Crime and Punishment with Habit Formation

doi:10.4236/tel.2011.13021

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Theoretical Economics Letters, 2011, 1, 99-104

doi:10.4236/tel.2011.13021 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

E-mail: [email protected]; jand [email protected]

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).

π(1 π)[(, )(1(,))]Pfkoo



 o

(1)

where



,0,

if oo

ooifoo

if oo

















Thus, (,)

ko represents the production function,

where is the capital stock and o the number of

hours spent on criminal activity. On the other hand,

(, )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]).

The production function may be represented by,

 

,1fkoAko





 (2)

where agents devotes the fraction of his non-

leisure time to current production,

(1 )o

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 :



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





, 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:





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.

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



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















 

 



 



 



 

  (6)



 









vc c

ooo







 







 

 





 

 

 







 

 



 



(7)









 





(8)

101

V. K. TELES ET AL.

ooh









 







 





 





 













(9)

where k



and h



are the co-state variables of and

respectively.

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





 (10)





 (11)

π(1 π)Pfc P (12)

oh (13)

where,



  

Υ1

1/11 oo

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



, the importance of habit in the utility function. This

brings us to the following results:



















(14)

where,

(1 π)( )[(1π)() ]

kok o

ffP fvvf



 

()BvfP

{[(1 π)](1π)}[ ]

(1 π)

kkoo

Cfvoff



 



[(1 π)][

(1 π)(1π)

ooo o

koo

Dof vf]



 



The value of is clearly positive, and the value of

is negative. The signs of

B D

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.

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

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

()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

takes

a negative value, and that

goes up when



goes up.

At the same time, Equation (14) indicates that /ddo



may be negative if

is big enough when

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



















becomes positive.

Henceforth, when



takes values between zero and



an increase of



reduces the equilibrium value of

. Since we are considering a representative agent

model the value of



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

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.

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.



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.

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



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.

B C

Secondly, the slope in the top of Figure 2 of the rela-

tionship between

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



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.

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.

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,

ddπdo





. 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.

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.

5. References

[1] G. S. Becker and K. M. Murphy, “A Theory of Rational

Addiction,” The Journal of Political Economy, Vol. 96,

No. 4, 1988, pp. 675-700. doi:10.1086/261558

[2] G. Becker, “Crime and Punishment: An Economic Ap-

proach,” The Journal of Political Economy, Vol. 76, No.

2, 1968, pp. 169-217. doi:10.1086/259394

[3] C. Banks, E. Maloney and H. Wittrock, “Public Attitudes

toward Crime and the Penal System,” British Journal of

Criminology, Vol. 15, No. 3, 1975, pp. 228-240.

[4] G. Jensen, “Sociology of Delinquency: Current Issues,”

Sage, London, 1981.

[5] L. Ellis and J. Peterson, “Crime and Religion: An Inter-

national Comparison among Thirteen Industrial Nations,”

Personality and Individual Differences, Vol. 20, No. 6,

1996, pp. 761-768. doi:10.1016/0191-8869(96)00023-2

[6] A. Gaviria, “Increasing Returns and the Evolution of

Violent Crime: The Case of Colombia,” Journal of De-

velopment Economics, Vol. 61, No. 1, 2000, pp. 1-25.

doi:10.1016/S0304-3878(99)00059-0

[7] W. Pareto, “Cours D´Economie Politique,” Droz, Genève,

1964.

[8] A. Marshall, “Principles of Economics,” 8th Edition, Ma-

cmillan, New York, 1898.

[9] C. Carroll, J. Overland and D. Weil, “Saving and Growth

with Habit Formation,” American Economic Review, Vol.

93, No. 3, 2000, pp. 341-355. doi:10.1257/aer.90.3.341

[10] E. J. Dockner and G. Feichtinger, “Cyclical Consumption

Patterns and Rational Addiction,” American Economic

Review, Vol. 83, No. 1, 1993, pp. 256-263.

[11] R. Alessie and A. Lusardi, “Consumption, Saving and

Habit Formation,” Economics Letters, Vol. 55, No. 1,

1997, pp. 103-108. doi:10.1016/S0165-1765(97)00061-X

[12] J. R. Faria and M. León-Ledesma, “Habit Formation,

Working Habits and Technological Progress,” Manches-

ter School, Vol. 72, No. 3, 2004, pp. 403-413.

doi:10.1111/j.1467-9957.2004.00399.x

[13] T. Ono, “Is Habitual Consumption Harmful to the Envi-

ronment?” Economics Bulletin, Vol. 17, No. 1, 2002, pp.

1-10.

[14] C. Burnside, M. Eichenbaum and J. Fisher, “Fiscal Shocks

and Their Consequences,” Journal of Economic Theory,

Vol. 115, No. 1, 2004, pp. 89-117.

doi:10.1016/S0022-0531(03)00252-7

[15] B. McCallum and E. Nelson, “Nominal Income Targeting

in an Open-Economy Optimizing Model,” Journal of Mo-

netary Economics, Vol. 43, No. 3, 1999, pp. 553-578.

doi:10.1016/S0304-3932(99)00005-7

[16] J. Amato and T. Laubach, “Implications of Habit Forma-

tion for Optimal Monetary Policy,” Journal of Monetary

Economics, Vol. 51, No. 2, 2004, pp. 305-325.

doi:10.1016/j.jmoneco.2003.05.002

[17] I. Ehrlich and F. Lui, “Bureaucratic Corruption and En-

dogenous Economic Growth,” The Journal of Political

Economy, Vol. 107, No. S6, 1999, pp. S270-S293.

doi:10.1086/250111

[18] V. Teles, “The Effects of Macroeconomic Policies on

Crime,” Economics Bulletin, Vol. 1, No. 1, 2004, pp. 1-9.

[19] J. Fender, “A General Equilibrium Model of Crime and Pu-

nishment,” Journal of Economic Behavior and Organiza-

tion, Vol. 39, No. 4, 1999, pp. 437-453.

doi:10.1016/S0167-2681(99)00050-5

[20] A. B. Abel, “Asset Prices under Habit Formation and Ca-

tching up with the Joneses,” The American Economic Re-

view, Vol. 80, No. 2, 1990, pp. 38-42.

[21] E. Glaeser and B. Sacerdote, “Why is There More Crime

in the Cities?” The Journal of Political Economy, Vol.

107, No. S6, 1999, pp. S225-S258. doi:10.1086/250109

[22] G. Akerlof and J. Yellen, “Gang Behavior, Law Enfor-

cement and Community Values,” In: H. Aaron, T. Mann

and T. Taylor, Eds., Values and Public Policy, Brookings

Institution Press, Washington D.C., 1994.

[23] E. Glaeser, B. Sacerdote and J. Scheinkman, “Crime and

Social Interactions,” The Quarterly Journal of Economics,

Vol. 111, No. 2, 1996, pp. 507-548. doi:10.2307/2946686

[24] F. Carneiro, P. Loureiro and A. Sachsida, “Crime and So-

cial Interactions: A Developing Country Case Study,”

The Journal of Socio-Economics, Vol. 34, No. 3, 2005,

pp. 311-318. doi:10.1016/j.socec.2004.09.015