An economic analysis of recidivism among drug offenders.

I. 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.

II. 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 Expression Omitted] (7) where [lambda.sub.i] = exp([V.sub.i] + x[delta]), and where m is the unknown number of support points [V.sub.i] each with probability of [P.sub.i]. Conditional upon m, the problem is once again a standard Maximum Likelihood estimation problem. Laird suggested that the parameters be estimated conditional upon some (small initial) choice of m, and that m be increased so long as the support points are distinct with positive probabilities, or until additional support points yield a negligible increase in the maximized value of the likelihood function. For purpose of comparison, v will also be assumed to have a degenerate distribution. That is, a single support point, [V.sub.1], is estimated, with P, = 1. In this case, no correction for heterogeneity is made, and the model reduces to the basic Weibull hazard model. These results will yield information regarding the severity of heterogeneity, and the sensitivity of the parameter estimates to that heterogeneity. Finally, after their initial incarceration, repeat offenders in our data may be sentenced to

either prison or probation.1 If individuals consider prison to be a more severe punishment than probation, then prison sentences may have a greater deterrent impact. Different time paths for the hazard functions of offenders sentenced to prison and those sentenced to probation may be obtained by replacing a in equation (7) with (a + [eta]J), where J is a binary variable indicating probation status (I for probation). In this case, a and a + q are the duration elasticities for those sentenced to prison and probation respectively. The parameter [eta] is the incremental change in the duration elasticity associated with probation. Because the time dependence parameters now differ for offenders sentenced to prison and probation, the hazard function at any point in time will be proportional for observations within, but not across, these groups. Estimation of the hazard function from observed duration times must control for the censoring of completed duration for individuals that have not yet recidivated. For these individuals, we can only infer that completed duration is not less than current duration. The log-likelihood function for this type of data is

ln L = [sigma].sub.R] ln[g (T; x)] + [sigma].sub.N] ln[1 - G (t; x)] (8) where T is completed duration for the set R of individuals that have recidivated, and t is current duration for the set N of individuals that have not yet recidivated. The density function for completed duration times, denoted g( ), is the partial derivative of the distribution function in equation (7) with respect to t. The partial derivatives of the log-likelihood function in (8) are a system of non-linear equations in the parameters [alpha], [eta], [delta], and the m-vector V. This system can be solved with the method of Berndt, Hall, Hall, and Hausman (BHHH) [7]. The m vector of probabilities P will be estimated with the EM algorithm, as suggested by Lindsay [18]. Standard efforts will be obtained with a final zero-step BHHH iteration.

III. The Determinants of Recidivism

The determinants of recidivism used in this study can be divided into four types: 1) opportunity cost; 2) disincentives and enforcement; 3) control variables; and 4) duration effects. Opportunity cost variables are those that reflect the earnings foregone while incarcerated. According to the economic theory of crime, higher opportunity costs win, on the margin, tend to deter criminal activity in general. We explore the question of whether this holds for drug offenders.(4) Disincentives are generated by the criminal justice system via the probability of arrest, the probability of conviction once arrested, and the length of prison sentences. Individual and community attributes, many of which have economic interpretations, are included among the control variables. The duration effects measure how the conditional probability of recidivism changes with time away from prison, given that the person has not already returned to Department of Corrections (DOC) control.

Before turning to a discussion of the specific variables included in these general categories, let us consider the impact of age on recidivism. Many studies have shown a strong negative correlation between criminal behavior and age. This is not surprising for age can affect criminal behavior through increasing opportunity cost as one gets older, more severe sentences for older criminals with more extensive criminal records, and changing preferences. One prominent theory in this regard holds that the preference for criminal activity falls with age, sometimes attributed to criminal burn-out [13] and maturity [11]. From an economic perspective "maturity" might involve an increase in risk aversion. These diverse impacts of age on criminal activity cannot be separated and the relative strengths of the various interpretations of this variable cannot be evaluated.

Opportunity cost cannot be directly measured for individuals in our sample, so we employ several individual and labor market measures that together may proxy the individual's earning capacity.(5) First is a measure of employment status at arrest, a dummy variable indicating whether the individual reported that he was employed (full time or part time) at the time of arrest. Indicating a clear attachment to the labor force, this variable should be positively related to legal earnings. Average wage and salary earnings in the county of release may reflect labor market conditions and therefore opportunities for legal earning.(6) Another opportunity cost variable is that of race, reflecting the fact that blacks tend to earn less income than whites with similar attributes and in the same location. Thus it complements the average earnings measure by providing some evidence about whether the individual's potential earnings are likely to be above or below the average. Age also may be a proxy for opportunity cost, although for the reasons discussed above, other interpretations of this variable are possible.

Punishment that provides a disincentive for illegal activity is the second pillar of the economic analysis of crime. The literature employs four variables to capture disincentive effects: 1) the probability of arrest, measured as the arrest/reported offense ratio; 2) the probability of conviction given arrest; 3) the probability of incarceration given conviction; and 4) length of incarceration. Analyzing the impact of these disincentives on recidivism is complicated by the fact that return to DOC custody is observed only when a crime is committed and the offender is caught and convicted. Two potentially offsetting effects are implied. To the extent that disincentives reduce the probability of committing a crime, they will be associated with a decline in recidivism. But when greater law enforcement resources result in a higher chance of arrest and conviction given an offense, we may observe a positive correlation between these variables and the probability of recidivism.

Since drug offenses are not typically reported to the police, it is impossible to directly measure the probability of arrest. Indirect measures employed here, for the county of release, are the number of sworn officers per thousand population, and the allocation of these police resources to drug enforcement as reflected by the proportion of all arrests that are for drug offenses. Data on the probability of conviction given arrest are available, however, so no indirect measures are required. These variables provide a general deterrence in that all potential offenders in a county face the same effort devoted to stopping drug activity. Of course, they may also reflect enforcement effects and may be positively correlated with the recidivism rate.

The rational criminal should consider the probability and severity of future punishment in decisions to recidivate. However, most recidivism studies have used the time spent in prison for the previous offense as a measure of "specific" deterrence. This would appear to be relevant if the previous time served influences expected future sentences.(7)

Some offenders in our sample recidivated but were put on probation rather than reincarcerated for the subsequent offense. Probation might generally be regarded as a less stringent sentence than incarceration and would therefore be expected to be a less effective specific deterrence. Perhaps more important, violations of probation conditions and law may be easily detected among this group since probationers are more closely monitored than persons released from prison. These data allow us to compare the recidivism rates among this population of drug offenders for persons placed on probation and for those incarcerated.

Other variables in the model control for environmental and personal characteristics that may influence the propensity to recidivate. Most prominent among these variables are two measures of the individual's criminal history, the number of prior probations and prison sentences. These control variables may reflect the individual's risk preferences, or imply that his legal opportunities are relatively limited. Alternatively, they may have implications for the severity of potential future punishment, if sentences reflect criminal record. Thus, these variables may have off-setting effects and no priori expectation about signs is possible. Other personal attributes that may influence the propensity to engage in illegal drug activity are marital status and gender.(8) Supervised release, proportion of prior sentence served, and crime committed under the influence of drugs are also included as control variables. The first two may influence the individual's perception of the severity of punishment, while the last might imply that drugs can change risk perceptions. Larger urban counties, as reflected by greater population density, may be associated with fewer social constraints, resulting in greater drug use. Offsetting this possibility is the fact that larger counties may be associated with more economic opportunity that is not adequately captured in the other opportunity cost variables. Finally, we use dummy variables to identify offenders convicted for possession, sale of drugs, and other drug offenses to compare their recidivism rates with that of offenders whose most recent primary offense was a non-drug crime.(9) A list of variables, their definition, and data sources are shown in Table I.

The hazard model estimates how time since release affects the probability of recidivism. Negative duration dependence for drug offenders, a negative value for [alpha] and/or ([alpha] + [eta]J), would be evidence that offenders, ceteris paribus, become "better risks" the longer they are out of prison. Positive duration-dependence, in contrast, would suggest that persons with a drug conviction are increasingly likely to be caught and convicted as time passes, given that they have not yet been reconvicted.

Table 1. Definition of Variables

Variables                      Definitions

Individual [Variables.sup.a]

 GENDER                        male = 1, 0 otherwise
 RACE                          white =  1, 0 otherwise
 MARRIED                       married = 1, 0 otherwise
 AGE                           age at admission to DOC control
 EMPLOYED                      employed at time of arrest = 1,
                               0 otherwise
 POSSESS                       charged with possession = 1, 0
                               otherwise
 SALE                          charged with sale = 1, 0
                               otherwise
 OTHER DRUG                    1 if charged with drug offense
                               other than sale or possession,
                               0 otherwise
 TIME SERVED                   number of days served in prior
                               incarceration
 % SERVED                      proportion of prior sentence
                               served
 PRIOR PRISON                  number of prior prison
                               commitments
 PRIOR PROBATION               number of prior probations
 SUPERVISED                    supervised release = 1, 0
                               otherwise
 CUI                           crime under the influence of
                               drugs = 1, 0 otherwise
 PROBATION                     previous conviction to
                               probation = 1, 0 otherwise

County [Variables.sup.b]

 SWORN                         number of sworn officers per
                               1000 population
 P CONVICTION                  probability of conviction given
                               arrest for drugs
                               (conviction/arrest)
 % DRUG ARRESTS                proportion of drug arrests to
                               total arrests
 POP DENSITY                   population per square mile
 EARNINGS                      average wage and salary
                               earnings

(a.) The individual variables are from the Florida Department of Corrections dat
abase.
(b.) County variables come from the Florida Statistical Abstract. The probabilit
y of conviction
given arrest is calculated
from The Florida Supreme Court Summary Reporting System and FDLE arrest statisti
cs.

IV. Empirical Results

The sample for this study was drawn from the Sentencing Guidelines/Florida Department of Corrections Matching Database. The sentencing Guidelines Commission database, which includes about 80 percent of all persons adjudicated guilty since March 1983, was matched with data from DOC for all persons incarcerated as of May 29, 1989 and all admissions since 1985.(10) Since our focus is on recidivism among drug offenders, 16,788 observations were drawn whose initial primary offense involved drugs. The sample used in this study, consisting of 4,398 individuals, includes all the drug offenders from the group of 16,788 who had been released prior to May 28, 1989. These data were updated to include subsequent returns to DOC control (via probation or incarceration) up to April 2, 1990. Since some individuals have been incarcerated more than once, the number of releases from prison, 7,171, exceeds the number of persons in the sample. Of these records, 3,534 offenders (50.7 percent) had returned to DOC control while 49.3 percent remained outside the prison system.

Duration, the length of stay out of prison, was calculated in the following way. First, every drug offender with a release date prior to April 2, 1990 was identified. Two variables were created. (1) A dummy variable (SPELL) takes a value of 1 if the offender has returned to DOC custody; if the person remains free the variable takes a value of zero. (2) Duration of completed spells is the difference between the release date and the readmission date. For incomplete spells (SPELL = 0) duration is the difference between the release date and the survey date (April 2, 1990). The average duration of release for the entire sample is 389 days, ranging from one day to 1,676 days. Completed spells averaged 272 days while incomplete spells average 509 days. Summary statistics for the variables included in the regression model are in Appendix A.

The first pair of columns in Table II report point estimates and test statistics, respectively, for the Weibull model (with degenerate heterogeneity distribution). The second pair of columns in Table II report point estimates and test statistics, respectively, for the Weibull model with non-parametric estimates of the heterogeneity distribution. The coefficients in [delta], which determine the baseline hazard rate, are labeled with the name of the corresponding explanatory variable. These estimates are fairly robust to the correction for heterogeneity.

[TABULAR DATA OMITTED]

Two variables related to opportunities for legal income, AGE and RACE, have the expected sign and are highly significant. Whether a person was employed at the time of the prior arrest is not a significant determinant of recidivism among this sample of drug offenders. EARNINGS, a county-wide variable representing the average earnings in wage and salary jobs, was not significant. Other county-wide variables, the unemployment rate and per capita income, were included in other specifications of the model but were never statistically significant. Opportunity costs appear to be more significantly influenced by individual characteristics (e.g., age, race) than local labor market characteristics. This is not surprising, of course, since individuals with marketable attributes can move to locations where those attributes are in demand. The disincentive and enforcement variables capture three possible effects: deterrence specific to the offender, general deterrence, and measured recidivism caused by more diligent law enforcement efforts. Our measure of specific deterrence, time served in the previous sentence (TIME SERVED), is not significant at customary test levels. This is a robust result in that no alternative specification revealed a deterrent effect for time served.(11) The number of sworn officers per capita (SWORN) and the probability of conviction given arrest (P CONVICTION) are county variables that capture general deterrence effects. More sworn officers and a higher probability of conviction might discourage would be offenders from engaging in illegal activity. In the model reported here, the significantly negative coefficient for sworn officers per capita (SWORN) suggests a deterrent effect resulting in less recidivism among drug offenders. The probability of conviction (P CONVICTION) coefficient is not significantly different from zero."

Drug offenders released from prison who recidivate are sometimes given probation rather than a prison sentence. About 7.8 percent of our sample, 559 observations, were given probation upon reconviction. This group, indicated by the variable PROBATION, has a very high propensity to recidivate. It is apparent that probation has little deterrent value.

Recidivism is the product of offender behavior and enforcement activity. As more police resources are allocated to the control of drug offenses, more drug offenders will be caught and we might expect a greater measured rate of recidivism (e.g., the deterrent effect of reallocating police resources to control drug crimes may be dominated by the increased probability of arrest). This important aspect of recidivism has been noted by some authors but has not been emphasized or included in empirical estimates of recidivism [23; 26]. Drug enforcement activity in a county is measured by %DRUG ARRESTS and has a substantial impact on the recidivism rate, as reflected by the highly significant positive coefficient.

Three variables reflect the individual's past inclination to choose crime and are expected to be positively correlated with recidivism. Such individuals may have relatively low opportunity costs, or they may have relatively low degrees of risk aversion. These variables are prior prison commitments (PRIOR PRISON), prior probation sentences (PRIOR PROBATION), and proportion of the prior sentence served (%SERVED). Similarly, drug consumption may reduce risk aversion. Alternatively, it may make individuals easier to catch because their cognitive faculties are reduced and they are more likely to make mistakes. Therefore, another individual variable employed in the model was crime committed under the influence of drugs (CUI). With the exception of PRIOR PROBATION these variables have highly significant positive coefficients.

Males (GENDER) are more likely to recidivate than females, a result consistent with other studies. Persons convicted for drug possession (POSSESS) are less likely to recidivate than those convicted of drug sales (SALES), and both groups are significantly less likely to recidivate than are persons convicted of other drug crimes (trafficking, smuggling, production, delivery and distribution), and non-drug crimes in the sample. These results suggest that drug offenders who have no record of non-drug criminal activity are different from the population of drug offenders who have also committed crimes against persons and property. Other control variables in the model, MARRIED, SUPERVISED, and POP DENSITY are not significant at customary test levels.

While the estimates of the baseline coefficients in [delta] are robust to the correction for heterogeneity, the estimated duration elasticities are not. Of course, this is to be expected. As Heckman and Singer [12] have shown, a failure to control for heterogeneity leads to a downward bias in estimated duration elasticities. A comparison of the estimated duration elasticities before and after the heterogeneity correction reflects this bias. While the estimated duration elasticity for those sentenced to prison ([alpha]) is positive and significant in each case, the correction for heterogeneity increases the estimated duration elasticity from .232 to .297. Perhaps of greater interest, the estimated duration elasticity for those sentenced to probation ([alpha] + [eta]) is negative and significant prior to the correction for heterogeneity, but is positive and significant after the correction for heterogeneity. The correction for heterogeneity increases the estimated duration elasticity from -.453 to .434. Since a failure to control for heterogeneity leads to a downward bias in the estimated duration elasticities, these results suggest: 1) that offenders sentenced to probation are a relatively heterogeneous group, and 2) that the estimates from the Weibull/Non-Parametric model are likely to be more reliable.(13)

Offenders sentenced to probation have both a higher baseline rate of recidivism (reflected in the positive and statistically significant coefficient in [delta] for the Probation binary discussed earlier) and a higher duration elasticity (reflected in the positive and statistically significant estimate of [eta]). Consequently, offenders sentenced to probation have a higher probability of recidivating over the range of duration values encompassed by our sample.

While there is little direct interest in the estimates of the heterogeneity distribution, they do provide some insight into the nature of the heterogeneity

correction. The standard Weibull model is a restricted case of the Weibull/Non-Parametric model that assumes [P.sub.1 = 1. That is, the standard Weibull model assumes a common intercept. The estimated intercept for this model is labelled [V.sub.1] in Table II. The correction for heterogeneity employed in the Weibull/Non-Parametric model treats the sample as a probabilistic mixture of Weibull models with differing intercepts. In Table II, the estimated intercepts are labeled [V.sub.1, V.sub.2], and [V.sub.3], and have corresponding probabilities [P.sub.1, P.sub.2], and 1 - [P.sub.1] - [P.sub.2], respectively. It is interesting to note that the sample distribution of these intercepts is bimodal. (The implicit sample probability corresponding to [V.sub.3] is .10.) It would be difficult to model this sample with a parametric form for the heterogeneity distribution, since most simple parametric distributions are unimodal.

Positive duration dependence among this sample of drug offenders can be interpreted in two ways. One is that duration dependence corrects for a problem in measuring the enforcement effort against drug offenders, a significant determinant of the recidivism rates. Drug enforcement in the model (%DRUG ARRESTS) is 1987 data, approximately in the middle of the time period analyzed. Between 1985 and 1989 drug arrests as a proportion of all arrests rose substantially, from 7.8 to 12.5 percent, a 60 percent increase. Even if offender behavior remained constant over this period, continuing drug offenders would be more likely to be arrested and reconvicted the longer they were out of prison.

Positive duration dependence is also consistent with a theory of rational addiction [6]. In their model addiction arises when past consumption of a commodity has a strong complementary impact on current consumption. Previous consumption creates a stock of "consumption capital" that increases the utility of current use, but this effect depreciates over a period of abstinence. Imprisonment of drug offenders may develop an analogous stock of "deterrent capital" which depreciates with time away from prison. Assuming abstinence from drug use in prison, a released prisoner has a depreciated stock of consumption capital that reduces the utility of drug use and a relatively high stock of deterrence capital due to the recent period of incarceration. The combination may result in a relatively low probability of drug use immediately after release. This could change over time. If the depreciation rate of consumption capital is slower than that of deterrence, drug consumption may tend to rise with time away from prison, suggesting positive duration dependence.

V. Drug Recidivism and Public Policy

This study of recidivism among drug offenders implies that these individuals do appear to respond to the incentives and constraints created by the criminal justice system. The size of the police force in their jurisdiction of residence provides a significant general deterrence. Furthermore, imprisonment is much more likely to deter recidivism for drug crimes than is probation. In addition, opportunity costs appear to matter. Reliable direct measures of opportunity costs were not available, but age and race proxies had the expected signs and significance. Thus, we can conclude the economic theory of crime can appropriately be applied to drug criminals. However, since drug offenders who commit other crimes are more likely to recidivate than other drug offenders, this study implies that it may be appropriate to consider drug offenders who are not committing other crimes separately in empirical models of criminal decision-making.

The rate of recidivism among drug offenders (and other offenders), defined by either re-arrest or return to DOC control, is partly determined by public policy. Measured recidivism is the result of illegal actions by the offender and the enforcement policies of the criminal justice system. Our results show clearly that the probability of recidivism among drug offenders rises when they are released to a county that allocates more of its police resources to arresting drug offenders.

Getting tough on drugs may be good politics but inefficient public policy. It appears to be good politics because candidates for public office can offer voters a "solution" to the "drug problem", e.g., longer sentences for drug offenders. Of course, part of the perceived "drug problem" is the apparent belief that the vast majority of drug offenders also commit property and/or violent crimes. Evidence presented here suggests that this belief may not accurately reflect reality. There are drug offenders who commit other crimes, but they can be statistically distinguished from the majority of drug offenders whose criminal activities appear to be restricted to participation in the drug market. This perception is reinforced by an examination of the history of persons having at least one misdemeanor or felony drug arrest in Florida. Of the 45,906 people arrested for drug possession during 1987, for instance, 76 percent had no previous arrests for violent felony crimes [25]. Similarly over 80 percent of the persons arrested for drug possession had no burglary or grand larceny arrests; and over 90 percent had not been arrested for stolen vehicle or stolen property offenses.(14) It appears that the costs of an indiscriminate war on drugs are either not recognized or they are ignored by voters and the politicians responsible for its creation. The cost of longer prison sentences for drug abusers who do not commit other crimes is not just the cost of a prison bed, but also, and perhaps more importantly, the risks associated with releasing other criminals early. In this context, the optimal degree of drug enforcement will not maximize the number of drug convictions.(15)

References

[1.] Barnett, Arnold, Alfred Blumstein and David Farrington, "A Prospective Test of a Criminal Career Model." Criminology, May 1989, 373-88. [2.]_____,_____, and _____, "Probabilistic Models of Youthful Criminal Careers." Criminology, February 1987, 83-107. [3.] Benson, Bruce L. and David W. Rasmussen, "The Relationship Between Illicit Drug Enforcement Policy and Property Crimes." Contemporary Policy Issues, October 1991, 106-15. [4.]_____, Il-joong Kim, David W. Rasmussen, and Thomas W. Zuehlke, "Is Property Crime Caused by Drug Use or Drug Enforcement Policy?" Applied Economics, July 1992, 679-92. [5.] Becker, Gary, "Crime and Punishment: An Economic Approach." Journal of Political Economy, March/April 1968, 169-217. [6.]_____ and Kevin M. Murphy, "A Theory of Rational Addiction." Journal of Political Economy, August 1988, 675-700. [7.] Berndt, E. R., B. N. Hall, R. E. Hall, and J. E. Hausman, "Estimation and Inference in Non-Linear Structural Models." Annals of Economic and Social Measurement, October 1974, 653-65. [8.] Block, M. K. and J. M. Heineke, "The Allocation of Effort under Uncertainty: The Case of Risk-Averse Behavior." Journal of Political Economy, March/April 1973, 376-85. [9.] Ehrlich, Isaac, "Participation in Illegitimate Activities: A Theoretical and Empirical Investigation." Journal of Political Economy, May/June 1973, 521-65. [10.]_____, "The Deterrent Effect of Capital Punishment: A Question of Life and Death." American Economic Review, June 1975, 397-417. [11.] Gottfredson, Michael R. and Travis Hirschi. A General Theory of Crime. Stanford: University Press, 1990. [12.] Heckman, James J. and Burton Singer, "Econometric Duration Analysis." Journal of Econometrics, January 1984, 63-132. [13.] Hoffman, Peter and James Beck, "Burnout-Age at Release from Prison and Recidivism." Journal of Criminal Justice, 1984, 617-23. [14.]_____ and _____, "Recidivism among Released Federal Prisoner." Criminal Justice and Behavior, December 1985, 501-07. [15.] Kiefer, J. and J. Wolfowitz, "Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Nuisance Parameters." Annals of Mathematical Statistics, 1956, 887-906. [16.] Laird, Nan, "Nonparametric Maximum Likelihood Estimation of a Mixing Distfibution." Journal of the American Statistical Association, December 1978, 805-11. [17]. Lancaster, Tony, "Econometric Methods for the Duration of Unemployment." Econometrica, July 1979, 939-56. [18.] Lindsay, Bruce G., "Properties of the Maximum Likelihood Estimator of a Mixing Distribution." in Statistical Distributions in Scientific Work, edited by C. Taillie et al. Boston: Kluwer Academic, 1981, pp. 95-109. [19.] Myers, Samuel L., "The Rehabilitation Effect of Punishment." Economic inquiry, July 1980, 353-66. [20.]_____, "Estimating the Economic Model of Crime: Employment versus Punishment Effects." Quarterly Journal of Economics, February 1983, 157-66. [21.] Rasmussen, David W., Bruce L. Benson, Iljoong Kim and Thomas Zuehlke. "An Economic Analysis of Drug Crime in Florida," A report for the Joint Legislative Management Committee of the Florida Legislature. Tallahassee; Florida State University, December 1990. [22.] Rhodes, William, "The Criminal Career: Estimates of the Duration and Frequency of Crime Commission." Journal of Quantitative Criminology, March 1989, 3-32. [23.] Schmidt, P. and Ann D. Witte. Predicting Recidivism Using Survival Models. New York: Springer-Veriag, 1988. [24.] Stigler, George J., "The Optimal Enforcement of Law." Journal of Political Economy, May/June 1970, 526-30. [25.] Trager, Kenneth and Michael Clark. Florida Drug Offender Profile. Florida Department of Law Enforcement, Statistical Analysis Center, February 1989. [26.] Trumbull, William, "Estimation of the Economic Model of Crime Using Aggregate and Individual Level Data." Southern Economic Journal, October 1989, 423-39. [27.] Witte, Ann D., "Estimating the Economic Model of Crime with Individual Data." Quarterly Journal of Economics, February 1980, 57-84. [28.]_____, "Estimating the Economic Model of Crime: Reply." Quarterly Journal of Economics, February 1983, 158-75.

(1.) Economic models of criminal activity based on rational individual choice [5; 9; 10] have derived comparative static results that can be used for empirical testing of the economic model of crime. Block and Heineke [8] show that comparative static results are ambiguous under the usual preference restrictions in a labor supply type model. Witte [27] creates a time allocation model which, like Block and Heineke, does not allow signing of the principal variables. Rasmussen et al. [21] present comparative statics for a model in which drug addiction is included in the utility function and report that the sign of the variables is an empirical rather than theoretical question. Witte [27] argues that the value of these models is that they indicate the major variables that should be included in the analysis. Since this function has already been provided, we do not introduce another model of criminal choice here, but instead rely on previous studies to guide the choice of variables affecting recidivism. (2.) In this model, one must impose either the condition [delta].sub.1 = O (and interpret the elements of V as different intercepts) or the condition [mu].sub.v] = O (and interpret the elements of V as increments the intercept [delta].sub.1). We will impose the former condition. (3.) In Florida, offenders are generally sentenced to probation and not put on probation after release from prison. Furthermore, early release programs do not involve release to probation. (4.) Witte's [27] results suggest that recidivism among released prisoners varies depending on the crime in which they specialize. Crime types are categorized by degree of seriousness, consumption offenses and income offenses. Serious income offenses, for example, include robbery and sale of narcotics. (5.) A direct measure of opportunity cost, self reported monthly earnings, is reported in our data set for individuals but this was deemed unreliable because of a high non-response rate and uncertainty as to whether some respondents included illegal income in their estimate of earnings. (6.) The county of release is assumed to be the county of residence for all place specific variables used in the model. When the county of release was not reported, the place of residence was assumed to be the county of arrest. (7.) The Florida's Sentencing Guidelines call for longer sentences based on the number of previous convictions. In the context of the early release program, required to avoid prison overcrowding, however, the correlation between sentence served and prior convictions is uncertain. This uncertainty is reflected in the fact that some persons in our sample were sentenced to parole rather than prison on subsequent offenses. (8.) Marital status may be relevant because a married person has a higher non-monetary opportunity cost of time. Gottfredson and Hirschi [11] imply this variable involves a self selection process, i.e., persons who stay married may have more self control and are more responsible, attributes not commonly associated with illegal activity. (9.) Other drug crimes include offenses such as forging a prescription and possession of drug paraphernalia. (10.) These data were provided by the Economic and Demographic Research Division of the Florida Legislature and updated by the Bureau of Planning, Research and Statistics, Florida Department of Corrections. Since there is no common ID number for persons in these data bases, a match was assumed correct when the person's name, date of birth, and primary offense code were identical. (11.) This result is in contrast to Trumbull's [26] evidence of specific deterrence among a sample of all felons released from the North Carolina prison system. This difference in results may reflect a difference between drug offenders and other felons. It is also possible that Trumbull's specific deterrence effect would not be confirmed if the sample were disaggregated into major crime categories. Schmidt and Witte [23] use the North Carolina data in a variety of models, however, and consistently find a positive significant relationship between time served and recidivism. Schmidt and Witte use all Index I crime categories, but unlike Trumbull, do not use any county enforcement variables which are important determinants of recidivism. Thus, their model may be mis-specified. Myers [20], using different data, reports that legal earnings has an important impact on recidivism while finding no deterrence effects. (12.) These general deterrence effects are sensitive to model specification. When population size is added to the specification reported in Table II the results reported here are changed. Other coefficients are robust and not affected by this alternative specification. Population size and the county variables are probably related via scale in a way that prevents a precise estimate of these general deterrence effects. (13.) That duration dependence among probationers becomes positive with the correction for heterogeneity is consistent with the fact that probation by its nature is a less severe punishment and should have a smaller deterrent impact than incarceration. Increasing duration dependence among probationers may also reflect the practice of Florida courts over this period to sentence more offenders with longer criminal records to probation due to prison over crowding. Thus the composition of probationers changed over the sample period, possibly explaining the greater effect of unobserved heterogeneity for probationers than those sentenced to prison. (14.) Perhaps it should be noted that if there are differences among the drug consuming population, then the "drugs cause crime" argument may be perfectly valid for a subset of the population. After all, those who commit property crime are generally doing so to finance their consumption of the goods and services they desire. Such desires might be said to "cause" the criminal activity. If these criminals are drug users, it follows that they are committing crime to finance their drug consumption, so in a sense, the desire to consume drugs "causes" crime. Of course, the desire to own a television set may "cause " crime in the same sense. However, addiction and the apparent accompanying increasing tolerance for drugs can lead to an increased demand for drugs [6; 21], and this could in turn "cause" the criminally inclined addict (e.g., someone already committing crimes or someone facing incentives for which some marginal change in incentives due to addiction can induce entry into property crime) to commit more crime. But even if this "drug causes crime" argument is valid for a subset of the drug consuming population, a policy that focuses on the indiscriminate arrest and imprisonment of drug users in order to lower property crimes may be misguided if a substantial portion of the users who are arrested are not actually committing such crimes. For one thing, even though some of those for whom the marginal impacts of addiction induces criminal behavior might be deterred from trying drugs and therefore not become criminals, many others who might be deterred from drug use would probably be criminals anyway, in order to finance their consumption of other goods and services. After all, the decision to choose crime as a source of income is driven by considerations of opportunity costs, along with the risks and severity of punishment. Eliminating drug use is not likely to change all the incentives relevant to this choice. (15.) We use "optimal" in an efficiency sense as defined by Stigler [24]. He explained that optimal enforcement policy must incorporate the principle of marginal deterrence wherein higher penalties and conviction probabilities should apply for more serious offenses. Property crimes and crimes of violence are presumably more serious offenses than drug use because they affect other parties (victims).