An economic analysis of recidivism among drug offenders.

• Author: Il-Joong Kim

1. Introduction

   Following Becker [5], economists have modeled decisions to commit crimes as rational choices in reaction to incentives and constraints. In particular, potential criminals consider the probability of being caught and the severity of punishment in their decision calculus, as well as the opportunity cost of their time. But arguments are frequently made by police and other public officials which imply that drug users are "different" than the rational criminal of the economic model. Therefore, this study investigates factors affecting recidivism among Florida's drug offenders from the perspective provided by the economic theory of crime in order to see if drug offenders do react to incentives and constraints, and to explore the implications of alternative policies toward this large and growing segment of the criminal population.

   The relationship between drug use and non-drug crime may have very important policy implications. For example, Florida's prison population has increased dramatically in the past decade, growing from 19,681 at the end of the 1980-81 fiscal year to 38,059 by June 30, 1989, and a substantial portion of these increased admissions are for drug offenses. Drug offenses accounted for only 21.5 percent of Florida's prison admissions during the first half of FY86-87, but during the same period in FY89-90 drug offenses accounted for 36.6 percent of all admissions. New admissions to prison have forced a more rapid turnover of inmates through early release programs resulting in a declining proportion of sentences served. In January 1987 Florida's prisoners served an average of 52 percent of their sentence. By December 1989 this proportion had fallen to 33 percent, a 37 percent decline in two years. These trends are largely a result of an increasingly tough policy toward drug offenders. Similarly, police resources have been reallocated to focus on the control of drug offenses, thereby reducing the ability of police to deter non-drug crimes [3; 4]. This reallocation of resources has led to rising property crime rates. Therefore, the opportunity costs of drug control are substantial. If drug criminals do not respond to incentives (e.g. are not deterred) then this criminal justice policy is clearly questionable (even if they are, of course, the costs should be compared to the benefits).

   Theoretical models in the economics of crime literature start with individuals making decisions in an effort to maximize expected utility. Therefore, studies of recidivism provide a natural application of the economic model of crime because they allow the analysis of individual behavior. Major studies of recidivism use individual data [1; 2; 13; 14; 19; 20; 22; 23; 26; 27; 28]. Most of these studies have not accounted for the impact of the public sector's law enforcement efforts on observed recidivism (re-arrest or re-conviction), however, despite the emphasis on the interaction between the supply of criminal offenses and the public sector's law enforcement efforts in the economic theory of crime. Furthermore, none have focused on drug offenders. Therefore our study draws together the economics of crime and recidivism literatures to explore the determinants of recidivism among drug offenders in Florida. Section II develops a model which explicitly takes into account the probabilistic nature of arrest and conviction. In section III we discuss the empirical specification of this model, focusing on the principal factors that affect the probability of conviction and incentives to engage in drug crime that are prerequisite to our definition of recidivism. The empirical results are presented in section IV which is followed by concluding comments.

2. Modeling the Probability of Recidivism

   When defined as the return to criminal activity, recidivism is the result of an individual decision made by the offender. Typically, however, the return to criminal activity is observed only when recidivists are arrested or, as in the case of this study, arrested and re-convicted. When recidivism is defined in this fashion, the probability of recidivism is the product of the probability of a return to criminal activity for the individual and the probability of arrest and re-conviction by the criminal justice system conditional upon a return to criminal activity. This is a function of both individual and law enforcement decisions which cannot be separately identified statistically. For example, an increase in enforcement efforts against drug offenses may reduce the probability that an individual will engage in these activities, given that drug offenders can be deterred, but the increase in enforcement effort is also likely to increase the probability of arrest and conviction given a drug offense. Consequently, the effect of increased enforcement activity on measured recidivism is ambiguous, depending on the relative strengths of the deterrence and apprehension effects.

   There is an extensive literature in economics examining the decision to engage in crime.(1) While differing in focus and generality, a common thread among these models is the assumption that the decision is a rational one involving a comparison of expected utilities from alternative "bundles" of criminal and non-criminal activities. That is, an individual chooses to engage in a crime (along with other activities) at time t if [U.sup.l.sub.t] > [U.sup.l.sub.t] where denotes the expected value of the indirect utility function associated a bundle of activities which includes that crime and [U.sup.o.sub.t] is an alternative bundle without the criminal activity.

   In this study we model the probability of recidivism for drug and non-drug offenses among a sample of persons initially convicted of a drug offense. While the individual's choice is a deterministic one, and the theoretical literature models it as such, no econometric model is capable of replicating this decision with certainty. Even if all variables relevant to the individual's decision were observed, the inability to model individual variation in preferences would preclude perfect prediction of the individual's choice. Consequently, an individual's utility under the alternative choices is typically modeled as linear in a set of explanatory variables with an additive error term. That is,

   [U.sup.o.sub.t] = [X.sub.t.beta.sup.o] + [epsilon.sup.o.sub.t]

   and

   [U.sup.1.sub.t] = [X.sub.t.beta.sup.1] + [epsilon.sup.1.sub.t] (1) where the individual chooses criminal activity at time t if

   [U.sup.1.sub.t] > [U.sup.o.sub.t]

   or

   [epsilon.sup.o.sub.t] - [epsilon.sup.1.sub.t] < [X.sub.t]([beta.sup.1 - beta.sup.0] (2)

   Denoting the distribution function of [epsilon.sup.o.sub.t] - [epsilon.sup.l.sub.t] by ( ), the probability of an offense is [beta.sup.1] - [beta.sup.o]. In a static framework, this structure is the foundation for the Probit and Logit models. Assuming that the probability of arrest and conviction given an offense may be represented as a non-homogeneous Poison process with parameter [theta]([X.sub.t]), then the hazard probability of recidivism, in a small time interval dt, is

   [Mathematical Expression Omitted] (3)

   As implied above, this hazard function is too general to be of empirical use, since the separate effects of variation in [X.sub.t] on the probability of an offense and the probability of arrest and re-conviction given an offense cannot be identified in the absence of information on offenses committed without apprehension. The most useful econometric approach is one that starts by specifying the empirical hazard function, recognizing the reduced form nature of the parameter estimates.

   The hazard models employed in this paper are based upon the reduced-form Weibull models of Lancaster [17]. Therefore, assume a log-linear relationship between the hazard function and current duration:

   [Mathematical Expression Omitted] (4)

   where v represents the effect of heterogeneity (unmeasured individual-specific, but time-invariant, explanatory variables), x is a vector of observed explanatory variables (which includes measures of individual characteristics, law enforcement activity, and legal sanctions), and t is current duration. The duration elasticity of the hazard function is [alpha]. The hazard function exhibits positive (negative) duration dependence when a is positive (negative).

   Because the likelihood function for duration data is expressed in terms of the distribution and density functions for completed duration, we must know the relationship between a given specification of the hazard function and the resulting distribution of duration times. For the specification given in equation (4) above, the distribution function for completed duration, conditional upon v and x, is

   [Mathematical Expression Omitted] (5)

   If v were observed, there would be no special problem in estimation.

   Estimation when v is unobserved was originally considered by Kiefer and Wolfowitz [15], who demonstrated that the parameters can be estimated consistently by the method of Maximum Likelihood, as long as v can be represented by a common distribution function, not necessarily of known parametric form. Estimation proceeds by maximizing the unconditional likelihood function, which corresponds to the expectation of the conditional likelihood over the distribution of v. The distribution function for completed duration, conditional upon x (but not v), is thus

   [Mathematical Expression Omitted] (6)

   When the distribution of v is of known parametric form, the problem is a standard Maximum Likelihood estimation problem.

   Rather than make an assumption about the distribution of v, it will be estimated by non-parametric methods. Laird [16] illustrated that, for cases of practical interest, the non-parametric estimator treats v as having a discrete distribution with a finite, but unknown, number of support points.(2) In this case, the expectation in equation (6) reduces to a simple weighted sum, and the distribution function becomes

   [Mathematical...