2015 by iván g. lópez cruz. all rights reserved. short ... · iván g. lópez cruz . indiana...
TRANSCRIPT
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.
©2015 by Iván G. López Cruz. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.
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.
Iván G. López Cruz 3
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.
4 Policing, Schooling and Human Capital Accumulation
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
Iván G. López Cruz 5
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.
6 Policing, Schooling and Human Capital Accumulation
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
Iván G. López Cruz 7
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.
8 Policing, Schooling and Human Capital Accumulation
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-
Iván G. López Cruz 9
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.
10 Policing, Schooling and Human Capital Accumulation
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:
Iván G. López Cruz 11
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):
12 Policing, Schooling and Human Capital Accumulation
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
Iván G. López Cruz 13
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)
14 Policing, Schooling and Human Capital Accumulation
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
Iván G. López Cruz 15
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.
16 Policing, Schooling and Human Capital Accumulation
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:
Iván G. López Cruz 17
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.
18 Policing, Schooling and Human Capital Accumulation
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
Iván G. López Cruz 19
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:
20 Policing, Schooling and Human Capital Accumulation
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:
Iván G. López Cruz 21
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,
22 Policing, Schooling and Human Capital Accumulation
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
Iván G. López Cruz 23
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:
24 Policing, Schooling and Human Capital Accumulation
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.
Iván G. López Cruz 25
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.
26 Policing, Schooling and Human Capital Accumulation
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
Iván G. López Cruz 27
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
28 Policing, Schooling and Human Capital Accumulation
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
�!
Iván G. López Cruz 29
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
30 Policing, Schooling and Human Capital Accumulation
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 !:
Iván G. López Cruz 31
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.
REFERENCES
Anderson, E., 1999. Code of the Street: Decency, Violence, and the Moral Life of the Inner
City. W. W. Norton and Company.
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:
32 Policing, Schooling and Human Capital Accumulation
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.
Foco Económico, <http://focoeconomico.org/2014/12/07/mas-y-mejor-educacion-tambien-
pueden-contribuir-a-la-reduccion-del-crimen/>.
Michelle R. Gar�nkel & Stergios Skaperdas, 2006. Economics of Con�ict: An Overview.
Working Papers 050623, University of California-Irvine, Department of Economics, revised
September 2006.
Glomm, G., Kaganovich, M., 2003. Distributional E¤ects of Public Education in an Economy
with Public Pensions. International Economic Review, 44, 917-937, August.
Gonzalez, F.M., 2005. Insecure property and technological backwardness. Economic Journal
115, 703-721.
Gonzalez, F.M., 2010. The use of coercion in society: insecure property rights, con�ict and
economic backwardness. The Oxford Handbook of the Economics of Peace and Con�ict.
Herschel I. Grossman & Minseong Kim, 2003. Educational Policy: Egalitarian or Elitist?
Economics and Politics. Wiley Blackwell, vol. 15(3), pages 225-246, November.
Heckman J., 2011. The Economics of Inequality: The Value of Early Childhood Education.
American Education, 35, 31-35 and 47. Spring
Hirshleifer, J., 1988. The analytics of continuing con�ict. Synthese 76, 201-33.
Hirschleifer J., 1991. The Paradox of Power. Economics and Politics 3, 177-200.
Hirshleifer, J., 2000. The Macrotechnology of Con�ict. Journal of Con�ict Resolution 44,
773-792.
Hjalmarsson, R., Lochner, L., 2012. The Impact of Education on Crime: International Evi-
dence. CESifo DICE Report 10, 49-55.
León, G., 2012. Civil Con�ict and Human Capital Accumulation: The Long-Term E¤ects of
Political Violence in Perú. Journal of Human Resources 47, 991-1022.
Lochner, L., Moretti, E., 2004. The E¤ect of Education on Crime: Evidence from Prison
Inmates, Arrests and Self-Reports. The American Economic Review 94, 155-189.
Mocan, H., Billups, S., Overland, J., 2005. A Dynamic Model of Di¤erential Human Capital
and Criminal Activity. Economica 72, 655-681.
Munyo, I., 2014. The Juvenile Crime Dilemma. Review of Economic Dynamics 18, 201-211.
Justino, P. Violent Con�ict and Human Capital Accumulation. MICROCON Re-
search Working Paper No. 54. Available at SSRN: http://ssrn.com/abstract=1948651 or
http://dx.doi.org/10.2139/ssrn.1948651
Iván G. López Cruz 33
Schneider B., Atteberry A., Owens A., 2005. Family Matters: Family Structure and Child
Outcomes. Alabama Policy Institute.
Skaperdas, S. and C. Syropoulos 1997, The distribution of income in the presence of appro-
priative activities. Economica 64, 101-117.
Skaperdas, S., 1992. Cooperation Con�ict and Power in the absence of property rights. Amer-
ican Economic Review 82, 720-39.
Swee, E. L., 2009. On War and Schooling Attainment: The case of Bosnia and Herzegovina.
Households in Con�ict Network Working Paper, 57.
Sylwester, K., 2002. Can Education expenditures reduce income inequality. Economics of
Education Review 21, 43-52.
Thurow, L., 1972. Education and Economic Equality. Public Interest 78, 66-81.
Martins P. and Pereira P., 2004. Does education reduce wage inequality? Quantile regression
evidence from 16 countries. Labor Economics 11, 355-371.