ALTERNATIVAS V. ABSOLUTO V RELATIVO
DEBILIDADES AMENAZAS
Verbal description of the model
A key characteristic of drugs such as cocaine is that addiction and tolerance make ‘demand’ a function of the current level of use. Therefore, it is important to
differentiate between users who will consume at a greater or lesser rate depending on the current price and non-users who do not participate in the market at all until they undergo a discrete transition (called ‘initiation’) and become users.
Practically, the number of non-users is so large that it is effectively constant, so we simply track the number of drug users over time.
There are clearly different intensities of use, and people do not become addicted the minute they initiate, so more refined state spaces are possible (see the next section). We restrict ourselves to a single ‘state’ variable, because we want to include price effects without making the model too complicated (8).
Initiation is modelled to be increasing in the current number of users, reflecting the fact that most people start using when a friend or sibling introduces them to the drug. In addition, it is assumed that initiation is modulated by price as described later.
Over time, users quit, many of their own accord, but some with the assistance of treatment. That is, treatment can be seen as augmenting the flow out of the population of users. Unfortunately, not all treated users cease use. Relapse is common. Some types of users are more likely to relapse than others, and the treatment system has some capacity to target interventions at those for whom the prognosis is most favourable. Hence, we follow the lead of Rydell et al. (1996) in assuming that treatment’s marginal effectiveness diminishes as its scale increases. Enforcement is quite different. In the first place, most enforcement efforts are directed at dealers, not users (9). In the second place, enforcement against black
markets, when they are mature, does not work primarily by removing people from the population. Imprisoned dealers are easily replaced. Rather, enforcement is believed to act more like a tax; it raises risks and, as a result, the costs of
distributing drugs, which in turn drives up prices. This ‘risks and prices’ aspect of enforcement encompasses not only domestic enforcement, but also interdiction and source-country control operations (Reuter and Kleiman, 1986). Finally, whereas diminishing returns mean that treatment becomes inefficient if it is too large relative to the target, ‘enforcement swamping’ (Kleiman, 1993) implies that
enforcement is ineffective if it is too small relative to the size of the market. For any given level of enforcement spending, the larger the market over which that effort is spread, the lower the risk and, hence, the smaller the effect on price.
By focusing on prices, we ignore two other, less important, ways in which enforcement helps to control use. First, retail enforcement can drive up ‘search
(8) In an optimal control framework, ‘state’ variables are those for which differential equations describe their dynamics.
The standard solution method for optimal control models, Pontryagin’s maximum principle (see, for example, Feichtinger and Hartl, 1986; Leonard and Long, 1992), requires the introduction of one additional differential equation (the so-called co-state equation) for each of the state equations. Consequently, the introduction of just one additional state variable into the model increases the system to be analysed by two differential equations, etc., which, in general, makes the analysis much harder.
time’ by reducing availability (Moore, 1973; Kleiman, 1988). Second, interdiction can occasionally generate temporary conditions of physical scarcity. The former is probably of second-order importance (Caulkins, 1998). The latter has generated some successes (10), but suppliers can adapt their routes and methods fairly quickly
to restore equilibrium conditions (Caulkins et al., 1993).
Analysts used to reason that, since drugs are addictive, consumption must be relatively unresponsive to price, but four recent empirical studies (reviewed by Caulkins and Reuter, 1998) suggest that the price elasticity of demand for cocaine in the US is about – 1. That is, if prices go up by 1 %, consumption will go down by 1 %. Some of the reduction occurs in the short run, as current users reduce their consumption. Some accrues in the longer term, as higher prices suppress initiation and promote cessation.
We formulated the objective as minimising the discounted sum of the costs associated with drug use plus the costs of drug control. ‘Quantity consumed’ has merits as a general purpose measure of the magnitude of a drug problem (Rydell et al., 1996; Caulkins and Reuter, 1997), so we assume that the societal costs of drug use are proportional to the quantity consumed, where consumption is given by the number of users times the price-modulated consumption rate.
The final part of our model pertains to control spending. We consider both
unrestricted control (any non-negative level of treatment and enforcement spending is feasible) and a restricted model in which total spending must be proportional to the number of users (11). The latter is a crude way of recognising that budgeting is
often reactive, i.e. responding to the current size of the problem. In some cases, the unrestricted model calls for enormous levels of spending when there are relatively few users in order to prevent future initiation. While such a proactive, aggressive approach can be optimal, it might not be possible to convince
taxpayers to spend a lot of money on a potential future problem that is currently small. Likewise, we consider a variant of the restricted problem in which not only the level but also the mix of spending is fixed, not optimised dynamically. That is, the decision-maker chooses, once and for all, what fraction of drug control spending goes to enforcement. The more restricted the set of controls, the worse the objective function value, but the easier it would be to implement optimal control.
(10) The French connection/Turkish opium ban of the early 1970s and the 1989–90 cocaine price spike.
(11) A more realistic approach would use ‘quantity consumed’ as a proxy variable for the size of the current drug
This concludes the verbal description of the dynamic optimisation model as given by Tragler (1998) and Tragler et al. (2001). The next section presents the
equations and mathematical expressions that describe the optimisation problem, to give a better picture of the ingredients and the structure of the model.
Mathematical formulation of the model
Table 1 presents the functions, parameters, and variables that are needed for a complete characterisation of the dynamic model of illicit drug control described above. Like any optimal control model, our optimisation problem consists of the following components (Leonard and Long, 1992).
Table 1:
Functions, parameters and variables for the one-state modelDescription Type
t Time Independent
variable
A(t) Number of users at time t State variable
u(t) Treatment spending Control variable
v(t) Enforcement spending Control variable
a Absolute value of the elasticity of initiation with
respect to price Parameter
b Elasticity of desistance with respect to price Parameter
c Treatment efficiency proportionality constant Parameter
k Constant governing the rate of initiation Parameter
r Time discount rate Parameter
α Constant in the initiation term Parameter
κ Social cost per unit of consumption Parameter
µ Baseline rate at which users quit without treatment Parameter
ϖ Absolute value of the short-run price elasticity
of demand Parameter
θ Per capita rate of consumption at baseline prices Parameter
p(A(t), v(t)) Retail price Function
β(A(t), u(t)) Outflow rate due to treatment Function
J(A(t), u(t), v(t)) Discounted weighted sum of the costs of drug use Objective
Some variables can be identified that describe the state of the system: they are called ‘state variables’. In our case, there is only one state variable, which is the number of users at time t,A(t) . The rate of change over time in the value of a state variable may depend on the value of that variable, time itself, or some other variables, which can be controlled at any time by the operator of the system (often referred to as the ‘decision-maker’, e.g. the government in our case). These other variables are called control variables, which in our case are treatment and enforcement spending, denoted by u(t) and v(t), respectively. The equations describing the rate of change in the state variables are usually differential
equations and are called state equations. For each state variable, there is exactly one state equation, so for our model, there is one state equation for the single state A(t), given by
Once values are chosen for the control variables (at each date), the rates of change in the values of the state variables are thus determined at any time and, given the initial value for the state variables, so are all future values.
The object of controlling a system is usually to contribute to a given objective. As described above, the objective in our case is to minimise the discounted sum of the costs associated with drug use plus the costs of drug control over some fixed time horizon. In mathematical terms,
where J(A(t), u(t), v(t)) is called the objective functional (12).
Finally, the control and/or state variables are in general subject to constraints. For instance, as a minimum requirement, we need the control variables to satisfy
u t
( )
≥ 0,v t( )
≥ 0. min u t( ),v t( )J A t(
( )
,u t( )
,v t( ))
= minu t( ),v t( ) e −rt discounting{ κ θA t( )
p A t(
( )
,v t( ))
−ω consumptioncosts of drug use
144424443+ u t
( )
treatment spending { + v t( )
enforcement spending { dt , 0 ∞∫
A t( )
= kp A t(
( )
,v t( ))
−aA t( )
−α initiation 14444244443− cβ(
A t( )
,u t( ))
A t( )
outflow due to treatment
144424443−µp A t
(
( )
,v t( ))
b
A t
( )
outflow without treatment
144424443.
In the constrained budget variant, in which total spending must be proportional to the number of users, we have the additional constraint
,
for a positive constant, G. When the mix of interventions is also constrained, we have
with a constant f between 0 and 1 (13).
Summarising, our optimal control problem consists of the minimisation of the objective functional subject to the state equation and constraints on the controls and/or state. Before we get to some policy conclusions, we still need to complete the model description.
Modelling consumption as θAp–ω is consistent with a constant elasticity model of
per capita demand. The state equation describing the dynamics of the number of users has terms for initiation, outflow due to treatment, and the background rate of desistance (Figure 1). The rate of initiation is an increasing function of the current number of users (k Aα) modulated by price. The per capita rate of desistance is
assumed to be a constant (µ) modulated by price. High prices suppress initiation and encourage desistance. In the absence of controls, the elasticity of the steady- state number of users with respect to price is –a–b. The overall, or long-term, elasticity of demand is the sum of the elasticity of demand per capita and the price
u t
( )
= fGA t( )
,v t( )
= 1− f( )
GA t( )
, u t( )
+ v t( )
= GA t( )
Figure 1: Flow diagram for the one-state model
Sources: Tragler (1998); Tragler et al. (2001).
Number of users Initiation
Outflow due to treatment
elasticity of the number of users. Hence, we set –(a + b + ω) equal to the overall elasticity of demand.
Outflow due to treatment is modelled as being proportional to treatment spending per capita raised to an exponent (z) that reflects diminishing returns, with a small constant in the denominator (δ) that prevents division by zero. In particular,
We take our model of enforcement’s effect on price from Caulkins et al. (1997):
where ε is an arbitrarily small constant that avoids division by zero. The
parameter d captures the fact that prohibition itself forces suppliers to operate in inefficient ways (what Reuter, 1983, calls ‘structural consequences of product illegality’). Because of enforcement swamping, the marginal effectiveness of enforcement (e) is multiplied by enforcement effort relative to market size (v(t)/A(t)), not total enforcement effort.
Conclusions from the analysis of the model
For reasons of brevity, we restrict ourselves to the most interesting insights that are provided by the analysis of the model, for which parameter values were chosen that describe the actual cocaine epidemic in the US (see Tragler, 1998, for the analysis in full detail).
The basic conclusions of this model are that, if one initiates control early, when there are relatively few users, and the problem is truly an epidemic in the sense that initiation into drug use is driven by contact with current users, then one should apply both enforcement and treatment very aggressively to short circuit the
epidemic spread. Otherwise, the optimal policy is not to stop the growth of the epidemic, but rather to moderate it. Initially, this should be done primarily by enforcement, to keep prices high and suppress initiation to a maximum extent. Over time, enforcement spending should increase, but not nearly as fast as treatment spending. Hence, treatment should receive a larger share of control resources when a drug problem is mature than when it is first growing. If initiation rates subsequently drop, for example because the drug develops a negative
reputation, and the problem shrinks, then treatment funding should be reduced, p A t
(
( )
,v t( ))
= d + e v t( )
A t( )
+ε , β(
A t( )
,u t( ))
= u t( )
A t( )
+δ z .The model generates a variety of other insights, including: (i) a quick detection of the onset of a drug epidemic is valuable because total costs are much lower if control begins early; (ii) people who perceive drug use to be costly for society should favour greater drug control spending and allocating a greater proportion of that spending to enforcement; and (iii) sharp price declines, such as those observed in the 1980s for cocaine in the US, do not necessarily imply a policy failure; indeed, it can even be optimal to have such declines.