Universidad de los Andes
Facultad de Economía
Model of Criminal Entropy
Daniel Mejía Londoño
(Asesor)
Eduardo García Echeverri
(Código: 200821874)
1
Model of Criminal Entropy*
Abstract
This paper proposes a theoretical model, inspired heuristically by the definition of entropy, which aims to explain the diverging evolution of criminal trends in different societies. In order to do this, the model relaxes two of the most common assumptions in the literature: an indictment probability independent from the social amount of crime and an omniscient agent who knows this probability with absolute precision. The model consists of a society composed by agents that estimate the indictment probability from the observation of their environment, and a government that is only capable of prosecuting a fixed proportion of the total population, not of the delinquents. Under this approach, the model performs notably well in predicting criminal stylized facts such as the large interurban variance in criminal rates, the hysteretic nature of criminal trends, recidivism’s determinants, and the effects of reserve utilities and the state’s prosecuting capacity upon aggregate crime levels, among others.
Keywords: Criminal dynamics, hysteresis, social interaction, entropy, recidivism.
JEL Classification Numbers: K42, D83, D84.
* I would to thank Çağatay Kayi, Oskar Nupia, Paula Jaramillo and Daniel Mejía for their helpful comments and suggestions. I am also grateful with the participants and organizers of the “Semillero de economía teórica” at the Faculty of Economics at Universidad de los Andes.
2
1
Introduction
“Entropy: […]Irreversible tendency of a system, including the universe, towards increasing disorder”
Conceptually, entropy has been widely used in various disciplines to study the transformation and evolution of diverse types of systems. From thermodynamics (where it originated) to information theory, passing through computational science, cryptography, statistics, cosmology and even linguistics, the concept has been adapted to explain the increased randomness and disorder that some very different sets of phenomena exhibit. This success proves the idea’s versatility, raising the question whether social systems could be analyzed under a
similar perspective. Some societies may suffer from a progressive and irreversible decay into chaos, triggered only by a temporal disruption that, nonetheless, inflicts permanent consequences. This article presents an economic model of criminality and three subsequent extensions inspired heuristically by the definition of entropy. In the model, we will conceive society as mass of individuals that decide to commit crime depending on their previously observed indictment rates, and a government with a limited capacity to prosecute them.
This approach proves to be successful because the resulting model is very capable in explaining an extensive range of stylized facts concerning crime. First, the entropy model below predicts one of criminal trends’ most puzzling features: its hysteretic nature. Two identical societies differing only in their initial crime rate may evolve to completely divergent stationary states in the long run. This proves that not only structural parameters matter but history also does. Different beliefs in the state’s capacity to prosecute law violators may be the only cause of the
initial difference in crime rates, and thus, over skeptical societies with their governments may end up in a self-fulfilled prophecy of high crime rates and low indictment probabilities. The model is also able to predict the better-understood features of criminal trends such as the negative effects on delinquency levels that an enhancement of the prosecuting capacity, an increase in fines, or a raise in the agent’s reserve utility all have.
Predictions of the model are not restricted to the societal level only, but also shed light on individual criminal behavior, specifically on recidivism. The model shows how societal variables which are seldom included in econometric models on the subject, such as the crime rate itself or
3 the state’s prosecuting capacity, affect an individual’s probability of relapsing in criminal activities. For further detail on these econometric models refer to Carvalho and Bierens (2002), Rasmussen (1999) and Worthington, Higgs and Edwards (2000). These omitted variables most certainly imply a bias in the estimated coefficients of idiosyncratic characteristics and could alter conclusions in these empirical studies. Another important recidivism’s determinant, according to the entropy model, is the level of social interaction of the studied crime. This former result is empirically supported by the recidivism model in Sirakaya (2006).
The baseline two-period entropy model consists of a world inhabited by an infinite population of measure 1 and composed entirely by rational agents that drive their cars to work each period1. The model controverts the assumption of an indictment probability independent from the amount of crime in the society by introducing a maximum state prosecuting capacity. This capacity is represented as a fixed proportion of the total population, not only of the delinquents, that authorities are capable of capturing and punishing in case they violate the law. The independent indictment probability paradigm is one of the most popular assumptions in economic literature regarding criminality since Becker’s pioneering work in 1968. With the axiom of a limited state’s prosecuting capacity introduced in the entropy model, the system may
experience, after an exogenous temporal disturbance that raises delinquency in one period, a sequential deterioration of its security standards. This happens because the shock induces a positive externality among all potential and active criminals by augmenting the degree of impunity in the system. As a consequence, crime will increase steadily as time passes until, eventually, it will converge to a much worse stationary state. This hypothetical scenario is coherent with the general definition of entropy as the tendency of a system to increased disorder.
A situation as the one described above has occurred many times before, especially in developing countries. Using data from the Colombian national police and Medicina Legal, Gaviria (1999), shows how, after the entrance of the drug cartels in the late 1970s, the ratio of indictments over homicides dropped persistently as congestion in the Colombian law enforcement apparatus increased. Consequently, other illicit activities not related with narco-trafficking, such as car thefts, bank robberies and kidnappings began to rise. His set of data suggests a fixed prosecuting capacity by the authorities. In 1980, roughly 46% of homicides were
1
Traffic infractions provide an excellent illustrative example but do not imply that the model is only restricted to this type of offenses.
4 indicted in Colombia whereas in 1992 the rate descended to about 15% (approximately one third of its original value). During this same period, the murder rate rose from 23 per 100000 inhabitants to 73 (roughly three times its original value). This implies that there were approximately 10.58 homicide indictments per 100000 inhabitants in 1980 and approximately 10.95 in 1992, an insignificant increase. The state’s capacity to prosecute murderers, as a proportion of the population, remained nearly constant for the period of study. The current situation in Mexico is another good example of an unresponsive prosecuting capacity when dealing with an unforeseen and sudden rise in homicides. According to the United Nations Office on Drugs and Crime (UNODC), the homicide rate nearly tripled from 2007 to 2012, rising from 7.8 to 21.5 murders per 100000 inhabitants. During this same period, according to the Instituto Nacional de Estadísticas y Geografía (INEGI), homicide indictments by the Mexican authorities barely increased from 7911 cases in 2007 to 8333 in 2012. This implies that the indictment rate, as a proportion of the total population, actually diminished from approximately 7.27 to 7.24 indictments per 100000 inhabitants during the period of study.
We would like to pose a caveat on the previous analysis since it is utterly impossible to control these statistics for corruption in the judicial system. Judicial corruption increases with the consolidation of the drug cartels and consequently affects convictions, as judges are bribed or intimidated more often. Nevertheless, the analysis is conclusive in proving that the indictment probability is not independent from the crime rate in developing countries. Thus, we conclude that a state that can prosecute a fixed proportion of the population is a more realistic assumption than an independent indictment probability.
There have been pioneer theoretical works that explore positive externalities among criminals. Sah (1991) develops a model in which he assumes a general functional form for the probability of being caught that depends negatively on the delinquency rate. As a result, he is able to predict crime series’ hysteretic behavior, as the entropy model presented here does. However, due to the generality of the functional form, his model is able to accurately predict the effects caused by changes in other relevant variables upon crime rates only with the aid of auxiliary axioms. An example of these axioms is the assumption that the indictment rates are relatively “small” for most crimes. This may not be the case for many types of crimes such as murder and forcible rape, in which, according to the FBI, clearance rates are as high as 62.5%
5 and 40.1% respectively, and denounce rates are close to 100%. A main contribution of this paper is the specification of the indictment probability’s functional form (derived directly from the
fixed prosecuting capacity introduced before) as well as empirical support for this precise formulation. This specification of the functional form allows for the construction of the extensions and the straightforward proofs of the theorems without auxiliary axioms. Additionally, this paper introduces the possibility that agents divide resources continuously between criminal and honest activities; depicting the fact that intensity varies greatly among criminals. To the extent of our knowledge, even though the literature of positive externalities among criminals is abundant, Sah (1991) has been the only theoretical work to explore this particular positive externality.
The second assumption the baseline two-period entropy model relaxes is the omniscience attributed (most times implicitly) to the economic agent in his or her maximization programs. In most theoretical models, the probability of indictment is known by the potential criminal with complete accuracy. Our entropy model proposes a mechanism through which individuals
estimate, based on what they see in their immediate environment, the probability of being captured while performing illicit activities (specifically ignoring a stoplight). The process is simple: each period the agent, from a sample of constant size , takes the number of captures and the number of criminals and calculates the observed indictment ratio. The size of this sample is called the agent’s exposure. With this ratio, the individual estimates the indictment probability
and thus decides to become a criminal or not. As a result, all individuals end up with different indictment beliefs; something more compatible with mundane observation than a unique and perfectly known indictment belief common to all people.
Again, both theoretically and empirically, some works have introduced the notion of sample estimation, or the social network’s influence, on individual decision making. Glaesser et al. (1996), calculated an index of social interaction for several types of crimes in different U.S. cities. They find that this indicator is negatively associated with the offense’s severity and, more importantly, that the notion of social interaction captured by it helps to explain the high degree of variance in criminal rates between urban centers. The entropy model below, and particularly its extension to infinite horizon, predicts that the variance of crime rates between groups of cities may grow larger the more populated they are (using population as a proxy for the agents’
6 exposure). From a purely theoretical perspective, Sah (1991) proposes a general functional form for the individual estimation of the indictment probability that increases with the number of observed captures and decreases with the number of observed criminals. He also assumes a constant sample size from which the agent obtains this data.
The theoretical contributions introduced by the entropy model are various. First, a specific functional form that guarantees the stability of the system (delinquency participation rates in the interval [ ]). Second, the study of the effect of changes in agent exposure upon the stationary crime rates. Third, the study of criminal recidivism’s determinants. Fourth, in order to expand the entropy model’s predictive power under more complex and realistic scenarios, two extensions are
developed. One that studies the effects of extremely well-informed agents and another that allows agents to continuously allocate time to criminal activities. The first extension aims to depict the effect of highly socially interactive agents on the evolution of crime in a society. Examples of this type of agents could include taxi and bus drivers in cities (since they drive on average four times more than the regular driver) or drug lords in cartels. The model predicts that in high impunity rate environments, these agents will induce regular agents to more crime by reducing the indictment probabilities and by dwindling the agents’ observed indictment rates. The second extension’s objective is to depict a more realistic approach of criminal decision making that
allows the exploration of an intensive dimension of delinquency rather than a purely extensive
one. It is well-known that there is a significant difference between criminals in their time and effort allocations to crime.
This paper is divided into seven main sections including this introduction. Section 2 presents the construction of the model, from the basic set of assumptions, but restricted to only two periods. Sections 3, 4, and 5 extend the model to infinite horizon, agents’ exposition heterogeneity and continuous time allocation to illicit activities, respectively. Section 6 includes concluding remarks and open questions for future research. Finally, Section 7 is a mathematical appendix that contains the proofs of all propositions and corollaries.
7
2
Baseline Two-Period Model
This model consists of a two-period world inhabited by an infinite population of measure 1. This population is composed by rational agents that drive their cars to work each period and remains constant through time. The objective of this two-period model is to explain the crime rate in the second period , as a function in terms of the initial crime rate and other relevant parameters.
During their way, agents must decide whether they respect the traffic light or, on the contrary, pass it while in red. Each agent has a utility loss for waiting for the light to change of
, a utility gain of for passing without being caught, and a fine of otherwise.Suppose this utility valuations satisfy , , and .2 An agent decides to trespass the light according to a simple expected utility maximization process involving his/her subjective indictment belief ( for agent ). Let be a binary variable that equals if the traffic light is not respected by agent or otherwise. Then, the expected utility of is:
( ) {( ) ( )
Define ̃ as the indictment belief that leaves the agent indifferent between waiting and ignoring the stoplight (referred to hereafter as “threshold belief”). This “threshold belief” must
equate the reserve utility with the expected utility of performing crime:
( ̃) ̃( ) ̃
Only drivers with an indictment belief strictly above this threshold will stop and wait.3 Since utility valuations are the same for all agents, this threshold belief is identical among them.
Additionally, notice that ̃ ( ). In the first period, indictment beliefs are distributed following an exogenous probability density function ( ). Thus, the proportion of the population that initially passes the red light ( ) is given by:
2 For the particular example of traffic lights illustrated above the natural assumption is , but in a more general context can be interpreted as a reserve utility that could be positive. Consequently, there is no explicit restriction in the model upon its sign.
3
Agents which have exactly the threshold belief, by assumption, do not wait for the green light. The adoption of this or the opposite is innocuous since results and propositions remain the same.
8
∫ ( )
̃
∫ ( )
( )
In the second period, suppose agents form their respective indictment beliefs from a simple empirical calculation: number of observed captures C over number of observed trespassers T in the previous period. Notice both are random variables that may take different values for each driver. If this fraction is not well defined, , then assume the agent remains with its prior belief. All agents have the same exposure ; this means that they observe exactly p cars (including themselves) during their travel in the first period. From this sample, the agent makes statistical inference about the infinite population that surrounds him. The best analogy to understand this scenario is to think of this society as composed by econometricians which have a consistent and unbiased estimator of the indictment probability. Thus, for an agent with this exposure, the probability that , i.e. ( ), is given by the following binomial distribution:
( ) {( ) ( )
( )
For the distribution of C, we make the final assumption that authorities have a fixed capacity of prosecuting trespassers, expressed as a proportion of the total population, that satisfies ( ]. 4 This parameter can be interpreted as a function of institutional factors such as number of policemen, number of judges, technology and efficiency in the judicial system that are supposed static in the short run.Thus, the probability of fining a trespasser, , is given by the amount of captures over the amount of criminals. Since population is normalized to 1 this equals to:
( )
Therefore the probability of observing captures from trespassers ( | ) is given by:
( | ) {( ) ( )
( )
4
Arbitrarily, we will suppose that no matter how undeveloped the prosecuting institutions are they can fine at least one positive fraction of the total population. Additionally, we assume no innocents are captured.
9 For simplicity, we will consider the case in which first. Thus, in this case, the above is equivalent to:
( | ) {( ) ( ) ( )
( )
The bivariate probability mass function of and , ( ), must be calculated in order to determine . This is the function that assigns the probability of observing captures and trespassers for an agent with an exposure of . Notice that since ( | ) is a conditional probability, then ( ) must be equal to: 5
( ) ( ) ( | )
Or, equivalently:
( ) {( ) ( ) ( ) ( ) ( )
Simplifying the above expression we obtain:
( ) {( ) ( ) ( ) ( )
Then, it only remains to add the probabilities of the particular points of C and T for which the driver will have a belief less than or equal to the threshold belief ̃ calculated before. These points are illustrated by the area in red in Figure 1. Notice that a driver for which
, will always wait for the light to change. Since the exposition includes oneself then the only logical conclusion is that this driver respected the stoplight in period 1. This implies that his exogenous prior belief had to be large enough to dissuade him of trespassing. As it was assumed previously, this agent remains with this belief that inevitably dissuades him again of ignoring the traffic light.
Finally, with the aid of Figure 1 and the bivariate probability mass function we can conclude that:
∑ ∑ ( ) ( ) ( ) ( ) ⌊ ̃⌋
( )
5 Remember the definition of a conditional probability: ( | ) ( )
10 Figure 1:
Figure 1 shows all possible realizations of the random variables C and T. For points below the dotted line (in red) the agent has an indictment belief below the threshold and thus decides to become a criminal.
Equation 5 holds under the assumption that . If , then all criminals in the first period get captured (due to the definition of ) and thus all agents observe an indictment probability of 1 or remain with their prior belief. As a consequence, crime in the second period will reduce to 0. Thus, more generally:
( ̃)
{
∑ ∑ ( ) ( ) ( ) ( )
⌊ ̃⌋
( )
Results:6
This subsection presents the comparative statics of the model with respect to the exogenous parameters. After each proposition comments are made concerning the relevant empirical literature and the model’s prediction’s relation with it. Additionally, policy measures to
combat delinquency and their effectiveness are discussed.
6
Most of the proofs in this article are found in the mathematical appendix in Section 7. Some, due to its brevity, are presented directly after the proposition.
11 Proposition 1.1: An enhancement in the prosecuting capacity of the state does not increase the
crime rate in the second period, i.e., .7
Proposition 1.2: An increase in the crime rate in the first period, caused by an exogenous shift in the initial distribution of beliefs such that ( ) is stochastically dominated by ( ), does not
decrease the crime rate in the second period, i.e., .
These two propositions depict very basic stylized facts of criminal trends that have been widely confirmed by empirical Vector Autoregressive models. First, the crime rate responds negatively to improvements in the prosecuting capacity measured as the arrest rate over all adult population (considering only apprehensions for the specific studied crime). Rosenfeld and Fornango (2007), Saridakis (2003), Koskela and Viren (1997) and Levitt (2004) all ratify this same conclusion. Second, the crime rate today positively affects the crime rate tomorrow. According to the model presented in this paper, two effects occur. First, the increase in criminality exerts a positive externality among potential and active criminals by reducing the indictment probability. Second, this increment tends to reduce indictment beliefs and thus promotes delinquency. Jantzen (2011) proves this temporal persistency for non-negligent man slaughter and assault, showing that statistical significance and positive sign only hold for the immediate previous period (year), as the model presented here predicts. Joyce and Lovitch (1987) reaffirm this result for property-related felony crimes in New York City from 1970 to 1984, though not only for the immediate previous year. More remarkably, Proposition 1.2 shows the importance of people’s beliefs in tangible outcomes such as crime rates. An over skeptical society with its authorities may end up in a self fulfilled prophecy of high crime rates and low indictment probabilities, as these low beliefs raise delinquency in the first period.
Proposition 1.3: A rise in the threshold belief does not reduce the crime rate in any period, i.e.,
̃ ,
̃ . Intuitively:
i) It is clear that an increase in ̃ will add to the area under ( ) that equals by shifting the upper limit of the integral further to the right.
7 Notice that if , very probably the most common scenario, inequalities in propositions 1.1 and 1.2 become strict. Refer to the Mathematical Appendix in Section 7 for further details.
12 ii) There are two effects of an increase of ̃ upon . First, it increases and, as it was
seen in Proposition 1.2, this increases . Second, the increment in ̃ will pivot counterclockwise the dotted line in Figure 1 and this may aggregate new possible points of C and T to . This is shown in Figure 2.
Corollary 1.3.1: Rises in fines , rises in the reserve utility , or falls in the utility gain of succeeding in crime E, do not increase the crime rate in any period.
Figure 2:
Figure 2 shows how a raise in the threshold belief pivots counterclockwise the threshold line and causes more realizations of C and T to become criminals.
Proposition 1.4: As the agents’ exposition grows to infinity, crime in the second period increases
if ̃ and decreases if ̃ . More precisely the function ( ̃) converges
pointwisely as to the function:
( ̃) {
̃
̃
Figure 3 gives an illustration of how ( ̃) converges pointwisely to the
13 Figure 3:
Figure 3 shows how as exposure tends to infinity the future crime rate tends to 1 if the initial crime rate is sufficiently high. The opposite occurs if the initial crime rate is sufficiently low.
This proposition shows that, in societies with relatively high participation rates in criminality, the more informed the agents are deteriorates even further delinquency levels. Contrary to what it is sometimes assumed, more information is not always better. This occurs because their estimator is statistically consistent and, consequently, when their sample tends to infinity, it becomes more and more precise. An agent with such an accurate estimation, in a scenario of low probabilities of indictment, will most likely choose to be a criminal. In a situation with high indictment rates, the opposite will happen.
Criminal recidivism:
In this subsection, we study how the variables introduced so far impact the probability of an individual relapsing in criminal activities. In most economic models regarding criminality, recidivism is seldom studied, and if it is, it is mainly explained by idiosyncratic variables under a purely statistical approach. The entropy model presented here predicts that general societal variables, including the crime rate itself, also have an effect upon the recidivism rate and should be included in the econometric estimations on the subject.
Notice that, in the model, the recidivism rate is the probability that an individual participates in crime conditional to the fact that he was already participating in the previous
14 period. A precision should be made: this conditional probability is the real recidivism rate, the proportion of all criminals that recurs in this behavior, and should be differentiated from the
observed rate which is calculated only among the captured criminals.
If the offender was apprehended in the previous period and he or she is to remain in criminality, the number of observed captures ( ) and the number of observed trespassers ( ),
from a sample of , must satisfy ̃. On the other hand, if he was not apprehended and
he is to continue in criminal activities, and must satisfy ̃ (again from a sample of
)8. Thus, adding these two conditional probabilities pondered by the probability of the two
possible scenarios, we obtain the real recidivism rate (abbreviated hereafter):
( ̃) ∑ ∑ ( )
⌊( ) ̃ ⌋
( ) ∑ ∑ ( )
⌊( ) ̃⌋
( )
Proposition 1.5: An increase in the initial crime rate does not reduce the recidivism rate, i.e.
.
9
Proposition 1.6: An enhancement in the prosecuting capacity of the state does not increase the
recidivism rate, i.e. .
Proposition 1.7: As the agents’ exposition tends to infinity, the recidivism rate raises if
̃
and falls if ̃ . Formally, ( ̃) {
̃ ̃
These theoretical findings are coherent with the results obtained by the empirical recidivism model of Sirakaya (2006). In that model, the author proposes an individual hazard function of relapsing in criminal activities that depends both on idiosyncratic and neighborhood characteristics. Social interaction is measured as the percentage of recidivists in the neighborhood and the mean time to be rearrested among them. Under this perspective, the author finds that social interactions are the most significant factor determining recidivism among all ethnic and
8 Notice that, since is positive integer,
̃ ⌊( )̃ ⌋and ̃ ⌊( ) ̃⌋.
9Notice that if , very probably the most common scenario, inequalities in propositions 1.5 and 1.6 become
15 racial groups in the studied sample. Additionally, recidivism varies considerably between different types of crimes; not fortuitously, the most commonly committed crimes according to the FBI, are those where the individual, ceteris paribus, will most probably recidivate. This result is coherent with Proposition 1.5. Unfortunately, Sirakaya (2006) doesn’t include in her estimations the state’s prosecuting capacity or the agent exposure specifically, to validate or refute
Propositions 1.6 and 1.7.
3
Infinite Horizon
This extension introduces infinite periods to allow for the study of criminal time series’ evolution given certain initial conditions. It is also used in order to calculate stationary states and study the appearance of hysteresis. This last is defined as the dependence of a system or variable on its past history or own lagged values, and implies that two identical systems differing only on initial conditions may evolve to diverging outcomes. We suppose that in each of these periods the agent updates his or her indictment belief according to the number of trespassers and captures observed only in the immediately previous period. The calculation of this belief is the same as the one in the two period model. Additionally, suppose in this extension that the crime rate in the first period is completely exogenous in order to simplify the comparative statics. Thus, crime rate tomorrow relates with the crime rate today by the formula:
( ̃)
{
∑ ∑ ( ) ( ) ( ) ( )
⌊ ̃⌋
( )
A stationary state occurs when the function ( ̃) crosses the line .
Figure 4 shows a hypothetical criminal entropic system with two stable stationary states (b and zero) and an unstable stationary state between them (a).
From this hypothetical scenario, it is clear that the initial crime rate may lead two societies which are identical in all other aspects (threshold belief, prosecuting capacity and agents’ exposition) to very different crime rates in the long run. This hysteretic behavior may be
summarized by the function ( ̃) which takes as input the initial crime rate and
16 is a non-decreasing function of . Figure 5 depicts this function for the hypothetical scenario illustrated in Figure 4.
Figure 4:
Figure 4 shows the transitions of three different systems to their respective stationary states. These systems only differ in their initial crime rates.
Figure 5
17 In order to formally define ( ̃) we need to find the fixed points of the
function ( ̃) on the domain [ ]. Notice that a fixed point of the function ( ) is a
root of the function ( ) ( ) . With this is mind, let be the set of roots of the polynomial ( ) in the domain[ ]. Where:
( ) ∑ ∑ ( ) ( ) ( ) ( )
⌊ ̃⌋
Then:
( ̃) { { { } ( } ( ) )
The graphical example depicted in Figure 6 may be useful to understand the expression on the left. Notice that, in this example, and the polynomial ( ) has clearly various roots in the domain [ ]. Points and satisfy ( ) and ( )
, respectively. Point also satisfies ( ) . Point , on the other hand,
does not satisfy any of the two, since . Consequently using the definition above we obtain:
( ̃) ( ̃) ( ̃) ( ̃)
Figure 6:
18
Results:
Proposition 2.1: An enhancement of the prosecuting capacity does not increase the stationary
crime rate regardless of the initial crime rate, i.e. for each [ ] ( ̃) .
Proposition 2.2: A decrease in the threshold belief ( ̃) does not increase the stationary crime rate
regardless of the initial crime rate, i.e., for each [ ] ( ̃ ̃) .10
Corollary 2.2.1: Rises in fines , rises in the reserve utility , or falls in the utility gain of succeeding in crime E, do not increase the stationary crime rate regardless of the initial crime rate.
Propositions 2.1, 2.2 and Corollary 2.2.1 prove that the comparative statics of the baseline two-period entropy model transmit to the infinite horizon extension. This is important because effects on stationary states are more persuasive for the development of public policy measures than only the immediate effects on the subsequent period. Additionally, these statements show that, no matter how grievous initial conditions are, improvement can be made by improving the prosecuting capacity or increasing fines.
Proposition 2.3: As the agents’ exposition tends to infinity, the stationary state increases if
̃ and decreases if ̃ . More precisely the function ( ̃) converges
pointwisely as to the function:
( ̃) {
̃
̃
Proposition 2.3 offers a possible explanation to what is, according to Glaeser et al. (1996), one of the most puzzling features of crime rates: its high variance between urban centers. According to them, the differences in costs and benefits of delinquency are not enough to explain
10 This proposition is still valid if is considered endogenous as in the baseline two-period model. A
reduction of ̃ diminishes . Thus, since ( ̃) is an increasing function in , the fall in the threshold belief may decrease the level of crime in the long run even further, by shifting to an inferior stationary state. These two effects (the reduction of the functional form and the decrease in the initial level of crime) complement each other.
19 such a high cross-city variance in criminal rates. They propose that social interactions are the underlying reason to explain this phenomenon and estimate an index of social interaction for several types of crimes that depends positively on the variance of their respective rates. They find that crimes such as rape, arson and murder have a very small variance when compared to larceny and auto theft (after controlling for urban characteristics and unobserved attributes). Assault, robbery and burglary have an intermediate variance (between that of rape and arson and that of auto theft).
Notice that, in the model, these differences in variances can be explained by the differences in agent exposures between crimes. It is reasonable to suppose that agent exposure is much higher in petty crimes such as robbery and burglary than in more serious offenses such as rape and murder. For an individual, it is much easier to observe if another person stole something or ignored the stoplight than it is to verify if he is an assassin or a rapist. As the exposure grows, Proposition 2.3 predicts that the stationary states will grow apart, as they approach 0 or 1. If the initial levels of crime are sufficiently disperse in the sample of cities, this will imply that crimes with a higher exposure will present a higher inter-city variance.
4
Agents’ Exposure Heterogeneity
In this extension, we introduce a proportion of agents that are extremely well informed of the social crime rate and of the state’s ability to respond against it (referred hereafter as omniscient agents). To model this greater experience we will assume that omniscient agents have a much higher exposition than the regular agent. In fact, we will assume that their exposition is large enough to calculate very precisely the indictment probability, as if they observed the entire population. Agents know their own type but are unable to distinguish between them in the rest of the agents. Additionally is unknown to all individuals. Regular agents have a finite exposition (p) and both kinds of agent share the same utility valuations as in the previous model (consequently they share the same threshold belief ̃). In the first period, beliefs are distributed following the same exogenous distribution used before identical for both types of agents. Thus,
20
∫ ( )
̃
( ) ∫ ( )
̃
∫ ( )
The amount of criminal omniscient agents in period is equal to if ̃ or if
the opposite occurs. This introduces a discontinuity in the function ( ̃ ) at the
point ̃. The amount criminal regular agents in period , assuming they form beliefs in
the same way as in the previous models, equals to ( ) ∑ ∑⌊ ̃⌋ ( ) ( ) (
) ( ) if or otherwise. Thus, adding for both omniscient and regular
agents we obtain:
( ̃ )
{
( ) ∑ ∑ ( ) ( ) ( ) ( ) ⌊ ̃⌋
̃
( ) ∑ ∑ ( ) ( ) ( ) ( ) ⌊ ̃⌋
̃
( )
The stationary state as a function of the initial crime rate, ( ̃ ), is found
analogously as in the standard version. The discontinuity of ( ̃ ) does not add any
subtleties to the analysis. Let be the set of all roots of ( ) ( ̃ ) on
the domain [ ]. Then:
( ̃ ) { { { } ( } ( ) )
Figure 7 compares the evolution of delinquency rates with the presence of omniscient agents and without. It also depicts its effect upon the stationary state.
Results:
All propositions concerning the effect of the agents’ exposition, the threshold belief and the prosecuting capacity upon the crime rate in (propositions 1.1, 1.3 and 1.4) and the stationary state (propositions 2.1, 2.2 and 2.3) are valid for this extension and the proofs are too similar to present them again. This section will only focus its attention on the impact of changes
21 in the proportion of omniscient agents on the crime rate in period and on the stationary state.
Figure 7:
Figure 7 shows how the presence of omniscient agents affects crime rate´s evolution in a system by introducing a
discontinuity at the point ̃ .
Proposition 3.1: A raise in the proportion of very informed agents increases the crime rate in the
next period if
̃, reduces it if ̃ , or leaves it the same if .
It is important here to notice that omniscient agents have a dual effect upon criminal rates
if ̃. First, they cause a direct increase in criminality due to the fact that they all become
criminals in the next period. Second, in subsequent stages, they induce regular agents into more
criminality since they reduce the indictment probability for all the population. If ̃ , both
effects occur in the opposite direction. Agent exposition heterogeneity helps to explain, for instance, why taxi drivers in Bogotá are more involved in traffic infractions than the average driver. Differences in utility valuations cannot certainly explain this result. Taxis, perhaps, have a higher utility loss for waiting for the traffic light to change but, if their vehicle is immobilized by the authorities, they cannot work any longer so their “fine” is greater. With the extension
22 presented here we can conclude that as taxis drive much more than the average private vehicle, they calculate more precisely the probabilities of indictment. Therefore, in a system with high impunity rates for traffic infractions, such as Bogotá, they tend to violate traffic regulations more frequently. Moreover, this behavior induces all other common drivers to behave in a similar way.
5
Continuous Allocation of Time to Criminal Activities
For a more realistic illustration, we introduce a variation of the first model to include the possibility that individuals choose the amount of time dedicated to criminal activities from a fixed provision equal between them all (normalized, for simplicity, to 1). This permits the exploration of an intensive margin in criminal behavior, depicting the well-known empirical fact that there is very significant variance in optimal crime allocations among criminals.
The first difference with the purely extensive models appears in the expected utility function for agent , with an allocation of units of time to crime and an indictment belief . This is:
( ) ( ) ( ) ( ) ( )
Where is the fine per unit invested in criminal activities and represents the fact that punishments rise with the severity of crimes, ( ) is the return obtained from time invested in the legal labor market and ( ) is the return from time invested in criminal activities. This second return, however, is uncertain, and thus pondered by the subjective probability of not being caught. Nevertheless, we will assume that successful criminal activity is always more profitable than honest work (i.e. ( ) ( ) ( ]) and, for both functions, no time dedicated implies no gain (i.e. ( ) ( ) ). Additionally, assume that the functions are smooth, strictly increasing and strictly concave (i.e. ( ) ( ) and ( ) ( ) ( ]).
Agent is considered a criminal if , and this happens if the expected marginal return of criminal activities is greater than the marginal cost at point . Notice that beyond this point the net marginal return of crime falls, since, by construction, the expected utility is strictly concave in . This implies that the net benefit of further investment in illegal activities would be negative if it was already zero or less in . Thus, an individual is criminal if and
23 only if ( ) . From this fact, we may deduce that there exists a threshold belief ( ̅) that
divides the criminals (agents with a belief strictly beneath) and the non criminals. This threshold must satisfy:11
( ̅) ( ) ̅ ( ) ̅ ( ) ( )
( )
The endogenous process of formation of beliefs is identical to the discrete case. Each period, the agent estimates the indictment probability by dividing the number of observed captures over the number of observed criminals in the immediately previous period. If this fraction is not well defined, , the agent remains with its prior belief. Similarly as in the infinite horizon extension, we will suppose the participation rate in crime in the first period as exogenous.
In order to find the total level of crime in this society, we must first obtain ( ), the optimal amount of time allocated to criminal activities by an individual with an indictment belief . The explicit expression of this function depends on the functional forms of ( ) and ( ) but, unless very specific definitions of these two are chosen, it cannot be obtained algebraically. It can be defined, nonetheless, implicitly from the first order conditions and this allows us to derive the effects of a change in the indictment belief or the fine in the optimal time invested.
Let ( ( ) ) ( ) ( ( )) ( ( )) . Then, using
implicit differentiation, for a criminal with an interior solution we obtain:
( )
( ( ) )
( ( ) )
( )
( ( ))
( ) ( ( )) ( ( ))
( )
( ( ) )
( ( ) )
( ) ( )
( ( )) ( ( ))
For a full time criminal ( ) the previous inequalities are not strict.
11
Notice that, since ( ) ( ) ( ), ̅ ( () ( )
24 Assume the prosecuting capacity operates identically as in the previous models: the state can only capture a fixed proportion of the population regardless of their individual levels of crime. Thus, the participation rate in crime evolves similarly to the extensive case, except for a different threshold belief ( ̅) and the fact that now an individual with exactly this belief cannot be
considered a criminal, since ( ̅) . So,
( ̅) {
∑ ∑ ( ) ( ) ( ) ( )
⌈ ̅⌉
( )
The amount of crime at stage , , is found by adding the individual time investments in crime and multiplying them by their respective probability of occurrence found in the previous expression. This is:
( ( ) ( ))
{
∑ ∑ ( ) ( ) ( ) ( ) ( ) ⌈ ̅⌉
( )
From the expression above, it is clear that the level of crime reaches a stationary state if, and only if, the participation rate does. As in the previous models, we resume the hysteretic
behavior of the participation rate by the function ( ̅), which returns the value of this
variable at the stationary state given any initial level of participation. The formal definition is identical to the one presented in Section 3. Finally, we define the stationary state of the level of crime as:
( ( ) ( )) ( ( ̅) ( ) ( )) ( )
Results:
All propositions regarding the effects of the prosecuting capacity, the threshold belief and the agents’ exposition on the participation rate and its stationary state (propositions 1.1, 1.3, 1.4,
2.1, 2.2 and 2.3) are valid for this extension and the proofs are too similar to present them again. This section will only focus on the effects of these variables on the level of crime and its stationary state.
25 Proposition 4.1: An enhancement in the prosecuting capacity of the state today does not raise the
level of crime in the next period, i.e. .
Two important effects occur. First, this increase reduces the participation rate as a result of Proposition 1.1, and second, it causes a decrease in the optimal allocations of time to criminal activities. This second effect can only be modeled by this extension and proves that upgrades in the prosecuting capacity of the state trigger, as well as an extensive reduction of criminality, an
intensive one. People that persist in delinquency will dedicate less effort to it. Proposition 4.2 proves that this result transmits to the stationary state.
Proposition 4.2: An improvement in the prosecuting capacity of the state today does not raise the
level of crime in the stationary state, i.e. .
Proposition 4.3: As the agents’ exposition grows to infinity, the level of crime in the stationary
state increases if ̅ and decreases if ̅ . More precisely the function
( ( ) ( )) converges pointwisely as to the function:
( ( ) ( )) {
( )
̅
̅
Intuitively, as agents estimate with more precision the probability of indictment, they will
all become criminals in the second period if this probability is low ( ̅ ) and therefore their
optimal allocation to crime will be ( ). Since, under this scenario of high participation rates,
( ̅) , their optimal allocation will be ( ) from the third period onwards. If
the probability of capture is high ( ̅ ) none of the agents will become a criminal and thus the
level of delinquency falls to zero in the steady state. This proposition proves that hysteretic behavior persists under the scenario of continuous time allocation to criminal activities.
26
6
Concluding Remarks
A model inspired heuristically in the definition of entropy may at a first glance seem too abstract to be able to predict much of criminality’s stylized facts. However, the model
constructed in this paper and its extensions prove quite the opposite. When societies are studied as systems that may undergo an irreversible deterioration in their indictment rates, due to the steady increase of delinquency participation rates and the state’s limited prosecuting capacity, the model performs outstandingly well in explaining empirical facts concerning crime.
On a societal level, the model predicts criminal trends’ key features such as the positive persistency in crime rates and the reduction of these rates due to an enhancement in the prosecuting capacity. All Vector Autoregressive Models consulted confirm these two conclusions. It also clarifies crime series’ hysteretic nature and the way in which a society may end up suffering a self-fulfilled prophecy of poor security standards. Additionally, the model puts forward an explanation to one of crime’s most puzzling aspect: the yet not fully explained high interurban variance in crime levels. As population is larger it is reasonable to suppose that agent exposure increases, and this makes stationary states grow further apart between cities.
On an individual level, the model sheds light on recidivism’s determinants and, more importantly, suggests that unheeded social variables in econometric estimations regarding this subject could alter their conclusions. Idiosyncratic variables’ estimated impact might be biased because of the omitted social variables the entropy model above highlights as important. Specifically, the state’s prosecuting capacity, the agents’ exposure and the crime rate itself could
all influence an individual’s decision to relapse in criminal behavior.
Predictions made by the model on a wide range of topics are very accurate and hopefully sound policy measures should be devised from them. One example of this, for instance, is the model’s dissuading argument against punitive populism. Moreover, all functional forms’ relative
simplicity allow the model for further extensions, as the ones presented, that could be implemented to address more complex delinquency scenarios and to formulate more adequate policies for them. A state that responds to crime epidemics by increasing its prosecuting capacity or an individual that ponders his own experience relative more than that of other individuals, are extensions that should be explored.
27
7
Mathematical Appendix
Proposition 1.1:Proof: Notice that :
∑ ∑ ( ) ( ) ( ) ( ) ( ) ⌊ ̃⌋ ∑ ( ) ( ) ∑ ( ) ( ) ( ) ⌊ ̃⌋
The cumulative distribution function of a binomial distribution ( ( )) can be expressed as a Regularized Incomplete Beta function of the form: 12
( ) ( ) ∫ ( ) Thus: ∑ ( ) ( ) ∫ ⌊ ̃⌋ ( )⌊ ̃⌋
By the Fundamental Theorem of Calculus:
∑ ( ) ( )
( ) ( ) ⌊ ̃⌋ ( )⌊ ̃⌋
If ,
□
Proposition 1.2:Proof: First notice that if : 13
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
12 With the probability of success, the number trials and the maximum number of successes
considered. 13( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
28 Thus, adding first by rows instead of columns we obtain:
∑ ∑ ( ) ( ) ( ) ( ) ( ) ( ) ⌈ ⁄ ⌉ ̃ Or, equivalently: ∑ ( ) ( ) ∑ ( ) ( ) ( ) ( ) ( ) ⌈ ⁄ ⌉ ̃ ( )
The second summation is the cumulative distribution of a binomial distribution with a
success probability of , total trials and ⌈
̃
⁄ ⌉ maximum number of successes. As
in the previous proof, this can be expressed by means of a Regularized Incomplete Beta function. Thus, ∑ ( ) ( ) ∫ ⌈ ⁄ ⌉ ̃ ( ) ⌈ ⁄ ⌉ ̃ ( )
By the Fundamental Theorem of Calculus:
∑ ( ) ( ) ( ) ( ) ⌈ ⁄ ⌉ ̃ ( ) ⌈ ⁄ ⌉ ̃ ( )
If ,
□
Proposition 1.3: Proof: Notice that, for the first period, we have:
29 As a result of Proposition1.2, for the second period we have:
( ̃) ( ̃) ( ̃)
□
Corollary 1.3.1: Proof: Notice that,
̃ ( ) ( ) , ̃ ( ) , ̃ .
□
Proposition 1.4:Proof: Due to the properties of a probability limit we have14:
Thus, if ̃ we obtain:
̃
( ̃) ( ̃)
If ̃ we obtain:
̃
( ̃) ( ̃)
□
Proposition 1.5:Proof: Notice that:
∑ ∑ ( ) ⌊( ) ̃ ⌋ ∑ ∑ ( ) ⌈ ̃ ⌉ ∑ ( ) ( ) ∑ ( ) ( ) ( ) ( ) ( ) ⌈ ̃ ⌉
30 ∑ ( ) ( ) ∫ ⌈ ⁄ ⌉ ̃ ( ) ⌈ ̃ ⌉
Thus, as a result of the Fundamental Theorem of Calculus:
∑ ∑⌊( ) ̃ ⌋ ( ) ∑ ( ) ( ) ( ) ( ) ⌈ ⁄ ⌉̃ ( ) ⌈ ̃ ⌉ Similarly, ∑ ∑⌊( ) ̃⌋ ( ) Implying that, ( ̃)
□
Proposition 1.6: Proof: Notice that:
∑ ∑ ( ) ⌊( ) ̃ ⌋ ∑ ( ) ( ) ∑ ( ) ( ) ( ) ⌊( ) ̃ ⌋ ∑ ( ) ( ) ∫ ⌊( )̃ ⌋ ( )⌊( )̃ ⌋
Thus, as a result of the Fundamental Theorem of Calculus:
∑ ∑⌊( ) ̃ ⌋ ( ) ∑ ( ) ( ) ( ) ( ) ⌊( ) ̃ ⌋ ( ) ⌊( ) ̃ ⌋ Similarly, ∑ ∑⌊( ) ̃⌋ ( )
31 Notice that, ( ̃) ∑ ∑⌊( )̃ ⌋ ( ) ∑ ∑ ( ) ⌊( )̃ ⌋ ( ) ∑ ∑ ( ) ⌊( )̃⌋ ∑ ∑ ( ) ⌊( )̃⌋
□
Proposition 1.7: Proof: If the criminal was apprehended in the previous period,
Analogously, if he or she was not apprehended:
In either case,
If ̃
̃ ( ̃) ( ̃) ( ̃) If
̃ we obtain:
̃ ( ̃) ( ̃) ( ̃)
□
32 Proposition 2.1: Proof: Case 1: ,
Case 1.1: Let be any initial level of crime such that ( ) and
let ( ̃). Since ( ) (as a result from Proposition 1.1) then, after an
increase in c, ( ) . If, after the increase of to ( ), ( ) is still greater than 0 then, by a direct cause of the Intermediate Value Theorem, there exists a ( ) such that
( ) . This implies that ( ̃) . If, after the increase in c, ( ) , then, by definition, ( ̃) . In either case ( ̃) falls when c augments.
Case 1.2: Let be any initial level of crime such that ( ) and
let ( ̃). Since ( ) (as a result from Proposition 1.1) then, after an
increase in c, ( ) . This means that, after the increase in c,
( ̃) ( ̃) . Again, ( ̃) falls when c augments.
Case 1.3: Let . Then, by definition, ( ̃) , before and after
the increase in c. In this subcase, the function ( ̃) does not grow.
Case 2: . Then for all [ ], ( ̃) regardless of the level of c. Thus, after an increase in c, the function does not grow.
□
Proposition 2.2:Proof: Notice that a decrease in the threshold belief ( ̃), decreases ( ) for all
[ ] similarly to the effect of an increase in c. I.e.: ( ̃)
( ̃) as a result of
Proposition 1.3. Thus, all the cases and their respective reasoning are identical to the ones exposed in the previous proof.
□
Proposition 2.3: Proof: By Proposition 1.4 we know that:
( ̃) {
̃
33 Thus implying that, as , all stationary states are arbitrarily close to 1 if ̃ or
if the opposite occurs.
Proposition 3.1:Proof: Notice that:
{
∑ ∑ ( ) ( ) ( ) ( )
⌊ ̃⌋
̃
∑ ∑ ( ) ( ) ( ) ( )
⌊ ̃⌋
̃
Since, by construction, ∑ ∑⌊ ̃⌋ ( ) ( ) ( ) ( ) , we have
that if ̃, if ̃ or if .
□
Proposition 4.1: Proof: Notice that, if , the total level of crime in period can be rewritten as:
( ( ) ( )) ∑ ( ) ( ) ∑ ( ) ( ) ( )
( )
⌈ ̅⌉
∑ ( ) ( ) ( ( ) | )
As increases to ( ) the new conditional distribution of ( | ) ( )
stochastically dominates the previous conditional distribution ( | ) ( )Thus we have
that, for all decreasing real valued functions of , ( ( )) ( ( )).15 Particularly,
( ( ) | ) ( ( ) | ) { }. This implies that,
( ( ) ( )) ( ( ) ( ))
15For further reference on the univariate (usual) stochastic dominance and its properties consult “Stochastic Orders and its Applications”, Chapter 1.
34
If , the level of crime remains the same:
( ( ) ( )) ( ( ) ( ))
In both cases the function is non-increasing.
□
Lemma 1: An exogenous increase in the participation rate today does not decrease the level of
crime in the next period, i.e. .
Proof: Notice that, if , the total level of crime in period can be rewritten as:
∑ ( ) ( ) ∑ ( ) (
)
(
)
⌊ ̅⁄ ⌋
( )
∑ ( ) ( )
( ( ) | )
As increases to ( ) the new conditional distribution of
( | ) ( ) stochastically dominates the previous conditional
distribution ( | ) ( ). Thus we have that, for all increasing real valued
functions of ,
( ( )) ( ( )).
Particularly ( (( ) ) | ) ( (( ) ) | ) { } . This
implies that,
( ( ) ( )) ( ( ) ( ))
If , the level of crime remains the same:
( ( ) ( )) ( ( ) ( ))
35 Proposition 4.2:
( )
( ( ̅) ( ) ( ))
( )
( )
( )
Propositions 4.1 and Lemma 1 imply that and . Proposition 2.1
implies that ( ) .
Thus,
( )
□
Proposition 4.3: Proof:
( ( ) ( )) ( ( ̅) ( ) ( ))
( ( ̅) ( ) ( )) (
( )
)
Proposition 2.3 we know that
( ) {
̅
̅. Thus,
( ( ) ( )) {
( ) ̅
̅
36
8
References
Becker, Gary. (1968). “Crime and Punishment: An Economic Approach”. Chicago:
Journal of Political Economy Vol. 76. Pages 169-217.
Carvalho, Jose R., & Bierens, Herman J. (2002). “A Competing Risk Analysis of Recidivism”, Philadelphia: Working Paper, Penn State University.
Gaviria, Alejandro. (2000). “Increasing Returns and the Evolution of Violent Crime: The Case of Colombia”.Journal of Development Economics, Elsevier Vol. 61 Pages 7-11.
Glaeser, Edward., Sacerdorte, Bruce., & Scheinkman, Jose. (1995). “Crime and Social Interactions”, Oxford: Quarterly Journal of Economics Vol. 111. Pages 507-548.
Jantzen, Robert. (2011). “Relative Crime Series in New York City”, New Rochelle: Working Paper, Iona College University.
Koskela, Erkki., & Viren Matti. (1997). “An Occupational Model of Crime Switching”,
Applied Economics, Vol. 29. Pages 655-660.
Levitt, Steven. (2004). “Understanding Why Crime Fell in the 1990s: Four Factors That Explain the Decline and Six That Do Not”, Pittsburgh: Journal of Economic Perspectives, Vol. 18. Pages 163-190.
Lovitch, Norman., Joyce, Theodor., & Corman, Hope. (1987). “Crime, Deterrence and the Business Cycle in New York City: A VAR Approach”. Cambridge: The Review of Economics and Statistics, MIT Press. Pages 695-700.
Rasmussen, Lucinda A. (1999). “Factors Related to Recidivism in Juvenile Sexual Offenders”. Sexual Abuse: A Journal on Research and Treatment Vol.11. Pages 69-85.
Rosenfeld, Richard. & Robert., Fornango. (2007). “The Impact of Economic Conditions on Robbery and Property Crime: The Role of Consumer Sentiment”. Criminology Vol. 45, Pages 735-770.
Sah, Raaj K. (1991). “Social Osmosis and Patterns of Crime”. Chicago: Journal of Political Economy Vol. 6. Pages 1272-1295.
37 Saridakis, George. (2003). “Violent Crime in The United States: A Time Series Analysis Between 1960-2000”, Leicester: Discussion Papers in Economics 03/14, Department of Economics, University of Leicester.
Shaked, Moshe., & Shanthikumar George (2007),, Stochastic Orders, Springer Editorial 2007.
Sirakaya, Sibel., (2006). “Recidivism and Social Interactions”. Alexandria: Journal of the American Statistical Association Vol. 101 Issue 475. Pages 863-877.
Worthington, Andrew C., Higgs, Helen., & Edwards, Glyn. (2000). “Determinants Of Recidivism In Paroled Queensland Prisoners: A Comparative Analysis Of Custodial And Socioeconomic Characteristics”. Australian Economic Papers, 39(3). Pages 312-326.
