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Article Excerpt
I. INTRODUCTION
A body of literature spanning from medical ethics to public economics has amassed regarding the nonmarket rationing of public resources. A dominant focus has surrounded the merits and shortcomings of alternative rationing mechanisms as gauged by their effectiveness, distributive justness, and allocative efficiency. These instruments include administrative criteria such as need, merit, or seniority-based rules; queues or waiting lines, whereby access to the rationed good or service is determined by arrival times; and lotteries, whereby chance dictates the distribution of the good. While nonmarket rationing in any case may lead to the misallocation of resources compared to rationing solely by price (e.g., Glaeser and Luttmer 2003), empirical observation reveals that public agencies commonly distribute underpriced goods and services to individual consumers and firms.
The current study takes nonmarket rationing as given and investigates the properties of alternative lottery mechanisms. The use of lot teries dates to biblical times (Boyce 1994; Elster 1992), with modern examples including the military draft (Fienberg 1971), medical services (Cookson and Dolan 2000), recreational privileges on public lands and waterways (Scrogin 2005), residential housing following Hurricane Katrina (The White House, Office of the Press Secretary 2005), and public school admission and voucher programs (Berry-Cullen, Jacob, and Levitt 2006). From a design perspective, lotteries are intriguing because they can take a variety of forms, with each possessing unique incentive and revenue-generating properties. However, given the purported inefficiencies of lottery rationing, it is natural to question how they have sustained in allocative situations entailing public resources. Two explanations have emerged in the literature.
The first, historic explanation is attributed to the "fairness" of the classic lottery mechanism, in which the probability of being drawn is uniform and endogenous to the number of participants. Because entrants are equally likely to be drawn, the mechanism is perceived as being nondiscriminatory. Empirical evidence presented in Kemp and Bolle (1999) supports this hypothesis in the rationing of some types of goods and services to individual consumers. The fairness of lottery rationing is also consistent with the distributive objective of"specific egalitarianism" put forth by Tobin (1970) and discussed later by Elster (1992), whereby specific goods and services are allocated such that a greater degree of equality is attained relative to that of a competitive market. While egalitarianism and efficiency would appear to be at odds, Che and Gale (2007) demonstrated that lotteries can be more efficient than markets when individuals face binding wealth constraints. In a similar vein, Wijkander (1988) demonstrated that social welfare can be increased by lottery rationing under certain conditions.
A second, more objective explanation for the sustained use of lotteries is that public agencies may be legally constrained by price controls, particularly when rationing to individuals and households (Taylor, Tsui, and Zhu 2003). Acting as a self-sustaining regulated monopolist, the agency is confronted with selecting a rationing mechanism and a pricing policy that are both politically feasible and financially viable. Compared to need, merit, and seniority-based allocation rules, lotteries and queues are administratively simple and low cost to implement. Alternatively, lotteries eschew the waiting time and coordination costs confronted by potential entrants under queuing; however, the initial distribution under lottery rationing may prove allocatively inefficient. Collectively, the literature suggests that agency objectives may deviate from revenue maximization in distributing some types of scarce public resources, with lotteries proving to be superior in some settings to alternative nonmarket rationing mechanisms.
In addition to the variety of contexts in which lotteries have historically been employed, they have taken a variety of forms that can be distinguished by two fundamental characteristics. The first rests with the structure of the random component: the probability of being drawn. Lotteries are designed such that the probability is uniform over entrants, unique to particular groups of entrants, or entrant specific, whereby the probability is a function of individual merit, need, or seniority. The present study focuses upon the second distinguishing characteristic: the pricing arrangement. While underpricing is inherent to public lotteries, the price structure may range from uniform one-part tariffs to more elaborate two-part schemes consistent with discriminatory pricing. However, despite the growing role of lotteries in public resource and congestion management, price has received only tangential consideration in the literature (Boyce 1994; Mumy and Hanke 1975; Wijkander 1988), with many studies simply setting the price of entry and awarded units of the good to zero (Porter 1977; Seneca 1970; Taylor, Tsui, and Zhu 2003). (1) Similar to the probability of being drawn, price is a critical component of lottery design because its structure will affect entry and the associated probability of being drawn, the net benefits realized by entrants, and the revenues collected by the managing agency.
Two commonly used one-part pricing arrangements are investigated: the user-pay (UP) format, whereby a uniform tariff is incurred solely by the entrants who are awarded units of the good, and the all-pay (AP) format, whereby the tariff is required to enter and is nonrefundable to those who are not drawn. UP and AP pricing arrangements have an extensive history spanning from public finance (Buchanan 1960) to auction theory (Holt and Sherman 1982; Klemperer 2003) and are extended here to rationing by lottery. The UP and AP lotteries are formalized in the next section. In both cases, comparative statics indicate that across the price range over which the lotteries are binding, the expected value of the rationed good may increase (decrease) when price increases (decreases) if aggregate entry is sufficiently responsive though the AP entry function is inelastic at all prices. Further, under revenue-equivalent prices, the UP and AP lotteries are shown to attract an equal number of entrants and yield equal individual and aggregate expected consumer surplus. Selected results are numerically evaluated in Section II! with simulations (Taylor, Tsui, and Zhu 2003) performed across a broad class of distributions describing the population of individual private values. Section IV concludes.
II. UP AND AP LOTTERIES
To begin, several characteristics are assumed common between the lottery mechanisms. First, the supply of the rationed good is perfectly inelastic and the available quantity is divisible into Q* homogenous units. An individual may enter the lottery once and receive no more than one unit of the good, and awarded units are nontransferable. The inefficiency of lotteries is commonly attributed to their randomness; however, it is the underlying restriction of nontransferability that leads allocative inefficiency on average. The UP and AP lotteries possess a uniform and exogenously determined price denoted by P*, where P* < P(Q*]). The underpricing of units relative to the clearing rate, P* = P(Q*), leads to a shortage...
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