Below are the first 10 and last 10 pages of uncorrected machineread text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapterrepresentative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 356
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 WisconsinMad
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
WisconsinMadison.
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.
OCR for page 356
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
individuallevel 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 decisiontheoretic 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
OCR for page 356
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 utilitymaximization
OCR for page 356
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 rationalchoice 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
pectedutilitymaximization 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
constantwage 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
changingwage 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
variablesentencelength
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.
OCR for page 356
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,"fulltime"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 timeallocation 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 higheror(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) = ~ USBit~w + Y]
dFi(YI ~it), (1)
OCR for page 356
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 righthand 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 ConstantWage 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.
OCR for page 356
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 timeallocation 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 timeallocation problem
yields a corner solution. If B* = 0, the
OCR for page 356
DYNAMIC MODELS OF CRIMINAL CAREERS
individual is always engaging in legiti B* =  bo/b
mate activity; if B* = 1, he is a "fulltime"
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
closedform 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 closedform 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 anumber 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
OCR for page 356
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 finitelength 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 constantwage 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
crimespecific human capitalthat 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
OCR for page 356
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 (1b~) 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 freehe accumu
lates no market experience. The amount
of marketrelated 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 = ~
Marketrelated 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 timeallo
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 changingwage 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
OCR for page 356
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 timeallocation
problem depends on the set of parame
CRIMINAL CAREERS AND CAREER CRIMINALS
ters in the constantwage model, plus the
wagegrowth parameter a. Unlike the
constantwage moclel, it is not possible to
finct closec3form 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 changingwage model, and,
in aciclition, d~/3cY < 0the 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(lualleve] 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 beginningofperioc! wage level for
both population groups. Note that both
OCR for page 356
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 highcrime 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 lowcrime type
receiving a wage of .5 will spend only
about onethird 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 lowcrime types.
369
TABLE 3 Time Allocation to Criminal
Activity Given Wage Growth with
Varying Sentence Length
LowCrime
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
HighCrime
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
OCR for page 356
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 variablesentence
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
LowCrime 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
OCR for page 356
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 fulItime
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
decisiontheoretic 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,
OCR for page 356
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);
s0, n = 1, 2, . . . .
then we can write the joint distribution of
the first n spells as
OCR for page 356
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
w0 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 timevarying, 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
OCR for page 356
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 kdimensional 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 inclividtlalspecific, 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
OCR for page 356
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 onestate 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 crimespecific 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 individualspecific, spell
anc3 timeinvariant heterogeneity compo
nent; r)ij is a spellspecific, timeinvariant
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
onefactor specification of unobserved
heterogeneity
Vij = Cj ~i' J
i= 1,...,J,
OCR for page 356
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 ~iji
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 twocrime
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.
OCR for page 356
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 Ok1 ] ]
This term enters the likelihood function
for incomplete spells at least T in length.
Say we have access to eventhistory 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
OCR for page 356
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 personspecific
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*
OCR for page 356
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 closedform 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
1year 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.: JosseyBass.
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.