- Email: [email protected]
When haste makes sense: Cracking down on street markets for illicit drugs
Socio-Econ. Plann. Sci. Vol. 31, No. 4, pp. 293-306, 1997 © 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain P I I : S0038-0121(97)00002-5 0038-0121/97 $17.00 + 0.00
Pergamon
When Haste Makes Sense: Cracking Down on Street Markets for Illicit Drugs ALOK BAVEJAt School of Business, Rutgers University, Camden, NJ, 08102, U.S.A. J O N A T H A N P. C A U L K I N S H. John Heinz IIl School of Public Policy and Management, Carnegie Mellon University, Pittsburgh, PA, 15213, U.S.A. WENSHENG LIU Department of Mathematical Sciences, Rutgers University, Camden, NJ, 08102, U.S.A. R A J A N B A T T A and M A R K H. K A R W A N Department of Industrial Engineering, 342 Lawrence D. Bell Hail, State University of New York at Buffalo, Buffalo, NY, 14260, U.S.A. Abstract--This paper presents an analytical approach to the tactical question: 'What level of enforcement over time allows one to eliminate a street market for illicit drugs while expending the least possible total effort?' The analysis is done in the context of Caulkins' model [6] which predicts the rate of change of dealers as a function of enforcement level and several market parameters. Our analysis suggests that the simple strategy of using the maximumavailableenforcementintensity until the market has been eliminated minimizes the total enforcement effort required. © 1997 Elsevier Science Ltd
INTRODUCTION The consumption and distribution of illicit drugs impose significant costs on American society, as do drug control efforts. U.S. drug control policy has taken a criminal justice approach, with considerable attention devoted to local drug enforcement (in contrast to source country control, interdiction, and high-level domestic enforcement). This paper addresses one of the most popular local drug enforcement tactics: cracking down on street level markets. Police crackdowns, a type of focused enforcement, entail an abrupt increase in police activity, especially proactive enforcement, which is intended to dramatically increase the perceived and/or actual threat of apprehension for specific types of offenses in certain places or situations and, hence, produce a general deterrent effect [23]. However, crackdowns invariably entail high costs, particularly those involving personnel costs [15]. Kleiman [14] discusses crackdowns at length, but a simple working definition is: 'An intensive local enforcement effort directed at a particular geographic target.' Crackdowns are distinguished from day-to-day enforcement operations, which generally spread resources more or less uniformly over the 'problem,' and other special operations that may target individuals, organizations, a particular drug, a class of users, or an ethnic group rather than a geographic target. The efficacy of crackdowns has not been definitively established. (For competing views see, e.g., Chaikan's N I J Issues in Brief [1, 5, 14]. While there are strong theoretical arguments favoring local enforcement in general [16, 19], some crackdowns [24] have dramatically improved the quality of life in target areas, while others [14] have been dismal failures. Despite this mixed track record, crackdowns continue to be implemented in cities across the country. tAuthor for correspondence. 293
294
Alok Baveja et al.
Crackdowns raise a number of management issues, including selecting the sequence of target neighborhoods [17], deciding which tactics to employ, and coordinating the activities of police with both other governmental and community-based groups. One important management problem that has received little attention and essentially no formal analysis is: 'What is the best rate at which to crack down on a market?' Should one start slowly and build up gradually? Should one begin aggressively and then ease back? This paper suggests that in many cases 'as forcefully as possible. starting immediately and continuing until the job is done' may be a sound heuristic solution. In the next section, we review some relevant literature, we then develop the intuition underlying our results via two extreme cases. Subsequently, the question is formulated as an optimal control problem and its solution presented. Finally, we illustrate the results with a numerical example. PRIOR W O R K Crackdowns have been conducted in numerous cities, some have been analyzed, but empirical evidence on their effects is scarce [14, 23, 24]. Indeed, it is rare for conditions to resemble a controlled experiment; police and other officials are understandably reluctant to subordinate the immediate needs of their constituencies to the long-term interests of research. Occasionally, natural experiments can be exploited (e.g. [7]), and there are now several studies under way that have gone to extraordinary lengths to conduct valid experiments. Even in these exceptional examples, however, the dependent variables of greatest interest, e.g. dollar volume of sales, number of dealers in the market, etc., are exceedingly difficult to estimate even approximately, and surrogate measures are weak. In the face of these limitations, a complementary way to develop insight about crackdowns is to explore relevant mathematical models of them and present analytical results. Beginning with the perceived risks and profits of being a drug dealer, Caulkins [6, 8] presents a model for quantifying the rate of growth (or decay) of a drug market under crackdown enforcement. The rate of change of dealers with respect to time in a given market is postulated to be
(l) where N = t= Q ( N ) = ~N a = ct, fl = 7= E =
the number of dealers in the market, time, number of sales per unit time, demand parameters, risk aversion parameter, increment in enforcement pressure, above and beyond the baseline level, that is placed on the market during the crackdown, C~ = proportionality constant, rc = generalized profit per transaction, and w0 = a dealer's reservation wage.
In words, the rate of change of dealers is proportional to the difference between the utility available to dealers in the market and their reservation wage. The dealer's utility is his/her generalized profit (profit per sale net of pecuniary and other costs times the number of sales per dealer per unit time) minus his/her perceived risk from crackdown enforcement (modeled as the enforcement risk per dealer raised to an appropriate power). The reservation wage is simply what the dealer could earn in alternative employment (including dealing elsewhere). Thus, the model treats drug markets as an economic entity where there is the potential for profit (from drug sales), the risk of being caught (by enforcement), and an alternative means of employment. A dealer will be motivated to enter the market if and only if his/her expected utility exceeds that offered by the next best alternative. The model considers police crackdowns which target particular geographical areas where concentrated drug dealing takes place, referred to as 'Drug Markets.' The sales volume depends
Cracking down on street markets for illicit drugs
295
directly on the number of dealers, N; enforcement, E, is assumed to affect sales only indirectly by changing N. The model also assumes that the number of drug markets is large enough that cracking down on one does not affect the return available to dealers in other markets. Drug dealers are modeled as identical and indistinguishable. Analyzing the steady state behavior of this model under constant enforcement, Caulkins [6, 8] obtains a number of results; notably, that for an enforcement operation to have lasting effects, the market should be driven below a minimum viable size. An argument is put forth for having enough resources to prevent the market from re-emerging. Baveja et al. [2] analyze the dynamic behavior of drug markets using the above model, and generalize the steady state results by showing that (i) the general shape of the plot of dN/dt as a function of N is all that is required to derive those results, (ii) the shape of the plot of dN/dt is generalizable to any parameter values, i.e. its shape is the same for different drug markets, and (iii) this shape holds for any function that satisfies a set of intuitively appealing properties. However, none of the work described thus far has addressed the question of finding the time trajectory of enforcement effort, E(t), i.e. that which eliminates a drug market while expending the least possible enforcement effort. Sherman [22] considers this issue but does so only qualitatively, without the aid of mathematical models or a formal analysis. His paper argues that a crackdown-backoff strategy, in which crackdowns are conducted for a limited duration and rotated across targets, could capitalize on the benefit of a free bonus due to residual deterrence. Feichtinger et al. [11] develop a continuous time model using a fixed enforcement level that incorporates addicted and dealer populations. They suggest the existence of two stable equilibria for each enforcement level and propose a two-step control policy. However, their model is more relevant at the national level rather than for local crackdowns. The question we raise for street-level drug markets has not been investigated using a formal, quantitative analysis. The next section thus develops a mathematical model to provide guidance on this issue.
ANALYSIS We here formulate the problem of finding E*(t), the optimal time trajectory of enforcement, that which eliminates a drug market while using the least amount of enforcement resources. This problem is solved using optimal control theory. The results are then extended to more general objective functions. However, to develop intuition about the optimal strategy, we first consider a simplified version of the problem.
Constant-enforcement policy In the context of Caulkins' model, we look for a constant enforcement strategy that minimizes the total amount of resources used. In order to do so, we consider two specific extreme markets called the sellers' and the buyers' markets [2, 6]. An actual drug market would probably be between these two extremes and, therefore, the results should provide insights relevant to determining optimal policy for the general case. Sellers' market Informally, a sellers' market is one in which demand for illicit drugs is abundant. The implication of abundant demand is that another arriving dealer will expand total market sales and not reduce individual sales for any of the original dealers. This can be modeled by setting fl = 1 in (1). For sake of simplicity and analytical convenience, we assume ~ = 1, i.e., dealers are risk neutral. Let T~(E) denote the time needed to eliminate the market when a constant enforcement, E, is used; let Emaxbe the maximum level of enforcement possible at any instant of time. The objective is to find the enforcement that minimizes the total resources used to eliminate the market, where total resources are the integral, over time, of enforcement at each instant of time. Lemma 1: Under constant enforcement, using the maximum available enforcement minimizes the total enforcement resources used to eliminate the market.
296
Alok Baveja et al.
Proof: Under constant enforcement, the total resources used are ETa(E). We need to show that Min
Et,.(E) = Em,~T,(Em,0.
0 < E ~< Ema x
Solving for T,(E) using (1), and incorporating boundary conditions, we obtain:
~(E)-
-No C,0t~ - Wo) + C,(rtct-L w0)2 In E - (net - wo)No
"
So, objective function Z = E T a ( E ) is:
Z=E
G(~Two)
+ G ( r t ~ - w o ) 21n E - ( m T L W o ) N o
"
(2)
From (1), by setting d N / d t = 0, we first note that the enforcement required to maintain a sellers' market of size N is E h ( N ) = (rt~ - wo)N. Therefore, to eliminate a market of size No, we need an enforcement E > (net - wo)No. If we let (n~ - w0) = b, this condition can be restated as E / b = m must be greater than 1. Using this, we can write (2) as:
m~m
(3)
Taking the derivative of Z with respect to m, we get:
dm -
C,
~
-
+ m--Z-l--1
"
For m > 1, d Z / d m is strictly negative. Since m is proportional E, Z is a strictly decreasing function of m. Therefore, E = Emax is optimal. A plot of Z vs m [3] shows that Z decreases sharply as E increases for enforcement less than 1.5b. Thus, little is gained by increasing enforcement beyond two times the minimum enforcement required to eliminate the market. The minimum value of the enforcement that does not compromise on the total enforcement resources used will therefore be approximately 1.5b. Buyers' market
In the context of drug markets, a buyers' market is one in which there is a fixed number of sales with dealers simply fighting for market share. Increasing the number of dealers would not increase the number of sales since there is already a surplus of dealers. This scenario can be modeled in (1) by setting fl = 0. As before, ? is assumed to be equal to one. By setting b = net = woNo, a similar argument to that in Lemma 1 shows that Z is a decreasing function of m and, hence, E. A plot of Z v s . m for a buyers' market [3] gives similar inferences. In this case, the objective function does not significantly improve ( < 10%) for an enforcement level greater than 1.2b. Therefore, if one is constrained to use constant enforcement, for both extreme cases the optimal strategy is to use the maximum possible enforcement intensity. This result can easily be extended to policies where enforcement can change at discrete points in time [3, 4]. Under such circumstances, using the maximum enforcement is still optimal for both the buyers' and the sellers' markets. However, these results cannot be verified for an intermediate market using this method. The next section considers that more general case within a continuous-time enforcement policy scenario. Generalized problem formulation
Police/policy-makers are concerned with (1) budgetary costs of enforcement and (2) a variety of problems associated with the markets they are seeking to eliminate (e.g. drug use, disorder,
Cracking down on street markets for illicit drugs
297
market related violence, etc.). Other concerns such as officer safety, equity and public relations, though important, are beyond the scope of this paper and are thus not considered in the model. Initially, we analytically address the simpler problem of minimizing the amount of enforcement expended in the process of eliminating a street market. Later, we discuss the implications of considering more general objective functions that include the size of the market. The basic problem considered here is the following. Suppose the police wish to reduce the number of drug dealers in a particular street market from an initial level No to a final level of NI = 0. (Note, the problem with any general level NI > 0 is no more difficult to solve and yields similar results [4], but typically the goal is to eliminate a market, not just reduce it.) The objective is to accomplish this reduction while expending the least possible amount of enforcement resources, subject to a constraint on the maximum intensity of enforcement (E(t) < Emax) and the assumption that the market behavior is as described by eqn (1). We thus have: Minimize Z =
~0TE(t)dt,
subject to:
l, N(O) = No,
N ( T ) = O, 0 < E(t) < Emax N(t) >_ O, where 0 < fl < 1,7 _> I(OPC) This problem is set up for application of optimal control theory. Optimal control theory has been used for a variety of problems in the fields of economics, environment etc. For example, Sethi [21] used it to determine the optimal pilfering policy of a profit-maximizing thief. The terminal time, T, is the time when the market is eliminated. From an optimal control theory perspective, the formulation is one in which the terminal time is not specified. One aspect of this formulation might seem artificial at first, but is not in fact constraining. Indeed, it is analytically convenient to express the problem in continuous time, but the reader might justifiably be concerned about the relevance of that exercise, since enforcement pressure can only be modified at a finite number of discrete points in time. However, the optimal continuous-time solution is to apply the maximum possible enforcement effort throughout, which is a simple policy to implement. That is, even when one admits a broad range of possible solutions, it turns out that the optimal solutions happen to fall in a restricted class of policies that can practically be implemented. Furthermore, similar results can be obtained for a formulation that explicitly considers discrete-time policies.
Solution In order to show that using the maximum enforcement is optimal we need to establish both the appropriate necessary and sufficient conditions. The necessary condition will be arrived at using the Maximum Principle while the sufficient condition will be demonstrated using transformations of the optimal control problem (OPC) to an equivalent problem known to have a solution. Before exploring the solution methodology, we present two important conditions that will be used in the analysis. First, for (OPC) to have a feasible solution, the constant control, E = Em~x, should be able to steer No to 0. That is, dN/dtlE~E~,, must be negative for all values of N: 0 < N < No; otherwise, the problem has no feasible solution. We therefore restrict our attention to markets (with parameters n, ~t, fl, ?, w0, No) that can be eliminated if the maximum possible enforcement is used.
298
Alok Baveja et al.
Second, from eqn (1) it is clear that if
I dN/dt will be negative even when no enforcement is applied. Therefore, the least costly way to begin
to reduce the size of such a market is simply to let it shrink on its own with E = 0. However. mathematically, the market will approach N,. asymptotically and never reach the equilibrium value in finite time. From a theoretical standpoint, the problem will have no optimal control since, to reach -Arcin finite time, we will have to apply a non-zero enforcement that will not be optimal. From a practical perspective, it makes no sense to use expensive enforcement and punishment resources against dealers if the dealers are exiting on their own. Therefore, we restrict our attention to markets where
No < N~
\ w'o /
Note that for the case No = Nc, a market can stay at its equilibrium level without expenditure of any resources. Mathematically, you can wait indefinitely before undertaking a crackdown. The rest of the analysis in this section considers the start time as that instant when the crackdown operation is to be initiated with a non-zero enforcement level. For convenience, we let y = N :+' and m ( y ) = C,(7 + l)[ltcz), ~
- woy:,~ i
We can therefore rewrite (OPC) more simply as: Min Z ( E ) =
~0TE(t)dt
subject to:
ay dt-
re(y) - C,(7 + I)E%
y(0) = Y0, y ( T ) = O,
0 < E < Emax.
Theorem 1: For any 0 < Y0 < Yc, the control E = E,,ax on [0, T~] is optimal for (OPC). Proof: Let U = E/E~,,x and consider the following control problem, (OPC)I, equivalent to (OPC). Min Z ( U ) =
~0TU(s)ds
subject to: ely = re(y) - CU'; dt y(O) = Yo,
y(T) =
O,
O
Cracking down on street markets for illicit drugs
299
T o prove the theorem, we need to show that U = 1 on [0,T~] is optimal for (OPC)~. In order to establish this, we first look at the special case when 7 = I. F o r this case of ~ = 1, it is fairly easy to show that (OPC)~ has a solution, i.e. has an absolute m i n i m u m . This existence result also follows directly from existence theorems in optimal control theory (e.g. T h e o r e m 9.3.i in [10]). Let U = U(t) on [0,T] be an optimal control for this special case o f (OPC)~. Further, let y = j7(0 be the solution on [0,T] o f y' = re(y) - CU(t), y(O) = Yo, where 0 < iT(t) < y0.(4) By the Pontryagin m a x i m u m principle [18], we know that there exists a co-vector, 2(0, on [0,T] that satisfies 2 ' ( 0 = - 2(t)m'(fi(t)) and a constant 20 > 0 such that: (i) (2(0,20) ~ 0 for all t q 0 , T ] , (ii) F o r almost all t~[O,T],O(t) minimizes
H(.~(t),U,2(t),2o) = 2(t)[m(.~(t)) - CU] + 2oU = 2(t)m(fi(t)) + (20 - C2(t))U and (iii)
H(~(t),8(t),2(t),2o =- O. Note that (20 - C2(t)) ~ 0 on (0,T). Indeed, if there is a ? ~(0,T) such that 20 - C2t" = 0 then (iii) implies that =
o
(for t~(O,T), m07(t)) :/: 0). So 2 ( 0 - 0 to [0,T]. But then 20~:0 by (i), contradicting 20 - C2t" = 20 = 0. Therefore, if20 - C2(t) > 0, U(t) = 0 for almost all t([0,T], and if20 - C2(t) < 0, then O(t) = 1 for almost all t by (ii). Clearly then, 0(t) must be equal to one almost everywhere on [0,T]. This shows that if U = 0 ( t ) is optimal for the special case of (OPC)~ when ), = 1 and if (4) holds, then U(t) = 1. The optimal cost is equal to T~. Having established that U(t) = 1 is optimal for the case when ~ = 1, we next use this to prove it for any general ~ value. Let U = U(t) on [0,T] be any control such that, 0 < U(t) < 1, and the corresponding solution of y' = m(y) - U'~(t),y(O) = Yo satisfies y ( T ) = 0. Then we know that SorU"(s)ds >_ T~ from above. Since 0 < U(t) < 1 and ~, > 1, we have SorU(s)ds > ~U)'(s)ds > T~. This implies that U = 1 on [0,T~] is optimal for (OPC)~. Therefore, E = Emax is optimal for (OPC). Thus, if the m a r k e t is such that No < No, the optimal strategy is to move with the most force possible at the earliest possible time, a policy that is easy to implement. An i m p o r t a n t side-benefit of this optimal policy is that the m a r k e t will continuously decrease in size since dN/dt IE= ~ma, is always negative. It is fortuitous that the optimal policy generates a shrinking market, because if this were not so, it would be very difficult indeed to justify it politically.
Numerical example We now consider a numerical example to illustrate both the importance of using an optimal enforcement strategy and some m o r e subtle insights*. T o do so, we need to find reasonable estimates for the parameters. Reuter et al. [20] estimate that, in 1988, Washington, D C had about 14,000 full-time-equivalent retail dealers, and G a r r e a u [12] reports that there were about 90 distinct street markets, suggesting a ratio o f 155 dealers per market. Presumably, not all retail sellers operate out o f open-air street markets, since the average m a y be biased by a few big markets, 100 dealers m a y therefore be a reasonable estimate for a typical street market. Values for the p a r a m e t e r fl lie between 0 and 1 while ~, is at least 1. F o r convenience, we take as our base case fl = 1/2 and ? = 1 and report the results of some sensitivity analyses with respect to these parameters. *Examples were run using a differential equation subroutine of the IMSL math library on a UNIX-based mainframe computer.
300
Alok Baveja et al.
The reservation wage, w0, can be estimated by the legitimate employment wage for a drug dealer. Reuter et al. [20] estimated this to be about $1,046 per month, or approximately $50 per work day. Based on the equilibrium condition that the dNidt = 0 when E = 0, we obtain ~.~ = 500. With these parameter values, an enforcement level of E > 1250 is needed to eliminate the market [2]. We are interested first and foremost in the relationship between the intensity of enforcement effort applied and both how long it takes to eliminate the market and the cumulative amount of effort exerted in doing so. Figure la shows the former relation for three values of fl, When B = 1/2,
(a)
~
o.9
lg
p=o.]
p--o.~
10 5 0250---
i
750
lifo
t250
1450 1 7 0 0 1950 22100 2 4 5 0 2 7 0 0 2 9 5 0 1
3450 3650 I
4150 1
I
Enforcement Intqmtlty
(b)
:t T= 1.75 I 15
10
i . 20S
i -¸ 7oo
~50
12?5
1~
1"r50
20OO
EnfarcemmM I ~ I ~ F
Fig. 1. (a) Time to eliminate market as a function of enforcement intensity and deraand parameter ft. (b) Time to eliminate market as a function of enforcement intensity and dealers' risk aversion parameter ?.
Cracking down on street markets for illicit drugs
301
1.25l i
8=4
1.2
ff~ .,/'J
1.15 0=3
1.1
1.85
i
~
J /
8=1 6=4
J.q tI-
*5
0.85¸
o.o
0.85 0.S
100
I 2OO
J 3OO
I I I 40O S00 e00 Amplitude of PulN In Enforcement
I
B
7OO
eo0
Intensity
I 900
I 10~0
(/1)
Fig. 2. Pulsed crackdowns are usually less efficientthan constant intensity crackdowns. for enforcement intensities below 1250 no amount of time is sufficient to eliminate the market. With E = 1875, or just 50% more than the minimum required, the time needed for elimination is less than five days. As enforcement intensity increases beyond 1875, the time to eliminate the market is reduced further, but there are sharply diminishing returns. Indeed, it is striking how 'kinked' the curves are in Fig. 1a. Furthermore, plots of total effort required vs. enforcement intensity (not shown) are even more sharply 'kinked'. Figure la also shows that, for different markets with the same number of dealers in equilibrium before a crackdown, the higher fl is, the easier it is to eliminate the market. This makes intuitive sense. Thus, if fl is high and enforcement drives away a few dealers, the number of customers is also reduced, which, in turn, makes the market less attractive for the remaining dealers. This feedback generates a negative spiral. In contrast, when fl is small, even if enforcement reduces the number of dealers, the number of customers remains relatively high. This increases the ratio of customers to dealers, helping to offset the additional enforcement pressure experienced by the smaller number of dealers. Figure l b also shows the relationship between enforcement intensity and time to eliminate a market, but it does so for various values of ~. Two observations deserve mention. First, the larger is, the more kinked the curve. Since Fig. la assumed 7 was unity, it was conservative in the sense of showing the 'least kinked' case. Second, the larger ~, is, the easier it is to eliminate a market. This makes intuitive sense since ~ is a measure of how risk-averse the dealers are. Figure 2 pertains to Sherman's [22] recommendation of a 'crackdown backoff strategy which alternates periods of intense pressure with periods of more modest enforcement. The goal of such a strategy is to take advantage of residual deterrence. Informally, the hope is that during the relative lulls in enforcement, dealers will remember and make judgments based not on the current enforcement risk, but on the enforcement risk operative during the intense portion of the crackdown. This model cannot evaluate the upside potential of Sherman's strategy because it is predicated on the assumption that dealers correctly deduce and react to the actual, current enforcement intensity. That is, it assumes a priori that there is no such thing as residual deterrence. However, we can quantify how badly Sherman's strategy might stumble if its hope of residual deterrence proves elusive. To do so, we need to model both Sherman's strategy and an appropriate referent and then compare their results. We model a 'crackdown backoff' strategy as alternating between SEPS 31/4--C
302
Alok Baveja
et al.
E = 1400 + A and 1400 - A with a period of 20 days. We choose, as a foil, applying a fixed level of enforcement equal to the average enforcement intensity applied by the crackdown, backoff strategy. For example, if applying E = 1700 and E = 1100 every other day would eliminate the market in exactly three days, then we take as a foil applying a constant effort of E = (1700 + 1100 + 1700)/3 = 1500 for however long it took to eliminate the market. Figure 2 shows the ratio of how long it takes a crackdown, backoff strategy to eliminate a market relative to the time required by its constant intensity foil. There are three observations worthy of note. First, there is a rather complicated, nonlinear relationship between the amplitude (A) and period (0) of the oscillations in enforcement intensity and the effectiveness ratio. Second, at least for the cases we examined, in no case was the crackdown, backoff strategy markedly less effective than its foil. Third, there are instances in which the crackdown, backoff strategy is superior to its foil (e.g., 0 = 3 and A small). At first glance, this last result might seem to contradict the paper's main finding. However, what Theorem 1 finds is not that any constant enforcement strategy is optimal, but, rather, that applying the maximum possible enforcement throughout is optimal. By definition, if it is feasible to alternate between some base intensity plus A and that intensity minus A, then the base intensity cannot be the maximum feasible intensity and there is no reason to expect that maintaining an intensity at that level will be optimal. Figure 3 addresses the question of whether a crackdown that ramps up intensity over time is more or less effective than one that ramps it down. One can construct 'arm-chair' arguments in favor of either approach. Ramping enforcement intensity down over time allows one to focus the greatest effort on when the market is largest (at the beginning of the crackdown). On the other hand, ramping enforcement up reserves the greatest effort for the end of the crackdown when the market has been made more vulnerable by earlier efforts. To explore the issue quantitatively, we examined strategies that vary enforcement intensity linearly between E = 1275 and E = (1 + K)*1275 for various values of K. Ramp up strategies start with E--- 1275 and linearly increase enforcement intensity at a rate such that they just reach E = (1 + K)*1275 at the moment the market is eliminated. Ramp down strategies start with E = (1 + K)*1275 and decrease enforcement linearly to 1275 at the moment the market is eliminated. Obviously, such contrived strategies cannot be implemented in practice because one
t
+| i!i ~.
-
i!+ I~|11 I It o.~, 15 J
0.9 0.85
0.8
I
I
0.2
0,4
- - 4 0.6
z
I
I
I
I
I
I
0.8
1
1.2
1,4
1,6
1.8
2
2.2
[
i
2,4
2.6
Magnitude of Ramp as Mulllple of Base (K)
Fig. 3. Ramp-down strategies are less effective than ramp-up strategies.
- i 2.8
Cracking down on street markets for illicit drugs
303
~
0.9
'~o.e 0.7
J
~ 0.5 ~0.5
~
0.4 .Q ~S0.3 0
~ 0.2 a ~.~0.1 I
0.5
2.5
4.5
05.
85.
10.5
12.5
14.5
I
16.5
15.5
20.5
22.5
24.5
26.5
28.6
11m (aa~) Fig. 4. If crackdown will eliminate market, it usually does so quickly.
does not know a priori how long it will take to eliminate a market. However, their symmetry facilitates comparison. Figure 3 plots the ratio of time to eliminate the market for the ramp down strategy relative to that for the ramp up strategy. (Note: by construction, this is also the ratio of the total amounts of enforcement used.) For every value of K, the ramp down strategy took more time and more resources than did the ramp up strategy. This suggests that strategies wherein enforcement intensity is proportional to market size as it decreases over time may be ill-advised. Figure 4 addresses one final issue: how persistent should the organizers of a crackdown be? Above, we argued that all available resources should be deployed. Elsewhere [2, 8] we have argued that crackdowns should be pursued long enough to drive a market below its minimum viable size; otherwise, the market will re-emerge to its pre-crackdown size after the crackdown has ended. Figure 1 makes clear, however, that if the resources available are insufficient (E < 1250 in our base case), then no matter how long the crackdown is maintained, the market cannot be eliminated. So, suppose all available resources have been summoned, a crackdown has been instituted, and after a week the market has not been eliminated. Should the crackdown be continued in hopes that a little more time and patience are all that is needed? How about after a month, should the crackdown be continued? After a year? At the blackboard, these are easy questions. If the enforcement available exceeds the critical level (e.g. E > 1250) then one should continue; otherwise, abort. Practically, however, these are difficult questions because the critical level of enforcement is not known. Furthermore, it is typically difficult to directly observe changes in the number of dealers, and, in some cases, (markets with small fl's) the number of dealers will not decline much before it takes a sudden, precipitous plunge to zero. To solve this problem formally, one would need to define a prior probability distribution on the relationship between E and the minimum enforcement needed to eliminate the market. That probability distribution would then need to be updated based on observations of how long the market has withstood the crackdown. Doing this market by market is daunting, but performing a similar exercise for our illustrative market is informative. Suppose the available enforcement intensity is equally likely to be any value between 0 and three times the minimum necessary to eliminate our base case market. Then, Fig. 4 shows the cumulative distribution of the time necessary to eliminate the market. There is a one-third chance the market will never be eliminated. There is a 60% chance the market will be eliminated within a week. There is only a chance of about 2% that a market will survive for a month but eventually succumb.
304
Alok Baveja et al.
The particular probabilities depend on the parameter values chosen, especially the prior range for the available enforcement intensity relative to the minimum needed. The basic message, however, is less equivocal. Persistence and patience may be virtues, but generally if a crackdown is going to succeed, it will succeed relatively quickly. If, after a time, the market still stands, it might be wise to obtain additional resources, change tactics, or pursue other, smaller and more manageable targets. Extension to more general objective functions Policy makers are concerned not just with the budgetary costs of enforcement but with the societal costs imposed by drug markets. Thus, if the former are related to enforcement effort (E) and the latter to the number of dealers (N), one might prefer to minimize a more general cost function C(E,N) = f ( E ) + g(N). Note, though, that the optimal solution in the very special case of C(E,N) = E involves reducing N as rapidly as possible. Hence, afortiori this same strategy is optimal for any cost function C(E,N) = E + g(N) for which g(N) is an increasing function of N. The societal costs associated with drug markets are almost certainly increasing in the number of dealers, N. The strategy of cracking down with the maximum possible effort as soon as possible will thus still be optimal for this more general class of objective functions. A similar argument can be made for generalizations to C ( E , N ) = f ( E ) where f ( E ) is concave in E. However, because of diminishing returns, it seems more likely that the cost of imposing a particular enforcement effort would be convex in E, not concave. For example, the marginal cost of additional enforcement might increase with the amount of effort already imposed because of the need to pay overtime or to employ less efficient personnel. If one allows f ( E ) to be convex in E, it is trivial to construct examples for which it is no longer optimal to apply the maximum possible effort as soon as possible. Hence, the principal result carries an important caveat. When f ( E ) is convex, one's best hope might be to use the model as a simulation for what-if analysis. Note that even if the cost function is convex in E beyond some threshold level, the analysis can still yield insight. For example, suppose that f ( E ) is linear in E up to a particular threshold value and convex in E thereafter. Although one cannot find the overall optimal enforcement trajectory without estimating 'difficult-to-measure' parameters, one can still conclude that it would never be optimal to employ any level of effort below that threshold value. DISCUSSION This paper addresses the problem: 'What is the optimal time trajectory of enforcement that should be applied during a crackdown on a local drug market?' It concludes that deploying the maximum possible effort from the earliest possible time minimizes the total effort (integral over time of as a function of time) required to eliminate a market. Crackdowns that reduce enforcement intensity as the market shrinks may be particularly inefficient. Full-intensity crackdowns that have made little progress after a moderate amount of time might best be revised or abandoned: usually, if a crackdown will suppress a market, it will do so relatively quickly. A point that may seem obvious but which nevertheless deserves mention is that the level of effort employed must be one that can be sustained long enough for the market to actually be eliminated. Because of certain economies of scale, street markets for drugs have a minimum viable size. It is not difficult to keep markets below this size from growing. In particular, once a market has been completely eradicated, it is relatively easy to keep it from springing back; no one wants to be the first dealer to challenge the police. Conversely, if the market is still above its minimum viable size when the crackdown ends, it will grow back. Too often in practice police have begun crackdowns intensely but have been unable to maintain the pressure long enough. The resulting gains are thus only temporary, and the market soon returns to its original size. Of course, the above result is subject to a number of caveats. First, as noted in the previous section, the result may not hold at all if the marginal cost of applying additional effort increases significantly with the amount of effort deployed. Second, the results are predicated on the reasonableness of the underlying model. Models are inevitable simplifications of reality. However, in this case an extra 'grain of salt' is appropriate since data limitations imply that validation of models of illicit activity is extremely difficult. While we
Cracking down on street markets for illicit drugs
305
believe that the model captures fundamental dynamics of the prototypical street corner market for illicit drugs, we would caution practitioners to review the assumptions underlying the model and to think carefully about the idiosyncrasies of the particular situation they confront before adopting the results. Third, there are some subtleties associated with interpreting the objective function, particularly the quantity E, which is referred to as the enforcement effort. Returning to eqn (1), one sees that this enters the dealers' utility function as a cost. Since this utility function governs a future decision (namely, to enter or leave this market), it is actually the expected future cost from enforcement of dealing in the market in question. Hence, the true lesson of this analysis is to raise the expected cost of dealing as quickly as possible to the greatest extent possible. The most natural way to raise expectations of costs associated with enforcement is by actually applying enforcement pressure. Indeed, the model assumes that these are essentially identical, but there may be ways of raising the dealers' expectations about those costs without actually imposing them. For example, in Houston [13], a planned crackdown received so much media attention that by the time the police arrived, the market had already disappeared. The credibly communicated threat of enforcement was sufficient to eliminate the market. At the other extreme, in theory a crackdown could exert so much effort that the dealers would conclude that it could not be sustained and, hence, expect the future risk of enforcement to be relatively low. The point is that it is the expectation of future costs, not the actual costs imposed, that ultimately govern the flows of dealers in and out of the market. Fourth, there is the important issue of displacement. The underlying model focuses on a single local market even though most cities have a dozen or more distinct open-air markets. The existence of these alternative markets suggests that the outcome of the crackdown may not be to force dealers out of business, but rather to force them to relocate their operations. Baveja [3] developed a model that incorporates dealer displacement in which interaction between different markets is explicitly considered. Using control theory analysis, similar to the one developed in the paper, he demonstrated that extreme point solutions are still optimal. Practically, this implies a policy using the maximum possible enforcement on a given market and collapsing markets sequentially. The mere possibility of displacement does not imply that cracking down on a single market is not beneficial. A successful crackdown would still remove the blight from one neighborhood. More importantly, if the crackdowns are strategically planned, the displacement might create forms of dealing that impose fewer negative externalities on society. For example, displacing dealing from street corners to indoor locations might reduce the incidence of drive-by shootings and other street violence. Caulkins [9] discusses these considerations at length; but, in summary, although the possibility of displacement should certainly never be ignored, it also is not a prima facie case for the futility of crackdowns. Finally, note that this paper in no way argues that crackdowns, or even local enforcement, are efficacious. One might object to the very concept of local crackdowns for reasons that are beyond the scope of this model (e.g. because of the likelihood that a disproportionate fraction of the arrests will be imposed on minorities). This paper merely suggests that, if the decision has been made to crack down on a market, then the most efficient way to reduce the size of the market is to apply the maximum possible effort from the earliest possible moment. This result is important in and of itself, however, precisely because, in so many cities across the country, the decision to crack down on local markets has already been made. Acknowledgement--The authors thank anonymous refereesfor their many helpful suggestions.
REFERENCES I. Barnett, A., Drug crackdowns and crime rates: a comment on the Kleiman paper, Street-level Drug Enforcement: Examining the Issues, ed. M.R. Chaiken. National Institute of Justice, Washington, DC, 1988, pp. 35-42. 2. Baveja, A., Batta, R., Caulkins, J. P. and Karwan, M. H., Modeling the response of illicit drug markets to local enforcement. Socio-Economic and Planning Sciences, 1993, 27, 73-89. 3. Baveja, A., Optimal strategies for cracking down on illicit drug markets, Unpublished Ph.D. Dissertation, State University of New York, Buffalo,NY, 1993.
306
Alok Baveja et al.
4. Baveja, A., Batta, R., Caulkins, J. P. and Karwan, M. H., Collapsing street markets for illicit drugs: the benefits of being decisive. Working Paper 93-39, H. John Heinz II1 School of Public Policy and Management, Carnegie Mellon University, Pittsburgh, PA, 1993. 5. Bouza, Anthony V., Evaluating street-level drug enforcement. Street-lez~el Drug Enforcement: Examining the Issues, ed M. R. Chaiken. National Institute of Justice, Washington, DC, 1988. 6. Caulkins, J. P., The distribution and consumption of illicit drugs: some mathematical models and their policy implications, Unpublished Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1990. 7. Caulkins, J. P., Larson, R. C. and Rich, T. F., Geography's impact on the success of focused local drug enforcement operations. Socio-Economic and Planning Sciences, 1993, 27, 119-130. 8. Caulkins, J. P., Local drug markets' response to focused police enforcement. Operations Research, 1993, 41, 846-863. 9. Caulkins, J. P., Thinking about displacement in drug markets: Why observing change of venue isn't enough. Journal of Drug Issues, 1991, 22, 17-30. 10. Cesari, L., Optimization theory and applications: problems with ordinary d!fferential equations, (Edited by Balakrishnan). Springer-Verlag, New York, 1983. 11. Feichtinger, G., Granani, A. and Rinaldi, S., Dynamics of drug consumption: a theoretical model, in submission, 1995. 12. Garreau, J., The invisible hand guides D.C.'s visible menace, The Washington Post, A1 (continued on A14), December 29, 1988. 13. Kennedy, D., Fighting the drug trade in link valley, John F. Kennedy School of Government, Harvard University, Working Paper C16-90-935.0, 1990. 14. Kleiman, M. A. R., Crackdowns: the effects of intensive enforcement on retail heroin dealing. Street-Level Drug Enforcement: Examining the Issues, ed. M. R. Chaiken. National Institute of Justice, Washington, DC, 1988, 3-26. 15. Kleiman, M. A. R. and Smith, K., State and local drug enforcement: in search of a strategy. Drugs and Crime, eds. M. Tonry and Q. W. Wilson. University of Chicago Press, Chicago, Illinois, 1990. 16. Moore, M. H. and Kleiman, M. A. R., The police and drugs. Perspectiv,es on Policing, NCJ 117447, National Institute of Justice, Washington, DC, September, 1989. 17. Naik, A., Baveja, A., Batta, R. and Caulkins, J. P., Scheduling crackdowns on illicit drug markets. European Journa/ of Operational Research, 1996, g$, 231-250. 18. Pontryagin, L. S., Boltyanski, V. G., Gamcrelidze, R. V. and Mishchenko, E.F., The mathematical theory ol optimal processes, Pergamon Press, New York, 1964. 19. Reuter, P. and Kleiman, M. A. R., Risks and prices: an economic analysis of drug enforcement. Crime and Justice': An Annual Review of Research, eds. Tonry, M. & Morris, N., University of Chicago Press, Chicago, Illinois, 7. 1986, 289-340. 20. Reuter, P., MacCoun, R., Murphy, P., Abrahamsen, A. and Simon, B., Money from crime: a study of the economics of drug retailing. The RAND Corporation, Santa Monica, Calif. R-3894-RF, 1990. 21. Sethi, S. P., Optimal pilfering policies for dynamic continuous thieves. Management Science. 1979, 25, 535 542. 22. Sherman, L.W., Police crackdowns: Initial and residual deterrence. Crime and Justice: A Review o["Research. eds. Tonry. M. & Morris, N., University of Chicago Press, Chicago, Illinois, 12, 1990. 23. Worden, R. E., Bynum, T. S. and Frank, J., Police crackdowns. Drugs and (?rime: Evaluating Public Polio3' Initiatives, (Edited by Mackenzie and Uchida). Sage, Thousand Oaks, CA, 1994. 24. Zimmer, L., Operation pressure point: the distribution of street-level drug trade on New York's lower east side. Occasional Papers from the Center for Research in Crime and Justice, New York University School of Law, New York, 1987.
Pergamon
When Haste Makes Sense: Cracking Down on Street Markets for Illicit Drugs ALOK BAVEJAt School of Business, Rutgers University, Camden, NJ, 08102, U.S.A. J O N A T H A N P. C A U L K I N S H. John Heinz IIl School of Public Policy and Management, Carnegie Mellon University, Pittsburgh, PA, 15213, U.S.A. WENSHENG LIU Department of Mathematical Sciences, Rutgers University, Camden, NJ, 08102, U.S.A. R A J A N B A T T A and M A R K H. K A R W A N Department of Industrial Engineering, 342 Lawrence D. Bell Hail, State University of New York at Buffalo, Buffalo, NY, 14260, U.S.A. Abstract--This paper presents an analytical approach to the tactical question: 'What level of enforcement over time allows one to eliminate a street market for illicit drugs while expending the least possible total effort?' The analysis is done in the context of Caulkins' model [6] which predicts the rate of change of dealers as a function of enforcement level and several market parameters. Our analysis suggests that the simple strategy of using the maximumavailableenforcementintensity until the market has been eliminated minimizes the total enforcement effort required. © 1997 Elsevier Science Ltd
INTRODUCTION The consumption and distribution of illicit drugs impose significant costs on American society, as do drug control efforts. U.S. drug control policy has taken a criminal justice approach, with considerable attention devoted to local drug enforcement (in contrast to source country control, interdiction, and high-level domestic enforcement). This paper addresses one of the most popular local drug enforcement tactics: cracking down on street level markets. Police crackdowns, a type of focused enforcement, entail an abrupt increase in police activity, especially proactive enforcement, which is intended to dramatically increase the perceived and/or actual threat of apprehension for specific types of offenses in certain places or situations and, hence, produce a general deterrent effect [23]. However, crackdowns invariably entail high costs, particularly those involving personnel costs [15]. Kleiman [14] discusses crackdowns at length, but a simple working definition is: 'An intensive local enforcement effort directed at a particular geographic target.' Crackdowns are distinguished from day-to-day enforcement operations, which generally spread resources more or less uniformly over the 'problem,' and other special operations that may target individuals, organizations, a particular drug, a class of users, or an ethnic group rather than a geographic target. The efficacy of crackdowns has not been definitively established. (For competing views see, e.g., Chaikan's N I J Issues in Brief [1, 5, 14]. While there are strong theoretical arguments favoring local enforcement in general [16, 19], some crackdowns [24] have dramatically improved the quality of life in target areas, while others [14] have been dismal failures. Despite this mixed track record, crackdowns continue to be implemented in cities across the country. tAuthor for correspondence. 293
294
Alok Baveja et al.
Crackdowns raise a number of management issues, including selecting the sequence of target neighborhoods [17], deciding which tactics to employ, and coordinating the activities of police with both other governmental and community-based groups. One important management problem that has received little attention and essentially no formal analysis is: 'What is the best rate at which to crack down on a market?' Should one start slowly and build up gradually? Should one begin aggressively and then ease back? This paper suggests that in many cases 'as forcefully as possible. starting immediately and continuing until the job is done' may be a sound heuristic solution. In the next section, we review some relevant literature, we then develop the intuition underlying our results via two extreme cases. Subsequently, the question is formulated as an optimal control problem and its solution presented. Finally, we illustrate the results with a numerical example. PRIOR W O R K Crackdowns have been conducted in numerous cities, some have been analyzed, but empirical evidence on their effects is scarce [14, 23, 24]. Indeed, it is rare for conditions to resemble a controlled experiment; police and other officials are understandably reluctant to subordinate the immediate needs of their constituencies to the long-term interests of research. Occasionally, natural experiments can be exploited (e.g. [7]), and there are now several studies under way that have gone to extraordinary lengths to conduct valid experiments. Even in these exceptional examples, however, the dependent variables of greatest interest, e.g. dollar volume of sales, number of dealers in the market, etc., are exceedingly difficult to estimate even approximately, and surrogate measures are weak. In the face of these limitations, a complementary way to develop insight about crackdowns is to explore relevant mathematical models of them and present analytical results. Beginning with the perceived risks and profits of being a drug dealer, Caulkins [6, 8] presents a model for quantifying the rate of growth (or decay) of a drug market under crackdown enforcement. The rate of change of dealers with respect to time in a given market is postulated to be
(l) where N = t= Q ( N ) = ~N a = ct, fl = 7= E =
the number of dealers in the market, time, number of sales per unit time, demand parameters, risk aversion parameter, increment in enforcement pressure, above and beyond the baseline level, that is placed on the market during the crackdown, C~ = proportionality constant, rc = generalized profit per transaction, and w0 = a dealer's reservation wage.
In words, the rate of change of dealers is proportional to the difference between the utility available to dealers in the market and their reservation wage. The dealer's utility is his/her generalized profit (profit per sale net of pecuniary and other costs times the number of sales per dealer per unit time) minus his/her perceived risk from crackdown enforcement (modeled as the enforcement risk per dealer raised to an appropriate power). The reservation wage is simply what the dealer could earn in alternative employment (including dealing elsewhere). Thus, the model treats drug markets as an economic entity where there is the potential for profit (from drug sales), the risk of being caught (by enforcement), and an alternative means of employment. A dealer will be motivated to enter the market if and only if his/her expected utility exceeds that offered by the next best alternative. The model considers police crackdowns which target particular geographical areas where concentrated drug dealing takes place, referred to as 'Drug Markets.' The sales volume depends
Cracking down on street markets for illicit drugs
295
directly on the number of dealers, N; enforcement, E, is assumed to affect sales only indirectly by changing N. The model also assumes that the number of drug markets is large enough that cracking down on one does not affect the return available to dealers in other markets. Drug dealers are modeled as identical and indistinguishable. Analyzing the steady state behavior of this model under constant enforcement, Caulkins [6, 8] obtains a number of results; notably, that for an enforcement operation to have lasting effects, the market should be driven below a minimum viable size. An argument is put forth for having enough resources to prevent the market from re-emerging. Baveja et al. [2] analyze the dynamic behavior of drug markets using the above model, and generalize the steady state results by showing that (i) the general shape of the plot of dN/dt as a function of N is all that is required to derive those results, (ii) the shape of the plot of dN/dt is generalizable to any parameter values, i.e. its shape is the same for different drug markets, and (iii) this shape holds for any function that satisfies a set of intuitively appealing properties. However, none of the work described thus far has addressed the question of finding the time trajectory of enforcement effort, E(t), i.e. that which eliminates a drug market while expending the least possible enforcement effort. Sherman [22] considers this issue but does so only qualitatively, without the aid of mathematical models or a formal analysis. His paper argues that a crackdown-backoff strategy, in which crackdowns are conducted for a limited duration and rotated across targets, could capitalize on the benefit of a free bonus due to residual deterrence. Feichtinger et al. [11] develop a continuous time model using a fixed enforcement level that incorporates addicted and dealer populations. They suggest the existence of two stable equilibria for each enforcement level and propose a two-step control policy. However, their model is more relevant at the national level rather than for local crackdowns. The question we raise for street-level drug markets has not been investigated using a formal, quantitative analysis. The next section thus develops a mathematical model to provide guidance on this issue.
ANALYSIS We here formulate the problem of finding E*(t), the optimal time trajectory of enforcement, that which eliminates a drug market while using the least amount of enforcement resources. This problem is solved using optimal control theory. The results are then extended to more general objective functions. However, to develop intuition about the optimal strategy, we first consider a simplified version of the problem.
Constant-enforcement policy In the context of Caulkins' model, we look for a constant enforcement strategy that minimizes the total amount of resources used. In order to do so, we consider two specific extreme markets called the sellers' and the buyers' markets [2, 6]. An actual drug market would probably be between these two extremes and, therefore, the results should provide insights relevant to determining optimal policy for the general case. Sellers' market Informally, a sellers' market is one in which demand for illicit drugs is abundant. The implication of abundant demand is that another arriving dealer will expand total market sales and not reduce individual sales for any of the original dealers. This can be modeled by setting fl = 1 in (1). For sake of simplicity and analytical convenience, we assume ~ = 1, i.e., dealers are risk neutral. Let T~(E) denote the time needed to eliminate the market when a constant enforcement, E, is used; let Emaxbe the maximum level of enforcement possible at any instant of time. The objective is to find the enforcement that minimizes the total resources used to eliminate the market, where total resources are the integral, over time, of enforcement at each instant of time. Lemma 1: Under constant enforcement, using the maximum available enforcement minimizes the total enforcement resources used to eliminate the market.
296
Alok Baveja et al.
Proof: Under constant enforcement, the total resources used are ETa(E). We need to show that Min
Et,.(E) = Em,~T,(Em,0.
0 < E ~< Ema x
Solving for T,(E) using (1), and incorporating boundary conditions, we obtain:
~(E)-
-No C,0t~ - Wo) + C,(rtct-L w0)2 In E - (net - wo)No
"
So, objective function Z = E T a ( E ) is:
Z=E
G(~Two)
+ G ( r t ~ - w o ) 21n E - ( m T L W o ) N o
"
(2)
From (1), by setting d N / d t = 0, we first note that the enforcement required to maintain a sellers' market of size N is E h ( N ) = (rt~ - wo)N. Therefore, to eliminate a market of size No, we need an enforcement E > (net - wo)No. If we let (n~ - w0) = b, this condition can be restated as E / b = m must be greater than 1. Using this, we can write (2) as:
m~m
(3)
Taking the derivative of Z with respect to m, we get:
dm -
C,
~
-
+ m--Z-l--1
"
For m > 1, d Z / d m is strictly negative. Since m is proportional E, Z is a strictly decreasing function of m. Therefore, E = Emax is optimal. A plot of Z vs m [3] shows that Z decreases sharply as E increases for enforcement less than 1.5b. Thus, little is gained by increasing enforcement beyond two times the minimum enforcement required to eliminate the market. The minimum value of the enforcement that does not compromise on the total enforcement resources used will therefore be approximately 1.5b. Buyers' market
In the context of drug markets, a buyers' market is one in which there is a fixed number of sales with dealers simply fighting for market share. Increasing the number of dealers would not increase the number of sales since there is already a surplus of dealers. This scenario can be modeled in (1) by setting fl = 0. As before, ? is assumed to be equal to one. By setting b = net = woNo, a similar argument to that in Lemma 1 shows that Z is a decreasing function of m and, hence, E. A plot of Z v s . m for a buyers' market [3] gives similar inferences. In this case, the objective function does not significantly improve ( < 10%) for an enforcement level greater than 1.2b. Therefore, if one is constrained to use constant enforcement, for both extreme cases the optimal strategy is to use the maximum possible enforcement intensity. This result can easily be extended to policies where enforcement can change at discrete points in time [3, 4]. Under such circumstances, using the maximum enforcement is still optimal for both the buyers' and the sellers' markets. However, these results cannot be verified for an intermediate market using this method. The next section considers that more general case within a continuous-time enforcement policy scenario. Generalized problem formulation
Police/policy-makers are concerned with (1) budgetary costs of enforcement and (2) a variety of problems associated with the markets they are seeking to eliminate (e.g. drug use, disorder,
Cracking down on street markets for illicit drugs
297
market related violence, etc.). Other concerns such as officer safety, equity and public relations, though important, are beyond the scope of this paper and are thus not considered in the model. Initially, we analytically address the simpler problem of minimizing the amount of enforcement expended in the process of eliminating a street market. Later, we discuss the implications of considering more general objective functions that include the size of the market. The basic problem considered here is the following. Suppose the police wish to reduce the number of drug dealers in a particular street market from an initial level No to a final level of NI = 0. (Note, the problem with any general level NI > 0 is no more difficult to solve and yields similar results [4], but typically the goal is to eliminate a market, not just reduce it.) The objective is to accomplish this reduction while expending the least possible amount of enforcement resources, subject to a constraint on the maximum intensity of enforcement (E(t) < Emax) and the assumption that the market behavior is as described by eqn (1). We thus have: Minimize Z =
~0TE(t)dt,
subject to:
l, N(O) = No,
N ( T ) = O, 0 < E(t) < Emax N(t) >_ O, where 0 < fl < 1,7 _> I(OPC) This problem is set up for application of optimal control theory. Optimal control theory has been used for a variety of problems in the fields of economics, environment etc. For example, Sethi [21] used it to determine the optimal pilfering policy of a profit-maximizing thief. The terminal time, T, is the time when the market is eliminated. From an optimal control theory perspective, the formulation is one in which the terminal time is not specified. One aspect of this formulation might seem artificial at first, but is not in fact constraining. Indeed, it is analytically convenient to express the problem in continuous time, but the reader might justifiably be concerned about the relevance of that exercise, since enforcement pressure can only be modified at a finite number of discrete points in time. However, the optimal continuous-time solution is to apply the maximum possible enforcement effort throughout, which is a simple policy to implement. That is, even when one admits a broad range of possible solutions, it turns out that the optimal solutions happen to fall in a restricted class of policies that can practically be implemented. Furthermore, similar results can be obtained for a formulation that explicitly considers discrete-time policies.
Solution In order to show that using the maximum enforcement is optimal we need to establish both the appropriate necessary and sufficient conditions. The necessary condition will be arrived at using the Maximum Principle while the sufficient condition will be demonstrated using transformations of the optimal control problem (OPC) to an equivalent problem known to have a solution. Before exploring the solution methodology, we present two important conditions that will be used in the analysis. First, for (OPC) to have a feasible solution, the constant control, E = Em~x, should be able to steer No to 0. That is, dN/dtlE~E~,, must be negative for all values of N: 0 < N < No; otherwise, the problem has no feasible solution. We therefore restrict our attention to markets (with parameters n, ~t, fl, ?, w0, No) that can be eliminated if the maximum possible enforcement is used.
298
Alok Baveja et al.
Second, from eqn (1) it is clear that if
I dN/dt will be negative even when no enforcement is applied. Therefore, the least costly way to begin
to reduce the size of such a market is simply to let it shrink on its own with E = 0. However. mathematically, the market will approach N,. asymptotically and never reach the equilibrium value in finite time. From a theoretical standpoint, the problem will have no optimal control since, to reach -Arcin finite time, we will have to apply a non-zero enforcement that will not be optimal. From a practical perspective, it makes no sense to use expensive enforcement and punishment resources against dealers if the dealers are exiting on their own. Therefore, we restrict our attention to markets where
No < N~
\ w'o /
Note that for the case No = Nc, a market can stay at its equilibrium level without expenditure of any resources. Mathematically, you can wait indefinitely before undertaking a crackdown. The rest of the analysis in this section considers the start time as that instant when the crackdown operation is to be initiated with a non-zero enforcement level. For convenience, we let y = N :+' and m ( y ) = C,(7 + l)[ltcz), ~
- woy:,~ i
We can therefore rewrite (OPC) more simply as: Min Z ( E ) =
~0TE(t)dt
subject to:
ay dt-
re(y) - C,(7 + I)E%
y(0) = Y0, y ( T ) = O,
0 < E < Emax.
Theorem 1: For any 0 < Y0 < Yc, the control E = E,,ax on [0, T~] is optimal for (OPC). Proof: Let U = E/E~,,x and consider the following control problem, (OPC)I, equivalent to (OPC). Min Z ( U ) =
~0TU(s)ds
subject to: ely = re(y) - CU'; dt y(O) = Yo,
y(T) =
O,
O
Cracking down on street markets for illicit drugs
299
T o prove the theorem, we need to show that U = 1 on [0,T~] is optimal for (OPC)~. In order to establish this, we first look at the special case when 7 = I. F o r this case of ~ = 1, it is fairly easy to show that (OPC)~ has a solution, i.e. has an absolute m i n i m u m . This existence result also follows directly from existence theorems in optimal control theory (e.g. T h e o r e m 9.3.i in [10]). Let U = U(t) on [0,T] be an optimal control for this special case o f (OPC)~. Further, let y = j7(0 be the solution on [0,T] o f y' = re(y) - CU(t), y(O) = Yo, where 0 < iT(t) < y0.(4) By the Pontryagin m a x i m u m principle [18], we know that there exists a co-vector, 2(0, on [0,T] that satisfies 2 ' ( 0 = - 2(t)m'(fi(t)) and a constant 20 > 0 such that: (i) (2(0,20) ~ 0 for all t q 0 , T ] , (ii) F o r almost all t~[O,T],O(t) minimizes
H(.~(t),U,2(t),2o) = 2(t)[m(.~(t)) - CU] + 2oU = 2(t)m(fi(t)) + (20 - C2(t))U and (iii)
H(~(t),8(t),2(t),2o =- O. Note that (20 - C2(t)) ~ 0 on (0,T). Indeed, if there is a ? ~(0,T) such that 20 - C2t" = 0 then (iii) implies that =
o
(for t~(O,T), m07(t)) :/: 0). So 2 ( 0 - 0 to [0,T]. But then 20~:0 by (i), contradicting 20 - C2t" = 20 = 0. Therefore, if20 - C2(t) > 0, U(t) = 0 for almost all t([0,T], and if20 - C2(t) < 0, then O(t) = 1 for almost all t by (ii). Clearly then, 0(t) must be equal to one almost everywhere on [0,T]. This shows that if U = 0 ( t ) is optimal for the special case of (OPC)~ when ), = 1 and if (4) holds, then U(t) = 1. The optimal cost is equal to T~. Having established that U(t) = 1 is optimal for the case when ~ = 1, we next use this to prove it for any general ~ value. Let U = U(t) on [0,T] be any control such that, 0 < U(t) < 1, and the corresponding solution of y' = m(y) - U'~(t),y(O) = Yo satisfies y ( T ) = 0. Then we know that SorU"(s)ds >_ T~ from above. Since 0 < U(t) < 1 and ~, > 1, we have SorU(s)ds > ~U)'(s)ds > T~. This implies that U = 1 on [0,T~] is optimal for (OPC)~. Therefore, E = Emax is optimal for (OPC). Thus, if the m a r k e t is such that No < No, the optimal strategy is to move with the most force possible at the earliest possible time, a policy that is easy to implement. An i m p o r t a n t side-benefit of this optimal policy is that the m a r k e t will continuously decrease in size since dN/dt IE= ~ma, is always negative. It is fortuitous that the optimal policy generates a shrinking market, because if this were not so, it would be very difficult indeed to justify it politically.
Numerical example We now consider a numerical example to illustrate both the importance of using an optimal enforcement strategy and some m o r e subtle insights*. T o do so, we need to find reasonable estimates for the parameters. Reuter et al. [20] estimate that, in 1988, Washington, D C had about 14,000 full-time-equivalent retail dealers, and G a r r e a u [12] reports that there were about 90 distinct street markets, suggesting a ratio o f 155 dealers per market. Presumably, not all retail sellers operate out o f open-air street markets, since the average m a y be biased by a few big markets, 100 dealers m a y therefore be a reasonable estimate for a typical street market. Values for the p a r a m e t e r fl lie between 0 and 1 while ~, is at least 1. F o r convenience, we take as our base case fl = 1/2 and ? = 1 and report the results of some sensitivity analyses with respect to these parameters. *Examples were run using a differential equation subroutine of the IMSL math library on a UNIX-based mainframe computer.
300
Alok Baveja et al.
The reservation wage, w0, can be estimated by the legitimate employment wage for a drug dealer. Reuter et al. [20] estimated this to be about $1,046 per month, or approximately $50 per work day. Based on the equilibrium condition that the dNidt = 0 when E = 0, we obtain ~.~ = 500. With these parameter values, an enforcement level of E > 1250 is needed to eliminate the market [2]. We are interested first and foremost in the relationship between the intensity of enforcement effort applied and both how long it takes to eliminate the market and the cumulative amount of effort exerted in doing so. Figure la shows the former relation for three values of fl, When B = 1/2,
(a)
~
o.9
lg
p=o.]
p--o.~
10 5 0250---
i
750
lifo
t250
1450 1 7 0 0 1950 22100 2 4 5 0 2 7 0 0 2 9 5 0 1
3450 3650 I
4150 1
I
Enforcement Intqmtlty
(b)
:t T= 1.75 I 15
10
i . 20S
i -¸ 7oo
~50
12?5
1~
1"r50
20OO
EnfarcemmM I ~ I ~ F
Fig. 1. (a) Time to eliminate market as a function of enforcement intensity and deraand parameter ft. (b) Time to eliminate market as a function of enforcement intensity and dealers' risk aversion parameter ?.
Cracking down on street markets for illicit drugs
301
1.25l i
8=4
1.2
ff~ .,/'J
1.15 0=3
1.1
1.85
i
~
J /
8=1 6=4
J.q tI-
*5
0.85¸
o.o
0.85 0.S
100
I 2OO
J 3OO
I I I 40O S00 e00 Amplitude of PulN In Enforcement
I
B
7OO
eo0
Intensity
I 900
I 10~0
(/1)
Fig. 2. Pulsed crackdowns are usually less efficientthan constant intensity crackdowns. for enforcement intensities below 1250 no amount of time is sufficient to eliminate the market. With E = 1875, or just 50% more than the minimum required, the time needed for elimination is less than five days. As enforcement intensity increases beyond 1875, the time to eliminate the market is reduced further, but there are sharply diminishing returns. Indeed, it is striking how 'kinked' the curves are in Fig. 1a. Furthermore, plots of total effort required vs. enforcement intensity (not shown) are even more sharply 'kinked'. Figure la also shows that, for different markets with the same number of dealers in equilibrium before a crackdown, the higher fl is, the easier it is to eliminate the market. This makes intuitive sense. Thus, if fl is high and enforcement drives away a few dealers, the number of customers is also reduced, which, in turn, makes the market less attractive for the remaining dealers. This feedback generates a negative spiral. In contrast, when fl is small, even if enforcement reduces the number of dealers, the number of customers remains relatively high. This increases the ratio of customers to dealers, helping to offset the additional enforcement pressure experienced by the smaller number of dealers. Figure l b also shows the relationship between enforcement intensity and time to eliminate a market, but it does so for various values of ~. Two observations deserve mention. First, the larger is, the more kinked the curve. Since Fig. la assumed 7 was unity, it was conservative in the sense of showing the 'least kinked' case. Second, the larger ~, is, the easier it is to eliminate a market. This makes intuitive sense since ~ is a measure of how risk-averse the dealers are. Figure 2 pertains to Sherman's [22] recommendation of a 'crackdown backoff strategy which alternates periods of intense pressure with periods of more modest enforcement. The goal of such a strategy is to take advantage of residual deterrence. Informally, the hope is that during the relative lulls in enforcement, dealers will remember and make judgments based not on the current enforcement risk, but on the enforcement risk operative during the intense portion of the crackdown. This model cannot evaluate the upside potential of Sherman's strategy because it is predicated on the assumption that dealers correctly deduce and react to the actual, current enforcement intensity. That is, it assumes a priori that there is no such thing as residual deterrence. However, we can quantify how badly Sherman's strategy might stumble if its hope of residual deterrence proves elusive. To do so, we need to model both Sherman's strategy and an appropriate referent and then compare their results. We model a 'crackdown backoff' strategy as alternating between SEPS 31/4--C
302
Alok Baveja
et al.
E = 1400 + A and 1400 - A with a period of 20 days. We choose, as a foil, applying a fixed level of enforcement equal to the average enforcement intensity applied by the crackdown, backoff strategy. For example, if applying E = 1700 and E = 1100 every other day would eliminate the market in exactly three days, then we take as a foil applying a constant effort of E = (1700 + 1100 + 1700)/3 = 1500 for however long it took to eliminate the market. Figure 2 shows the ratio of how long it takes a crackdown, backoff strategy to eliminate a market relative to the time required by its constant intensity foil. There are three observations worthy of note. First, there is a rather complicated, nonlinear relationship between the amplitude (A) and period (0) of the oscillations in enforcement intensity and the effectiveness ratio. Second, at least for the cases we examined, in no case was the crackdown, backoff strategy markedly less effective than its foil. Third, there are instances in which the crackdown, backoff strategy is superior to its foil (e.g., 0 = 3 and A small). At first glance, this last result might seem to contradict the paper's main finding. However, what Theorem 1 finds is not that any constant enforcement strategy is optimal, but, rather, that applying the maximum possible enforcement throughout is optimal. By definition, if it is feasible to alternate between some base intensity plus A and that intensity minus A, then the base intensity cannot be the maximum feasible intensity and there is no reason to expect that maintaining an intensity at that level will be optimal. Figure 3 addresses the question of whether a crackdown that ramps up intensity over time is more or less effective than one that ramps it down. One can construct 'arm-chair' arguments in favor of either approach. Ramping enforcement intensity down over time allows one to focus the greatest effort on when the market is largest (at the beginning of the crackdown). On the other hand, ramping enforcement up reserves the greatest effort for the end of the crackdown when the market has been made more vulnerable by earlier efforts. To explore the issue quantitatively, we examined strategies that vary enforcement intensity linearly between E = 1275 and E = (1 + K)*1275 for various values of K. Ramp up strategies start with E--- 1275 and linearly increase enforcement intensity at a rate such that they just reach E = (1 + K)*1275 at the moment the market is eliminated. Ramp down strategies start with E = (1 + K)*1275 and decrease enforcement linearly to 1275 at the moment the market is eliminated. Obviously, such contrived strategies cannot be implemented in practice because one
t
+| i!i ~.
-
i!+ I~|11 I It o.~, 15 J
0.9 0.85
0.8
I
I
0.2
0,4
- - 4 0.6
z
I
I
I
I
I
I
0.8
1
1.2
1,4
1,6
1.8
2
2.2
[
i
2,4
2.6
Magnitude of Ramp as Mulllple of Base (K)
Fig. 3. Ramp-down strategies are less effective than ramp-up strategies.
- i 2.8
Cracking down on street markets for illicit drugs
303
~
0.9
'~o.e 0.7
J
~ 0.5 ~0.5
~
0.4 .Q ~S0.3 0
~ 0.2 a ~.~0.1 I
0.5
2.5
4.5
05.
85.
10.5
12.5
14.5
I
16.5
15.5
20.5
22.5
24.5
26.5
28.6
11m (aa~) Fig. 4. If crackdown will eliminate market, it usually does so quickly.
does not know a priori how long it will take to eliminate a market. However, their symmetry facilitates comparison. Figure 3 plots the ratio of time to eliminate the market for the ramp down strategy relative to that for the ramp up strategy. (Note: by construction, this is also the ratio of the total amounts of enforcement used.) For every value of K, the ramp down strategy took more time and more resources than did the ramp up strategy. This suggests that strategies wherein enforcement intensity is proportional to market size as it decreases over time may be ill-advised. Figure 4 addresses one final issue: how persistent should the organizers of a crackdown be? Above, we argued that all available resources should be deployed. Elsewhere [2, 8] we have argued that crackdowns should be pursued long enough to drive a market below its minimum viable size; otherwise, the market will re-emerge to its pre-crackdown size after the crackdown has ended. Figure 1 makes clear, however, that if the resources available are insufficient (E < 1250 in our base case), then no matter how long the crackdown is maintained, the market cannot be eliminated. So, suppose all available resources have been summoned, a crackdown has been instituted, and after a week the market has not been eliminated. Should the crackdown be continued in hopes that a little more time and patience are all that is needed? How about after a month, should the crackdown be continued? After a year? At the blackboard, these are easy questions. If the enforcement available exceeds the critical level (e.g. E > 1250) then one should continue; otherwise, abort. Practically, however, these are difficult questions because the critical level of enforcement is not known. Furthermore, it is typically difficult to directly observe changes in the number of dealers, and, in some cases, (markets with small fl's) the number of dealers will not decline much before it takes a sudden, precipitous plunge to zero. To solve this problem formally, one would need to define a prior probability distribution on the relationship between E and the minimum enforcement needed to eliminate the market. That probability distribution would then need to be updated based on observations of how long the market has withstood the crackdown. Doing this market by market is daunting, but performing a similar exercise for our illustrative market is informative. Suppose the available enforcement intensity is equally likely to be any value between 0 and three times the minimum necessary to eliminate our base case market. Then, Fig. 4 shows the cumulative distribution of the time necessary to eliminate the market. There is a one-third chance the market will never be eliminated. There is a 60% chance the market will be eliminated within a week. There is only a chance of about 2% that a market will survive for a month but eventually succumb.
304
Alok Baveja et al.
The particular probabilities depend on the parameter values chosen, especially the prior range for the available enforcement intensity relative to the minimum needed. The basic message, however, is less equivocal. Persistence and patience may be virtues, but generally if a crackdown is going to succeed, it will succeed relatively quickly. If, after a time, the market still stands, it might be wise to obtain additional resources, change tactics, or pursue other, smaller and more manageable targets. Extension to more general objective functions Policy makers are concerned not just with the budgetary costs of enforcement but with the societal costs imposed by drug markets. Thus, if the former are related to enforcement effort (E) and the latter to the number of dealers (N), one might prefer to minimize a more general cost function C(E,N) = f ( E ) + g(N). Note, though, that the optimal solution in the very special case of C(E,N) = E involves reducing N as rapidly as possible. Hence, afortiori this same strategy is optimal for any cost function C(E,N) = E + g(N) for which g(N) is an increasing function of N. The societal costs associated with drug markets are almost certainly increasing in the number of dealers, N. The strategy of cracking down with the maximum possible effort as soon as possible will thus still be optimal for this more general class of objective functions. A similar argument can be made for generalizations to C ( E , N ) = f ( E ) where f ( E ) is concave in E. However, because of diminishing returns, it seems more likely that the cost of imposing a particular enforcement effort would be convex in E, not concave. For example, the marginal cost of additional enforcement might increase with the amount of effort already imposed because of the need to pay overtime or to employ less efficient personnel. If one allows f ( E ) to be convex in E, it is trivial to construct examples for which it is no longer optimal to apply the maximum possible effort as soon as possible. Hence, the principal result carries an important caveat. When f ( E ) is convex, one's best hope might be to use the model as a simulation for what-if analysis. Note that even if the cost function is convex in E beyond some threshold level, the analysis can still yield insight. For example, suppose that f ( E ) is linear in E up to a particular threshold value and convex in E thereafter. Although one cannot find the overall optimal enforcement trajectory without estimating 'difficult-to-measure' parameters, one can still conclude that it would never be optimal to employ any level of effort below that threshold value. DISCUSSION This paper addresses the problem: 'What is the optimal time trajectory of enforcement that should be applied during a crackdown on a local drug market?' It concludes that deploying the maximum possible effort from the earliest possible time minimizes the total effort (integral over time of as a function of time) required to eliminate a market. Crackdowns that reduce enforcement intensity as the market shrinks may be particularly inefficient. Full-intensity crackdowns that have made little progress after a moderate amount of time might best be revised or abandoned: usually, if a crackdown will suppress a market, it will do so relatively quickly. A point that may seem obvious but which nevertheless deserves mention is that the level of effort employed must be one that can be sustained long enough for the market to actually be eliminated. Because of certain economies of scale, street markets for drugs have a minimum viable size. It is not difficult to keep markets below this size from growing. In particular, once a market has been completely eradicated, it is relatively easy to keep it from springing back; no one wants to be the first dealer to challenge the police. Conversely, if the market is still above its minimum viable size when the crackdown ends, it will grow back. Too often in practice police have begun crackdowns intensely but have been unable to maintain the pressure long enough. The resulting gains are thus only temporary, and the market soon returns to its original size. Of course, the above result is subject to a number of caveats. First, as noted in the previous section, the result may not hold at all if the marginal cost of applying additional effort increases significantly with the amount of effort deployed. Second, the results are predicated on the reasonableness of the underlying model. Models are inevitable simplifications of reality. However, in this case an extra 'grain of salt' is appropriate since data limitations imply that validation of models of illicit activity is extremely difficult. While we
Cracking down on street markets for illicit drugs
305
believe that the model captures fundamental dynamics of the prototypical street corner market for illicit drugs, we would caution practitioners to review the assumptions underlying the model and to think carefully about the idiosyncrasies of the particular situation they confront before adopting the results. Third, there are some subtleties associated with interpreting the objective function, particularly the quantity E, which is referred to as the enforcement effort. Returning to eqn (1), one sees that this enters the dealers' utility function as a cost. Since this utility function governs a future decision (namely, to enter or leave this market), it is actually the expected future cost from enforcement of dealing in the market in question. Hence, the true lesson of this analysis is to raise the expected cost of dealing as quickly as possible to the greatest extent possible. The most natural way to raise expectations of costs associated with enforcement is by actually applying enforcement pressure. Indeed, the model assumes that these are essentially identical, but there may be ways of raising the dealers' expectations about those costs without actually imposing them. For example, in Houston [13], a planned crackdown received so much media attention that by the time the police arrived, the market had already disappeared. The credibly communicated threat of enforcement was sufficient to eliminate the market. At the other extreme, in theory a crackdown could exert so much effort that the dealers would conclude that it could not be sustained and, hence, expect the future risk of enforcement to be relatively low. The point is that it is the expectation of future costs, not the actual costs imposed, that ultimately govern the flows of dealers in and out of the market. Fourth, there is the important issue of displacement. The underlying model focuses on a single local market even though most cities have a dozen or more distinct open-air markets. The existence of these alternative markets suggests that the outcome of the crackdown may not be to force dealers out of business, but rather to force them to relocate their operations. Baveja [3] developed a model that incorporates dealer displacement in which interaction between different markets is explicitly considered. Using control theory analysis, similar to the one developed in the paper, he demonstrated that extreme point solutions are still optimal. Practically, this implies a policy using the maximum possible enforcement on a given market and collapsing markets sequentially. The mere possibility of displacement does not imply that cracking down on a single market is not beneficial. A successful crackdown would still remove the blight from one neighborhood. More importantly, if the crackdowns are strategically planned, the displacement might create forms of dealing that impose fewer negative externalities on society. For example, displacing dealing from street corners to indoor locations might reduce the incidence of drive-by shootings and other street violence. Caulkins [9] discusses these considerations at length; but, in summary, although the possibility of displacement should certainly never be ignored, it also is not a prima facie case for the futility of crackdowns. Finally, note that this paper in no way argues that crackdowns, or even local enforcement, are efficacious. One might object to the very concept of local crackdowns for reasons that are beyond the scope of this model (e.g. because of the likelihood that a disproportionate fraction of the arrests will be imposed on minorities). This paper merely suggests that, if the decision has been made to crack down on a market, then the most efficient way to reduce the size of the market is to apply the maximum possible effort from the earliest possible moment. This result is important in and of itself, however, precisely because, in so many cities across the country, the decision to crack down on local markets has already been made. Acknowledgement--The authors thank anonymous refereesfor their many helpful suggestions.
REFERENCES I. Barnett, A., Drug crackdowns and crime rates: a comment on the Kleiman paper, Street-level Drug Enforcement: Examining the Issues, ed. M.R. Chaiken. National Institute of Justice, Washington, DC, 1988, pp. 35-42. 2. Baveja, A., Batta, R., Caulkins, J. P. and Karwan, M. H., Modeling the response of illicit drug markets to local enforcement. Socio-Economic and Planning Sciences, 1993, 27, 73-89. 3. Baveja, A., Optimal strategies for cracking down on illicit drug markets, Unpublished Ph.D. Dissertation, State University of New York, Buffalo,NY, 1993.
306
Alok Baveja et al.
4. Baveja, A., Batta, R., Caulkins, J. P. and Karwan, M. H., Collapsing street markets for illicit drugs: the benefits of being decisive. Working Paper 93-39, H. John Heinz II1 School of Public Policy and Management, Carnegie Mellon University, Pittsburgh, PA, 1993. 5. Bouza, Anthony V., Evaluating street-level drug enforcement. Street-lez~el Drug Enforcement: Examining the Issues, ed M. R. Chaiken. National Institute of Justice, Washington, DC, 1988. 6. Caulkins, J. P., The distribution and consumption of illicit drugs: some mathematical models and their policy implications, Unpublished Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1990. 7. Caulkins, J. P., Larson, R. C. and Rich, T. F., Geography's impact on the success of focused local drug enforcement operations. Socio-Economic and Planning Sciences, 1993, 27, 119-130. 8. Caulkins, J. P., Local drug markets' response to focused police enforcement. Operations Research, 1993, 41, 846-863. 9. Caulkins, J. P., Thinking about displacement in drug markets: Why observing change of venue isn't enough. Journal of Drug Issues, 1991, 22, 17-30. 10. Cesari, L., Optimization theory and applications: problems with ordinary d!fferential equations, (Edited by Balakrishnan). Springer-Verlag, New York, 1983. 11. Feichtinger, G., Granani, A. and Rinaldi, S., Dynamics of drug consumption: a theoretical model, in submission, 1995. 12. Garreau, J., The invisible hand guides D.C.'s visible menace, The Washington Post, A1 (continued on A14), December 29, 1988. 13. Kennedy, D., Fighting the drug trade in link valley, John F. Kennedy School of Government, Harvard University, Working Paper C16-90-935.0, 1990. 14. Kleiman, M. A. R., Crackdowns: the effects of intensive enforcement on retail heroin dealing. Street-Level Drug Enforcement: Examining the Issues, ed. M. R. Chaiken. National Institute of Justice, Washington, DC, 1988, 3-26. 15. Kleiman, M. A. R. and Smith, K., State and local drug enforcement: in search of a strategy. Drugs and Crime, eds. M. Tonry and Q. W. Wilson. University of Chicago Press, Chicago, Illinois, 1990. 16. Moore, M. H. and Kleiman, M. A. R., The police and drugs. Perspectiv,es on Policing, NCJ 117447, National Institute of Justice, Washington, DC, September, 1989. 17. Naik, A., Baveja, A., Batta, R. and Caulkins, J. P., Scheduling crackdowns on illicit drug markets. European Journa/ of Operational Research, 1996, g$, 231-250. 18. Pontryagin, L. S., Boltyanski, V. G., Gamcrelidze, R. V. and Mishchenko, E.F., The mathematical theory ol optimal processes, Pergamon Press, New York, 1964. 19. Reuter, P. and Kleiman, M. A. R., Risks and prices: an economic analysis of drug enforcement. Crime and Justice': An Annual Review of Research, eds. Tonry, M. & Morris, N., University of Chicago Press, Chicago, Illinois, 7. 1986, 289-340. 20. Reuter, P., MacCoun, R., Murphy, P., Abrahamsen, A. and Simon, B., Money from crime: a study of the economics of drug retailing. The RAND Corporation, Santa Monica, Calif. R-3894-RF, 1990. 21. Sethi, S. P., Optimal pilfering policies for dynamic continuous thieves. Management Science. 1979, 25, 535 542. 22. Sherman, L.W., Police crackdowns: Initial and residual deterrence. Crime and Justice: A Review o["Research. eds. Tonry. M. & Morris, N., University of Chicago Press, Chicago, Illinois, 12, 1990. 23. Worden, R. E., Bynum, T. S. and Frank, J., Police crackdowns. Drugs and (?rime: Evaluating Public Polio3' Initiatives, (Edited by Mackenzie and Uchida). Sage, Thousand Oaks, CA, 1994. 24. Zimmer, L., Operation pressure point: the distribution of street-level drug trade on New York's lower east side. Occasional Papers from the Center for Research in Crime and Justice, New York University School of Law, New York, 1987.