Theoretical Economics Letters, 2011, 1, 99-104
doi:10.4236/tel.[phone] Published Online November 2011 (http://www.SciRP.org/journal/tel)
Crime and Punishment with Habit Formation
Vladimir Kühl Teles1, Joaquim Andrade2
1Getulio Vargas Foundation, Sao Paulo School of Economics (FGV-EESP), Sao Paulo, Brazil
2University of Brasilia, Brasilia, Brazil
Received August 5, 2011; revised September 30, 2011; accepted October 8, 2011
Abstract
Moral concepts affect crime supply. This idea is modelled assuming that illegal activities is habit forming.
We introduce habits in an intertemporal general equilibrium framework to illegal activities and compare its
outcomes with a model without habit formation. The findings are that habit and crime presents a non linear
relationship that hinges upon the level of capital and habit formation. It is possible to show that while the
effect of habit on crime is negative for low levels o habit formation it becomes positive as habits goes up.
Secondly habit reduces the marginal effect of illegal activities return on crime. Finally, the effect of habit on
crime depends positively on the amount of capital. This could explain the relationship between size of cities
and illegal activity.
Keywords: Crime, Habit Formation, Punishment.
1. Introduction
“Those, on the contrary, who have had the misfortune to
be brought up amidst violence, licentiousness, falsehood,
and injustice; lose, though not all sense of the impropri-
ety of such conduct, yet all sense of its dreadful enormity,
or of the vengeance and punishment due to it. They have
been familiarized with it from their infancy, custom has
rendered it habitual to them, and they are very apt to re-
gard it as, what is called, the way of the world, some-
thing which either may, or must be practised, to hinder
us from being the dupes of our own integrity.” (Adam
Smith, 1759, The Theory of Moral Sentiments, Part V,
Chapter 2).
In the seminal paper of [1] habit was included in the
utility function to describe consumption behavior of
harmful goods, notably drugs. In the present paper we
merge the Becker and Murphy insight about this link
between illegal behavior and habit with the traditional
crime and punishment approach due to [2] in a general
equilibrium framework.
The existence of habit formation due to factors such as
social interactions may affect the behavior of agents in
crime supply. Factors such as culture or religion provide
social incentives that may induce habits in illegal active-
ties. For example, surveys in Britain and the United
States have indicated that at least a third of the citizens in
both countries believe that religion provides a sociocul-
tural and/or spiritual foundation for curtailing criminal
behavior [3,4]. [5] finds that more religious countries have
lower crime rates than less religious countries, at least
regarding property crimes, using data from 13 industrial
nations.
On the other hand [6] demonstrates, using a myriad
empirical evidence—both statistical and anecdotal—that
the daily contact of youth with criminal adults and cri-
minal peers results in the erosion of morals and hence in
a greater predisposition toward crime.
The insight that through the process of habit formation,
one’s own past decisions might influence the utility
yielded by current decisions is hardly new; see, for exa-
mple [7,8].
In fact the habit formation hypothesis has been applied
in many issues such as endogenous growth models [9],
cyclical consumption [10], aggregate savings [11], mo-
ney and growth [12], environment [13], fiscal policy [14]
and monetary policy [15,16], to mention a few. All these
papers introduce habit in consumption. Nonetheless, [12]
uses habits in number of hours worked to study labor
supply. In fact, it is not only in consumption that habits
may occur. [1], for example, explain that: “Not only ci-
garettes, alcoholic beverages, and cocaine are obviously
addictive, but many other goods and activities have ad-
dictive aspects”.

V. K. TELES ET AL.
100
In this paper we assume that social incentives create an
ethic that affects the number of hours allocated to criminal
activities by a representative agent. This is modelled by
assuming that crime is habit forming. The idea is quite
intuitive: past crime forms a stock of habits that affect
agents’ disposition towards present crime.
This paper is organized as follows. Section 2 develops
the model and characterizes equilibrium. Section 3 shows
the main result and section 4 provides concluding re-
marks.
2. The Model
While agents are stimulated to engage in criminal active-
ties according to the expected positive returns, they are
also subject to the effects of crime, with loss in income.
According to this idea, we may argue that the total ex-
pected income () of a representative agent of this
economy will be given by (1).
Y
π(1 π)[(, )(1(,))]Pfkoo   o (1)
where

0,
,0,
0,
if oo
ooifoo
if oo








Thus, (,)
f
ko represents the production function,
where is the capital stock and o the number of
hours spent on criminal activity. On the other hand,
k
(, )oo  represents the net income function of the crimi-
nal activity, where the agent chooses the number of
hours that will be dedicated to crime, when faced with
the average number of hours of the other agents, o. This
type of function is commonly used in illicit activity
models such as in [17] and [18]. Complementarily, is
the probability of punishment, and is the payoff of
the punishment. In fact, may represent the consump-
tion supplied for criminals by the society in prisons, for
example. (see [19]).
π
P
P
The production function may be represented by,
 
1
,1fkoAko



 (2)
where agents devotes the fraction of his non-
leisure time to current production,
(1 )o
A
is the level of tech-
nology, and

is the capital-share.
If we consider that criminal activity (o) directly af-
fects the well-being of an agent, and if we incorporate
this in his utility function, and that the individual cares
not only about consumption () and the instantaneous
flow of offenses (), but also takes into account his past
criminal activities, captured by his stock of habits (),
then the instantaneous isoelastic utility function proposed
by [20] is adapted to introduce :
c
oh
o


1
11
,,
ho c
Ucoh









1


where

is a positive parameter that lies in the unitary
interval,

is the coefficient of relative risk aversion,
and [0,1)


indexes the importance of habits. If
0


, then habit stock has no relevance, and the utility
function reduces to the traditional case. While if 1


,
crime relative to habit stock is very important.
Following [9] it is assumed that the stock of habits is a
weighted average of past offenses. The stock of habits
evolves according to:

o

hv h


 (4)
where is a positive parameter determining the rela-
tive weights of offenses at different times. The smaller is
, the less important is offenses in the recent past.
v
v
Equations (3) and (4) provides a picture of the rela-
tionship between crime and habits. The agents have a
disutility of practice crimes, but when he does it the
stock of habit increase, diminishing, in a second moment,
the disutility of crime. Then while the agent engage in
criminal activities he “accustoms” to them, what dimin-
ish his disutility in the practice of the crime. When the
value of

is high this effect occurs quickly, however,
if is equal to zero, this effect simply does not occur.
The individual maximizes a discounted, infinite stream
of utility:

, ,o
c
h

,,
0
max max()et
coco Uc hdt
kY P
hvo







(5)
By substituting (1) and (3) in (5) and solving the pro-
blem, and applying the equilibrium condition oo

, the
following first-order conditions are obtained,
1(1 )
khh
oo

cc







 
 

 

 

 
  (6)

 

 
00


11
1
1
h
k
hh
vc c
ooo
ff




 



 
  

   
 



 
 

 

(7)

k

1
k
k
f


 


(8)

101 V. K. TELES ET AL.
1
1
hh
h
hh
c
ooh
v



1
c


 




 
 
 
 
 







(9)
where k

and h

are the co-state variables of and
respectively.
k
h
By substituting (6) in (7), and then this result in (9) and
considering that in this model’s steady-state the per capita
variables remain constant, meaning that the shadow-price of
capital and habits remains constant, we will have that the
following equations establishes the steady-state condi-
tion.
 
1πk

f
  (10)
Υ

  (11)
π(1 π)Pfc P (12)
oh (13)
where,

  
1/
Υ1
1/11 oo
hc
ocf f







  





It is possible to analyze, through these equations, the
main effects of habit formation in the illegal activity and
assess the efficacy of the policies used to bend the crime.
3. Results
3.1. Crime and Habit
To examine how the habit formation affects the criminal
activities dynamics we should solve the system (10) to
(13) and calculate the first derivative of in relation to
o

, the importance of habit in the utility function. This
brings us to the following results:

d
1
d
oB
A
CD








(14)
where,
(1 π)( )[(1π)() ]
kok o
A
ffP fvvf

 
()BvfP
{[(1 π)](1π)}[ ]
(1 π)
o
kkoo
f
Cfvoff

 

[(1 π)][
(1 π)(1π)
ooo o
koo
ff
Dof vf]


 

The value of is clearly positive, and the value of
is negative. The signs of
B D
A
and however depend on
the values of the parameters of the model. For instance,
an increment of the importance of habit in the model
does not imply necessarily a positive or negative effect
on crime.
C
It is important to note that the key parameters and
variables to determine the sign of the relationship be-
tween the habit formation and the criminal activity are

and , and the last one determines the value of k
f
,
o, k and ko. Henceforth it is possible to use Equa-
tion (14) to understand how these parameters affect this
relationship.
f ff
As an example we will consider a value for such
that
k
()CD 0

. In this case the relationship between

and crime can be illustrated by Figure 1. It should be
noted that, initially, when

is equal to zero
A
takes
a negative value, and that
A
goes up when

goes up.
At the same time, Equation (14) indicates that /ddo

may be negative if
A
is big enough when
A
is nega-
tive. In this case, Figure 1 tells us that, when

over-
shoot the value 0

the value of d do

becomes posi-
tive, since when this value is reached
Figure 1. Crime and habit.

V. K. TELES ET AL.
102

1

A
CD







 becomes positive.
Henceforth, when

takes values between zero and
0

an increase of

reduces the equilibrium value of
. Since we are considering a representative agent
model the value of
o

is the same for all individuals.
Consequently, increases of

in this stage may be taken
as the cause of the negative impact on the worked hours
of society as a whole, reducing therefore the return of
crime. At the same time, in this stage, the habit does not
reduce significatively the disutility of crime, and, as a
result, the habit does not lead to an increase in criminal
activity. This explain the sign of ddo

in this first
stage.
On the other hand, when

pass over 0

, the func-
tion becomes convex up to the value of 1

. In this in-
terval ddo

is positive and increases in

raise the
value of ddo

. The disutility of crime falls strongly
when crime is practiced because of habit and any varia-
tion in

leads to increases in the number of hours
spent in crime activity more than proportional.
Finally, when

pass over the point 1

relationship
becomes concave. Now
A
becomes positive, and
ddo

positive as well. Nonetheless increases in

lead to smaller ddo

. This happens because, when
increases the number of hours spent on work fall, in-
creasing the opportunity cost of crime. Thus, while the
disutility of crime falls quickly, the value of tends to
a limit value. This is due to the existence of a trade off
between time spent in crime activity and work.
o
o
Though this example was designed for specific values
of it is a general case about the possibilities of the
relationship between the importance of habit and the
crime dynamics. Changes in the value of that implied
changes in the sign or magnitude of would
only eliminate some of the three stages discussed above.
k
k
C

D

3.2. Crime, Habit and Capital Stock
The equilibrium value of o does not depend on

only, but fundamentally on the value of , as can in-
ferred from the previous section. Therefore it is neces-
sary to analyze how this relationship is affected by . It
follows from Equation (14) that an increase in affects
the marginal effect of the importance of habit on the
number of hours spent on criminal activity. This effect is
illustrated in Figure 2.
k
k
k
The economic effects of the capital stock on crime can
be observed from the analysis of the behavior of crime
when

is equal to zero. Nonetheless let us focus on
the differences in the relationship of habit-crime of an
increase in . Figure 2 displays this net effect in such a
way that of the economy with high and of the
with low are the same when
k
o
kk

is zero.
Two effects are presented. Firstly, the point where
ddo

becomes positive is smaller in the economy with
high than in the economy with low k, since the
shift of the constant in the diagram on the top of Figure
2 is larger for the economy with hgh k than for the
with w k. To see this it is sufficient to note that larger
values ofk change the values of and in Equa-
tion (14), raising their absolute values.
k
lo
i
B C
Secondly, the slope in the top of Figure 2 of the rela-
tionship between
A
and

changes in a way that in-
creases in

have a marginal impact successively strong-
Figure 2. Crime and habit in big cities.

103 V. K. TELES ET AL.
er on in the economy with high than in the other.
Henceforth, the difference between the behavior in rela-
tion to crime in the economy with high vis a vis the
economy with low increases as
ok
k
k

increases.
These two points present a practical result well estab-
lished: two persons with the same moral behavior, that is,
with the same

, can have different conducts in relation
to crime depending on the value of . Simplifying, the-
same person that is a peaceful worker in a small city be-
comes a violent an criminal one in a big city, since the
habit dynamics would be quite different because of dif-
ferences in , considering that on average an increase in
the size of the cities is followed by an increase in the
stock of capital per capita.
k
k
The increase in the criminal rate that accompanies the
increase in the size of the cities is a well established fact
in the empirical literature. A very good reference on the
subject is [21] that brings a comprehensive discussion on
these empirical facts. This study points out the three
main reasons: for the increase in the criminality rate in
the big cities. Two of the reasons are based in economics:
a lower probability of punishment and a greater expected
value of crime in the big cities. The third reason is essen-
tially moral. Based in empirical evidences raised by
studies made by criminologists, the argument is that in
big cities a moral degradation reduces the disutility of
crime and consequently leads to an increase in the
criminal activity of big cities vis a vis small cities.
Therefore, the crime rate will be determined by the in-
teraction of moral and social values that determine

and by economic incentives represented by different
values of . Such result is confirmed by several em-
pirical and anedoctical evidences. Empirical results con-
firm that the criminal behavior has a strong relationship
with social interactions (e.g. [22-24]). This idea explains
not only the relation of habit formation to criminal activ-
ity but also its relation to criminal behavior. It is note-
worthy that when an individual enters a criminal activity
by external influence, as in a gang, for instance, his dif-
ficulty to exit is due to the external coercion of the peers
group. This interaction raises the importance of habit in
criminal activity.
k
3.3. Crime and Punishment
An important question to be answered is: How the im-
portance of habit in criminal behavior affects the efficacy
of punishment? To answer this question we should go to
Equation (14) and analyze the sign of 2
dddoP

. In this
case its is clear that dd 0AP and that dd 0BP

,
what implies that 2
ddd 0oP 

. Thus, when

in-
creases, the marginal effect of punishment on crime is
reduced. In other words, more important is habit to hu-
man behavior towards crime, less efficacious is punish-
ment to reduce criminality. As a result of that more pun-
ishment will be necessary to contain criminality.
3.4. Crime and Law Enforcement
Finally, it is possible to verify the effects of public poli-
cies, as for instance the increase in investment to combat
the crime in the model. In this case, going again to equa-
tion (14) it is possible to show that ddπ0A and
ddπ0D

. Henceforth, if ddπ0C or small enough,
2
ddπdo  0

. This result implies that the investment in
the combat of criminal activity may reduce its effective-
ness greater is the stock of habit towards criminal activ-
ity similarly to the effect of punishment. Indeed, this
result does not depend on the size ofdd0Cpi.
4. Conclusions
The tradition of the models addressing the economics of
crime defines the decision of an agent participating in an
illicit activity as a rational one, since it is an economic
decision in which the benefits and costs of crime are
weighted along with the alternatives [19]. From this per-
spective, this study has introduced habit in an in-
tertemporal general equilibrium framework and demon-
strates that habit affects the rationality of crime. It was
shown that the importance of habits towards criminal
activity depends on the opportunities of criminal active-
ties. These on the other hand are affected by the amount
of capital of the economy. The relationship of crime to
the importance of habit is non linear. For small values of
the importance of habit is more than compensated by
the disutility of crime, after a critical the opposite
occurs and the habit affects positively criminal activity.
On the other hand, the effect of punishment and law en-
forcement become less efficacious greater the impor-
tance of habit.
kk
Comparing these results with the traditional result due
to [1], that built a model of drugs consumption with habit
formation some considerations may be due. In Becker
and Murphy framework, an increase in permanent pun-
ishment implies an increase on price of drugs, and, con-
sequently, a decrease on its long run demand. Thus, the
drugs traffic will fall, as crime related. In our model,
where another type of crime (property crime) is consid-
ered, this relationship will not occur. If the punishment
rises, the effects on crime level may not change signify-
cantly, as demonstrated in proposition 3.
Considering education programs, Becker and Murphy
argues that greater efforts to educate the population on
the harms of the use of drugs may not offset the effects
of the reduction on dugs price on the long run. Contrar-

V. K. TELES ET AL.
104
ily, according to our model education policies may be
important to break increases on long run crime level if it
is able to build an ethics pattern to avoid illegal activities.
Summarily what this paper shows is that different kinds
of crime may have opposite forms of combat, and the
theoretical dynamic path of crime will change drastically
if we consider habit formation in alternative ways.
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