Cartel warfare, drug supply, and policy∗ Juan Camilo Castillo† September 2015

Abstract This paper explores under which conditions drug trafficking cartels wage war or stay in peace, and finds the effect of policy on drug supply and violence. I model cartels as profit maximizers that would attack each other in a one-period game, but reduce the resources they commit to warfare in a repeated game. If cartels are sufficiently forward looking, a peaceful equilibrium arises. Policies that lead to impatience or fragmentation increase violence without an effect on supply. Policies that reduce the productivity of cartels curb supply, although they increase violence if demand is too inelastic.

Keywords: War on Drugs, Illegal Drug Markets, Violence, Supply Reduction JEL Classification Numbers: D74, K42 ∗ Earlier

versions of this paper were circulated under the title “Should drug policy be

aimed at cartel leaders? Breaking down a peaceful equilibrium". I am especially grateful to Daniel Mejía for his guidance. I would also like to thank David Bardey, Leopoldo Fergusson, Ana María Ibáñez, Hernán Vallejo, and Andrés Zambrano for many helpful comments. † Economics Department, Stanford University. E-mail: [email protected]

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The illegal drug trade has shown an exceptional capability to transform itself according to the conditions it faces. Whenever authorities eliminate one major trafficking route, a new one arises, and whenever one form of trafficking organization is suppressed, new types of cartels are born. Despite good intentions when designing policy, negative unintended consequences arise frequently. For instance, the frontal war against cartels started by the Mexican government in late 2006 was followed by a doubling in the homicide rate between 2006 and 2010 (Guerrero, 2011; Castillo et al., 2015). And after 2005, when the Colombian government led the demobilization of paramilitary groups, multiple small bands emerged to control the drug trade, causing a new period of violence in regions that were previously peaceful (Camacho, 2009, 2011). The Colombian and Mexican cases are examples of a common trend that takes place when governments turn to strategies that fragment the drug trade, based on the belief that the illegal drug trade cannot be eliminated if large, powerful cartels subsist. However, the outcome of such strategies has been that the drug trade continues, led by former lieutenants, with an important increase in violence which is driven by fights between cartels (and not between the government and cartels). Various works have described this behavior (Guerrero, 2011; Camacho, 2009, 2011; Dell, 2015) 1 , but so far the only attempt to understand the economic incentives behind it is Castillo et al. (2015), which shows how supply shortages increase the stakes cartels fight over, thus increasing violence. The purpose of this paper is to provide a theoretical framework to understand the incentives behind cartels’ decisions about trafficking and engaging in violence, and how policy affects such decisions. I go in line with 1 Guerrero

(2011) and Camacho (2009, 2011) argue that violence increases because be-

heading cartels leads to fights between former lieutenants in order to fill the voids of power. Dell (2015) shows empirical evidence that government crackdowns in Mexico did increase violence.

2

a relatively recent trend that draws from industrial organization and explains drug cartels as profit maximizers in an industry (Poret, 2003; Poret and Téjédo, 2006; Burrus, 1999; Bardey et al., 2013; Baccara and Bar-Isaac, 2008).2 I model cartels’ behavior in two parts. First, they engage in a conflict with other cartels over trafficking routes, in which they decide the amount of resources they spend in the conflict. Second, they use those routes to transport drugs to consumer markets while trying to avoid government forces. The works I mentioned have focused on the static interaction between cartels. I show that a dynamic setting is essential. Cartels would ideally be in a peaceful state in which they split routes according to some agreement and drug trafficking takes place with no fighting at all. However, every individual cartel would benefit by attacking others in order to control more routes and increase profits. In a dynamic setting, punishment strategies can lead to an agreement in which cartels engage in a milder conflict than is possible in a static setting. Even a peaceful equilibrium with no fighting at all can be sustained if cartels care enough about the future and the number of cartels is low enough. Some papers on the economics of conflicts describe how peace can arise if one side in the conflict is more powerful than others, or if the conflict destroys part of the prize being fought over (Grossman and Kim, 1995). My model is the first one to show how a peaceful equilibrium can arise due to incentives that only exist with dynamic interaction. The approach I take to evaluating policy contrasts with previous works in two ways. First, I compare two different forms of drug policy. Governments can aim at trafficking operations of cartels (which I call enforce2 Poret

(2003), Poret and Téjédo (2006), and Burrus (1999) model the horizontal and

vertical structure of drug markets. Bardey et al. (2013) propose a model of entry and exit of cartels, and Baccara and Bar-Isaac (2008) study the efficiency of organizational structures of criminal groups.

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ment), reducing their efficiency as drug traffickers. But they can also attack cartel leaders, as in Mexico after 2006. Most previous works had only analyzed the efficiency of policy against the productive structure of cartels (Chumacero, 2008; Bogliacino and Naranjo, 2012; Grossman and Mejía, 2008), without assessing the convenience of direct attacks against cartel bosses. The second way in which my approach improves on previous works is that I assess policies based on two different goals: reducing supply and violence. The bulk of the theory on illegal drug markets focuses on consumer nations, where the main goal of policy is to curb supply (Reuter and Kleiman, 1986; Lee, 1993; Poret, 2003). Some recent works have taken into account the whole production chain, also including producer and trafficking nations, (Mejía and Restrepo, 2011a,b) but they still evaluate policy in terms of supply reduction. The literature has largely ignored the huge impact of violence due to the drug trade, especially in consumer and trafficking nations.3 I thus take a more global approach and also look at the effect of policies on violence. I show that supply does not depend on the outcome of the conflict over routes. Regardless of who holds each route, there is an optimal amount to be taken through it to consumer markets. Thus, governments can only curb supply through enforcement: attacking leaders does not affect production unless it disrupts trafficking operations. Enforcement, however, usually comes at the cost of more violence. Drug prices increase as a consequence of reduced supply, so if demand is sufficiently inelastic4 the total value of the drug trade increases and cartels spend more resources in a conflict over a more valuable prize. Thus, enforcement implies a tradeoff between reducing supply and violence. Attacks on drug leaders also increase vi3 All

seven countries with the highest homicide rates lie in the main drug trafficking

routes from South America to North America according to the UNODC (2013) 4 Nisbet and Vakil (1972), Roumasset and Hadreas (1977), DiNardo (1993), and Saffer and Chaloupka (1999) show empirical evidence that it is.

4

olence. The risk of being captured or killed makes cartel leaders shortsighted, and leaderless cartels break up into smaller factions. Peaceful equilibria are harder to sustain with impatience and fragmentation (since deviating yields a higher benefit), so the equilibrium leans towards the violent equilibrium of a one-period setting. Hence, there is no justification for attacks on leaders from an economic perspective5 : they increase violence, while achieving no result on supply. Finally, I argue that my model fits the Colombian and Mexican cases. As the Colombian government succeeded in signing a demobilization agreement with the paramilitaries, they drastically increased the number of operating cartels, breaking down the peaceful equilibrium and increasing violence. I also argue that the Mexican government’s strategy of beheading cartels led to cartel bosses being more short-sighted, and to an increase in the number of cartels, inducing wars between them. This paper is organized as follows. Section 1 describes the model of the trafficking industry. In section 2 I show that the supply of drugs is independent of how the conflict over routes is resolved, and I show how supply depends on enforcement. Section 3 solves for the equilibria of the conflict over routes, and determines how policies affect the level of violence. I show how the results from the model match the Colombian and Mexican cases in section 4. Finally, section 5 concludes. 5 There

may be, however, some alternative motivations that are beyond the scope of

this work, such as political considerations. For instance, one of the main reasons the Mexican government started attacking cartel leaders was because powerful cartels had significantly weakened institutions. Therefore, Felipe Calderón ran for president in 2006 with a campaign based on attacking drug traffickers, and started the war agains drug cartels once his term started.

5

1

The trafficking industry

A fixed number n of cartels participate in drug trafficking. The number is constant because incumbents’ military power constitute barriers to entry: any group attempting to enter would be quickly wiped out by incumbents. I denote the set of indices for cartels by I. The behavior of cartels can be divided in two parts. In its productive behavior, it buys drugs in producer markets and takes them through routes leading to a consumer market. In order to do so, it must evade the government, which spends an exogenously set amount of resources in enforcement to seize the drugs.6 Cartels’ operations also involve their military behavior, in which they engage others in a fight over routes which they later use to transport drugs.

1.1

Productive behavior

Cartel i buys xi drugs in producer markets at a price p p , transports them through Ri routes they control, and sells them in consumer markets at a price pc . The government is able to seize a fraction of xi which depends on e, the amount it spends in enforcement. I define enforcement as any type of activity aimed against the productive behavior of cartels. Some examples are seizing drugs in transit, patrolling routes, or seizing submersibles or airplanes for drug transportation. On the other hand, I do not consider capturing or killing leaders or gunmen to be enforcement: such activities disrupt cartels’ military behavior, but not their productive behavior. The amount the cartel is able to sell in consumer markets, qi , is given by function q( xi , Ri , e). An alternative description is given by the survival rate wi = w( xi , Ri , e), the fraction of drugs that reach their destination, so 6I

will characterize the equilibrium reached given some value of e, and I will analyze

the comparative statics of such equilibrium with respect to changes in enforcement.

6

that q( xi , Ri , e) = w( xi , Ri , e) xi . I make the simplifying assumption that all cartels are equally efficient, so this function holds for every cartel. Assumption 1. The production function q( xi , Ri , e) has the following properties:

1. It is twice-differentiable, increasing in both factors of production, and de∂q

creasing in enforcement ( ∂x > 0, i

∂q ∂Ri

> 0, and

∂q ∂e

< 0). ∂2 q

2. The marginal productivity of both factors of production is decreasing ( ∂x2 < 0 and

∂2 q ∂R2i

i

< 0).

3. The marginal productivity of both factors of production decrease with en∂2 q

forcement ( ∂e∂x < 0 and i

∂2 q ∂e∂Ri

< 0).

The survival rate w( xi , Ri , e) has the following property: 4. It is homogeneous of degree zero in ( xi , Ri ), Hence, it can be rewritten as w( xi , Ri , e) = w(ri , e), where ri =

Ri xi

is the inverse saturation of routes.

Assumptions 1.1 and 1.2 are standard. Assumption 1.3 matches the intuition that more enforcement decreases the fraction of every additional unit of drugs bought that reaches consumer markets. In order to justify assumption 1.4, suppose that both xi and Ri increase in the same proportion. The routes will be equally saturated, and it will be neither easier nor harder for the government to seize drugs from any given shipment. The fraction seized does not change and neither does the survival rate, which thus depends only on ri =

Ri xi .

This means that it is homogeneous of degree

zero. These assumptions directly imply some other useful properties that are useful throughout this paper:

7

Lemma 1. The production function q( xi , Ri , e) has the following properties:7 1. It is homogeneous of degree one in ( xi , Ri ). 2. It is concave in ( xi , Ri ). ∂2 q

3. Routes and drugs are complementary production factors ( ∂x ∂R > 0). i

i

The survival rate w( xi , Ri , e) has the following property: 4. It is increasing in ri ( ∂w ∂r > 0). i

2

5. The marginal productivity of ri is decreasing ( ∂∂rw2 < 0). i

Each cartel’s share of the total market is small, so it has no market power and takes prices as fixed8 . The whole trafficking region, however, may involve an important share of the total drug trade, so the total amount of drugs has an effect on drug prices. The elasticity of demand of drugs in the consumer market is ec .9 I assume that prices in the producer market are fixed, which corresponds to an elasticity e p = ∞. This greatly simplifies the final expressions that I obtain, without losing any important insight. I 7 Proof:

1 is straightforward to check. For 2, the conditions on the first derivatives,

on both second derivatives, and on the cross derivatives (which I will soon state) imply that the function is quasiconcave. Quasiconcavity and homogeneity of degree one ∂2 q ∂2 w ∂w ∂xi ∂Ri = x ∂xi ∂Ri + ∂Ri . ∂w ∂2 ri ∂ri ∂xi ∂Ri . The derivatives of ri

imply concavity. For 3, ∂2 w ∂xi ∂Ri

=

∂2 w ∂ri2

∂ri ∂ri ∂Ri ∂xi

+

By the chain rule,

∂w ∂Ri

=

∂w ∂ri ∂ri ∂Ri

and

can be readily calculated. Substituting

everything in the initial expression for the cross derivative of q yields

∂2 q ∂xi ∂Ri

2

= − xR2i ∂∂rw2 , i

i

which is positive due to the decreasing marginal productivity of ri . 4 and 5 can be easily checked by finding the derivatives of w with respect to Ri , holding xi fixed. 8 As an example, the Herfindahl index for Mexican cartels is around 0.15, suggesting that this is not a very strong assumption. I made the calculation based on the data from Castillo et al. (2015). 9 This is the elasticity of residual demand, after taking into account the demand satisfied by other sources of drugs not being analyzed.

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analyze the trafficking industry with e p ∈ (0, ∞) in appendix D, which in no way changes the main results. Cartels’ productive behavior determines the aggregate supply of drugs in the consumer region, Q = ∑i∈ I qi , which consumer regions would like to minimize. I also define X = ∑i∈ I xi , the aggregate amount of drugs bought in the producer market.

1.2

Conflict over routes

There is a continuum of routes normalized to one, i.e., ∑i∈ I Ri = 1. In order to control routes, cartel i invests gi in the conflict, which includes the salaries of gunmen, the cost of guns, losses associated with dead gunmen, etc. At the end of the conflict, the amount of routes held by the cartel is a contest success function Ri ( gi , g−i ) that depends on its expenditure on the total amount g−i = ∑ j6=i g j spent by all other cartels. Aggregate spending in the conflict, G = ∑i∈ I gi , is a proxy for the level of violence, which the trafficking region would like to minimize. Assumption 2. The function Ri ( gi , g−i ) has the following properties: 1. It is continuous and homogeneous of degree zero. 2. It is the same for every cartel (Ri ( g, G − g) = R j ( g, G − g)

∀i, j)

3. Spending no resources in the conflict results in controlling zero routes unless all cartels have zero expenditure (Ri (0, g−i ) = 0 for g−i 6= 0). 4. Every route is controlled by one of the cartels (∑i Ri ( gi , g−i ) = 1) Assumption 2.1 matches the intuition that if all cartels increase consumption proportionately the amount of routes controlled does not change. It is a introduced for tractability and it is standard for contest success functions in the economic theory of conflicts. For a detailed discussion, see 9

Skaperdas (1996). Assumption 2.2 means that all cartels are a priori equally efficient in the way they use resources in the conflict. There might be institutional reasons for heterogeneity between cartels10 , but studying them goes beyond the scope of this paper. These assumptions are enough to pin down function Ri : Lemma 2. The contest success function Ri has the following form11 : R i ( gi , g − i ) =

gi gi + g − i

(1)

This implies the following properties: i 1. It is increasing in own expenditure in the conflict ( ∂R ∂g > 0) i

2. The marginal productivity of expenditure in the conflict in decreasing and it goes to zero in the limit ( ∂g∂

2R

i −i ∂gi

< 0,

limgi →∞

∂R ∂gi

= 0)

In the end, since cartel i sells an amount qi of drugs in the consumer market at a price pc , the profit it obtains is π i = p c q ( x i , R i ( gi , g − i ) , e ) − gi − p p x i

(2)

Note that cartels have two choice variables, but they interact strategically only through expenditure in the conflict gi : they are price takers, so the amounts of drugs bought in the producer market and sold in the consumer market do not affect rival cartels.12 10 For

instance, one cartel having ties to corrupt government officials that make it easier

for them to control routes. 11 Proof: If m cartels spend g and every other cartel spends zero, 1 = ∑ R i ( gi , g − i ) = mR( g, (m − 1) g) = mR(1/m, 1 − 1/m) =⇒ R(1/m, 1 − 1/m). If p cartels spend g and one cartel spends qg, the first cartels get R( g, ( p − 1 + q) g) = R(1/( p + q), 1 − 1/( p + q)) = 1/( p + q) routes, so the other cartel gets R(q/( p + q − 1), 1 − 1/( p + q − 1)) = 1 − p/( p + q) = q/( p + q). This pins down the function for every rational s ∈ (0, 1) as R(s, 1 − s) = s. By continuity, this also has to be true for every real s ∈ (0, 1). And by homogeneity of degree zero, R( g, G − g) = R( g/G, 1 − g/G ) = g/G. 12 If cartels had some market power, interaction through drug quantities becomes relevant. This is now the widely treated problem of a traditional Cournot oligopoly in IO.

10

2

Productive behavior

Before turning to the various types of equilibria that may arise, I show that the aggregate productive behavior of cartels (that is, the total amount of drugs bought at producer markets and sold at consumer markets) does not depend on the equilibrium that arises from the conflict. Then I will determine the effect of enforcement on aggregate productive profit.

2.1

Independence of aggregate productive behavior

Suppose that in some equilibrium the amount of routes controlled by cartel i is Rˆ i . Since the amount of drugs bought at the initial market does not affect others, i chooses the quantity that maximizes its profit, i.e.,   xi∗ = argmax pc q( xi , Rˆ i , e) − gi − p p xi ,

(3)

xi

which can be solved from the following first order condition: ∂q = pp pc ∂x |{z} | {z }i MgCx MgBxi

(4) i

which equals marginal benefit and cost. I assume that

pc pp

is large enough

that there is an interior solution, since otherwise the illegal drug market would not even exist.13 Lemma 3. Aggregate demand from the producer market and supply to the consumer market are determined by pc 13 The

infinity:

∂q( X, 1, e) = pp ∂x

Q = q( X, 1, e),

(5)

solution is bounded since the marginal productivity drops to zero as xi goes to ∂r ∂x

= − xR2 , and

∂q ∂X

∂w ∂r ∂w = w + x ∂w ∂x = w + x ∂r ∂x = w − r ∂r . And w, r and

∂w ∂r

drop to zero as xi goes to infinity. Also note that cartels decide the amounts of xi and gi simultaneously, so this is only a necessary condition at equilibrium.

11

which are independent of the distribution of routes. The share of drugs bought and sold by each cartel is equal to the fraction of routes it controls: xi = Ri X

qi = Ri Q

Proof. For two different cartels i and j,

∂q( xi∗ ,Ri∗ ,e) ∂xi

(6)

=

∂q( x ∗j ,R∗j ,e) . ∂x j

Since q is

homogeneous of degree one, its derivative is homogeneous of degree zero,

∂q( x ∗j /R∗j ,1,e) ∂q( xi∗ /Ri∗ ,1,e) = , and since this derivative is strictly decreasing, ∂xi ∂x j ∗ xj xi∗ xi∗ qi∗ Rˆ i Ri∗ = R∗j . Thus, x ∗j = Rˆ j = q∗j , where I used homogeneity of q again for the x∗ last step. Summing over i yields xX∗ = R1ˆ = qQ∗ , so X = Rˆi . Substituting in j j j i

so

(3), taking into account the homogeneity of q, yields the condition for X. For Q, Q = ∑i q( Rˆ i X, Rˆ i , e) = ∑i Rˆ i q( X, 1, e) = q( X, 1, e). For the shares of production, qi = q( xi , Rˆ i , e) = Rˆ i q( X, 1, e). Lemma 3 is a very general result. It is a consequence of the production technology having constant returns to scale, and of the fact that the conflict is a zero-sum game, at the end of which the amount of routes controlled by all cartels is the same. It is true regardless of how the conflict over routes is solved. There are clear implications in terms of public policy: actions taken by the government to affect the conflict between cartels may affect violence, but they have no effect on the amount of drugs reaching final consumer markets. I now define aggregate productive profit, π A = pc Q − p p X, the sum of the profits obtained by all cartels, without taking into account their expenditure in the conflict (taking into account productive behavior but not military behavior). This is useful because the problem is equivalent to one in which cartels fight over a share of aggregate productive profit: Lemma 4. Cartel i’s profit can be restated as π i = π A R ( gi , g − i ) − gi 12

(7)

Proof. Cartel i invests gi in the conflict, buys an amount of drugs xi = R( gi , g−i ) X and sells an amount qi = R( gi , g−i ) Q. Substituting in (2) yields π i = p c q i − p p x i − gi = ( p c Q − p p X ) R ( gi , g − i ) − gi . In other words, cartels fight over the aggregate productive profit, π A = pc Q − p p X, and they spend resources in the conflict in order to obtain a share of the prize. After the conflict, they end up receiving a fraction of π A that is equal to the fraction of routes they control. The effect of enforcement on the aggregate amount of drugs bought can be found by applying the implicit function theorem on (5). Writing

dpc dQ

in

terms of elasticities yields   ∂X  = ∂e



1 Qec

(a)

(b)



{ z }| { z }| 2  −1 ∂ q    1 ∂q ∂q  +   2  Qec ∂X ∂e ∂X∂e  ∂2 q ∂q  + ∂X2  ∂X

(8)

All derivatives of q are evaluated at ( X, 1, e). The term in parentheses is positive, so the sign is determined by the term in square brackets. Two mechanisms are at work. First, in term (a), enforcement decreases the amount of drugs that reach the consumer market, if the amount of drugs bought from the producer market were held fixed. This leads to an increase in prices, which increases the marginal benefit of drugs and encourages cartels to take more drugs from the producer to the consumer region. Second, in term (b), enforcement reduces the marginal productivity of X, and thus its marginal benefit. In response to this, cartels buy a smaller amount of drugs so that marginal benefits and costs of X are again equal. Enforcement thus increases or decreases the amount of drugs bought depending on which effect is larger. This depends, in particular, on whether demand is inelastic enough that effect (a) is larger. Note that if the trafficking region has a small share of supply, ec → ∞, and mechanism (a) has no effect on X. 13

The chain rule can be used to find the effect on the supply of drugs, which yields ∂Qe = ∂e

2

∂2 q ∂q ∂X 2 ∂e 1 Qec

∂q ∂ q − ∂X ∂X∂e  2 ∂2 q ∂q + ∂X 2 ∂X

(9)

By looking at the individual derivatives, it is clear that enforcement reduces supply. Proposition 1. The comparative statics on the amount of drugs supplied to the consumer region is: •

∂Q < 0: Increasing enforcement reduces the supply of drugs. ∂e

•

∂Q = 0: The number of cartels has no effect on the supply of drugs ∂n

Actions taken by the government should aim against productive behavior (i.e., enforcement) if they are to reduce the supply of drugs in consumer markets.

2.2

Elasticity and productive profit

It is not clear whether enforcement increases or decreases productive profit π A = pc Q − p p X. This is a very important question because, as I will show in the rest of this paper, this also determines whether violence increases or decreases, both in a one-period equilibrium and in a dynamic model. The effect of enforcement on productive profit can be expanded in terms of costs and revenues:

∂π A ∂e

=

∂pc ∂Q ∂Q ∂e Q

∂X + pc ∂Q ∂e − p p ∂e . Rewriting

terms of the elasticity of demand leads to   ∂π A 1 ∂Q ∂X = pc 1 + − pp ∂e ec ∂e ∂e Substituting

∂Q ∂e

and

∂X ∂e

∂pc ∂Q

in

(10)

from (8) and (9) allows me to find a threshold for

the elasticity of demand such that productive profit increases if ec > eˆ c : 14

enforcement causes an increase in prices large enough that aggregate productive profit increases. Thus, the prize cartels fight for increases, increasing the marginal benefit of expenditure in the conflict. The expression for this threshold is:  eˆ c = −1 −

 ∂q 2 ∂X

∂q ∂2 q ∂e ∂X 2



∂q  ∂e

| {z } | (a)

Q



−

∂2 q ∂X∂e  ∂q ∂X

{z

(b)

(11)

}

Proposition 2. The comparative statics on aggregate productive profit is: ∂π A < 0: If demand is sufficiently elastic, enforcement ∂e reduces aggregate productive profit.

a) If ec < eˆc , then

∂π A > 0: If demand is sufficiently inelastic, enforcement ∂e increases aggregate productive profit.

b) If ec > eˆc , then

This is an improvement on the threshold of −1 that determines whether revenues (equal to the market size, pc Q) increase or decrease in response to reduced supply. The analysis by (Becker et al., 2006) that leads to this threshold measured the decrease in welfare in consumer nations under competitive markets, where market size is a good measure of the harm caused by illegal drugs. However, subsequent works have taken this analysis out of context and concluded that it also applies to the profit of drug traffickers. This result is not right in this context since it does not take into account cartels’ costs, which may increase or decrease in response to supply, with opposing effects on violence. Equation (11) is thus the threshold of −1, plus a correction that tells if it is easier or harder for enforcement to increase violence than with the original threshold. In order to find the sign of the correction, note that term (a) is negative, so the sign is determined by term (b), which can be written as ∂ log q ∂e

∂q

−

∂ log ∂X ∂e :

whether enforcement has a larger effect on productivity or 15

on marginal productivity of drugs. I will now explain why this difference determines the sign of the correction. Since the correction comes from costs p p X, and p p is fixed, we are interested in the effect of enforcement on X. As in section 2.1, enforcement has two effects on X. Due to its effect on productivity (

∂ log q ∂e ),

X increases,

as well as costs, which means that it is possible for profit to decrease while revenue increases. Hence, the effect of enforcement on productivity makes it more difficult for profit to increase, so the threshold is higher14 . On the ∂q

other hand, due to enforcement’s effect on marginal productivity (

∂ log ∂X ∂e ),

X decreases, which makes it easier for profit to increase, and the threshold is lower. So the correction depends on the relative sizes of the effect through productivity or the effect through marginal productivity. If costs are low the correction is smaller. From first order condition (5), ∂q ∂X

=

pp pc .

Thus, when prices in the producer market are low in comparison

with the prices at which cartels sell drugs, costs take a smaller share of revenues and the correction becomes less important. Enforcement reduces supply, increasing prices. Thus, enforcement makes the correction to the threshold less important. 2.2.1

Elasticity threshold for some functional forms

I analyzed the threshold as far as possible without specifying a particular functional form for q. Now I assume one particular functional form. Although the results of this section are not general, I show in appendix C that they hold more generally for functional forms satisfying the condition that the survival rate w is not greater than one. For the most traditional functional forms for q (such as a Cobb-Douglas 14 eˆ

c

can be positive: the effect of enforcement on productivity would be so large that

even with perfectly inelastic demand enforcement would decrease profits.

16

or CES function), the correction to the threshold vanishes15 . But with these functional forms holding the amount of drugs x fixed and increasing R results in q( x, R, e) increasing without bound, meaning a survival rate greater than one. I will now consider another production function which better fits this particular conflict, a standard contest-success function (CSF) that takes into account both parties’ resources to determine which fraction of the prize is obtained by each one of them16 : w(r, e) =

r r + ϕe

(12)

The production function is q( x, R, e) = xw( x/R, e). A straightforward calculation shows that with this production function the effect on marginal productivity is always larger than the effect on productivity, and the correction means that the range of elasticities for which violence increases with enforcement is greater than previously thought. A simple expression for the threshold can be found by noting that homogeneity of degree zero of w means that r is uniquely determined by the p

ratio of the prices γ = ppc . Substituting everything in the threshold yields   √ γ eˆ c = − 1 + 2(1−√γ) . Figure 1 plots this function. Its effect is the fact that as the gap between consumer and producer prices widens, and γ goes down to zero, costs become a smaller share of revenues and the threshold gets closer to −1. For high γ, the threshold goes well below −1, but empirical evidence shows that each step in the production chain of drugs, from 15 For

example, for a Cobb-Douglas function whose productivity parameter depends

on enforcement, q( x, R, e) = A(e) x α R1−α . This results in

∂ log q ∂e

∂q

=

∂ log ∂X ∂e

correction vanishes. A similar result holds for a CES function q( x, R, e) =

A0 (e) , so the A(e) r A(e)( ax + (1 −

=

1

a ) Rr ) r . 16 The functional form that I present depends on the ratio of the resources committed by each party, and it is the most commonly used function in the literature of the economic theory of conflicts. Hirshleifer (1989) analyzes the implications of this type of function and some alternatives.

17

producers to consumers, implies a large increase in prices (Mejía and Rico, 2010), and the results for high values of γ are therefore not very relevant. 0 -1 ` Εc

-2 -3 -4 -5 -6 0.0

0.2

0.4

0.6

0.8

1.0

Γ

Figure 1: Variation of the threshold as a function of γ

3

Equilibria in the war over routes

I first solve the stage game of the conflict over routes, which serves as a benchmark for comparison, and will be useful to prove some of my main results with repeated interaction.

3.1

Stage-game Nash equilibrium (SGNE)

First, let’s find cartels’ best response function. Interpreting the conflict as in lemma 4, the problem cartel i faces is: max πi = π A R( gi , g−i ) − gi gi

(13)

which leads to the following first order condition: ∂R = |{z} 1 πA ∂gi | {z } MgCgi

(14)

MgBgi

Marginal benefit of investment in the conflict must thus equal its marginal cost (one). There cannot be a corner solution with gi = 0 and xi > 0, since 18

the marginal productivity of expenditure in the conflict tends to infinity if all cartels spend zero resources in the conflict. The solution is bounded because the marginal productivity of expenditure in the conflict goes to zero as gi goes to infinity. Thus, every cartel has an interior solution. I check the second order conditions in appendix A.17 First order conditions (14), one for each cartel, give the best-response functions for gi in terms of the quantities g−i chosen by all other cartels. The Nash equilibrium of the stage game occurs when the first order conditions are satisfied simultaneously for all cartels, i.e., when all best-response functions are consistent. The symmetry of the problem means that the oneperiod Nash solution has gi = g j = g N ,

xi = x j = x N

∀i, j ∈ I, and

g−i = (n − 1) g N . Furthermore, every cartel controls an equal amount of routes Ri = n1 . I already showed that there exists a solution for each cartel. Furthermore, it is unique because the marginal productivity of gi is strictly decreasing. Proposition 1 stated the comparative statics on the amount of drugs taken to the consumer region. The following proposition states the comparative statics on violence: Proposition 3. Under a symmetric stage-game Nash equilibrium, the comparative statics on the level of violence in the region is as follows: ∂G N < 0: If demand is sufficiently elastic, enforcement ∂e reduces the level of violence.

a) If ec < eˆc , then

∂G N > 0: If demand is sufficiently inelastic, enforcement ∂e increases the level of violence.

b) If ec > eˆc , then

c) 17 It

∂G N > 0: An increase in the number of cartels increases violence. ∂n seems like it is satisfied, since

∂2 R ∂gi2

< 0, but I am maximizing in two steps, (3) and

(13), without taking into account that the maximization is simultaneous. Thus the need for appendix A.

19

Proof. Individual expenditure is determined by (14). Enforcement has no direct effect on

whereas g N has no direct effect on π A . This, and the

∂Ri ∂gi ,

∂g N ∂e

implicit function theorem, lead to negative:

∂2 R ∂g N ∂gi

=

∂2 R ∂gi2

= −

∂π A ∂e   ∂ ∂R ∂g N ∂gi

. The denominator is

2

+ (n − 1) ∂g∂−iR∂gi < 0. Thus, the sign of the effect of

enforcement is the same as the sign of the effect on π A . Now take (14). Since π A does not depend on the number of cartels, ∂Ri ∂gi

cannot depend on it either, so its derivative with respect to n must   2 ∂g N ∂g N ∂2 R ∂2 R ∂2 R be zero: ∂n∂g = ∂g2 + (n − 1) ∂g ∂g + g N ∂g∂∂gR = 0. Isolating ∂n ∂n i i −i i −i i   −1 2 ∂g N ∂2 R ∂2 R yields ∂n = − g N ∂g∂∂gR + ( n − 1 ) . 2 ∂g ∂g ∂g i

−i

i

i

−i

In order to find the comparative statics on aggregate violence G, I use    −1 N N 2 2 2 2 N ∂g ∂ R ∂G N N ∂ R − ∂ R + (n − 1) ∂g∂∂gR the fact that ∂G , ∂n = g + n ∂n to obtain ∂n = g ∂g ∂g ∂g2 ∂g2 i

−i

whose sign is undetermined, since it depends on whether

∂2 R ∂gi2

i

i

i

or

∂2 R ∂gi ∂g−i

is

greater. However, it is negative in a symmetric equilibrium: R( g, (n − 1) g) = n1 , so

( n − 1)2

∂2 R( g,(n−1) g) 2 ∂g− i

d2 R( g,(n−1) g) ∂dg2

=0=

∂2 R( g,(n−1) g) ∂gi2

+ 2( n − 1) ∂

2 R ( g,( n −1) g )

∂gi ∂g−i

. Let S( gi , g−i ) = 1 − R( gi , g−i ), the fraction of routes

held by all cartels other than i. In a symmetric equilibrium S = (n − 1) R, ∂2 (1− S ) ∂2 ((n−1) R) 1 ∂2 R ∂2 R = − n− = = − 2 2 1 ∂g2 . Substitut∂((n−1) gi )2 ∂g−i ∂g−i i ∂R( g,(n−1) g) ( n −1) g ) ∂2 R ∂2 R = 2 ( n − 1 ) , so < − 2) ∂R( g,∂g 2 2 2 ∂gi ∂g−i ∂g−i ∂gi i

and g−i = (n − 1) gi , so ing above yields (n and

∂G N ∂n

> 0.

The key in parts a) and b) is that the effect on violence is the same as the effect on the prize being fought over. Thus, the conflict intensifies as the stakes increase.

20

+

−i

3.2

Repeated interaction and collusion

I will now consider a repeated game. The total profits obtained by a cartel are the discounted sum of the profit obtained in each of the periods: Πi =

∞

∑ βt πi,t

(15)

t =0

where πi,t is the profit obtained by cartel i in period t, and β ∈ (0, 1) is the discount factor. The discount factor depends on the monetary discount factor related to the interest rate, which I call δ, and the probability p that the current leader of the cartel will still be in charge in the next period. This means that cartel leaders are selfish agents that do not value the future of their organization after they are captured or killed. The discount factor is then β = δp. The probability p depends on the government’s actions, since policies aimed at capturing or killing leaders decrease the probability that they will be standing during the next period. In the repeated game a very large number of equilibria arise. I take into account three of them. The baseline equilibrium is repeating the stageπN for each cargame Nash equilibrium perpetually, with profit Π N = 1−β tel. Since most previous works only focused on the stage game, this is a direct point of reference to compare the new equilibria I analyze. The other two equilibria I analyze arise from punishment strategies: if some cartel deviates by increasing its expenditure, in the next period all other cartels punish it. For each punishment strategy I analyze the equilibrium that maximizes payoffs for cartels in the equilibrium path. The first type of punishment is Nash reversion, in which cartels return to the SGNE. I look at it for its simplicity, and because it is useful to prove some results. Second, with optimal punishment cartels punish deviators as harshly as possible (Abreu, 1983). I focus on this punishment strategy because it achieves the greatest possible payoff for cartels. In this problem the harshest pun-

21

ishment is making sure that any cartel that betrays receives zero profits from that moment on, as in Rotemberg and Saloner (1986), since any lower profits would encourage the deviator to exit from the industry. Assumption 3. Optimal punishment is feasible. I.e., there exists some punishment against one cartel that is strong enough to induce zero profits. Furthermore, it is subgame perfect.18 Optimal punishment is not a reasonable assumption in collusion models in oligopolies for three reasons. First, firms only observe prices, so they can only see that some firm deviated, but they cannot determine which one. Second, they punish by starting price wars that do not have a specific firm as target, so all firms are equally punished, regardless of whether they deviated or not. This makes harsh punishments difficult to achieve because the harsher the punishment the greater the incentives not to participate in it. Third, firms have no mechanism outside the game to punish deviators. On the other hand, it is a reasonable assumption for conflicts between cartels. First, it is clear which cartel deviated from the collusive equilibrium. Second, cartels are violent organizations that are willing to gang up against whoever broke the agreement. Thus, coordinating a harsh punishment against the lone deviator is easy. And third, they have mechanisms outside the game to punish deviators: instead of just punishing by investing more resources in the fight over routes, they can choose to attack the leadership structure of the deviator. If a cartel leader is thinking about deviating, the threat of other cartels trying to kill him is a powerful deterrant. 18 Once

the optimal punishment exists, it is not too hard to justify that it is subgame

perfect. If any cartel decides not to join the punishment, every other cartel would gang up against it, so it would be suboptimal for cartels not to punish the deviator.

22

3.2.1

Peaceful equilibrium

In an ideal agreement, cartels split routes evenly without any expenditure in the conflict. Each cartel controls

1 n

routes, and, from lemma 4, each cartel

obtains profits 1 A π (16) n where π a (0) stands for the profit obtained under an agreement with zero π a (0) =

expenditure in the conflict. If all cartels invest zero resources in the conflict, they do not have the means to defend their routes. Thus, any single cartel can invest an infinitesimal amount η in the conflict and be able to control all routes. For a single period, the deviator takes all the aggregate productive profit, π d (0) = π A − η = nπ a (0) − η, where π d (0) is the profit when deviating from an agreement with zero expenditure in the conflict. From the next period on it will receive profits π p due to others’ punishment. With Nash reversion, π p = π N , and with optimal punishment, π p = 0. The incentive constraint (IC) is thus

1 a 1− β π (0 )

≥ π d (0) + 1−β β π p . Isolating β yields:

Proposition 4. A peaceful collusive equilibrium can be sustained if β ≥ For optimal punishment, this reduces to β ≥

n−1 . n

n−1 p . n − ππA

Under the right circumstances cartels can coexist without any violence. This depends on two conditions: there must be a low number of cartels, and they must place a high value on future earnings. Even though an individual cartel could seize all routes for one period, the punishment will decrease its long run profits, so cartels do not deviate if they are sufficiently fearful of the future punishment. And the potential gain from deviating is greater if there is a large number of cartels, requiring a high discount factor to sustain peace. It may be surprising that enforcement has no effect on whether peace 23

can be sustained with optimal punishment. The reason is that enforcement A

reduces both the profits of an agreement ( πn ) and deviating (π A ) through productive profits, so it does not tilt the balance either way in the incentive constraint. 3.2.2

Equilibrium with violence

If peace cannot be sustained (β <

n −1 n ),

one would not expect cartels to

wage all-out war, as in the SGNE. They can still agree on spending g¯ < g N on the conflict, after which they end up controlling the same amount Ra =

1 n

of routes as in the SGNE but with a higher profit19 . The purpose of

g¯ > 0 is to make betrayal more costly than with zero expenditure, when an infinitesimal expenditure was enough to grab all routes. Thus, punishment and g¯ work together as deterrents against deviating. I therefore call g¯ deterrent expenditure. The profit obtained by each cartel is then π a ( g¯ ) = π A R a − g¯ =

1 A π − g¯ n

(17)

Productive behavior is the same as in the stage game, so the only difference between this agreement and the SGNE is lower investment in the conflict. Comparing the profit from colluding from the profit at the SGNE yields ¯ π a ( g¯ ) = π N + g N − g. ¯ since the agreement means Cartels would benefit if they could reduce g, that regardless of the amount spent they all hold the same fraction of routes. But if g¯ is too low, any cartel can break the agreement, invest a larger amount of resources to attack other cartels and grab a larger fraction of the routes, thereby increasing its profit. Thus, g¯ cannot be arbitrarily low: it is as low as possible while still working as a deterrent. 19 This

was not possible in the stage game because a lower level of expenditure by all

other cartels meant an increase in the marginal utility of expenditure for every cartel (since ∂2 q ∂g∂g−i

< 0), implying an incentive to deviate unilaterally and increase expenditure.

24

Since the benefit of deviating from the cooperative strategy only lasts for one period, the deviating cartel would want to take as much profit as it can for that single period, i.e., the optimal one-period behavior given that ¯ Thus, the profit obtained by the deviator i is: all other cartels spend g. h i π d ( g¯ ) = max π A R( gi , (n − 1) g¯ ) − gi gi

(18)

The first order condition is the same as for the SGNE, equation (14), but ¯ 20 with expenditure by other cartels evaluated at g−1 = n g. Solving (18) gives the optimal expenditure for the deviator gd , which determines its optimal amount of routes Rd = R( gd , (n − 1) g¯ ). The deviator’s profit is then π d ( g¯ ) = π A R a − gd The total profits of the deviator are π d + it complies with the agreement are

1 a 1− β π ,

β p 1− β π ,

(19) and the total profits if

so the new incentive constraint

is: π a ( g¯ ) ≥ (1 − β)π d ( g¯ ) + βπ p

(20)

If there exists some deterrent expenditure such that (20) is satisfied, an equilibrium with g¯ can be sustained. Proposition 5. An equilibrium with less violence than the SGNE (i.e., g¯ < g N ) always exists, both with Nash reversion and optimal punishment. Proof. Setting g¯ = g N , π a ( g N ) = π N , since all cartels spend the amount corresponding to the SGNE. A deviator’s one-period optimal response is g N , so π d ( g N ) = π N . Thus, with Nash reversion the IC is satisfied with equality at g¯ = g N . 20 As

in the SGNE, I stated the problem as two maximizations over different variables,

but the cartel solves a joint maximization problem over both variables. Thus, I must check the optimality of the joint maximization problem, but it leads to the exact same second order conditions from appendix A.

25

From (17),

∂π a ∂ g¯

= −1. From the envelope theorem on (18),

∂π d ∂ g¯

=

∂R π A (n − 1) ∂g∂R . From the betrayer’s first order condition, π A ∂g = 1, so − i i   d ∂π N A ∂R + ( n − 1) ∂R ∂ g¯ = π ∂gi ∂g−i − 1. For g¯ = g the term in parentheses is ¯ (n−1) g¯ ) ∂R( g, , which is zero because all cartels increase their expenditure by ∂ g¯ ∂π d the same amount. Thus, ∂ g¯ N = −1. This means that at g¯ = g N the g¯ = g

derivative on the left hand side of the IC is −1, which is lower than the derivative on the right hand side, −(1 − β). So the IC holds strictly for some g¯ < g N with Nash reversion.21 With optimal punishment, the profit from betraying is lower than with

Nash reversion. Thus, any level of g¯ that satisfies the IC with Nash reversion also satisfies it with optimal punishment. There always exists an agreement that results in lower levels of violence than in the SGNE. Previous works that focused on one-period interaction missed an important part of the behavior of cartels: if they are able to negotiate an agreement, the do not engage in a conflict with the level of violence predicted by one-period models, resulting in a less violent conflict. I will now analyze the precise level of violence that allows collusion. In order to maximize profits, cartels agree on the minimum amount that ensures that the IC is fulfilled. I denote these amounts with g a,N for Nash reversion, and g a for optimal punishment. They are defined by π a ( g a,N ) = (1 − β)π d ( g a,N ) + βπ N

π a ( g a ) = (1 − β ) π d ( g a )

(21)

When compared with the stage game, this equation replaces second order condition (22). This can be understood graphically. Consider the profit from deviating with Nash reversion. Figure 2 shows how two different cases can arise. If 21 This

is a particular case of a general theorem in Mas-Colell et al. (1995), chapter

12 appendix A, that states that any SGNE can be improved by using Nash reversion strategies.

26

(� - β) Π�

(� - β) Π� ������

���������

���������

������

���������� ���� ��������

���������� ���� �������� ���������� ������� ����������

���������� ������� ����������

����

����

��

��

����

��

(a) High β: The future is important

(b) Low β: deviating is relatively more

enough that with zero investment

profitable, so cartels would deviate if

in the conflict the agreement can be

they were in a peaceful equilibrium,

sustained.

but there still exist levels of deterrent expenditure g¯ < g N that enable an agreement.

Figure 2: Determining the equilibrium. the discount factor is high, Πd ( g¯ ) crosses π a ( g¯ ) only once, at g¯ = g N : even with zero expenditure in the conflict cartels would prefer not to deviate, so there exists a peaceful equilibrium. But if the discount factor is low, there is a second crossing, which determines the level of investment in the conflict by each cartel, since it is the minimum value for which the IC is satisfied22 If cartels use optimal punishment, the profits from deviating are (1 − β)π N lower than with Nash reversion. Figure 2 shows these profits as the curve with Nash reversion displaced downwards. A peaceful equi22 It

would seem that a third possibility exists. If the derivative of the right side of the

IC at the SGNE were lower than the derivative of the left side, g N would be the lowest level for which the IC is fulfilled, i.e., g¯ = g N . However, the derivative of the left side is greater than the derivative of the right side, regardless of the functional forms used. Nothing precludes the profit of betrayal from being concave, which would only mean that it would be easier for the peaceful equilibrium to exist, and would not change the analysis.

27

librium and an equilibrium with some violence can also arise, depending on the discount factor. It is clear that when there is an agreement with some violence the amount each cartel spends in the conflict with optimal punishment, g a , is lower than the amount spent with Nash reversion, g a,r . There remains the possibility that for intermediate values of β a peaceful equilibrium exists with optimal punishment but not with Nash reversion. The quantity of drugs that reach the final market is still the same, from lemma 3. Therefore, the comparative statics is the same as with the SGNE. It is also interesting to find the comparative statics with respect to the discount factor. Since the aggregate productive profit is the result of a single-period maximization, the discount factor does not affect it. This justifies the following proposition: Proposition 6. Under an agreement with less violence than the SPNE, the comparative statics on the total amount of drugs taken to the consumer market is as follows: •

∂Q a < 0: Enforcement reduces supply. ∂e

•

∂Q a = 0: The number of cartels has no effect on supply. ∂n

•

∂Q a = 0: The discount factor has no effect on supply. ∂β

Before finding the comparative statics on the level of violence, I assume that cartels use optimal punishment, and therefore they achieve the best agreement they can attain. The equation that determines the level of conflict is then π a = (1 − β)π d . The impact of policies can be found by determining how the level of violence has to adjust in order for this constraint to hold. Thus, the effect of exogenous changes on violence depends entirely on how they affect the relative profits of complying and deviating. The main results are summarized in the following proposition: 28

Proposition 7. If peace cannot be sustained (i.e. β <

n −1 n ),

the comparative

statics on the level of violence in an agreement with optimal punishment is as follows: a) If ec < eˆ t , then reduces violence.

∂G a < 0: If demand is sufficiently elastic, enforcement ∂e

∂G a > 0: If demand is sufficiently inelastic, enforcement ∂e increases violence.

b) If ec > eˆ t , then

c)

∂G a > 0: An increase in the number of cartels increases the level of violence. ∂n

d)

∂G a < 0: More forward-looking cartels decreases the level of violence. ∂β

Proof. See appendix B. The first three statements in proposition 7 (about

∂G a ∂e

and

∂G a ∂n )

are not

new: some very similar results were found for a SGNE. The intuition behind them, however, is very different, since cartels’ expenditure in the conflict is not determined by a first order condition but by an incentive constraint. Results a) and b) can be understood as follows. Depending on the elasticity of demand, enforcement increases or reduces violence. Just as with the SGNE, violence increases for elasticities above eˆ c since in that case enforcement increases the total profits π A cartels fight over, but the mechanism from greater profits to more violence is different. Looking at the IC, the question is whether enforcement has a larger impact on π a or (1 − β)π d , two quantities that were initially equal. This is equivalent to asking whether enforcement causes a larger percentage increase in profits from colluding or from betraying. Percent changes in productive profit are equal, since every cartel’s productive profit is a constant fraction of the aggregate productive profit (π A ). Therefore, the impact 29

of such changes on total profit depends on how efficient is expenditure in the conflict when complying or deviating. A cartel must increase its expenditure in the conflict to deviate, but 2

since the marginal benefit of expenditure is decreasing ( ∂∂gR2 < 0), the peri

cent increase in routes is not as large as the percent increase in expenditure. This means that expenditure in the conflict is a larger share of the fraction of the productive profit when deviating than when complying. So final profit (prodctive profit minus expenditure in the conflict) is a smaller share of productive profit when deviating than when complying, and a fixed percent change in productive profit causes a larger percent change in profit when betraying than when colluding. Result c) has a straightforward interpretation. In the SGNE, if violence remained equal after an increase in the number of cartels, every cartel would reduce its expenditure in the conflict, thus increasing the marginal productivity of expenditure. But the marginal cost does not change, so expenditure has to increase to get back to the optimum. The mechanism for the increase in violence under an agreement is very different. A greater number of cartels means that the potential prize when deviating becomes greater: a cartel can attempt to take away all other cartels’ routes. Cartels must therefore spend more in the conflict in order to deter potential deviators, which leads to an increase in violence. Result d) is new, as it played no role in a SGNE. It is perhaps the simplest result to grasp intuitively: more forward-looking cartels are more fearful of punishment. This makes it easier to dissuade them with the threat of punishment, allowing cartels to reduce deterrent expenditure in the conflict and decreasing violence, while still maintaining a collusive equilibrium.

30

3.3

Comparison with models of collusion

The model I just described shares many elements with collusion models from industrial organization (IO). In both cases, higher discount factors make it easier for treaties to hold. A lower number of players is also a facilitator of collusion in IO, as antitrust authorities are well aware: they check permanently bellwethers of collusion, the first of which are the number of firms operating in an industry and more advanced indicators of concentration such as the Herfindahl index. An agreement requires a high level of information in both cases. Players must hold two important pieces of information. First, they must know rivals’ characteristics in order to determine how far down production (in IO) or expenditure in the conflict (in drug markets) can go while still fulfilling the IC. Obtaining this information is equally complicated in both cases. Second, players must monitor whether rivals comply with the agreement in order to punish them if they do not. In IO, this is complicated since there is no way to know the quantity produced by each firm. Firms only know prices, which can hint that somebody broke the agreement, but it says nothing about who broke it. On the other hand, monitoring is much easier in drug markets, since it is clear which cartel decided to increase resources spent in the conflict in order to increase its share of the routes. In both cases theory predicts that players always comply with the agreement. Hence, the punishment stage is never reached. This is not the case in reality: firms wage price wars, and cartels wage war against each other. But this is the case if the model considers imperfect information. Uncertainty in the optimal level of production or expenditure in the conflict leads to violations of the agreement: underestimates make it too easy for rivals to deviate, whereas overestimates can be misinterpreted as deviation. Error in measuring others’ actions can also lead to punishing them when they actually complied. Additionally, coordination is harder for a larger num31

ber of players, an additional reason why more fragmented markets make it harder for an agreement to hold. All these issues point to information and communication between cartels as a potential extension to this model. An important difference lies in the punishment strategies. Nash reversion is feasible and beneficial for players in both cases, but as argued by Abreu (1983) this is far from optimal. In IO, even if firms knew who betrayed, they have no mechanism to punish the single firm that deviated. Abreu thus looked for punishments within strongly symmetric strategies, in which the punishment is equal for all firms, regardless of having betrayed or not. During the punishment phase, all firms tolerate some losses for the prospect of returning to the collusive agreement after a number of periods. But this complication is not necessary in drug markets, since which cartel betrayed is public information, and cartels can join forces against the lone traitor in order to give it a harsh punishment. Thus, the problem of giving an optimal punishment to deviators is much less complicated in the context of illegal drug markets than in IO. There is an important difference between my model and collusion in IO when designing policy: collusive agreements in IO are negative for society, since they move away from the efficient equilibrium that would be attained under perfect competition. On the other hand, treaties between cartels are positive for society, since they decreases violence without having any effect on the productive behavior of cartels. The role of governments is thus reversed: Antitrust authorities’ main goal is to prevent collusive agreements. Governments in drug trafficking nations should instead abstain from following policies that make collusion harder (see section 4).

32

4

Enforcement, attacks on leaders, and fragmentation

The analysis that I have developed so far shows that there is a misconception regarding the traditional methods that policymakers have promoted in order to curb supply: they do not achieve their aim, while causing important increases in violence in trafficking countries. High-profile operations whose aim is to capture or kill bosses or to demobilize previously existing cartels involve large political gains for governments, both because of popular support and because the international community, led by the U.S. government, has always promoted them. But like any policy that focuses on cartels’ military operations, they have no consequence on the quantity of drugs supplied to consumer regions, since aggregate productive behavior is independent of the war being fought over the control of routes, as shown in section 2. These policies are not only inefficient; they are precisely the type of government action that may increase violence. Both the demobilization of previous organizations, as in Colombia, and successful attacks on cartel leaders, as in Mexico, create voids of power that are usually filled by more than one group, thereby increasing the fragmentation of cartels. Additionally, operations against bosses instill a feeling of restlessness and impatience in cartel leaders, who come to believe that their tenure is about to end. Cartels’ strategies will then focus on short-term operations that may bring temporally large profits, without much concern for future operations, and this is precisely the type of strategic planning that leads to higher violence. If cartels were initially able to form an agreement in which each one controls some routes without any violence between them, fragmentation and impatience can trigger the breakdown of the peaceful equilibrium. And even if cartels could not form a peaceful agreement at the beginning,

33

fragmentation and impatience increase the level of violence under which they operate. These policies may bring no benefits, but a large repertoire of alternative policies is available to governments. If their main purpose is to decrease the amount of drugs reaching consumer regions, policies that reduce the productivity of cartels fulfill their aim: patrolling routes and borders in order to increase the rate of seizures, attacking key lieutenants that coordinate the shipment of drugs through routes, and seizing assets used to transport drugs, such as boats, submersibles and airplanes. This kind of policy, however, may have an adverse effect. Drugs are addictive, which means that demand is inelastic (see, for instance, Nisbet and Vakil, 1972; Roumasset and Hadreas, 1977; DiNardo, 1993; Saffer and Chaloupka, 1999), and elasticity is certainly below the threshold for an increase in cartels’ profits and violence. Authorities thus face a tradeoff: if they succeed in curbing supply, they will increase violence in trafficking regions. Some governments justify attacks on cartel bosses by arguing that such attacks disrupt the operations of cartels, but it is not clear that attacking cartel bosses reduces the productivity of cartels: past experience has shown that bosses can be readily replaced by former lieutenants with a seamless transition in the productive operations of cartels. What matters is whether such attacks affect cartels’ productivity as drug traffickers, i.e., how efficient they are in taking the drugs they buy at producer nations to consumer nations through the land they control. Although drug leaders are a vital part of cartels, they usually play a larger role in the conflict over routes, since they act as warlords. The lesson of this paper is that authorities should focus on raids that have the largest impact on the productive operations of cartels. This discussion fits very closely the Colombian and Mexican cases I mentioned in the introduction. The government-led demobilization of the

34

AUC in Colombia was followed by the emergence of a number of criminal bands, with large fragmentation of cartels. The areas where these bands operate have been among the most violent regions in Colombia in recent years. Mexican President Calderón’s war against drugs is perhaps an even clearer example. Before his war, a few cartels operated in Mexico in a state of peace. Mexican homicide rates were among the lowest in Latin America, despite the fact that most of the cocaine that went from Andean nations to the U.S. was shipped through Mexico. After Calderón started attacking drug leaders frontally, some cartels broke into smaller pieces, and new cartels were able to grab some portion of the illegal drug business. Most importantly, the level of violence doubled between 2006 and 2010. Both cases agree with an initial situation in which cartels were patient enough and their number was low enough that they could collude in peace, but the government’s actions induced the breakdown of the peaceful equilibrium and led to a new state of war.

5

Conclusions

In this paper I extend the analysis of cartels as single-period profit maximizers to a repeated-interaction setting. Cartels are modeled as profit maximizers that buy drugs in a producer region and attempt to take them to consumers through a trafficking region. In the process, they engage in two conflicts: they fight against other cartels over who control routes in the trafficking region, and they engage government forces who try to seize drugs on their way to consumers. If cartels are able to coordinate an agreement, they do not end at the stage-game Nash equilibrium (SGNE) that previous works analyze. Instead, they agree to decrease the amount of resources spent in the conflict against other cartels, resulting in less bloodshed than predicted by the SGNE. A peaceful equilibrium without any violence be-

35

tween cartels can be sustained if there are only a few powerful cartels that are interested in maximizing the present value of their profits with a high enough discount factor. Cartels’ productive behavior (the amount of drugs bought from upstream markets and the amount of drugs sold to consumer markets) remains unchanged if governments attack cartel leaders or if cartels are more fragmented; this is a consequence of the fact that productive behavior is independent of the conflict over routes. Thus, some traditional policies fostered by the U.S.-led war on drugs do not accomplish their purpose of curbing supply. As an unintended consequence, such policies increase violence between cartels: they harm trafficking regions while achieving no positive effect on consumer regions. Governments do have the means to reduce supply: enforcement activities, focused on reducing the productivity of cartels, decrease the amount of drugs reaching final markets. However, enforcement is not totally beneficial, as it increases drug prices. Hence, it increases cartels’ profits if demand is not too elastic, after which cartels fight for higher stakes in trafficking regions with increased levels of violence. Previous analyses that only took into account cartels’ revenues suggested that this happens if demand is inelastic. By also taking costs into account, I present an improved criterion with the implication that enforcement may increase violence even if demand is elastic. Hence, governments willing to decrease supply face a tradeoff, since they may do so through enforcement, but this usually comes at the cost of increasing violence in trafficking nations.

References Abreu, Dilip, “Repeated games with discounting: A general theory and an application to oligopoly.,” Ph.D. thesis, Princeton University, October

36

1983. Baccara, Mariagiovanna and Heski Bar-Isaac, “How to Organize Crime,” Review of Economic Studies, 2008, 75 (4), 1039–1067. Bardey, David, Daniel Mejía, and Andrés Zambrano, “The endogeneous dynamics of crime structure: Heracles’ lessons on how to fight the Hydra,” MIMEO, Universidad de los Andes, 2013. Becker, Gary S., Kevin M. Murphy, and Michael Grossman, “The Market for Illegal Goods: The Case of Drugs,” Journal of Political Economy, 2006, 114 (1). Bogliacino, Francesco and Alberto J. Naranjo, “Coca Leaves Production and Eradication: A General Equilibrium Analysis,” Economics Bulletin, 2012, 32 (1), 382–397. Burrus, Robert T., “Do Efforts to Reduce the Supply of Illicit Drugs Increase Turf War Violence? A Theoretical Analysis,” Journal of Economics and Finance, 1999, 23 (3), 226–234. Camacho, Álvaro, “Paranarcos y narcoparas: trayectorias delicuenciales y políticas,” in Álvaro Camacho, ed., A la sombra de la guerra. Ilegalidad y nuevos órdenes regionales en Colombia., Ediciones Uniandes - CESO, 2009. , “Narcotrafico: mutaciones y política,” in Alejandro Gaviria and Daniel Mejía, eds., Políticas antidroga en Colombia: éxitos, fracasos y extravíos, Ediciones Uniandes, 2011. Castillo, Juan Camilo, Daniel Mejía, and Pascual Restrepo, “Scarcity without Leviathan: The Violent Effects of Cocaine Supply Shortages in the Mexican Drug War,” Working Paper, 2015. Chumacero, Rómulo A., “Evo, Pablo, Tony, Diego and Sonny: General Equilibrium Analysis of the Market for Illegal Drugs,” World Bank Policy Research Working Paper No. 4565, 2008. Dell, Melissa, “Trafficking Networks and the Mexican Drug War,” American Economic Review, 2015, 105 (6), 1738–79.

37

DiNardo, John, “Law enforcement, the price of cocaine and cocaine use,” Mathematical and Computer Modeling, 1993, 17 (2), 53–64. Grossman, Herschel I. and Daniel Mejía, “The war against drug producers,” Economics of Governance, 2008, 9 (1), 5–23. and Minseong Kim, “Swords or Plowshares? A Theory of the Security of Claims to Property,” Journal of Political Economy, 1995. Guerrero, Eduardo, “La raíz de la violencia,” Nexos, 2011. Hirshleifer, Jack, “Conflict and rent-seeking success functions: Ratio vs. difference models of relative success,” Public Choice, 1989, 63 (101-112). Lee, Li Way, “Would Harassing Drug Users Work?,” Journal of Political Economy, 1993, 101 (5), 939–959. Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green, Microeconomic Analysis, Oxford University Press, 1995. Mejía, Daniel and Daniel Rico, “La microeconomía de la producción y tráfico de cocaína en Colombia,” Documento CEDE 2010-19 - Universidad de los Andes, 2010. and Pascual Restrepo, “The War on Illegal Drug Production and Trafficking: An Economic Evaluation of Plan Colombia,” Documento CEDE 2008-19, 2011. and

, “The war on illegal drugs in producer and consumer countries:

A simple analyticial framework,” Documento CEDE 2011-02 - Universidad de los Andes, 2011. Nisbet, Charles T. and Firouz Vakil, “Some Estimates of Price and Expenditure Elasticities of Demand for Marijuana Among U.C.L.A. Students,” The Review of Economics and Statistics, 1972, 54 (4), 473–475. Poret, Sylvaine, “Paradoxical effects of law enforcement policies: the case of the illicit drug market,” International Review of Law and Economics, 2003, 22, 465–493. and Cyril Téjédo, “Law enforcement and concentration in illicit drug

38

markets,” European Journal of Political Economy, 2006, 22, 99–114. Reuter, Peter and Mark Kleiman, “Risks and Prices: An Economic Analysis of Drug Enforcement,” Crime and Punishment, 1986, 7. Rotemberg, Julio J. and Garth Saloner, “A Supergame-Theoretic Model of Price Wars during Booms,” American Economic Review, 1986, 76 (3), 390–407. Roumasset, James and John Hadreas, “Addicts, Fences, and the Market for Stolen Goods,” Public Finance Review, 1977, 5 (2), 247–272. Saffer, Henry and Frank Chaloupka, “The Demand for Illicit Drugs,” Economic Inquiry, 1999, 37 (3), 401–411. Skaperdas, Stergios, “Contest Success Functions,” Economic Theory, 1996, 7, 283–290. UNODC, Global Study on Homicide, UNODC Research and Trend Analysis Branch, 2013.

Appendix A

Second order conditions for the cartel

The cartel’s problem is max( gi ,xi ) πi = pc q( xi , Ri ( gi , g−i ), e) − gi − p p xi , with first order conditions (4) and pc

∂q ∂Ri =1 ∂Ri ∂gi

(22)

∂2 q pc ∂x2i i

∂2 π i ∂xi2



∂2 π i ∂gi2

∂2 q i ∂R2i

∂qi ∂2 Ri ∂Ri ∂g2 i



∂Ri ∂gi

The second-order conditions are = < 0, = pc +    2 2  2 2 2  2 2  2 q ∂q 2R ∂ ∂ q ∂ q qi ∂2 π i ∂2 π i ∂ πi ∂ ∂R i i 0, and ∂x2 ∂g2 − ∂x ∂g = p2c ∂x2i ∂Ri ∂g2i + ∂g i − ∂x∂ ∂R > ∂x2 ∂R2 i

i

i

i

i

i

i

i

i

i

i

i

0. Strict concavity of Ri and concavity of qi ensure that all three conditions are satisfied. It still remains to show that (14) and (22) are equivalent. Homogeneity of degree one means that derivatives are homogeneous of degree zero, so ∂q ∂Ri

is the same if it is evaluated at ( xi , Ri , e) or ( X, 1, e). Euler’s theorem 39

2 

<

means that Q = X π A,

∂q( X,1,e) ∂X

+

∂q( X,1,e) , ∂R

∂q

and from (5), pc ∂R = pc Q − p p X = i

so both first order conditions are indeed equivalent.

Appendix B

Comparative statics for a collusive equilibrium

The implicit function theorem on (21) yields the following derivatives: ∂g a πd =− a ∂π ∂π d ∂β ∂ g¯ − (1 − β ) ∂ g¯ " #" # −1 ∂π a ∂π d ∂π a ∂π d ∂g a = − + (1 − β ) − (1 − β ) ∂e ∂e ∂e ∂ g¯ ∂ g¯ " #" # −1 ∂g a ∂π a ∂π d ∂π a ∂π d = − + (1 − β ) − (1 − β ) ∂n ∂n ∂n ∂ g¯ ∂ g¯

(23)

(24)

(25)

The term in the denominators is positive: it is the difference between the derivatives of the profit from colluding and the profit from betraying. I am analyzing the collusive equilibrium, at which the profit from colluding is less negatively sloped (see figure 2). The profit from betrayal is positive, so ∂g a ∂β

< 0. Multiplying it by n yields

∂g a ∂β

< 0.

The sign of (24) is the sign of its numerator. The analysis from 3.2 concludes that this is equivalent to comparing ∂π A a ∂e R A π Ra − ga

From lemma 3,

xa xd

=

Ra Rd

=

and

qa , qd

∂π a a ∂e /π

∂π A d ∂e R A π Rd − gd

and

∂π d d ∂e /π ,

i.e., (26)

which allows me to multiply both the

numerator and denominator on the right side by

Ra , Rd

so the two quantitates

to compare are ∂π A a ∂e R π A Ra − ga

and

∂π A a ∂e R a π A Rd − RRd gd

(27)

¯ (n − The amount of routes when complying and deviating are R a = R( g, 1) g¯ ) and Rd = R( gd , (n − 1) g¯ ). Since Rd > R a , and R has decreasing 40

marginal productivity,

gd ga

>

Rd Ra

=⇒

Ra d g Rd

> g a . This means that the

denominator on the left side is greater than the one on the right side. The absolute value of the expression on the right side is thus greater, and in a ∂π d d , ∂g < 0, /π ∂e ∂e ∂g a ∂π A n yields ∂e < 0. If, on the other hand, ∂e is positive, a ∂π a a < ∂π d /π d , ∂g > 0, and ∂G a > 0. /π ∂e ∂e ∂e ∂e

conclusion, if by so

∂π A ∂e

is negative,

∂π a a ∂e /π

>

and multiplying all signs change,

The sign of (25) is undetermined, and it depends not only on the functional forms used, but also on the values of the parameters e, n and β. Not everything is lost, however, since I am primarily interested in finding the comparative statics on total violence, G a . Thus, the derivative of interest is ∂G a ∂n

∂ng a ∂g a a ∂n = g + n ∂n . a n ∂g a g ∂n > −1.

=

that

I will now show that it is positive, or equivalently, ¯ (n−1) g¯ ) ∂q( x a ,R a ,e) ∂R( g, ∂π a ¯ g, ∂n = pc ∂Ri ∂g−i a d d ∂q( x ,R ,e) ∂R( g ,(n−1) g¯ ) pc (n − ∂R ∂g−i

From the envelope theorem, the derivatives of profits are ∂π d ∂n

a

= pc ∂q(x∂R,Ri

d ,e )

1). Note that

a ∂R( gd ,(n−1) g¯ ) ∂π d ¯ ∂π g, ∂g−i ∂ g¯ = −1, and ∂ g¯ = d d a a ∂q( x a ,R a ,e) = ∂q(x∂R,Ri ,e) , since xxd = RRd ∂Ri

(lemma 3) and q is

homogeneous of degree one. From the first order condition for betrayal that relates the marginal benefit and cost of investment in the conflict, i h d ( n −1) g¯ ) −1 ∂q( x d ,Rd ,e) . Finally, since R is homogenous of degree pc ∂R = ∂R( g ,∂g i

i

zero, Euler’s homogeneous function theorem means that

∂R( gd ,(n−1) g¯ ) ∂gi

=

−(n − 1) ggd ∂R( g,∂g(n−−i 1) g) . Substituting all these expressions in (25) yields ¯

¯

¯

a

d

∂R ∂R n ∂g a ∂g−i − (1 − β ) ∂g−i h i =− n−1 g¯ ∂Rd ∂Rd gc ∂n − (1 − β ) n

where

∂R a ∂g−i

=

¯ (n−1) g¯ ) ∂R( g, ∂g−i

and

∂g−i

πa . πd

(28)

∂g−i

∂R( gd ,(n−1) g¯ ) . By ∂g−i ∂R( g a ,(n−1) g¯ ) again, = ∂gi

∂Rd ∂g−i

geneous function theorem once and from the IC, (1 − β) =

gd

=

using Euler’s homo-

−(n − 1) ∂R( g,∂g(n−−i 1) g) , ¯

¯

After some manipulation, this allows

me to rewrite the last expression in terms of ratios between quantities for collusion and betrayal. For instance, g˜ = 41

ga gd

is the ratio of expenditure in

collusion to expenditure when betraying. Using a similar notation, R˜ = π˜ =

πa πd

and R˜ i =

∂R a ∂gi ∂Rd ∂gi

Ra , Rd

. I thus obtain the following expression: n g˜ R˜ i − π˜ n ∂g a = gc ∂n n − 1 π˜ − g˜

(29)

˜ In order to do so, reI now want to express R˜ i and R˜ in terms of g. call that since the conflict is symmetric, R(1, n − 1) = n1 , which means that 1 y +1

R(1, y) =

for any value of y. Homogeneity of degree zero of R means 1 , and after some ¯ d +1 (n−1) g/g ¯ (n−1) g¯ ) ∂R( g, = − n12 , the other hand, ∂n ¯ ¯ (n−1) g¯ ) ¯ (n−1) g¯ ) ∂R( g, = − n−g 1 ∂R( g,∂g = g¯ ∂g−1 i

¯ d) = that R( gd , (n − 1) g¯ ) = R(1, (n − 1) g/g manipulation, R˜ =

Ra Rd

=

and from the chain rule

n −1 1 n g˜ + n . On ¯ (n−1) g¯ ) ∂R( g, = ∂n

1 ∂R(1,n−1) − n− . In the last two steps I used Euler’s homogeneous function 1 ∂gi

theorem and the fact that the derivatives of R are homogeneous of degree minus one. From the last expressions,

∂R(1,n−1) ∂gi

=

n −1 , which n2 ¯ (n−1) g¯ ) ∂R( g, ∂gi

means

∂R(1,y) −1 = (y+y1)2 . I am now in a position to find = ngn ∂gi ¯ 2 ¯ d) ¯ d ∂R( gd ,(n−1) g¯ ) −1) g/g ) g/g and . By dividing both = g1d ∂R(1,(n∂g = g1d ((n(−n1−)1g/g ∂gi ¯ d +1)2 i 2 (n−1+1/ g˜ ) derivatives, I finally find that R˜ i = . n2 Note that gd > g a and π d > π a , since the betrayer increases its expen-

that

diture in the conflict in order to increase its profits. Thus, both g˜ and π˜ are less than one. However, increasing expenditure in the conflict does not ˜ This means increase the profit proportionally, which means that g˜ < π. that R˜ i − R˜ i >

R˜ g˜

R˜ g˜

+

(1− g˜ )n+ g˜ + 1g˜ −2 > 0, and π˜n−g˜1 < 0. Both n2 g˜ π˜ −1 n −1 π˜ n g˜ = n + n g˜ . Some straightforward

=

expressions imply that algebra, in which one

must be careful to change the direction of the inequality when dividing by ˜ yields g˜ − π,

n g˜ R˜ i −π˜ n−1 π˜ − g˜

> −1. Therefore,

42

a n ∂g gt ∂n

> −1, and

∂G a ∂n

> 0.

Appendix C

Elasticity threshold for more general functions

Suppose that w depends on the ratio of effective routes r to enforcement e. In order to allow for different efficiencies and increasing or decreasing returns to scale, I assume that w is a function of ρ =

r ϕeη :

ϕ is a parameter that

captures the relative efficiency of enforcement, and η is a parameter that captures whether the returns to scale of enforcement decrease faster than the returns to scale of effective routes. Thus, w(r, e) = w(ρ). The conditions set on the derivatives of q in section 1 mean that w0 > 0 and w00 < 0. This kind of function includes a variety of production technologies. For instance, if w(ρ) = ρ1−α the production function is q = eη (1−α) x α R1−α , a Cobb-Douglas function, and the same CSF used in section 2.2.1 results if w(ρ) =

ρ 1+ ρ

with η = 1.

I will now show that such functions result in a correction that lowˆ ˆ ers the elasticity threshold  ec . In terms of w, the threshold is ec = −1 −  2w ∂w 2 ∂w ∂w ∂ (w−r ∂r ) ∂e ∂e −r ∂r∂e The term in parentheses, which determines its ∂w w − 2 ∂w ∂2 w r

∂e ∂r2

sign, is now

w−r ∂r

rww00 ρr ρe +rww0 ρre −r (w0 )2 ρr ρe . w(w−rw0 ρr )

The denominator is positive, and by

substituting the derivatives of ρ, its numerator is −ρww00 − ww0 + ρ(w0 )2 , which is positive if θ=

w00 ( w 0 )2 w

−

w0 ρ

>1

(30)

The numerator is clearly negative, and the numerator is also negative since the conditions on w imply that w > ρw0 . If (30) is satisfied, the effect of enforcement on marginal productivity is greater than the effect on productivity, so the threshold is lower than −1. Condition (30) has the advantage that it is scale free: θ does not change by substituting w(ρ) with wˆ (ρ) = w( aρ), where a is an arbitrary constant. It is also independent of η. Setting wCD = ρ1−α , a Cobb-Douglas technology, yields θCD = 1. But as I argued in the main text, this is not a very reasonable form for w since

43

it increases without bound. For it to be bounded above, given some value 0 , w00 should be less than for a Cobbw = wCD and some value of w0 = wCD 00 ) so that the function curves downward fast Douglas function (w00 < wCD

enough that it does not go past w = 1. This implies that θ > 1. The relevance of θ being scale-free now becomes clear: the scale parameter a 0 , allowing comparison of θ can be chosen so that w = wCD and w0 = wCD 00 . and θCD only in terms of w00 and wCD

1.0 0.8 0.6

0.4

wHΡL = 1-expH-ΡL wHΡL = Π2 arctanHΡL

0.2 0.0

Θ

w

wHΡL = 0.4Ρ Ρ wHΡL = 1+Ρ

0.4

0

2

4

6

8

10

7 6 5 4 3 2 1 0 0.0

wHΡL = 0.4Ρ0.4 Ρ wHΡL = 1+Ρ wHΡL = 1-expH-ΡL wHΡL = Π2 arctanHΡL

0.2

0.4

Ρ

0.6

0.8

1.0

w

(a) Four functional forms for w(ρ).

(b) Relation between derivatives (θ).

0 -1

` Εc

-2 -3 -4

wHΡL = 0.4Ρ0.4 Ρ wHΡL = 1+Ρ

-5

wHΡL = 1-expH-ΡL wHΡL = Π2 arctanHΡL

-6 0.0

0.2

0.4

0.6

0.8

1.0

Γ

(c) Elasticity threshold.

Figure 3: Comparison of different functional forms. Figure 3 illustrates my argument graphically with three functions that

44

fulfill the conditions for w(ρ):23 w = 2 π

ρ 1+ ρ ,

w = 1 − exp(− 21 ρ), and w =

arctan ρ. I also show w = 0.4ρ0.4 for comparison. The particular values

of the parameters were chosen so that the functions are relatively similar, although this does not change my conclusions. Figure 3a shows the general form of the functions. Figure 3b shows how θ behaves as a function of the value of w, and, in particular, that for all three functional forms θ > θCD . Finally, figure 3c shows the threshold that results for each functional form in terms of γ =

pp pc .

Comparison with figure 1 shows that the conclusions

from section 2.2.1 are not a peculiarity of the functional form that I chose for w.

Appendix D

Variable prices in the producer market

In this section I relax the assumption that prices in the producer market are fixed. Since cartels are price takers, their individual behavior does not change in any way, and their maximization problem is the same, both in the SGNE and with repeated games. The comparative statics, however, must now take into account that changes in policy will have an effect in the producer market, thus changing p p . This effect is described by the elasticity of supply e p . D.1

Aggregate productive behavior

From proposition 3, the number of cartels has no effect on productive behavior, which means that it does not affect the amount of drugs bought from the producer region, and p p . Thus, hand,

∂X ∂e

and

∂X ∂e

∂Q ∂n

stays the same. On the other

do change. The analysis must now take into account that

prices in producer markets are increasing in X, so marginal cost is increasing. The implicit function theorem yields the following expression, which 23 w (0)

= 0, w > 0, limρ→∞ w(ρ) = 1, w0 > 0, and w00 < 0

45

replaces (8): ∂2 q ∂X∂e

∂X =− ∂e



1 Qec

+

 ∂q 2 ∂X

1 ∂q ∂q Qec ∂X ∂e

+

∂2 q ∂X 2

−

(31)

1 pp Xe p pc

The only change is a new term in the denominator, which does not change the sign, although the magnitude of the effect is less. From the chain rule, the new expression that replaces (9) is ∂2 q ∂q ∂X 2 ∂e

∂q ∂2 q 1 p p ∂q ∂X ∂X∂e − Xe p pc ∂e  2 p ∂q ∂2 q 1 − Xe1 p ppc + Qec ∂X ∂X 2

∂Qe = ∂e

−

(32)

The sign of this expression does not change either. The comparative statics thus remains the same. D.2

Threshold for the elasticity of demand

The effect of enforcement on violence depends on the effect it has on the aggregate productive profit. The new dependence of producer prices on quantities means that ∂pc ∂Q

and

∂p p ∂X

∂π A ∂e

∂pc ∂Q ∂Q ∂e Q

=

+ pc ∂Q ∂e −

∂p p ∂X ∂X ∂e X

− p p ∂X ∂e . Rewriting

in terms of elasticities leads to     1 ∂Q 1 ∂X ∂π A = pc 1 + − pp 1 + ∂e ec ∂e e p ∂e

instead of (10). Substituting

∂Q ∂e

∂X ∂e

and

(33)

from (31) and (32) and isolating ec

yields the following threshold for the elasticity of demand:  eˆ c = −1 −

∂2 q ∂q ∂X 2 ∂e

1+

+

1 ep

1 ∂q e p ∂X

 

 ∂q 2 ∂X

∂2 q ∂X∂e

−

 1 ∂q X ∂e



∂q  ∂e

Q



−

∂2 q ∂X∂e  ∂q ∂X

(34)

Two new terms arise. First, the correction is smaller, since increasing marginal cost means that changes in X are smaller (the new term in the

46

denominator)24 . On the other hand, any change in X induces a larger   change in costs, since p p changes with X (see 1 + e1p in the numerator). The sign of the correction is still determined by the sign of

24 The

∂ log q ∂e

∂q

−

∂ log ∂X ∂e .

sign of the correction could actually change if supply is very inelastic and

1 ∂q X ∂e , but ∂2 w ∂e∂r > 0.

∂2 q ∂X∂e

>

expanding this in terms of the derivatives of w shows that this would imply

47