1 “ Pumpkin pies and public goods: The raffle fundraising strategy ” BRIAN DUNCAN (2002) Sandra Orozco Contest and Tournaments November, 2007

2 Charitable organizations commonly use raffles to raise money. The probability of winning is equal to the number of tickets owned divided by the total number of tickets sold. They are generally very unfair gambles. Yet, many people buy tickets to show support. To raise money there exist two strategies: ask for voluntary contributions or raffle off a prize. Introduction

3 Without a prize, spectators buy tickets motivated by increasing the supply of public good. With a prize, spectators buy tickets motivated by increasing the supply of public good, and by winning the prize. For a raffle to outperform voluntary contributions the increase in ticket sales must be greater than the value of the prize. Introduction

4 Buying a raffle ticket introduces uncertainty as ticket- buyers trade consumption for expected consumption. How a ticket-buyer feels about this trade-off depends on risk-preference. Raffles lower the marginal price of the public good. Offering a prize can increase the supply of public good even among risk-adverse ticket-buyers. The size of the prize matters. Depending on risk- preference, the public good may not increase for all prizes. Motivation

5 Using general utility functions a raffle can increase or decrease the supply of a public good, depending on the value of the prize. With general utility functions, the raffle affects the supply of public good in two ways. First, the raffle lowers the marginal price of giving, which leads to an increase in the supply of public good. Second, the raffle introduces uncertainty in the marginal utility of private consumption. Motivation

6 With general utility functions, a raffle will increase the supply of public good only if the fundraiser chooses an appropriate raffle prize. Motivation

7 Outline 1. The private supply and efficient supply of a public good 2. A raffle with quasi-linear utility 3. A raffle with general utility 4. Conclusions

Model Set of individuals N = {l,2,..,i,..,n) Each member has preferences over combinations of two items: private consumption xi and a public good G. Endowed wealth wi, which he may spend on private consumption or contribute to the public good gi. Preferences are represented by the utility functions U i [xi,G], where U i xi > 0,U i G > 0,U i xixi < 0,U i GG < 0, and U i xiG ≥ 0.

The private supply without a raffle When a fundraiser finances a public good through voluntary contributions the equilibrium supply of public good will be less than socially efficient. Consumer’s utility maximization problem: Max {G} U i [wi + G −i − G,G] ….. (1) s.t. G − G −i > 0 Where G −i = G − gi represents the level of public good without i’s contribution.

The private supply without a raffle The FOC to choosing G that maximizes (1) is: U i xi − U i G ≥ 0, with equality if gi > 0 ……(2) A NE allocation of private consumption and public good solves (2) for every individual. Let G ∗ (0) be equilibrium public good derived from (2). Let N ∗ (0) be the set of individuals contributing to the public good. mrs i (G ∗ (0), x ∗ i (0)) = 1, if i N ∗ (0).……(3)

A raffle with quasi-linear utility With quasi-linear utility, tying contributions to a raffle will increase the supply of public good. Consider the following quasi-linear utility function: U i [xi,G] = α i x i + f i (G),……………… (4) αi ≥ 0 (marginal utility of private consumption) A ticket-buyer is always certain of his marginal utility. With quasi-linear utility, the fixed prize raffle lowers the marginal price of the public good which increases the public good.

A raffle with quasi-linear utility The marginal price of contributing to the public good is less than one and, mrs i (G ∗ (r), x ∗ i (r)) < 1 for all i….. (8) Equations (3) and (8) imply that mrs i (G ∗ (r), x ∗ i (r)) < mrs i (G ∗ (0), x ∗ i (0)) for all i N ∗ (0). Therefore, G ∗ (r) > G ∗ (0) whenever r > 0.

A raffle with quasi-linear utility

14 With general utility functions, the fixed prize raffle incorporates two effects: Lowers the marginal price of contributing to the public good and Introduces uncertainty in the marginal utility of private consumption A raffle with general utility

A raffle with general utility The consumer’s von Neumann-Morgenstern utility maximization problem is:

16 Each of the heuristic raffles captures one of the two effects incorporated in the fixed prize raffle, while ignoring the other. The expected prize raffle removes uncertainty, so it captures the raffle’s effect of lowering the marginal price of the public good. The door prize raffle fixes the price of contributing, so it captures the raffle’s effect of introducing uncertainty in the marginal utility of private consumption Two heuristic raffles

17  Fixed prize raffle: fundraiser randomly selects a single winner.  Expected prize raffle: every participant wins a share of the prize.  Difference: no risk in the expected prize raffle; each ticket-holder consumes his expected consumption with certainty.  Door prize raffle: only one participant wins.  Difference: all the participants in a door prize raffle have an equal probability of winning the prize, regardless of any additional contributions to the public good Two heuristic raffles

18 Proposition 1. With quasi-linear utility functions, the fixed prize raffle is equivalent to the expected prize raffle The expected prize raffle

The expected prize raffle The expected prize raffle and the fixed prize raffle with quasi-linear utility increase the public good: they lower the marginal price of contributing to the public good. mrs i (G ∗ (r), x ∗ i (r)) < mrs i (G ∗ (0), x ∗ i (0)) for all i N ∗ (0). Therefore, G ∗ (r) > G ∗ (0) whenever r > 0. The public good is greater with an expected prize raffle than without one. Increasing the raffle prize will always increase the supply of public good.

20 Proposition 2. With an expected prize raffle (or a fixed prize raffle with quasi-linear utility), a larger prize will result in a higher level of public good until one individual spends his or her entire wealth on raffle tickets The expected prize raffle

21 Corollary 2.1. If every individual has identical preferences and equal wealth, then the raffle prize that maximizes the supply of public good under an expected prize raffle (or a fixed prize raffle with quasi-linear utility) has every individual spending his or her entire wealth on raffle tickets The expected prize raffle

The expected prize raffle Supply of public good with an expected prize raffle

23 A participant’s probability of winning the prize is fixed. Both the door prize and fixed prize raffles introduce uncertainty in the consumption of private good; a participant may win or loose the raffle The door prize raffle

The door prize raffle An individual’s von Neumann-Morgenstern utility maximization is

The door prize raffle FOC: EU i xi − EU i G = 0 In a door prize raffle, participants equate their expected marginal utilities of public and private consumption.

The door prize raffle With quasi-linear utility, a door prize raffle will not influence the supply of public good. With general preferences, a door prize raffle can increase or decrease the supply of public good. If participants’ MgU of consumption is decreasing and concave in xi, then the door prize raffle will increase the public good. If the MgU of consumption is decreasing and convex in xi, then the door prize raffle will decrease the public good.

The door prize raffle Marginal utility of private consumption

The door prize raffle FOC: EU i xi = EU i G……………. (20) The vector of gifts g(r) is called the offset allocation because G(r) = G ∗ (0) If the MgU of consumption is decreasing and convex in x i, then the EU i xi is greater at the offset allocation than at the original equilibrium. Therefore, the offset allocation cannot be an equilibrium. Every participant desires more private consumption and less public good.

The door prize raffle Proposition 3. If every person’s marginal utility of private consumption is decreasing and convex, then a larger door prize raffle will result in less public good.

The door prize raffle The fundraiser maximizes the supply of public good when he sets the raffle prize at zero. Supply of public good with a door prize raffle

The fixed prize raffle with general utility With general utility, a fixed prize raffle incorporates the properties of both the expected prize and door prize raffles. That is, buying a raffle ticket lowers the marginal price of giving, but it also introduces uncertainty.

The fixed prize raffle with general utility The left-hand side is identical to the FOC of a door prize raffle. The raffle introduces uncertainty in the marginal utility of private consumption. The right-hand side represents the marginal price effect.

33 A fixed prize raffle incorporates both the marginal price effect of the expected prize raffle, which leads to an increase in public good, and the uncertainty effect of the door prize raffle, which leads to a decrease in public good. Whether a fixed prize raffle increases or decreases the supply of public good depends on preferences and on the size of the raffle prize The fixed prize raffle with general utility

34 Supply of public good with a fixed prize raffle The fixed prize raffle with general utility

35 When the raffle prize is small, the supply of public good increases. As the fundraiser increases the raffle prize, the uncertainty effect increases relative to the marginal price effect, eventually resulting in a decrease in the supply of public good. The maximum attainable supply of public good decreases as the community size (n) increases The fixed prize raffle with general utility

The fixed prize raffle with general utility % change in the supply of public good as a the value of the prize increases

The fixed prize raffle with general utility The % change in public good is greater for larger economics. Thus, the raffle does better compared to private supply as the number of people in the economy grows. The % decrease in the supply of public good compared to the efficient level gets larger as n increases. Therefore, the raffle does better as n increases relative to private supply, but it does worse relative to the efficient supply.

The fixed prize raffle with general utility Deviation from the efficient supply of public good as a function of the raffle prize

39 In general, a raffle can increase or decrease the supply of public good, depending on the preferences of ticket-buyers and on the value of the raffle prize. When a person buys a raffle ticket he or she redistributes expected private consumption away from other ticket-buyers. The redistribution of expected private consumption lowers the marginal price of contributing to the public good and, therefore, leads to an increase in its supply. However, the redistribution trades certain consumption for expected consumption. Depending on how ticket-buyers view uncertainty, that tradeoff can lead to a decrease the supply of public good Conclusions