Two models of institutional design in crime and education
Soin, Sikandar Singh
This thesis consists of two chapters looking at elements of optimal institutional design. In Chapter two I present a model of Crime Deterrence. One of the most ubiquitous aspects of modern society is crime. Adherence to the law is an important component of the social contract. Not impinging on the rights of others is the price all citizens pay for the protection of their own rights. I consider the problem of choosing rewards and punishments to incentivize adherence to the law and deter crime. From the point of view of the policy maker this is a moral hazard problem. Whether or not someone has committed a crime can never be perfectly observed. This necessarily implies that some criminals may be falsely acquitted and some innocent citizens falsely convicted. This fact must necessarily temper the magnitudes of both rewards and punishments as well as the level of suspicion regarding unobserved citizens. This is similar to such classic moral hazard problems as of providing insurance when the level of caution is unobserved or of choosing wages for the manager of a firm when their effort is unobservable. However, unlike most classic moral hazard problems, crime deterrence pertains to multiple agents whose actions are unobservable. The first thing I do in this paper is extend the workhorse moral hazard model of Grossman and Hart  to consider the scenario with many agents. I find that the optimal contracts chosen in this scenario are identical to those in the single agent case in every important respect. My framework considers a very simple form of crime. I use the example of fare evasion on a train. The train firm would like everyone to buy a ticket however they can’t monitor everyone’s action. They choose rewards for agents who they find bought tickets, punishments for those caught evading and rewards/punishments for those whom they are unable to monitor. These must be such that every traveller would like to buy a ticket rather than risk getting caught. Another key difference between crime and many of the other issues studied in the moral hazard literature is that of externalities. The actions of any criminal agent induces externalities on other agents. In a society with a large amount of crime, individual criminals may be hard to find, and vice versa. This may induce various agents to collude with each other to commit crimes together. Furthermore, there may be complementaries between different agents, i.e. an agents payoffs from crime may be increasing in the intensity of crime in society. This too would induce collusion. If the policy maker fails to take this into account when choosing rewards and punishments, they may fail to deter crime efficiently. I explore the implications of this by allowing for collusion in my model. Due to the monitoring structure I choose, there are very strong externalities that arise from an agents action. If an agent chooses not to buy a ticket, they make it less likely that any other agent gets fined and conversely, buying a ticket makes it more likely for other agents to be rewarded. Given this, it is possible for agents to collude with each other. I demonstrate that the scope of collusion changes the magnitudes of rewards and punishments, but that the nature of the optimal contracts remains unchanged. Implying that, fixing the number of agents, if the scope for collusion exists, it is costlier to deter wrongdoing. The above conclusion leads naturally to the question of how the number of agents affects the nature and magnitude of the optimal incentives. This is the final thing I examine in this paper. I find that the magnitude of the incentives must increase as the number of agents increases. The third chapter deals with the question of how agents make decisions in a dynamic framework when their preferences are reference dependent. We consider the case of a student who must choose how much effort to put into studying for a course that is evaluated on the basis of two exams. The student chooses how much effort to put in to studying for the mid term, observes the midterm score and then chooses how much effort to put in to studying for the final. The result depends on the ability of the student, the effort they put into studying as well as luck. The student initially knows only the parameters of the distribution from which their ability is drawn, not the ability itself. They also know the distribution of the random component. We have used a considerably simplified version of aspirations in our model. We assume that exerting greater effort leads to higher aspirations, i.e. that aspirations are a function of effort. This seems reasonable on the grounds that someone who has worked very hard would feel disappointed with a low result. The utility framework we consider has two components. The first component which we term the ‘achievement effect’ depends on the score on the exam only. Higher scores give higher utility. The second component, the ‘aspiration effect’ is reference dependent, and depends on the ‘aspiration gap’ i.e. the difference between the aspiration and the actual result. This framework admits two possibilities, that the student is disappointed (a negative aspiration gap) or that they are elated(a positive aspiration gap). This paper looks at what happens in the second period, once the result of the midterm is observed. We begin by deriving the students newly induced distribution over their ability. We assume that everything is normally distributed, for computational ease as well as because this seems like a reasonable assumption. We use a reference dependent utility function. We then set up and solve an expected utility maximization problem on the basis of these beliefs. We find that the optimal effort is unique, given parameter values. We compare the choice of optimal effort in our reference dependent framework with one that would arise from a ‘fully rational’ set up with no reference dependence. We find that the effect of aspirations on the optimal effort depends entirely on whether exerting more effort grows aspirations faster or (expected) results. We then investigate the effect of a better midterm result on the optimal effort in the second exam. We find that it is almost always the case that a better first result leads to decreased effort. So, in the majority of circumstances, a fear of failure trumps ambition. The cases when the opposite is true are quite interesting as well. A better first result only induces greater effort if two conditions are true: Firstly, the marginal expected aspirations gap must be positive, i.e. it must be the case that exerting effort increases the expected result more than it increases the aspiration. Secondly, the aspiration gap must be small and negative. So, if the aspiration induced by the optimal effort is slightly higher than the expected result then a better first result would induce more effort. Another interesting finding is that there is a magnitude effect. The higher the aspiration, the larger the set of aspiration gaps that correspond to increased optimal effort.