BIAS AND BELIEFS IN DETERRENCE AND DETECTION
3.2 Related Literature
Becker (1968) is generally recognized as having inaugurated the modern theoretical economic study of crime. In this literature, the act of committing a crime is modeled as a risky gamble that imposes harm on third parties, and potential criminals as selfish expected utility maximizers. The objective of law enforcement is generally taken to be the maximization of social welfare, and the primary policy instruments are the magnitudes and types of punishments levied against offenders, and the quantity of costly effort expended toward detection and apprehension. Becker (1968) argues that the socially optimal policy is to maximize punishment magnitude and reduce detection effort, since increasing punishment magnitude is costless while detection effort is costly, and these variables are substitute determinants of the expected
punishment for offending.
Subsequent studies identify conditions under which maximal punishments are so- cially undesirable. Since imprisonment is socially costly for many reasons, the maximal prison term length is generally not optimal, and less costly punishments should be used when possible (Becker (1968), Posner (1980)). When agents are risk averse in wealth, reducing fines may improve social welfare by reducing risk born by agents (Polinsky and Shavell (1979), Kaplow (1992)). If agents are uninformed of the legality of an illegal act (Kaplow (1990)), or if they are unsure of the probabil- ity of apprehension (Bebchuk and Kaplow (1992)), optimal fines are generally not maximal. Many other theoretical justifications of non-maximal punishments have been offered, Polinsky and Shavell (2000) and Garoupa (1997) give useful reviews of these and other related extensions of the model. In general, when there are some costs associated with increasing punishment severity, non-maximal punishments are optimal and some amount of costly investment into detection and apprehension is socially preferable, and it is the use of this policy instrument that is the focus of this study.
This previous literature on optimal law enforcement generally takes as known the distribution of benefits from crime in the population and the detection capabilities of enforcement agents. To extend the analysis to the case in which these primitives are, more realistically, unknown, I adopt a modeling approach related to several strands of recent theoretical study.
First, the equilibrium notion I use roughly corresponds to a notion of self-confirming equilibrium, introduced by Fudenberg and Levine (1993). In a self-confirming equilibrium, each agent best responds given his beliefs, and beliefs are correct along the equilibrium path but may be incorrect off-path. Similarly, I will say the law enforcement agency is in equilibrium when its beliefs about the crime data it observes are correct given its actual investment decision.
Secondly, there is a multitude of studies on the behavior of agents with incorrect beliefs about, or misspecified models of, their environment. For example, Kagel and Levin (1986) among others document a winners curse in common value auctions in the lab, especially among inexperienced bidders. Eyster and Rabin (2005) introduce the cursed equilibrium solution concept inspired by this phenomenon, in which agents do not understand how the actions of other agents depend on their private information. In another example, Rabin (2002) studies a model in which agents suffer from a belief in the “law of small numbers", the tendency to over-infer from
small samples.
Most relevant for this study is the behavioral literature on overconfidence, an un- realistic belief in personal ability. Heidhues et al. (2018) study an overconfident agent that chooses an action in some production process, receives payoff feedback, and learns from this about a primitive of the production process. In my model, the mapping from detection investment to the probability of detecting crime can be interpreted as the enforcement agency’s ability, and the incorrect beliefs about this mapping held by the agency can be interpreted as an institutional over- or under- confidence. Thus I contribute to the overconfidence literature by showing how an institution that is over- or underconfident at the organizational decision making level can be confirmed in its incorrect beliefs while making suboptimal decisions.
Alongside the large body of theoretical work on crime, is the large empirical liter- ature in economics that seeks to estimate the deterrent effect of law enforcement.
Chalfin and McCrary (2017) provides a helpful, recent review of methods and evi- dence. At least two key impediments to the estimation of the causal effect of policing on crime are generally acknowledged.
First, variance in enforcement efforts is likely not exogenous to other determinants of the quantity of crime. For example governments may increase police funding and manpower in response to increases in crime. Many early panel data studies using data on police manpower and crime found very small or incorrectly signed relationships between policing and crime. More recent instrumental variable and natural experiment studies, including Levitt (2002), Evans and Owens (2007) and Lin (2009) find large negative effects of police on crime. I am aware of little econometric work on the other direction of causation, the response of police behavior to crime.
Second, some authors have recognized that errors in the measurement of police activity or quantity of crime may influence empirical results. Levitt (1998) discusses random errors in the measurement of the quantity of crime, and dismisses the possibility that it has a major influence on empirical estimates of the deterrence effect. Chalfin and McCrary (2013, 2018) argue that mis-measurement of the quantity of police is a better explanation than simultaneity bias for the small estimates of the deterrent effect in cross-sectional and panel data studies. After correcting for measurement error they find a strongly negative relationship between policing and the quantity of violent crime.
However, to the best of my knowledge, measurement error in the quantity of crime resulting from variation in policing intensity has not been addressed in the empirical literature. Crime data collected by police agencies are likely to suffer from a missing data problem. Only those crimes detected by police show up in the data, and the proportion of crime detected depends on the allocation of police resources. It is possible that, holding the actual amount of crime constant, more heavily policed areas show up as having more crime in the data. Without a more thorough under- standing of the relationship between crime data and police behavior, it is unclear what its implications are for empirical estimates of the deterrent effect.