would imply a police protection level, p; below

the threshold �p pointed out in Proposition 3. Hence, further increments in E would induce the

agents to accumulate more street capital. Therefore, the overall bene�ts of improving public

education would fall in comparison with the case when d!=dE = 0: Conversely, if we assume

that d!=dE < 0; for high values of E and values of p below the threshold �p; further increments

in E would imply more incentives to accumulate human capital, favoring a corner solution

with no police protection.

19The parameter values in panel A and B of Figure 1 are �E = 1, Bh = 4 and Bl = 1: The low value of m is:2, while the high is 5


5.1.2. Beyond budget competition

Finally, it is important to relax the budgetary competition assumption. This is relevant

because so far it seems that overinvesting in education is only possible when the resources

invested on it have to be taken from the public safety line. But, this is not true. Indeed,

if public education reduces the incentives to study via the enlargement of income inequality

between types, and the level of policing remains �xed at a low level, increasing the schooling

spending can reduce total output. Of course, this e¤ect depends on the functional forms being

used. More speci�cally, suppose that the type-relative return to education can be written as

follows20 :

! = E�Bh

Bl; � 2 R

Hence, with the right selection of the parameter values, it is possible to obtain the scenario

represented in the panels C and D of Figure 421 . In particular, when the quality of public

education rises the type-relative return to education and the policing services are rather limited,

public spending in schooling will dilute the agent�s incentives to study. Such a scenario is

described in Panel C of Figure 4. The reduction in the time that the agents spend in school

is so large that it outweighs the bene�ts of higher returns to education. As a result, the total

output falls. On the contrary, panel D shows that if the �xed level of policing is set beyond the

threshold �p pointed out in Proposition 2, the total output increases in response to investments

in public education.

Summarizing, the incentives to study depend on the level of public safety. If increasing

the quality of education requires cutting down the resources allocated to policing, then the

government needs to breakdown the budget optimally. But even in the case where both polices

are budgetary independent, they need to be coordinated. Indeed, if the level of public safety is

too low and the improvements in the educational system induce higher income inequality, the

severity of the con�ict between types rises and promotes the incentives to accumulate more

street capital. The overall e¤ect of improving the quality of education can be counterproduc-

tive.

5.2. Policy Implications of the Distribution of Types

Next, I solve the government�s problem discussed in Section 5.1.1, this time in the context

of the general case model. Before moving forward, it is convenient to recall some details

mentioned earlier. As before, the policy maker has to decide which is the vector of policies

(p;E) that maximizes the expected output �Y by choosing an intermediate point along the

convex combination � (0; 1)+ (1� �)��E; 0

�: In the equilibrium of the general case model, the

20The underlying function de�ning the return to education is given by

f (E;Bi) =��1Bi�Bh (Bi)

�E +

�1� 1Bi�Bh (Bi)

�E��Bi

Where 1Bi�Bh (Bi) is an indicator function that takes the value of 1 when Bi � Bh and 0 otherwise. I alsode�ne � = � �; whre and � are positive real numbers.21 In both panels (C and D), I set � = 2; Bh = 4, Bl = 1 and increase E from 1.5 to 1.7 in 1000 steps of the

same length. In panel C there is no police protection (p = 0) and in panel D, p equals .5:


expected output, �Y , is a weighted average of the following values22 :

Yh = f (E;Bh) (1� g)

Yl = f (E;Bl) (1� g)

Yhh =2f (E;Bh) (1� g)

1 +m

Yll =2f (E;Bl) (1� g)

1 +m

Yhl =[f (E;Bh) + f (E;Bl)] (1� g)

1 +m

The �rst two expressions denote the outputs produced by each type of agent in the peaceful

production scenario. The second two, correspond to the overall output produced by a couple

of agents engaged in a symmetric match, and �nally, the last expression stands for the overall

output produced by the members of an asymmetric match. All these formulas can be derived

from equations (C.20) in the Appendix and (3.2). Then, the expected output can be presented

as:�Y = p [�Yh + (1� �)Yl] + p

h�2Yhh + (1� �)2 Yll + 2� (1� �)Yhl

iFigure 5 plots the expected output as a function of �, the fraction of government�s resources

allocated to public education, for di¤erent proportions of types. Interestingly, the optimal

breakdown of the budget seems to be neutral to changes in �; the fraction of high type agents.

As one would anticipate, the value of the expected output increases as a result of an increment

in �; because the average productivity of the system improves when there are more high type

agents.

Figure 5: Policy experiment for di¤erent distributions of types

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

µ

Exp

ecte

d O

utpu

t

�: � = :05 j - - : � = :2 j... : � = :7

parameters: m = 5; Bl = 1; ! = !�; Bh = !Bl; �E = 10

Finally, Figure 6 plots the expected output, �Y ; as a function of the fraction � for di¤erent

22Here we assume, as in section 4, that f (E;B�) = EB� . Hence, ! = Bh=Bl; value that is independent ofE:


values of m: As can be observed, when the e¤ectiveness of violence in shaping the distribution

of the output among contestants is high (m = 5), an interior solution to the government�

problem exists. Indeed, it results optimal to allocate about 39% of the resources to policing.

But if the violence e¤ectiveness is low enough (m = :1), a corner solution arises, implying that

the optimal policy consists in spending the entire government�s budget on schooling.

Summarizing, the types� distribution does not in�uence the optimal breakdown of the

government�s budget. But, as before, the e¤ectiveness of violence, m, can induce a corner

solution for the government�s problem. That is to say, that there are instances in which

spending in schooling is a superior policy when compared to policing, but this only depends

on the importance of the redistributive power of violence.

Figure 7: Policy experiment for di¤erent violence e¤ectiveness

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

µ

Exp

ecte

d O

utpu

t

�: m = 5 j - - : m = :1

parameters: � = :4; Bl = 1; ! = !�; Bh = !Bl; �E = 10

Overall, the analysis contained in this section indicates that the policy implications emerg-

ing from the baseline model and the general case coincide. However, it is worth noting that so

far no dependency between the composition of types and the quality of public education (E)

was assumed. But in a more complex model where the fraction of high types (�) positively

depends on E and when police surveillance (p) is limited, a policy maker should take into

account that modest increments in educational spending may increase the type heterogeneity

and thus, the overall con�ict.

6. CONCLUSION

Existing evidence reveals that education is a powerful instrument to �ght criminality and

violence. Also, recent empirical contributions suggest that the provision of public safety has

large e¤ects on educational outcomes. Hence a formal discussion on how to balance the pro-

vision of both public services becomes relevant.

By means of a simple and stark model, I analyzed the incentives to accumulate human

and street capital when property rights protection is not ensured. The framework I used,


allowed me to make inferences about the optimal blend of spending on policing and on public

education. Speci�cally, if both policies compete for the budget, that is, if more spending

on education means a lower level of public safety, then an optimal breakdown of the budget

arises, supporting the idea that public safety in�uences educational outcomes. However, if the

power of violence in establishing property rights over the output is too low, a corner solution

involving no spending in policing emerges. Such result could explain the conclusions coming

from the empirical literature that favors the use of education over policing as a mean to increase

productivity and �ght criminality.

The model discussed in this paper concludes that both policies should remain coordinated

even in the absence of budgetary competition between them. The reason is that if the level

of policing is su¢ ciently low and the government�s spending in education expands the income

inequality between agents, then the use of violence during adulthood increases, inducing young

agents to accumulate more street capital. The overall e¤ect can consist in a contraction of the

available output.

The analysis of the model presented in this paper opens several lines of research. Firstly,

data measuring the returns of public investments in education in areas di¤ering in the level

of public safety is required. Also, it would be useful to measure the e¤ectiveness of educa-

tion in reducing crime in developing countries. On theoretical grounds, investigating the role

of education in expanding the fraction of agents with high learning capacity might help to

understand better the consequences of education in unsafe places.

APPENDIX A: INTERIOR SOLUTION FOR THE ADULTHOOD PROBLEM

As the value of ! increases, the least productive type will enjoy a stronger advantage in

�ghting (see equation (??)). Thus, it is natural to anticipate that if ! is high enough, zl, thetime that the low type devotes to exert violence, will eventually hit the constraint zl � 1:

In fact, using the equilibrium expression for zl and zh (eq. (??)) it is easy to verify thatlim

!�!1zl =1 and that zh < zl for any m > 0 and ! > 1: Now the problem is to compute the

value of !, say !�; such that zl < 1 i¤ 1 < ! < !�: To pin down this threshold for a given

m > 0; it su¢ ces to solve the inequality zl < 1 using (??), which yields the condition:�m

m+ 1

�! � ! m

m+1 <1

m+ 1(A.17)

Denote the LHS of (A.17) by h(!): It is easy to verify that h(1) < 0 8m; implying thatthe adult agents�maximization problem will have exclusively interior solutions whenever we

consider only one type. It is also obvious that h is increasing without bound in ! and that it

is convex on its domain: Since it is not possible to o¤er an explicit formula of ! in terms of

m; one needs to use numerical methods to compute the threshold !�(m) for di¤erent levels of


violence e¤ectiveness.

Figure A1: Maximum type-relative return to education

Figure A1 shows that the threshold !� decreases in the e¤ectiveness parameterm: The intuition

is that as m goes up, the marginal bene�t of exerting violence is greater for any !; inducing

both agents to allocate more resources towards appropriation. In particular, agents of type l

will now be closer to get fully specialized in violence and hence, the value of !� has to decrease.

APPENDIX B: NUMERICAL EXPERIMENTS FOR THE COMPLETE VERSION

Street capital accumulation and police protection. Notice that in terms of how p

interacts with the incentives to accumulate street related skills, nothing has really changed

given that more police protection still reduces the likelihood of being forced to use violence,

and hence, a higher p still induces a lower value of g�. Figure 2 plots g� as a function of p

con�rming the negative relation between both.23

Figure 2: Police protection vs. accumulation of street skills

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

p

g*

Street capital accumulation and the type-relative return to education. Next, let�sexamine the behavior of g� with respect to ! in the the current setting. To begin with, notice

that the key distinction between the baseline model and the extended version is the inclusion

of instances where agents of the same type are matched to each other. In such matches,23The values of the parameters used in Figure 3 are m = :99; � = :5; ! = :9� !� (m) ; where !� (m) is the

maximum value that can be assigned to the type-relative return to education so that an interior equilibriumexists for the adulthood optimization problem. Many other values of � and ! were used, obtaining always anegative relation between g� and p:


the ratio of the agent�s adulthood labor productivity, f (E;Bi) (1� gi) =f (E;Bj) (1� gj) ; isequal to 1 in any symmetric equilibrium. Hence, when the agents possess the same type, a

change in ! does not modify their strategic positions since both share the same opportunity

cost of �ghting, namely, the marginal product of their labor. As a result, if we only consider

symmetric matches (that is if � = 0 or � = 1 and p = 0); we obtain an expression of g� that

does not depend on !24 :

g� =m

m+ 1

Since the adult agents�consumption, when trapped in a symmetric match, does not depend

on the relative return to education, the only channels through which increments in ! can alter

the equilibrium value of g� are exclusively the asymmetric matches. Hence, the conclusions

presented in Proposition 3 can be extrapolated to the current setting. That is to say, the

sign of the derivative dg�=d! will still be determined by the features described in Section 4.3.

Hence, if p is low enough, ! and g� will be positively related. On the �ip side, if p is low

enough the sign of the relation between ! and g� will be reversed. Figure 3 plots g� as a

function of ! for di¤erent values of p; reinforcing our claim that Proposition 3 remains valid.

Figure 3: Accumulation of street skills vs. type-relative return to education

A) Low level of police protection (p = :1) B) High level of police protection (p = :2)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60.344

0.345

0.346

0.347

0.348

0.349

0.35

0.351

0.352

0.353

ω

g*

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60.246

0.2465

0.247

0.2475

0.248

0.2485

0.249

0.2495

0.25

0.2505

0.251

ω

g*

Interaction between the relative return to education and the proportion oftypes. Finally, as Figure 5 shows, the value of the type relative return to education !; alsoplays a role in shaping the equilibrium relation between g� and �: Speci�cally, the higher the

value of !; the lower the value of � for which g� reaches its maximum or its minimum, as the

case may be. This is observed because a greater value of ! makes street related skills more

critical to consume during adulthood.

Indeed, as � starts growing, the probability of being in the weak side of an asymmetric

con�ict becomes strictly positive. Again, let�s take the case when p is low. Then, if we

signi�cantly increase !, the event of being in disadvantage with respect to �ghting skills will

represent a serious danger for any agent. This is the case because the consumption attached to

such a situation, which the low value of p makes it more likely to happen, would get drastically

24As the previous expression indicates, the usage of street capital is also relvant in symmetric confrontations.In fact, when living in a unsafe neighborhood, the agents are always trapped in a prisoner�s dilemma that forcesthem to use violence regardless the nature of the match that was randomly assigned to them.


reduced. Hence, increasing the human capital stock at the expense of the street capital one,

will be less appealing than before, and the value of � above which g� begins to increase will

be smaller with respect to the original value of !25 :

Figure 5: Accumulation of street skills vs. fraction of high types

Low level of police protection (p = :15)

A) Low relative return to education B) High relative return to education

(! = :2� !�) (! = :8� !�)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2931

0.2931

0.2931

0.2931

0.2931

0.2931

0.2931

0.2931

0.2931

0.2931

0.2931

α

g*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.2931

0.2932

0.2933

0.2934

0.2935

0.2936

0.2937

0.2938

0.2939

0.294

0.2941

α

g*

High level of police protection (p = :5)

C) Low relative return to education D) High relative return to education

(! = :2� !�) (! = :8� !�)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.0999

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

α

g*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.094

0.095

0.096

0.097

0.098

0.099

0.1

0.101

α

g*

APPENDIX C: PROOFS

Lemma 1 De�ne �i = � (Sizi; Sjzj) and �j = � (Sjzj ; Sizi) : The problem that agent

i solves consists in selecting the value of zi 2 [0; 1] to maximize the value of �iY; where

Y =P

k2fi;jgHk (1� zk) : Combining the �rst order conditions associated to i0s and j0s

problems yieldsGjGi

=

�(Hi=Si)

(Hj=Sj)

� 1m+1

; (C.18)

25Again, !� (m) is the maximum value that can be assigned to the type-relative return to education so thatan interior equilibrium exists for the adulthood optimization problem.


that can be substituted in �i to obtain (4.7). Next, adding the �rst order conditions for i0s

and j0s problems yields Hi

SiGi +

Hj

SjGj = mY , that can be written exclusively in terms of Gi

after using (C.18). Equating Gi in terms of the parameters yields:

Gi =

�m1+m

�(Hi +Hj)

Hi

Si

�1 +

�Hj=SjHi=Si

� mm+1

� (C.19)

Using Sizi = Gi allows us to obtain the equilibrium expression of zi, namely equation (4.5).

Finally, one can compute agent i0s consumption (equation (4.6)) and the total output producedjointly by i and j:

Y = (Hi +Hj)1

1 +m(C.20)

Proposition 1 First, lets introduce some notation26 . Let F� = f (E;B� ), Hh (g) =

Fh (1� g) ; Hl (g) = Fl (1� g) ; �� (g) = F� (1� g) =g; H (h; l) = Hh (gi) +Hl (gi) ; H (l; h) =Hh (gj) +Hl (gi) and �nally denote the consumption of agent i when he adopts the high type

and his opponent the low type by Ci(h; l); whereas Ci(l; h) represents again i0s consumption

but with the opposite allocation of types. Now, agent i0s problem can be written as follows:

Max0�gi�1

p

�1

2Hh (gi) +

1

2Hl (gi)

�+

(1� p)

8>>>>>>><>>>>>>>:12

h1

1+m

iH (h; l)

1 +h�h(gi)�l(gj)

i mm+1| {z }

Ci(h;l)

+

12

h1

1+m

iH (l; h)

1 +h�h(gj)�l(gi)

i mm+1| {z }

Ci(l;h)

9>>>>>>>=>>>>>>>;Take the �rst order conditions to obtain:

p [�Fh �Fl] + (1� p)�dCi (h; l)

dgi+dCi (l; h)

dgi

�= 0

Divide the last equation by Fl! to get:

p

��1� 1

!

�+ (1� p)

�dCi (h; l)

dgi+dCi (l; h)

dgi

�1

!Fl= 0 (C.21)

Now, notice that:

dCi (h; l)

dgi=

h1

1+m

i1 +

h�h(gi)�l(gj)

i mm+1

8><>:H 0h (gi)�

h1

1+m

iH (h; l)

h�0h(gi)�l(gj)

ih�h(gi)�l(gj)

i 1m+1

+h�h(gi)�l(gj)

i9>=>;

dCi (l; h)

dgi=

h1

1+m

i1 +

h�l(gi)�h(gj)

i mm+1

8><>:H 0l (gi)�

h1

1+m

iH (l; h)

h�0l(gi)�h(gj)

ih�l(gi)�h(gj)

i 1m+1

+h�l(gi)�h(gj)

i9>=>;

26Recall that � is used to indicate the type of any agent. Thus � 2 fh; lg


In a symmetric equilibrium, we have gi = gj = g�; in which case the last two equations boil

down to:

dCi (l; h)

dgi= �

h1

1+m

iFh

1 + !m

m+1+

hm

(m+1)2

i[Fh (1 + !)]

h1g�

i�1 + !

mm+1

� �!

1m+1 + !

�dCi (h; l)

dgi= �

h1

1+m

iFl

1 + !�mm+1

+

hm

(m+1)2

i �Fl�1 + !�1

�� h1g�

i�1 + !

�mm+1

��!

�1m+1 + !�1

�After a few algebraic steps one gets:

dCi (h; l)

dg

1

Fl!=

��zl + zlzh

1

g�

� �1

m (1 + !)

�dCi (l; h)

dg

1

Fl!=

��zh + zlzh

1

g�

� �1

m (1 + !)

�Hence, equation (C.21) can be written as follows:

p

��1� 1

!

�+ (1� p)

�� (zl + zh) + 2zlzh

1

g�

�1

m (1 + !)= 0

Solving the last equation for g� yields:

g� =(1� p) 2zlzh

phm(!+1)2

!

i+ (1� p) [zl + zh]

Proposition 2: Part 1 First notice that one of the assumptions in Lemma 2 is that zhand zl form an interior solution to the adult�s problem. Since zh = (1=!)

1m+1 zl and ! > 1;

it follows that zh < zl: Hence, to obtain an interior solution we only need zl < 1 which is

equivalent to equation (A.17). Next, taking the �rst derivative of zl with respect to ! yields:

dzld!

=

�1 + !

mm+1

� �mm+1

��

mm+1

�(1 + !)

�mm+1!

� 1m+1

��1 + !

mm+1

�2Now, dzl=d! > 0 if and only if:

mm+1 (1 + !)

!1

m+1 + !< 1

The last inequality is equivalent to(A.17) which as discussed above holds in any interior equilib-

rium. Hence, in any interior equilibrium zl is increasing in !: Next, notice that dzh=d! < 0 i¤

!1=(m+1)

+ ! � (! + 1)�(1= (m+ 1))!�m=(m+1) + 1

�< 0

Which is equivalent to

!1

m+1+ ! � (! + 1)

�1

m+ 1!�

mm+1 + 1

�< 0


The last inequality holds i¤ !1

m+1 � 1 � [1= (m+ 1)]!1

m+1 < [1= (m+ 1)]!�mm+1 and can be

written asm

m+ 1! � ! m

m+1 <1

m+ 1

which is equivalent to (A.17). Hence, in any interior equilibrium zh is decreasing in !:

To prove the last part of the lemma, de�ne F (!) = zh + zl = (!) zl; where (!) =�1 +

�1=!1=m+1

��:Hence, it su¢ ces to show that F (!) is an increasing function in !: Next,

notice that F 0 (!) > 0 if and only if -� (!) = 0 (!)

�> zl=z

0l; which can be presented as�

!1

m+1 + 1�!

1m+1

>

�!

1m+1 + !

�(1 + !)

!1

m+1 + ! � mm+1 (1 + !)

After rearranging terms, one obtains:h!

m+3m+1 � !

i+

1

m+ 1

h!

2m+3m+1 � ! 1

m+1

i> 0

Given that m > 0 and ! > 1; we have that both terms in the last inequality are strictly

positive. Hence, we conclude that F (!) = zl + zh is increasing in !:

Proposition 2: Part 2 Let A = 2zlzh; C = m (! + 1)2=! and B = zl + zh; so that g�

can be written as:

g� =(1� p)A

pC + (1� p)B

Lemma 2. B < C

Proof of the Lemma. First, substitute the equilibrium expressions of zl and zh in terms

of ! to obtain:

zl + zh = zh

�1 + !

1m+1

�=

�mm+1

�(1 + !)

�1 + !

1m+1

�! + !

1m+1

Now, since m and ! are both positive, we have:

! + !m+2m+1 >

1

m+ 1

�! + !

m+2m+1

�Hence,

! + !m+2m+1 + !

1m+1 + !2 >

1

m+ 1

�! + !

m+2m+1

�The last inequality can be written as follows:

(1 + !)�! + !

1m+1

�>

1

m+ 1

�1 + !

1m+1

�!


which is equivalent to,

(1 + !)

!>

1m+1

�1 + !

1m+1

��! + !

1m+1

�

Proof of part 2. Take the �rst derivative of g�with respect to p to obtain:

dg�

dp=[pC + (1� p)B] [�A]� [(1� p)A] [C �B]

[pC + (1� p)B]2

The sign of this derivative only depends on the sign of the numerator. Given that C; B and

A are always positive and provided that (C �B) > 0; we conclude that the sign of dg�=dp isalways negative.

Proposition 2: Part 3

Lemma 3. Denote by g�0 the value of g� when p = 0: Then,

dg�0d!

> 0

Proof of the Lemma. We proceed by contradiction. Suppose that dg�0

d! j!<!�< 0 then:

(zh + zl) (zhz0l + zlz

0h)� zhzl (z0h + z0l)

(zh + zl)2 < 0

()z2hz

0l + z

2l z0h < 0

Where z0 = dzd! . Now, using zh = (1=!)

1m+1 zl; we get that

dg�0d! j!<!�< 0 i¤

�1!

� 2m+1 z0l < �z2l .

But proposition 1 ensures that z0l > 0 and in any second-stage interior equilibrium z2l > 0; so

0 <�1!

� 2m+1 z0l < �z2l < 0:

Lemma 4. Following the same notation as in proposition 4, we have that

d(A=C)

d!< 0

Proof of the Lemma. First notice that

A

C=

2zlzh!

m (! + 1)2

And recall that zlzh = (1=!)1

m+1 z2l to obtain

A

C=

2z2l !m

m+1

m (! + 1)2


Using the equilibrium expression of zl we get:

A

C=

2!m

m+1

m (! + 1)2

�mm+1

�2(1 + !)

2�1 + !

mm+1

�2=

2h

m

(m+1)2

i!

mm+1�

1 + !m

m+1�2

But this last expression is decreasing in ! when m > 0 and ! > 1: To see this, notice that

if 1 < ! then 1 < !M where M = m= (m+ 1) : Hence, 1 + !M < 2!M which leads to�1 + !M

�2< 2!M

�1 + !M

�: Now it follows that

2 [M= (m+ 1)]h�1 + !M

�2M!M�1 � 2!M

�1 + !M

�M!M�1

i(1 + !M )

2| {z }d(A=C)d!

< 0

Proof of part 3. Recall from proposition 4, that27

g� = (1� p)A (!) = [pmC (!) + (1� p)B (!)]

Next, take the �rst derivative of g� with respect to ! and rearrange terms to obtain:

dg�

d!=mp

hd(A=C)d!

i+ (1� p)

hdg�0d! B

2 (!)i

�11�p

�[pmC (!) + (1� p)B (!)]2

Let N1 (!) = m [dA (!) =dC (!)] and N2 (!) = (dg�0=d!)B2 (!) : By the lemmas, we know

that N2 (!) > 0 and N1 (!) < 0 for any given ! > 1: Now notice that sign [dg�=d!] =

sign [pN1 (!) + (1� p)N2 (!)] : But the argument of the LHS sign function is a convex com-bination of a negative and a positive number so there must be a number p 2 (0; 1) such

thatdg�

d!> 0 if p < p

anddg�

d!< 0ifp > p

Proposition 3 Denote by �Y0 the output that is produced when � = 0 and use �Y� for

the output produced when � > 0: Also denote r (�) = [(�m+ 1) = (1 +m)] [1� g (�)] and�B = Bh +Bl: In this way, the government�s problem can be written as follows,

Max0��1

�B (1� �) �Er (�)

27Here we are emphasizing the dependence that A; B and C have on !:


As indicated in the text, the only relevant corner solution is � = 0:

Lemma 5. Following the notation used in Proposition 1, we have A = 2zlzh; B = zl + zh

and C = m (! + 1)2=!: Furthermore, denote ~� = � (1�m+ �m) : For any m > 0; there is

an interior solution to the government�s problem if there is 0 < � < 1 st.

C [A� ~�B] > ~�

�B (1� �) [B �A] (C.22)

Proof of the Lemma. Given m > 0; the solution � = 0 is optimal only if �Y0 > �Y� for any

� with 0 < � < 1: Hence, the solution is interior whenever there is � in the interval (0; 1) such

that �B �Er (0) < (1� �) �B �Er (�) which is equivalent to the following condition:

1� g (0)1� g (�) < 1� ~� (C.23)

Where g (0) = A=B and g (�) = (1� �)A=(�C + (1� �)B): Also notice that in any interiorequilibrium for the baseline model we have that 0 < A < B; so that 0 < g (0) < 1: Next,

rewrite (C.23) as follows,

(B �A) [�C + (1� �)B]B [�C + (1� �)B � (1� �)A] < 1� ~�

After some basic algebraic steps one can rewrite the last expression as (C.22).

Lemma 6. There exists m� < 1 st. if m > m�; then CA > (1�m)B (B �A)

Proof of the Lemma. First notice that A < B; to see this, recall a couple of results: �rst

that in any interior equilibrium of the baseline model we have 0 < g < 1 for any value of m and

second, g (0) = A=B; then it follows that A < B: Hence, (1�m)B (B �A) converges to zerofrom above as m approaches 1 from below. Next, observe that CA =

hm (! + 1)

2=!i2zlzh is

increasing inm since zl and zh are both increasing inm:Also, whenm = 1 we have CA = CA =h(! + 1)

2=!i2zl (1) zh (1) > 0; hence, CA converges to CA from below. Therefore, we can set

m su¢ ciently close to 1 so that CA is close enough to CA which is a positive number, and

(1�m)B (B �A) is close enough to zero, which altogether yields that CA > (1�m)B (B �A)for such a value of m28 :

Finally, notice that C [A� ~�B] �! CA and (~�=�)B (1� �) (B �A) �! (1�m)B (B �A)as � �! 0: Then, if m is su¢ ciently high we can be sure that there is �� > 0 such that (C.22)

holds, which proofs the proposition.

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Dionissi Aliprantis, 2013. "Human capital in the inner city," Working Paper 1302, Federal

Reserve Bank of Cleveland.28 It is straightforward to check that CA is a continuous function in m:


Damm, A.P., Dustmann, C., 2014. Does Growing Up in a High Crime Neighborhood A¤ect

Youth Criminal Behavior? American Economic Review, 104, 1806-1832.

Galiani S., 2014. Más y mejor educación también pueden contribuir a la reducción del crimen.

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pueden-contribuir-a-la-reduccion-del-crimen/>.

Michelle R. Gar�nkel & Stergios Skaperdas, 2006. Economics of Con�ict: An Overview.

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