Buying, Shipping, and Selling Illegal Goods

Cartels’ productive activities can be described in one sentence: Cartel $$ i\in I $$ buys a quantity $$ {x}_i $$ of illicit drugs in producer markets at a price $$ {p}_p $$, transports those drugs through $$ {R}_i $$, routes it controls, and sells them in consumer markets at a price $$ {p}_c $$.

We assume that a fixed number $$ n $$ of cartels participates in drug trafficking. The principal cost of this assumption is that it precludes analyzing how government policy affects cartel consolidation and/or fragmentation, which are perpetual features of illegal markets (e.g., Durán-Martínez Reference Durán-Martínez2018). For example, we do not study the conditions under which arresting crime bosses would split a cartel into smaller factions (Phillips Reference Phillips2015, 326), nor do we study the conditions under which higher profits could draw new cartels into the market. If beheadings and/or interdiction were to change the number of cartels, these policies would also affect violence in ways we do not consider. In Appendix A.10, however, we do discuss comparative statics on the number of cartels $$ n $$.Footnote ⁹ Endogenizing $$ n $$ would enrich the model but add considerable complexity.

The government seizes a fraction of all drug shipments; the size of the fraction depends on government expenditures on interdiction, which we denote $$ e $$. By interdiction, we mean any operation targeting drug shipments: seizing drugs in transit, patrolling routes, or attacking drug boats or airplanes. Interdiction excludes capturing or killing cartel leaders (beheadings), which we model as a separate policy.

The amount that the cartel sells in consumer markets ($$ {q}_i $$) increases with the amount it purchases ($$ {x}_i $$) and the routes it controls ($$ {R}_i $$) and decreases with the government’s interdiction expenditures ($$ e $$): $$ {q}_i=q\left({x}_i,{R}_i,e\right) $$. Interdiction shrinks the fraction of drugs that reaches consumer markets, thereby reducing the marginal productivity of both drug purchases ($$ {x}_i $$) and routes ($$ {R}_i\Big) $$. There are decreasing marginal returns both to drug purchases ($$ {x}_i $$) and to control of routes ($$ {R}_i $$). We state and prove additional properties of the production function $$ q $$ in Appendix A.1.

We make the simplifying assumption that all cartels are equally efficient, so that the function $$ q\left({x}_i,{R}_i,e\right) $$ holds for every cartel. Introducing asymmetry would make for an interesting extension.Footnote ¹⁰

In our baseline analysis, we also assume that the production function $$ q\left({x}_i,{R}_i,e\right) $$ has constant returns to scale in routes $$ \Big({R}_i $$) and quantity purchased $$ \left({x}_i\right) $$. This does not imply constant returns to scale in all cartel operations. We do not assume constant returns to conflict investment, nor do we preclude the possibility that there are, for example, fixed costs such as hiring bodyguards or, say, inefficiencies in running a large organization. Rather, constant returns to scale in routes $$ \left({R}_i\right) $$ and drugs purchased $$ \left({x}_i\right) $$ amounts to assuming that drug smuggling is additive across routes.Footnote ¹¹ This would be violated if, for example, controlling a large swath of territory made it more difficult for the government to monitor certain routes—in which case doubling routes $$ \left({R}_i\right) $$ might allow the cartel to more than double drugs sold.

Introducing complementarities across routes would make for an interesting, but complex, extension to the model. Therefore, we focus on production with constant returns to scale, but we provide intuition throughout for how our results would change if we relaxed this assumption.

Each cartel controls a small share of the total market, so it has no market power and takes both upstream and downstream prices as fixed.Footnote ¹² But all traffickers together account for an important share of the total drug trade, so the total quantity of drugs supplied—that is, $$ Q={\sum}_{i\in I}{q}_i $$—affects prices. We denote the elasticity of demand for drugs in the consumer market $$ {\upepsilon}_c $$.Footnote ¹³

We assume that drug cartels cannot collude to reduce quantity. For one thing, monitoring compliance would be exceptionally difficult. Cartels’ sales are less visible to rivals even than quantity or price choices of legal firms, which are themselves difficult to observe. For another, we are not aware of any examples of this type of collusion in the qualitative literature on illegal markets (Kenney Reference Kenney2007, 234–5).

Fighting for Trafficking Routes

In order to move drugs from producers to consumers, traffickers need to control smuggling routes. We model the conflict over smuggling routes as a repeated contest: in each period, routes are redivided in proportion to each cartel’s investment $$ \left({g}_i\right) $$ in firearms, salaries of gunmen, and related costs. This conflict expenditure $$ \left({g}_i\right) $$automatically generates violence according to a function $$ v\left({g}_1,\dots, {g}_n\right) $$ that (a) increases in the expenditure of every cartel and (b) is zero when all cartels have zero conflict expenditure. Because conflict expenditure is symmetric in equilibrium, we use conflict expenditure and violence interchangeably.

There is a continuum of drug-trafficking routes normalized to one: $$ {\sum}_{i\in I}{R}_i=1. $$Footnote ¹⁴ Each cartel’s share of routes is determined by a contest success function $$ R\left({g}_i,{G}_{-i}\right) $$ that depends both on own conflict expenditure and on the total amount $$ {G}_{-i}={\sum}_{j\ne i}{g}_j $$ spent by all other cartels:

$$ R\left({g}_i,{G}_{-i}\right)=\frac{g_i}{g_i+{G}_{-i}}. $$(1)

This function implies diminishing marginal returns to own conflict expenditure: as cartel $$ i $$ increases conflict expenditure, $$ {g}_i $$, its share of routes increases, but more slowly as $$ {g}_i $$ grows. This assumption is common in the literature (Fearon Reference Fearon2018; Skaperdas Reference Skaperdas1996). Moreover, it is well-motivated: the marginal return to the first firearm should be greater than the marginal return to the thousandth. A second feature of this contest success function is homogeneity of degree zero: if all cartels increase conflict expenditure proportionally, route shares stay the same. Throughout, we highlight how departures from this contest success function affect our results.

Three features of drug wars motivate our use of a repeated contest model. First, cartel turf boundaries change frequently—much more frequently than do international borders. Therefore, it would be inappropriate, in our view, to model turf war as, for example, a costly lottery in which the victor locks in her gains for all future periods (e.g., Fearon Reference Fearon2018; Powell Reference Powell1993, Reference Powell2006). Indeed, Powell (Reference Powell1993) notes that the notion of a permanent victor is unrealistic even in the context of interstate war (121); in our context, it would be yet more unrealistic. Our model allows turf boundaries to be redrawn in each period.

Second, conflict among cartels often involves intermediate outcomes: multiple cartels share the contested territory rather than one cartel winning it all. Our model naturally accommodates these divisions.

Third, our repeated contest model assumes that purchasing arms mechanically entails using them. This is better suited to our setting than the assumption that purchasing and using arms constitute separate decisions (e.g., Fearon Reference Fearon and Elhanan2018; Jackson and Morelli Reference Jackson and Morelli2009; Powell Reference Powell1993). For one thing, cartels’ arms purchases are less visible (to competitors) than states’ military expenditures; it is thus harder to make a case for arms as deterrence. For another, the notion of continuous variation in violence—that is, that competing traffickers could engage in low- or medium-intensity skirmishes, not just all-or-nothing war—constitutes a reasonable description of cartel relations (but might be hard to justify among states).Footnote ¹⁵

This choice also evades the problem posed by Jackson and Morelli (Reference Jackson and Morelli2009), who show that arms levels high enough to deter war are not stable: “given that war is costly, if one of the countries deviates and slightly lowers its arms level, then the countries will still not go to war and the deviating country will save some resources” (282). In other words, temporary spending cuts save money without drawing attack, meaning that there is no pure-strategy peaceful equilibrium.Footnote ¹⁶Fearon (Reference Fearon and Elhanan2018) provides one solution to this problem, arguing that arms may provide leverage in bargaining over international issues. In our setting, the problem does not arise: when purchasing arms mechanically entails using them, any cut in conflict expenditure immediately reduces that cartel’s share of routes.

Profit

Since cartel $$ i $$ sells a quantity $$ {q}_i $$ of drugs in the consumer market at a price $$ {p}_c $$, it ultimately obtains profit given by

$$ {\displaystyle \begin{array}{cc}{\uppi}_i={p}_cq\left({x}_i,R\left({g}_i,{G}_{-i}\right),e\right)-{g}_i-{p}_p{x}_i& \end{array}} $$(2)

While cartels make two decisions—the quantity of drugs to buy $$ \Big({x}_i $$) and how much to invest in conflict, $$ \left({g}_i\right) $$, only conflict expenditure $$ \left({g}_i\right) $$ affects strategic interaction among cartels. Since cartels are price takers, the quantity of drugs sold by cartel $$ i $$ does not affect its rivals.Footnote ¹⁷

Why Does Violence Increase with the Stakes?

For reasons described above, we characterize conflict among traffickers using a contest model—that is, a model in which profits are divided in proportion to cartels’ conflict expenditure. In a single-shot game, it is a well-known property of contest models that conflict increases with the stakes (Garfinkel and Skaperdas Reference Garfinkel, Skaperdas, Sandler and Hartley2007, 661). In a repeated contest model, conflict also increases with the stakes, for reasons we discuss below, but this result is less well-known (though see Fearon Reference Fearon and Elhanan2018, 532).

Other models of conflict produce the reverse outcome: that conflict abates as the pie grows, or equivalently, that hard times drive conflict. This can arise from lack of information. When firms collude on prices, for example, unobserved negative demand shocks can trigger price wars because firms wrongly suspect their coconspirators of undercutting a deal (Green and Porter Reference Green and Porter1984). In politics, similarly, asymmetrically observed economic shocks require a less-informed opposition to discipline a more-informed government by fighting whenever the government makes a low offer, which occurs when times are bad (Dal Bó and Powell Reference Dal Bó and Powell2009). Neither of these mechanisms strikes us as especially relevant to conflict among traffickers, who neither collude on price nor bargain over profits. Moreover, the information problems in these models arise from short-term shocks, whereas we analyze long-term shifts that stem from policy changes.

In another set of models, positive shocks to certain economic sectors increase wages, raising the opportunity cost of fighting and thereby decreasing conflict (Dal Bó and Dal Bó Reference Dal Bó and Bó2011). But the core logic actually works in the same direction as in our model: in Dal Bó and Dal Bó (Reference Dal Bó and Bó2011), conflict declines as the economy grows only because the size of the appropriable pie shrinks, in relative terms (relative to wages). Similarly, in Fearon (Reference Fearon and Elhanan2008), conflict abates as the economy grows because development reduces the fraction of resources that are appropriable.Footnote ²² Thus, in these models, a growing pie reduces violence only by lowering the stakes of conflict relative to the stakes of other activities. Similarly, in our model, shrinking profits reduce violence by lowering the stakes of conflict.

Stage Game

Recall from Lemma 1 that cartel $$ i $$’s one-period problem can be written as follows:

$$ {\displaystyle \begin{array}{cc}\underset{g_i}{\max }{\uppi}_i={\uppi}^A\mathrm{R}\left({g}_i,{G}_{-i}\right)-{g}_i& \end{array}} $$(5)

This resembles a problem faced by actors in other one-period models in which a prize is divided according to a contest success function (like $$ R $$), and the results are analogous: cartels’ investments $$ {g}_i $$ in the conflict rise with the stakes $$ {\uppi}^A $$ (Garfinkel and Skaperdas Reference Garfinkel, Skaperdas, Sandler and Hartley2007, 661).Footnote ²³

This, in turn, tells us how violence in the unique symmetric stage-game Nash equilibrium changes with interdiction. When demand is sufficiently inelastic (that is, when $$ {\epsilon}_c>{\hat{\epsilon}}_c $$), violence increases with interdiction; otherwise, violence declines with interdiction (Appendix A.6).Footnote ²⁴

The level of violence in the stage-game equilibrium, which we denote $$ {g}^N $$, is the level of violence we would expect in the repeated game in the absence of cooperation. It is thus the upper bound of the set of levels of violence that can be sustained in a repeated game.

Setup of the Repeated Game

Of course, cartels do not interact in a one-period setting. They interact repeatedly, which enables less violent solutions to the conflict over routes—just as repeated interaction enables cooperation among states (e.g., Fearon Reference Fearon1995, Reference Fearon2018; Powell Reference Powell1993, Reference Powell2006). A key difference between our model and previous literature lies in the role of the third party. Rather than serve as mediator or peacekeeper (Fey and Ramsay Reference Fey and Ramsay2010; Kydd Reference Kydd2003; Walter Reference Walter2002), the state shapes strategic interaction among traffickers by changing the size of the pie (profits), by changing traffickers’ time horizons, and, in some cases, by explicitly creating incentives for violence reduction.

Cartel $$ i $$’s total profits are the discounted sum of the profits obtained in each period:

$$ {\displaystyle \begin{array}{cc}{\Pi}_i=\sum_{t=0}^{\infty }{\beta}^t{\uppi}_{i,t}& \end{array}} $$(6)

where $$ {\uppi}_{i,t} $$ is the profit obtained by cartel $$ i $$  in period $$ t $$ and $$ \beta \in \left(0,1\right) $$ is the discount factor. This discount factor depends both on a monetary discount (related to the interest rate), which we call $$ \updelta $$, and the probability $$ p $$ that the cartel leader will still be in charge in the next period, such that $$ \beta =\updelta p $$.

The probability $$ p $$ of a cartel leader staying in power depends on the government: we assume that policies that are directed at capturing or killing capos decrease $$ p $$, thereby decreasing the value of the future for current bosses (i.e., making leaders more impatient). In other words, we assume that when a government begins aggressively targeting kingpins, remaining cartel leaders become more pessimistic about their own survival.

Punishment Strategies

The baseline equilibrium repeats the stage-game Nash equilibrium perpetually, with profit $$ {\Pi}^N={\uppi}^N\divslash \left(1-\beta \right) $$ and conflict expenditure $$ {g}^N $$ for each cartel; this is the equilibrium that arises in the absence of cooperation. The comparative statics are exactly as in the stage-game Nash equilibrium: interdiction increases violence if demand is sufficiently inelastic and decreases it otherwise.

But cartels can do better through cooperation. Repeated interaction allows cartels to sustain low-violence pacts by threatening to punish any cartel that deviates. Punishment yields profits $$ {\uppi}^p $$ for all subsequent periods.Footnote ²⁵

We consider two punishment strategies. With Nash reversion, cartels punish by moving to the stage-game Nash equilibrium, with $$ {\uppi}^p={\uppi}^N $$ in every subsequent period. With maximal constant punishment, cartels punish by moving to an even more violent equilibrium with even lower profits, $$ {\uppi}^p={\uppi}^m<{\uppi}^N $$. We define maximal constant punishment (or just maximal punishment, for short) as the harshest punishment possible within the set of subgame-perfect strategies in which nondeviating cartels punish the deviator by spending a constant conflict expenditure $$ \overset{\sim }{g} $$  for all subsequent periods; during the punishment phase, the deviator can choose its conflict expenditure freely. Our results also hold if punishment lasts only for $$ T $$ periods, after which cartels return to the low-violence agreement (Appendix A.11).Footnote ²⁶

Let $$ \overline{g} $$ be the level of conflict expenditure under the low-violence agreement; in a slight abuse of notation, $$ \overline{g} $$ also denotes the one-period strategy profile in which every cartel spends $$ \overline{g} $$. The agreement can be sustained if cartels prefer to honor the agreement instead of deviating for one period and then incurring punishment thereafter.

This implies one of two incentive constraints (IC1): $$ \frac{1}{\;1-\beta }{\uppi}_i\left(\overline{g}\right)\ge \underset{g_i}{\max}\kern0.3em {\uppi}_i\left({g}_i,{\overline{g}}_{-i}\right)+\frac{\beta }{1-\beta }{\uppi}^p $$, which simplifies to

$$ {\displaystyle \begin{array}{cc}{\uppi}_i\left(\overline{g}\right)\ \ge\ \left(1-\beta \right)\underset{g_i}{\max}\kern0.3em {\uppi}_i\left({g}_i,{\overline{g}}_{-i}\right)+\beta {\uppi}^p.& \end{array}} $$(7)

For maximal punishment, a second incentive constraint must be satisfied: it must be advantageous for cartels to actually punish the deviator. This requires that they, in turn, be punished if they renege on the punishment strategy.

Let $$ {\overset{\sim }{g}}^i $$ refer to a one-period strategy profile to punish cartel $$ i $$. All cartels except for $$ i $$  spend some quantity $$ {\overset{\sim }{g}}_{-i}^i=\overset{\sim }{g} $$, while $$ i $$ spends the quantity that maximizes its profits, $$ \tilde{g}_{i}^i={\mathrm{argmax}}_{g_i}{\uppi}_i\left({g}_i,{\overset{\sim }{g}}_{-i}^i\right) $$. A punishing cartel $$ j\ne i $$ thus sticks to the punishment strategy if the following incentive constraint (IC2) holds:

$$ {\displaystyle \begin{array}{cc}{\uppi}_j\left({\overset{\sim }{g}}^i\right)\ge \left(1-\beta \right)\underset{g_j}{\max\ }{\uppi}_j\left({g}_j,{\overset{\sim }{g}}_{-j}^i\right)+\beta {\uppi}_j\left({\overset{\sim }{g}}^j\right).& \end{array}} $$(8)

For cartel $$ j\ne i $$, punishing the deviator and obtaining profits $$ {\uppi}_j\left({\overset{\sim }{g}}^i\right) $$ must be preferable to deviating and getting profits $$ \underset{g_j}{\max}\;{\uppi}_j\left({g}_j,{\overset{\sim }{g}}_{-j}^i\right) $$ for one period, after which $$ j $$ is punished by all other cartels—including the original deviator $$ i $$—obtaining profits $$ {\uppi}_j\left({\overset{\sim }{g}}^j\right) $$ thereafter.

Maximal punishment is the maximum expenditure $$ \overset{\sim }{g} $$ such that IC2 holds, leading to punishment profits $$ {\uppi}^p={\uppi}_i\left({\overset{\sim }{g}}^i\right) $$ in the first incentive constraint.

Beheadings, Impatience, and Intercartel Violence

Figure 1b illustrates what happens when the shadow of the future shortens.Footnote ³² The (normalized) profits under complying with the agreement don’t change, but the normalized profits under deviating from the agreement increase, as cartels put more weight on the one-period spree in which they enjoy an outsized share of profits. In other words, more forward-looking cartels are more easily deterred by the threat of future punishment, which facilitates low-violence pacts:

Formally, this is a straightforward result, but it has an important substantive interpretation. If jailing or killing cartel leaders shortens other capos’ time horizons, this policy—while perhaps politically popular—exacerbates intercartel violence. This suggests a possible mechanism driving empirical findings like those of Calderón et al. (Reference Calderón, Robles, Díaz-Cayeros and Magaloni2015), who find that the Mexican government’s aggressive campaign to arrest and execute cartel leaders increased the homicide rate. If this “kingpin strategy” made cartel leaders more short-sighted, it also would have strengthened their temptation to break low-violence pacts.

Of course, targeting kingpins could drive violence through mechanisms other than shortening the shadow of the future. Calderón et al. (Reference Calderón, Robles, Díaz-Cayeros and Magaloni2015) mention four possible mechanisms: that removing kingpins could trigger wars of succession, that it could shift the offense-defense balance in favor of offense, that it could spur internal disciplinary violence, and that it could prompt cartels to attack the state. While there are perhaps too few cases to rigorously distinguish among these mechanisms (nor are they mutually exclusive), our story does entail different empirical implications. While wars of succession or internal disciplinary violence would likely generate conflict within the targeted cartel’s territory, our proposed mechanism would generate violence elsewhere as well. Similarly, while the other proposed mechanisms would explain intracartel violence, cartel–state violence, or violence against the beheaded cartel, our mechanism predicts intercartel violence more broadly. Moreover, our model suggests that the mere publication of the Mexican government’s list of most-wanted kingpins could work against intercartel pacts by affecting all cartel leaders’ expectations, while other possible mechanisms rely on the actual arrest or execution of cartel leaders.

Overall, our result on beheadings, impatience, and violence underscores the danger of kingpin strategies. Our model implies that this policy can spark violence not only locally (near the targeted cartel’s territory) but also globally—and not only in the wake of arrests or killings, but also in anticipation of them.

The Unintended Consequence of Interdiction

We now turn to the relationship between interdiction and violence. Under what conditions does interdiction intensify violent conflict among cartels? Under what conditions does interdiction mitigate violence? Under what conditions does it not matter one way or the other?

To answer these questions, note first that aggregate productive behavior (that is, cartels’ purchase and sale of drugs) in the repeated game is identical to that in the stage game. This means that, under an agreement with less violence than in the stage-game Nash equilibrium, interdiction still reduces supply—and the discount factor does not affect it (Appendix A.3).

Proposition 1 established that when demand is sufficiently inelastic, interdiction boosts traffickers’ productive profit. Figure 2a depicts how this increase in profits affects intercartel agreements: total profits under the agreement and total profits under deviation both increase, but profits under deviation increase more. The logic is straightforward. Under a low-violence agreement with conflict expenditure $$ {g}^{a,p} $$, complying and deviating entail equal profits but from different sources. Complying cartels control a small number of routes but also have low conflict expenditure; a deviating cartel controls more routes but has higher conflict expenditure. Therefore, interdiction that raises productive profit will confer larger benefits on the deviating cartel than the complying cartel because the deviating cartel controls more routes and thus a larger share of productive profits (see Appendix A.10 for additional discussion).

Because interdiction raises the benefits of deviating more than the benefits of sticking to the agreement, the original level of conflict expenditure $$ \overline{g} $$ will be insufficient to deter deviators. As a consequence, conflict expenditure—and thereby violence—must increase (The effect is reversed if demand is sufficiently elastic). We find the following:

FIGURE 2.

The Effect of Interdiction on Violence

Note: Panel (a) illustrates an unintended consequence of interdiction (or, more generally, a consequence of an increase in profits): higher productive profits raise the benefits of breaking an agreement more than the benefits of sticking to it, which increases the level of violence required for an agreement to hold. Panel (b) illustrates why an increase in profits does not necessarily break a peaceful agreement, when cartels are patient enough to share the market with no violence at all.

Proposition 5 says that violence follows productive profit. When demand is sufficiently inelastic, interdiction raises the stakes of the conflict (productive profit) and fuels violence.Footnote ^33, Footnote ³⁴

This result helps explain empirical patterns of violence in illegal markets. That alcohol prohibition in the United States fueled violence is firmly established both in the academic literature and in Hollywood films (e.g., Miron Reference Miron1999; Owens Reference Owens2011, Reference Owens2014). We also know that the violence stemmed largely from conflict over the illegal alcohol market (e.g., Miron Reference Miron1999; Owens Reference Owens2011, Reference Owens2014). Equally apparent is that gangs sometimes divided the alcohol business peacefully. Okrent (Reference Okrent2010) provides numerous examples both of gang treaties (“you take the north side, I’ll take the south”) and also of “escalating arms races” among competing criminal organizations (275).

What accounts for this variation? García-Jimeno (Reference García-Jimeno2016) collected data on the intensity of the enforcement of Prohibition, which varied both across cities and over time. In Okrent’s (Reference Okrent2010) simplification, local enforcement “took on one of two humors—either a vigor that outshone federal efforts or something close to torpor” (255). García-Jimeno (Reference García-Jimeno2016) estimates the elasticity of crime to prohibition enforcement, finding “that the Prohibition-related homicide rate was increasing with the level of law enforcement” (513).

Our model illuminates a possible mechanism: that enforcement reduced the supply of alcohol, driving prices up and increasing gangs’ incentives to fight rather than abide by treaties. As Okrent (Reference Okrent2010, 274) observed, “To secure a cash flow like [3.6 billion untaxed dollars], murder could seem like bookkeeping.”

Of course, there are other mechanisms through which interdiction could drive intertrafficker violence. For one thing, an intense interdiction campaign in one location could displace cartels, which could generate conflict (Dell Reference Dell2015). For another, a cartel might commit to deliver a certain quantity of drugs to consumer markets; if the government then seized part of that shipment, the cartel might attack a rival in desperation. But these alternative mechanisms entail empirical implications different from those implied by our model. Conflict arising from displacement or desperation would spark local and short-run violence in the wake of specific seizures, whereas our model implies a global increase in violence—cartels everywhere know that the new equilibrium means higher profits, which fuels conflict.

Castillo, Mejía, and Restrepo (Reference Castillo, Mejía and Restrepo2020) observe this outcome in Mexico. Around 2008, the US and Colombian governments moved from an ineffective strategy of coca crop eradication—which one writer compared to “trying to drive up the price of fine art by raising the cost of paint” (Wainwright Reference Wainwright2016)—to the more effective approach of drug interdiction (Mejía and Restrepo Reference Mejía and Restrepo2016). This policy change had an unintended consequence: higher prices and higher profits for the Mexican drug cartels that bring Colombian cocaine to consumers. Simultaneously, violence in Mexico doubled. Castillo, Mejía, and Restrepo (Reference Castillo, Mejía and Restrepo2020) estimate that a 1% decrease in the supply of cocaine (because of interdiction) drove a 0.12% to 0.16% increase in homicide rates in the Mexican municipalities most exposed to drug trafficking. Our model provides a possible explanation for this result: negative supply shocks increased the stakes of conflict and thus violence among cartels.

Government efforts to reduce supply through interdiction have often been criticized for being ineffective. Our model provides another cause for concern: if they are effective, they may spur violence in illegal markets.

The Durability of the First-best Outcome

Proposition 5 raises an empirical question: If interdiction (or more generally, profits) fuel intercartel violence, why do we observe periods of high profits and yet minimal violence in illegal markets? And why does this peace sometimes appear immune to changes in interdiction?

This section provides an answer. When cartels are patient enough that they can share the market with no violence at all, interdiction does not necessarily break the peace. Figure 2b visualizes the logic. Even if interdiction boosts total profits, shifting both curves outward from the origin, patient-enough cartels can still share the market peacefully. In other words, interdiction can narrow the gap between the profits from deviating and the profits from complying, without entirely closing that gap.Footnote ³⁵ As long as the discount factor remains above the threshold in Equation 9, changes in interdiction will not break intercartel peace.Footnote ³⁶

This may explain how a peaceful agreement among Colombian cartels survived a surge in profits in the late 1970s and early 1980s (Lessing Reference Lessing2018, 129). It may also explain how, for many years, Mexican cartels ran a massive cocaine trafficking operation with few turf-war homicides.Footnote ³⁷In Appendix E, using the same research design as Castillo, Mejía, and Restrepo (Reference Castillo, Mejía and Restrepo2020), we show that interdiction did not drive violence among Mexican cartels in the pre-2006 period. Why did interdiction generate violence in Mexico after 2006 but not before? Our model suggests an explanation. Prior to the Mexican government’s “decapitation strategy,” cartel leaders were patient enough to abide by a peaceful agreement. The peaceful agreement could survive fluctuations in profit, but targeting kingpins shortened capos’ time horizons. The peaceful equilibrium broke down, and in the absence of peace, the logic of our model prevailed. Profits, and thus violence, rose with interdiction.

The fact that profits only affect violence when traffickers cannot sustain peace underpins the danger of jailing or killing high-profile traffickers. The risk of being captured or killed makes capos shortsighted, which makes peace harder to sustain. But once a peaceful equilibrium is infeasible, the level of violence begins to respond to profits. Therefore, targeting kingpins and ramping up interdiction—which often go hand-in-hand as part of a crackdown—constitute a fatal combination: taking out capos breaks a peaceful equilibrium, and then rising profits fuel violence.

Indiscriminate Conditionality

Consider first a policy in which the state sets the overall level of interdiction in response to cartels’ behavior. In an attempt to keep violence below some level $$ \overline{v} $$, the state increases interdiction from $$ e $$ to $$ \overset{\widetilde }{e} $$ whenever total violence rises above $$ \overline{v} $$. We call this indiscriminate conditional interdiction because the policy conditions interdiction on overall violence, but it does not discriminate among cartels: as in the baseline model, the policy treats all cartels equally.

This policy might appear to encourage cartels to abide by low-violence agreements. If breaking the agreement would provoke the government to step up interdiction, cartels might be more likely to stick with the deal. We show in Appendix A.12 that this logic is incorrect. When demand is sufficiently inelastic, more interdiction actually boosts cartel profits.Footnote ³⁸ Paradoxically, then, indiscriminate conditionality makes deviation more attractive: for a cartel considering breaking out of a low-violence pact, the government choosing $$ \overset{\widetilde }{e}>e $$ looks like a reward—not a punishment.

Of course, this implies that the government could reduce violence by decreasing interdiction in response to conflict: that is, by setting $$ \overset{\widetilde }{e}<e $$ whenever violence rises above $$ \overline{v} $$. Such a policy strikes us as implausible. First, the government would have to intentionally and explicitly abandon the goal of supply reduction. Second, a promise to ease up on interdiction whenever cartels start fighting would be a tough sell to the public.

One might think that more arrests or killings of cartel leaders in response to violence—conditional beheading—would unambiguously facilitate low-violence pacts. Under indiscriminate conditional beheading, the government would more aggressively target kingpins whenever a low-violence agreement broke down, thus reducing cartels’ discount factor $$ \beta $$ in the (off-path) post-deviation period. Unlike conditional interdiction, which could potentially fuel violence through boosting profits, conditional beheading does not affect aggregate supply one way or the other.

We nevertheless find that indiscriminate conditional beheading can entail its own competing forces. While indiscriminate conditional beheading indeed reduces violence in equilibrium when cartels use Nash reversion as a punishment strategy, the consequences are ambiguous under maximal punishment (Appendix A.12). On one hand, the returns to deviating decline because deviation induces more arrests and killings; this facilitates pacts (just as under Nash reversion). On the other hand, additional beheadings in the (off-equilibrium path) postdeviation setting make cartels more impatient, hampering their ability to enforce harsh punishments. This works against low-violence agreements.

Targeted Conditionality

Indiscriminate conditionality is a blunt tool. What if the state instead targeted the cartel that deviates from a low-violence pact?

Consider a policy in which the state sets interdiction at the default level $$ e $$ if all cartels abide by the low-violence agreement. Whenever some cartel deviates from the agreement, the state sets a higher level of interdiction $$ \overline{e} $$ for that cartel while keeping the default level $$ e $$ for every other cartel. We call this targeted conditional interdiction.

Unlike indiscriminate conditional interdiction, targeted conditional interdiction exerts opposing forces on low-violence pacts. On one hand, raising $$ \overline{\mathrm{e}} $$ (interdiction against the punished cartel) can reduce overall supply, raising profits (if demand is sufficiently elastic) and making deviation more tempting. On the other hand, raising $$ \overline{e} $$ reduces the punished cartel’s “productivity.” A smaller fraction of the punished cartel’s drug purchases makes it to consumers. Under targeted conditionality, the punished cartel’s routes lose value. This discourages deviation.

Under what conditions will the latter force dominate? In other words, when will targeted conditionality facilitate low-violence pacts? Targeted conditionality is more difficult to analyze than are the policies studied thus far. The principal complication is that Lemma 1 no longer holds: cartels no longer split aggregate profits according to their shares of smuggling routes. Because the targeted cartel (the one subject to higher interdiction $$ \overline{e} $$) now uses its routes less efficiently than other cartels do, its share of profits will be smaller than its share of routes. As a result, routes are divided asymmetrically: routes shift away from the targeted cartel toward other cartels. This, in turn, complicates analysis of the relationship between interdiction and aggregate supply reaching consumers (and thus prices) because cartels no longer purchase the same quantities of drugs per route.

To make the analysis tractable, we impose an additional assumption on the production function $$ q $$. Rather than allowing $$ q\left(x,R,e\right) $$ to depend freely on $$ e $$, we assume that interdiction affects production according to $$ q\left(x,R,e\right)=\overset{\sim }{q}\left[x,\uptheta (e)R\right] $$, where $$ \uptheta (e) $$ is a decreasing function.Footnote ³⁹ This restriction rules out the possibility that the same government investment in interdiction has different consequences for cartels that control different numbers of smuggling routes.

Under this restriction, we show in Appendix A.12 that targeted conditional interdiction will facilitate low-violence pacts—and thus reduce the equilibrium level of violence—unless demand is extremely inelastic. Specifically,

Result 1. Let $$ \overline{s} $$denote the share of supply provided by the targeted (punished) cartel, and let $$ S $$denote the ratio of aggregate productive profit to aggregate revenue ( $$ S={\uppi}^A\divslash {p}_cQ $$). A necessary condition for violence to increase with $$ \overline{e} $$ is

$$ {\displaystyle \begin{array}{cc}{\upepsilon}_c>-\frac{\overline{s}}{S}& \end{array}} $$(10)

We prove in Appendix A.12 that Result 1 holds under Nash reversion, and we verify numerically that it holds under maximal punishment.

In order for targeted conditionality to backfire—in other words, in order for targeted conditionality to hamper low-violence pacts—demand would have to be more inelastic than $$ -\overline{s}\divslash S $$. Consider some plausible magnitudes for this threshold. In drug markets, $$ S $$ is typically close to one because the cost of drug purchases is small relative to the revenue from drug sales (i.e., cartels’ main cost is smuggling) (Reuter Reference Reuter and Vellinga2004). And $$ \overline{s} $$, the share of supply provided by the targeted cartel, will certainly be less than $$ 1\divslash n $$, where $$ n $$ is the number of cartels in the market. For a market with six cartels, then, $$ {\upepsilon}_c $$ would have to exceed $$ \approx -1\divslash 6 $$ in order for targeted conditionality to backfire. Even the most pessimistic estimates of the price elasticity of demand in cocaine markets do not exceed this threshold (Gallet Reference Gallet2014). This means that targeted conditional interdiction is very likely to facilitate low-violence pacts among cartels.Footnote ⁴⁰

Similarly, targeted conditional beheading evades the problems of indiscriminate conditional beheading. Consider a policy in which the government steps up beheadings only against the cartel that breaks an agreement, thereby reducing that cartel’s discount factor to a level $$ \overset{\check{}}{\beta } $$ after deviation (but leaving other cartels’ patience unaltered). Kleiman (Reference Kleiman2011) argued that this would “condition the traffickers’ ability to remain in business on their willingness to conduct their affairs in a relatively nonviolent fashion” (101). In Appendix A.12, we show that this policy also facilitates cooperation among traffickers. Targeted conditional beheading unambiguously discourages deviation, lowering the level of violence in equilibrium (under the agreement) and also reducing the discount rate required to maintain peace.

These findings provide a possible mechanism behind recent empirical results about the use of conditional repression in Latin America. In Mexico, Trejo and Ley (Reference Trejo and Ley2018) find that state-government agents served as a “third-party enforcer” for intercartel agreements (915). Gubernatorial political alternation disrupted this enforcement, requiring cartels to “develop their own private militias to protect their drug trafficking routes.” In Brazil, the government of Rio de Janeiro rolled out a security policy that was explicitly directed at violence reduction. The city’s security secretary repeatedly emphasized that the goal of the new Pacification program was not to eliminate drug trafficking but rather to retake territory from armed gangs and “bring peace to the residents” (Lessing Reference Lessing2018, 195). Magaloni, Franco-Vivanco, and Melo (Reference Magaloni, Franco-Vivanco and Melo2020) find that this policy also reduced intergang conflict—but only in areas where multiple gangs contested turf (36). Our model suggests an explanation. By threatening to arrest or kill only those gang leaders who broke low-violence pacts, the Pacification program facilitated gangs’ efforts to share the retail drug market peacefully.

Table 1 summarizes our results for the six policies we consider. When cartels are forward-looking enough that they are able to sustain a peaceful agreement (that is, share the market with no violence), that peace is surprisingly resilient to changes in profits, including those changes induced by interdiction. But when cartels are too impatient to sustain peace, the traditional policy tools of prohibition enforcement—interdiction and the pursuit of high-profile traffickers—are counterproductive in that they fuel intercartel violence. The same problem plagues naively conditional policies in which the government simply cracks down on all cartels in response to rising violence. Only by targeting interdiction and/or beheadings against the cartel that breaks a low-violence agreement can the government facilitate the reduction of violence through intercartel cooperation.

Table 1.

The Effect of Policy on Intercartel Turf War

Note: This table summarizes the policies we analyze and their consequences for intercartel violence. Under the “indiscriminate conditional” policies, the government steps up interdiction and/or beheadings against all cartels whenever violence rises. Under the “targeted conditional” policies, the government steps up enforcement only against the deviator. If the condition on demand elasticity is not met, the policy has the opposite effect on violence.