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9 Dynamic Models of Criminal Careers Christopher Flinn BEHAVIORAL MODELS IN CRIMINAL CAREER RESEARCH Economists have Tong been interested in the determinants of criminal activity (e.g., Bentham, 1780), but only in the past few decacles have economic applications in this field! of inquiry become something of a growth industry (see, for example, Schmidt anal Witte, 1984, and references therein). A number of models of individual decision making have been applied to the problem of criminal activity, and those models share several common features. First, they all posit rational behavior on Me part of indi- viduals, in that, subject to a set of con- straints facing the individual, a function Christopher Flinn is associate professor, Depart- ment of Economics, University of Wisconsin-Mad- ison. The author is indebted to his colleagues Arthur Goldberger and Charles Manski for many valuable discussions and comments. Detailed discussions with Alfred Blumstein, Jacqueline Cohen, and John Lehoczky were extremely helpful in preparing this revision. Glen Cain and Ariel Pakes also provided helpful comments. This research was partially sup- ported by a grant from the Sloan Foundation to the Institute for Research on Poverty at the University of Wisconsin-Madison. 356 characterizing the inclividuaT's preferences is maximized. Second, all models recog- nize that risk is an essential component of the decision to engage in criminal activity. In contrast to the purchase of a can of soup, which has a virtually certain level of ulti- mate satisfaction associated with consump- tion of the procluct, the eventual level of satisfaction associated with the decision to undertake criminal activity can only be cle- scribecI probabilistically. All moclels of criminal activity, ~en, must inclu(le some method by which the potential outcomes of risky activities can be evaluated. Third, attention is typically restricted to mone~y or monetarizecl yields from criminal activ- ity. In particular, the "psychic" rewards (whether positive or negative) obtained from criminal activity are not explicitly modeled. The aversion that many neocIas- sical economists have to explaining differ- entials in behavior through differences in preferences is reflected in the strong and controversial assumption Mat indivicluals have identical preferences;) all differences iAltematively, it is assumed that differences may be captured in some simple, paramedic manner.

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DYNAMIC MODELS OF CRIMINAL CAREERS in behavior arise through differences in the choice sets inclivicluals face. Finally, the theoretical moclels that have been formulated are essentially static in na- ture; they do not take account of how the criminal and legitimate opportunities ex- pectect to prevail in the future affect current decisions about criminal activity. Owing to the neglect of these inter- temporal considerations, it might be claimed, no theory of rational criminal choice has as yet been rigorously formu- lated. The report of the Panel on Deterrence and Incapacitation (Blumstein, Cohen, and Nagin, 1978) cited a need for in- creased behavioral and statistical moclel- ing at the individual level of analysis. In the second part of this paper an econo- metric model of the criminal career is presented that is designed for use with individual-level data. While this econo- metric mode] is not explicitly derived from a behavioral moclel, it floes provide a relatively general statistical representa- tion of criminal careers, and the parame- ters of the moclel may be interpreted in the context of stanclarc3 behavioral theo- ries of criminal activity choice. In the first part ofthis paper, behavioral models of criminal activity are developed to begin to address the issue of what type of criminal careers these models might generate. To this end, analytic results are presented when possible; alternatively, some limited simulation experiments are presented when analytic results are not available. These behavioral models are also used as a baseline against which some of the statistical models used in this field of inquiry can be evaluated. (Some discussion along these lines is contained in the second part of this paper.) Many behavioral assumptions are implicit in the statistical descriptions of criminal ca- reers, ant! it may be of some interest to assess the value of various statistical mod- els not only in terms of their ability to 357 predict behavior (which is typically quite Tow, see Chaiken and Chaiken, 1981, for example), but also in terms of the degree of correspondence between characteris- tics of the statistical moclel and character- istics of a consistent, dynamic model of decision making ant! criminal activity. The converse is also obviously true; cur- rent empirical knowledge regarding the dynamics of criminal careers must be used as a guicle in the construction and evaluation of any theoretical model that purports to describe the criminal activity clecision over time. Structural models of decision making also serve a relatecl purpose. They are often required for an assessment of the effects of changes in the distributions of rewards and punishments associated with criminal activity on the amount of time spent on those activities. The practical need for structural moclels was insight- fully presenter! by Marschak (19531. To paraphrase Marschak's argument, say we are interested in the development of a mo(lel to explain some measure of the degree or intensity of criminal activity, clenotecT by x. Generally speaking, incli- viclual differences in x may arise from differences in earnings potentials in legit- imate activities (e), background character- istics (b), the distributions of rewards as- sociated with criminal activities (R), ant! (distributions of penalties if apprehended (P). Then we assume there exists a func- tional relationship among these charac- teristics x = x (e, b, R. P.; Qj, where Q is the vector of parameters that, in conjunc- tion with the functional form x~ ), com- pletely characterizes the relationship be- tween x and the characteristics e, b, R. P. In this case a decision-theoretic moclel may be of use in guiding our choice of a functional specification of x; ); but once the function is selected the determination of the effects of the exogenous variables on x is simply an empirical matter. The qualitative and quantitative effects of all

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358 exogenous variables are contained in the parameter estimates 0. Such an empirically based strategy has at least one advantage over a highly struc- tured approach to the problem. By speci- fying a flexible functional form for xt ), we are likely to be able to capture the observer! relationships among the vari- ables well that is, we wit! be able to fit the ciata. We wouIc3 then be able to assess the effects of changes in the distributions of punishments on the level of criminal activity, for example, comparing x = Be, b, R. P.; {L) with x = Be, b, R. P.; 0), where P denotes the "new" punishment distri- bution. This evaluation is straightforward even if x is a highly nonlinear function. This approach runs into one major problem in practice, however. If we are to estimate the parameters associated with the exogenous variables, those attributes must exhibit a sufficient degree of sample variability. If we want to assess the effects of the distribution of punishments on criminal activity, the sample members cannot all be subject to the same set of punishment distributions. If all inclividu- als are subject to the same P. at least one element of the parameter vector Q will not be estimable. Even if a few different values of P are present in the sample, thus making it possible to estimate all ele- ments in Q. sample variability in P may be so Tow as to preclude precise estima- tion of Q. The choice the analyst has is to ignore the effects of characteristics that vary little or not at all across sample members or to formulate a behavioral mode} in which those characteristics ap- pear as parameters. For example, assume R and P vary little or not at all in the sample. Following the first option, we wouIcI estimate a function of the form x = cafe, b; ey, where xa is the new functional form and e is the new parameter vector. It is impossible to say anything concern- ing the effect of changes in R and P on x. Following the second option, we wouIcI CRIMINAL CAREERS AND CAREER CRIMINALS estimate a function of the form x = able, b; R. P. A), where we treat R and P as parametric to the problem, ant! ~ is a vector of other parameters. The func- tional form of xb will be derived from an explicit behavioral moclel. Using this ap- proach it will be possible to perform con- ceptual experiments in which the effects of changes in R and P on x are analyzed. Thus, this "structural" approach to mocl- eling behavior is not pursued for reasons of aesthetics; it enables the analyst to perform conceptual experiments that are not possible with models less closely linked with behavioral theory. Dynamic Mo(lels of Criminal Behavior In this section three moclels of the pro- portion of time allocated to criminal activ- ity are clevelope(1 to analyze how this allocation of time changes as a function of the individual's age and as a function of criminal career. All models are definition- ally simplifications of and abstractions from the "real" world. It may be disqui- eting to some to view criminal behavior simply as the outcome of a rational calcu- lus. However, if behavior is a manifesta- tion of conscious choice, it seems neces- sary to posit that individuals make decisions in a way that is consistent with some underlying set of preferences or view of the alternatives facing them. In the models discussed below, individuals are assumed to act rationally.2 Their pref 2In our legal system, individuals charged with crimes are "punished" when found guilty at least partially because the commission of the crime is held to have been an outcome of conscious choice. Only when individuals are adjudicated to have keen noncompetent at the time of the crime are they not held legally responsible for the crime they are found guilty of committing. Thus, rationality only requires that individuals make consistent choices with re- spect to some objective and given the choice sets they face. It is a large leap from the assumption of rationality, per se, to the simple utility-maximization

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DYNAMIC MODELS OF CRIMINAL CAREERS erences and choices are specified in a deliberately limited way In terms of ar- eas of potential application, these models may be useful in the analysis of the rates at which various types of property crimes ,`3 are committed. (The symbols used in this section are listed below for easy refer ence.) fit wt Ht Proportion of time devoted to V criminal activity in period t (O C TV 1~. Legitimate work wage rate in period t. The indivicluaT's criminal recorc! as of time t (e.g., arrests, time in prison). Consumption flow from incar ceration. cat P(0t) Probability of arrest in period t. Total monetarizec3 returns V(St) from criminal activity in pe riocl t for individual i. hi The conclitional distribution function of criminal rewards. The utility of consumption level c. The parameter describing the conditional expectation of re- wards in criminal activity for individual itEi(Y~ 8) = Hi ~ ~ models developed below. Unfortunately, it is often the case that discussions of the manner in which criminal behavior should be modeled conclude with the claim that rational-choice models are too simplistic to be useful. The point is not whether rationality is a reasonable assumption; no social science investigation can be attempted without it. The correct point is that current attempts at behav- ioral modeling of criminal behavior using the ex- pected-utility-maximization principle are unques- tionably overly simplistic. Realistically, to capture the dynamics of criminal behavior adequately, structural models will have to evolve substantially. 3S9 The distribution function of in the population. Sentence length if arrested. Discount factor (O c l] < 11. The parameter describing the probability of incarceration function [P(~) = 7761 Value of being free at the be- ginning of any period in the constant-wage model. The increment to wage rates for each period of nonincar- ceration. V(w~) Value of being free for individ- ual with current wage we in changing-wage model. Previous number of arrests as of period t. USED Sentence length function. Value of being free for indivicl- ual with arrest record So in variable-sentence-length model. All three models have a number of common features. In(livicluals are as- sumed to be infinitely lived, or, equiva- lently, to have an unknown length of life (T) which is clistributed as an exponential random variable. Since the vast majority of in(livicluals seriously engaged in crim- inal activity are inactive after age 40, the assumption of infinitely lived individuals is not artificial for purposes of analysis.3 Within the context of these dynamic behavorial models, the inclividual's time- allocation decision will be investigated. The proportion of time spent in crime in period t is clenoted A. The total amount of time in each period of life is normalized Explicitly incorporating finiteness of life would considerably complicate the analysis, and the sub- stantive results would be unchanged.

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360 to 1. The remainder of time in each pe- riod (1 - ~) is spent in "legitimate" market work, which is compensated at a rate we. Leisure is ignorer] in what fol- Tows, or, equivalently, the leisure deci- sion is assumed to be exogenous to the criminal activity decision, and the time to be allocated between market work and criminal activity is the residual (total time in period minus leisure). It is also assumer! that no capital mar- kets exist so that individuals cannot bor- row or lend money in any period. Total consumption in any given period, then, is purchased solely with contemporaneous income if the individual is not incarcer- ated at any time cluring the period. This lack of the existence of capital markets is a limitation of the model; however, for purposes of studying behavior in the criminally active subpopulation, it may not be entirely unrealistic. Unlike legitimate activity, criminal be- havior is "risky" in a particular sense. If an individual is caught engaging in crim- inal activity, he or she is incarcerated for a total of gHt _ i) periods beginning with the perioc! in which apprehension occurs, where H~ _ ~ denotes the indivicluaT's criminal record through period t - 1. Thus, if apprehension occurs in period t, the indiviclual will be incarcerated for periods t, t + 1, t + gHt _ i) - 1. Note that sentence length is a deterministic func- tion of the inclividuaT's criminal history, which at the beginning of period t is summarized by Ht _ i. In general, it is reasonable to assume that the sentence length is an increasing function of the number of previous arrests, past time served in prison, or other observable characteristics of previous criminal activ- ity. While incarcerated, the inclividual has a consumption level c* each period. The probability of being apprehended for criminal action in a period is a func- tion of the amount of criminal activity engaged in over the period. This func CRIMINAL CAREERS AND CAREER CRIMINALS tional relationship is expresser] as Pi = P(~), where Pf ~ is monotonically in- creasing in ~ and P(O) = 0, that is, if the incliviclual is not criminally active in the period, there is a zero probability of ap- prehension. It is not necessarily the case thatP(l)= l;thatis,"full-time"criminals are not necessarily certain to be appre- henclecI. In general, P(1) c 1. Note that inctiviclual apprehension probabilities are a function of current period activities only, not of criminal activities in previous periods. To complete the specification of the choice set in(livicluals face, we next con- sicler the potential rewards from criminal activity. Let the total monetary and psy- chic rewards from criminal activity in period t for individual i be denoted Yin. When the time-allocation decision is made in period t, the final outcome or realization of Ye is unknown. Each indi- viclual floes know the distribution of re- wards he or she faces conditional on the time clevoted to criminal activity. The conditional distribution function for indi- vi(lual i is given by FRAYS. Unlike the other parameters of the problem, these conditional (listribution functions differ across population members. This varia- tion is meant to capture, in an admittedly limited way, the notion that individuals differ in their valuation of rewards from criminal activity. For all indivicluals, we assume that increases in 0, criminal activ- ity, will increase the expected value of criminal rewards in the period. By the assumptions below, we do not need to consider the effect of ~ on higher-or(ler moments of the distribution. Finally, we must consider the total val- uation of rewards from legitimate activi- ties. Con(litional on not being appre- hended in period t, the expected utility of individual i in period t is given by E Uit(dit, S) = ~ US-Bit~w + Y] dFi(YI ~it), (1)

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DYNAMIC MODELS OF CRIMINAL CAREERS where it is assumed that Ei(Y~ city is bouncled for Bit in the unit interval, and where S (success) inclicates that the indi- vidual was not apprehended in the pe- riod. In what follows we will assume that individuals are risk neutral, so that U(x) = x. This is done for reasons of tractability and because there seems to be no com- pelling reason to make differences in at- titudes toward risk the basis of a mocked of clifferential criminal activity. Then Equa- tion 1 becomes E Uit(8it, S) = (1 - Bit~wt + r Y ~Fi(Y~dit). (2) The last term on the right-hand sicle of Equation 2 is the expectation of criminal rewards in period t conditional on an activity level Bit. We will consider the case in which conditional expectation is linear, Ei(Y~ Bill = Bi fib. This wouIcl be true, for example, if the distribution of rewards was normal. The heterogeneity in individual valuations of criminal re- wards is reflected in the fact that di in the conditional expectation function varies across individuals in the population The population distribution of ~ is given by G(~), defined over the interval [3, Bl. Now we can state for the current period the expecter! utility associated with a level of criminal activity bit First, note that, given success, the expected utility from action Bit is given by (1 - Bi~)w~ + limit and the probability of not being ap- prehendec] is 1 - P(di~) If the individual is apprehended and incarcerated, the util- ity yield is a certain c*, and the probabil- ity of this occurring is Pang. Then ex- pected utility in period t is E Ui~( Bill = t 1 - Pt dial ] [ ~1 - Bill w ~ + Bi Bit] + PI Bit) c*. Before proceeding to the three dy- namic models, a few obvious restrictions on the parameters in this mode] shouIc3 36] be noted. First, if c* > we, there is no incentive not to engage in criminal be- havior, for even if incarcerated, the indi- viclual wouIc3 have a higher consumption value that when engaged in any level of market work. Second, assuming c* < we, it must be the case that hi > we for at least some individuals in the population or no criminal activity wouIc3 be undertaken. These restrictions are we > c* b> we. (4a) (4b) Note that for any individual with a value of ~ that satisfies the inequality ~ c we, no criminal activity will be uncler- taken in period t.4 The analyses below pertain only to individuals with ~ > we; all others will optimally choose not to engage in criminal activity. Let us turn to the consideration of dynamic behavior under three specifications of constraints on criminal choices. The Constant-Wage Monet To begin, we consider the case in which the wage of each individual in the population is fixed over time: we = w, t = O. 1, .... We will also begin by assuming that conditional on apprehension, sen- tence length is the same for all inclividu- als, regardless of criminal history, so (Oh _ ~) = A, t = 1, 2, .... Since we assume inclividuals are infinitely lived and that the choices individuals face are constant over time (but may differ across indivicluals), each individual will devote the same amount of time to criminal ac- tivity in each perioc! in which not initially incarcerated. For an individual, the con- stant rate of criminal activity, B*, will be a function of the parameters characterizing (3) preferences and constraints. In this first This condition is strictly correct only if the wage sequence wl, w2, . . . is increasing, which is the case in all models considered here.

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362 simple model, B* = d*(c*, 6, P( ), in, a). (The individual subscript i has been dropped for notational simplicity.) We now turn to an investigation of the function B*. Denote the value of being free (not incarcerated) at the beginning of any pe- riod by V. Conditional on choice of ~ in the period, an inclividual's expected util- ity given that he is not incarcerated is (1 - B)w + bB + ,`3V. The term ,BV is inter- pretec! as follows. If the incliviclual is not incarcerated in this period, he will be free to make a time-allocation decision next period. By the structure of this problem, the value of the decision is given by V. But rewards in the future are not per- ceived by individuals to be as valuable as rewards today. The rate at which individ- uals discount future rewards is given by the discount factor ,ll (O c ,l3 < 1). (If,B = 0, individuals completely ignore the eject of their current actions on future choices. As ,B approaches 1, individuals consider current and future rewards as virtually perfect substitutes.) Thus the value of being free next period, evaluated as of this periocl, is ,l3~. The probability of not becoming incarcerated is 1 - Pa). The "value" of becoming incarcerated during the period is determined in the following way. If incarcerated, the indi- vidual will serve ~ periods in prison, be- ginning toclay. The value of being in prison in the current period is c*; as of today, the value of being in jail next period is ,Bc*; and for m periods from now, it is j3mc*. Then the utility yield cluring the period of incarceration is T- ~ ~ ,lBic*. In addition, the individual will i = 0 be free to allocate time optimally in ~ periods the value of this is TV. Then the total value of incarceration is T- ~ ~ ,Bic* + ,BW. The probability of in i = 0 carceration is Pie). CRIMINAL CAREERS AND CAREER CRIMINALS When we combine all the elements dis- cussed above, the maximum value of the individual's time allocation problem in all periods when he is not incarcerated as of the beginning of the period is given by V= max :~1 - P(~)l[(1 - B)w + 30 ones 1 ~- 1 ~ ~ To simplify discussion, we make a further assumption about functional form. Let the conditional probability of apprehen- sion EP(~)] be given by Pa) = lid, O < 7' ' 1. Then we have V= max :(1 - nd)[(1 - 0)w + BB OF. tic 1 + PV] + ~ ~ [C ~+ ~ V ] ~ . Denote by 0* the amount of time de- voted to criminal activity not taking into account the restriction that this is a pro- portion lying in the unit interval. Then B* . is given Dy 6* = [2~(~- W)] 1 [a- ~(1 + A) T- 1 7~,BV + No ~/3i + 7~p7V (6) i= 0 The solution to Equation 5' is denoted B*. Then (Oif0* c 0 d* = ~ B* if O < d* < 1 (7) Cliff*- 1. If d* = 0 or d* = 1, we say that the individual's time-allocation problem yields a corner solution. If B* = 0, the

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DYNAMIC MODELS OF CRIMINAL CAREERS individual is always engaging in legiti- B* = - bo/b mate activity; if B* = 1, he is a "full-time" criminal. An interior solution exists if O c B* < 1; in this case the individual devotes some time to criminal activity and some time to legitimate activity. For this model it is possible to find a closed-form solution in the following manner. Note that V is defined by V= (1 - ~*~1 - B*)w + BB* + ~V] + *c* ~ pi + SAVE . (8) i = 0 Solving for V, we obtain V= t1 - b(1 - ~*) - nd*~-] - *1 - b*)w + So*] + Ad* c* I- pi] . (9) i = 0 This can be written as V aO + ale* + a240*)2 be + bid* where a0 = w, al = ~ - w(1 + ~) ~-~ + races, ,A, a2 = rat /(w - a), ho = 1 - /3, i=o and be = 7~(~p - ,B-). From Equation 6, write B* = C + TV, (6') where c = tSr1~3 - w)~-i t~ - w(1 + r') + ~-~ 71 c ~ ~] and (1 = t243 - w)~-i (A - ,(~). i=o Substituting Equation 9' into 6', (~*42+ ed* + q = 0 where e = 2bo/bl and q = (b! a2)-1 (Al be - b1 aO). Thus the solution for 0* is given by 363 bO a2 - a bO bl + b2 ao 1/2 (Jo) bit a2 1 Since a closed-form solution is avail- able for the proportion of time spent in criminal activity, determining the quaTita- tive effect of changes in the parameters (71, ,B, w, 3, c*, a) on behavior is straight- forward. Qualitatively, the following re- sults hold: - - - de* 60* 60* 2 O. ~ O- C 0 bb TIC* bT Ad* -' 0; ''O. (11) ~LOW That is, an increase in the expected mar- ginal rate of return to criminal activity results in an increase in the rate of crim- inal activity. The rate of criminal activity also is increasing in the utility associated with "failure" (incarceration), which is given by parameter c*. As punishments increase in length (a), criminal activity declines. Increases in the marginal arrest rate (71) result in decreases in crime rates. An increase in the direct opportunity cost of crime, the wage rate in legitimate work (w), causes decreases in the rate of crime. Results ofthis type have been obtained previously in a-number of static rational- choice models of criminal behavior. In fact, if sentence length ~ is equal to 1, this model reduces to a series of static optimi- zation problems. By allowing ~-2, indi- viduals face one of two choice problems at the beginning of each period. If they are not incarcerates! at the beginning of the period, they choose the amount of time to engage in criminal activity ~ and as noted above, in this model, they will always set ~ to the same value. If they begin the period incarcerated, their util

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364 ity for the current period is predeter- minecT at the value c*. The only parameter that reflects the dynamics of the problem, aside from the sentence length a, is the discount factor ,B. In this model the sign of dd*/d,B is ambig- uous. This partial derivative can be com- puted in a straightforward manner, but the result is not particularly enlightening. The intuition is basically the following. In any period in which indivicLuals are free initially, their expected current pe- riocT utility is given by Equation 3 and by the assumptions of the moclel, E U(~*) > c* for each individual in the population. If the sentence length is ~ periods, the difference in expected utility of freedom versus incarceration is (1 + ,B + ,B2 + . . . + IT- i) tE U(~*) - cod. Holding constant B*, an increase in ,B increases this cost. However, for any finite-length sentence a, an infinitely liver! individual (or an individual with a sufficiently Tong but finite life) will eventually be releasecI. The value of being free at the time of the release is ,B7V. As ,ll approaches 1, ,B~ ~ ,B so the value of being free at the beginning of the period ~ periods from the present (~7) approaches the value of being free next period (,l3V). At the same time, as ,`3- ~ 1, the value of the optimization problem goes to infinity. Thus, the penalty (1 + ,l3 + ... + by- i) tE U(~*) - c*] becomes insignificant, and this results in increases in criminal activity. Which effect will dominate depends on the values of all the parameters in the model. By the assumptions of this model, indi- viduals commit a constant rate of crime over their lifetime, which is contradictory to the empirical evidence that exists. In the model in the next section, criminal activity decreases, on average, as individ- uals age. Accumulation of Human Capital in Legitimate Activity Using the constant-wage model, the proportion of time nonincarcerated indi CRIMINAL CAREERS AND CAREER CRIMINALS viduals devote to criminal activity re- mains constant as they age. This simple model can be modified in several ways so as to produce the result that the crime is a decreasing function of age. One obvious modification is to allow the returns from legal and illegal activity to be age depen- dent. Intuitively, if the difference be- tween returns from legitimate work and expected returns from criminal activity diminishes over time, other things equal, the crime rate will decrease with age, (Recall that it was necessary to assume that the expected returns from crime were strictly greater than the legitimate wage if we were to observe any criminal activity. As the legitimate wage ap- proaches the expected returns from crime, we will observe a continuous de- cline in the crime rate of an individual.) The approach taken in this section is to hold the expected returns from criminal activity constant but to allow the legiti- mate wage to change systematically over the life cycle as a result of individual behavior and random events. While it would be desirable to allow the expected returns from criminal activity to vary sys- tematically over the life cycle also, such an extension would add greatly to the complexity of the model. Furthermore, what is really of interest is the difference between expected rewards from criminal activity and legitimate work. Thus, it is somewhat inconsequential whether we model the change in this difference as resulting from shifts in the legitimate wage, the expected returns from crime, or both. There exists a voluminous literature on the subject of human capital accumula- tion. For a statement of the general the- ory, see Becker (19751. We will assume here that there is no accumulation of crime-specific human capital-that is, in- dividuals do not become more proficient criminals as they acquire criminal experi- ence. Market wage rates do increase as individuals acquire market experience, however. We will characterize this de

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DYNAMIC MODELS OF CRIMINAL CAREERS pendency in the following way. In period t, when the amount of criminal activity is given by A, we will say that the indivic3- ual accumulates a total of (1-b~) units of experience if not incarcerated during the period. If incarcerated during the period, he accumulates no market experience. Similarly, if incarcerated at the beginning of the period he is not free-he accumu- lates no market experience. The amount of market-related human capital the indi- vidual has at the beginning of period t will be denoted by ht. The wage rate an individual faces in the period will be assumed equal to ht~wt = kit). The amount of human capital the individual possesses at the beginning of period t is defined in the following way. First, define a vari- able: ok = ok* if the individual was not in- carcerated during period k; ok = 1 if incar- cerated during period k. Then, the total amount of market experience the individ- ual has at the beginning of period t is - ~ (1 - ski k = ~ Market-related human capital is assumed to be a simple transformation of market experience, kit= g: -1 ~ (1 - Ski k= 1 , (12) where g is a monotonically increasing function; human capital is increasing in labor market experience. The choices an individual can make at any time t depend on his past allocation of time, {add, - it, in all periods when free, and on luck- that is, how often he was incarcerated in the past. These are the sources of variation in the sequence {0k~k _ i, which determine beginning of period t human capital, and hence the period t wage rate. At any age, t = 1, 2, , individuals will in general be differentiated accord- ing to their stock of human capital. Con- sider an individual making a time-allo 36S cation decision in period t. His choice ot a rate of criminal activity will depend on wage rate for the current period ht. This wage rate changes over time and is a state variable. An individual in state ht faces the optimization problem Vowel= max t(1- 71~1- howl 0 s 0, c ~ b~ + Avows + ~ by, wig) ~- ~ c* ~ hi + b\(w~) i = 0 , (13) where we + Ace, wit denotes the fact that given the wage rate in period t twit, the time allocated to criminal activity ~ oh, and the fact that the individual was not incar- cerated during the period, the period t + 1 wage is known with certainty. The func- tion we + Ace, wit is decreasing in the amount of time spent in criminal activity and increasing in the previous wage rate. Note that, if an individual is incarcerated in period t, when he is released in period t + ~ he will be able to work at the same wage as in period t. Thus we have as- sumed an absence of stigma the effect of jail on wages is simply an absence of growth, not a decline. In the changing-wage model there ex- ist a number of additional costs of crimi- nal activity. To review the structure ofthe model, the costs are as follows: 1. In the current period t, if the indi- vidual is not incarcerated, the opportu- nity cost of crime is simply forgone mar- ket work, which is remunerated at rate we. 2. In period t, increased criminal activ- ity increases the probability of incarcera- tion. The difference between the level of expected utility as a free individual and that obtained as a prisoner, multiplied by the increase in the probability of being incarcerated, is an additional cost of in- creased criminal activity. 3. Conditional on the current wage rate we, increases in criminal activity de

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366 crease next period's wage wt + 1 tglven no incarceration in period t) owing to for- gone human capital accumulation. Since the expected utility of all individuals is an increasing function of the market wage in all periods, the Tower future wage rate must Tower expected utility levels in fu- ture periods. 4. Increased criminal activity leads to an increased probability of incarceration, and, while incarcerated for ~ periods, the individual does not accumulate any mar- ket capital. This represents a permanent wage reduction in future periods, or a persistent effect of incarceration. Solving Equation 13 turns out to be quite difficult in practice, even for simple foes of the human capital accumulation function g. Therefore, for the remainder of this section the discussion is confined to the following special case. We will assume that as Tong as an individual is not incarcerated during period t, his wage will increase by ~ in period t + 1. Then, we + ~ = we + car (given no incarceration in period t). Given that the individual is not incarcerated in period t, the period t + 1 wage is independent of Cowl + ~ by, wit = we + I. (wig. The cost referred to in Point 3 above is absent. However, Point 4 is still operative increased crime increases the probability of incarceration, which is as- sociated with forgone human capital ac- cumulation. With this simplification, Equation 13 can be rewritten V(wt) = max ~ (1 - 71~1 - howl 0 ~ 8, ~ ~ + be, + pV(w' + a)] ?- ~ +~ c* ~ i = 0 hi + p7V(w') 11 Now the inclivicluaT's time-allocation problem depends on the set of parame CRIMINAL CAREERS AND CAREER CRIMINALS ters in the constant-wage model, plus the wage-growth parameter a. Unlike the constant-wage moclel, it is not possible to finct closec3-form solutions for d*(w~) or V(w~), so numerical methods must be used to investigate quantitative proper- ties of these functions. However, all the comparative static results in Equation 11 hoIcl for the changing-wage model, and, in aciclition, d~/3cY < 0-the larger the wage increment, the Tower the crime rate, for the larger is the opportunity cost of incarceration. Finally, some numerical examples wit! illustrate the indivi(lual-leve] and ag- gregative characteristics of this model. These computations do not constitute an exhaustive study of the Function 13'; rather they demonstrate the types of crim- inal careers that can be generates] by this simple model. The parameter values se- lectec] for this illustration were not cho- sen after an exhaustive search. It appears that this moclel can generate "interest- ing" career patterns (i.e., not all corner solutions) without extensive search over the parameter space. The actual parameter values chosen are arbitrary. The initial wage level twit is set to .5. Then Conclition 4a is imposed by setting we > c* anti, in particular, setting c* = 0. The wage increment Acid is set to .05. The discount factor (,B) is equal to .8, the arrest parameter (71) is set to .5, and the sentence length (~) is set to three periods. All inclividuals face these same parameters; however, two distinct values of ~ are assumed to exist in the popula- tion. The conclitional expectation param- eter is given the value 3 for 50 percent of the population, and the value 2 for the other 50 percent. The us = 3 indivicluals are "high crime" types and the ~ = 2 ( 13 ~ inclivicluals are "low crime" types. In Table 1 the amount of time devoted to criminal activity is shown as a function of the beginning-of-perioc! wage level for both population groups. Note that both

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DYNAMIC MODELS OF CRIMINAL CAREERS V(St)= max i(1- r'0t)[(l- Bt)w ocols 1 ~ ads,)- 1 ~ Apt + TV] + ~ C* ~ pi i = 0 +~St)V(St+l) (14) Corresponding to this problem there ex- ists a solution B*(S~). The ordering of the solutions is B*(0) 2 B*(1) 2 8*(2) 2.... The larger are the differences in the sen- tence length function Ok) - Ok - 1), k = 1, 2, ..., the larger are the differences B*(k) - 8*(k - 1). (Note thatIarge changes in ~ as a function of sentence length may result in inclividuals who originally de- vote a substantial amount of time to crim- inal activity eventually switching out of crime completely.) An example similar to the one used above illustrates the characteristics of criminal careers generated by this model. All parameter values are exactly the same with the exception of the sentence length a. In this model, ~ is set to 1 if the individual has no prior convictions and is set to 5 if the individual has any prior . . convictions. The decision rules are presented in Table 3. The amount of criminal activity for the Tow- and high-crime types is greater than was the case in Table 1, conditional on no previous arrests. This increased activity results in increased ar- rest probabilities, however, and, after one arrest, individuals devote less time to criminal activity than was the case in Table 1. After one arrest, a low-crime type receiving a wage of .5 will spend only about one-third as much time in criminal activity as was previously the case. High- crime types also substantially reduce criminal activity after one arrest but not to the same degree as low-crime types. 369 TABLE 3 Time Allocation to Criminal Activity Given Wage Growth with Varying Sentence Length Low-Crime Types Wage No Level Arrests .5711 .5525 .5374 .5242 .5098 .4937 .4755 .4544 .4297 .4004 .3652 .3222 .2697 .2058 .1291 .0392 0.0 Some Arrests High-Crime Types No Some Arrests Arrests 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 .2176 .1326 .0406 0.0 .6693 .4786 .6513 .4466 .6330 .4117 .6145 .3734 .5962 .3313 .5786 .2851 .5621 .2345 .5472 .1796 .5343 .1204 .5233 .0574 .5137 0.0 .5042 .4941 .4832 .4714 .4587 .4449 .4298 .4131 .3945 .3736 .3499 .3228 .2915 .2553 .2132 .1646 .1089 .0463 0.0 Aggregate statistics are presented in Table 4. Compared with Table 2, we see that a greater amount of crime occurs in the first few periods given varying sen- tence lengths, but eventually total crime is recluced as more indivicluals are subject to the stiffer sentence ~ = 5. The jai] population is substantially smaller over the life of the cohort in this model. Identification of Structural Moclels The models proposed above were pri- marily clesignecT to illustrate how various

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370 empirical regularities, such as the clecTine of crime rates with age, can be generated from dynamic behavioral moclels. As dis- cussed earlier, such structural models may be preferable to less behaviorally motivated statistical models in that all parameters have relatively clear interpre- tations. Structural models are probably not useful when their structure precludes them, a priori, from reproducing salient empirical regularities. If these structural models are to prove useful empirically, we must of course be able to obtain consistent estimates of all or most of the parameters in the decision rules. The first consideration is one of iclentification. What types of data are re- quired to estimate one of these models? Let us consider the variable-sentence CRIMINAL CAREERS AND CAREER CRIMINALS length model presented above as an ex- ample. The model win increasing sentence lengths is described by the following set of parameters: 71, w, 3, A, c*, T( ). Identi- fication may proceed in the following way. First, recognize that the consump- tion value of being in prison (c*) is arbi- ~ary. Setting it to a given value essen- tially fixes the location of the utility index. It seems most natural to set cat = 0. The rates of arrest, conviction, and incarcera- tion may be computed from victimization surveys, which give an estimate of the total number of crimes committed (of a particular type). These, combined win the number of individuals incarcerated for the crime, will yield an estimate of 7~. Computing the sentencing function T( TABLE 4 Aggregate Crime Statistics in Simulated Population Given Wage Growth with Varying Sentence Length, 1,000 Individuals in Each of High- and Low-Crime Groups Total Crime Arrests Jail Population Period 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 35 40 45 50 Low 571.1 463.7 351.3 228.2 152.4 112.1 87.7 67.7 45.1 31.2 20.5 14.8 10.8 7.0 3.7 1.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 High 669.3 587.8 488.9 386.8 302.3 229.3 209.4 207.8 189.6 164.6 139.2 105.4 103.9 102.4 87.4 72.3 54.6 52.0 48.1 40.9 19.2 9.2 4.1 1.2 0.4 0.1 Low 265 213 190 103 72 55 43 38 24 22 12 o o o o o o o o o o o High 368 292 264 195 158 122 91 103 111 74 79 52 45 52 44 30 23 35 31 18 10 3 :0 Low o 27 61 70 70 43 12 11 11 8 3 o o o o o o o o o o o High ' O o 81 193 296 399 410 364 346 337 311 315 278 226 210 183 164 145 128 116 61 30 9 2 o o

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DYNAMIC MODELS OF CRIMINAL CAREERS is also relatively straightforward. Either actual sentencing records may be used or official guiclelines, when available. The parameters ,`3, w, and ~ present more of a challenge. Often the value ,`3 is not estimates! in analyses ofthis sort it is merely set to a "reasonable" value. tYDi- cally .9 or .95. Since criminally active individuals are often thought to discount the future rather heavily (that is, they have Tow values of,(~), in an analysis such as this it may be of interest to estimate ,(3. This parameter is in principle identified in this model, at least if we are able to observe ~ the proportion of criminal activity in period t. The other inclividual- level ciata neecled for purposes of iclenti- fication are wage rates. Wage rates are obviously not identical over time and in- clividuals nor are they identical over time for the same individual. We can incorporate this observation by assuming wit ~ N(X'it 7, By, for example, where X'i~ is a vector of individual characteristics at time t and ~ and 3 can be estimated. In periods when individuals are fulI-time criminals no wage will be observed, but by making a distribution assumption re- garding wit, (lata from such periods will still be informative for A, a2. Estimation of the parameter ~ or its distribution in the population is the most cliff?icult. It is not necessary to measure the returns from criminal activity to esti- mate this parameter, however. One could proceed in the following fashion. First, assume a form for the population distribu- tion of 5, say M(3, id, where ~ is a param- eter vector that characterizes M. The like- lihood of observing wit and Bib in a period can be constructed conditional on a value of b. By taking the expected value of this conditional likelihood with respect to the distribution of &, we can form an uncon- ditional likelihoocl that depends on the parameters (y, 3, ,`3, s) By conjecture, for identification of A, ,l3 must be fixed. But note that in this analysis it is possible to 371 estimate a rather abstract but interesting distribution M(3, (), even if we assume that criminal rewards are not measurable or even operationally cleanable. ECONOMETRIC MODELS OF CRIMINAL CAREERS The dynamic moclels of the criminal career developed above are based on op- timizing behavior. As discussed, there are advantages and disadvantages to the esti- mation of such highly structured models. In short, the principal advantage is unam- biguous interpretation of parameter esti- mates and statistical tests. The principal disadvantages are the complicated com- putational algorithms that are required for estimation and the typically poor "ex- planatory" power of such models. Given the current level of understanding of the simple statistical properties of the crimi- nal career process, perhaps it is beneficial to work with econometric moclels that are less closely linked with a specific behav- ioral moclel, but that allow for statistical associations precluded in any tractable decision-theoretic model. Actually, the choice is not between one approach or the other. Both can and should be used in any systematic study of the criminal career. In this section the relevant theory is outlined ant! a relatively general frame- work is presented in which parameters of continuous time behavioral moclels may be estimated. The focus of the discussion is the econometric and statistical proper- ties of continuous time models. To fix ideas, consider a continuous time, discrete state space stochastic proc- ess X, where the state space consists of the nonnegative integers S = Z+ anc! where the parameter set T = (0, so). The state of the process at time t Isis indicates the number of times some event has oc- curre<1 from We origin of the process, normalized at 0 without Toss of generality,

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372 through time t. For example, say life be- gan at time 0, and there is only one type of crime individuals can commit. Then so indicates the number of times an individ- ual has committed this crime as of age t. Let the times at which crimes occur be 4~'lV~ll LJ' ~ i' amp. . . - The number of crimes previously committed as of time t Isis is equal to so if and only if as* ' t < As* + ~y Also, define the duration until the first crime as we = ;~-0 = Hi. The duration of the kth spell, i.e., the elapsed time be- tween crimes k and k - 1, is wk = ~- Hi-- ~ The stochastic process can be cnarac- terized in a number of alternative ways. For the most part in this section the dis- cussion is focused on the interval specifi- cation ofthe process. Starting from time 0, the process is characterized by the joint distribution of intervals between events, [;n(Wi' W2, ...' Wn)' for n = 1, 2, .... In terms of a specific application to the analysis of criminal careers, the formal- ism above reduces to the following. Say an individual is born at time 0 and lives to age T (T ' oo). At each instant of life t (0 ' t ' T) the individual either commits a crime or does not. The length of time between successive criminal acts is, in general, not constant. Loosely speaking, the individual's propensity to commit a crime at some particular time t depends not only on his or her "normal" rate of crime commission and the elapsed time since the last crime was committed but also on the opportunities for crime com- mission that exist at that particular mo- ment. Thus, even if an individual would "normally" be highly likely to commit a crime at t, the fact that a police officer happened to be in close proximity would probably induce a postponement to a later date. Or the fact that an attractive "mark" appears may induce a crime be- fore we would normally expect one. Un- anticipated or anticipated changes in the choice sets of individuals will cause vari- ations in criminal behavior over the life ~i`7~n hat =` ~ CRIMINAL CAREERS AND CAREER CRIMINALS cycle, as the preceding section demon- strated. To the extent that those changes are not anticipated by the individual or observable to the analyst, the process of crime commission must be considered to be random. Then, the length of time from the beginning of life to the time the first crime is committed is we, which is a ran- dom variable. The distribution of we is given by Flows). Analogously, the length of time between the first and second commis- sion (w2) has distribution F2(w2), and in general the duration oftime between crime (i - 1) and i is Fiji). The joint distribution over the first n crimes is given by knit, w2, . . ., we), as stated above. To be useful for purposes of statistical (or theoretical) analysis of the criminal career, some structure must be imposed on the general joint distribution Fn(W~, we, . . ., we). A natural starting point is to assume that, for a given individual, all spell lengths are independently diskib- uted, i.e., the joint distribution of dura tions we, w2,. ., wn can be written as n F(wl, w2' . . . ~ Wn) = II Fi~wi)' i = 1 for n = 1, 2, .... Simply stated for the case n = 2, this implies that the length of interval 1 does not alter our assessment of the likelihood of observing any particular value for the duration of the second spell. By adding another assumption con- cerning the joint distribution of the spell lengths, we can produce a class of models often used in engineering and increas- ingly in the social sciences. If we assume that the distribution functions have the same (identical) form, Fits) = F2(s) = ... = Fats); s-0, n = 1, 2, . . . . then we can write the joint distribution of the first n spells as

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DYNAMIC MODELS OF CRIMINAL CAREERS n Fn(Wl' w2' . . ~ ~ Wn) = II F(wi) i = 1 A point process in which the spell lengths are independently ant! iclentically clistrib- utec! (i.i.~.) is a renewal process. If the common (to all spells) duration distribution is everywhere cTifferentiable (as is assumed throughout this paper), there exists an associated probability den- sity function f(W). A renewal process can be completely characterized by F(w) or f(w), if it exists. Alternatively, it can be characterized by its hazard function h(w), which is defined as h(W)= few) 1 - F(w) 373 many econometric advantages, which are cliscussec3 below. One of these is that information from incomplete spells, those which began during the sample period but hacT not been completed when the sample period enclect, can be incorpo- ratec3 into the estimation procedure in a straightforward way. In actuality, individ- uals are only observed over some portion of their lifetime. Let the sampling period! be the interval (0, I) and assume that over this interval the incliviclual is observer] to commit m crimes. For the pure renewal process described above, we know that the mth event occurred at time Tm; how- ever, we clic3 not observe the time at w-0 which the (m + l)st event occurred. We do know that this event had not occurred by the end of the sample I. This occurs with probability 1 - F(! - Tm). It is easy to show that this quantity, referred to as the survivor function, is equal to exp [- TOSh(u)clul,where s = ~ - Tm. Environments are, of course, highly nonstationary, ant! at a single point in time there exist substantial amounts of heterogeneity with respect to budget sets and preferences. Renewal processes can still provide a useful framework for ana Tyzing dynamic behavior if we generalize them so as to incorporate some forms of nonstationary and heterogeneity. We may retain the i.i.~. assumptions regarding the density of duration times, but make the parameters describing the duration den sity functions of observable and unobserv able in(liviclual characteristics. These characteristics may change over time. For example, we may write the conditional hazard function as htWik ~ Zi~-(7i~ + wit); Hi, where k indexes the serial order of the spell, wil is the duration of the kth spell for individual i, Zi( ) is an individual specific vector of observable and unob servable sources of heterogeneity that can be time-varying, and ~ is a conformable parameter vector. In the case of the pure, unconditional renewal process first described, the den The hazard function is the conditional den- sity of duration times given the individual has not committed a crime for a period of length w. The hazard function h is used in the econometric model formulated below. One of the most important characteris- tics of a duration density from both a behavioral and statistical perspective is the degree and type of duration depen- dence exhibited. Duration dependence is most easily investigated through the haz- ard function. Simply differentiate in(w) with respect to w, dh/dw. If dh(w) dw s - ~ o' we say that the hazard function (or den- sity) exhibits positive, no, or negative du- ration dependence when evaluated at du- ration s. If the sign of the derivative is the same for all s c (0, so), we say that the hazard or density exhibits monotonic clu- ration dependence. If the signs switch at least once, duration dependence is nonmonotonic. The only duration density that exhibits no duration dependence over the entire interval (0, so) is the expo- nential f(w) = ~ exp (- tow), ~ > 0. Parameterizing the hazard directly has

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374 sity of duration times fits couIc3 be esti- matecI by parametric or nonparametric methods simply from a sufficiently large number of completed spells for one indi- vidual. Once we allow for conditioning on a set of incliviclual characteristics Zi, some of which may be time invariant, it is clear that to estimate all elements of ~ we will need observations for many individ- uals The econometric model developed below is designed for use with event- history data (dates of criminal actions for large numbers of individuals). Most dynamic moclels of behavior im- ply restrictions as to the form of the con- ditional hazard function. This is also true for moclels of criminal behavior. For ex- ample, a popular moclel of the criminal career assumes that inctivicluals commit crimes at some constant rate A over the course of a criminal career (0, T*), where To is random. This implies that the dura- tion times between successive criminal acts over the period (0, T*) are clistributed exponentially with parameter A. As al- ready discussed, the exponential clistribu- tion exhibits no duration depenclence. An individual is equally likely to commit a crime in the next small interval of time no matter how Tong it has been since the last criminal action. Alternative moclels of criminal activity would not be consistent with an exponential distribution of times between successive crimes. For example, if the opportunity costs associates] with committing a crime increased in the length of time since the last crime was committed, while the distribution of potential rewards from criminal actions was constant, the duration distribution of intervals between crimes would ex- hibit negative duration depenclencc the greater the duration since the last crime, the lower the instantaneous rate of com- mitting a crime. The flexible econometric moclel pre- sensed in Flinn and Heckman (1982a) controls for observed and unobserved CRIMINAL CAREERS AND CAREER CRIMINALS heterogeneity in the population by pa- rameterizing the hazard function in a gen- eral way. If we assume that spell lengths for an incliviclual are i.i.~. conditional on observed and unobserved heterogeneity and that only one spell is observed for each individual (for notational simplic- ity), we can write the hazard function as hi(W) = exp [Zi(W)3 + A(w)y + View)]' (15) where we have assumed for notational simplicity that the start of the observa- tional period corresponds to calendar time 0. The vector of observable, exogenous individual characteristics at time w is denoted Zinc), and ,B is a con- formable parameter vector. The vector A(w) consists of polynomial terms in clu- ration, that is, A(w) = (w, w2, . . ., wk), and ~ is a k-dimensional parameter vector. An unobservable variable View) is permitted to be a function of duration. Exponentia- tion of the term in brackets ensures that hi(W) is nonnegative, as is required, since hi(W) is a conditional density function. Many stochastic models ofthe duration between crimes can be nested within this moclel as special cases. In many models the role of inclividtlal-specific, unob- served heterogeneity is stressed the View) in Equation 15. Conditional on Views, these moclels typically restrict fly to be a zero vector; thus they posit no clura- tion clepenclence. Where (luration clepen- dence is allowed, functional forms are estimated that restrict the hazard function to be monotonically increasing or cle- creasing in time since the last criminal event. By using a polynomial "approxi- mation," expLA(w)~], we allow for non- monotonic patterns of duration clepen- dence. In the absence of a behavioral mode! that gives the analyst a strong rea- son to restrict his or her attention to spe- cial cases, it can be argued that as general a form of estimating the equation as is

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DYNAMIC MODELS OF CRIMINAL CAREERS feasible should be usecl. Computationally it is straightforward to introduce the term expLA(w)^y], as is clone in what follows. The one-state renewal moclel can be generalized in several ways that may prove useful in the study of criminal ca- reers. The assumption that the criminal career is a conditional renewal process (i.e., conditional on other exogenous sto- chastic processes) can be dropped. Flinn and Heckman (1982b) discuss several forms of departure from the basic renewal process that may be relevant for the anal- ysis of dynamic behavior. First, consider a case in which crimi- nals acquire crime-specific human capital in the course of engaging in criminal behavior. Experienced criminals may be better at avoiding detection or identifying profitable targets than nonexperienced criminals. Then, if the rewards from legit- imate market activity remain approxi- mately constant over the life cycle' we wouIcI expect both the frequency with which crimes are committed and the yielc! from criminal activity to change over the career. We should unambigu- ously expect the yields from crime to increase; the frequency with which crimes are committed may increase or decrease as criminal experience is ac- quired. Even if it were possible to mea- sure criminal human capital or yields from crime sufficiently precisely, by con- ditioning on those characteristics the criminal career could still not be consid- ered a renewal process, since the level of those characteristics clepencTs on the past history of the process. We can mocle] this departure in a rel- atively straightforward way. Consider the intervals between crimes for an in- diviclual who has committed n crimes. Conclitional on all observable exogenous characteristics, we can consider the clu- rations we, we,..., wn to be inclepen- clently but not identically distribute~l. Then, 375 F(w1, W2' ~ En) n = ~ Fi(Wi), i = 1 but it is not the case that F1 - F2 = = Fn. Consider a multiple spell version of Equation 15. Let j inclex the serial order ofthe spell (i = 1 corresponds to the spell beginning at time O and ending with the first crime, j = 2 is the spell between the first and second] crimes, and so on). Then we can write the hazard function for in- terval j for indiviclual i as hij(W) = expEZi(;ij + w),Bj + A(w)>j + Vij(~ij + w)l, (16) where rij is the calendar time at which individual i committed his jth crime, ,{3 and lye are parameters associates! with the hazard function for the jth spell, and Vij is the unobserved heterogeneity compo- nent associated with the jth spell for in- clividual i. By analogy with the variance components moclel often used in the anal- ysis of discrete time pane] data, we write Vij(7ij + w) = Hi + Bij + e(Tij + w), where hi is an individual-specific, spell- anc3 time-invariant heterogeneity compo- nent; r)ij is a spell-specific, time-invariant heterogeneity component; and S(t) iS white noise [that is, (t)-8(S) iS normally distributed with mean O and variance (t - s) for t > s]. In what follows, we neglect continu- ously varying components of unobserved heterogeneity. While it wouIc3 be highly desirable to mocle! such components ex- plicitly, their inclusion in the economet- ric model does not seem computationally feasible. We assume that unobserved het- erogeneity components are constant within spells, i.e., Vij(7ij + w) - Vij. To simplify calculations further, we aclopt a one-factor specification of unobserved heterogeneity Vij = Cj ~i' J i= 1,...,J,

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376 where the Cj are parameters or tne model and ~ is the maximum number of spells observed in the sample. Thus, individual heterogeneity is constant over time and spells, although the relationship between hi and the rate of exit from the spell depends on the serial order of the spell through the parameter Cj. The rate of criminal activity will, in general, depend not only on the length of time since the previous crime was com- mitted, but also on the inclividual's age, and, more important, his previous record of crime commission. Consider spell j. The previous history of inclividual i's criminal career consists of twit, Wi2,--e, wi,j _ i; Z(t), O ' t ' ri j _ il. Suppose certain characteristics of this history are of interest to us, for example, the mean, variance, or some other moments of the sample clistri- bution of (Wit, Wi2, . . ., Wi,j _ i). These char- acteristics are simply functions of the his- tory, StHi(ri,j _ i)], where Hi(ri,j _ i) is incliviclual i's history up to time ~ij-i Then we can estimate the conditional hazard function for the jth interval as hij(W) = exp~zi(rij + w)pj + A(w)7j + StHi(rij_ i)] fj + Vij), where fj is the parameter vector associ- ated with characteristics of the history up through crime j - 1. In this version of the model, spells between crimes are neither identically nor inclepenclently distrib- uted; thus, the criminal career is modeled as a point process rather than as a strict renewal process. Because the process evolves uniclirectionally in time, the time dependence is recursive. Presumably, a moclel along these lines is required to assess the degree of state clepenclence in criminal careers that is, the extent to which the current commission rate de- pencIs on the criminal history after concli- tioning on both observed ant] unobserved exogenous processes. Up to this point we have assumect tnat only one type of crime is committed in CRIMINAL CAREERS AND CAREER CRIMINALS r.l l l the population or, at the least, that each individual commits only one type of crime, although different individuals may specialize in different crimes. It is rela tively straightforward to generalize the econometric model presented above to cover the possibility of crime switching when each individual may commit any one of a number of types of crimes. Say there are K types of crime, K > 1. We will initially restrict our attention to (condi tional) renewal processes. Imagine that an individual commits a crime oftype k at time r. Then, we are interested in esti mating the parameters of the length of time between the commission of a type k crime and the commission of all other crimes, for k = 1, 2, . . ., K. For simplicity, assume K = 2. At time ~ a type 1 crime is committed. The "latent" time to commis sion of another type 1 crime will be de noted tot. The density of these latent times is assumed to exist and to be given by gate. If type 2 crimes did not exist, this density could be directly estimated using observed durations between suc cessive type 1 crimes. Denote the "la tent" duration between type 1 crimes and type 2 crimes by tt2 and its associated density by g~2(tt21. It is necessary to as sume that the random variables tat and tt2 are independent. In terms of the ob served outcome of the criminal process, a type 1 crime will be the next type ob ~ ~ ~ served if tot = minutia, tt2), and a type 2 crime will be observed if tt2 = minutia, tI2). Then if to = minutes, tt2), we will observe a type j crime at time ~ + tfj. Similarly conditional on a type 2 crime at time a, there will exist latent duration densities g21(t2~) and g22(t2*2) generating times until the next crime, so t2*j = mint, thy. Then, in this two-crime world, we would be interested in estimat- ing the parameters of the four latent den- sities gii, gi2, g2i, and g22. These densi- ties constitute a complete description of the criminal history.

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DYNAMIC MODELS OF CRIMINAL CAREERS For the general K state case, we will need a total of K2 latent density functions to describe the crime process Gil, i, j = 1, . . ., K. (In addition, we will have to esti- mate densities go, j = 1, . .., K, which correspond to the latent duration densi- ties from initial entry into the population at risk of committing a crime, which we will denote by state 0, until a crime of type j is committed.) For each latent den- sity Gil, i = 0, 1, . . ., K; j = 1, . . ., K, there is a corresponding hazard function hij. The joint density of the k latent durations . . IS given Dy hij(tj) exp - hij(u) du, -1 (Jo 1 i= 1,...,K. An individual is observed to commit a type j' crime after the type i crime if the latent time to is the smallest of the K possible latent times, tip, . . ., talk. Let the probability that an individual commits a type j' crime after a type i crime be denoted Pij. Then, roe ~r~ rx r K Pij = ~ ~ ~ ~ ~ hij(tij) O _ to' i., Nisi' expel-: hij(u)du~dti*j I x thij,(ti,)exp~-: hij(U)dU]~:dtt*j' rx Jo his (tiff ) P [ J ~ ~ hik(u) | due dt,*j,. The conditional density of exit times from state i into state j' given that ti*j < to (Nij: i 7ij')is 377 g(ti*j,~ti*j, < t*,) A: j 7t j' ~ J ''J ~ K ] Pij' It follows that the marginal density of exit times from state i can be written K gi(ti*) = A, Pij,g(tl7,lt*~, < tail; j = 1 ~ K ] k = 1 exp t J [ ~ hik(u) ] du The probability that the spell is not com plete by some time T. where T is the end of the observation period, is prob (to > -T) -- 1 - Gi. (T), where Gi. is the cumulative distribution function associated with gi.. ~] . ~ nits expression Is r x Prob(ti~ > T) = | gi.(ti*)dt~ J T = exp - T K hik(u) du [ io Ok-1 ] ] This term enters the likelihood function for incomplete spells at least T in length. Say we have access to event-history data for I individuals. For a given individual i, we observe his or her criminal career from time of entry into the criminal process t~o(i)] until some termination time T(i), which corresponds to the end of the sample pe- riod or the time of death (both events are assumed unrelated to criminal activity). In general we observed a total of m(i) crimes over the sample period. Denote the calen- dar time of each criminal event by ~(i), ~ = 1, 2, ..., m(i). Now, define a function of

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378 s[7~(i)]-sl(i), which gives the type of crime committed at calendar time ~(i).5 Then, conditional on a set of unknown parame- ters, Q and unobserved person-specific heterogeneity component Vi, the likeli- hood of observing the recorded criminal history for individual i is m`vi'- ~ i(Q~Vi) = II gs/(ils/+ lfi)Et*/(i~s/+ Bait I=0 t*/(i~s/+ I(i) < t*/(i~j; j = 1, . sit+ l(i) - 1, sit_ 1(i) + 1, . . ., K; Vi] x Ps/(i~s/ + ~(i~)(Vi) ~ Gsm(ix) ET(i) - M(i) Ovine where t* can be obtained under stan- dard regularity conditions as the solution to max L(Q, 4?). (17) Oil) Given that the distributional assumptions regarding h ant] B are correct, the maxi- mum likelihood estimator defined by Equation 17 has optimal statistical prop- erties asymptotically (as the number of individuals grows large). This model is relatively general and has been used to estimate the stochastic structure of labor market attachments. The generality of the model, however, seems to preclude treatment of compli- cated initial-conditions problems or com- mon forms of sample selection. The solu- tions to those problems seem only to be tractable when sufficient stationarity is imposed as when the underlying crime process is exponential-see for example Rolph, Chaiken, and Houchens (1981). The difficult choice for the analyst ap- pears to be either to use relatively general econometric models, which require a type and quality of data rarely available to students of criminal behavior, or to tailor the econometric models to the data cur- rently available. This latter option results in stationarity assumptions that are not consistent with the spirit of the dynamic behavioral models presented above and, more importantly, are not testable. It is essential, first, to estimate general models for stochastic processes on some "ideal" data set (no doubt yet to be collected) so that we can determine what types of sta- tionary assumptions are reasonable. Until that time, we should remain cautious in interpreting the results from the empiri- cal analyses of criminal careers.

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DYNAMIC MODELS OF CRIMINAL CAREERS CONCLUSION In this paper various approaches to the mocleling of criminal careers were pre- sentecI. A number of dynamic behavioral models of criminal activity were clevel- oped, and characteristics of the solutions were cliscussecI. Although closed-form so- Intions are not typically available for cly- namic optimization models, numerical methods may be used in a relatively straightforward way. The behavioral models were designed to illustrate the fact that the effect of current choices on future options has po- tentially important deterrence effects. Thus,.the fact that an incliviclual facing a 1-year sentence if caught committing a crime will face stiffer sentences in the future if caught committing acIditional crimes will, in general, affect criminal behavior at all points over the life cycle. The static models usually employed in empirical research are not capable of cap- turing these dynamic deterrence effects. It was also shown that personal character- istics, such as race, age, or drug usage, may not be simple indicators of an indi- viduaT's "inherent" propensity to commit criminal acts but instead may merely re- flect the relative rewards to criminal ver- sus noncriminal actions that the individ- ual faces. Thus these characteristics may be better thought of as indicators of dif- ferences in choice sets than of differences in preferences. While these interpreta- tions may seem indistinguishable for pur- poses of conducting empirical analysis, they imply very different policy actions in dealing with criminal behavior. Econometric models of the duration of time between criminal activities (differ- entiatect by type) were also presented. These models are capable of capturing _ the dyr~amics of the criminal career more 379 adequately than the behavioral models from a strictly empirical perspective. One is left with the clifficulty of substantive interpretation of parameter estimates, however since no explicit behavioral . . . model is used to generate the function estimated. It should be possible to learn something interesting, even if clescrip- tive, about the dynamics of criminal ca- reers from the estimation of such models. REFERENCES . .. ,, Becker, G. 1975 Human' Capital. 2nd ed. New York: Colum- bia University Press. Bentham, J. 1780 An Introduction to the Principles of Morals and Legislation, I. Burns and H. Hart, eds. London, England: Athlone Press (1970). Blumstein, A., Cohen, J., and Nagin, D.j eds. 1978 Deterrence and Incapacitation: Estimating the Effects of Criminal Sanctions on Crime Rates. Report of the Panel on Deterrence and Incapacitation. Washington, D.C.: Na- tional Academy Press. Chaiken, I., and Chaiken, M. 1981 Varieties of Criminal Behavior. Santa Monica, Calif.: Rand Corporation. Flinn, C., and Heckman, J. 1982a Models for the analysis of labor force dynam- ics. Pp. 3~95 in R. Basmann and G. Rhodes, eds., Advances in Econometrics I. Green- wich, Conn.: JAI Press. 1982b New methods for analyzing individual event histories. Pp. 9~140 in S. Leinhardt, ea., Sociological Methodology 1982. San Francisco, Calif.: Jossey-Bass. Marschak, J. 1953 Economic measurements for policy and pre- diction. Chapter 1 in W. C. Hood and T. C. Koopmans, eds., Studies ire Econometric Method. Cowles Commission Monograph 14. New York: John Wiley & Sons. Rolph, J., Chaiken, J., and Houchens, R. 1981 Methods for Estimating Crime Rates of In- dividuals. Santa Monica, Calif.: Rand Corpo- ration. Schmidt, P., and Witte? A. 1984 An Economic Analysis of Crime and Justice. Orlando, Fla.: Academic Press.