Thirteen mathematical theorems about the war on drugs = - Las matemáticas de la guerra contra las drogas

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(1)Thirteen Mathematical Theorems about. The War on Drugs. Pascual Restrepo Mesa Universidad de los Andes DEPARTAMENTO DE MATEMÁTICAS. Bogotá, Colombia Febrero 15, 2011. 1.

(2) 1. Introduction. Interventions in drug markets tend to occur at the national or global levels. Hence, they are likely to affect prices, which in turn feed back into markets. It is common that governments design policies ignoring these feedback effects, or general equilibrium effects. For instance, Plan Colombia was designed to halve the Colombian supply of cocaine by eradicating 50% of the land with coca crops. However, in practice, things have worked quite differently: while the land with coca crops fell almost to a half -from 160.000 hectares to 80.000 hectares between 2000 and 2008- domestic production only fell by 12%. This paradoxical outcome can be understood once general equilibrium effects are taken into account: farm gate prices rose from US$1.500 to about US$2.200 per kilo in the same period, because supply shifted inwards, and producers responded by increasing their productivity, from 4.3 kilos per hectare a year to about 7 kilos per hectare a year. Basically, this occured because supply is upward slopping, and because higher prices provide incentives for producing more cocaine per hectare of land. These feedback effects are also absent in the political discussions around the subject. In particular, many politicians and commentators have argued that increasing the interdiction rate would decrease domestic production, as trafficking becomes less profitable. However, imagine that the interdiction rate increases from 15% to 20%. The inmediate effect would be a reduction of 5% in wholesale supply outside Colombia, but this would increase wholesale prices by more than 5% if demand is inelastic. The increase in the price of drugs outside of the producer country, relative to cocaine in the producer country, would encourage traffickers to send more, not less drugs, partially counteracting the improvements in interdiction. Traffickers would end up demanding more domestic drugs to send abroad, even if a higher fraction gets interdicted, because the increase in wholesale prices more than compensates their losses. Thus, the interdiction policy could end up increasing domestic production, contrary to standard predictions (see Mejı́a and Restrepo (2008) for more on these effects or, as they call them, “Strategic Responses”). Finally, localized small experiments can be used to estimate the impact of prevention and treatment policies in drugs demand. But this is a partial equilibrium estimate, in the sense that these interventions do not affect prices. However, if such experiments were to be taken to a national scale, where they are likely to affect prices, results would be different: on the one hand, the inward shift in demand would lower prices, increasing consumption and offsetting the effects of the policies, but on the other hand, producers and traffickers would. 2.

(3) also react to the lower prices by supplying less drugs and reinforcing the effects of the policy. Both effects change the equilibrium results of prevention and treatment, and hence should be considered when designing and evaluating these policies. In short, the objective of this paper is to understand the effect of the endogenous change in prices on drug markets outcomes. In the second section, I will deduce formulas showing how to calculate the endogenous effect on prices, and the implied effect on quantities of anti-drug policies. Then, I will explore the determinants of the size of this effect. In particular, I will show that the equilibrium effect in quantities is smaller when demand is inelastic; the agents involved in the illegal trade are more homogeneous; there are possibilities for innovation in the industry; and there are bigger returns to scale. The first two determinants require a small demand elasticity in order to reduce the effect on quantities. Otherwise, the sign of their effect would be reversed. Finally, I show how this formula can be used to calculate the marginal cost of reducing retail quantities by targeting an input required for drug production, trafficking and distribution. In the third section I explore the reactions of people and firms involved ind rug production and trafficking, to the price change in more detail. In particular, I will show that many puzzling facts about drug markets can be understood as reactions to the endogenous increase in relative prices. For instance, if demand is inelastic, improving interdiction efforts motivates traffickers to ship more drugs, even if less are successfully send abroad; and eradication is counteracted by farmers by cultivating coca bushes more extensively or by demanding more technological capital goods, increasing the productivity of land. Finally, I will show that this reactions are present even if one observe downward trends in market prices, but have to be understood as deviations from the original variable trends. In the fourth section I investigate the relation between prices and violence. More precisely, I show in a series of models that market size increases the violence associated with the conflict for rents in drug markets. I will argue that prohibition does not allow well defined property rights over these rents, or destroys cheaper non-violent alternatives that could be used instead of violence, generating violence in illegal markets. Moreover, I wills show that if demand is inelastic, the endogenous increase in prices raise these rents, and hence violence. I will end this section by presenting some preliminary empirical evidence, suggesting a causal relation from coca cultivation’s value and homicides in Colombia. My estimates suggest that a 10% increase in the value of coca cultivation in a municipality, increases the homicide rate by 1.2-2%. The last section summarizes my findings. I will present my conclusions in each section as theorems, but this does not mean they 3.

(4) are strict rules about the way the world behaves. They are called theorems because they can be deduced analytically following a sequence of logical steps starting from some assumptions. But they are not intended to provide strong and precise rules, but to give some insight about the forces operating in drug markets, and that one should keep in mind when designing or evaluating anti-drug policies.. 2. Equilibrium Multipliers. In essence, feedback effects, strategic responses or general equilibrium effects, as I have called them, occur because both supply and demand depend on prices. Thus, to quantify them in the most general setting, I will start with a generic and reduced-form supply function Qs (P, S), in which P is the price and S is a measure of supply reduction programs. The two basic assumptions about this general supply function are ∂Qs ∂Qs > 0 and < 0, ∂P ∂S. (1). which simply mean that supply is upward sloping, and that supply reduction programs shift supply inwards. This simple framework contains all cases in which supply is not horizontal. In this case, supply reductions must be modeled as an upward shift in the constant production cost. However, this special case can be treated with similar tools to the ones developed here, and in fact it has been widely explored and analyzed (see Becker et al. (2006) for a very complete analysis). The results in this section can be presented in an alternative fashion, using the inverse supply curve. In this case, supply reduction policies are interpreted as increasing the marginal production cost for any given quantity, or simply shifting supply upwards. Although both formulations are equivalent in most cases, I think it is more informative to think of supply reduction policies as reducing the quantity supplied for any given price, as long as supply is not horizontal. Although this is a matter of taste, the current formulation allows one to interpret the direct effect as a partial equilibrium effect with fixed prices, or the effect one would get from localized interventions that dont affect prices. The equilibrium multipliers are the formulas telling us how this effect changes once one takes into account the endogenous change in prices. On the demand side, I will start with a generic and reduced-form demand function d. Q (P, D), in which P is the price and D measures demand reduction programs. The two. 4.

(5) basic assumptions about this general demand function are ∂Qd ∂Qd < 0 and < 0, ∂P ∂D. (2). which simply mean that demand is downward sloping and that demand reduction policies shift demand inwards. In this case, none of the assumptions are controversial. d s From now on, let ε > 0 be the demand elasticity, ε > 0 the supply elasticity, and dy x εyx = dx , the elasticity of y with respect to x. All this elasticities will be taken in absolute y. value as long as the sign of the effect is clear. For instance, I will assume that the effect of prices on demand is negative and that the effect of prices on supply is positive. The s. d. Q elasticities εQ S , εD define the partial equilibrium effects of policies on supply and demand,. and would be the expected effects on quantities, if the policies did not affect prices. Let P ∗ (S, D) and Q∗ (S, D) be the equilibrium prices and quantities given implicitly by the market clearing condition (see the implicit function theorem in the appendix for details on this methodology, and also Spivak (1971).): Q∗ (S, D) = Qs (P ∗ (S, D), S) = Qd (P ∗ (S, D), D).. (3). To understand the effects of the endogenous change in prices, it is necessary to differentiate the market clearance condition with respect to S. Using the chain rule and the implicit function theorem, we get: ∂Qs ∂P ∗ ∂Qs + ∂P ∂S ∂S ∂Qs ∂P ∗ ∂Qd ∂P ∗ − ∂P ∂S ∂P ∂S ∂P ∗ ∂Qs ∂Qd − ∂S ∂P ∂P ∗ ∗ ∂P Q εd + εs ∗ ∂S P. ∂Qd ∂P ∗ ∂P ∂S ∂Qs = − ∂S ∂Qs = − ∂S ∂Qs = − . ∂S =. (4) (5) (6) (7) (8). Thus, we get the formula s. εQ ∂Qs P∗ ∂P ∗ P∗ =− → εS = d S s ∂S ∂S Q∗ (εd + εs ) (ε + ε ). (9). Using the elasticity of demand definition, we also get that s. d ∂Qs εd εQ ∂Q∗ Q∗ S ε → ε = = S ∂S ∂S εd + εs (εd + εs ). 5. (10).

(6) These identities are summarized in the following theorem:. Theorem 1: If supply reduction programs shift supply inwards by. ∂Qs ∂S. for some given prices, the general. equilibrium effect in prices and quantities is given by: ∂Q∗ ∂Qs εd = ∂S ∂S εd + εs. ∂P ∗ ∂Qs P∗ =− . ∂S ∂S Q∗ (εd + εs ). and. (11). Equivalently, these formulas are given in terms of elasticities by s. ∗. εQ S =. s. εSQ εd (εd + εs ). ∗. and εPS =. εQ S , (εd + εs ). (12). s. with εQ S the elasticity of supply with respect to supply reduction at fixed prices. The sign of the effects is as expected: supply reduction reduces quantities and increases prices.. . In an analogous way, one can calculate the effects of demand reduction programs, which are summarized in the following proposition:. Theorem 2: If demand reduction programs shift demand inwards by. ∂Qd ∂D. for some given prices. Then,. the general equilibrium effect in prices and quantities is given by: ∂Q∗ ∂Qd εs = ∂D ∂D εd + εs. ∂P ∗ ∂Qd P∗ = . ∂D ∂D Q∗ (εd + εs ). and. (13). Equivalently, these formulas are given in terms of elasticities by d. ∗ εQ D. εs εQ = dD s (ε + ε ). d. and. ∗ εPD. εQ = dD s (ε + ε ). (14). s. with εQ D the elasticity of demand with respect to demand reduction at fixed prices. Once again, the sign of the effects is as expected: demand reduction lowers quantities and prices. These formulas can be better understood using a diagram. Panel (a) of Figure 1 shows a relative inelastic demand, together with a relative elastic supply. If supply is shifted to the left, then the effect on quantities is modest (panel (a)). But if demand is shifted to the left by the same amount, its effect is bigger (panel (b)). Panel (a) of Figure 2 shows a relative elastic demand together with a relative inelastic supply. If supply is shifted to the left, then the effects on quantities is significant (panel (a)). But if demand is shifted to the left by the same amount, its effects are modest (panel (b)). 6.

(7) Figure 1: Elastic supply and inelastic demand. Figure 2: Inelastic supply and elastic demand. In fact, from the formulas in Theorems 1 and 2, we get that the elasticity of supply, relative to the elasticity of demand, determines the general equilibrium effects of both supply and reduction policies. Defining x = εs /εd , we get that as x increases, the general equilibrium effect on quantities, of supply reduction policies becomes smaller, relative to the effect of demand reduction policies, although both become smaller. Thus, a bigger supply elasticity relative to the demand elasticity, implies that taking the partial equilibrium effects as given, supply reduction programs are less effective, relative to demand reduction programs. For instance, suppose that two groups of researchers evaluate small drug market interventions. The first group evaluates a demand reduction program that finds out that it costs 50.000 dollars to reduce the consumption of users by 1 kg in using treatment and prevention programs in some area. While the second group finds that it costs 40.000 dollars to reduce. 7.

(8) the supply of drugs by 1 kg using supply reduction programs in other place. If both studies focused on small interventions not affecting prices (or with small negligible effects), and the government is planning to undertake both interventions to a bigger scale, it must consider the general equilibrium effects of both policies and not simply their partial equilibrium effect, which is the one reported by the researchers. If, for instance, x = 2, so that supply is elastic relative to demand, the marginal costs of reducing quantities by 1kg would be 120.000 for supply reduction (multiply 40.000 by the inverse of the equilibrium multiplier 1/3), and 75.000 for demand reduction policies (multiply 50.000 by the inverse of the equilibrium multiplier 2/3). Thus, demand reduction programs would be more cost-effective once the effect of prices is taken into account. As shown by these formulas, the general equilibrium effect of supply reduction policies depends on the elasticity of demand, εd , the elasticity of supply, εs , and the elasticity of s. enforcement, εQ S . Thus, it becomes crucial to understand the determinants of the equilibrium ∗. ∗. P multipliers εQ S and εS . The elasticity of demand has been widely studied and mentioned. in various works (see, for instance Grossman et al. (1998) and Grossman et al. (2002) for thorough reviews of the undertaken studies). Thus, I will focus on the elasticity of supply and enforcement. Unfortunately, as will be clear in the following sections, the analysis of both elasticities cannot be separated, because both are theoretically related. However, this relation can also be quantified, and some determinants of the equilibrium multipliers can be exposed. One could do the same analysis for the demand reduction equilibrium multipliers, but for the purposes of this work, I will limit myself to the supply reduction ones.. 2.1. Determinants of the equilibrium multipliers ∗. In the previous section I have shown that a small equilibrium multiplier, εQ S , makes supply reduction policies ineffective. The determinants of this multiplier include the supply elasticity, the demand elasticity and the elasticity of enforcement. Thus, as a first attempt to understand the size of these multipliers, I will study the determinants of the elasticity of supply, and how these determinants also affect the elasticity of enforcement. A classical determinant of the supply elasticity are the returns to scale exhibited by the production function, or equivalently, the convexity of the cost function. However, I will propose alternative determinants of the elasticity of supply and enforcement, which could be operating in drug markets, even with CRS. A source of supply elasticity, and thus a determinant of the equilibrium multipliers, is. 8.

(9) the heterogeneity among the actors engaged in the trade of illegal goods. These markets require very specific abilities and assets that are scarce, and this implies that even if each firm or producer faces CRS, but has a fixed capacity, supply will be elastic because prices would affect entry or exit of new firms or agents. Even in the long run, heterogeneity implies upward sloping supplies. The first subsection explores the consequences of heterogeneity on the equilibrium multipliers. Innovations are also likely to affect supply elasticity, and hence equilibrium multipliers. This occurs because a higher price implies bigger potential profits for firms supplying intermediate goods, encouraging innovation. The recent news regarding the use of submarines, or genetically modified varieties of coca bush suggest that this is more than a theoretical curiosity, and may be a relevant issue for drug markets. The second subsection studies the impact of innovation on the equilibrium multipliers. Finally, the last determinant is related to the fact that many of the intermediate inputs required for drug production, trafficking or even distribution, cannot be purchased in markets because prohibition does not allow a good definition of their property rights. This could be the case for the land used in drug production, the routes used in trafficking, or the territories used for distribution. All these factors must be appropriated through conflict against the state and other producers, and enforcement must be targeted towards gaining the effective control of these inputs. These missing markets create a source of decreasing returns to scale, affecting the elasticity of supply. Also, directing enforcement at inputs affects the elasticity of enforcement, implying different effects on the equilibrium multipliers. The last subsection will examine the consequences of introducing enforcement as a conflict for intermediate goods which cannot be purchased in markets, but must be appropriated via conflict, on the equilibrium multipliers. In all of these cases, the effects of the analyzed factor affect not only the supply elasticity, but also the enforcement elasticity. For instance, the enforcement elasticity increases when there are more heterogeneous marginal individuals, which are dissuaded by enforcement; enforcement may also reduce innovation in intermediate products, amplifying its effect; and finally, enforcement has different effects when modeled as a conflict for one of the inputs required for drug production. However, the equilibrium multipliers formula allows me to focus on the net effect of these factors on the equilibrium multiplier and not simply on their effect on the elasticity of supply. On the other hand, the elasticity of demand is easier to handle, since its determinants are not related to supply elasticity, neither to enforcement elasticity. Thus, in the analysis that follows I will assume an exogenous fixed value for this 9.

(10) parameter.. 2.2. Heterogeneous agents. Another determinant of supply elasticity and the equilibrium multipliers, even in the long run, is the heterogeneity of firms, or agents. In this case, even if each agent is able to supply inelastically certain supply, after aggregation one gets an upward sloping supply. To introduce this idea consider a mass M of individuals endowed with one unit of labor (or any other input) which they supply inelastically. Individuals must allocate their labor between trafficking and production, on the one hand, and a legal activity on the other hand. The legal activity gives them a fixed wage of w per hour worked. Trafficking and production gives them a return qθp, per unit of time. Here, q is the probability of not getting caught, or an inverse measure of enforcement, p is the price of the drugs successfully produced and trafficked, θ is the individual productivity or ability in the drug market and x is the time allocated to this activity. One could introduce heterogeneity in many dimensions, but in this case I will do it by assuming that θ is drawn from a distribution f , with support [0, ∞). Given the constant returns for both uses of time, individuals will allocate all of their time to the more productive sector. Thus, there will be a marginal individual whose opportunity cost of engaging in the drug business (his wage) is exactly equal to the expected benefits (qθp). Thus, individuals engaged in drug trafficking and production would be those with θ > involved in legal activities would be those with θ ≤. w . qp. w , qp. while individuals. To simplify notation, let θ =. w . qp. It follows that the supply of drugs is given by ∞ Qs = q θf (θ)M dθ,. (15). w qp. and the price elasticity of supply with respect to prices, p, is given by (using the fundamental theorem of calculus and the chain rule, see Spivak (1971) and the appendix for details) ∂Qs q 2 = s θ f (θ)M > 0. ∂p Q. (16). Thus, heterogeneity implies an upward slopping supply, even if each agent has CRS technologies, or produces at its minimal operating price. Supply would be more inelastic the more heterogeneous its agents are (when there is a small mass f (θ)M of agents around the indifference threshold), and more elastic the more homogeneous its agents are (when there is a big mass f (θ)M of agents around the indifference threshold). Thus, from the viewpoint of 10.

(11) supply reduction policies, equilibrium multipliers are bigger for markets with diverse agents s. and smaller when agents are very similar, leaving the enforcement elasticity, εQ q , fixed. However, heterogeneity also affects the enforcement elasticity, because enforcement also has an extensive margin effect (get people in or out of the market). In particular, we have ∂Qs P ∂Qs q = 1 + . ∂q Qs ∂P Qs. (17). Therefore, supply reduction is more effective precisely when supply is more elastic, because there are more individuals at the margin, which supply reduction can take out of the market. Thus, the role of heterogeneity is not clear, because when it creates an elastic supply, it also implies more effective supply reduction policies. To see how both forces weight against each other, we can calculate the resulting equilibrium multiplier in this model, given by ∗. Q εQ q = εq. q 2 θ f (θ)M εs ) Qs εd +. εd q 2 θ f (θ)M Qs. .. (18). In order to see if it is still more ineffective to target markets with less heterogeneity and more elastic supplies, we must calculate the derivative of the equilibrium effect with respect to a measure of the homogeneity f (θ). Doing so gives ∗. ∂εQ q. ∂f (θ). =. q 2 εd θ M (εd − 1). Qs (εd + εs )2. (19). Thus, if demand is inelastic (εd − 1 < 0), markets with more homogeneous agents (bigger f (θ)), are harder to disrupt because they have a smaller equilibrium multiplier. Otherwise, if demand is elastic, these markets are easier to disrupt.. Theorem 3: If agents involved in drug production and trafficking are heterogeneous, and can produce a fixed quantity of drugs at a fixed cost (or opportunity cost), the equilibrium multiplier is given by ∗. Q εQ q = εq. q 2 θ f (θ)M εs ) Qs εd +. εd q 2 θ f (θ)M Qs. .. (20). In this case, homogeneity increases the supply elasticity, but it also increases the enforcement elasticity. When demand is inelastic, the former effect dominates the latter, and it becomes harder to disrupt homogeneous markets. On the other hand, if demand is elastic, the higher elasticity of enforcement dominates the higher supply elasticity and disrupting homogeneous . markets becomes easier. 11.

(12) The intuition of this theorem, is that with an inelastic demand, enforcement increases entry in equilibrium, because prices rise more than the fall in q, making the risk adjusted price, qP , increase. If the market is very homogeneous, there is a big mass of new entrants, whose entry counteracts the initial effect of the policy, implying a smaller equilibrium multiplier. On the other hand, with an elastic demand, enforcement increases exit, because q falls more than the increase in prices, making the risk adjusted price, qP , fall. If the market is very homogeneous, there is a big mass of agents indifferent between illegal production and legal work. These agents would exit the market, reinforcing the effect of the policy, and implying a bigger equilibrium multiplier.. 2.3. Technical change. Another factor that increases the elasticity of supply, the elasticity of enforcement, and hence determines the equilibrium multipliers, is technical change. The basic idea is that innovations in intermediate products, used in drug production and trafficking, also respond to final prices. This occurs because producers and traffickers raise their demand for hightech intermediate goods when prices increase in final-goods markets. This increases potential profits for successful R&D firms, fostering innovation and shifting supply outwards. In order to formalize this idea consider one aggregate drug producer and trafficker, with a Cobb-Douglas technology Qt = q(LAt )1−α xαt . The Cobb-Douglas technology is used to simplify the algebra, but the argument in this section holds more generally. Here, Qt is the quantity of drugs produced, L is a fixed factor of production, 1 − q is the enforcement rate, and xt is an intermediate good with a L-augmenting technology embodied on it, At . Following Aghion and Howitt (1997), suppose the intermediate good is produced each period by a monopolist, at a unitary cost and sold at a price of rt . The inverse demand for the input by producers and traffickers, is given by rt = αqPt. LAt xt. 1−α ,. in which Pt are the drug prices. Thus, the monopolist benefits are given by 1−α LAt πt = αqPt xt − xt . xt. (21). (22). The monopolist sets an optimal quantity of the intermediate good equal to 1. x∗t = (α2 qPt ) 1−α qLAt , 12. (23).

(13) in order to maximize profits, which will be given by 1+α. 1. πt∗ = (1 − α)α 1−α LAt (qPt ) 1−α .. (24). Finally, the equilibrium supply is given by α. 1. Qs = (α2 Pt ) 1−α q 1−α LAt .. (25). Without innovations, and assuming that At is constant, the supply elasticity would be given by. α . 1−α. However, the price also increases the incentives to innovate by increasing. πt∗ , and therefore, probably appears with a positive sign in At , making supply even more elastic. To show this explicitly, consider a world with two periods, t, and t + 1. In t ∗ potential monopolists expect a profit πt+1 if they are able to innovate and create a technology. At+1 = γAt , with γ > 1. Let the probability of innovating be given by φt+1 = φ(Rt ), with φ and increasing and concave function of the resources invested in research, Rt . Thus, expected ∗ − Rt . The concavity of φ implies that optimal profits for innovators are given by φ(Rt )πt+1. investments in innovation are given by ∗ φ (Rt )πt+1 = 1.. (26). ∗ , increase innovation expenAgain, the concavity of φ implies that expected profits, πt+1. ditures, and hence the expected technology level in t + 1, calculated using the law of big numbers as At+1 ≈ (1 + (γ − 1)φt+1 )At .. (27). ∗ , to the technology level At+1 , implying Thus, there is a positive elasticity ≥ 0 of profits, πt+1. that a 1% increase in potential profits, increases the technology level by %. This elasticity is an exogenous parameter measuring the extent of innovation. It approaches zero when there are no possibilities to innovate, or γ ≈ 1, and becomes bigger when it is possibile to innovate. ∗ As can be seen from the profits formula in equation 24, πt+1 increases with the expected. price Pt+1 . Moreover, the elasticity of potential profits to future prices is given by. 1 . 1−α. This. elasticity increases with α, because it implies that intermediate goods are relatively more important for production, and hence, a larger fraction of the price increase would be used in them. Thus, the elasticity of supply is given by 1 α + . 1−α 1−α. (28). From this formula, it follows immediately that the extent of innovation increases supply elasticity. One would be tempted to conclude, that the possibility to innovate implies smaller 13.

(14) equilibrium multipliers, because they depend negatively on the elasticity of supply. However, enforcement also has an effect on innovation, implying that we must also consider the effect on the enforcement elasticity. To see this, notice that the elasticity of enforcement is given by 1 1 + 1−α 1−α. (29). The first term captures the direct effect of enforcement on Qs , holding At+1 constant, that is, without innovations. The second term captures the fact that enforcement reduces expected monopolists profits with an elasticity of 1, reducing innovation by . To see how both effects weight against each other in equilibrium, let’s compute the equilibrium multiplier for this market, given by (1 + )εd . εd (1 − α) + α + . (30). Thus, we get the following theorem:. Theorem 4: As can be seen from the equilibrium multiplier formula in equation 30, the possibility to innovate, measured by a bigger elasticity , makes supply reduction policies less effective if demand elasticity is smaller than. 1 . 1−α. In particular, this is always the case if demand is. inelastic. The intuition is that innovation increases with more enforcement, counteracting the effect of supply reduction, because the rise in prices more than compensates innovators and producers. On the other hand, if demand is very elastic, incentives to innovate fall with more enforcement, and prices do not increase enough to compensate innovators. In this case, the resulting lack of innovation reinforces the supply reduction policy. Thus, markets with small demand elasticity and a greater room for innovation have smaller equilibrium multipliers. Consequently, introducing innovation generates an additional determinant of the equilibrium multipliers. The formula in equation 30, implies that markets in which producers and traffickers can use land or routes, together with other intermediate inputs with a big participation in the total product (α big), and with significant possibilities for innovation ( big), are harder to disrupt with supply reduction policies, always that the elasticity of demand is smaller than. 1 . 1−α. In the extreme case when → ∞ (for instance, in the very very long run),. the equilibrium multiplier tends to εd . Finally, notice that in this case, it is also true that s. s the supply elasticity, and the enforcement elasticity are related by the formula εQ q = 1+ε .. 14.

(15) 2.4. Missing markets. Suppose one of the inputs required for dug production and trafficking is not traded in a competitive market, but it has to be appropriated via conflict, because there are no well defined property rights. The canonical example is land in Colombia, which is necessary for producing coca, but is not usually traded in markets, specially in remote areas where coca bushes are grown. Thus, one does not observe land market dynamics, but illegal producers fighting for the control of this land, against each other and against the government. This is equivalent to assuming production exhibits CRS and requires several inputs: land, which must be appropriated via conflict from a total supply of L, and another input, x, traded in a competitive market at a price w. I am referring here to the input in dispute as land, but it could equally be the trafficking routes, or the distribution city corners, among others. Let q be the share of L under the effective control of the producer, and assume production is given by Q = f (qL)g(x). This assumption implies that the elasticity of substitution between land, and the other input is 1. This greatly simplifies the analysis. The profit maximization conditions, holding q constant, implies that P f (qL). ∂g = w → εgx P Q = xw ∂x. (31). With, εgx the degree of homogeneity of g in x by Euler’s theorem (see the appendix, or Escobar (2005)). Thus, If g is homogeneous of degree s < 1, producers spend a fraction s of their product on the market inputs, x. In order to calculate the supply elasticity, we differentiate the above expression with respect to prices, assuming that ∆s ≈ 0. That is, approximating g by a Taylor expansion of the first order. By doing so, we get sQ + sP. ∂Q ∂x ∂x ≈ w. ∂x ∂P ∂P. (32). Rewriting this in terms of elasticities, and isolating εxp , we get εxp ≈ 1/(1 − s), which implies an elasticity of supply of. s . (33) 1−s Thus, the elasticity of supply, holding q constant, increases with the participation of inputs εs ≈. purchased in well functioning markets, s. The intuition behind this result is that producers or traffickers cannot increase their production by using more of the appropriable input, and must rely heavily on those goods purchased in markets. The increase in production depends on the relative importance of this good in their technology, captured by s. Thus, when enforcement disrupts the market for an input and it cannot longer by purchased in markets, it generates 15.

(16) an extra source of decreasing returns to scale, which is bigger when the input participation in total product is bigger (s is smaller). In general, firms can increase production, holding q constant, as long as land and other inputs are not perfect complements. In the general case, a smaller elasticity of substitution would imply a less elastic supply because land cannot be easily replaced by toher inputs. On the other hand, the enforcement elasticity can be calculated as s. f s εQ q = εq (1 + ε ) = 1.. (34). Here I have used εfq = 1 − s, implied by CRS, and the fact that an increase in q increases g which has a direct effect (captured by the 1 term) and an indirect effect through the choice of inputs, equal to the effect of prices (captured by the term εs ). Thus, the equilibrium multiplier can be approximated as ∗. εQ q =. εd (1 − s) . s + εd (1 − s). (35). Theorem 5: Assume drug production and trafficking exhibit CRS and requires an input L which is not traded in a competitive market. If producers control a fraction q of this input, the equilibrium effect of a reduction in q is given by equation 35. This equation implies that supply reduction policies, in markets where the participation of inputs purchased in well functioning input markets is bigger (s is big), are less effective. This occurs because these markets have a more elastic supply, since producers can increase production by using market inputs.. . The basic idea from this theorem is that supply is more elastic when the participation of inputs purchased in well functioning markets is big, or equivalently, when enforcement is directed at less relatively important inputs. This insight is very helpful because the value for the share s can be calculated for the current quantities and prices as the share of the product used in market inputs. The equilibrium multipliers imply that supply reduction policies have larger effects when directed at markets with a small participation of inputs purchased in markets, or equivalently, with a large participation of the inputs being targeted. The previous theorem and arguments, only consider the effectiveness of such policies, but do not consider the costs of them. If producers invest in avoiding enforcement (increasing q), then the cost of reducing q would depend on the effect of q on profits, which is smaller when s is big. Therefore, enforcement is less effective, but also less costly when s increases. However, it is easy to see that if demand is very inelastic, the effectiveness effect dominates, and 16.

(17) disrupting markets with a smaller participation of market inputs becomes a more attractive option. The idea in this theorem is used by Mejı́a and Restrepo (2008) who modeled the war on drugs as disrupting two different stages of drug production: an upstream market (drug trafficking) and a downstream market (drug production in Colombia). In each market, the actors involved use two inputs, one obtained in a competitive market, and the other obtained via conflict (modeled with a contest success function) against the government for a fixed supply. For instance, drug trafficking requires routes (obtained via conflict) and domestic drugs (obtained in a competitive market), while drug production requires land (obtained via conflict, which seems like a realistic assumption for Colombia, where land property rights are hardly defined) and complementary factors (obtained in a competitive market). The authors approximate the supply elasticity of each market as. s , 1−s. where s is the participation of the. inputs purchased in competitive markets at each stage, and find a trafficking supply elasticity of about 0.48, and a domestic supply elasticity of about 1.5. This difference in elasticities is at the heart of Mejı́a and Restrepo conclusion, suggesting that inward supply shifts in the trafficking market (namely, interdiction) are more effective than inward supply shifts in the production market (namely, eradication), because the equilibrium multiplier of the former is bigger than the latter. There is another feature of this theorem worth mentioning. When the government tries to disrupt this market by investing in the conflict for L, and manages to reduce q (the fraction controlled by producers or traffickers), supply shifts inwards and prices rise. Suppliers must react to the higher prices by increasing their use of inputs purchased in markets, precisely because supply is upward slopping and this is the only way to increase production. As a consequence, the productivity of the appropriable input must increase (and this is true as long as there is some degree of complementarity among inputs). Mejı́a and Restrepo (2008) call this responses in the choice of inputs “Strategic responses”. In the production front, these strategic responses help to explain the sharp increase in land productivity between 2000 and 2008, while in the trafficking front, these strategic responses are associated with an increase in the demand for domestic drugs. The argument is that higher interdiction rates are counteracted by traffickers by sending more drugs when demand is inelastic (that is, purchasing more of the input obtained in markets). The last point can be formalized mathematically. Using the equilibrium multipliers, we get that the general equilibrium elasticity of quantities with respect to enforcement results (measured by 1 − q), equals. εd , εd +εs. and this implies that the productivity, per unit of land, 17.

(18) qL, must increase by. εs , εd +εs. offsetting the initial effects of the policy. For drug production,. using εs ≈ 1.5 (s is about 0.6), and εd = 0.5 (similar to the value used in Mejı́a and Restrepo (2010)), we get that a fall of 63% in the eradication rate should lead to an increase in land productivity of about 44%, which is the average documented increase in land productivity between 2000 and 2008 (from 4.26 to 6.66).. 2.5. Consequences for anti-drug policies. The insights and formulas presented above can be used to calculate important measures that policy makers should keep in mind when designing anti-drug policies. The first measure is the marginal monetary cost of reducing quantities traded in illegal markets by 1kg. In order to calculate this, in the most general framework, suppose drug production and trafficking targets an intermediate good, Q, used in the production of retail illegal drugs in consumer countries, Qr . For instance, Q could be coca bush, drugs at transit countries, wholesale drugs in the U.S. or any intermediate form of drugs during the whole production and trafficking chain. Let P be the price of Q and Pr the retail price of drugs. Let εd be the elasticity of demand for the input Q, and εs its supply elasticity. Assume there is an aggregate production and trafficking network (the P&T network from now on), with a CRS technology Qr = T (Q, y), which uses other inputs, y, and the intermediate good Q to produce retail drugs. Given CRS, the P&T network produces at a constant marginal cost, equal to the retail price, and given by Pr = min Q1 P + y1 s.t:F (Q1 , y1 ) = 1. Q1 ,y1. (36). For simplicity, I have normalized the prices of other inputs to 1. Using the envelope theorem (see the appendix or Escobar (2005) for a complete proof), it follows that ∂Pr P PQ ∂Pr = = s(P ). = Q1 → ∂P ∂P Pr Pr Qr. (37). Thus, the elasticity of final prices with respect to the intermediate good price, equals its participation in total product. This result has led many to argue that interventions in consumer countries are more effective than interventions in source or transit countries, because distribution has a bigger participation in total product than trafficking or production. This is a direct consequence of the so called “additive model”. However, this insight ignores the costs of enforcement, which might be lower for markets with small participations, in which total income is smaller and hence, the stakes for the agents involved lower. 18.

(19) The share function, s(P ), satisfies the following properties: If the input can be easily substituted, then s < 0. Otherwise s > 0. When the marginal rate of substitution is 1 (as s (P )P s(P ). be the. elasticity of s with respect to the intermediate good relative price. We have s(P ) =. Q1 P , y1 +Q1 P. in the Cobb Douglas case), s = 0, and s is constant. To see this, let κ = . and therefore κ = (1 − s(P )). ∂ ln x2 /x1 − 1 = (1 − s(P ))(1 − σ), ∂P. (38). with σ the elasticity of substitution between the intermediate good, Q, and other inputs, y. The above observations follow easily from this formula (see Bronfenbrenner (1960) for more on this matter). Now, suppose the intermediate industry has a production technology qF (x), with x a vector of inputs, F a concave technology of production, and 1 − q measuring enforcement. Let x(P, q) be the unconditional demand for inputs (ignore the input prices in the arguments), and Q(P, q) = qF (x(P, q)) be the supply function. It follows that the adjusted price, qP , determines the input’s demand as long as firms are risk neutral and hence Q εQ q = 1 + εP .. (39). Thus, the elasticity of Q with respect to enforcement, q, is bigger than the supply elasticity of Q because enforcement affects both the extensive (less inputs are used) and intensive margin (less supply for a given choice of inputs). The extensive margin effect is of the same size, both for prices and enforcement, as long as firms are risk neutral. Setting the enforcement rate at 1 − q is costly because firms in the intermediate industry try to avoid it. In order to incorporate this argument, let q = c(i/e), with c a conflict technology that depends positively on the ratio i/e, where i are the resources used by firms in this industry to avoid enforcement and, e are the resources used by the government in enforcing prohibition. The basic assumption here is that the enforcement rate depends on the relative ratio of resources invested by the parties engaged in conflict. This is a natural assumption to start with, but at the end of the section I will discuss the consequences of relaxing it. However, the homogeneity of c has a very appealing consequence: it guarantees that the government must outbid intermediate firms investing in enforcement if it wants to reduce q. Thus, this will allow me to calculate the cost of setting an enforcement rate, 1 − q, by computing the firms’ optimal expenditures in avoiding it. In the more general case this can also be done implicitly, and the cost would be an increasing function of the firms’ expenditures 19.

(20) in avoiding enforcement. The advantage with the form q = c(i/e), is that the cost function will be proportional to such investments, which greatly simplifies the analysis. To see this, notice that intermediate firms optimal choice of anti-enforcement expenditures, i, satisfies i=. 1i c P Q. qe. (40). Let c−1 be the inverse function of c, defined using the implicit function theorem, explained in the appendix or in Spivak (1971). Then, if the government wants to set an enforcement level of q it costs him C(q, P ) = Let f (q) =. c (c−1 (q)) . q. c (c−1 (q)) i = P Q(P, q). c−1 (q) q. (41). This function measures the percent change of q, when the ratio. i/e changes. Thus, this function is positive and decreasing in q. Equivalently f (q) is the slope of ln c(q) evaluated at q. Any reasonable conflict technology satisfies the following limit conditions: lim c(r) = 0 and lim c(r) = 1,. r→0. r→∞. (42). therefore, assume ln c(q) is a concave function such that lim ln c(r) = −∞ and lim ln c(r) = 0.. r→0. r→∞. (43). Generic forms for both functions are shown in the following figures (see Figure 3). Figure 3: Conflict technologies. The above cost expression shows that setting the enforcement rate at 1 − q, costs the government f (q)s(P )Pr Qr . Thus, this expression shows one of the points I mentioned above: 20.

(21) it is true that directing enforcement to inputs with bigger participations is more effective at increasing final prices, but doing so is more costly. Thus, the government must balance both forces when deciding the enforcement level of each input market. This result depends crucially on the assumption q = c(i/e), which guarantees that the intermediate good market size, s(P )Pr Qr , appears with a constant elasticity equal to 1. Before proceeding to calculate the marginal cost, let’s gain some intuition on how these calculations are going to work. Imagine the government wants to reduce q by a small amount. On the market size, this would shift the intermediate good supply inwards, as shown in figure 4, panel A. The demand for the input, leaving the retail market size constant, is given by ) , which has an elasticity of κ(p) − 1 = (1 − s(P ))(1 − σ) − 1 < 0, and is downward Pr Qr s(P P. slopping. This would increase intermediate prices, P , which in turn increases retail prices Pr . Therefore, retail supply shifts upwards (panel B), increasing retail prices and decreasing retail quantities. Depending on the elasticity of demand at the retail level, market size would increase, or decrease, which will shift the intermediate good demand to the left or to the right. The locus of the equilibrium points P, Q determines a “endogenous” demand curve which incorporates the change in the retail market size, implied by a change in intermediate prices. The resulting demand is plotted in figure 4, panel A. The downward slopping lines plotted with dots, are the demands that do not incorporate retail price movements. In this case, if retail demand is inelastic, the resulting demand is more inelastic than the dotted one, because the increase in market size counteracts the fall in the demand implied by the relative price of the input. Figure 4: Effects on intermediate and retail markets.. (A). (B). 21.

(22) At the end, supply and demand elasticity determine the increase in intermediate prices using the equilibrium multipliers formula for this market. In turn, the intermediate good price increase determines the change in retail prices. Finally, the elasticity of demand, at the retail level, determines the fall in quantities implied by the reduction in q. On the cost side, reducing q affects cost in two ways. First, the market size of the input changes and hence the total anti-enforcement expenditures by firms operating on it (the term P Q in C(q, P ) = f (q)P Q). Second, holding the market size constant, the government must invest more on enforcement, relative to the firms expenditures in avoiding it, in order to reduce q (the term f (q) in C(q, P ) = f (q)P Q). Both effects determine the increase (or decrease in some cases) in the cost of reducing q. The marginal cost of reducing retail quantities by 1kg can be calculated as the ratio between the increase in costs and the fall in retail quantities implied by a small reduction in q. In order to formally calculate these terms, lets compute the change in retail quantities, Q∗r , when q falls by ∆q ≈ 0. This is given by: ∂Qr ∂Pr ∂P ∗ ∂Q∗r = − ∂q ∂Pr ∂P ∂q Pr εQ P Qr q = b s(P ) s d Pr P ε +ε q εQ Qr q = bs(P ) s . d ε +ε q Here, I have used the chain rule in the first line, the formula together with the equilibrium multipliers formula (to find elasticities. This formula implies that the elasticity of ∗. r εQ q = bs(P ). Q∗r. εQ q . s ε + εd. ∂y ∂x. ∂P ∗ ), ∂q. = εyx xy in the second line and the previously found. with respect to q is (44). This formula corresponds to the equilibrium multipliers for inputs, and it allows me to calculate the change in final quantities produced by enforcement directed at any of its inputs. In this formula, we can assume that enforcement targets the whole input market (if it targeted s an input, we could simply apply it for the input!). Therefore, I will assume εQ q = 1 + ε in. what follows. Again, the term s(P ) shows that targeting inputs with bigger participations in total product is more effective at reducing retail quantities, but it may also be more costly. The elasticity of retail demand b reduces the effectiveness of reductions in q because quantities do not fall by much as prices increase. Finally, a more elastic demand for the intermediate good 22.

(23) and a more elastic supply, imply a smaller equilibrium multiplier, which implies that the increase in intermediate prices will be modest. The term εQ q measures the impact of reducing q in intermediate supply taking prices as given (the partial equilibrium effect of reducing q). On the cost side, the marginal cost of reducing q by ∆q ≈ 0, is given by −. ∂P Q ∂C(q, P ) = −f (q) − f (q)P Q ∂q ∂q ∂P Q ∂P ∗ = −f (q) − f (q)s(P )Pr Qr ∂P ∂q ∂P ∗ = −f (q)Q(1 − εd ) − f (q)s(P )Pr Qr . ∂q. The first term shows the fact that, for a fixed enforcement level q, the cost of enforcing it increases with the market size P Q as explained above. Using the identity. ∂P Q ∂P. = Q(1 − εd ),. we get that this effect is positive if the input demand is inelastic, because the market size increases with enforcement, 1 − q, which increases equilibrium prices, P ∗ . The second term captures the fact that, for a fixed market size, the government must invest more resources in enforcement, relative to the expenditures of firms in avoiding it, if it wants to reduce q. The magnitude of this effect depends on the input market size s(P )Pr Qr , which is the size of the prize for firms engaged in conflict against the government, and the concavity in the conflict technology f (q). This effect is always positive because ln c(q) is concave, and hence, f (q) is decreasing in q. To save space, let. −qf (q) f (q). = θ(q) > 0, be a function measuring the curvature. of the conflict technology. Calculating the quotient of both expressions, we get εs + εd 1 d . M C = f (q)Pr 1 − ε + θ(q) b 1 + εs. (45). However, to complete the analysis one needs to understand the parameter εd , because it is related to other fundamentals. As the graphical analysis presented above suggests, this elasticity depends on s(P ), retail market demand, and the elasticity of substitution. In order to calculate it, let’s begin with the formula P Q = Pr Qr σ → ln P + ln Q = ln Pr + ln Qr + ln s(P ).. (46). If ln P increases by ∆ ≈ 0, ln Pr must increase by s(P )∆. Input quantities ln Q fall by εd ∆, and retail quantities fall by bs(P )∆. Finally, ln s(P ) changes by. s (P )P ∆ s(P ). = (1−s(P ))(1−σ)∆.. Thus, ∆ − εd ∆ = s(P )∆ − bs(P )∆ + (1 − s(P ))(1 − σ)∆. 23. (47).

(24) Isolating εd , we get the formula: εd = 1 − (1 − s(P ))(1 − σ) − s(P )(1 − b) = s(P )b + (1 − s(P ))σ,. (48). which is a famous formula, introduced by John Hicks, in his 1963 book “The Theory of Wages”.1 Replacing this result in the marginal cost expression, we get the formula εs + s(P )b + (1 − s(P ))σ 1 . M C = f (q)Pr 1 − s(P )b − (1 − s(P ))σ + θ(q) b εQ q. (49). The analysis of this formula is done in the following theorem. In this analysis, I will refer to ∆f (q)P Q the term ∆Q∗ as the effect on budget through market size. That is, the increase r f (q)=f (q). (or decrease) in the cost of keeping fixed the enforcement rate q, as the intermediate good market expands (or contracts), depending on the intermediate good elasticity of demand. (q)P Q as the effect through enforcement. On the other hand, I will refer to the term ∆f∆Q ∗ r. P Q=P Q. That is, the increase in costs necessary to reduce Qr by 1kg, by improving enforcement efforts, 1 − q, and holding the market size constant. This effect is the quotient between the cost of reducing q and the effect of reducing it on retail quantities, leaving P Q constant. From the above derivation of the marginal cost expression, it is clear that 1 ∆f (q)P Q = f (q)Pr (1 − s(P )b − (1 − s(P ))σ) ∆Q∗r f (q)=f (q) b and. 1 ∆f (q)P Q εs + s(P )b + (1 − s(P ))σ = f (q)θ(q)P . r ∆Q∗r P Q=P Q b εQ q. (50). (51). Theorem 6: The marginal cost of reducing retail quantities by 1kg, by increasing enforcement in an intermediate market, can be calculated with equation 49. According to this formula, the marginal cost satisfies the following properties : 1. The M C is bigger when the retail market is more inelastic, that is, b is small. The elasticity of demand has several channels: First, it makes the intermediate good market demand more inelastic, increasing the market size effect implied by a price increase. Second, the increase in retail and input prices has a smaller effect on quantities, making 1. In this calculation, I assumed other inputs are supplied at a fixed price. If the supply elasticity for these. inputs is e, the above formula can be modified to εd =. 24. σ(b+e)+s(P )e(b−σ) b+e−s(P )(b−σ) ,. as also shown by Hicks (1970)..

(25) the effect of reducing q on retail quantities, Qr , smaller. Finally, a more inelastic demand for the intermediate good implies a sharper increase in input prices, when supply shifts inwards, but this effect is dominated by the smaller effect of this price increase in quantities. Thus, the enforcement effect is also smaller. 2. The M C increases with retail prices Pr , which determine the size of the prize available for the firms engaged in the illegal trade. Price appears linearly, both in the market size and enforcement effect. 3. The M C increases with f (q) and θ(q), which measure the curvature of the conflict technology. In particular, if c(r) is more convex, the M C increases, because the intermediate firms have a marginal advantage when defending against enforcement. On the contrary, if c(r) is more concave, the M C decreases. 4. The intermediate good elasticity of demand has an ambiguous effect. On the one hand, a bigger elasticity implies that the market size effect is smaller. In fact, if this demand is elastic, market size actually falls with more enforcement, making these policies less costly. On the other hand, a more elastic demand for the input, leaving market demand constant, implies that the increase in input prices is modest, and hence the effect on retail quantities and prices. The elasticity of demand is determined by Hick’s formula, and it depends positively on the elasticity of substitution, σ. 5. The participation of the intermediate good being targeted has an ambiguous effect on the marginal cost. Moreover, this parameter only has an effect through the intermediate good elasticity of demand. The fact that directing enforcement against markets with bigger participation has a more significant effect on final prices (the so called additive model), is not important, because these bigger markets are also more costly to disrupt, and both effects counteract each other when enforcement results depend on the ratio i/e. 6. The inverse of the equilibrium multiplier for intermediate goods,. εs +εd , εQ q. increases the. marginal cost. This occurs because a smaller multiplier implies a modest increase in input prices. This, in turn, implies a smaller increase in retail prices, and a smaller fall in retail quantities. By theorems 3,4 and 5, as long as demand elasticity is small; the input market is more homogeneous; there is more room for innovation; and the input technology exhibits higher returns to scale, the marginal cost of reducing quantities by . targeting this input increases.. 25.

(26) Let’s see this formula in action with some examples. Suppose one is interested in determining the M C of reducing retail quantities in consumer countries with a source country intervention, such as Plan Colombia. The following table (see Table 1), shows these calculations assuming that q is given by a standard contest success function (this implies f (q) = and θ(q) =. 1+q ). 1−q. (1−q)2 , q. Moreover, this calculation assumes q ≈ 0.8 × 0.7 (20% of shipments are. seized by Colombian authorities and 30% of crops are eradicated). This value for q implies an expenditure level of 1.1 billions a year by the U.S. and Colombia, which is very similar to the expenditure figures reported by GAO and DNP (see Mejı́a and Restrepo (2008), DNP (2006) and GAO (2008)). Also, I assume εs = 1. Mejı́a and Restrepo (2008) reports an elasticity of production of 1.5 and trafficking in Colombia of 0.48, so I used a simple average. Finally, I show different values for the MC, assuming different demand elasticities, b, and different elasticities of substitution, σ. Table 1: Marginal costs estimations 1 Required data Retail price, Pr. $ 150,000. Retail quantities, Qr. 300000. Intermediate price, P. $ 8,000. Intermediate quantities, Q. 400000. Input participation, s(P ). 7.11%. Enforcement rate, q. 0.56. σ. b. 0.5. 1. 1.5. 0.25. $ 652,436. $ 726,880. $ 801,324. 0.5. $ 327,643. $ 364,865. $ 402,087. 0.75. $ 219,378. $ 244,193. $ 269,008. As shown in this table, the marginal cost decreases with the demand elasticity b. This is a classical point, first noticed by Becker et al. (2006). Importantly, this table also shows that the marginal cost increases with the elasticity of substitution (in this case, the enforcement effect dominates the market size effect of this variable). This has a interesting interpretation: Imagine that the cocaine supplied by Colombia has close substitutes (Cocaine from Perú, or Bolivia, for instance). Supply reduction in Colombia shifts demand for the input by P&T 26.

(27) networks to other source countries, implying that the increase in Colombian prices is small, and hence, the increase in retail prices would be even lower. Thus, in this case, the marginal cost of reducing retail quantities by targeting one input, increases when this input can be easily substituted. Let’s compare these estimates with others. Mejı́a and Restrepo (2008), find a marginal cost of about $13,000, for reducing Colombian supply using eradication and interdiction efforts. This marginal cost refers to a reduction of Q, not Qr , as the ones in the previous tables. To make it comparable, let εd be the elasticity of demand for Colombian cocaine at international transit markets. Then, reducing Colombian supply by 1% is equivalent to increasing prices by. 1 %. εd. This is equivalent to an increase in retail prices of. 1 σ(P )%, εd. and. a decrease in retail quantities of εbd σ(P )%. Thus, a reduction of Colombian supply by 1kg implies a reduction in retail quantities of εbd σ(P ) QQr = εbd PPr . Therefore, this estimate can be d adjusted as $13, 000 × εb PPr ≈ $316, 875, using b = 0.5, and εd = 0.65 (as assumed byMejı́a and Restrepo (2008)). This is very similar to the second row estimates in table 1. Let’s show another application of the formula by estimating the marginal cost of reducing retail quantities by 1kg , by enforcement against distribution in the U.S.. Assume q ≈ 0.6, which implies a total expenditure of 8 billion dollars per year in domestic enforcement. The participation of domestic distribution networks in total product can be calculated as s(P ) =. Pr Qr −Pw Qw Pr Qr. ≈ 0.67, with w indexing wholesale prices and quantities. Using a contest. success function for c(i/e), and assuming εs = 1 as before, it is possible to construct the following table, which shows estimates for this marginal cost (see table 2). The implications of this table are similar to the ones mentioned above: a more inelastic demand increases the marginal cost of reducing retail quantities, and the possibility to substitute the input also increases it. There are several possible extensions. For instance, one could include anti-enforcement expenditures as part of the cost, because these resources are a dead weight loss for society, and cannot be used in other productive alternatives. Mathematically, this would increase f (q) and θ(q) and would imply much larger estimates for the marginal cost of reducing retail quantities. In other applications, the elasticity of q can be normalized to 1, by assuming that the intermediate industry has CRS, and enforcement is directed against one of its inputs. In this case, the elasticity of supply, holding enforcement constant, could be approximated by εs =. s , 1−s. like in theorem 3. However, this approach also requires f (q) to be adjusted, because. firms would invest less in avoiding enforcement when it targets one input, compared to when 27.

(28) Table 2: Marginal costs estimations 2 Required data Retail price, Pr. $ 150,000. Retail quantities, Qr. 300000. Wholesale price, Pw. $ 30,000. Wholesale quantities, Qw. 500000. Input participation, s(P ). 66.67%. Enforcement rate, q. 0.6. σ. b. 0.5. 1. 1.5. 0.25. $ 533,333. $ 560,000. $ 586,667. 0.5. $ 280,000. $ 293,333. $ 306,667. 0.75. $ 195,556. $ 204,444. $ 213,333. it targets the whole product. This is the approach used in Mejı́a and Restrepo (2008), and the estimates are similar to mine. Related to this is the fact that we have included enforcement in a multiplicative structure, which implies that the elasticity of substitution between q and other inputs is constant and equal to 1. This implied that 1 + εs = εQ q , a restriction that could be relaxed by introducing q as another input in the intermediate good production technology. In this case, the relationship between the elasticities εs and εQ q is more flexible, but the function f (q) becomes more complicated. The only interesting feature from this extension is that the M C becomes more responsive to the elasticity of supply when q and other inputs are gross substitutes in the production of the intermediate good. The intuition is that in this case, the elasticity of supply with respect to q falls relative to the price elasticity of supply. Further extensions could explore cost functions of the form c (c−1 (q))c−1 (q) C(q, P ) = g 1 − q, P Q(P, q) , q. (52). with g an increasing function and 1 − q capturing the cost of increasing enforcement, independently of the strategic reaction by firms involved in the intermediate market, and c (c−1 (q))c−1 (q) P Q(P, q) q. the costs arising from this reaction. In this case, the participation of. the input in the total production could have a direct effect on the marginal cost, but the 28.

(29) direction of this effect would be ambiguous and depends on the elasticity of C with respect to market size P Q. Again, the usual implications of the additive model are far from obvious. In the derivation of the above theorem, I incidentally proved a formula for the equilibrium multipliers of inputs. The formula can be rewritten as ∗. r εQ q =. b 1 + εs Q∗ s(P )ε = bs(P ) , q εd εs + s(P )b + (1 − s(P ))σ. (53). ∗. Therefore, the input equilibrium multiplier, εQ q , determines the retail good equilibrium mulQ∗. tiplier, εq r . This formula shows that, if the input demand is inelastic, the market equilibrium multiplier falls with all the determinants of the input equilibrium multiplier. Namely, the homogeneity among producers, the possibilities to innovate, and the returns to scale in proQ∗. duction, as shown in theorems 3,4, and 5. Moreover, εq r increases with the participation of the input (as in the additive model), and decreases with the elasticity of substitution σ, so the inputs that can be easily substituted, have smaller equilibrium multipliers. Market demand reduces the equilibrium multiplier because a more inelastic demand implies smaller effects on quantities. Using this equilibrium multiplier, and any cost function one can also estimate marginal costs in the way I did it above.. 3. It is all about the Prices. In the previous section, I introduced the equilibrium multipliers formulas, and used them to calculate estimates for the marginal cost of reducing drug production. In this section, I will use them in order to prove some interesting theorems about supply reduction policies. The equilibrium multipliers essentially suggest that firms, producers or traffickers, react to prices. Since prices change in equilibrium so do people’s reactions. In this section, I will explore the “form” such reactions can take, and show that many puzzling facts about drug markets can in fact be understood as manifestations of these reactions. Let’s start with the following puzzling stylized facts. The first one is shown in figure 5. Panel (A), shows cocaine production in Colombia before eradication (right axis, in blue), that is, current potential production plus the potential production of eradicated coca crops, plotted against the interdiction rate in Colombia (left axis, in red). As the figure shows, during the last decade interdiction efforts have intensified, increasing the interdiction rate from 5% to about 24%. However, despite these efforts, traffickers have reacted by simply shipping abroad more cocaine, as the increase in potential production suggests. I have added the eradicated crops to the potential production because Colombia also underwent strong 29.

(30) eradication campaigns during this period that reduced net domestic production and could confound the interpretation of this figure. Figure 5: Stylized fact one.. (A). (B). A similar example, with a clearer interpretation, comes from Peru and Bolivia. These source countries also increased their domestic production when the interdiction rate in transit countries increased from 30% to 45% during the last decade. Panel (B) in figure 5, shows Peru’s and Bolivia’s potential production (right axis, in blue), plotted against the interdiction rate in transit countries (left axis, in red). The figure suggests that traffickers counteracted the higher interdiction rate of the last years by demanding more domestic drugs from these source countries. This interpretation is clearer from this figure, since Perú and Bolivia did not implement strong eradication campaigns during this time, and since the interdiction rate in transit countries is exogenous to the source countries production. These figures suggest that, seizing a bigger fraction of the drugs coming from one source country, tends to increase its domestic production. In fact, a simple calculation shows that, while the interdiction rate increased by about 50% for Peruvian and Bolivian cocaine, production increased by 45%, leaving a net reduction of 5% in successfully trafficked cocaine. This is the first stylized fact, which I will explain in the section “Trafficking and Conventional Wisdom”. The intuition behind it is that the traffickers respond to the policy change by sending more drugs, because the increase of prices outside the source country, relative to the prices inside the country, compensates them for the extra losses. The second stylized fact is shown in figure 6. Panel (A) shows the total land with coca cultivations before eradication (right axis, in blue), plotted against the eradication rate (left 30.

(31) axis, in red). The figure suggests that, contrary to what one would expect, eradication does not have a dissuasive effect on coca growers. On the contrary, growers respond to eradication by growing more crops, counteracting part of the policy. This stylized fact was noticed before by Moreno-Sanchez et al. (2003). The authors provide some time series evidence suggesting that eradication is an ineffective mean for disrupting cocaine production because farmers compensate by cultivating the crop more extensively. The intuition behind this result is that the inward shift in supply increases prices, compensating farmers and motivating them to cultivate the crop more extensively. Panel (B) shows that, in fact, the recent increase in the eradication rate has increased farm gate prices. The sharp increase in domestic prices may be partly responsible for the increased cultivation, because although producers can only sell a smaller fraction of their crops, they do it at a higher price, which counteracts the dissuasive effect of eradication by compensating producers for the associated losses. I will explain this stylized fact in the section “Why has cultivation increased with more eradication?”. Figure 6: Stylized fact two.. (A). (B). The third and last stylized fact is the relation between cultivation, prices and land productivity. Many authors have shown that, despite the sharp decrease in cultivation from 1999 to 2008 in Colombia, potential production has remained relatively stable because productivity has increased (see Mejı́a and Posada (2008), Mejı́a and Restrepo (2008) and Mejı́a and Rico (2010)). Figure 7, panel (A) shows this phenomenon, while panel (B) shows that there seems to be a positive relation between prices and productivity. Thus, these figures and previous findings suggest that more eradication efforts reduce total land with coca crops, but this land 31.

(32) becomes more productive because prices are higher, increasing technical change. I will study this stylized fact in the section “Why has land productivity increased so sharply?”. Figure 7: Stylized fact three.. (A). 3.1. (B). Trafficking and Conventional Wisdom. In an academic seminar in 2008, Alfredo Rangel, one of the panelists, claimed that increasing seizures of Colombian cocaine was a very effective mean for reducing supply, because it also decreased domestic production. The panelist claimed that more interdiction would reduce traffickers’ profits, reducing the demand for domestic drugs. This argument would be valid if interdiction did not affect prices, but this is hardly the case. Using the equilibrium multipliers it is possible to show that, after considering the effect on prices, the argument holds only if demand is elastic, and otherwise the effect on domestic quantities is the opposite. That is, if demand is inelastic, interdiction actually increases domestic production, because traffickers intend to smuggle more drugs, motivated by the higher prices outside the source country. In order to formalize these ideas, imagine an aggregate trafficker and assume that drugs successfully trafficked are given by Qf = hT (Qd ). (54). where 1 − h is the interdiction rate and T is a concave trafficking technology that determines the number of shipments the traffickers can generate with Qd kilos of domestic drugs. Trafficking profits are given by π = Pf hT (Qd ) − Pd Qd . 32. (55).

(33) The maximization of profits with respect to the choice of domestic drugs, Qd , gives the trafficker’s demand for domestic drugs, Qd (h, Pf , Pd ), and drug supply outside of the producer country by the trafficker, Qf (h, Pf , Pd ). Drug demand, Qd , is given implicitly by the following equation: T (Qd ) =. Pd , hPf. (56). and the implicit function theorem implies: −T (Qd ) ∂Qd = > 0. ∂h hT (Qd ). (57). Also, by the concavity of T , Qd is an increasing function of hPf , and a decreasing function of Pd . Thus, the direct effect of tougher interdiction efforts (lower h), holding prices constant, would be to reduce the demand for domestic drugs and shipments, as suggested by Rangel. Drug supply is given by Qf = hT (Qd (h, Pf , Pd )), which implies that the direct effect of interdiction is given by ∂Qd [T (Qd )]2 ∂Qf = T (Qd ) + hT (Qd ) = T (Qd ) − > 0. ∂h ∂h T (Qd ) Here, in the last step, I have substituted. ∂Qd ∂h. (58). from equation 57. Both terms of the above. expression are positive, and the first captures the fact that more interdiction (lower h) implies less successfully trafficked shipments, while the second term captures the fact that less shipments are sent. To find the general equilibrium effect of interdiction, one also needs to take into account the effect of h on final prices Pf . In fact, we want to find the total effect on the risk adjusted price hPf which determines domestic demand. Using the equilibrium multipliers formula we get ∂hPf ∂h. = h. ∂Pf + Pf ∂h . 1 T (Qd )2 Pf + Pf = −h T (Qd ) − T (Qd ) Qf εs + εd −hT (Qd ) hT (Qd )2 Pf s d + = s +ε +ε ε + εd Qf Qf T (Qd ) Pf T 2 s d = s −1 + +ε +ε ε + εd T T . (59) (60) (61) (62) (63). 2. and since − TTT = εs , we get. ∂hPf εd − 1 = s Pf . ∂h ε + εd 33. (64).