Bulletin on Narcotics
Volume LIII, Nos. 1 and 2, 2001
Dynamic drug policy: Understanding and controlling drug epidemics
The need for dynamic models of drug markets1P. REUTER
Professor, School of Public Affairs, University of Maryland, College Park, Maryland,
and Senior Fellow, RAND Drug Policy Research Center,
Santa Monica, California, United States of America
Markets for cocaine and heroin are created out of epidemics and are best analysed in terms of dynamic models rather than comparative statics. The author presents the following three problems requiring dynamic models and sketches a potential approach to each one: (a) why prices have declined so persistently in the face of intensified enforcement; (b) informing the choice between buyer and seller targeting for enforcement; and (c) explaining the extremely high variability of retail prices in such markets. For the first two problems, critical issues are the modelling of the path of sources of earnings (such as other crime or legitimate activities) and the drug-selling labour supply as consumption becomes dominated by drug users with increasingly long criminal histories and fewer legitimate employment opportunities. For the third problem, models need to take into account both the difficulty of ascertaining quality at the time of retail purchase and the instability of use and sale opportunities within those markets.
Illicit cocaine and heroin markets in the United States of America2 appear to be different from legal markets in a number of ways: high levels of violence; rapid turnover of participants; the association, at the individual level, of frequent use and selling; and the large variation of prices and quality in narrowly defined geographic markets at a given point in time. These are not universal characteristics of other illegal markets in the United States. For example, the market for illegal gambling services in the 1970s was characterized by stable participation and uniform prices; bookmakers tended to be heavy bettors themselves but that was not true of numbers operators. Furthermore, the gambling markets were subject to light enforcement .
Moreover, violence and instability, for example, are not fixed characteristics of the market for cocaine or heroin but may be sensitive to characteristics of users and sellers (such as age and relative number of light and heavy users) that change in a systematic fashion, as well as to the intensity and form of law enforcement. That suggests the need for dynamic models, which take into account the fact that those markets evolve in the course of an epidemic and the multidimensional responses of those markets to policy interventions. Yet the model of risks and prices  most used in empirical policy models  is a strictly comparative static model, aiming only to describe the long-run adjustment of a market whose principal distinguishing characteristics are the centrality of enforcement risk and violence by other participants in the determination of prices.
The present article outlines three topics that go beyond the framework of risks and prices and seem to lend themselves to dynamic models. They each represent a class of problems that are explanatory, policy-aiding and conceptual. The first topic focuses on an explanation for the persistent long-term decline in prices as the illicit cocaine and heroin markets develop. The second topic addresses whether enforcement resources should be allocated over the epidemic between customer sanctions and seller sanctions. The focus of the third topic is the determinants of variations in price and purity. Each topic requires its own modelling approach. The discussion below begins with a synopsis of the evolution of the cocaine market in the United States.
The evolution of drug markets
Drug epidemics, at least for expensive3 and dependency-creating drugs,4 are characterized by sharp peaks in population incidence rates followed, with a lag, by a plateauing at a new high in the number of dependent users. The pattern reflects the fact that a portion of new users become dependent within a few years, that incidence is partly driven by the extent of perceived problematic use and that exit from dependence is slow. Everingham and Rydell  offer a now classic representation of that phenomenon, while Behrens and others [6, 7] have explored the dynamics in more detail. Drug markets vary over the course of a drug epidemic in the ratio of heavy users to light users, the mean age of users and, in a predictable fashion, the opportunity cost of sellers' time. Those in turn affect the level of property crime generated by drug use.5
In the early stages of the cocaine epidemic in the United States, drug users were not predominantly poor. The image of the drug was relatively benign, its dangers were little known and its attractions were great. Most users were inexperienced and did not at that time consume large quantities or suffer significant problems. Low-income users could earn substantial incomes selling to users who are not poor.6 Such conditions are likely to be common in the early stages of drug epidemics in which the drugs are not well known to the population.
In the late 1980s, frequent users made up a much larger fraction of all cocaine users and accounted for a larger fraction of total cocaine consumption. Cocaine users were then poorer and had acquired both a criminal history and a record of treatment. More educated cocaine users were likely to have responded to messages about the adverse consequences of the drug and to experience better outcomes in treatment. Evidence from the National Household Survey on Drug Abuse shows that the negative correlation of current cocaine use (that is, use in the previous month) and education increased substantially after 1985 .
The continued decline in prices
Law enforcement responded to increased cocaine use with a lag, as would be expected with any problem that emerges rapidly. Reuter  suggests that the stringency of law enforcement by various metrics (arrests or incarcerations as the numerator; number of users, transactions or sellers as the denominator) may have fallen during the first half of the 1980s as the market expanded. The stringency of law enforcement then increased after 1985 as the market stabilized, at least in terms of the quantity consumed. The early decline in cocaine prices is consistent with enforcement swamping ,7 as well as with the framework of risks and prices, with “learning by doing” reinforcing the effect of lower pressure from law enforcement . In both analytic frames, it is difficult to account for the continued sharp decline until 1989 and the more modest declines of the 1990s, since the stringency of law enforcement rose sharply over an extended period.
One possible explanation for the time pattern of prices focuses on the changing income of cocaine users as well as on risks. Occasional and affluent users may have a low price elasticity, since cocaine expenditures account for a small share of their total expenditures. On the other hand, they may be very sensitive to any increase in risks associated with purchase, such as use of “sell and bust” operations,8 or simply higher arrest risks. The opposite patterns may be found with poor dependent users: high price elasticity9 (reflecting the dominance of drugs in their consumption baskets)10 and little sensitivity to changes in arrest and other purchase risks.
Everingham and Rydell  developed a model with light and heavy cocaine users who differed only in their intensity of use. The outline above suggests a model in which users are classified not simply as light and heavy users in terms of consumption per unit time, but also by income or education. Each group has a potentially different elasticity of demand with respect to price and to other kinds of risks. The simplest useful version of the model has three demand components: the light user group is divided into poor and non-poor, while those in the heavy user group are all poor. The segments are linked in that it is the non-poor buyers whose purchases finance consumption by low-income users. The decline in the demand curve of the non-poor then generates a similar downward shift in the demand by poor heavy users, since their income falls.
With a fixed supply curve, reduced consumption and lower prices are generated; however, what is observed is relatively flat consumption (quantity) and declining prices . The explanation for this may lie in the dynamics of the supply curve. Poor frequent users have three sources of income: legitimate work, non-drug crime and selling to non-poor users.11 The predicted declines in their income from legitimate work may be sufficient to explain the downward shift in the supply curve, given that the marginal return to property crime is presumably declining for any individual.
Turning such verbal conjecture into a formal model that takes account of the changing population of users is a major undertaking. It will require not only many of the heuristics underlying the models developed at RAND to deal with links between demand and supply , but also a more complex dynamic structure for those links. For example, both the probability of moving from light use to abstinence and the elasticity of demand may need to be modelled as a function of user income. However such a model may help to generate a parsimonious account of one of the most puzzling aspects of drug markets in the last decade.
Choosing between buyers and retailers as enforcement targets
The outline of market dynamics may also help in developing a model for guiding decisions about allocations between buyer and seller sanctions. It is widely assumed that sellers are more culpable, hence the heavier sanctions for those convicted of selling rather than buying or using and the greater intensity of law enforcement aimed at sellers. However, it is easy to argue for a reverse hierarchy of culpability when the seller is an impoverished dependent user and the buyer is non-dependent and non-poor.12 Furthermore, sellers can be replaced; buyers cannot. There is regular reporting, for example, of “sell and bust” operations, aimed at closing down specific geographic markets.
The change in the population of frequent users may have increased the share of total expenditures derived from crime other than drug selling through three mechanisms: (a) declines in the fraction of users who are non-poor light users; (b) declining employment opportunities for frequent users with low levels of education, as their addiction and criminal histories made them less attractive to employers; and (c) decreased willingness of families and friends to provide monetary or any other kind of support. Since the market had contracted in terms of the number of users, who were on average poorer than they had been in the past, potential drug market earnings of the growing pool of dependent users declined.
It is assumed that the goal of drug law enforcement is to minimize the total harm caused by drug use and drug control. A first approximation can be represented as a function of the number of light users, the number of heavy users and drug-related crime other than drug selling or drug use: crime is an increasing function of the number of heavy users and the ratio of the number of heavy users to the number of light users.
Each light user is at risk of becoming a heavy user. Again, the assumption is that light users have low price elasticity but are highly sensitive to the risk of arrest. A shift away from seller enforcement, assumed to raise prices, towards buyer enforcement will reduce the number of light users in the next period; the notional budget here is a fixed total number of arrests. The change in ratio of the number of heavy users to the number of light users in that next period will be a function of two parameters: (a) the probability that a light user will become a heavy user and (b) the increased shift towards abstinence resulting from the shift in arrest activity. Crime will be affected by the same two parameters.
Those parameters are not fixed: they change in systematic ways in the course of an epidemic. For example, the shift away from light drug use towards abstinence may be low in the earliest part of the epidemic, increase rapidly and then decline, since later recruits have more knowledge of the drug than their predecessors. The optimal allocation of effort may also vary, with a larger share of the budget going to “sell and bust” operations in the early stages of the epidemic when there are many non-criminal light users at risk of becoming criminal heavy users. Later in the epidemic, the primary effect on crime may be changes in the ratio of the number of light users to the number of heavy users, leading to a shift towards conventional seller enforcement.
This is a highly simplified model. The objective function for enforcement decisions is broader than crime and prevalence; for example, buyer-oriented enforcement may generate markets with lower disorder. Similarly, the budget representation as a fixed number of arrests is a convenient simplification; budgets are more likely to be financial and the costs of the two types of arrests will differ. However, decisions about enforcement strategies, if they are to be made on more than an impressionistic basis, require that kind of dynamic modelling.
Variation of prices and purity
A prominent feature of drug markets is the extraordinary variation in price and purity. Weatherburn and Lind  report the most detailed data on prices from a single market for heroin (in a suburban area of Sydney, Australia): about 300 observations over a two-year period. The price per gram in a two-year period ranged from 118 to 11,667 Australian dollars. Even the average price per pure gram for a “fortnight” (a two-week period) showed dramatic changes, for example, collapsing from about $A 6,000 (in “fortnight 7”) to $A 2,000 (in “fortnight 11”). Similar variation appears to characterize prices in the United States, though no similarly detailed analysis of a specific local market has been published.13 Models of user and seller behaviour should accommodate the enormous uncertainty in the cost and purity of transactions, but the paradigm of risks and prices focuses on expected value calculations.
This price variation is made possible by an odd feature of the market: fixed prices but variable quantity that cannot be readily assessed by the buyer at the time of purchase. A dime bag of heroin always costs US$10; it may always weigh 0.1 gram, but whether it contains 50 milligrams or 2 milligrams of pure heroin cannot be ascertained at the time of purchase. It may not be ascertainable even after consumption. The diluents may mimic the drug's effects and the user may have only a general notion of how much of the drug was actually consumed. The user will make an assessment of the quality of the experience, but it will have an uncertain relationship to the actual quantity consumed.
If variation is anticipated, users can adapt. One mode of adaptation is to identify sellers who provide predictable quantities of the drug. Such sellers may charge higher average prices per pure milligram of heroin as a consequence. Prices may still seem confusing then because two classes of sellers emerge: one selling at a high mean price with low variance and the other selling at a low mean price with a high variance. The basis for segregation may include the time of day of the transaction, the circumstances and/or the place. For example, late-night purchases may be lower-priced as dealers seek to unload stocks because the market is thinning out and their risks per unit time are increasing. Purchases in which buyer and seller are known to each other may have more predictable prices, in part to avoid retaliation by disappointed buyer against seller.14
Buyers' attitudes towards risk and buyer experience would affect the distribution between the two. Ethnographic reports of continual information exchange among users about the quality of different dealers' drugs suggest the existence of such heterogeneity. For example, Simon and Burns  describe how some heroin dealers from Baltimore distribute samples early in the day to a few experienced users in order to increase demand for that day's product.
There are more complications. Buyers and sellers frequently leave the market unexpectedly, as the result of arrest, incarcerations, injury or death. In such circumstances, it is not clear whether the optimal strategy is to develop a strong reputation for reliable selling. In strategic games of repeated interaction, whether it is optimal to cooperate (that is, sell high-quality drugs) or defect (that is, sell low-quality drugs) depends on the probability that the game terminates after any given move . A seller does not want to stop selling with any reputational asset remaining: reputations may not be well disseminated because of “churning” (movement or change) among buyers. This does not imply that the optimal strategy is to defraud every customer all the time. If a dealer knows with certainty that he will leave the drug market immediately after the next sale, it would be to his advantage to cheat the next customer. A strategic investment decision needs to be made about how much reputation to maintain.
Varying attitudes towards risk, as well as variations in assessing those risks, will lead to price dispersion. That may drive the market to an equilibrium in terms of low price and high variance, with no sellers choosing to provide quantity that is predictable. Tougher enforcement may shift that equilibrium to still lower price and higher variability, since the dealer, with a lower probability of being able to reap the returns on his reputation for, say, the next three months, will choose to take advantage of more opportunities to defraud buyers.15
Although limited, the empirical literature shows that buyers of cocaine and heroin usually have substantially more than one supplier [19, 20]. That is likely to be an optimal strategy, given the turnover of suppliers and the fact that so many of them work part-time. For example, perhaps as many as half of all heroin retailers are arrested in the course of a year; injury and their own drug habit may make them unavailable at other times. That diversification of sources may limit the feasibility of establishing local monopolies, since customers will always be seeking diversification of sources to ensure reliable supplies.
The latter point requires elaboration. The customer may seek to avoid dependence on a single seller because sellers, not organizations, are incarcerated or injured. The customer's additional sellers may work for the same organization in the neighbourhood, so having multiple sources is not incompatible with buying from a single retail organization. However, the retail agents themselves also seek diversification of sources. Incarceration of their principal forces them to seek supplies elsewhere, mitigating against broad retail monopolies.
Prices (per pure gram) are determined competitively. If buyers and sellers were anonymous and the buyer could not ascertain the content of purchases at the time of the transaction, then the profit-maximizing strategy would be to defraud every purchaser by providing zero-content bundles. The market would collapse, since buyers would seek alternative sources of intoxication.
The market exists; thus, the model is too simple. Even those who buy in truly anonymous transactions and who obviously pose no threat to the seller (for example, the suburban user purchasing nervously in a drive-by inner-city market) receive on average enough of the drug to induce return. One possible explanation is that apparently independent sellers are retail agents for a single organization. The organization has an incentive to encourage users to return to that location; locations rather than sellers develop reputations. There is evidence of localized territoriality, perhaps not enough to generate market power (given the ease with which buyers can test other locations) but sufficient to induce competition through predictability.
That, however, is a model to be tested, not an assumption to be made. Organizational forms evolve over the epidemic, again perhaps because of variations in enforcement risk and dealer characteristics. In the mid-1980s, when the market for crack cocaine first appeared in many cities in the United States, it was characterized by three factors: (a) relatively modest law enforcement intensity; (b) new users who perhaps were not well known to each other; and (c) a low rate of dependence among primarily young users. By the mid-1990s, everything had changed: the probability of incarceration had increased substantially, the buyers and sellers had formed a stable group (at least over a period of, say, two years, which was long enough to allow for the average incarceration) and a large portion of purchases were made by dependent users who had been in the market for at least 10 years. Under such circumstances, the optimal price and purity strategy for dealers may vary, but it is not obvious how. The greater “churning” in the later stages of the epidemic creates incentives for cheating, but the loss of seller anonymity in a market of experienced participants counteracts that. Modelling such a phenomenon is likely to require use of dynamic game theory.
Drugs are sold in markets. The prices are determined systematically: notwithstanding the high rate of observed variation, they have clear patterns. Moreover, those prices have important implications both for participants and for others, including potential users (because prices affect the incentive to start using drugs) and society more generally (through crime and the generation of criminal income). Understanding how the price system functions should help to inform those who make drug policy decisions in a number of areas, including in the allocation of control resources between law enforcement and treatment and prevention and in the allocation of law enforcement resources for targeting buyers and sellers.
Markets for some illegal drugs appear to differ from conventional markets in fundamental ways. While the comparative statics methods developed for markets generally may carry over to those drug markets, the dynamic behaviour of the latter may require new tools.
Analysis of drug law enforcement and drug markets has been something of an intellectual desert, that is, a large territory with few occupants in widely scattered settlements. The body of theoretical literature is limited and is detached from any empirical work. There are only a few modelling efforts with an empirical base . Drug markets are more difficult to model than conventional markets because data on drugs are more difficult to obtain; however, more sophisticated modelling of the kind being undertaken by some researchers, namely, Tragler, Feichtinger, Caulkins and Behrens [6, 7], is an important step in the right direction.
- P. Reuter, Disorganized Crime: the Economics of the Visible Hand (Cambridge, MIT Press, 1983).
- P. Reuter and M. Kleiman, “Risks and prices: an economic analysis of drug enforcement”, Crime and Justice: an Annual Review, M. Tonry and N. Morris, eds., vol. 9 (Chicago, University of Chicago Press, 1986), pp. 128-179.
- C. Peter Rydell and Susan S. Everingham, Controlling Cocaine: Supply vs. Demand Programs (Santa Monica, California, RAND, 1994).
- G. Becker and K. Murphy, “A theory of rational addiction”, Journal of Political Economy, 1988.
- Susan Everingham and C. Peter Rydell, Modeling the Demand for Cocaine (Santa Monica, California, RAND, 1994).
- D. A. Behrens and others, “A dynamic model of drug initiation: implications for treatment and drug control”, Mathematical Biosciences, vol. 159, 1999, pp. 1-20.
- D. A. Behrens and others, “Optimal control of drug epidemics: prevent and treat—but not at the same time?”, Management Science, vol. 46, No. 3 (2000), pp. 333–347.
- Peter Reuter, Robert J. MacCoun and Patrick J. Murphy, Money from Crime: the Economics of Drug Selling in Washington, D.C. (Santa Monica, California, RAND, 1990).
- P. Reuter, P. Ebener and D. McCaffrey, “Patterns in drug use”, in When Drug Addicts Have Children: Reorienting Society's Response, D. Besharov, ed. (Washington, D.C., American Enterprise Press, 1994), pp. 3-31.
- P. Reuter, “On the consequences of toughness”, Searching for Alternatives: Drug Control Policy in the United States, Edward Lazear and Melvyn Krauss, eds. (Stanford, California, Hoover Institution Press, 1991), pp. 138-162.
- M. Kleiman, “Enforcement swamping”, Computer and Mathematical Modeling, vol. 17, No. 2 (1993).
- J. Cave and P. Reuter, The Interdictor's Lot: a Model of Drug Interdiction (Santa Monica, California, RAND, 1988).
- J. Caulkins and P. Reuter, “What can we learn from drug prices?”, Journal of Drug Issues, vol. 28, No. 3 (1998), pp. 593-612.
- R. Needle, Expenditure Patterns of Out-of-Treatment Drug Users (Rockville, Maryland, National Institute on Drug Abuse, 1994).
- Abt Associates, Illicit Drugs Price/Purity Report (Washington, D.C., United States Government, Office of National Drug Control Policy, 1998).
- Don Weatherburn and Bromwyn Lind, “The impact of law enforcement activity on a heroin market”, Addiction, vol. 92, No. 5 (1996).
- David Simon and Edward Burns, The Corner (New York, Broadway Books, 1997).
- R. Axelrod, The Evolution of Cooperation (New York, Basic Books, 1989).
- A. M. Rocheleau and D. Boyum, “Heroin users in New York, Chicago, and San Diego”, Washington, D.C., United States Government, Office of National Drug Control Policy, 1994.
- K. J. Riley, Crack, Powder Cocaine, and Heroin: Drug Purchase and Use Patterns in Six U.S. Cities (Washington, D.C., United States Department of Justice, National Institute of Justice and Office of National Drug Control Policy, 1997).
- J. Caulkins and others, Mandatory Minimum Drug Sentences: Throwing Away the Key or the Taxpayers' Money? (Santa Monica, California, RAND, 1997).
1The research reported in the present article was supported by the National Institute of Justice of the United States Department of Justice. The author is grateful to Jonathan Caulkins for his helpful comments.
2The cocaine and heroin markets are the most important illicit drug markets in the United States, in terms of both the income generated and the consequent social costs.
3Price is usually measured per unit weight. Thus, heroin, selling at 500-1,000 United States dollars per pure gram, is more expensive than cocaine, selling at approximately US$ 100 per gram. However, in terms of dose (that is, the amount taken in a typical session in which a drug is used), there may be little difference. Yet another possible metric, consistent with the rational addiction model of Becker and Murphy , is per annum expenditure or expected lifetime expenditures: cocaine may be more expensive in the first of those and substantially cheaper in the latter.
4Marijuana is both dependency-creating (with about 10 per cent of users self-reporting dependence at some time in their lives) and expensive (US$ 5-10 per gram); however, the dependency appears to be short-lived, that is, it lasts rarely more than a few years and the annual expenditures are much smaller for marijuana than for cocaine or heroin.
5The issue of violence warrants separate treatment. Discussions of violence in drug markets focus on the incentives of sellers as independent agents, either resulting from competition (seeking territorial control) or transactional uncertainty (disagreement about the appropriate quantity of drugs or money). Intra-organizational violence, however, may also have important consequences for the markets. There are two forms of violence generated within organizations: (a) disciplinary violence directed by managers against agents who can either defect with goods or serve as informants against more senior figures; and (b) successional violence, the means by which a junior member of the organization may attain leadership.
6Retailers were reported to be earning US$ 30 per hour in 1988 in the Washington, D.C., market .
7Enforcement swamping grows out of the framework of risks and prices. If enforcement risk is the dominant source of costs for drug dealers, then increased volume, other things being equal, lowers cost; that generates the perverse phenomenon of a downward sloping supply curve.
8"Sell and bust" reverse the usual "buy and bust" tactics. Instead of using undercover agents or informants to apprehend sellers through controlled buys, the police pose as sellers and catch intended buyers.
9Caulkins and Reuter  report a number of studies with price elasticities of one or more in absolute value. The highest values were obtained with a data system associated with heavy users (entitled "Drug Use Forecasting"). Lower values were obtained with data systems associated with broader populations of users (the National Household Survey on Drug Abuse and the data system entitled "Monitoring the Future", sponsored by the United States National Institute on Drug Abuse).
10Needle  reports on the high share of income going to cocaine purchases.
11Clearly, the poor sell to each other as well; however, for the group as a whole, that is merely redistributive.
12It is interesting to note that for another illegal market, prostitution, Sweden has recently instituted legal reforms that shift all criminal penalties from the prostitute to the customer.
13The data from the System to Retrieve Information from Drug Evidence (STRIDE) allow only analysis at the city level, which is much broader. The Domestic Monitor Program of the Drug Enforcement Administration of the United States Department of Justice, providing data only for heroin, contains the geographic identifiers that would allow analysis of more localized markets, but rarely provides more than about 10 observations per quarter.
14In recent reports of price and purity, it was noted that the heroin market was characterized by a bi-modal distribution of pure gram prices .
15The discount rate of a given user or seller may also vary over time. When the need for drugs is urgent, discounted prices for drugs may be hyperbolic, with the buyer or seller sacrificing all future gains in order to obtain cocaine or heroin immediately. That would also generate price and purity dispersion.