Show Summary Details
More options …

# The B.E. Journal of Economic Analysis & Policy

Editor-in-Chief: Jürges, Hendrik / Ludwig, Sandra

Ed. by Brunner, Johann / Fleck, Robert / Mendola, Mariapia / Requate, Till / Schirle, Tammy / de Vries, Frans / Zulehner, Christine

4 Issues per year

IMPACT FACTOR 2017: 0.306
5-year IMPACT FACTOR: 0.492

CiteScore 2017: 0.50

SCImago Journal Rank (SJR) 2017: 0.414
Source Normalized Impact per Paper (SNIP) 2017: 0.531

Online
ISSN
1935-1682
See all formats and pricing
More options …
Volume 15, Issue 4

# Turf and Illegal Drug Market Competition between Gangs

Alberto J. Naranjo
• Corresponding author
• Department of Economics, Universidad de La Sabana, Puente del Comun, Km 7, Chia 107 32, Colombia
• Email
• Other articles by this author:
Published Online: 2015-08-04 | DOI: https://doi.org/10.1515/bejeap-2013-0161

## Abstract

Street-level illegal drug markets generate much of the violence and intimidation that local communities face nowadays. These markets are mainly driven by territorial gangs who finance their activities through the sale of drugs. Understanding how the existence of both turf and drug market competition may have unintended consequences of law enforcement policies on violence is the main contribution of the paper. We propose a two-stage game-theoretical model where two profit maximizing gangs compete in prices and invest in guns. We find that policies such as traditional or community policing can have different and unexpected effects on the level of violence.

## 1 Introduction

Retail illegal drug markets have become increasingly influential in the life of many communities around the globe. This influence is not only due to the drug selling itself but to the violence and intimidation these markets create in local communities.

Gangs, in their primary definition, were confined to specific territories 1 but later on, due to their control over these territories, became street dealers of illegal drugs 2 so as it is stated by UNODC (2009) in its World Drug Report, “[R]emoval of the territorial element may take drug markets out of the hands of street gangs” (p. 172). Therefore, since low-level drug markets in inner cities and rural areas are mainly run by street gangs the impact of these markets on the local communities comes from both drug market competition and territorial disputes (Howell and Decker 1999).

As it is stated by the International Narcotics Control Board (2011) in its report, “Drug abuse, drug trafficking and organized criminality have become everyday occurrences within these communities. There are areas […] where heavily armed, well-financed criminal gangs have taken on the role of providing local governance, shaping the lives of local people through the combination of intimidation and short-term reward” (pp. 1–2).

According to NDIC (2010) in its National Drug Threat Assessment, 69% of law enforcement agencies in the United States report gang involvement in illegal drug distribution in 2010, whereas according to NGIC (2011) in its National Gang Threat Assessment, gangs in the United States are responsible for an average of 48% of violent crime in most jurisdictions due to their control over drug distribution and disputes over territory. This is also true in other parts of the world (INCB 2011; Baird 2012; Carvhalo and Soares 2011). According to the Matrix Knowledge Group (2007), in the United Kingdom retail drug market dealers own territories and any attempt to move to someone else’s areas result in gang violence. This violence and intimidation tend to be higher at the lower levels of the market (i.e., street) than in other levels, motivated by the control of customer bases and territories. Furthermore, in Brazil and Colombia, many of the violence/intimidation associated with gangs in favelas/comunas arise also from the struggle to control drug markets and territories (Zaluar 2004; Baird 2012).

From this evidence, it is then clear that drug market and turf competition are interrelated issues whose consequences over local communities on the generation of violence must be better understood. In this paper we ask how the level of competition in both the drug market and the territory affect total level of violence. This is our first contribution.

The model focuses on three important characteristics of the drug market: imperfect competition, presence of switching costs and consumer loyalty and gang intimidation. Three other stylized facts from illegal street-level drug markets are also taken into account in the model. According to ONDCP (2000–2012) in its Arrestee Drug Abuse Monitoring Program I and II (ADAM), (i) over 62% of arrestees who bought drugs frequently did it the last time directly from a regular dealer and over 20% did it from an occasional or new dealer, (ii) they bought the drug through a cash transaction and most of the time in open markets and (iii) 50% of the arrestees bought at least one transaction inside their own neighborhood. 3 We assume that two gangs supply drugs at the street level, share a given territory, maximize profits, and choose the drug price and the level of guns investments to secure their own turf and intimidate users.

Due to their negative impact on local communities, governmental authorities implement law enforcement policies with the aim of disrupting drug markets and diminishing the level of violence. For instance, law enforcement policies in the United States, such as traditional and community policing, 4 have been implemented to diminish the impact of drug markets and the level of gang violence. While programs like Community Oriented Policing Services (COPS), Project Safe Neighborhoods, the Comprehensive Anti-Gang Initiative, the Violent Crime Reduction Partnership Initiative, or the Violent Crime-Focused Task Forces are part of a comprehensive plan to combat gangs across America, others like Organized Crime Drug Enforcement Task Force or Mobile Enforcement Team Program target violent gangs involved in significant drug trafficking (NGIC 2009). In Brazil and Colombia, the governments have carried out high-profile raids to arrest gang leaders at the favelas/comunas and set a rule of law, together with community programs and urban upgrading in order to decrease gang violence and disrupt illegal drug markets; and in the UK, some projects involve taking vulnerable young people off the streets at night and return them to their parents and others combine law enforcement with community policing initiatives to achieve the same goals (INCB 2011).

However, the empirical evidence on the effect of law enforcement policies on violence has had mixed results. For instance, in a study based on data from the National Institute of Justice in the United States, Resignato (2000) finds that more intense drug law enforcement appears to increase violent crime. Other studies have found that traditional policing strategies concentrating on resources to areas where crime is high have had mixed results in reducing violence. In some cases, violence is reduced, in others the effect is negligible or even the opposite (Svidiroff and Hillsman 1994; Weisburd and Green 1995; Miron 2001; Rasmussen and Benson 2001; Mazerolle, Soole, and Rombouts 2007; GAO 2003, 2005; Mastrofski 2006; MacDonald 2002; Sherman and Eck 2006; Skogan 2006; Weisburd and Eck 2004; Zhao, Scheider, and Thurman 2002; Zhao and Thurman 2004).

This paper might contribute to shed some light on how some law enforcement policies affect the level of violence in drug markets since the level of violence will depend on both drug market and turf competition. Understanding this relationship will then help to clarify the circumstances under which law enforcement policies could either work or not. This is the second contribution of the paper.

We propose a two-stage game-theoretical model to study the interplay of drug market and turf competition between two gangs, the effect on the total level of violence and the impact of law enforcement policies. Gangs first invest in guns, which is assumed to be positively correlated to violence, as a form of securing their territory and intimidating potential costumers, and then compete in drug prices. Drug users buy a fixed amount of drugs from either gang (market is covered 5), live in one of the two gang territories and face disutility from switching drug supplier and/or gang intimidation. On one hand, the disutility from switching supplier depends on two elements. First, the idiosyncratic random switching cost which represents the level of drug addiction for each user, and second, the switching parameter that scales up or down the impact of the idiosyncratic cost due to external factors in the turf. On the other hand, the disutility from gang intimidation also depends on two elements. First, the level of gun investments by a gang, and second, the witness intimidation parameter that scales up or down the impact of gun investments depending on the relationship established between drug users and the gang in a territory.

Therefore, we model explicitly the witness intimidation parameter that a gang controlling a turf generates on drug users living in that territory by the fear of having some retaliation if they are caught acting against the gang’s interests. In our case this act is to buy drugs from the other gang. Ethnographic reports and evidence on the number of illegal transactions and drug-related homicides make clear that gang confrontations that escalate to actual violence are a small proportion of all potential ones (Caulkins, Reuter, and Taylor 2006). Hence, threat and intimidation is a larger problem than homicides, implying that gangs’ intimidation is a more problematic behavior than the actual use of violence (Reuter and Truman 2004; NGIC 2011).

Three main results are found. First, total level of violence in a drug market depends on relative gangs’ shares of drug market and turf, and witness intimidation and switching parameters. Second, law enforcement policies such as traditional policing (e.g., police presence) or community policing (e.g., trust enhancement) may have unintended consequences. Since the empirical evidence readily available has demonstrated inconclusive results on traditional and/or community policing, the reasons exposed here can be possible explanations of these facts. And third, we extend the benchmark model by endogenizing the gangs’ turf sizes which implies that the use of violence by gangs can also produce gang wars that change the turf sizes. In this extension, law enforcement policies such as community policing decrease the total level of violence in a market, whereas traditional policing can have opposite results depending on drug market shares and both switching and witness intimidation parameters. Furthermore, comparing the results between the benchmark model and its extension we find that traditional policing might have unexpected consequences in a turf when gangs decide to fight for turf relative to the case when gun investments are made under the shadow of conflict (i.e., exogenous turf shares).

To my knowledge, the relationship between turf and drug market competition and the level of violence/intimidation in illegal drug markets has not yet been studied, neither the consequences of different law enforcement policies under this context. There are two related literatures: switching costs and (behavior-based) price discrimination by purchase history with imperfect competition, and the use of violence in illegal drug markets.

Klemperer (1987) studied the effects of the existence of customer bases and switching costs on prices for legal markets, whereas the application for illegal drugs has been studied by Skott and Jepsen (2003). They present a stylized model of the market for hard drugs under imperfect competition and focus on the impact of drug policies on marketing strategies by drug suppliers and the number of addicts and total drug consumption. As opposed to them, our paper focuses on the effect of drug policies on total violence and the influence of drug market and turf competition between drug suppliers. Furthermore, one approach to modeling imperfect competition and purchase-history price discrimination, used by Nilssen (1992), Chen (1997) and Taylor (2003) among others, assumes that the goods are initially homogenous but after purchase the users are partially locked in with their sellers and exogenous costs must be incurred to switch to different firms in the future. This approach will also be taken in the present paper but unlike them the modeling of violence in an illegal drug market will be done explicitly within this approach.

The link between illegal drug markets and violence has been normally studied through the “tripartite framework” developed by Goldstein (1985). A psychopharmacological, economic-compulsive or systemic link are the three possible explanations to this relationship. In this paper, the systemic link is the one assumed to be the most important in understanding the relationship between drug markets and violence. Assuming the systemic link between violence and illegal drug markets, in the past decade, the literature has made emphasis on the effect of violence over drug prices trying to understand the puzzle about lower drug prices together with large investments in supply-oriented anti-drug policies (Donohue and Levitt 1998; Burrus 1999; Poret 2003; Caulkins, Reuter, and Taylor 2006; Naranjo and Jacobsson 2009; Naranjo 2011) but has left the issue of the interplay between drug market and turf competition and its impact on violence unstudied.

In the next section we describe the basic model and show the main findings in equilibrium. Section 3 studies the comparative statics of the model, focusing on the relevant policy-induced changes in the parameters to show the impact of different law enforcement policies. Section 4 makes an extension of the model by allowing endogenous turf shares, Section 5 compares the main results between the basic model and its extension and Section 6 concludes.

## 2 The Model

In this section, we examine a simple two-stage model where two profit-maximizing gangs (drug suppliers), $i$ and $j$, belonging to two different territories, compete in drug prices and gun investments on a mature drug market, and where drug users face a switching cost whenever they change suppliers. Thus, the drug market refers to a territory divided between both gangs where drug users live and choose a drug supplier.

Since gangs $i$ and $j$ (i.e., territories $i$ and $j$) have already customer and territorial bases, initial drug market and turf shares are $\mathrm{\alpha }$ and $\left(1-\mathrm{\alpha }\right)$ and $\mathrm{\beta }$ and $\left(1-\mathrm{\beta }\right)$, respectively. Drug market shares are formed by users who previously bought a unit of drug from one of the gangs, independently of where they live, whereas the turf shares are formed by users who live in the gang’s territory, independently of whom they previously bought a unit of drug.

In the first stage, gangs simultaneously decide on the level of gun investments, ${z}_{i}$ and ${z}_{j}$, they will maintain in their own turf, given the initial drug market and turf shares. In the second stage, gangs set their drug prices, ${p}_{mn}$, where $m$ identifies the user’s part of a gang’s market share and $n$ identifies the new gang supplier, since users can switch between suppliers. After prices are set, drug users choose a gang supplier. Drugs are homogenous and, for simplicity, can be produced at a zero marginal cost.

Drug users are assumed to each demand one unit of drugs and they all derive an intrinsic utility $r$ from this. 6 However, the level of gun investments by gangs when purchasing the unit of drug generates a source of disutility. Specifically, the disutility experienced by a drug user living in a gang’s territory and buying a unit of drug in the other gang’s territory (i.e., from the other gang) is given by ${\mathrm{\theta }}_{k}{z}_{k},$ where ${z}_{k}$ is the level of gun investments by a gang in territory $k,$ and ${\mathrm{\theta }}_{k}$ the witness intimidation level users living in territory $k$ perceive from gang $k$, where $0<{\mathrm{\theta }}_{k}\le 1$ and $k=i,j$. Witness intimidation comes in many forms such as explicit threats, physical violence, implicit intimidation and/or community-wide intimidation, 7 and it is also affected by the level of trust between the community and the law enforcement authorities. This witness intimidation parameter closer to $1$ means that a higher share of gun investment by a gang is transferred into the user’s disutility.

On the other hand, the cost of switching supplier, 8 $s$, may vary among users and is assumed to be given by a random variable uniformly distributed on $\left[0,1\right]$. 9 This cost is idiosyncratic and can represent, for example, the addiction level each consumer has. The more addictive a consumer is the lower cost of switching gang supplier it faces since more addictive users are more willing to buy from an unknown dealer (i.e., switch supplier) than less addictive users, implying the existence of different switching costs among consumers. This switching cost is also a source of consumer’s disutility. Specifically, the disutility experienced by a drug user who previously bought a unit of drug from a gang and buys it now from the opponent is given by ${\mathrm{\gamma }}^{k}s,$ where $s$ is the idiosyncratic switching cost by a drug user and ${\mathrm{\gamma }}^{k}$ the switching parameter indicating external costs present in the territory $k$, where ${\mathrm{\gamma }}^{k}\ge 1$ and $k=i,j$. This switching parameter represents all external costs associated with the territory such as physical disorder or police presence in the locations where transactions take place. 10 The closer to $1$ the less external costs can be associated with switch drug supplier.

The analysis is confined to covered markets where all drug users buy drugs. The actual demand faced by a gang consists of its loyal customers plus buyers switching from the competitor, either from the turf it controls or from its competitor’s turf. Depending on whose customer base the user initially belongs to, where the user lives and who is the new supplier, the user faces different utility levels of buying a unit of drug.

First, the utility of a loyal buyer of gang $i$ and living in $i$’s turf is ${U}_{ii}^{i}=r-{p}_{ii}$where the first position in the subscript denotes the buyer’s original supplier and the second position denotes the current supplier, whereas the overscript denotes the turf where the user lives. By symmetry, a loyal buyer of gang $j$ who lives in $j$’s turf becomes ${U}_{jj}^{j}=r-{p}_{jj}.$ We can then compare these utilities with the ones of a switcher buyer living in $i$’s turf, ${U}_{ij}^{i}=r-{p}_{ij}-{\mathrm{\gamma }}^{j}s-{\mathrm{\theta }}_{i}{z}_{i},$ and a switcher buyer living in $j$’s turf, ${U}_{ji}^{j}=r-{p}_{ji}-{\mathrm{\gamma }}^{i}s-{\mathrm{\theta }}_{j}{z}_{j}.$

And second, the utility of a loyal buyer of gang $i$ and living in $j$’s turf is ${U}_{ii}^{j}=r-{p}_{ii}-{\mathrm{\theta }}_{j}{z}_{j}$By symmetry, a loyal buyer of gang $j$ who lives in the $i$’s turf becomes ${U}_{jj}^{i}=r-{p}_{jj}-{\mathrm{\theta }}_{i}{z}_{i}.$ We can then compare these utilities with the ones of a switcher buyer living in $j$’s turf, ${U}_{ij}^{j}=r-{p}_{ij}-{\mathrm{\gamma }}^{j}\phantom{\rule{thinmathspace}{0ex}}s,$ and a switcher buyer living in $i$’s turf, ${U}_{ji}^{i}=r-{p}_{ji}-{\mathrm{\gamma }}^{i}\phantom{\rule{thinmathspace}{0ex}}s$. The hypothesis behind this argumentation relies on the fact that a drug user cannot resell the drug it buys. Due to the nature of these markets where gangs use violence to secure their profits against anyone who could potentially diminish them (not only other gangs), it can be very difficult for a user to buy and resell the drug since the cost of doing must be very high (including its own life). 11

Therefore, a drug user living in the gang $i$’s turf is indifferent between being loyal or switching to gang $j$ if its critical switching cost is $s=\frac{{p}_{ii}-{p}_{ij}-{\mathrm{\theta }}_{i}{z}_{i}}{{\mathrm{\gamma }}^{j}}.$ In addition, we can establish that a buyer originally buying from gang $i$ and living in gang $j$’s turf is indifferent between being loyal or switching to gang $j$ if its critical switching cost is $s=\frac{{p}_{ii}-{p}_{ij}+{\mathrm{\theta }}_{j}{z}_{j}}{{\mathrm{\gamma }}^{j}}.$ Under our assumptions, gang $i$’s current demand, ${Q}_{i}$, consists of loyal users who live both in its turf and in the opponent’s turf, ${Q}_{ii}={q}_{ii}^{i}+{q}_{ii}^{j},$ and switchers from the rival gang who live in either turf, ${Q}_{ji}={q}_{ji}^{i}+{q}_{ji}^{j}$, i.e., ${Q}_{i}={Q}_{ii}+{Q}_{ji}$.

Given initial turf and drug market shares, and the uniform distribution of switching costs, the demands for gang $i$ become 12 ${Q}_{ii}={q}_{ii}^{i}+{q}_{ii}^{j}=\mathrm{\alpha }\mathrm{\beta }\left(\frac{{\mathrm{\gamma }}^{j}-{p}_{ii}+{p}_{ij}+{\mathrm{\theta }}_{i}{z}_{i}}{{\mathrm{\gamma }}^{j}}\right)+\mathrm{\alpha }\left(1-\mathrm{\beta }\right)\left(\frac{{\mathrm{\gamma }}^{j}-{p}_{ii}+{p}_{ij}-{\mathrm{\theta }}_{j}{z}_{j}}{{\mathrm{\gamma }}^{j}}\right)$[1] ${Q}_{ji}={q}_{ji}^{i}+{q}_{ji}^{j}=\left(1-\mathrm{\alpha }\right)\mathrm{\beta }\left(\frac{{p}_{jj}-{p}_{ji}+{\mathrm{\theta }}_{i}{z}_{i}}{{\mathrm{\gamma }}^{i}}\right)+\left(1-\mathrm{\alpha }\right)\left(1-\mathrm{\beta }\right)\left(\frac{{p}_{jj}-{p}_{ji}-{\mathrm{\theta }}_{j}{z}_{j}}{{\mathrm{\gamma }}^{i}}\right)$[2]From the equations above increases in the level of gun investments by a gang will increase the number of loyal users and switching users living in the gang’s turf due to a higher intimidation. In addition, increases in these investments by the opponent will have the opposite effect.

The profit for a gang $i$ is simply the price times the number of users who buy from it minus the cost of gun investments. Using eqs [1] and [2], the profit function for a gang $i$ is $\begin{array}{rl}{\mathrm{\pi }}_{i}=\phantom{\rule{thickmathspace}{0ex}}& \mathrm{\alpha }{p}_{ii}\left(\frac{{\mathrm{\gamma }}^{j}-{p}_{ii}+{p}_{ij}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}}{{\mathrm{\gamma }}^{j}}\right)\\ & +\left(1-\mathrm{\alpha }\right){p}_{ji}\left(\frac{{p}_{jj}-{p}_{ji}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}}{{\mathrm{\gamma }}^{i}}\right)-{\left({z}_{i}\right)}^{2}\end{array}$[3]where the cost of the investment in guns is assumed to be quadratic 13 and a sunk cost at the second stage.

We solve the game by backward induction, starting from the gangs’ decisions in the second stage.

## 2.1 Stage 2

Taking the first-order conditions in eq. [3] with respect to drug prices and solving for ${p}_{ii}$ and ${p}_{ji}$ yields the following reaction functions 14 ${p}_{ii}=\frac{1}{2}\left({\mathrm{\gamma }}^{j}+{p}_{ij}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right)$[4] ${p}_{ji}=\frac{1}{2}\left({p}_{jj}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right)$[5]As would be expected in a Bertrand model, drug prices are strategic complements, i.e., an increase in a gang’s price results in a higher price from its rival. In addition, increased gun investments by one gang shifts its price-reaction function upward, reflecting that it will enjoy part of the demand increase in terms of higher price, while shifting downward its opponent’s price-reaction function. Therefore, there are forces working in opposite directions, since the gang’s own drug price is increased by a direct effect of gun investments while it is reduced by an indirect effect through the decrease in the price of its opponent.

Furthermore, initial drug market shares do not affect drug prices while both turf shares $\mathrm{\beta }$ and $\left(1-\mathrm{\beta }\right),$ and the witness intimidation level parameter ${\mathrm{\theta }}_{k}$ do it directly through the factor $\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}$ and indirectly through the opponent’s drug price. From the direct effects, the higher the gang’s turf share and the intimidation level parameter, the higher its drug price since more users live in the gang’s territory so gun investments intimidate more users. Finally, the higher the opponent’s switching cost parameter ${\mathrm{\gamma }}^{k}$ the higher the drug price set to loyal drug users leaving unaffected drug prices to switchers.

Using both reaction functions, eqs [4] and [5], and the symmetric results from the opponent’s problem it is straightforward to solve for the respective drug prices for gang $i$: ${p}_{ii}^{\ast }=\frac{1}{3}\left(2{\mathrm{\gamma }}^{j}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right)$[6] ${p}_{ji}^{\ast }=\frac{1}{3}\left({\mathrm{\gamma }}^{i}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right)$[7]Hence, increased gun investments by a gang ${z}_{i}$ allows it to raise its own price but reduces that of its opponent for any type of drug user (either loyal or switcher). Moreover, note that a higher gang’s turf share $\mathrm{\beta }$ increases its drug price for both loyal and switcher customers (i.e., coming from the opponent) while higher switching cost parameter ${\mathrm{\gamma }}^{i}$ raises the price only for switchers since drug prices are strategic complements.

Furthermore, by substituting eqs [6] and [7] into eqs [1] and [2], the total number of drug users (or drug quantities) for both gangs become ${Q}_{ii}^{\ast }={q}_{ii}^{\ast i}+{q}_{ii}^{\ast j}=\frac{\mathrm{\alpha }}{3{\mathrm{\gamma }}^{j}}\left(2{\mathrm{\gamma }}^{j}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right)$[8] ${Q}_{ji}^{\ast }={q}_{ji}^{\ast i}+{q}_{ji}^{\ast j}=\frac{\left(1-\mathrm{\alpha }\right)}{3{\mathrm{\gamma }}^{i}}\left({\mathrm{\gamma }}^{i}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right)$[9]From eqs [8] and [9], the number of drug users for gang $i$ increases with its turf $\mathrm{\beta }$ and gun investments ${z}_{i}$ but decreases with the opponent’s. Having a larger combination of both gun investments and turf share relative to its opponent’s (i.e., $\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}>0$), increases the number of drug users buying from gang $i$, either from the loyal or the switcher users living in any turf. Moreover, higher switching cost parameters ${\mathrm{\gamma }}^{i}$ will have a non-monotonic effect on the number of drug users for gang $i$ depending on the sign of $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right),$ whereas a higher drug market share by a gang will increase (decrease) the number of loyal (switcher) users.

In the subsections below, we will examine the equilibrium level of gun investments. However, we first need to determine the objective function of the gang $i$. Inserting eqs [6]–[9] into the profit function [3] yields the following function: ${\mathrm{\pi }}_{i}^{\ast }=\frac{\mathrm{\alpha }}{9{\mathrm{\gamma }}^{j}}{\left(2{\mathrm{\gamma }}^{j}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right)}^{2}+\frac{\left(1-\mathrm{\alpha }\right)}{9{\mathrm{\gamma }}^{i}}{\left({\mathrm{\gamma }}^{i}+\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}\right)}^{2}-{z}_{i}^{2}$[10]

## 2.2 Stage 1

At this stage, given initial market and turf shares and switching cost and intimidation level parameters, gangs simultaneously decide the level of gun investments.

Taking the first-order conditions in eq. [10] with respect to ${z}_{i}$ and solving for the level of gun investments, we have the following reaction functions 15: ${z}_{i}=\mathrm{M}\mathrm{a}\mathrm{x}\left\{a-b{z}_{j},0\right\}$[11] ${z}_{j}=\mathrm{M}\mathrm{a}\mathrm{x}\left\{c-d{z}_{i},0\right\}$[12]where $a=\frac{\mathrm{\beta }{\mathrm{\theta }}_{i}\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}}{9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A},b=\frac{\mathrm{\beta }{\mathrm{\theta }}_{i}\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}A}{9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A},c=\frac{\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\left(2-\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}}{9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A}$ and $d=\frac{\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\mathrm{\beta }{\mathrm{\theta }}_{i}A}{9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A}$ are all positive, and $A=\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right).$

These reaction functions imply that increased gun investments by one gang is met by a reduction in these investments by the opponent. More gun investments produce higher intimidation to drug users living in a gang’s turf but buying from the other gang which in turn secures a greater number of users. If this happens, independently of the size of market share, the opponent will need to decrease its gun investments since it becomes relatively more expensive. In addition, increased switching cost parameter implies a non-monotonic effect on these investments depending on the level of the opponent’s gun investments. When the opponent’s gun investments are sufficiently low increases in the gang’s switching parameter lower its investments but when the opponent’s gun investments are sufficiently high the increase in the switching parameter raises the gang’s gun investments.

The next section studies these reaction functions and looks for the interior equilibrium in the game.

## 2.3 The Interior Equilibrium

For an interior equilibrium to be unique and stable, the following conditions must hold: 16 $a<\frac{c}{d},c<\frac{a}{b}$ and $\frac{1}{b}>d$. These conditions always hold.

Below, we derive the levels of gun investments by each gang in an interior equilibrium using expressions [11] and [12]: ${z}_{i}^{\ast }=\frac{3\mathrm{\beta }{\mathrm{\theta }}_{i}\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\mathrm{\beta }{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}^{2}A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}$[13] ${z}_{j}^{\ast }=\frac{3\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\left(2-\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\left(1-\mathrm{\beta }\right){\mathrm{\beta }}^{2}{\mathrm{\theta }}_{j}{\mathrm{\theta }}_{i}^{2}A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}$[14]where ${z}_{i}^{\ast }>0$ and ${z}_{j}^{\ast }>0$ in an interior equilibrium and $A=\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right)$.

The following lemma sums up the effect of both intimidation and switching parameters on the total level of gang violence in the drug market.

(i) A higher switching parameter ${\mathrm{\gamma }}^{i}$ has a non-monotonic effect on gun investments by both gangs. When gang $i$ has a sufficiently large (low) turf share ($\mathrm{\beta }>\left(<\right)\phantom{\rule{thinmathspace}{0ex}}\stackrel{˜}{\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{\beta }}$) its gun investments decrease (increase) and its opponents increase (decrease). (ii) A higher intimidation parameter ${\mathrm{\theta }}_{i}$ increases gun investments by gang $i$ but decreases its competitors.

The intuition is as follows. A sufficiently large turf share for a gang implies that users living in its territory are very important relative to the users from the other gang’s turf. We also know that users living in a territory buy drugs from both gangs, either as loyal users or as switchers. When a gang decreases its gun investments both loyal users and switchers living in its territory face lower costs to buy a unit of drug from the opponent. However, the fact that the gang has a large territory implies it can save resources lowering its gun investments even though there will be some users in its territory buying now the drug from the opponent. This does not happen when the turf share is sufficiently small since the opponent’s turf share becomes more relevant and the gang needs to avoid the users living in its territories to buy from the opponent.

Moreover, increasing the marginal impact of gun investments, independently of the turf size shares, implies that the benefit of increasing these investments is higher than its cost. These investments will bring higher costs to users living in a territory and buying in the other.

In addition, the total amount of gun investments in the market is $\begin{array}{rl}{Z}^{\ast }& ={z}_{i}^{\ast }+{z}_{j}^{\ast }\\ & =\frac{3{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left(\mathrm{\beta }{\mathrm{\theta }}_{i}\left(1+\mathrm{\alpha }\right)+\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\left(2-\mathrm{\alpha }\right)\right)-\mathrm{\beta }\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}A\left(\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}+\mathrm{\beta }{\mathrm{\theta }}_{i}\right)}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}>0\end{array}$[15]Since increases in both witness intimidation and switching parameters have opposite effects on both gangs the final impact on violence in the drug market will depend on which effect is stronger. In terms of the witness intimidation parameter, the increase in ${\mathrm{\theta }}_{i}$ will increase gang $i$’s gun investments more than the decrease in gang $j$’s investments, increasing the total level of violence in the market. The intuition for this result says that a higher witness intimidation parameter will make drug users buying less from the opponent and more from the gang whose parameter increased which makes even more profitable for the gang to increase gun investments. However, the negative impact on the opponent’s revenues will have the consequence of diminishing its gun investments. In the aggregate, the incentive for the gang to increase its gun investments is larger than the saving cost strategy by the opponent resulting in a higher level of violence (i.e., gun investments) in the market.

In addition, in terms of the switching parameter, the increase in ${\mathrm{\gamma }}^{i}$ will result in $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}>0$ and $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}<0$ if $\mathrm{\beta }<\phantom{\rule{thinmathspace}{0ex}}\stackrel{˜}{\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{\beta }}$ and in $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}<0$ and $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}>0$ if $\mathrm{\beta }>\phantom{\rule{thinmathspace}{0ex}}\stackrel{˜}{\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{\beta }}$. In the aggregate, if $\mathrm{\beta }<\phantom{\rule{thinmathspace}{0ex}}\stackrel{˜}{\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{\beta }}$ then $\left|\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}\right|>\left|\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}\right|$ and if $\mathrm{\beta }>\stackrel{ˉ}{\mathrm{\beta }}$ then $\left|\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}\right|<\left|\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}\right|.$ The intuition for this result says that when the switching parameter increases in a turf drug users find less incentives to buy drugs from the gang controlling that turf, independently where they live. However, when that gang has a sufficiently large turf share drug users living in its turf becomes less important. This lower relevance makes the gang to decrease its gun investments in order to save costs. Moreover, the opponent will find optimal to increase its gun investments to increase the disutility for its users to buy drugs from the other gang securing some revenue since drug users in its turf become very important. In the aggregate, the decrease in gun investments by the gang whose switching parameter increases will be lower than the increase in these investments by the opponent, increasing the level of violence (i.e., total gun investments) in the drug market. 17 It is important to say that the size of the impact will depend on both drug market shares and witness intimidation parameters since the threshold value $\stackrel{ˉ}{\mathrm{\beta }}$ depends on these values. 18

Furthermore, inserting expressions [13] and [14] into eqs [6] and [7] gives the average drug price in equilibrium for both gangs 19: ${P}_{i}^{\ast }=\frac{1}{6}\left(2{\mathrm{\gamma }}^{j}+{\mathrm{\gamma }}^{i}+2\left(\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}^{\ast }-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}^{\ast }\right)\right)$ ${P}_{j}^{\ast }=\frac{1}{6}\left(2{\mathrm{\gamma }}^{i}+{\mathrm{\gamma }}^{j}-2\left(\mathrm{\beta }{\mathrm{\theta }}_{i}{z}_{i}^{\ast }-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{z}_{j}^{\ast }\right)\right)$And the average drug price in equilibrium for the market is 20 ${P}^{\ast }=\frac{{P}_{i}^{\ast }+{P}_{j}^{\ast }}{2}=\frac{{\mathrm{\gamma }}^{i}+{\mathrm{\gamma }}^{j}}{4}$[16]As we see, the average drug price is not affected by the gangs’ witness intimidation parameters. An increase in this parameter makes gang’s gun investments higher while decreases its opponent’s and both effects cancel out in the aggregate. However, increases in the switching parameters make the average drug price to increase, since it gives a higher market power for each gang. Therefore, if average drug prices fall (as it has been the case in many places) can be because gangs have strategies to make users to switch from the opponent.

## 3 Impact of Law Enforcement Policies

The model in section 2 can be used to examine the effects of law enforcement policies. Our approach is simple: law enforcement policies may change both the intimidation and switching parameters, ${\mathrm{\theta }}_{k}$ and ${\mathrm{\gamma }}^{k}$,and the comparative statics of the model will reveal the effects on total level of violence in the market proxied by the level of gun investments.

Hence, below we examine how the interior equilibrium level of gun investments in a drug market responds to changes in the key parameters in our model, i.e., the intimidation level, ${\mathrm{\theta }}_{k}$, and the switching cost parameter, ${\mathrm{\gamma }}^{k}$, where $k=i,j$. We will explicitly assume two different law enforcement policies affecting these parameters: community policing and traditional policing. First, community policing reflects the level of trust between a local community and the law enforcement authorities affecting the intimidation level ${\mathrm{\theta }}_{k}$ generated by gun investments for each gang. Hence, ${\mathrm{\theta }}_{k}=f\left({\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{k}\right)$ where $\frac{d{\mathrm{\theta }}_{k}}{d{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{k}}<0$ and $k=i,j$. And second, traditional policing such as patrol police presence in the streets within a gang’s turf will impact the gang’s switching cost, ${\mathrm{\gamma }}^{k}=f\left(\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}{\mathrm{e}}^{k}\right)$ where $\frac{d{\mathrm{\gamma }}^{k}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}{\mathrm{e}}^{k}}>0$ and $k=i,j$.

## 3.1 Community Policing

Trust level between a local community and the law enforcement authorities would affect the level of gang’s intimidation against drug users, ${\mathrm{\theta }}_{k}$, by lowering the effect of intimidation from gun investments by gangs. For instance, community policing implies a problem solving approach in terms of structural needs in the communities which enhance the dialogue between citizens and police.

Inserting the functional form ${\mathrm{\theta }}_{k}=f\left({\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{k}\right)$ where $\frac{d{\mathrm{\theta }}_{k}}{d{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{k}}<0$ and $k=i,j$ into expression [15] and taking the first derivative with respect to the policy parameter ${\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{i}$ we get $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{i}}=\left(\frac{d{\mathrm{\theta }}_{i}}{d{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{i}}\right)\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}+\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}\right)<0$since $\left|\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}\right|>\left|\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}\right|.$

The following proposition sums up these results.

Higher community policing in a territory decreases the level of violence in the drug market.

The intuition is as follows. Higher community trust in a gang’s turf will bring lower gang’s witness intimidation level increasing drug users’ utility for those living in $i$’s turf and buying from gang $j$ (either loyal or switchers). Hence, gang $i$ finds it optimal to increase its gun investments in order to protect its market. Since gang $j$ will decrease its gun investments taking advantage of the lower witness intimidation parameter faced by gang $i$, the net effect on the total level of gun investments in the market will depend on which effect is stronger. The result says that the increase in gun investment by gang $i$ is larger than the decrease in these investments by gang $j$ so total levels in the market decrease. A lower witness intimidation parameter in a turf will make gun investments, by the gang in that turf, more profitable in terms of the impact they have on drug users and relative to its cost, so gun investments increase. In addition, the extra income generated by the new users of the opponent gang will make gang $j$ to decreases gun investments but this does not compensate the increase from the other gang.

Traditional policing such as police presence in the streets within a gang’s turf affects positively the switching parameter, ${\mathrm{\gamma }}_{k}$, by making the encounter between drug users and dealers more difficult. Hence, more police presence makes the drug user to prefer being loyal and buying from the past gang supplier than trying to switch to the other.

Inserting the functional form ${\mathrm{\gamma }}^{k}=f\left({\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{k}\right)$ where $\frac{d{\mathrm{\gamma }}^{k}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{k}}>0$ and $k=i,j$ into expression [15] and taking the first derivative with respect to ${\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}$ we get $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}=\frac{\mathrm{\partial }{\mathrm{\gamma }}^{i}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}+\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}\right)+\frac{d{\mathrm{\gamma }}^{j}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{j}^{}}\left(\frac{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{j}^{}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\right)\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{j}}+\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{j}}\right)\lessgtr 0$[17]The following sections analyze the direction of this effect in the equilibrium of gun investments in the market under two different assumptions: (i) $\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}=0}$ and (ii) $\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{{dpolice}^{i}=-1.}$

## 3.2.1 No Correlation between Traditional Policing in$i$ and $j$$\left(\frac{d{police}^{j}}{d{police}^{i}}=0\right)$

The police presence in a turf can be the result of new police forces assigned to a turf. In this case more police presence in a turf should not affect police presence in the other turf. Hence, $\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}=0}$ and expression [17] becomes $\begin{array}{l}\frac{\partial {Z}^{\ast }}{\partial {\text{police}}_{i}}=\frac{d{\gamma }^{i}}{d{\text{police}}^{i}}\left(\frac{\partial {z}_{i}^{\ast }}{\partial {\gamma }^{i}}+\frac{\partial {z}_{j}^{\ast }}{\partial {\gamma }^{i}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{d{\gamma }^{i}}{d{\text{police}}^{i}}\left(\frac{9{\gamma }^{j}A\left({\left(1-\beta \right)}^{2}{\theta }_{j}^{2}\left(2-\alpha \right)-{\beta }^{2}{\theta }_{i}^{2}\left(1+\alpha \right)\right)\left(\beta {\theta }_{i}-\left(1-\beta \right){\theta }_{j}\right)}{9{\left(9{\gamma }^{i}{\gamma }^{j}-{\beta }^{2}{\theta }_{i}^{2}A-{\left(1-\beta \right)}^{2}{\theta }_{j}^{2}A\right)}^{2}}\right)\gtrless 0\end{array}$[18] Independently of both drug market and turf shares, increases in police presence in a gang $i$’s turf will increase violence in the drug market when the gang’s witness intimidation parameter is not too low or too large relative to the opponent’s. In addition, it will decrease violence in the drug market when the gang’s witness intimidation parameter is sufficiently low or large relative to the opponent’s.

When a gang $i$’s witness intimidation parameter is sufficiently low relative to its opponent’s, increase in its switching parameter due to the increase of police presence in its turf will increase its gun investments (and reduce its opponent’s). This happens because it needs to rise user’s disutility living in its turf and buying drugs from the opponent. However, gang $i$’s gun investments will be reduced if its witness intimidation parameter is sufficiently large relative to its opponent’s because the intimidation on drug users is sufficiently high to discourage users of buying from the opponent without a need of increasing gun investments, so saving in these investments becomes more relevant. The opposite result is found from the opponent. In the aggregate, total level of violence in the drug market will depend not only on the direction of the effect of the gang $i$’s switching parameter on gun investments from both gangs but also on the size of the effect. This size will depend on the factor $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)$. If gang $i$’s witness intimidation parameter is not too low then the increase in gun investments by gang $i$ is larger than the decrease in gun investments by gang $j$ so total level of violence in the drug market increases. However, when gang $i$’s witness intimidation parameter is sufficiently low then the increase in gun investments by gang $i$ is smaller than the decrease in gun investments by gang $j$ so total level of violence in the drug market decreases.

The following proposition sums up the main results.

When police presence in one turf is not correlated with its presence in the other turf, its increase in turf $i$ will decrease the level of violence in a drug market if gang$i$s witness intimidation parameter ${\mathrm{\theta }}_{i}$ is sufficiently low or sufficiently large relative to its opponents. When this parameter is neither too low nor too high within a range relative to the opponents then an increase in police presence in turf$i$will increase the total level of violence in the drug market.

## 3.2.2 Negative Correlation between Traditional Policing in $i$ and $j$$\left(\frac{d{police}^{j}}{d{police}^{i}}=-1\right)$

A police presence in a turf can be the result of a reallocation of a fixed amount of police assigned to both turfs. In this case more police presence in a turf implies less police presence in another. Hence, $\frac{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}{\mathrm{e}}^{j}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}{\mathrm{e}}^{i}}=-1.$ In the following, it is fair to assume that $\frac{d{\mathrm{\gamma }}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}=\frac{d{\mathrm{\gamma }}^{i}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}$ which means that police presence has equal effects on the switching parameters for both gangs. Hence, expression [17] becomes $\begin{array}{rl}& \frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\\ & \phantom{\rule{1em}{0ex}}=\frac{d{\mathrm{\gamma }}^{i}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\left(\frac{9A\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)\phantom{\rule{negativethinmathspace}{0ex}}\lessgtr 0\end{array}$[19]Since now an increase in police presence in one turf implies a decrease in the other, switching parameters in both turfs will need to be taken into account since they will increase in one turf and decrease in the other. We then see that when police presence in a turf is negatively correlated with police presence in the other then increases in police presence in a turf will have the same results as in Proposition 2 depending on the difference between gangs’ switching parameters. Even though the gang $i$’s intimidation parameter is too low or too high relative to the opponent’s, increases in police presence in its turf (that in this case decreases the police presence in the other) may end up increasing the level of violence in the drug market because the gang $i$’s switching parameter is larger than its opponent’s. When this happens the increase in police presence in turf $i$ will have a stronger impact on drug users’ disutility (relative to the opponent’s) if its switching parameter is higher than the other gang then the decrease in police presence in turf $j$ will impact gang $j$’s drug users disutility. For this reason, the decrease in gun investments by gang $j$ will be larger than the increase in gun investments by gang $i$ since gang $j$ have an advantage in switching parameters allowing more users to switch to them than the opponent. These results will also be opposite than in Proposition 2 if gang $i$’s witness intimidation parameter is not too low or too high but its switching parameter is larger than its opponent’s.

The following proposition sums up these results.

When police presence in one turf is negatively correlated with its presence in the other turf, having a larger switching parameter implies that increases in police presence in a turf, where witness intimidation is sufficiently low or large relative to the other, will increase total level of violence in the drug market. However, if the witness intimidation parameters in the turf whose police presence increased is within a range of the other turfs witness intimidation parameter then the increase in police presence will decrease total level of violence when the switching parameter is relatively larger.

## 4 Fighting: Endogenous Turf Shares

In this section we endogenize gangs’ turf shares by explicitly introducing a functional form for $\mathrm{\beta }$. In other words, we allow both gangs to affect their turf shares by choosing their level of gun investments so actual fighting is allowed. 21 In practice, endogenous turf shares implies the effective use of guns as opposed to the context of arming where there is a shadow of conflict. Practices such as drive-by shooting or actual exchange of fire are among the gangs’ strategies in the context of fighting.

The functional form will be represented by the standard contest success function 22 $\mathrm{\beta }=\frac{{\mathrm{\theta }}_{i}{z}_{i}}{{\mathrm{\theta }}_{i}{z}_{i}+{\mathrm{\theta }}_{j}{z}_{j}}$[20]This function says that increases in gun investments times the witness intimidation parameter by a gang will raise the gang’s turf share given a level of investment by the opponent. Having more gun investments and/or a higher witness intimidation in a community than the opponent will then be transferred into a higher probability of success, gaining a larger turf than the opponent. Therefore, inserting expression [20] into eq. [10] and using the values found in the stage 1 of the benchmark model, the new profit function from stage 2 becomes ${\mathrm{\pi }}_{i}^{\ast }=\frac{\mathrm{\alpha }}{9{\mathrm{\gamma }}^{j}}{\left(2{\mathrm{\gamma }}^{j}+{\mathrm{\theta }}_{i}{z}_{i}-{\mathrm{\theta }}_{j}{z}_{j}\right)}^{2}+\frac{\left(1-\mathrm{\alpha }\right)}{9{\mathrm{\gamma }}^{i}}{\left({\mathrm{\gamma }}^{i}+{\mathrm{\theta }}_{i}{z}_{i}-{\mathrm{\theta }}_{j}{z}_{j}\right)}^{2}-{z}_{i}^{2}$[21]Maximizing eq. [21] with respect to ${z}_{i}$ we get $\frac{\mathrm{\partial }{\mathrm{\pi }}_{i}^{\ast }}{\mathrm{\partial }{z}_{i}}=\frac{2\mathrm{\alpha }}{9{\mathrm{\gamma }}^{j}}\left(2{\mathrm{\gamma }}^{j}+{\mathrm{\theta }}_{i}{z}_{i}-{\mathrm{\theta }}_{j}{z}_{j}\right){\mathrm{\theta }}_{i}+\frac{2\left(1-\mathrm{\alpha }\right)}{9{\mathrm{\gamma }}^{i}}\left({\mathrm{\gamma }}^{i}+{\mathrm{\theta }}_{i}{z}_{i}-{\mathrm{\theta }}_{j}{z}_{j}\right){\mathrm{\theta }}_{i}-2{z}_{i}=0$Solving for ${z}_{i}$ we get 23 ${z}_{i}=\frac{{\mathrm{\theta }}_{i}{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left(1+\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}A{z}_{j}}{9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A}>0$[22]And by symmetry ${z}_{j}=\frac{{\mathrm{\theta }}_{j}{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}A{z}_{i}}{9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{j}^{2}A}>0$[23]where $A=\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right).$

From expressions [22] and [23] we get the interior equilibrium for the gun investments in the market as 24 ${z}_{i}^{\ast }=\frac{3{\mathrm{\theta }}_{i}{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left(1+\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}^{2}A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}$[24]And by symmetry ${z}_{j}^{\ast }=\frac{3{\mathrm{\theta }}_{j}{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{j}{\mathrm{\theta }}_{i}^{2}A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}$[25]where ${z}_{i}^{\ast }>0$ and ${z}_{j}^{\ast }>0$ which always hold.

The following lemma sums up the effect of both intimidation and switching parameters on the level of gang violence.

(i) A higher switching parameter ${\mathrm{\gamma }}^{i}$ has a non-monotonic effect on gun investments. When gang$i$has a sufficiently small (large) drug market share ($\mathrm{\alpha }<\left(>\right)\stackrel{ˉ}{\mathrm{\alpha }}$) its gun investments increase (decrease) and its opponents decrease (increase). (ii) A higher witness intimidation parameter ${\mathrm{\theta }}_{i}$ increases gun investments by gang $i$ but decreases its competitors.

Impacts of both witness intimidation parameters and switching parameters on the level of violence by each gang will depend on the size of the gangs’ market shares. Increasing gang $i$’s switching parameter implies that less drug users will switch to buy drugs from that gang which will increase gun investments so less users living in that turf will buy drugs from the other gang. This is true if gang $i$’s market share is sufficiently small. However, when gang $i$’s market share is sufficiently large the increase in switching parameter will not be that costly because of the number of drug users buying from that gang, living in either turf. So, users living in gang $i$’s turf are not that important as in the case where gang $i$’s market share is sufficiently small. Opposite results are found for the other gang.

In terms of a higher intimidation parameter, this will always increase gang $i$’s gun investments because the gang can save in gun investments due to the higher drug users’ disutility.

Inserting eqs [24] and [25] into expression [20], the gang $i$’s turf share becomes ${\mathrm{\beta }}^{\ast }=\frac{3{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}{\mathrm{\theta }}_{j}^{2}A}{3{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left({\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)+{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)\right)-2{\mathrm{\theta }}_{i}^{2}{\mathrm{\theta }}_{j}^{2}A}$[26]where $A=\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right).$

And the total level of dealers and gun investments in the market ${Z}^{\ast }={z}_{i}^{\ast }+{z}_{j}^{\ast }=\frac{3{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left({\mathrm{\theta }}_{i}\left(1+\mathrm{\alpha }\right)+{\mathrm{\theta }}_{j}\left(2-\mathrm{\alpha }\right)\right)-{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}\left({\mathrm{\theta }}_{j}+{\mathrm{\theta }}_{i}\right)A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}$[27]Therefore, in the aggregate, increases in the witness intimidation parameter ${\mathrm{\theta }}_{i}$ increases the level of violence in the drug market since the increase in gang $i$’s gun investments is not offset by the decrease in gang $j$’s investments. In addition, increases in the switching parameter ${\mathrm{\gamma }}^{i}$ will have an ambiguous effect on the level of violence in the market depending on the relative witness intimidation parameters and market shares. If ${\mathrm{\theta }}_{i}$ is either sufficiently large or sufficiently small relative to its opponent’s then an increase in ${\mathrm{\gamma }}^{i}$ will decrease the level of violence in the drug market. However, if ${\mathrm{\theta }}_{i}$ is within a range of the opponent’s witness intimidation parameter then an increase in ${\mathrm{\gamma }}^{i}$ will increase the level of violence in the drug market. It is important to say that the threshold for the parameter ${\mathrm{\theta }}_{i}$ will depend on the drug market shares.

The intuition behind these results says that when the witness intimidation parameters are sufficiently similar between gangs then increases in ${\mathrm{\gamma }}^{i}$ will reduce the number of drug users switching from the opponent and buying from gang $i$ which in turn will increase its gun investments to increase the disutility of buying from the opponent for those who live in its turf. This increase will not be offset by the decrease in the opponent’s gun investments. However, if the gang $i$’s witness intimidation parameter is sufficiently small or large the decrease in gun investments by the opponent is larger than the increase in these investments by gang $i$, diminishing the level of violence in the drug market.

Furthermore, inserting expressions [24] and [25] into eqs [6] and [7] give the average drug price in equilibrium for both gangs 25: ${P}_{i}^{\ast }=\frac{1}{6}\left(2{\mathrm{\gamma }}^{j}+{\mathrm{\gamma }}^{i}+2\left({\mathrm{\theta }}_{i}{z}_{i}^{\ast }-{\mathrm{\theta }}_{j}{z}_{j}^{\ast }\right)\right)$ ${P}_{j}^{\ast }=\frac{1}{6}\left(2{\mathrm{\gamma }}^{i}+{\mathrm{\gamma }}^{j}-2\left({\mathrm{\theta }}_{i}{z}_{i}^{\ast }-{\mathrm{\theta }}_{j}{z}_{j}^{\ast }\right)\right)$And the average drug price in equilibrium for the market is 26 ${P}^{\ast }=\frac{{P}_{i}^{\ast }+{P}_{j}^{\ast }}{2}=\frac{{\mathrm{\gamma }}^{i}+{\mathrm{\gamma }}^{j}}{4}$[28]As it was the case where turf shares were exogenous, average drug prices will depend on switching parameters and not on witness intimidation parameters. The higher disutility faced by a user who switches gang supplier the higher the drug price charged by the opponent gang.

The next sections study the impact of law enforcement policies on levels of violence in a drug market under the context of a gang war (i.e., actual fighting).

## 4.1 Community Policing

Trust level between a local community and the law enforcement authorities would affect the level of gang’s intimidation against drug users, ${\mathrm{\theta }}_{k}$, by lowering the effect of intimidation from gun investments by gangs. For instance, community policing implies a problem solving approach in terms of structural needs in the communities which enhance the dialogue between citizens and police.

Inserting the functional form ${\mathrm{\theta }}_{k}=f\left({\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{k}\right)$ where $\frac{\mathrm{\partial }{\mathrm{\theta }}_{k}}{d{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{k}}<0$ and $k=i,j$ into expression [27] and taking the first derivative with respect to the policy parameter ${\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{i}$ we get $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{i}}=\frac{d{\mathrm{\theta }}_{i}}{d{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{i}}\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}+\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}\right)<0$since $\left|\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}\right|>\left|\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}\right|.$ Therefore, under the context of a gang fighting (i.e., endogenous turf shares) the increase in community policing will reduce the total level of violence in the drug market as it was also the case of arming but not fighting (i.e., exogenous turf shares).

Traditional policing such as police presence in the streets within a gang’s turf affects positively the switching cost parameter, ${\mathrm{\gamma }}_{k}$, by making the encounter between drug users and dealers more difficult. Hence, more police presence makes the drug user to prefer being loyal and buying from the past gang supplier than trying to switch to the other.

Inserting the functional form ${\mathrm{\gamma }}^{k}=f\left({\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{k}\right)$ where $\frac{d{\mathrm{\gamma }}^{k}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{k}}>0$ and $k=i,j$ into expression [27] and taking the first derivative with respect to ${\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}$ we get $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}=\frac{d{\mathrm{\gamma }}^{i}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}+\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}\right)+\frac{d{\mathrm{\gamma }}^{j}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{j}^{}}\left(\frac{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{j}^{}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\right)\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{j}}+\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{j}}\right)\gtrless 0$[29]The following sections analyze the direction of this effect in the equilibrium of gun investments in the market under two different assumptions: (i) $\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}=0}$ and (ii) $\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}=-1.}$

## 4.2.1 No Correlation between Traditional Policing in $i$ and $j$$\left(\frac{d{police}^{j}}{d{police}^{i}}=0\right)$

A police presence in a turf can be the result of new police forces assigned to a turf. In this case more police presence in a turf does not affect the police presence in another. Hence, expression [29] becomes $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}=\frac{d{\mathrm{\gamma }}^{i}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\left(\frac{9{\mathrm{\gamma }}^{j}A\left({\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\left({\mathrm{\theta }}_{i}-{\mathrm{\theta }}_{j}\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)\gtrless 0$[30]When gang $i$’s market share is lower than its opponent’s, the increase in police presence in its turf (which increases its switching parameter) will increase the level of violence in the market if its witness intimidation parameter is higher than its opponent’s but not too high. Moreover, it will decrease the level of violence if this parameter is either sufficiently lower or sufficiently larger. However, when gang $i$’s market share is larger than its opponent’s, the results may be the opposite. For instance, if gang $i$’s witness intimidation parameter is higher but sufficiently similar to its opponent’s then the police presence will decrease the level of violence in the market, and if this parameter is smaller but sufficiently similar to its opponent’s then this police presence will increase the level of violence. The intuition of these results is as follows.

On one hand, a lower drug market share than the opponent means that both drug users living in the turf and switching from the opponent become more important to a gang. Higher police presence increases the switching parameter so the gang needs to compensate this loss by changing its gun investments. It will increase these investments if its witness intimidation parameter is sufficiently small relative to its opponent’s (which will in turn decrease these investments). If this witness parameter is larger but sufficiently similar than its opponent’s the increase in that gang’s gun investments will be larger than the decrease in its opponent’s and the total level of violence in the market increases. However, if the witness parameter is sufficiently small the impact of the increase in gun investments is not that large so the total level of violence in the market decreases.

On the other hand, a higher drug market share than the opponent will make witness intimidation parameters less relevant since the gang has more drug users. Hence, witness intimidation parameters that were necessary for increasing gun investments when the gang had a lower market share than its opponent are now not necessary so the gang will actually decrease its level of gun investments resulting in opposite effects on the level of violence in the market.

The following proposition sums up the main results.

Police presence in turf $i$ will increase gang $i$s gun investments if its witness intimidation parameter is sufficiently low and decrease them if the parameter is sufficiently large relative to its opponents. Opposite results are found if these parameters are more extreme (lower or larger). Gang $j$s gun investments will follow the other direction. The net effect on the level of violence in the market depends on the relative size of the witness intimidation parameters. If the gang$i$s witness intimidation parameter is either sufficiently low or sufficiently large then increases in police presence in turf $i$ decrease total level of violence in the market whereas if the parameter lies within a range relative to the opponents, police presence will increase violence. Finally, the results will also be affected by gangsdrug market shares. When the drug market share for gang $i$ is lower than the opponent then its witness intimidation parameter can be lower than the opponents to decrease the total level of violence in the market. However, when its drug market share is larger, the witness intimidation parameter needs to be larger than the opponents in order to get the same result.

Proof.See Appendix A.

## 4.2.2 Negative Correlation between Traditional Policing in $i$ and $j$$\left(\frac{d{police}^{j}}{d{police}^{i}}=-1\right)$

A police presence in a turf can be the result of reallocation of a fixed amount of police assigned to both turfs. In this case more police presence in a turf implies less police presence in another. Hence, $\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}=-1}$. In the following, it is fair to assume that $\frac{d{\mathrm{\gamma }}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}=\frac{d{\mathrm{\gamma }}^{i}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}$ which means that police presence has equal effects on the switching parameters for both gangs. Hence, expression [29] becomes $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}=\frac{d{\mathrm{\gamma }}^{i}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\left(\frac{9A\left({\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\left({\mathrm{\theta }}_{i}-{\mathrm{\theta }}_{j}\right)\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)\lessgtr 0$[31]In this case, increases in police presence in one turf decreases its presence in the other. Therefore, differences between the switching parameters between turfs become relevant. If the switching parameter of the gang whose turf has a higher police presence is larger than its opponent’s then opposite results of Proposition 3 are found. The intuition is that when a turf has a higher switching parameter police presence in this turf will reinforce the incentives by both gangs making the impact of the gang with a higher parameter and a higher police presence on the level of violence in the market being offset by the impact of the opponent whose switching parameter is lower and faces a lower police presence.

The following proposition sums up the main results.

Under Proposition 4s conditions, having a higher switching parameter for gang $i$ whose turf had a higher police presence implies that when gang$i$increases its gun investments and the opponent decreases it, the result in the aggregate on the level of violence in the market will be the opposite as in Proposition 4. However, if the gang $i$s switching parameter is lower than the opponents then the same results as in Proposition 4 are found.

## 5 Arming versus Fighting

Considering the case when police presence in one turf is not correlated with police presence in the other $\left(\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}=0\right)$, when we compare the results between Propositions 2 and 4, and between Propositions 3 and 5, one important question arises: How does the impact of police presence in a turf change between the context of fighting and the context of arming?

Since there is more times of peace than war, we will assume for the analysis that the context of arming (i.e., peace) is the benchmark relative to the context of fighting (i.e., war).

## 5.1 When Traditional Policing Increases Violence

Let us first study the case when police presence in one turf is not correlated to police presence in the other $\left(\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}=0\right)$. When fighting is possible only the size of gang $i$’s drug market share and the witness intimidation parameters matter for the impact on total levels of violence in the drug market. However, when there is arming without fighting, the size of turf shares will also matter.

Independently of the context (arming or fighting) and the drug market shares, police presence in turf $i$ will increase the level of violence in the drug market when its witness intimidation parameter lies within a range of the opponent’s parameter. Since the differential impact of police presence on violence between the contexts of arming and fighting depends on the size of the turf shares, when the gang $i$’s turf share is larger than the opponent’s its witness intimidation parameter space moves to the left implying that lower levels of that parameter are required under the arming context to get the same result on violence than under the fighting context. However, when the gang $i$’s turf share is smaller than the opponent’s then its witness intimidation parameter space moves to the right, implying that higher levels of that parameter are required under the arming context to get the same result on violence than under the fighting context.

The next graphs show this situation for the case of $\mathrm{\alpha }>\frac{1}{2}$ (it is similar for the other case 27) and where $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ and ${\stackrel{˜}{\mathrm{\theta }}}_{j}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)$. $\mathrm{I}\mathrm{f}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\beta }>\frac{1}{2}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{\theta }}_{i}\phantom{\rule{negativethinmathspace}{0ex}}:0------\left[-a--\ast \ast \ast \ast \ast \ast \ast \ast \right]\ast \ast \ast \ast --------1\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$ ${z}_{j}=\text{Max}\left\{c-d{z}_{i},0\right\}$ $\mathrm{I}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\beta }<\frac{1}{2}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{\theta }}_{i}\phantom{\rule{negativethinmathspace}{0ex}}:0----\ast \ast \ast \ast \left[\ast \ast \ast \ast \ast \ast \ast \ast --b--\right]---------\phantom{\rule{negativethinmathspace}{0ex}}1\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$ ${z}_{j}^{\ast }=\frac{3\left(1-\beta \right){\theta }_{j}\left(2-\alpha \right){\gamma }^{i}{\gamma }^{j}-\left(1-\beta \right){\beta }^{2}{\theta }_{j}{\theta }_{i}^{2}A}{3\left(9{\gamma }^{i}{\gamma }^{j}-{\beta }^{2}{\theta }_{i}^{2}A-{\left(1-\beta \right)}^{2}{\theta }_{j}^{2}A\right)}$Therefore, under the benchmark model of arming, increases in police presence in turf $i$ will increase the level of violence in the drug market if its witness intimidation parameter, ${\mathrm{\theta }}_{i}$, is within a range of the opponent’s parameter (graphed as the parenthesis). Hence, if the gang $i$’s witness intimidation parameter lies within that range, increases in police presence will increase the level of violence in the drug market. As we said before, when the gang has a larger turf share then lower levels of this parameter are required compared to the case where it has a lower turf share.

If the gang $i$ has a larger turf share and a witness intimidation parameter equal to $a$ then under the context of arming increases in police presence will increase violence. However, if fighting between gangs occurs then the parameter $a$ will lie outside the set of parameters represented by asterisks $\ast$ (which makes police presence increase violence under the context of fighting) so increases in police presence will actually decrease violence in the market. In addition, if the gang $i$ has a smaller turf share and a witness intimidation parameter equal to $b$ then under the context of arming increases in police presence will increase violence. However, if fighting between gangs occurs then the parameter $b$ will lie outside a set of parameters represented by asterisks $\ast$ (which makes police presence increase violence under the context of fighting) so increases in police presence will actually decrease violence in the market.

The conclusion from this analysis says that, in the context of fighting, police presence in a turf whose gang has a larger turf share will actually decrease violence in a drug market, as opposed to the context of arming, when the witness intimidation parameter in that turf is in the lower part of the range. Moreover, when gang’s turf share is lower, the witness parameter must be in the higher part of the range in order to get the same result as in the context of arming when gang fighting is present.

Lastly, when police presence in turf $i$ is negatively correlated with police presence in the other turf $\left(\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}=-1\right)$ the results in the context of arming or fighting depend equally on switching parameters so results will be the same as in the case of no correlation.

## 5.2 When Traditional Policing Decreases Violence

Let us first study the case when police presence in one turf is independent on police presence in the other $\left(\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}=0\right)$. When fighting is possible only the size of gang $i$’s drug market share and the witness intimidation parameters matter. However, when there is arming without fighting, the size of turf shares matter.

Independently of the context (arming or fighting) and the drug market shares, police presence in turf $i$ will decrease the level of violence in the drug market when its witness intimidation parameter is sufficiently small or sufficiently large relative to its opponent’s. Since the differential impact of police presence on violence between the contexts of arming and fighting depends on the size of the turf shares, when the gang $i$’s turf share is larger than the opponent’s then its witness intimidation parameter becomes smaller for low levels and larger for high levels than under the context of fighting. However, when the gang $i$’s turf share is smaller than the opponent’s then its witness intimidation parameter becomes smaller for high levels and larger for low levels than under the context of fighting.

The next graphs show this situation for the case of $\mathrm{\alpha }>\frac{1}{2}$ (it is similar for the other case) and where $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ and ${\stackrel{˜}{\mathrm{\theta }}}_{j}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)$. $\begin{array}{l}\text{If}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta >\frac{1}{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{then}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\theta }_{i}:0\left[\ast \ast c\ast \ast \ast \right]\ast \ast \ast \ast --------\left[---d--\ast \ast \ast \ast \ast \ast \ast \ast \right]1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{\hspace{0.17em}}}^{{\stackrel{^}{\theta }}_{j}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\stackrel{^}{\theta }}_{j}}{\left(\frac{1-\beta }{\beta }\right)}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{\hspace{0.17em}}}^{{\stackrel{˜}{\theta }}_{j}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\stackrel{˜}{\theta }}_{j}}{\left(\frac{1-\beta }{\beta }\right)}}\end{array}$ $\mathrm{I}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\beta }<\frac{1}{2}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{\theta }}_{i}:0\left[\ast \ast \ast \ast \ast \ast \ast \ast \ast --e--\right]--------\ast \ast \ast \ast \left[\ast \ast f\ast \ast \right]1\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{\hspace{0.17em}}}^{\frac{{\stackrel{^}{\theta }}_{j}}{\left(\frac{1-\beta }{\beta }\right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{\theta }}_{j}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{\hspace{0.17em}}}^{\frac{{\stackrel{˜}{\theta }}_{j}}{\left(\frac{1-\beta }{\beta }\right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˜}{\theta }}_{j}}$Therefore, under the benchmark model of arming, increases in police presence in turf $i$ will decrease the level of violence in the drug market if its witness intimidation parameter, ${\mathrm{\theta }}_{i}$, is either too small or too large relative to the opponent’s (graphed as the parenthesis). Hence, if the gang $i$’s witness intimidation parameter lies within any of both ranges, increases in police presence will decrease the level of violence in the drug market.

If the gang $i$ has a larger turf share and a witness intimidation parameter equal to $c$ or $d$ then under the context of arming increases in polices presence will decrease violence. However, if fighting between gangs occurs then the parameter $d$ will lie outside a set of parameters represented by asterisks $\ast$ (which makes police presence decrease violence under the context of fighting) so increases in police presence will actually increase violence in the market. In addition, if the gang $i$ has a smaller turf share and a witness intimidation parameter equal to $e$ or $f$ then under the context of arming increases in polices presence will decrease violence. However, if fighting between gangs occurs then the parameter $e$ will lie outside a set of parameters represented by asterisks $\ast$ (which make police presence decrease violence under the context of fighting) so increases in police presence will actually increase violence in the market.

The conclusion from this analysis says that, in the context of fighting, police presence in a turf whose gang has a larger turf share will actually increase violence in a drug market, as opposed to the context of arming, when the witness intimidation parameter in that turf is high and in the lower part of the range. Moreover, when gang’s turf share is lower and the witness parameter low and in the higher part of the range then violence will increase when fighting is present. Lastly, if the witness intimidation parameter is too low relative to the opponent’s and the gang’s turf share is larger, then increases in police presence in either context (arming or fighting) will bring lower levels of violence. In addition, if the witness intimidation parameter is too high relative to the opponent’s and the gang’s turf share is smaller, then increases in police presence in either context (arming or fighting) will also bring lower levels of violence.

Finally, when police presence in one turf is negatively correlated with police presence in the other $\left(\frac{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{j}}{d{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}=-1\right)$ the results in the context of arming and fighting will be affected in the same manner by gangs’ switching parameters so results are the same as in the case of no correlation.

The next proposition sums up the main results in this section.

When gang $i$s turf share is larger (smaller) than the opponents and its witness intimidation parameter is within a range of the opponents parameter, an increase in police presence in the turf during a context of arming will bring lower levels of violence during a context of fighting if the gang $i$s witness intimidation is in the lower (higher) part of the range. In addition, when gang$i$s turf share is larger (smaller) than the opponents and its witness intimidation parameter is sufficiently high (low) and in the lower (higher) part of the parameters range then an increase in police presence will bring higher levels of violence during a context of fighting, as opposed to the context of arming.

## 6 Conclusion

Drug markets and the violence and intimidation they produce continue to be one of the main problems of local communities. Violence may emerge in illegal markets for many different reasons. Several of these, such as violence/intimidation as a means of securing territories and threatening individuals, have been dealt with in the literature. 28 One aspect that has received less attention is that illegal drug market competition and territorial competition may interact with each other generating unexpected results of law enforcement policies. Therefore, understanding the relationship between the level of competition in drug markets and the level of territorial disputes between gangs becomes an important issue on explaining these problems.

The model in the paper suggests that this interaction could be traced to factors affecting drug market and turf shares, and switching and witness intimidation parameters. In the model, gangs invest in guns to intimidate and scare off users who live in the gang’s territory and buy from a competitor in the hope that they will change drug supplier. Higher gang’s witness intimidation parameter has a non-monotonic effect on its gun investments. When gang $i$ has a sufficiently large (low) turf share its gun investments decrease (increase) and its opponent’s increase (decrease). In addition, a higher switching parameter faced by a gang increases gun investments by the gang but decreases its competitor’s. This is the first of the four main results of the paper.

In addition, due to the relationship between drug market and turf competition, different law enforcement policies may produce unintended consequences on the level of violence in a market. Since some law enforcement policies may affect the impact of gun investments on the level of violence in the drug market, the model can also serve an important purpose highlighting possible unintended consequences.

Traditional and community policing have been implemented throughout the last decades with ambiguous results. This paper sheds some light in explaining why this can happen. On the one hand, the effect of higher community policing, such as the trust between community and police in a territory, decrease total level of violence in the market where both gangs compete. On the other hand, the effect of higher traditional policing, such as police presence in a territory, on the level of violence depends on the relative size of both the witness intimidation and the switching parameters. On one hand, when police presence in one turf is not correlated with its presence in the other turf, increases in police presence in one turf will decrease the level of violence in the drug market if the witness intimidation parameter in that turf is either sufficiently low or sufficiently large relative to the other turf. However, violence will increase if the witness intimidation parameter is within a range relative to the other turf. On the other hand, when police presence in one turf is correlated with its presence in the other turf then the effect of an increase in police presence on the level of violence in the drug market will depend on the relative switching parameters between turfs. Opposite results will be obtained compared to the case of no correlation between police presence in the turf if the switching parameter in the turf where police presence increased is larger than in the other. This is the second result of the paper.

Furthermore, when turf shares are endogenized so actual fighting occurs, community policing will reduce total level of violence in the drug market as in the case of exogenous turf shares. Moreover, increase in traditional policing will have different effect on the level of violence in the drug market depending on the relative witness intimidation parameters and the drug market shares. When police presence is not correlated between turfs, increases in police presence in a turf will increase violence in the market if its witness intimidation parameter is within a range of the other turf’s parameter but will decrease violence if it is either sufficiently low or sufficiently large relative to the other. It is also important to say that the relative size of the witness intimidation parameters depend on the size of the drug market shares. For instance, when the drug market share for the gang whose turf experienced the increase in police presence is lower than the opponent then its witness intimidation parameter can be lower than the opponent’s to decrease the total level of violence in the market. However, when its drug market share is larger, the witness intimidation parameter needs to be larger than the opponent’s in order to get the same result. This is the third result of the paper.

Finally, comparing the results between the contexts of arming (i.e., where turf shares are exogenous) and fighting (i.e., where turf shares are endogenous) we find that an increase in community policing will reduce total level of violence in both contexts but increases in traditional policing will have ambiguous results depending on the relative size of turf shares. Under the context of fighting, increases of police presence that decrease the total level of violence in the market under the context of arming will bring higher levels of violence in the market if the gang’s turf share that experienced the increase in police presence is larger (smaller) than the opponent’s and its witness intimidation is sufficiently large (small) but in the lower (higher) part of the parameter’s range. This is the fourth result of the paper.

The paper has also some limitations. When studying covered drug markets, we assume users will not exit the market and decide not to use drugs. Hence, all users have a level of addiction to make them buy a unit of drug under any circumstances. Allowing users to exit the market brings a more complex environment that will need to be addressed in the future.

In the paper, we have also abstracted from dynamic aspects of the analysis. While interesting, a formal analysis of, say, a repeated interaction over time becomes quite complex and is left to future research. Furthermore, endogenizing drug consumption can be the next step to follow so the effect of violence on drug use and drug prices can be better modeled. Nonetheless, the paper provides a starting point for analyzing the strategic use of violence in a context where gangs compete in a drug market and have territorial disputes.

## A.1.1 Uniqueness and Stability of the Interior Equilibrium

(a) The interior equilibrium: We need to prove that both best reaction functions cross once and only once. On the one hand, the inverse of the best reaction function for gang $i$, ${z}_{j}^{i}\left({z}_{i}\right)>0$, where $a>0$ and $b>0$ imply that $\frac{\mathrm{\partial }{z}_{j}^{i}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}<0$, given that ${z}_{j}^{i}\left({z}_{i}\right)$ is continuous and differentiable. Therefore, ${z}_{j}^{i}\left({z}_{i}\right)\in \left[\frac{a}{b},0\right]$ $\mathrm{\forall }{z}_{i}>0.$ On the other hand, the best reaction function for gang $j$, ${z}_{j}\left({z}_{i}\right)>0$ where $c>0$ and $d>0$ imply that $\frac{\mathrm{\partial }{z}_{j}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}<0$, given that ${z}_{j}\left({z}_{i}\right)$ is continuous and differentiable. Therefore, ${z}_{j}\left({z}_{i}\right)\in \left[c,0\right]$ $\mathrm{\forall }{z}_{i}>0.$ Hence, if $\frac{a}{b}>c$ and $\left|\frac{\mathrm{\partial }{z}_{j}^{i}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}\right|>\left|\frac{\mathrm{\partial }{z}_{j}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}\right|$, both reaction functions cross once. Replacing these values, we have $\frac{a}{b}>c$ $\to$ $3\left(2-\mathrm{\alpha }\right)>{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\frac{\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right)}{{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}}$ and $\left|\frac{\mathrm{\partial }{z}_{j}^{i}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}\right|>\left|\frac{\mathrm{\partial }{z}_{j}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}\right|$ $\to$ $\frac{1}{b}>d$ $\to 9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}>\left(1-\mathrm{\beta }{\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right)+{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right)$, which is always the case given ${\mathrm{\gamma }}^{i},{\mathrm{\gamma }}^{j}\ge 1;0<\mathrm{\alpha },\mathrm{\beta }<1;$ and $0<{\mathrm{\theta }}_{i},{\mathrm{\theta }}_{j}\le 1$. Note also that when $\left|\frac{\mathrm{\partial }{z}_{j}^{i}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}\right|>\left|\frac{\mathrm{\partial }{z}_{j}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}\right|$, an interior equilibrium is also stable. Q.E.D.

(b) The corner equilibria: There are also two corner equilibria when either $\frac{a}{b} or $\frac{c}{d}, which are stable if $\left|\frac{\mathrm{\partial }{z}_{j}^{i}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}\right|>\left|\frac{\mathrm{\partial }{z}_{j}\left({z}_{i}\right)}{\mathrm{\partial }{z}_{i}}\right|.$

## A.1.2 Lemma 1

Let $A=\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right)>0$ and ${z}_{i}^{\ast }=\frac{3\mathrm{\beta }{\mathrm{\theta }}_{i}\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\mathrm{\beta }{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}^{2}A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}$[32] ${z}_{j}^{\ast }=\frac{3\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\left(2-\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\left(1-\mathrm{\beta }\right){\mathrm{\beta }}^{2}{\mathrm{\theta }}_{j}{\mathrm{\theta }}_{i}^{2}A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}$[33]From expressions [32] and [33] we have that:

• (i) $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}^{i}}=\frac{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}^{i2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)\left(3\mathrm{\beta }\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\mathrm{\beta }{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)+6{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}A\left(3\mathrm{\beta }\left(1+\mathrm{\alpha }\right){\mathrm{\theta }}_{i}{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\mathrm{\beta }{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}^{2}A\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\lessgtr 0$

and simplifying we get

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}^{i}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\frac{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}+{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)\left(3\mathrm{\beta }\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\mathrm{\beta }{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\gtrless 0$

Since $3\mathrm{\beta }\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\mathrm{\beta }{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A>0$ from ${z}_{i}^{\ast }>0$ and $9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}+{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A>0$ since $9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A>0$ then $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}>0.$

And $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}=\frac{-6{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}A\left[3\mathrm{\beta }\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\mathrm{\beta }{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right]}{9{\left[9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right]}^{2}}<0$ so $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}<0.$ Q.E.D.

• (ii) $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\frac{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)\left(3\mathrm{\beta }{\mathrm{\theta }}_{i}\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right)-\left(3\mathrm{\beta }{\mathrm{\theta }}_{i}\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-\mathrm{\beta }{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}^{2}A\right)3\left(9{\mathrm{\gamma }}^{j}\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\lessgtr 0$

And simplifying we get $\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\frac{9{\mathrm{\gamma }}^{j}\mathrm{\beta }{\mathrm{\theta }}_{i}A\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\lessgtr 0$

If ${\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)>0$ so $\stackrel{ˉ}{\mathrm{\beta }}=\frac{{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}{{\mathrm{\theta }}_{i}+{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}>\mathrm{\beta }$ then $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}>0$ and if ${\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)<0$ so $\stackrel{ˉ}{\mathrm{\beta }}=\frac{{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}{{\mathrm{\theta }}_{i}+{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}<\mathrm{\beta }$ then $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}<0.$

And $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\frac{9{\mathrm{\gamma }}^{j}\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}A\left({\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\lessgtr 0$

If ${\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)>0$ so $\phantom{\rule{thinmathspace}{0ex}}\stackrel{˜}{\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{\beta }}=\frac{{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}{{\mathrm{\theta }}_{i}+{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}<\mathrm{\beta }$ then $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}>0$ and if ${\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)<0$ so $\phantom{\rule{thinmathspace}{0ex}}\stackrel{˜}{\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{\beta }}=\frac{{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}{{\mathrm{\theta }}_{i}+{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}>\mathrm{\beta }$ then $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}<0.$ Q.E.D.

## A.2.1 Proposition 1

$\frac{\partial {Z}^{\ast }}{\partial trus{t}_{i}}=\left(\frac{d{\theta }_{i}}{dtrus{t}_{i}}\right)\left(\frac{\partial {z}_{i}^{\ast }}{\partial {\theta }_{i}}+\frac{\partial {z}_{j}^{\ast }}{\partial {\theta }_{i}}\right)\lessgtr 0$[34]

From expression [34] we have that $\begin{array}{rl}\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}+\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}\right)=& \phantom{\rule{thickmathspace}{0ex}}\frac{3\mathrm{\beta }\left(3\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}+{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\\ & +\frac{-6{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}A\left(3\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\end{array}$ $=\frac{3\mathrm{\beta }\left(3\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}+{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A-2\mathrm{\beta }\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}A\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}$where $A=\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right).$ Since $3\left(1+\mathrm{\alpha }\right){\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A>0$ then $\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}+\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}\right)\gtrless 0$ depending on the sign of the factor $9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\lessgtr A\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}+2\mathrm{\beta }\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\right)$. Since ${\mathrm{\gamma }}^{i},{\mathrm{\gamma }}^{j}\ge 1;0<\mathrm{\alpha },\mathrm{\beta }<1;$ and $0<{\mathrm{\theta }}_{i},{\mathrm{\theta }}_{j}\le 1$ then $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}+2\mathrm{\beta }\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\right)$ can have a maximum value of $2$, whereas $\frac{A}{{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}}$ can have a maximum value of Hence, $9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}>A\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}+2\mathrm{\beta }\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\right)$ and $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{s}\mathrm{t}}_{i}}<0$. Q.E.D.

## A.2.2 Proposition 2

$\frac{\partial {Z}^{\ast }}{\partial {\text{police}}_{i}}=\frac{d{\gamma }^{i}}{d{\text{police}}^{i}}\left(\frac{\partial {z}_{i}^{\ast }}{\partial {\gamma }^{i}}+\frac{\partial {z}_{j}^{\ast }}{\partial {\gamma }^{i}}\right)$

where $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\mathrm{\beta }{\mathrm{\theta }}_{i}{\mathrm{\gamma }}^{j}A\left(\frac{{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)}{{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)$ and $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{\mathrm{\gamma }}^{j}A\left(\frac{{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)}{{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right).$

Therefore, $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}=\phantom{\rule{negativethinmathspace}{0ex}}\left(\frac{d{\mathrm{\gamma }}^{i}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\right){\mathrm{\gamma }}^{j}A\left(\phantom{\rule{negativethinmathspace}{0ex}}\frac{\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)}{{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\phantom{\rule{negativethinmathspace}{0ex}}\right)\phantom{\rule{negativethinmathspace}{0ex}}\lessgtr 0$[35]From expression [35] we have that $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\lessgtr 0$ depending on $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\gtrless 0$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)\lessgtr 0.$ Hence, rearranging both factors around ${\mathrm{\theta }}_{i}$ we have $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\gtrless 0\to {\mathrm{\theta }}_{i}\gtrless \stackrel{ˆ}{{\mathrm{\theta }}_{j}}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)\gtrless 0\to {\mathrm{\theta }}_{i}\gtrless {\stackrel{˜}{\mathrm{\theta }}}_{j}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right).$

Case 1: When $\mathrm{\alpha }<\frac{1}{2}$ so $\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}>1$ and $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}>{\stackrel{˜}{\mathrm{\theta }}}_{j}.$

(a) $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)>0\to {\mathrm{\theta }}_{i}<\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)>0\to {\mathrm{\theta }}_{i}>{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Hence, when ${\stackrel{˜}{\mathrm{\theta }}}_{j}<{\mathrm{\theta }}_{i}<\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ we have that $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}>0.$

(b) $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)<0\to {\mathrm{\theta }}_{i}>\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)<0\to {\mathrm{\theta }}_{i}<{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Hence, there is an empty set.

(c) $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)>0\to {\mathrm{\theta }}_{i}<\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)<0\to {\mathrm{\theta }}_{i}<{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Hence, when ${\mathrm{\theta }}_{i}<{\stackrel{˜}{\mathrm{\theta }}}_{j}$ we have that $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}_{i}}<0.$

d) $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)<0\to {\mathrm{\theta }}_{i}>\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)>0\to {\mathrm{\theta }}_{i}>{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Hence, when ${\mathrm{\theta }}_{i}>\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ we have that $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}<0.$

Case 2: When $\mathrm{\alpha }>\frac{1}{2}$ so $\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}<1$ and $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}<{\stackrel{˜}{\mathrm{\theta }}}_{j}.$

(a) $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)>0\to {\mathrm{\theta }}_{i}<\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)>0\to {\mathrm{\theta }}_{i}>{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Hence, there is an empty set.

(b) $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)<0\to {\mathrm{\theta }}_{i}>\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)<0\to {\mathrm{\theta }}_{i}<{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Hence, when $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}<{\mathrm{\theta }}_{i}<{\stackrel{˜}{\mathrm{\theta }}}_{j}$ we have that $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}>0.$

(c) $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)>0\to {\mathrm{\theta }}_{i}<\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)<0\to {\mathrm{\theta }}_{i}<{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Hence, when ${\mathrm{\theta }}_{i}<\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ we have that $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}<0.$

(d) $\left({\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)<0\to {\mathrm{\theta }}_{i}>\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ and $\left(\mathrm{\beta }{\mathrm{\theta }}_{i}-\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}\right)>0\to {\mathrm{\theta }}_{i}>{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Hence, when ${\mathrm{\theta }}_{i}>{\stackrel{˜}{\mathrm{\theta }}}_{j}$ we have that $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}<0.$

Q.E.D.

## A.2.3 Proposition 3

$\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}=\frac{d{\mathrm{\gamma }}^{i}}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}\left(\left(\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}-\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{j}}\right)+\left(\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}-\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{j}}\right)\right)$

where $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\mathrm{\beta }{\mathrm{\theta }}_{i}{\mathrm{\gamma }}^{j}A\left(\frac{{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)}{{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)$, $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{j}}=\mathrm{\beta }{\mathrm{\theta }}_{i}{\mathrm{\gamma }}^{i}A\left(\frac{{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)}{{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)$, $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\left(1-\mathrm{\beta }\right){\mathrm{\theta }}_{j}{\mathrm{\gamma }}^{j}A\left(\frac{{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)}{{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)$ and $\frac{\partial {z}_{j}^{\ast }}{\partial {\gamma }^{j}}=\left(1-\beta \right){\theta }_{j}{\gamma }^{i}A\left(\frac{{\beta }^{2}{\theta }_{i}^{2}\left(1+\alpha \right)-{\left(1-\beta \right)}^{2}{\theta }_{j}^{2}\left(2-\alpha \right)}{{\left(9{\gamma }^{i}{\gamma }^{j}-{\beta }^{2}{\theta }_{i}^{2}A-{\left(1-\beta \right)}^{2}{\theta }_{j}^{2}A\right)}^{2}}\right)$ $\frac{\partial {Z}^{\ast }}{\partial {\text{police}}^{i}}=\left(\frac{d{\gamma }^{i}}{d{\text{police}}^{i}}\right)\left(\frac{9A\left({\left(1-\beta \right)}^{2}{\theta }_{j}^{2}\left(2-\alpha \right)-{\beta }^{2}{\theta }_{i}^{2}\left(1+\alpha \right)\right)\left(\beta {\theta }_{i}-\left(1-\beta \right){\theta }_{j}\right)\left({\gamma }^{j}-{\gamma }^{i}\right)}{9{\left(9{\gamma }^{i}{\gamma }^{j}-{\beta }^{2}{\theta }_{i}^{2}A-{\left(1-\beta \right)}^{2}{\theta }_{j}^{2}A\right)}^{2}}\right)$[36]From expressions [35] and [36] we have that $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}$ has the same sign as in Proposition 2 if $\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right)>0$ and the opposite sign if $\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right)<0.$ Hence, the same proof for Proposition 2 can be applied plus the sign of the factor $\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right)$ to get the results for Proposition 3.

Q.E.D.

## A.2.4 Lemma 2

Let $A=\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right)>0$ and ${z}_{i}^{\ast }=\frac{3{\mathrm{\theta }}_{i}{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left(1+\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}^{2}A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}$[37] ${z}_{j}^{\ast }=\frac{3{\mathrm{\theta }}_{j}{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{j}{\mathrm{\theta }}_{i}^{2}A}{3\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}$[38]From expressions [37] and [38] we have $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}=\frac{3\left(3{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}\left(1+\mathrm{\alpha }\right)-A{\mathrm{\theta }}_{i}^{2}\right)\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}+{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}>0$ $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\theta }}_{i}}=\frac{-6{\mathrm{\theta }}_{i}{\mathrm{\theta }}_{j}A\left(3{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}_{j}\left(1+\mathrm{\alpha }\right)-A{\mathrm{\theta }}_{i}^{2}\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}<0$And

(i) $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\frac{9{\mathrm{\gamma }}^{j}A{\mathrm{\theta }}_{i}\left({\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\lessgtr 0$

Hence, if ${\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)>0\to \mathrm{\alpha }<\frac{2{\mathrm{\theta }}_{j}^{2}-{\mathrm{\theta }}_{i}^{2}}{{\mathrm{\theta }}_{i}^{2}+{\mathrm{\theta }}_{j}^{2}}$ then $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}>0$ and if ${\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)<0\to \mathrm{\alpha }>\frac{2{\mathrm{\theta }}_{j}^{2}-{\mathrm{\theta }}_{i}^{2}}{{\mathrm{\theta }}_{i}^{2}+{\mathrm{\theta }}_{j}^{2}}$ then $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}<0.$

(ii) $\frac{\mathrm{\partial }{z}_{i}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}=\frac{9{\mathrm{\gamma }}^{j}A{\mathrm{\theta }}_{j}\left({\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A-{\left(1-\mathrm{\beta }\right)}^{2}{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\gtrless 0$

Hence, if ${\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)>0\to \mathrm{\alpha }>\frac{2{\mathrm{\theta }}_{j}^{2}-{\mathrm{\theta }}_{i}^{2}}{{\mathrm{\theta }}_{i}^{2}+{\mathrm{\theta }}_{j}^{2}}$ then $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}>0$ and if ${\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)-{\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)<0\to \mathrm{\alpha }<\frac{2{\mathrm{\theta }}_{j}^{2}-{\mathrm{\theta }}_{i}^{2}}{{\mathrm{\theta }}_{i}^{2}+{\mathrm{\theta }}_{j}^{2}}$ then $\frac{\mathrm{\partial }{z}_{j}^{\ast }}{\mathrm{\partial }{\mathrm{\gamma }}^{i}}<0.$

Q.E.D.

## A.3.1 Proposition 4

$\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}_{}}=\frac{d{\mathrm{\gamma }}^{i}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}_{}}\left(\frac{9{\mathrm{\gamma }}^{j}A\left({\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\left({\mathrm{\theta }}_{i}-{\mathrm{\theta }}_{j}\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)\gtrless 0$[39]

From expression [39] we have that depending on the sign of the factors $\left({\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\gtrless 0\to {\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}\gtrless {\mathrm{\theta }}_{i}$ and ${\mathrm{\theta }}_{i}-{\mathrm{\theta }}_{j}\gtrless 0\to {\mathrm{\theta }}_{i}\gtrless {\mathrm{\theta }}_{j}$ we have $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\gtrless 0.$ In addition, we know that $\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}\gtrless 1$ depends on $\frac{1}{2}\gtrless \mathrm{\alpha }.$

Case 1: Where $\mathrm{\alpha }<\frac{1}{2}$ so $\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}>1:$

(a) If ${\mathrm{\theta }}_{j}<{\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ then $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}>0$

(b) If ${\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}<{\mathrm{\theta }}_{i}$ and ${\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}$ then there is an empty set.

(c) If ${\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}$ then $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}<0$

(d) If ${\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}<{\mathrm{\theta }}_{i}$ then $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}<0$

Case 2: Where $\mathrm{\alpha }>\frac{1}{2}$ so $\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}<1$:

(a) If ${\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}<{\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}$ then $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}>0$

(b) If ${\mathrm{\theta }}_{j}<{\mathrm{\theta }}_{i}$ and ${\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ then there is an empty set.

(c) If ${\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ then $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}<0$

(d) If ${\mathrm{\theta }}_{j}<{\mathrm{\theta }}_{i}$ then $\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}}^{i}}<0$

Q.E.D.

## A.3.2 Proposition 5

$\frac{\mathrm{\partial }{Z}^{\ast }}{\mathrm{\partial }\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}=\frac{d{\mathrm{\gamma }}^{i}}{d\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{e}{i}^{}}\left(\frac{9A\left({\mathrm{\theta }}_{j}^{2}\left(2-\mathrm{\alpha }\right)-{\mathrm{\theta }}_{i}^{2}\left(1+\mathrm{\alpha }\right)\right)\left({\mathrm{\theta }}_{i}-{\mathrm{\theta }}_{j}\right)\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right)}{9{\left(9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}-{\mathrm{\theta }}_{i}^{2}A-{\mathrm{\theta }}_{j}^{2}A\right)}^{2}}\right)\gtrless 0$[40]

From expression [40] and the proof from Proposition 4 we see that the results will depend on the sign of the factor $\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right).$ Therefore, same results as in Proposition 4 will be obtained when $\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right)>0$ and opposite results when $\left({\mathrm{\gamma }}^{j}-{\mathrm{\gamma }}^{i}\right)<0.$

Q.E.D.

## A.3.3.1 When Traditional Policing Increases Violence

$\mathrm{\alpha }<\frac{1}{2}\phantom{\rule{thickmathspace}{0ex}}\mathrm{s}\mathrm{o}\phantom{\rule{thickmathspace}{0ex}}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}>1$

Fighting context: ${\mathrm{\theta }}_{j}<{\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$

Arming context: ${\stackrel{˜}{\mathrm{\theta }}}_{j}<{\mathrm{\theta }}_{i}<\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$

Since $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}=\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right){\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ and ${\stackrel{˜}{\mathrm{\theta }}}_{j}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)$ when the gang $i$’s turf share is larger than the opponent’s then $\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)<1$ and the witness intimidation parameter space moves to the left, then lower levels of that parameter are required under the arming context to get the same result on violence than under the fighting context. However, when the gang $i$’s turf share is smaller than the opponent’s then $\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)>1$ and the witness intimidation parameter space moves to the right, then higher levels of that parameter are required under the arming context to get the same result on violence than under the fighting context: $\mathrm{\alpha }>\frac{1}{2}\phantom{\rule{thickmathspace}{0ex}}\mathrm{s}\mathrm{o}\phantom{\rule{thickmathspace}{0ex}}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}<1$Fighting context: ${\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}<{\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}$

Arming context: $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}<{\mathrm{\theta }}_{i}<{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Since $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}=\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right){\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ and ${\stackrel{˜}{\mathrm{\theta }}}_{j}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)$ when the gang $i$’s turf share is larger than the opponent’s then $\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)<1$ and the witness intimidation parameter space moves to the left, then lower levels of that parameter are required under the arming context to get the same result on violence than under the fighting context. However, when the gang $i$’s turf share is smaller than the opponent’s then $\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)>1$ and the witness intimidation parameter space moves to the right then higher levels of that parameter are required under the arming context to get the same result on violence than under the fighting context.

## A.3.3.2 When Traditional Policing Decreases Violence

$\mathrm{\alpha }<\frac{1}{2}\mathrm{s}\mathrm{o}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}>1$Fighting context: ${\mathrm{\theta }}_{i}>{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ or ${\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}$

Arming context: ${\mathrm{\theta }}_{i}>\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ or ${\mathrm{\theta }}_{i}<{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Since $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}=\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right){\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ and ${\stackrel{˜}{\mathrm{\theta }}}_{j}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)$ when the gang $i$’s turf share is larger than the opponent’s then $\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)<1$ and the witness intimidation parameter becomes smaller for low levels and larger for high levels than under the fighting context. However, when the gang $i$’s turf share is smaller than the opponent’s then $\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)>1$ and the witness intimidation parameter becomes smaller for high levels and larger for low levels than under the fighting context:

$\mathrm{\alpha }>\frac{1}{2}\text{\hspace{0.17em}}\mathrm{s}\mathrm{o}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}<1$ Fighting context: ${\mathrm{\theta }}_{i}<{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ or ${\mathrm{\theta }}_{i}>{\mathrm{\theta }}_{j}$

Arming context: ${\mathrm{\theta }}_{i}<\stackrel{ˆ}{{\mathrm{\theta }}_{j}}$ or ${\mathrm{\theta }}_{i}>{\stackrel{˜}{\mathrm{\theta }}}_{j}$

Since $\stackrel{ˆ}{{\mathrm{\theta }}_{j}}=\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right){\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}$ and ${\stackrel{˜}{\mathrm{\theta }}}_{j}={\mathrm{\theta }}_{j}\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)$ when the gang $i$’s turf share is larger than the opponent’s then $\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)<1$ and the witness intimidation parameter becomes smaller for low levels and larger for high levels than under the fighting context. However, when the gang $i$’s turf share is smaller than the opponent’s then $\left(\frac{1-\mathrm{\beta }}{\mathrm{\beta }}\right)>1$and the witness intimidation parameter becomes smaller for high levels and larger for low levels than under the fighting context.

Q.E.D.

## References

• Arrestee Drug Abuse Monitoring Program I and II – ADAM. 2000–2010. US National Institute of Justice, Office of Justice Programs, Washington. Google Scholar

• Baird, A.. 2012. “Negotiating Pathways to Manhood: Rejecting Gangs and Violence in Medelln’s Periphery.” Journal of Conflictology 3 (1):30–41.

• Burrus, R. T.. 1999. “Do Efforts to Reduce the Supply of Illicit Drugs Increase Turf War Violence? A Theoretical Analysis.” Journal of Economics and Finance 23:465–93.

• Carvhalo, L., and R. Soares. 2011. “A Walk on the Wild Side: Crime Entry and Exit among Brazilian Youth.” Working Paper. Google Scholar

• Caulkins, J., and P. Reuter. 1998. “What Price Tell Us about Drug Markets.” Journal of Drug Issues 28 (3):593–613. Google Scholar

• Caulkins, J., O. Reuter, and L. Taylor. 2006. “Can Supply Restrictions Lower Price? Violence, Drug Dealing and Positional Advantage.” Contributions to Economic Analysis & Policy 5 (1):1–20. Google Scholar

• Chen, Y.. 1997. “Paying Customers to Switch.” Journal of Economics and Management Strategy 6:877–97.

• Donohue, J., and S. Levitt. 1998. “Guns, Violence and the Efficiency of Illegal Markets.” American Economic Review 88 (2):463–7. Google Scholar

• Fox, K., J. Lane, and R. Akers. 2010. “Do Perceptions of Neighborhood Disorganization Predict Crime or Victimization? An Examination of Gang Member versus Non-gang Member Jail Inmates.” Journal of Criminal Justice 38:720–9.

• Government Accountability Office.2003. “Technical Assessment of Zhao and Thurman’s 2001 Evaluation of the Effects of COPS Grants on Crime.” June 2003, Washington, DC. Google Scholar

• Government Accountability Office. 2005. “Interim Report on the Effects of COPS Funds on the Decline in Crime during the 1990s.” June 2005, Washington, DC. Google Scholar

• Goldstein, P. J.. 1985. “The Drugs/Violence Nexus: A Tripartite Conceptual Framework.” Journal of Drug Issues 15:493–506. Google Scholar

• Hirschleifer, J. 1991. “The Technology of Conflict as an Economic Activity.” American Economic Review 81:130–4. Google Scholar

• Howell and Decker. 1999. “The Youth Gangs, Drugs and Violence Connection.” Juvenile Justice Bulletin, US Department of Justice, January 1999. Google Scholar

• International Narcotics Control Board – INCB. 2011. Report by the United Nations. ISBN 978-92-1-148269-0, New York, January 2012. Google Scholar

• Johnston, L. D., P. M. O‘Malley, J. G. Bachman, and J. E. Schulenberg. 2011. Monitoring the Future National Survey Results on Drug Use, 1975-2010. Volume I: Secondary School Students. Ann Arbor: Institute for Social Research, The University of Michigan, 744 pp. Google Scholar

• Kelly, D. 2006. “Witness Intimidation.” US Department of Justice, Office of Community Oriented Policing Services, No. 42, 2006. Google Scholar

• Klemperer. 1987. “Markets with Consumer Switching Costs.” Quarterly Journal of Economics 102 (2):375–94. Google Scholar

• Levitt, S., and S. Venkatesh. 2003. “An Economic Analysis of a Drug-Selling Gang’s Finance.” Quarterly Journal of Economics 115 (3):755–89.

• MacDonald, J. M. 2002. “The Effectiveness of Community Policing in Reducing Urban Violence.” Crime & Deliquency 48 (4):592–618.

• Mastrofski, S. 2006. “Community policing: a skeptical view.” In Police innovation: contrasting perspectives, edited by D. Weisburd and A. Braga. New York: Cambridge University Press. Google Scholar

• Matrix Knowledge Group. 2007. “The Illicit Drug Trade in the United Kingdom.” Report 20, U.K. Home Office. Google Scholar

• Miron, J. 2001. “Violence, Guns and Drugs: A Cross Country Analysis.” Journal of Law and Economics XLIV:615–33. Google Scholar

• Mazerolle, L., D. Soole, and S. Rombouts. 2007. “Crime Prevention Research Reviews No. 1: Disrupting Street-Level Drug Markets.” Washington, DC: US Department of Justice Office of Community Orientes Policing Services. Google Scholar

• Naranjo, A., and A. Jacobsson. 2009. “Counter-intuitive Effects of Domestic Law Enforcement Policies in the United States.” Economics of Governance 10 (4):323–43.

• Naranjo, A. 2011. “Spillover Effects of Domestic Law Enforcement.” International Review of Law and Economics 30 (3):265–78.

• National Gang Center. 2007. National Gang Center Bulletin, No. 1, February 2007. Google Scholar

• National Gang Threat Assessment. 2009. National Gang Intelligence Center. Google Scholar

• National Gang Threat Assessment. 2011. National Gang Intelligence Center. Google Scholar

• National Drug Threat Assessment. 2010. US Department of Justice, National Drug Intelligence Center. Google Scholar

• Nilssen, T.. 1992. “Two Kinds of Consumer Switching Costs.” RAND Journal of Economics 23:579–89.

• Poret, S.. 2003. “The Illicit Drug Market: Paradoxical Effects of Law Enforcement Policies.” International Review of Law and Economics 22 (4):465–93.

• Rasmussen, D., and B. Benson. 2001. “Rationalizing Drug Policy under Federalism.” Florida State University Review 30:679–733. Google Scholar

• Resignato, A. 2000. “Violent Crime: A Function of Drug User or Drug Enforcement?.” Applied Economics 32:681–88.

• Reuter, P., R. J., MacCoun, and P., Murphy. 1990. Money from crime: a study of the economics of drug dealing in Washington D.C.: Santa Monica, CA: RAND. 1–172.Google Scholar

• Reuter, P., F. Trautmann, R. Pacula, B. Kilmer, D. Gadeldonk, and D. Van der Gouwe. 2007. “Assessing Changes in Global Drug Problems 1998-2007.” Report, RAND Corporation.Google Scholar

• Reuter, P., and E. M. Truman. 2004. “Chasing Dirty Money: The Fight against Money Laundering.” Peterson Institute, Massachussetts, Washington DC: Institute for International Economics. Google Scholar

• Shaffer, G, and Z. Zhang. 2000. “Pay to switch or pay to stay: preference-based price discrimination in markets with switching costs.” Journal of Economics & Management Strategy, 9(3):397–424. Google Scholar

• Sherman, D., and J. E. Eck. 2006. “Policing for Crime Prevention.” In Evidence-Based Crime Prevention, edited by L. W. Sherman, D. P. Farrington, B. C. Welsch and D. L. Mac Kenzie, 331–403. New York: Routledge. Google Scholar

• Skaperdas, S.. 1996. “Contest Success Functions.” Economic Theory 7:283–90.

• Skogan, W. G.. 2006. Police and the Community in Chicago. New York: Oxford University Press. Google Scholar

• Skott, P., and G. T. Jepsen. 2003. “Paradoxical Effects of Drug Policy in a Model with Imperfect Competition and Switching Costs.” Journal of Economic Behavior & Organizations 48 (4):335–54.

• Svidiroff, M., and S. Hillsman. 1994. “Assessing the Community Effects of Tactical Narcotics Teams.” In Drugs and Crime: Evaluating Public Policy Initiatives, edited by D. MacKenzie and C. Uchida. New York, London: Vera Institute of Justice. Google Scholar

• Taylor, C. 2003. “Supplier Surfing: Price Discrimination in Markets with Repeat Purchases.” RAND Journal of Economics 34:223–46.

• Tullock, G.1980. “Efficient Rent Seeking.” In Toward a Theory of the Rent Seeking Society, edited by J.Buchanan, R.Tollison and G.Tullock. Texas: A & M University Press, College Station, TX. Google Scholar

• Weisburd, D., and L. Green. 1995. “Policing Drug Hot Spots: The Jersey City Drug Market Analysis Experiment.” Justice Quarterly 12 (4):711–35.

• Weisburd, D., and J. E. Eck. 2004. “What Can the Police Do to Reduce Crime, Disorder, and Fear?.” Annals of the American Academy of Political and Social Sciences 593:593–625.

• Wilson, L., and A. Stevens. 2008. “Understanding Drug Markets and How to Influence Them.” The Beckley Foundation Drug Policy Programme Report 14:1–13. Google Scholar

• World Drug Report. 2009. United Nations Office on Drugs and Crime. ISBN: 978-92-1-148240-9, New York. Google Scholar

• Zaluar, A. 2004. “Violence in Rio de Janeiro: Styles of Leisure, Drug Use and Trafficking.” UNESCO, 369–78. Google Scholar

• Zhao, J., Sheider, M. and Q., Thurman. 2002. “Finding community policing to reduce crime: have cops grants made a difference?.” Criminology & Public policy 2 (1):7–32.

• Zhao, J., and Q. Thurman. 2004. Funding Community Policing to Reduce Crime: Have COPS Grants Made a Difference?. Washington, DC: Office of Community Oriented Policing Service. Google Scholar

## Footnotes

• 1

Territorial competition between gangs seems to be confined to the neighborhood’s boundaries so the competition is most of the time between two gangs. For instance, Levitt and Venkatesh (2003) show that even though there were two rival gangs who had boundaries with the borders of the gang studied, there was only a war against one of them.

• 2

Drug market competition is usually between few gangs (if not two) since street-level markets are very specific geographically. Caulkins and Reuter (1998) and Wilson and Stevens (2008) find evidence on the important spatial variation of drug markets (particularly through drug prices) even between neighborhood’s within a city.

• 3

The question in the survey was: “Did you buy it [name of drug] in the neighborhood where you live or outside your neighborhood?”, so the concept of“neighborhood” reflects the arrestee’s perception. See Levitt and Venkatesh (2003) for further references.

• 4

Traditional policing means policies that focus on the level of police presence in the streets or the use of crackdowns or raids, whereas community policing involves the use of third parties to increase trust levels within the community that will help with the criminality levels.

• 5

According to Johnston et al. (2011), the proportion of 12th grade students who used any illicit drug in the last 12 months between 1990 and 2011 is fairly stable around 40% (and 20% for any drug other than marijuana). In addition, drug-dependent user buys larger quantities than recreational user so this assumption does not seem unrealistic and also simplifies the calculations.

• 6

This assumption is also used in Poret (2003). She supports it by referring to Reuter, MacCoun and Murphy (1990) who found that a drug user buys a fixed amount of drug in each transaction.

• 7

For further references on witness intimidation, see the National Gang Center (2007) and Kelly (2006).

• 8

See Skott and Jepsen (2003), Caulkins and Reuter (1998, 2006) for further references on the existence of switching costs on illegal drug markets.

• 9

All the results in the paper can be generalized assuming the upper bound equal to $\mathrm{\sigma }$. However, for simplicity we set $\mathrm{\sigma }=1.$ Under this upper boundary the market is covered (i.e., $\mathrm{\sigma }$ is large enough to make all users in the market to buy one unit of drug). See Chaffer and Zhang (2000) for a discussion on asymmetric distribution or consumer heterogeneity.

• 10

Places with better street lighting, for example, will increase user’s disutility for buying from the gang in that territory due to a higher risk to be caught. According to Fox, Lane, and Akers (2010)), social disorganization theory has been mainly used to explain how real and perceived community characteristics can increase fear of crime. In this theory, territorial factors affect the social perception of crime. Therefore, differences in social disorganization factors such as physical disorder, social disorder and collective efficacy may differ between territories and create different costs to consumers.

• 11

A user living in turf $j$ and buying from gang $i$ will first face the violence (or threat) by gang $i$ if it tries to resell the drug in its turf since it can be seen as belonging to the gang $j$’s organization. And second, it will face the violence (or threat) by gang $j$ since it clearly does not belong to its organization and it is easily detected since it lives in its turf.

• 12

The demands for gang $j$ will be calculated symmetrically taking into account that turf share by gang $j$ is $\left(1-\mathrm{\beta }\right)$ and drug market share by this gang is $\left(1-\mathrm{\alpha }\right)$. Since the problem is symmetric we will show only the results for gang $i$.

• 13

We assume this functional form because the cost of dealers and guns seems to grow faster the larger is this investment due to their direct cost (especially in guns) and also an indirect cost due to the effect this investment can have in increasing the number of casualties and caught dealers by law enforcement. Levitt and Venkatesh (2003) found that the wages paid to gang members increased by 70% drug gang wars, before a gang took over its opponent, as compared to a non-war periods. Since higher levels of gun investments increase the likelihood of a war, this assumption seems to be reasonable.

• 14

The second-order condition holds for concavity since ${f}_{11}{f}_{22}>0$ and ${f}_{12}={f}_{21}=0.$

• 15

By symmetry ${z}_{j}$ is also solved. The second-order conditions for gangs $i$ and $j$ are negative if $9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}>{\mathrm{\beta }}^{2}{\mathrm{\theta }}_{i}^{2}A$ and $9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}>\left(1-\mathrm{\beta }{\right)}^{2}{\mathrm{\theta }}_{j}^{2}A$, which are always the case given ${\mathrm{\gamma }}^{i},{\mathrm{\gamma }}^{j}\ge 1;0<\mathrm{\alpha },\mathrm{\beta }<1;$ and $0<{\mathrm{\theta }}_{i},{\mathrm{\theta }}_{j}\le 1.$

• 16

See Appendix A for the respective proof. Moreover, to ensure duopoly between the two gangs, besides the conditions for uniqueness and stability, the critical switching costs must be positive and lower than 1 in an interior equilibrium. Under our assumptions $0 and $0

• 17

The opposite interpretation can be given when the turf share is sufficiently small. If this is the case, it is important for the gang to secure drug users in its turf by increasing gun investments. The opponent will then save some costs by decreasing gun investments. In the aggregate, the increase in gun investments will be higher than the decrease by the opponent’s making violence in the drug market to be higher.

• 18

Since $\phantom{\rule{thinmathspace}{0ex}}\stackrel{˜}{\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{\beta }}=\frac{{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}{{\mathrm{\theta }}_{i}+{\mathrm{\theta }}_{j}\sqrt{\frac{\left(2-\mathrm{\alpha }\right)}{\left(1+\mathrm{\alpha }\right)}}}.$

• 19

Where ${z}_{i}^{\ast },{z}_{j}^{\ast }$ are equilibrium values, ${P}_{i}^{\ast }=\frac{{p}_{ii}^{\ast }+{p}_{ji}^{\ast }}{2}>0$ and ${P}_{j}^{\ast }=\frac{{p}_{jj}^{\ast }+{p}_{ij}^{\ast }}{2}>0.$ In addition, ${p}_{ii}^{\ast },{p}_{ji}^{\ast },{p}_{jj}^{\ast },{p}_{ij}^{\ast }$ are all positive for any value of $\mathrm{\alpha }$ and $\mathrm{\beta }$.

• 20

If the upper limit of the uniform distribution of the switching cost was $\mathrm{\sigma }$ this average drug price would depend (positively) on this parameter.

• 21

In our model there is no extra cost for fighting. Under actual fighting drug users may face for instance a different (higher) switching parameter than under the context of arming. This can be added to our model without changing the main results.

• 22

We refer to Tullock (1980) for more on this functional form. See also Hirschleifer (1991) and Skaperdas (1996) for a discussion on contests success functions.

• 23

The second-order conditions hold if ${\mathrm{\theta }}_{i}^{2}\left(\mathrm{\alpha }{\mathrm{\gamma }}^{i}+\left(1-\mathrm{\alpha }\right){\mathrm{\gamma }}^{j}\right)<9{\mathrm{\gamma }}^{i}{\mathrm{\gamma }}^{j}$ which will be assumed.

• 24

There are also two corner solutions and one equilibrium where the intimidation level by both gangs is equal to zero. Again, this feature is not realistic and we focus our attention on the interior equilibrium.

• 25

Where ${P}_{i}^{\ast }=\frac{{p}_{ii}^{\ast }+{p}_{ji}^{\ast }}{2}>0$ and ${P}_{j}^{\ast }=\frac{{p}_{jj}^{\ast }+{p}_{ij}^{\ast }}{2}>0.$ In addition, ${p}_{ii}^{\ast },{p}_{ji}^{\ast },{p}_{jj}^{\ast },{p}_{ij}^{\ast }$ are all positive for any value of $\mathrm{\alpha }$.

• 26

Drug prices do not change under this specification. Again, this result stands from the fact that the uniform distribution of switching costs lies between 0 and 1. If the upper limit of this distribution was $\mathrm{\nu }$ the average drug price would depend (positively) on this parameter.

• 27

See the proof of Proposition 4 in Appendix A.

• 28

See Goldstein (1985), Caulkins, Reuter, and Taylor (2006) and Burrus (1999) for further references.

Published Online: 2015-08-04

Published in Print: 2015-10-01

Citation Information: The B.E. Journal of Economic Analysis & Policy, Volume 15, Issue 4, Pages 1507–1548, ISSN (Online) 1935-1682, ISSN (Print) 2194-6108,

Export Citation