Crime, Protection, and Incarceration
by
Allen Wilhite*
and
W. David Allen
Department of Economics and Finance
University of Alabama in Huntsville
Huntsville, Al 35763
U. S. A
phone: (256) 824-6591
fax: (256) 824-1328
abstract
Criminals impose costs on society that go beyond the direct losses suffered by their victims—they cast a shadow of uncertainty over our daily economic and social activities. Consequently, individuals, and society as a whole, choose to direct resources to crime prevention. This study creates a virtual society facing such choices. Individuals, neighborhoods, and cities make crime prevention decisions and adjust their decisions over time as they attempt to balance the cost of crime with the cost of crime prevention. What emerges is a society with aggregate criminal characteristics that mimic crime in the natural world. This virtual society is then used to conduct a series of anti-crime policy experiments, and the effects of those experiments are observed over time.
classification codes: K42, C63, Z13
*corresponding author
Crime, Protection, and Incarceration
I. Introduction
The lingering apprehension that follows crime intrudes into the lives of us all. Victims lose their personal possessions and sense of well being, people in the vicinity of a crime feel less secure and entire neighborhoods can acquire reputations that suppress future economic and social welfare. To avoid and/or minimize these many effects we undertake a host of crime fighting and crime prevention activities. But anti-crime measures tend to be expensive and unpleasant, and so we want them to be effective and efficient. This paper develops an agent-based model of several cities to study how anti-crime policy affects the level and dispersion of crime. After constructing a society in which law-abiding citizens and criminals interact, we conduct a series of experiments that compares the consequences of anti-crime activities undertaken in these artificial cities.[1]
This artificial society does not attempt to replicate reality. Instead, our computational model represents a theoretical construct, a mathematical model that, like traditional analytical economic models, focuses our thinking, leads to testable hypotheses, and helps us understand a part of the world. However, because the model incorporates interacting heterogeneous agents that evolve over time, its complexity resists analytical solutions, and so we explore its explanatory power computationally. In the end, this study provides an explanation for certain patterns of crime observed in real cities and also illustrates how economists can use computational methods to construct theory that yields insight into policy.
II. A Simple Model of Crime
Suppose a society contains three types of agent: citizens, criminals, and convicts. Law-abiding citizens allocate their time to produce output that they then consume for their own satisfaction. We assume that leisure time yields no utility, so that an agent unconcerned with crime would work “full time.” But criminals, agents who gain satisfaction by stealing output from productive citizens, lurk in some neighborhoods, and their craft reduces citizens’ satisfaction. Formally a citizen earns satisfaction
,
where yi represents output produced by agent i in a given period and represents the crime rate in agent i’s neighborhood, n. Thus, agent i’s satisfaction depends on how much he produces minus the portion lost to criminal activity.
The crime rate has two components: the proportion of the neighborhood’s population made up of criminals,, and the velocity of crime, v, which reflects the number of crimes an individual criminal commits in a single time period (e.g., in a given day). The proportion of criminals in a neighborhood,, changes if agents switch occupations (citizens become criminals or criminals become citizens), if agents change neighborhoods, or if criminals get caught and spend time in prison (i.e., when criminals become convicts).
The velocity of crime, v, depends on how citizens allocate resources to fight crime. Citizens can choose from any of three types of protection: self-protection xi, where an individual spends time to protect his own output; communal protection cn, where agents in a particular neighborhood voluntarily pool resources to produce neighborhood-wide protection; and city-government protection gn, funded by taxes ti, where gn and ti are determined by election (described in Section III). These modes of protection differ in their effectiveness and in the extent to which citizens can share protection’s benefits and costs. City-wide government protection (such as police services) offers the most effective protection, as cities hire individuals who specialize and train for that specific purpose. But this protection may be unevenly distributed across a city. Communal neighborhood efforts offer the second-most effective protection (per unit of time) and present some economies of scale in the production of protection, but they also present opportunities for free-riding; a citizen might contribute nothing to communal protection and yet benefit from it. Private efforts at crime prevention offer the least effective protection, but their benefits and costs are not subject to free-riding abuses.
Protection affects the velocity of crime according to
,
where represents the maximum potential velocity of crime, i.e., the velocity that arises in the absence of any crime-fighting effort. The exponents in equation (2) tune the effectiveness of a particular type of protection, acting essentially as exogeneous technology parameters. Given the variation in effectiveness of the three protection measures, we assume that γ < β < α .
The subscript i in equation (2) indexes an agent-specific characteristic. For example, the level of self-protection expenditures, xi, can differ from agent to agent. The subscript n applies to neighborhood-specific attributes. Thus, as suggested in equation (2), every agent in a neighborhood receives the same level of communal protection, even though communal protection can differ across neighborhoods. City spending on police protection, gn, carries the neighborhood subscript because police protection is allocated neighborhood by neighborhood. As a result, one neighborhood might receive more city protection than another even though each agent within a neighborhood receives the same city protection. Aggregating, the velocity of crime becomes an individual attribute (subscripted i); that is, the likelihood that citizen i becomes a victim differs from the likelihood that another citizen j becomes a victim either because they live in different locations, choose different levels of self-protection, or both.
An individual agent’s production, yi, depends on the amount and type of protection he selects because spending time on protection takes time away from production. Production is also affected by location. Agents interact and pick up attributes, habits, and reputations from their acquaintances. Neighborhoods reflect the character of their residents, and that character impacts everyone’s ability to earn a legitimate living. Altogether, output for each agent (per unit of time) consists of the total potential output minus the output lost due to allocating time to protection, minus the individual- and neighborhood-level output lost due to consorting with other criminals. Setting the maximum per-period production equal to one, actual production can be expressed as
.
Equation (3) indicates that agent i’s output per period depends on his contributions to self-protection (xi), to communal protection in the neighborhood (ci), his taxes (ti), collected to support the city-wide policing effort, and the effects of his acquaintances and neighborhood, captured within the last two terms.
Each agent is endowed with a “criminal profile,” a tag-string made up of 100 entries, , wherein each “1” in the string indicates a criminal element and each “0” indicates the absence of that element. Each neighborhood is endowed with a 100-entry tag-string as well, , where each “1” indicates a more detrimental environment to legal activity. Thus, in the context of equation (3), as an agent acquires more entries equal to 1, he becomes less capable of producing output, at a rate. This decline in productive capability might reflect the decline in human capital, the acquisition of poor work habits, or a declining reputation. Similarly, as a neighborhood acquires more 1’s in its bit-string, all agents who live there earn less. This reflects the deterioration of the economic base whereby a greater criminal element leads to a degradation of the physical capital in the neighborhood, the attraction of less desirable establishments, and a less safe environment overall. The degradation of the neighborhood adversely impacts agent i’s output according to the rate.
Criminal satisfaction depends on the amount of production occurring in the legitimate segment of society, the level of protection and policing activities undertaken by citizens, and the economic environment of the neighborhood. Thus, a criminal will achieve satisfaction
.
The first term in the numerator of (4) represents the output stolen from the citizens of a neighborhood, an amount that depends on production and protection. The second term in the numerator represents the additional criminal income derived from illegal market activities occurring in this neighborhood. Note its resemblance to the last term in equation (3), which reflected the decrease in legitimate income in a neighborhood due to crime. As a neighborhood deteriorates, it experiences a loss in legitimate income and a rise in illegal income. For example, a growing presence of prostitution in a neighborhood would reduce the income of legitimate business but would increase the potential earnings of criminals. In general, we expect , i.e., that the decline in legitimate income outweighs the rise in illegal income.[2]
III. The Action
Agents receive satisfactionif they function as law-abiding citizens and if they function as criminals. Agents prefer more satisfaction to less, and so equations (1) and (4) drive the model, but agents do not explicitly maximize (1) and (4). Instead, they periodically compare themselves to others and perhaps change some of their decisions if they find their situation sufficiently grim. Thus, equations (1) and (4) act less as objective functions and more as “fitness” measures that drive selection. All agents are initially assigned a randomly-selected decision for all factors under their control. Twenty percent of the general population begin as criminals, the others as citizens. Those citizens are assigned a level of self-protection, a contribution to neighborhood protection, a starting level of city policing efforts (and its accompanying tax burden), and a neighborhood in which to reside.[3]
After initialization, agents make choices every time step using a random evolutionary decision process introduced by Wilhite (2006). First, agents are ranked according to their satisfaction, identifying the bottom 2% as agents interested in change.[4] Those agents have seven options: (i) switch occupations (change to a criminal or citizen), (ii) increase the level of self-protection, (iii) reduce the level of self-protection, (iv) increase their contributions to neighborhood protection, (v) reduce their contribution, (vi) move to another neighborhood, or (vii) do nothing. Agents choose a particular action randomly. After these agents have made their choices, all agents are ranked again according to their satisfaction, this round’s poorest 2% are given an opportunity to make a change, and so forth.
This evolutionary procedure leads to a simple and natural decision process. Rather than optimizing every choice, agents evaluate their lot by comparing themselves to others; the least successful agents (those whose past choices have rendered them least fit) simply decide to try something different. In this approach, the new strategy takes on the character of an experiment because an agent does not know whether his new action will yield more satisfaction or less satisfaction. Over many rounds of decision-making and re-evaluation, agents tend to converge on a strategy, and the society takes on a relatively stable set of macroeconomic characteristics that one could view as a steady state. Naturally, we see turmoil at the microeconomic level, because a least-fit portion of the population always exists, but eventually these micro-level adjustments consist mostly of agents switching among slightly different choices.
Our virtual agents use elections to settle the collective, city-wide decisions about spending on police protection and the taxes necessary to support that spending. As in other routines, the program starts with a randomly-determined amount of city spending in each neighborhood, with taxes set commensurate with that aggregate level of spending. Spending can differ from neighborhood to neighborhood, but taxes do not, and taxes equal spending (cities incur neither deficits nor surpluses). Every 20 periods, an election occurs on a randomly-generated ballot initiative, which unfolds as follows. With probability 0.5, each neighborhood is selected to receive an increase in police spending of a small amount, and then with probability 0.5 neighborhoods are selected for an identical decrement of spending. This leaves a “ballot” in which some neighborhoods are proposed to receive an increase in spending, others a decrease in spending, and others no change. The taxes required to finance this new level of spending are calculated and added to the ballot.[5]
In the ensuing election, citizens determine how to vote by calculating their expected satisfaction, substituting the ballots’ spending and tax proposals into equation (1). Citizens support the initiative if their satisfaction would increase and vote against it if their satisfaction would decline. Agents unaffected by the initiative do not vote, nor do criminals and convicts. If a simple majority supports the spending and tax changes proposed by the ballot, those changes are adopted and agent satisfaction changes accordingly. Over time these elections tend to settle on a pattern of spending and taxation in which the majority is sufficiently content with the city’s protection expenditures; most ballot initiatives fail and the status quo reigns.
The two tag-strings that stand in for personal and neighborhood characteristics also change over time. Three activities alter the tags in an individual’s string. In each round, an agent “meets” another randomly-selected agent in his neighborhood. A particular position {0, … , 99} is selected randomly, and the two agents compare the value of the tag at that position. If they differ, one agent “flips” his tag to match the other agent’s tag. No change occurs if the tags match. Through this random meeting, a “good-hearted” agent (with a predominance of tags equal to zero) who happens to reside in a neighborhood with many “ill-hearted” agents (with a predominance of tags equal to one) will eventually become corrupted because he usually interacts with bad guys. Similarly, a bad guy living in a neighborhood of angels will find the goodness rubbing off onto him. In this way, agents influence their neighbors.