...world, I show that no separating contract exists. Furthermore, if the proportion of Dares is large enough, then the pooling contract, the amount of fraud and the number of agents found to have committed fraud are independent of the Dares' exact proportion in the economy. Finally, I show that investment in prevention can be useless if the proportion of Dares is large enough, which means that investing in prevention becomes a waste of resources. This last result holds when the proportion of Dares is large. When their proportion is small, investing in prevention reduces fraud.

INTRODUCTION

Crime, crime prevention, and crime punishment have always represented a major concern of any society. In the United States, almost 7 percent of the male workforce is under the supervision of the American penal system: 2 percent of the male workforce is incarcerated whereas another 5 percent is on probation or parole. Freeman (1996) compares these numbers to the long-term unemployment rates in Western Europe. In December 2001 Scientific American reported that the prison population in the United States approached 2.1 million in 2000. Compared to the 50 states, the U.S. prison system ranks 17th in terms of population, next to Nevada.

Close to 3 percent of the U.S. prison population (63,000 Americans) was incarcerated subsequent to a fraud conviction. Fraud is a relatively common crime in the United States. In automobile insurance, the rule of thumb is that insurance fraud represents 10 percent of total claims (see Weisberg and Derrig, 1991; Dionne and Gagne, 2002). If this is correct, and given that the total automobile insurance premium written annually in the United States is 110 billion dollars (see NAIC, 2001), the total cost to American policyholders of insurance fraud amounts to almost 11 billion dollars annually. Assuming this amount remains constant over time, the present value (using a 5 percent discount rate) of automobile insurance fraud in the United States is 220 billions dollars, which is almost three times the stock market value of Enron at the end of 2000. Irrespective of unemployment insurance fraud, worker's compensation fraud and other types of fraud in life, health, and property insurance, automobile insurance fraud is one of the most costly fraud scandals in the history of the United States.

The sheer size of the dollar value associated with insurance fraud begs the question of how we can fight this type of behavior. If insurance fraud is a crime, Becker (1968) suggests that, as for any crime, the government set the penalty to be very large, so that the probability of anyone committing one would be very small (see also Ehrlich, 1973; Friedman, 1999). (1) In an insurance context, however, Becker's philosopher-king approach, with an infinite penalty and a commitment to a prespecified investigation policy, (2) cannot hold for two reasons.

Firstly, penalties are not set to infinity and are usually determined by the courts. It is thus inappropriate to assume that the penalty is decided by the principal. Secondly, as argued by Besanko and Spulber (1989), Picard (1996), and Boyer (2004) it is not be reasonable to assume that the principal can commit perfectly to verify the agent's action. The reason is that, after the agent has announced his type, both players have an incentive to renegotiate their agreement to save on the audit cost.

A consequence of the principal's inability to commit is that the informed agent's optimal strategy may be to misreport the state of the world. The models developed by Graetz, Reinganum, and Wilde (1986), Sanchez and Sobel (1993), Picard (1996), Khalil (1997), and Boyer (2001) reach similar conclusions. In these articles there are agents who successfully cheat and who extract a rent from the principal. The problems related to the sequential enforcement process is also studied in Shavel (1991) and Jost (1997).

To simplify the problem, I will use a one-period model similar to that of Townsend (1979), where privately informed agents have a monetary incentive to commit insurance fraud because investigation by the principal is costly. Moreover, and again for simplicity, all players have only two possible actions they can take: The agents may commit fraud or be honest, whereas the principal may investigate the agent or not investigate. Committing fraud exposes the agent to a penalty if he is caught by the principal. (3)

This modeling approach can be applied to many insurance frameworks such as automobile insurance, social security, health insurance, and unemployment insurance (see Mookherjee and Png, 1989; Picard, 1996; Bond and Crocker, 1997; Boyer, 2001). Reinganum and Wilde (1985) also use this approach to study income tax fraud.

The model includes implicitly an agent's propensity to commit fraud; the agents who never play the game (propensity is zero) are called the "Truths," and the agents who dare play the fraud-game are called the "Dares." My model is thus greatly inspired by that of Picard (1996) in an automobile insurance context. Contrary to Picard (1996), however, I do not find instances when the insurance market collapses because the proportion of criminal elements in the economy is too large. Moreover, I address the issue of the presence of a technology that reduces the incidence of insurance fraud. I model prevention as a device that turns Dares into Truths.

The results of the article are the following. First, if the principal cannot commit, then she will not be able to design a contract that separates the Truths from the Dares. Second, the pooling contract is such that, when the proportion of Dares (given by [xi]) is smaller than some [[xi].sup.*], the agents' expected utility decreases as the proportion of Dares increases. Moreover, this pooling contract has exactly the same structure as the contract observed when there is a proportional loading factor on the premium. Third, when [xi] > [[xi].sup.*], the pooling contract is independent of the exact proportion of Dares. As a result, the agents' expected utility is independent of [xi] provided that [xi] > [[xi].sup.*]. Fourth, when [xi] > [[xi].sup.*], not only is the sunk cost penalty irrelevant in determining the amount of fraud in the economy, prevention also has no impact on the amount of fraud.

The article is constructed as follows. In the next section, I present the setup of the game between the agents and the principal. In the section "Optimal Contract When the Type Is Known" I present the benchmark case where each agent's type is common knowledge. I let each agent's type be private information in the section "Optimal Contract When Types Are Unknown" and I derive and discuss the optimal contract as a function of the proportion of criminal elements in the economy. I introduce the particular fraud prevention technology in the section "Fraud Prevention." The section "Dynamic Considerations" briefly discusses dynamic considerations. The final section concludes and leaves room for further research.

ASSUMPTIONS AND SETUP

In an unemployment insurance framework--the setup for automobile fraud would be slightly different, but the articles main conclusions would be the same--suppose a one-shot game between a risk neutral principal and risk averse agents. An agent may be of two types: "Truth" and "Dare." Truths always tell the truth whereas Dares commit insurance fraud if they believe it is in their best interest. The proportion of Dares in the economy is given by [xi]. All agents, Truths and Dares alike, have the same VonNeumann-Morgenstern utility function over final wealth (with U'(*) > 0, U'(*) < and U'(0) = [infinity]), and the same initial wealth, Y. An agent may be employed or unemployed. If employed an agent receives labor income W, otherwise he has no income. The sequence of the game is presented in Figure 1.

[FIGURE 1 OMITTED]

In the first stage of the game agents are offered a menu of contracts that specify an unemployment insurance benefit [beta] and a premium p. I assume that the unemployment insurance contract is actuarially fair so that the premium (p) is exactly equal to the expected benefits paid in case of unemployment plus expenses due to fraud. As a result, the setup lets the principal design the contract so that she makes no profit in equilibrium. Moreover, the agents' utility is maximized.

An agent is unemployed with probability [pi] < 1/2. The agent's employment condition is unknown to the principal so that his possible actions are to request unemployment insurance benefits or not. The principal may then investigate the agent at cost c to acquire this information. (4) If caught committing fraud the agent must incur some penalty. The sunk cost penalty is represented as some fixed disutility k, (5) such as prison time or reputationnal loss. Finally, the payoffs are paid and the game ends.

OPTIMAL CONTRACT WHEN THE TYPE IS KNOWN

Truths

If each agent's type is known, then the principal can design a contract that targets each type of agent. It is then clear that the Truths will choose to be fully insured. Furthermore, their contract will be independent of the penalty k. I present this as my first proposition. (6)

Proposition 1: Under full information on the type, the optimal contract designed for the Truths is independent of the penalty.

The intuition behind this result is straightforward. Since the Truths always tell the truth, they can never be caught committing fraud. Therefore, they never incur the penalty.

Dares

The Dares' problem is more complicated. The payoffs to the Dares and to the principal contingent on all possible actions are displayed in Table 1.

The principal must specify a price-coverage contract pair that maximizes the Dares' expected utility subject to equilibrium strategy constraints.

It is clear from this setup that the equilibrium of the game is in mixed strategies. Moreover, the equilibrium is perfect Bayesian. Let [eta] be the probability that a Dare requests benefits when employed (i.e., the probability a Dare commits fraud), and let v be the probability of investigating an agent who requests benefits. In equilibrium, and v are given by

[eta] = ([pi]/1 - [pi])(c/[beta] - c) (1)

v = U(Y + W - p + [beta]) - U(Y + W - p)/ U(Y + W - p + [beta]) - U(Y + W - p) + k. (2)

Given those optimal strategies, (7) it is possible to find the price of an unemployment insurance policy that is the fairest to the agents. Given the cost of investigating an agent and the fact that some fraud goes undetected, the fair price of this contract is given by

p = [pi][beta] + (1 - [pi])[beta][eta]/(1 - v) + cv[[pi] + (1 - [pi])[eta]], (3)

where (1 - [pi])[beta][eta](1 - v) represents the expected amount that is extracted by agents who commit fraud, and cv[[pi] + (1 - [pi])[eta]] represents the amount spent on investigations.

The problem faced by the principal is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

subject to (1), (2), (3), and a participation constraint.

Clearly, the probability of fraud ([eta]) is independent of the premium (p). On the other hand, the probability the principal investigates (v) depends on the benefits and the premium. By designing the optimal contract (p, [beta]), the principal rationally anticipates its impact on the strategic behavior of the players. Although the parameter k appears in the function to maximize (4) as well as in constraint (2), Proposition 2 shows that it...

NOTE: All illustrations and photos have been removed from this article.




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