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Underpricing in public lotteries: a critique of user-pay and all-pay tariffs.

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 of units in both cases. Last, the probability of being drawn is assumed to be uniform over entrants.

The population of potential consumers comprises [Q.sub.N] risk-neutral utility maximizers. Individuals are assumed to possess unit demand for the good, and the private value of a unit to individual i is denoted by P([Q.sub.i]). Ordering individuals by their private values and assuming [Q.sub.N] is sufficiently large, the distribution of private values may be described by a continuous function P(Q), where P(0) is finite, P([Q.sub.N]) = 0, and dP/dQ < 0.

A. The UP Lottery

The UP lottery is characterized by a uniform tariff P* that is incurred solely by the Q* entrants who are drawn. For a given value of P*, equilibrium entry occurs such that zero expected value is attained by the lowest valued entrant. The entry function relating the equilibrium number of entrants to the tariff is denoted [Q.sup.UP.sub.D] (P*). This is equivalent to the inverse of the private value function P(Q) over the price range (0, P(Q*)). The uniform probability of being drawn is equal to [Q.sup.*]/[Q.sup.UP.sub.D]([P.sup.*]) and the expected value of a rationed unit to an entrant i is written:

(1) E([V.sup.UP.sub.i]) = [Q.sup.*]/[Q.sup.UP.sub.D] ([P.sup.*])]P([Q.sub.i] - [Q.sup.*]/[Q.sup.UP.sub.D] ([P.sup.*])] [P.sup.*].

The first term is the entrant-specific expected benefit, and the second term is the expected price. Integrating Equation (1) over entrants, the expected consumer surplus generated by the UP lottery is written:

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

The first term represents the aggregate expected benefits, and the second term represents the aggregate expenditures that flow as revenue to the public agency. Figure 1 depicts the UP lottery with P(Q) specified as a linear function P = P(0) - [alpha]Q and the probability of being drawn denoted by [phi]. The expected value of the lottery to entrant i is represented by the vertical segment (a, b), and expected consumer surplus is equal to the area of the triangle ([phi] P(0), [phi]P*, c).

Compared to rationing solely by price, the relationship between the price of the lottery and the individual benefits is less transparent. If P* decreases, then entry will increase, implying that positive expected values will be realized by a greater number of individuals. However, as entry increases, the probability of being drawn and the expected benefit derived by those who participated at the initial price will decrease. Alternatively, if P* increases, then negative expected values will be realized by some individuals who participated at the initial price, in which case entry will decrease, the probability of being drawn will increase, and those who remain in the lottery will experience an increase in expected benefits. In either case, whether or not the expected value rises or falls will depend upon the elasticity of entry with respect to price.

[FIGURE 1 OMITTED]

The individual effects of price changes can be examined with simple comparative statics. The partial derivative of the expected value function with respect to the price is written:

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

The first set of terms represents the effect on expected benefits. The second and third sets of terms comprise the effect on the expected price. Rewriting Equation (3), the effect of changes in P* can be summarized in terms of the elasticity of entry:

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

The expression states that individual expected values will increase (decrease) as the lottery price increases if the elasticity of entry is greater (less) than the reciprocal of the percentage difference between the individual private value and the lottery price. Intuitively, because the size of the final term is inversely related to the size of P([Q.sub.i]), if gains in expected value result from an increase in P*, then they are necessarily realized by individuals in the upper end of the private value distribution and diminish to zero as p(Q) [right arrow] P*. (2)

B. The AP Lottery

The AP lottery is characterized by a nonrefundable tariff P* that is required to participate. Equilibrium entry again occurs such that zero expected value is realized by the lowest valued entrant. Denoting the equilibrium number of entrants by [Q.sup.AP.sub.D]([P.sup.*]), the probability of being drawn is equal to [Q.sup.*]/[Q.sup.AP.sub.D]([P.sup.*]) and the expected value of the lottery to entrant i is written:

(5) E([V.sup.AP.sub.i] = [Q.sup.*]/[Q.sup.AP.sub.D]([P.sup.*])P([Q.sub.i]) - [P.sup.*].

[FIGURE 2 OMITTED]

Compared to the UP lottery, equilibrium entry occurs such that [Q.sup.AP.sub.D]<[Q.sup.UP.sub.D] [for all] [P.sup.*] < P([Q.sup.*]). Thus, while the private value function P(Q) characterizes entry in the UP lottery, it overstates entry in the AP lottery. The entry function in the AP lottery is implicitly defined by the equilibrium condition E([V.sup.AP.sub.i]) = 0, and its properties may be evaluated with the implicit function rule (see, e.g., Silberberg and Suen 2001). (3)

Integrating Equation (5) over entrants, the expected consumer surplus is written:

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

The AP lottery is depicted in Figure 2 with the linear private value function and values of Q* and P* depicted in Figure 1. The expected value of the lottery to entrant i is equal to the vertical segment (a, b), and expected consumer surplus is equal to the area of the triangle ([phi]bP(0), P*, c), where [phi] represents the probability of being drawn and c is one point on the entry function.

Similar to the UP lottery, the effects of changes in P* on individual expected values will depend upon the elasticity of entry. Formally, the partial derivative of Equation (5) is written:

(7) [partial derivative]E([V.sup.AP.sub.i])/[partial derivative][P.sup.*] = - [Q.sup.*]/([Q.sup.AP.sub.D]).sup.2]] (d[Q.sup.AP.sub.D]/d[P.sup.*])P([Q.sub.i] - 1.

The first set of terms represents the indirect effect on individual expected benefits that results from the response of aggregate entry, and the second term represents the direct effect of the price change. The effect of price changes on individual expected values is summarized as:

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

The expression states that the expected value will increase (decrease) following an increase in P* if the elasticity of entry is greater (less) than the ratio of the tariff to the expected benefit. Similar to the UP lottery, as the entrant-specific term is an inverse function of the private value, it follows that if gains in expected value are attainable following an increase in P*, they will necessarily be realized by individuals in the upper end of P(Q) and diminish to zero as P(Q) [right arrow] P*.

Note that Equation (8) also reveals a unique property of the AP entry function. Specifically, because participation occurs such that the expected benefit derived by the lowest valued entrant is equal to P*, the final entrant-specific term is equal to 1, and the expected value derived by the entrant necessarily declines if the tariff increases (i.e.,[partial derivative]E([V.sup.AP.sub.i])/ [partial derivative][P.sup.*] < 0). Given that points along the AP entry function correspond to the lowest valued entrant at each value of the tariff, Equation (8) implies that the price elasticity is less than 1. As the result holds at all prices between zero and the clearing rate, it follows that the AP entry function is inelastic at all points. Alternatively, using Q/dP derived from the implicit function rule (3), the elasticity of entry may be shown to be equal to P*/[P* (1 - P'(Q))] < 1. Revenues in the AP lottery are therefore an increasing function of the tariff. (4)

C. Revenue Equivalency and Lottery Outcomes

Thus far, the analysis has focused separately upon the absolute outcomes of the lotteries over the range of tariff rates (i.e., 0 < P* < P(Q*)). As an understanding of the relative outcomes of the lotteries will be invaluable for selecting a format in practice, for completeness this section concludes with a formal comparison of the mechanisms. To the extent that budget considerations may dictate the tariff rate imposed by the managing agency, the lottery outcomes are evaluated at revenue equivalent values of the tariffs. Letting [P.sup.*.sub.UP] and [P.sup.*.sub.AP] denote the UP tariff and AP tariff, respectively, revenue equivalency implies [P.sup.*.sub.UP] [Q.sup.*] = [P.sup.*.sub.AP] [Q.sup.AP.sub.D]. The revenue-equivalent tariff rates are written:

(9) [P.sup.*.sub.UP] = [P.sup.*.sub.AP]([Q.sub.AP.sub.D]/[Q.sup.*]) and [P.sup.*.sub.AP] = [P.sup.*.sub.UP] ([Q.sup.*]/[Q.sup.AP.sub.D]).

As [Q.sup.AP.sub.D]> Q., it follows that [P.sup.*.sub.UP] > [P.sup.*.sub.AP]. Thus, greater surplus value is necessarily derived ex post by any randomly drawn set of entrants in the AP lottery relative to the UP lottery. Whether the UP or the AP format yields greater ex ante individual and aggregate expected surplus values is dependent upon the relative number of entrants. Assuming revenue equivalency, the ex ante lottery outcomes are summarized in the following propositions.

PROPOSITION 1. Given the private value function P(Q) and assuming revenue equivalency, aggregate entry in the UP lottery equals aggregate entry in the AP lottery: [Q.sup.UP.sub.D] = [Q.sup.AP.sub.D].

Proof Let k and j reference, respectively, the lowest valued UP entrant and the lowest valued AP entrant under revenue-equivalent values of the tariffs. Thus, for individual k in the UP lottery:

(10) E([V.sup.UP.sub.k]) = ([Q.sup.*]/[Q.sup.UP.sub.D])P([Q.sub.k]) - ([Q.sup.*]/[Q.sup.UP.sub.D]) [P.sup.*.sub.UP] = 0.

Substituting the UP tariff [P.sup.*.sub.UP] from Equation (9), it follows that:

(11) E([V.sup.UP.sub.k]) = ([Q.sup.*]/Q.sup.UP.sub.D])[P([Q.sub.k]) - [P.sup.*.sub.AP]([Q.sup.AP.sub.D]/[Q.sup.*])] = 0.

The condition is satisfied if the term in brackets is zero. Hence:

(12) E([V.sup.UP.sub.k]) = 0 [right arrow], P([Q.sub.k]) = [P.sup.*.sub.AP]([Q.sup.AP.sub.D]/[Q.sup.*])or ([Q.sup.*]/[Q.sup.AP.sub.D])P([Q.sub.k]) = [P.sup.*.sub.AP].

Thus, the expected benefit derived by individual k under the AP format is equal to the tariff, which is true only for the AP entrant possessing the lowest private value (i.e., individual j).

Conversely, for individual j in the AP lottery:

(13) E([V.sup.AP.sub.j]) = ([Q.sup.*]/[Q.sup.AP.sub.D])P([Q.sub.j]) - [P.sup.*.sub.AP] = 0.

Substituting the AP tariff [P.sup.*.sub.AP] from Equation (9), it follows that:

(14) E([V.sup.AP.sub.j]) = ([Q.sup.*]/[Q.sup.AP.sub.D])[P([Q.sub.j]) - [P.sup.*.sub.UP]] = 0.

The condition is satisfied if the term in brackets is zero. Hence:

(15) E([V.sup.AP.sub.j]) = 0 [right arrow] P([Q.sub.j]) = [P.sup.*.sub.UP]

The expression states that individual j's private value is equal to the UP tariff, which is true only for the lowest valued entrant in the UP lottery (i.e., individual k). Therefore, k = j and it follows that [Q.sup.UP.sub.D] = [Q.sup.AP.sub.D].

PROPOSITION 2. Given the private value function P(Q) and assuming revenue equivalency, the expected value of the UP lottery is equal to the expected value of the AP lottery for all entrants: E([V.sup.UP.sub.i]) = E([V.sub.AP.sub.i])

[for all] i.

Proof For a given entrant i, the expected value of the UP lottery is:

(16) E([V.sup.UP.sub.i]) = ([Q.sup.*]/[Q.sup.UP.sub.D])P([Q.sub.i]) - ([Q.sup.*]/[Q.sup.UP.sub.D]) [P.sup.*.sub.UP].

Substituting the revenue-equivalent UP tariff [P.sup.*.sub.UP] p from Equation (9):

(17) E([V.sup.UP.sub.i]) = ([Q.sup.*]/[Q.sup.UP.sub.D])P([Q.sub.i]) - ([Q.sup.AP.sub.D]/[Q.sup.UP.sub.D)[P.sup.*.sub.AP].

AS [Q.sup.UP.sub.D] = [Q.sup.AP.sub.D] (Proposition 1), it follows that the UP expected value is equal to the AP expected value for an entrant i and thus for all entrants:

(18) E([V.sup.UP.sub.i]) = ([Q.sup.*]/[Q.sup.AP.sub.D])P([Q.sub.i]) - [P.sup.*.sub.AP] = E([V.sup.AP.sub.i]) [for all]i.

Conversely, the expected value of the AP lottery is:

(19) E([V.sup.AP.sub.i]) = ([Q.sup.*]/[Q.sup.AP.sub.D])P([Q.sub.i]) - [P.sup.*.sub.AP].

Substituting the revenue-equivalent AP tariff [P.sup.*.sub.AP] from Equation (9) and noting that [Q.sup.UP.sub.D] = [Q.sup.AP.sub.D], it follows that the AP expected value is equal to the UP expected value for an entrant i and thus for all entrants:

(20) E([V.sup.AP.sub.i] =([Q.sup.*]/[Q.sup.UP.sub.D])P([Q.sub.i]) - ([Q.sup.*]/[Q.sup.UP.sub.D]) [P.sup.*.sub.UP] = E([V.sup.UP.sub.i])[for all]i.

From Propositions 1 and 2, it follows that the aggregate expected consumer surplus generated by the UP lottery (Equation 2) is equal to that generated by the AP lottery (Equation 6). Further, given revenue equivalency and Proposition 2, [P.sup.*.sub.UP] and [P.sup.*.sub.AP] may be interpreted in terms of certainty equivalence. Specifically, [P.sup.*.sub.AP] represents the value of the AP tariffincurred with certainty that yields identical expected value to that attainable at the tariff rate [P.sup.*.sub.UP] incurred with probability [Q.sup.*][P.sup.*.sub.AD] in the UP lottery.

Given that [Q.sup.UP.sub.D] = [Q.sup.AP.sub.D] (Proposition 1), the probability an entrant is awarded a unit of the good is equal between the lottery formats. For any randomly drawn set of [Q.sup.*] entrants, the difference between the realized, ex post surplus value in the AP lottery relative to the UP lottery is a linear function of the difference between the UP and the AP tariffs:

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

It is straightforward to confirm that Equation (21) is equal to the tariff revenues collected from the AP entrants who are not drawn. Decomposing the AP entrants into sets of successful and unsuccessful entrants and multiplying by the tariff, it follows that:

(22) [P.sup.*.sub.AP] [Q.sup.AP.sub.D] = [P.sup.*.sub.AP] [Q.sup.*] + [P.sup.*.sub.AP]([Q.sup.AP.sub.D] - [Q.sup.*].

The first term on the right hand side represents the tariff revenues collected from the set of AP entrants awarded units, and the second term represents the tariff revenues collected from the set of unsuccessful entrants. Substituting the revenue-equivalent AP tariff from Equation (9) and rearranging:

(23) [Q.sup.*]([P.sup.*.sub.UP] - [P.sup.*.sub.AP]) = ([Q.sup.AP.sub.D] - [Q.sup.*].

The result states that the excess consumer surplus realized ex post in the AP lottery relative to the UP lottery is equal to the expenditures made by entrants who failed to be drawn. It follows that under revenue equivalency, the tariff payments made by unsuccessful entrants directly subsidize the net benefits derived by the random set of AP entrants awarded units of the good.

III. LOTTERY SIMULATIONS

To demonstrate how the lottery outcomes may be numerically evaluated across a broad class of distributions describing the population of individual private values, numerical simulations are performed similar to Taylor, Tsui, and Zhu (2003). Three findings from the previous section are investigated. First, comparative statics indicated that individual expected values may increase as price increases (Equations 4 and 8). While the case of linear private values was shown analytically to be an exception, (2,4) the effects in the general case of nonlinear value functions are examined. Second, as revenue equivalency implies [P.sup.*.sub.UP] > [P.sup.*.sub.AP], the relative values of the tariffs necessary to generate specific amounts of revenues are measured. Third, given that the realized ex post consumer surplus attained by a randomly drawn set of entrants in either lottery depends upon the respective tariff rate (Equation 21), the disparity between the surplus values produced by the AP lottery relative to the UP lottery is evaluated across the range of revenue-equivalent tariffs.

Similar to Taylor, Tsui, and Zhu (2003), the population of individual private values is assumed to be characterized by the beta density function. The density is defined over the unit interval (0, 1) and possesses two "shaping" parameters (a, b) that allow it to take a variety of forms, ranging from uniform to approximately normal to negatively (left) or positively (right) skewed (see, e.g., Wackerly, Mendenhall, and Scheaffer 1996). Formally, a random variable x is beta distributed if its density function is:

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

The density is symmetric if a = b, with uniformity resulting when a = b = 1, and the density is mound shaped if a = b > 1, positively skewed if a < b, and negatively skewed if a > b. Interpreted with respect to individual private values, as the distribution skews positively (negatively) from uniformity, the proportion of relatively low (high)--valued individuals in the population increases, and as a result, the total social value of rationed good decreases (increases).

To generate the private value function P(Q), random samples are drawn from Equation (24) evaluated at seven combinations of the shaping parameters (a, b). The benchmark is the uniform case (i.e., linear P(Q)) treated in the previous section; two negatively skewed, two positively skewed, and two mound-shaped densities are also evaluated. In each case, 1,000 random samples of 100,000 values were drawn. For a given sample, P(Q) is approximated by arranging the 100,000 sample points in descending order, assigning consecutive integers to the ordered points (i.e., [Q.sub.i] = 1, 2, 3, ....), and then scaling [Q.sub.i] to the unit interval (0, 1) by dividing the integers by 100,000. The function is linear in the uniform case, convex in the positively skewed cases, concave in the negatively skewed cases, and convex (concave) below (above) the midpoint in the nonuniform symmetric (i.e., mound-shaped) cases. Table 1 reports the means of the descriptive statistics calculated from the 1,000 x 7 random samples. With the symmetric densities (a = b), the mean private values are 0.50 as expected and decrease (increase) sharply as the density skews positively (negatively) from uniformity. To proceed, the quantity of the good to be rationed must be specified. I assume that [Q.sup.*] = 0.10, indicating that the available quantity will satisfy a maximum of 10% of the population (i.e., 10,000 entrants) at a price of zero. (5)

The earlier results (Equations 4 and 8) indicated that for a given distribution of private values, the expected value of the UP and AP lotteries may increase (decrease) for some entrants following an increase in the tariff if the exogenous response of aggregate entry is relatively large. The uniform case was shown to be an exception, and the simulation results support this finding with the UP lottery (Table 2) and AP lottery (Table 3) as the mean proportion of entrants who experience an increase in expected values is statistically insignificant ([alpha] < 0.01) at all values of the tariffs over which the lotteries are binding. Similar results are observed when the distribution of private values is left skewed (a > b). In contrast, as the distribution skews positively from uniformity (a < b), statistically significant increases in expected values are consistently attained by about one-third of entrants in the UP and AP lotteries. And when P(Q) possesses both convex and concave segments (a = b > 1), significant increases in expected values are realized over the entire range of tariffs in the AP lottery and the convex range of tariffs (i.e., P* [greater than or equal to] 0.50) in the UP lottery.

The UP and AP lotteries were shown to satisfy revenue equivalency when the AP tariff is equated to the expected UP tariff: [P.sup.*.sub.AP]= ([Q.sup.*]/[Q.sup.UP.sub.D])[P.sup.*.sub.UP]. While entry under either lottery format at a given tariff rate is dependent upon the distribution of individual private values, the tariffs converge in all cases as [P.sup.*.sub.UP] [right arrow] P([Q.sup.*]). To evaluate the disparity between the revenue-equivalent tariff rates resulting with the simulated populations, the relative value [P.sup.*.sub.UP/[P.sup.*.sub.AP] is calculated from each random sample across the range of tariffs between zero and P(Q*). The mean values are reported in Table 4. In all cases, the ratio approaches 1 at a decreasing rate as the tariffs approach the clearing rate. With linear private values (a = b = 1), the results indicate that the UP tariff will exceed the revenue-equivalent AP tariffby as much as five times. And relative to the linear case, the disparity between the tariff rates increases (decreases) with the degree of concavity (convexity) of P(Q). Considering Proposition 2, the results suggest that if the revenues targeted by the agency are small relative to the revenues generated under price rationing (i.e., Q*P(Q*)), then the UP tariff that satisfies the target will be comparatively large relative to when the target approximates the maximum revenue Q*P(Q*).

Last, the surplus values realized in the AP lottery relative to the UP lottery are evaluated assuming revenue equivalency. Given the distribution of private values and tariff rates, the ex post surplus values are a function of the private values possessed by the randomly drawn entrants (see Equation 21). As ([Q.sup.AP.sub.D]) combinations can be formed by sampling Q* individuals from the set of [Q.sup.AP.sub.D] (or [Q.sup.UP.sub.D]) entrants, consumer surplus is calculated as the mean over all combinations. For a given sample of individual private values drawn from Equation (24), the mean surplus value realized under lottery format L may be written:

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

Evaluated at the boundary values of the lottery prices (i.e., zero and P(Q*)), the surplus values are identical between formats, and at all prices within the boundary values, [[bar.CS].sup.AP] > [[bar.CS].sup.UP].

The relative surplus values [[bar.CS].sup.AP] / [[bar.CS].sup.UP] are calculated over the range of revenue-equivalent tariffs with each random sample. The mean values over the samples are reported in Table 5. In all cases, as [P.sup.*.sub.AP] increases from zero to P(Q*), the relative surplus values increase, reach a maximum, and then decrease to 1. In the uniform case (a = b = 1), the mean AP surplus exceeds that of the UP lottery by as much as five times. And the relative disparity between the AP and the UP surplus values increases (decreases) as the distribution skews negatively (positively) from uniformity. Most notably, the results indicate that if the distribution of private values is negatively skewed (i.e., P(Q) is convex), the AP lottery will generate between 10 and 20 times more surplus value than the UP lottery in the intermediate ranges of the AP tariff.

In summary, the results suggest that public agencies confronted with lottery adoption will necessarily face trade-offs in choosing between the UP and the AP formats. If the AP format is selected, then the [Q.sup.AP.sub.D] - [Q.sup.*] entrants who fail to be drawn necessarily derive zero benefits from the tax payment [P.sup.*.sub.AP]. Yet, relative to the UP lottery, the revenues collected from this randomly determined group of entrants directly subsidize the net benefits derived by the [Q.sup.*] entrants awarded units of the good. Alternatively, if the UP format is selected, then by design the tax payment [P.sup.*.sub.UP] is restricted to those individuals who derive benefits from the good. However, as achieving a given revenue target requires that [P.sup.*.sub.UP] > [P.sup.*.sub.AP] , the net benefits realized by successful UP entrants will necessarily be less than the net benefits attainable in the AP lottery.

IV. CONCLUSIONS

Taking nonmarket rationing as given, the present study examined the structure of underpricing in lotteries for rationing goods and services to the public. Focusing upon the UP and AP lottery formats, the theoretical results and accompanying simulation analysis indicated that the expected value of the lottery may increase for relatively high valued entrants when the tariff is increased but will always decline for a set of relatively low-valued entrants. Comparison of the lotteries under revenue equivalency reveals that the UP and AP formats are outcome equivalent with respect to aggregate entry, individual expected values, and aggregate expected consumer surplus. However, the surplus values realized ex post by successful entrants are necessarily greater in the AP lottery.

As a concluding note, considering the ongoing adoption of lotteries for rationing goods and services to the public and the recurrent budget shortfalls faced by many state and federal programs, continued research on the design of lottery rationing mechanisms is warranted. Three extensions follow directly from the present study. First, while the UP and AP revenue functions were evaluated here, introducing the costs of provision and lottery administration would permit the welfare-maximizing tariffs to be characterized similar to that of the optimal capacity (Q*) in Mumy and Hanke (1975). Second, two-part tariff arrangements associated with discriminatory pricing may readily be derived in the context of lotteries as a combination of the UP and AP tariffs. Accounting for the costs of provision and lottery administration, their properties may be characterized relative to the one-part formats examined in this study (e.g., for welfare maximization). Last, given that experimental investigation of lotteries to date has focused upon their use in funding public goods, rationing lotteries remain under investigated in the literature. Yet, considering their role in revenue generation and the outcome equivalence of the mechanisms proven here, experimental investigation of rationing lotteries may be a fruitful avenue for future research.

ABBREVIATIONS

AP: All Pay

UP: User Pay

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Porter, R. C. "On the Optimal Size of Underpriced Facilities." American Economic Review, 67, 1977, 753-60.

Scrogin, D. "Lottery-Rationed Public Access under Alternative Tariff Arrangements: Changes in Quality, Quantity, and Expected Utility." Journal of Environmental Economics and Management, 50, 2005, 189-211.

Seneca, J. J. "The Welfare Effects of Zero Pricing of Public Goods." Public Choice, 8, 1970, 101-10.

Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Approach. 3rd ed. Boston, MA: Irwin McGraw-Hill, 2001.

Taylor, G. A., K. K. Tsui, and L. Zhu. "Lottery or Waiting-Line Auction?" Journal of Public Economics, 87, 2003, 1313-34.

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Tobin, J. "On Limiting the Domain of Inequality." Journal of Law and Economics, 13, 1970, 263-77.

Wackerly, D. D., W. Mendenhall III, and R. L. Scheaffer. Mathematical Statistics w#h Applications. 5th ed. Belmont, CA: Wadsworth Publishing Company, 1996.

Wijkander, H. "Equity and Efficiency in Public Sector Pricing: A Case for Stochastic Rationing." Econometrica, 56, 1988, 1455-65.

(1.) In contrast, empirical analyses of public lotteries have consistently given the pricing issue explicit recognition (see, e.g., Buschena, Anderson, and Leonard 2001; Scrogin 2005), in part because price serves a critical role in estimating the demand for and benefits derived from nonmarket goods and services (e.g., compensating variation and consumer surplus).

(2.) Continuing t he example oflinear private values (see Figure 1), the expected value of the lottery to an entrant/may be written E([V.sup.AP.sub.i]) = {[alpha][Q.sup.*]/[P(0) - [P.sup.*]}[P([Q.sub.i]) - [P.sup.*]]. From either Equation (4) or Equation (5), it is straightforward to show [partial derivative]E([V.sup.UP.sub.i])/[partial derivative] [P.sup.*] = - [[alpha].sup.2][Q.sup.*][Q.sub.i]/[P(0) - [[P.sup.*].sup.2] < 0. Thus, for every entrant, the expected value will decrease (increase) following an increase (decrease) in P*, and therefore, the expected consumer surplus will vary inversely with P*.

(3.) For example, defining the equilibrium condition by the implicit function F(Q, P) = 0 and [F.sub.Q] and [F.sub.p] as its partial derivatives, it can be shown that dQ/dP = - [F.sub.p]/[F.sub.Q] = - [Q.sup.2]/[[Q.sup.*](p(Q) - QP'(Q))] < 0.

(4.) In the case of linear private values depicted in Figures 1 and 2 (i.e., P = P(0) - [alpha]Q), the AP entry function may be expressed by [Q.sup.AP.sub.D] = [Q.sup.*]P(0)/ ([P.sup.*] + [alpha][Q.sup.*]), [for all][P.sup.*] [less than or equal to] P([Q.sup.*]). The expected value function is written E([V.sup.AP.sub.i]) = [([P.sup.*] + [alpha][Q.sup.*])/P(O)]P([Q.sub.i]) - [P.sup.*] and dE([V.sup.AP.sub.i])/d[P.sup.*] = - [alpha][Q.sub.i]/P(0)<0. Thus, similar to the UP lottery, individual expected values and aggregate expected consumer surplus will necessarily decrease (increase) if [P.sup.*] increases (decreases) when private values are linear.

(5.) While this degree of scarcity may be viewed as being consistent with the public choice of a lottery over a pure pricing mechanism, it is intended here to permit the lotteries to be evaluated across a relatively wide range of tariff rates.

DAVID SCROGIN *

* I am grateful to two anonymous reviewers and a coeditor and to Michael Caputo, Glenn Harrison, Kyung So Im, and Lisa Rutstrom for helpful comments. An earlier version of the paper was presented at the University of Central Florida and the 2005 meetings of the American Agricultural Economics Association. All errors are my own.

Scrogin: Associate Professor, Department of Economics, University of Central Florida, Orlando, FL 328161400. Phone 1-407-823-4129, Fax 1-407-823-3269, E-mail dscrogin@bus.ucf.edu
TABLE 1
Beta-Distributed Individual Private Values: Summary Statistics
on 1,000 Random Samples

 a=1,b=1 a=3,b=1
Description Uniform Skewed negative

Number of samples 1,000 1,000
Sample sizes 100,000 100,000
Mean private values (a) 0.50 (0.001) 0.75 (0.001)
Standard deviation (a) 0.29 (0.0004) 0.19 (0.0005)
Skewness (a) 0.00003 (0.01) -0.86 (0.01)
Minimum (a) 9.85e-06 (0.00001) 0.02 (0.01)
Maximum (a) 1.00 (0.00001) 1.00 (3.17e-06)

 a=5,b=1 a=1,b=3
Description Skewed negative Skewed positive

Number of samples 1,000 1,000
Sample sizes 100,000 100,000
Mean private values (a) 0.86 (0.0004) 0.25 (0.001)
Standard deviation (a) 0.14 (0.0004) 0.19 (0.0004)
Skewness (a) -1.18 (0.001) 0.86 (0.01)
Minimum (a) 0.09 (0.02) 3.34e-06 (3.37e-06)
Maximum (a) 1.00 (1.91e-06) 0.98 (0.01)

 a=1,b=5 a=3,b=3
Description Skewed positive Mound shaped

Number of samples 1,000 1,000
Sample sizes 100,000 100,000
Mean private values (a) 0.17 (0.0004) 0.50 (0.001)
Standard deviation (a) 0.14 (0.0004) 0.19 (0.0003)
Skewness (a) 1.18 (0.01) -0.0002 (0.005)
Minimum (a) 1.99e-06 (2.01e-06) 0.01 (0.003)
Maximum (a) 0.91 (0.02) 0.99 (0.003

 a=5,6=5
Description Mound shaped

Number of samples 1,000
Sample sizes 100,000
Mean private values (a) 0.50 (0.0005)
Standard deviation (a) 0.15 (0.0003)
Skewness (a) -0.0002 (0.005)
Minimum (a) 0.04 (0.01)
Maximum (a) 0.96 (0.01)

(a) The reported results are means (standard deviations)
of the descriptive statistics calculated for each random sample.

TABLE 2
Effects of Changes in the UP Lottery Price on Individual
Expected Values

 Proportion of UP Entrants Deriving
 an Increase in Expected Value from
 an Increase in Price:
 [partial derivative]([V.sup.UP.sub.i]/
 [partial derivative] ([P.sup.*.sub.UP]) > 0

UP Price
([P.sup.*.
sub.UP]) a = 1, 6 = 1 a = 3, b = 1 a = 5, b = 1

O.10 0.04 (0.07) 0.00 (0.00) 0.00 (0.00)
0.20 0.05 (0.08) 0.00 (0.00) 0.00 (0.00)
0.30 0.05 (0.07) 0.00 (0.00) 0.00 (0.00)
0.40 0.04 (0.07) 0.00 (0.00) 0.00 (0.00)
0.50 0.04 (0.07) 0.00 (0.00) 0.00 (0.00)
0.60 0.05 (0.07) 0.00 (0.00) 0.00 (0.00)
0.70 0.04 (0.07) 0.00 (0.00) 0.00 (0.00)
0.80 0.04 (0.07) 0.003 (0.02) 0.00 (0.00)
0.90 0.05 (0.07) 0.03 (0.06) 0.003 (0.02)
0.96-0.98 -- 0.03 (0.06) 0.03 (0.06)

 Proportion of UP Entrants Deriving
 an Increase in Expected Value from
 an Increase in Price:
 [partial derivative]([V.sup.UP.sub.i]/
 [partial derivative] ([P.sup.*.sub.UP]) > 0

UP Price
([P.sup.*.
sub.UP]) a = 1, b = 3 a = 1, b = 5

O.10 0.29 ** (0.05) 0.32 ** (0.06)
0.20 0.29 ** (0.05) 0.32 ** (0.06)
0.30 0.30 ** (0.05) 0.32 ** (0.06)
0.40 0.29 ** (0.05) 0.32 ** (0.06)
0.50 0.29 ** (0.06) --
0.60 0.29 ** (0.06) --
0.70 -- --
0.80 -- --
0.90 -- --
0.96-0.98 -- --

 Proportion of UP Entrants Deriving
 an Increase in Expected Value from
 an Increase in Price:
 [partial derivative]([V.sup.UP.sub.i]/
 [partial derivative] ([P.sup.*.sub.UP]) > 0

UP Price
([P.sup.*.
sub.UP]) a = 3, b = 3 a = 5, b = 5

O.10 0.00 (0.00) 0.00 (0.00)
0.20 0.00 (0.00) 0.00 (0.00)
0.30 O.O1 (0.02) 0.001 (0.003)
0.40 0.09 (0.06) 0.07 (0.05)
0.50 0.17 * (0.07) 0.18 * (0.07)
0.60 0.22 ** (0.07) 0.25 ** (0.06)
0.70 0.25 ** (0.07) 0.28 ** (0.06)
0.80 0.25 ** (0.06) --
0.90 -- --
0.96-0.98 -- --

Notes: Numbers in parentheses are standard deviations.
** and * denote significance at the 0.01 and 0.05 levels,
respectively.

TABLE 3
Effects of Changes in the AP Lottery Price on Individual Expected Values

 Proportion of AP Entrants Deriving an Increase
 in Expected Value from an Increase in Price:
 [partial derivative]([V.sup.UP.sub.i]/
 [partial derivative] ([P.sup.*.sub.UP]) > 0
AP Price
([P.sup.*.
sub.AP]) a = l, b = 1 a = 3, h = 1 a = 5, 6 = 1

0.10 0.04 (0.07) 0.00 (0.00) 0.00 (0.00)
0.20 0.04 (0.08) 0.003 (0.02) 0.001 (0.01)
0.30 0.05 (0.07) 0.01 (0.04) 0.01 (0.03)
0.40 0.05 (0.07) 0.02 (0.05) 0.02 (0.04)
0.50 0.05 (0.07) 0.03 (0.06) 0.03 (0.05)
0.60 0.05 (0.07) 0.04 (0.06) 0.04 (0.06)
0.70 0.05 (0.07) 0.05 (0.07) 0.05 (0.07)
0.80 0.06 (0.07) 0.06 (0.07) 0.08 (0.09)
0.90 0.06 (0.07) 0.08 (0.09) 0.10 (0.09)
0.96-0.98 -- 0.08 (0.09) 0.12 (0.10)

 Proportion of AP Entrants Deriving an Increase
 in Expected Value from an Increase in Price:
 [partial derivative]([V.sup.UP.sub.i]/
 [partial derivative] ([P.sup.*.sub.UP]) > 0
AP Price
([P.sup.*.
sub.AP]) a = 1, b = 3 a = 1, h = 5

0.10 0.30 ** (0.05) 0.33 ** (0.05)
0.20 0.30 ** (0.05) 0.34 ** (0.05)
0.30 0.30 ** (0.05) 0.33 ** (0.05)
0.40 0.30 ** (0.05) 0.34 ** (0.05)
0.50 0.30 ** (0.05) --
0.60 0.31 ** (0.05) --
0.70 -- --
0.80 -- --
0.90 -- --
0.96-0.98 -- --

 Proportion of AP Entrants Deriving an Increase
 in Expected Value from an Increase in Price:
 [partial derivative]([V.sup.UP.sub.i]/
 [partial derivative] ([P.sup.*.sub.UP]) > 0
AP Price
([P.sup.*.
sub.AP]) a = 3, h = 3 a = 5, h = 5

0.10 O.18 ** (0.06) 0.19 ** (0.06)
0.20 0.23 ** (0.06) 0.26 ** (0.05)
0.30 0.25 ** (0.05) 0.27 ** (0.05)
0.40 0.26 ** (O.O5) 0.29 ** (0.05)
0.50 0.27 ** (0.05) 0.29 ** (0.05)
0.60 0.27 ** (0.06) 0.30 ** (0.05)
0.70 0.28 ** (0.06) 0.31 ** (0.05)
0.80 0.28 ** (0.05) --
0.90 -- --
0.96-0.98 -- --

Notes: Numbers in parentheses are standard deviations.
** denotes significance at the 0.01 level.

TABLE 4
Relative Values of the Revenue-Equivalent UP Lottery Price
([P.sup.*sub.UP]/[P.sup.*.sub.AP])

AP Price
(P.sup.*.
sub.AP]) a = 1, b = 1 a = 3, 6 = 1 a = 5, b = 1

0.10 5.00 (0.008) 6.82 (0.006) 7.55 (0.005)
0.20 3.33 (0.005) 4.16 (0.003) 4.46 (0.002)
0.30 2.50 (0.004) 2.96 (0.002) 3.10 (0.001)
0.40 2.00 (0.003) 2.29 (0.001) 2.37 (0.001)
0.50 1.67 (0.002) 1.87 (0.001) 1.92 (0.001)
0.60 1.43 (0.002) 1.57 (0.001) 1.61 (0.001)
0.70 1.25 (0.001) 1.36 (0.001) 1.39 (0.001)
0.80 1.11 (0.001) 1.20 (0.001) 1.22 (0.001)
0.90 1.00 (0.000) 1.07 (0.001) 1.09 (0.001)
0.96-0.98 -- 1.00 (0.008) 1.00 (0.000)

AP Price
(P.sup.*.
sub.AP]) a = 1, h = 3 a = 1, b = 5

0.10 3.18 (0.006) 2.45 (0.005)
0.20 2.05 (0.004) 1.55 (0.004)
0.30 1.54 (0.003) 1.16 (0.003)
0.40 1.25 (0.003) 1.00 (0.008)
0.50 1.05 (0.002) --
0.60 1.00 (0.000) --
0.70 -- --
0.80 -- --
0.90 -- --
0.96-0.98 -- --

AP Price
(P.sup.*.
sub.AP]) a = 3, b = 3 a = 5,6 = 5

0.10 5.00 (0.006) 5.00 (0.005)
0.20 3.04 (0.003) 2.94 (0.003)
0.30 2.20 (0.002) 2.10 (0.002)
0.40 1.73 (0.002) 1.64 (0.002)
0.50 1.43 (0.002) 1.35 (0.001)
0.60 1.22 (0.001) 1.15 (0.001)
0.70 1.07 (0.001) 1.00 (0.001)
0.80 1.00 (0.000) --
0.90 -- --
0.96-0.98 -- --

Notes: Numbers in parentheses are standard deviations. With the
exception of the left-skewed beta densities, the reported clearing
rates are rounded upward to the nearest incremental value of the
tariff.

TABLE 5
Average Realized Consumer Surplus in the AP Lottery Relative
to UP Lottery

AP Price
(P.sup.*.
sub.AP]) a = 1, b = 1 a = 3, 6 = 1 a = 5, b = 1

0.10 2.60 (0.01) 4.26 (0.01) 5.52 (0.02)
0.20 3.80 (0.02) 8.27 (0.04) 12.52 (0.06)
0.30 4.60 (0.03) 11.26 (0.07) 17.78 (O.11)
0.40 5.00 (0.04) 13.08 (0.09) 21.07 (0.16)
0.50 5.00 (0.04) 13.72 (0.11) 22.37 (0.19)
0.60 4.60 (0.04) 13.17 (0.12) 21.68 (0.20)
0.70 3.80 (0.04) 11.43 (0.12) 19.00 (0.18)
0.80 2.60 (0.03) 8.49 (0.09) 14.33 (0.15)
0.90 1.00 (0.00) 4.35 (0.05) 7.67 (0.09)
0.96-0.98 -- 1.00 (0.00) 1.00 (0.00)

AP Price
(P.sup.*.
sub.AP]) a = 1, b = 3 a = 1, 6 = 5

0.10 2.28 (0.01) 2.15 (0.01)
0.20 2.43 (0.01) 1.97 (0.01)
0.30 2.22 (0.01) 1.46 (0.01)
0.40 1.80 (0.01) 1.00 (0.00)
0.50 1.23 (0.01) --
0.60 1.00 (0.00) --
0.70 -- --
0.80 -- --
0.90 -- --
0.96-0.98 -- --

AP Price
(P.sup.*.
sub.AP]) a = 3, 6 = 3 a = 5, h = 5

0.10 3.56 (0.01) 4.25 (0.00)
0.20 4.55 (0.02) 5.20 (0.00)
0.30 4.71 (0.02) 5.13 (0.00)
0.40 4.39 (0.03) 4.53 (0.00)
0.50 3.73 (0.02) 3.58 (0.00)
0.60 2.81 (0.02) 2.38 (0.00)
0.70 1.68 (0.01) 1.00 (0.00)
0.80 1.00 (0.00) --
0.90 -- --
0.96-0.98 -- --

Notes: Numbers in parentheses are standard deviations. With the
exception of the right-skewed beta densities, the reported clearing
rates are rounded upward to the nearest incremental value of the
tariff.
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