Charity without altruism.

Spiegel, Menahem

I. INTRODUCTION

Evidence suggests that many charitable contributions are channeled to organizations (or groups of people) that produce excludable goods (or services) that are sold on the market. Some of the frequently noted examples are in the field of performing arts, where the philanthropic contributions of individuals and institutions constitute a traditional source of financing for the performers' "regular" activities. Organizations like orchestras, theaters, dance, and opera companies treat these donations as a stable stream of revenue and not as temporary funding or as a form of emergency relief resulting from an unexpected event.(1) For many of these nonprofit organizations, it is common practice to show in their financial statements a gap between the cost of producing their output and the revenue generated from its sale. This "income gap" is traditionally bridged by the flow of philanthropic contributions.(2)

In the market setting, a philanthropic contribution is defined as a voluntary payment made by a consumer to the organization that produces the good. This payment is additional to the market price paid by the consumer to purchase the good or service.

Two questions motivated this paper.

1. On the contributor side: Why would a selfish utility-maximizing contributor pay the producer of a good an amount (as contribution) exceeding the minimum price he has to pay to buy the product?(3)

2. On the recipient side: Why would a producer (the group of recipients) prefer to use the philanthropic contribution mechanism rather than the simple market mechanism of adjusting prices and quantities in order to break even and avoid the dependency on voluntary contributions?

In the literature, the answer to the first question is founded on the altruistic motive of donors. Two general forms of altruistic donations are widely discussed. The first form is contributions made for the production of a nonexcludable pure public good, as discussed by Andreoni [1988] Bergstrom et al. [1986] and Roberts [1987]. In general, this literature suggests that, due to free ridership problems, voluntary contributions cannot by themselves support Pareto-efficient provision of the public good.(4) The second form of altruistic donation is represented by Hansmann [1981], James and Rose-Ackerman [1986], Ben-Ner [1986], and others. They consider contributions made to producers of excludable public goods. In Hansmann [1981] and Ben-Ner [1986], the contributing activity is described as self-imposed price discrimination. Implicitly they argue that club goods have a special feature such that consumers, in particular charitable contributors, are willing to give away parts of their consumer surplus of that product. In this setting, again, contributions are motivated by altruism, as the contributors do not bargain for benefits in return for their loss of consumer surplus. This explanation implies that donors have a special utility function and that they derive utility from the action of contributing, whereas other people, the recipients in particular, do not share this characteristic. The second question regarding producers' motives has received much less attention in the literature.

The Coase theorem suggests the basic answer to both the above questions. The Coase theorem claims that when there are inefficiencies, the transition to an efficient allocation could generate economic rents.(5) Thus, utility-maximizing economic agents would negotiate their way towards the efficient allocation and share the rents created in the process. It is the contention of this paper that giving and receiving charitable contributions in a market setting is an example of that phenomenon. Charitable giving facilitates the move from an otherwise inefficient allocation to an efficient one. In the process, the participating agents (the contributors and the recipients) try to divert the rents created to their private benefit.

This paper differs from the aforementioned literature in considering a non-altruistic motive for contributing. According to our model, in the market setting charity is given by utility-maximizing donors to utility-maximizing recipients who have the same tastes and share the same utility function. The central assumption of the model is that the goods considered are typical club goods: privately produced and equally consumed by all club members who pay the same market price, and nonpayers can be excluded without cost.(6)

Following Berglas and Pines [1981; 1984], McGuire [1974a], Porter [1978], and Stiglitz [1977], we assume here that strong economies of scale in the consumption of the club good require a larger membership than possible with homogeneity of consumers. Therefore, it is not optimal to divide the heterogeneous population of the mixed club into several smaller, but more homogeneous, populations (and clubs).

When the club members have identical utility functions but differ in their incomes, they will differ in their reservation prices for a club good of a given quality and in their preferred choice of quality. When quality is a normal good, high-income consumers have higher reservation prices and prefer to consume a club good of higher quality "sold" at a higher price.(7) Therefore, the quality and price choices made by a subgroup of club members is not the socially optimal one. The important role charitable contributions can play is in transforming such socially nonoptimal solutions into optimal ones.

Contributions received by nonprofit organizations producing a public (club) good must create rents to the consumers of the good. As these donations will not be distributed as a dividend, they will be spent on increasing the benefits of club members. Each group of club members would like to see the donations being used in a way which maximizes their benefits.(8) Low-income consumers would like the contribution revenue spent in a fashion that reduces the price (p). The selfish contributor, considering his contribution as an "investment," would like the quality of the club good to rise by more than just the amount implied by his contribution. For illustration, consider the following simple scenario. Assume a nonprofit organization receives a $C contribution that causes the quality of the club good produced to rise. That change enables all club members to enjoy higher utility levels. The contributor's goal is to share the cost of producing the additional consumer surplus obtained by others. This could be done by raising the market price of the club good. The additional revenue generated would go to increase, once again, the quality of the club good. Eventually the quality rises by more than that generated by the amount of the original contribution ($C). The important conclusion of the process is that the selfish donors operate to both increase the price and quality of the club good by means of their contributions. In the market it will be observed that the selfish (high-income) consumer is paying more than a low-income consumer for the same club good. This is not the regular imposed price discrimination as the contributor is receiving (in terms of quality) more than he paid for (in terms of contribution).

II. THE MODEL AND SOME BASIC DEFINITIONS

There are two goods in the model: the first is an all-encompassing consumption good, and the second is a club good. There are N consumers with the following identical utility function:

(1)U(X,Z) = {X [multiplied by] Z if the club good is consumed {X [multiplied by] A if not a member of the club

where

X is the dollar amount of the all-encompassing good whose price is normalized to one;

Z is a one-dimensional measure of the quality of the club good consumed (equally) by all club members;

A is a non-negative constant.(9)

Consumers differ in their income. The assumed distribution of consumers according to their income is the following:

h = number of consumers with a high income of $R.

l = number of consumers with a low income of $I, with l[greater than]h and l + h = N.

In order to consume the club good, a consumer has to give up p units of his private good, which we will refer to as the club membership fee, or the market price of the club good. A low-income consumer will buy the club good of quality Z at a price p if his utility from doing so is higher than the alternative, i.e., (I-p) Z[greater than or equal to]AI. Clearly, if a low-income consumer prefers to participate in the club for the same p and Z, a high-income consumer will obtain that (R - p) Z [greater than or equal to] AR and will participate. Thus if low-income consumers are in the club, all N consumers will join the club.(10)

The club good is produced by a not-for-profit private club for the benefit of its members only, and all nonpayers are excluded at no cost. The quality (or the size in terms of output) of the club good (Z) depends on the amount of resources used in the process of its production. A constant returns technology of production with a coefficient of one is assumed - i.e., each unit of the quality of the club good requires one unit of input. Specifically, measured in dollar terms, the quality of the club good produced is equal to the amount of dollars used in the production process. Given a nonprofit organization maintaining its zero-profit constraint, the quality of the club good that emerges is equal to the revenue of the club.(11)

A club's revenue is generated from the following two sources: from general membership fees $N [multiplied by] p, and from the philanthropic contributions $C. Thus, the quality that will be produced is

(2) Z = N [multiplied by] p + C.

Efficient Provision of the Club Good

Following Samuelson's efficient provision condition, a club good is efficiently supplied when the sum over all consumers of their willingness to pay, as measured by their marginal rates of substitution, is equal to the marginal rate of transformation, as represented by the marginal cost of producing quality. The provision of Z will be efficient when

[Mathematical Expression Omitted].

From (1), substituting for the high-income consumers [MRS.sup.h] = (R - P)/Z, and for low-income consumers [MRS.sup.1] = (I - P)/Z. Solving for Z we get

(3) [Z.sup.s] = (hR + II)/2.

The constrained Pareto-efficient quality of the club good, [Z.sup.s], is achieved when one-half of the total income of all club members is spent on producing (and consuming) the club good.(12) The single price which supports that club size [Z.sup.s] is

(4) [p.sup.s] = (hR + lI)/2N,

where [p.sup.s] is equal to one-half of the average income.

The feasibility of the efficient solution ([Z.sup.s], [p.sup.s]) might be constrained by the dispersion of the income distribution and the utility value of the alternative use of income (A). Let [p.sup.-I] denote the reservation price of a low-income consumer for the club good with a given quality (Z). The term [p.sup.-I] is the value of p which solves (I - p). Z = A [multiplied by] I. Thus,

(5) [p.sup.-1] = {I(1-A/Z) for A [less than] Z

{0 otherwise (A [greater than or equal to] Z)

Clearly, the reservation price increases with the level of income (I) and the quality of the product (Z), and falls with the rise of the utility value of the alternative (A). Further, as A [right arrow] 0 = [p.sup.-1] [right arrow] I. In the limiting case of A = 0, in order to ensure that [P.sup.s] [less than or equal to] I the dispersion of the income distribution has to be limited to R [less than or equal to] I(N + h)/h. Otherwise, [p.sup.s] would be larger than the income of low-income consumers and they would not consume the club good at the socially optimal quality. In other words, if the dispersion of incomes is too wide and/or A is too large and/or h is too large, there is no single price [p.sup.s] such that [p.sup.s] [less than] I, and thus the socially optimal quality is not feasible under a single-price policy. This result indicates that the income diversity of club members is limited if the socially efficient quality of the club good is to be achieved.(13)

The Provision of the Club Good When Quality Decisions Are Made by One Group of Club Members (Without Contributions)

For the purpose of comparison, we consider here two potential outcomes that might result if one group of consumers was to control the club and set the quality and the (single) market price of the club good.

Majority Rule Decision. Under this rule, the quality of the club good is determined by the preferences of the majority of low-income (low-demand) consumers.(14) Thus it meets the solution to the choice problem of the representative consumer of that group. Formally,

[Mathematical Expression Omitted].

The first-order conditions imply

(7) [Z.sub.mj] = NI/2, [p.sub.mj] = 1/2, [Mathematical Expression Omitted] and [Mathematical Expression Omitted].

Subscripts mj denote equilibrium values under the majority-rule decision.

Clearly, the majority-rule provision of quality [Z.sub.mj] and the price set [P.sub.mj] are lower than is Pareto optimal. The utility level of the low-income consumers [Mathematical Expression Omitted] is higher, and that of the high-income consumers [Mathematical Expression Omitted] is lower, compared with the similar values reached when Z = [Z.sup.s] and p = [p.sup.s].

Minority Rule Decision. Under this rule, the representative consumer of the high-income (high-demand) group chooses the quality of the club good that is produced and purchased by all club members. That quality is a solution to the following problem.

[Mathematical Expression Omitted].

The first-order conditions imply

(9) [Z.sub.mn] = NR/2, [p.sub.mn] = R/2, [Mathematical Expression Omitted], and [Mathematical Expression Omitted].(15)

Subscripts mn denotes equilibrium values under minority-rule decision.

Comparing these to the Pareto efficient results reveals that, under minority rule, the quality of the club good and the price are too high.

III. CONTRIBUTION GAME WHEN THE PRICE IS AFFECTED BY CONTRIBUTORS(16)

A philanthropic contribution is defined as a voluntary payment made by a consumer to the producer of the club good. This voluntary payment is in addition to the common price (p) paid by all N club members.(17) A potential contributor, a high-income consumer, needs to choose his optimal contribution. We assume that the contributor is faced with an explicit price/contribution function that is solved for the optimal contribution. The most important characteristic of the price/contribution function is that the contributor's equilibrium choice will not be rejected by the recipients (low-income consumers). Thus, in equilibrium each group of consumers should be better off compared to their baseline option. For the sake of simplicity it is assumed that the price/contribution function is the low-income consumers' willingness to pay for quality of the club good.

A selfish contributor would like to see an increase in the level of his utility as a result of his contribution. Given the contributor's utility function represented by equation (1), a contribution amounts to a voluntary reduction of the quantity of the private good consumed. The contributor will be better off when, as a result of his contribution, the dollar value of the public good (z) consumed is increased by more than the dollar value of his contribution. Such an increase in z can result from an increase in the uniform price of the club good. A small increase in the price of the club good (p) will be welcomed by the low-income consumers when their willingness to pay for the club good has sufficiently increased due to the rise in the quality of the club good (Z).

We take the uniform price (p) paid by all members to be a variable determined endogenously in the model and related to the total amount contributed. Consumers of the different income groups would like to see a different (opposing) relationship between the contribution and the price of the club good. While contributors would like to see the highest price increase per dollar of contribution, low-income consumers would favor the least price increase which will "secure" for them the contribution.

The exact relationship between contributions and the price of the club good is determined in a bargaining process that, among other things, also depends on the objective function of the "club organizers."(18) In order to obtain positive contributions, the price/contribution function and, in particular, the equilibrium choice must be accepted by all consumer groups.

For the sake of simplicity of the example and in order to ensure that the price/contribution function will not be rejected by the majority of low-income consumers, it is assumed that the price/contribution function faced by contributors is the indifference set of the non-contributors. This function is derived from the low-income consumers' willingness to pay for different qualities of the club good (while holding their utility level constant at k). Given the break-even constraint Z = Np + C, and the utility [V.sup.1] = Z(I - P) = k, the following price/contribution function faces the contributors.

(10) p = [1/(2N)]

[NI - C + [square root of [(NI + C).sup.2] - 4Nk]],

with

[Delta]p/[Delta]C [greater than] 0, [[Delta].sup.2]p/[Delta][C.sup.2] [greater than] 0,

[Delta]p/[Delta]I [greater than] 0, [Delta]p/[Delta]N [greater than] 0,

and [Delta]p/[Delta]k [less than] 0.

This function (10) represents the assumed equilibrium relationship between the total amount of contributions and its impact on the uniform price of the club good paid by all members. Using this function, some simple scenarios of the behavior of the contributors are considered in the next section.

The Donors Game

Assume the following familiar scenario. A small group of high-income consumers (donors) gathers at the fund-raising function (e.g., at a dinner party organized by the club itself, or at a function produced by some other organization which is interested in the production of the club good) in order to discuss (negotiate) and to pledge their contributions.(19) The likely outcome of the negotiations is the determination of the financial "needs" of the club (the desired quality of the club good) and the means of financing it. The counterpart of the negotiation will address the determination, by source, of the revenues to be received. The two sources of revenue are the "general sales revenue" generated by the price paid by all members and the "contribution revenue" paid by the small group of contributors. These revenue negotiations (simultaneously) determine the price to be paid by all members and the amounts pledged by each contributor. Being a small group, with an assumed similar interest in the quality of the club good, the contributors play an important role in these negotiations.

A simplified version of the negotiation process can be formally described as an h [greater than or equal to] 1 players cooperative game. A strategy of a contributor is an amount of $[Alpha] he would want to contribute. The strategy set is given by

[S.sub.h] = {[[Alpha].sub.i]; 0 [less than or equal to] [[Alpha].sub.i] [less than or equal to] R - p}.

Player i of the group of high-income consumers (contributors) assumes that his action and contribution is observed and will be followed by the other (h - 1) players of the group, that is, they all will contribute the same amount he does.(20) That is, if a pledge of $[[Alpha].sub.i] has been declared by contributor i, and it is not unanimously accepted by the other contributors, then any contributor j, j [not equal to] i, can raise or lower the pledge. An equilibrium of the game is a contribution [[Alpha].sup.*] where no contributor would want to pledge [[Alpha].sub.j] [not equal to] [[Alpha].sup.*]. That is, [[Alpha].sup.*] will be the equilibrium of the contribution game if, for each contributor i [element of] h, we have [V.sup.h]([[Alpha].sup.*]) [greater than or equal to] [V.sup.h]([[Alpha].sub.i]) for all [[Alpha].sub.i] [element of] [S.sub.h]. As a contributor, each consumer i [element of] h considers the total contribution to the club to be C = [Alpha]h. Thus, the problem of an individual contributor in the group is to determine his optimal contribution. This can be formally described by

[Mathematical Expression Omitted].

Substituting Z, C, and p into [V.sup.h] and solving the first-order conditions for the optimal contribution [Alpha], we get

(12) [[Alpha].sup.*] = [[h.sup.2][(R - I).sup.2] - [(NI).sup.2] + 4Nk]

/[2[h.sup.2](R - I) + 2hNI]

where [[Alpha].sup.*] is the equilibrium contribution of a typical contributor, and with [Delta][[Alpha].sup.*]/[Delta]h, [Delta][[Alpha].sup.*]/[Delta]R, and [Delta][[Alpha].sup.*]/[Delta]k [greater than] 0. That is, the equilibrium contribution [[Alpha].sup.*] is increasing with the number of contributors (h),(21) with the income of the contributor (R), and with the constant utility level of low-income consumers (k). Also, [Delta][[Alpha].sup.*]/[Delta]I and [Delta][[Alpha].sup.*]/[Delta]N [less than] 0, meaning that the equilibrium level of contribution is decreasing both with the income of low-income consumers (I) and with the number of members in the club (N).(22)

From the formal setting of the model of (11) it is clear that both the equilibrium of the game and the determination of the amount contributed by each player ([[Alpha].sup.*]) are based on the assumption of matching behavior by the contributors. The literature which has focused on the question of whether this conjecture of matching behavior of contributors in a noncooperative Nash setting is plausible has strongly rejected this possibility.(23) It is important to note that the model in this paper is different as it considers a Coasian situation. In this model the small number of contributors, playing the cooperative game, are making explicit binding agreements with "the club" and with the noncontributing club members. As in a Coasian situation, the terms of the contracts are the equilibrium values of the game. These binding contracts are also self-enforced contracts as the product we are dealing with here is an excludable public good; those who don't pay cannot consume the club product.

The equilibrium amount of contribution of all contributors is

(13) [C.sup.*] = [Alpha]h = [[h.sup.2][(R - I).sup.2] - [(NI).sup.2] + 4Nk]

/[2h(R - I) + 2NI].

Similarly to [[Alpha].sup.*], [C.sup.*] is increasing with h, R, and k. Also, it is decreasing with I and N. Given [C.sup.*], the implied equilibrium price is

(14) [p.sup.*] = I - [2k]/[h(R - I) + NI].

The equilibrium price of the club good ([p.sup.*]) is increasing with the number of contributors (h), with the income level of consumers (I and R), and with the number of club members (N). The equilibrium price ([p.sup.*]) is decreasing with the constant utility level of low-income consumers (k).

The implied equilibrium quality of the club good is

(15) [Z.sup.*] = [hR + (N - h)I]/2.

The equilibrium quality of the club good ([Z.sup.*]) is equal to one-half of the total income of the club members. At the equilibrium of the game ([c.sup.*], [p.sup.*], [Z.sup.*]), the marginal rate of substitution for the different types of consumers are as follows. For the low-income consumer:

(16) [MRS.sup.l] = ([Delta][V.sup.l]/[Delta]Z)/([Delta][V.sup.l]/[Delta][x.sup.l])

= [x.sup.l]/Z = (I - [p.sup.*])/[Z.sup.*] = k/[([Z.sup.*]).sup.2].

The marginal rate of substitution for a contributor is

(17) [MRS.sup.h] = ([Delta][V.sup.h]/[Delta]Z)/([Delta][V.sup.h]/[Delta][x.sup.h])

= [x.sup.h]/Z = (R - [p.sup.*] - [[Alpha].sup.*])/[Z.sup.*]

= (R - I - k/[Z.sup.*] - [[Alpha].sup.*])/[Z.sup.*]

Equations (16) and (17) imply that

(18) [summation of] [MRS.sup.i] = 1 = MRT where i=1 to N.

The important characteristic of the equilibrium of the donors game is that the quality of the club good which is now being offered is also a Pareto optimal one. This result implies that contributions made by selfish contributors can serve as a vehicle to improve welfare and promote Pareto optimal resource allocation. That is, when the contributor and the recipient of the contribution are both selfish consumers, charitable contributions can increase the utility level enjoyed by all of the club participants. Thus, the main driving force is the contribution, which increases the quality of the club good and creates "new" rents. These rents are generated by utility-maximizing donors who will "invest" in their creation up to the point where their marginal social value is equal to their marginal cost, and thus a Pareto efficient allocation is achieved. In the framework of our model, these allocations are not attainable in the contribution game in the case of altruistic contributions or when the uniform price of the club good is fixed.

The equilibrium utility level enjoyed by a typical contributor is

(19) [V.sup.h*] = [Z.sup.s](R - [p.sup.*] - [[Alpha].sup.*]) = [[([Z.sup.s]).sup.2] - lk]/h.

This utility level, [V.sup.h*], enjoyed by all contributors, is negatively related to k, the utility level enjoyed by each of the non-contributing club members. Thus, from (13) and (19) it is implied that there is a tradeoff in utility between contributors and noncontributing club members. Given that all members are selfish, the determination of k is an important element in determining the conditions under which contributions will appear and of what scale. The parameter k is not endogenously determined in the simple model presented. In the following section we consider the two limiting cases of the lower and upper boundaries of k, their implications for the amount contributed, and the quality of the club good produced.

Case I: The Maximum Joy of Giving

First we consider the equilibrium of the donors game when the donors set the rules. It is assumed that the high-income club members are faced with the price/contribution function (10), and are free to set the relevant parameters and variables to their liking. When the rich guys are controlling the club, they choose k which maximizes their utility [V.sup.h*] of (19). Clearly, they will set k to the lowest level acceptable to low-income consumers and the implied contribution and the implied market price. For the low-income club members to accept the proposed regime of contribution, they need to obtain a utility level equal to or higher than their baseline solution, which is the maximum they can obtain otherwise. It is assumed that the baseline solution is the majority rule without contributions. In the baseline case, the high-income consumers are better off joining the club at the price p = I/2 rather than staying out and not consuming the club good. Therefore, the highest utility level low-income consumers can secure for themselves without contributions is given by (7), [Mathematical Expression Omitted], the utility level obtained under majority rule without contribution. Therefore, this is also the minimum utility level that will be accepted by the group of low-income club members under the contribution regime.(24) By the nature of this choice problem, the amount of contribution that will be donated is also the amount which will enable the contributors to reach their highest utility level, and therefore it will represent the maximum joy of giving. When k is set to its lowest acceptable level, [Mathematical Expression Omitted], the total amount of contribution will be

(20) C = [Alpha]h = [h.sup.2][(R - I).sup.2]/2(hR + lI).

The price of the club good will be

(21) p = I - [N[I.sup.2]/2(hR +lI)].

This is the highest uniform price of the club good which will support the Pareto efficient quality of the good. This price is higher than I/2, which is the price under majority rule without contribution.

The utility level realized by the different consumers is

(22) [V.sup.h] = [2RIN - N[I.sup.2] + h[(R - I).sup.2]]/4

and

[Mathematical Expression Omitted].

Low-income consumers would be indifferent between the choice of majority-rule equilibrium without contributions and the minority-rule equilibrium in the contribution game, as they enjoy the same utility level [Mathematical Expression Omitted] under both regimes. For the high-income consumers, [Mathematical Expression Omitted] is the utility they enjoy at equilibrium with majority rule without contributions (see (7)). As a result of the contributions, the utility of the contributors will rise by h[(R - I).sup.2]/4 [greater than] 0. As a result of the setting of the problem, this additional utility enjoyed by the contributor also represents the maximum joy of giving.(25)

Case II: The Maximum Joy of Receiving

Now consider the case where low-income consumers are running the club and thus are in the position of setting the value of the parameters to their liking. Their problem is to present the contributors with a price/contribution function which maximizes the utility level they will enjoy at the equilibrium of the donors game. Thus the low-income consumers would like to maximize k, subject to the constraint that the level of k chosen is acceptable to the contributors. With higher levels of k, the dollar amount contributed by each contributor, [[Alpha].sup.*], and the total amount contributed, [C.sup.*], will rise (see (12) and (13)). When the club is run by the low-income members, the contribution revenue will be used to lower the uniform price of the club good ([p.sup.*]) paid by all consumers (see (14)). From (19) we have that [V.sup.h*], the utility level enjoyed by the contributor, will continuously fall with the increase of k. Clearly, there is a level of k, say [Mathematical Expression Omitted], for which [V.sup.h*] is sufficiently small such that the high-income consumer will be better off by refusing to contribute; that is, set [[Alpha].sup.*] = [C.sup.*] = 0.(26) In the case where a "too" high level of k is chosen, that is, [Mathematical Expression Omitted], it is assumed that the equilibrium that will prevail is the noncontributing majority rule, and the utility level enjoyed by the high-income consumers will be [Mathematical Expression Omitted] (see (7)). Thus the highest utility level [Mathematical Expression Omitted] which is attainable by low-income consumers at equilibrium of the contribution game is that k which solves

(23) [[(hR + lI).sup.2] - 4kl]/4h

= (2RIN - N[I.sup.2])/4.

The solution for k is

[Mathematical Expression Omitted].

This value of [Mathematical Expression Omitted] represents the highest utility level attainable by consumers of the low-income group in the contribution game. Comparing this level of k from (24) with [Mathematical Expression Omitted], the equivalent value of majority rule without contribution, we see that the gain in utility to the majority of low-income club members from setting k in the contribution game is [[h.sup.2] [(R - I).sup.2]]/4 [multiplied by] l. This value also represents the maximum joy from the contribution to the noncontributors, that is, the maximum joy of receiving.

There are two sources of this additional utility to the low-income club members. First, the quality of the club good produced in this contribution game is [z.sup.*], the socially optimal quality, and it is higher than that of majority rule without contribution. Second, the price of the club good is the lowest (see (14)). Therefore, low-income consumers are getting a product of a higher quality at a lower price. This is also the lowest price low-income consumers can expect to pay for the Pareto efficient quality of club good [Z.sup.s].

In equilibrium of the donors game, when k is set at its highest value [Mathematical Expression Omitted], the value of [C.sup.*] (the amount of contribution that can be "extracted" from the potential contributors) is also maximized, while the quality of the club good produced is maintained at the Pareto efficient level.

The above analysis can be summarized using the diagrams of Figure 1. In panels I through IV the horizontal axis represents the utility level of the low-income consumers. The lowest utility level secured by the low-income consumers is the majority-rule equilibrium [Mathematical Expression Omitted]. The highest utility level low-income consumers can reach is [Mathematical Expression Omitted].

Panel I presents the relationship between the quality of the club good offered and the utility level of low-income consumers. The constant value implies that the quality of the club good that will prevail in equilibrium of the donors game is the socially optimal value [Z.sup.*] for all values of utility of the low-income consumers. Panel II represents the utility frontier, or the tradeoff in utility between low ([V.sup.l]) and high income ([V.sup.h]) consumers. This tradeoff represents the set of equilibrium solutions of the donors game. Point A represents the maximum joy of receiving to low-income consumers, while point B represents the maximum joy of giving to the contributors. Clearly, the utility level obtained at the maximum joy of giving [Mathematical Expression Omitted] exceeds the utility level enjoyed by the high-income consumers at the minority-rule equilibrium, [Mathematical Expression Omitted]. Similarly, the utility level of the low-income consumers at k, majority-rule level, [Mathematical Expression Omitted], is less than the maximum joy of receiving [Mathematical Expression Omitted].

Panel III traces the total amount contributed in equilibrium of the donors game as a function of the utility level of the low-income consumers. The positively sloped curve demonstrates that an increase in the utility level of low-income consumers above the majority-rule level will be achieved when the amount contributed rises. The explanation to this is given in Panel IV, which exhibits the equilibrium set of the pairs of (1) the value of the low-income consumers' utility and (2) the price of the club good. This negative relationship holds while keeping the quality of the club good constant at the socially optimal level. Thus the increased amount of contribution is being used for reducing the price of the club good.

IV. CONCLUSION

The charitable contributions observed most often today take the form of voluntary payments made by the consumer to the producer of an excludable and market-traded good (or service). The model presented in this paper, and the accompanying example, introduce a new element to the analysis of donations which can support charitable giving by selfish, utility-maximizing consumers. That instrument, the contribution price function, uses the market price as an explicit incentive to encourage contributions. This is achieved by using the market price as a rent-sharing device for the purpose of sharing the benefits resulting from the contributions between different consumer groups, the contributors and the noncontributors.

When contributions to the club are observed, high-income consumers (contributors) are paying more than low-income consumers for the same club good. This should not be interpreted as price discrimination, where part of the consumer surplus of the contributor is taken away. The model presented and the cases discussed in this paper, in particular the maximum joy of giving, show that the contributors end up gaining consumer surplus. The "higher price" (the contribution) paid by the contributor is for a higher quality product. Thus the contributor is receiving (in terms of quality) more than he paid for (in terms of contribution). Therefore, it is not income-related price discrimination.

In order to stress the importance of the mechanism in which the market price varies with the amount of contribution, we make the following comments. First, fixing the market price of the good at a zero level, or at any other level, combined with a contribution game identical to the one above, leads to an equilibrium characterized by the underprovision of the public good. Second, the introduction of the contribution/price mechanism into the commonly used Cournot-type contribution game creates an equilibrium that is Pareto superior to the equilibrium of contribution games when market prices are fixed (including fixing the price at zero).

The important policy implication emerging from this model is that a democratic government will face a difficult voting situation. If, on the one hand, the government enforces the single socially efficient price by means of a lump-sum tax, this government collects enough resources to balance its budget but will not get enough votes (from the majority of low-income consumers) needed to survive. If, on the other hand, the government supports the majority-rule price (by a lump-sum tax), it would create an undersupply of the public good and a non-efficient solution. In contrast, a progressive income tax might generate the appropriate and "acceptable" price variation when the government regulates the quality of the club good at the socially optimal level by regulating the total revenue (and expenditures) of the nonprofit club.(27)

It is well known that tax deductions for charity expenditures provide an additional incentive (subsidy) to contribute as the price (in terms of other goods) of a dollar contributed is lower. Similarly, within the framework of the model suggested in this paper, having the charitable contribution tax deduction adds an extra incentive to donate by reducing the contribution/price relationship. As the cost (in terms of other goods) of a dollar contributed is lower, the contributor will be willing to trade that dollar for a lower "return" (in terms of the increase in quality of the club good). That is, the contributor would accept a smaller increase of the uniform price (p) paid by all consumers. It should be noted that the tax deductibility of the charitable contribution is not a necessary condition to achieve efficiency.

As governments today are deeply involved with nonprofit organizations, the use of the instrument of "matching funds" might be politically wise and economically efficient.(28) A simple scenario is that of the government regulating the quality of the club good at a level higher than the majority-rule decision. This higher quality is financed by a majority-rule price (or lower) and the "income gap" is financed by charitable contributions and government "matching funds." The incentive for the selfish high-income contributors to donate, in this case, is the knowledge that their action will lead to an increase in the quality of the club good by more than their contribution (i.e., by doubling the amount of their contribution). In other words, the government is the source of the additional funds rather than the low-income club members. In this respect, matching funds might be used to alter the distribution of gains among the efficient equilibria.

The model discussed raises the important question of the incidence of the cost of providing the public good and the real cost of the contribution. Specifically, when holding the total cost (quality) of the public good constant at the Pareto efficient level, a change in the parameter k will affect the equilibrium level of the following variables: the market price, the individual contribution, and the total amount contributed. Therefore, the incidence of the cost of providing the public good for the different consumers will be affected. As a result we might observe a different sort of "crowding-out" of private contributions while the level of government funding is unchanged.(29)

The discussion so far has been limited to charitable contributions made by high-income consumers. An expansion of this model into one where contributions are not related to income could be accomplished by changing the present underlying structure, where all consumers share the same tastes (utility) and differ in their incomes, to a structure where consumers have the same income but differ in their tastes. This structure would enable one to extend the club-good explanation to nonmonetary charities, in particular to those dependent on donations of labor (time). For example, consider the American Youth Soccer Organization. As an organized team activity, the quality of the team output is affected by the quality of the practices, the quality of the equipment used, the quality of opponents and frequency of play. The gung-ho leaders of the team, those parents with a strong taste for soccer, meet and decide how ambitious the program will be in terms of practicing, frequency of games and quality of competing teams, etc. To accomplish these ambitions, the leaders invest (contributing) extra time (for organization and commuting) out of their self interest. The other parents, those with a weaker taste for soccer, are the recipients of the contribution and will be willing pay a higher price (e.g., to spend more on equipment for their children) once they recognize the high quality of the club good (output).

Thus, when utility-maximizing consumers act in a selfish manner, charitable contribution can provide a vehicle to promote economic efficiency.

1. The model presented below is not limited to explaining cash contributions in the performing arts. Other organizations which belong on this list are cultural institutions like art museums, zoological and botanical gardens, some kinds of nonprofit hospitals, and religious institutes. It should be emphasized here that organizations which have traditionally relied on volunteer labor (i.e., a contribution of time), such as the American Youth Soccer Organization, Little League Baseball and organizations promoting ice hockey, basketball and swimming, should also be included in this list. The nature of their internal organization and the negotiation process among their members can be explained by the same model.

2. For documentation in the case of the performing arts, see Baumol and Bowen [1966] for a detailed description of the income gap, its behavior over time, and its financing. For more recent evidence describing the sources of revenue of nonprofit organizations, see Weisbrod [1988].

3. Clearly, a selfish consumer who does not benefit from consuming the product will neither pay for it nor contribute. For examples see Baumol and Bowen [1966, part III].

4. Bagnoli and Lipman [1989] show that if the dollar size of the public good is fixed at the socially optimal level and cannot be reduced, free riding will be eliminated.

5. See Coase [1960].

6. The theory of clubs was first developed in Buchanan [1965]. For a survey of the literature see Sandler and Tschirhart [1980].

7. For a detailed discussion of similar quality choice problems of consumers see Phlips [1983], and Gabszewicz and Thisse [1979].

8. For the sake of simplicity it is assumed that these benefits are confined to the quality and the price of the club good only. Thus, in the model contributors cannot expect to receive any special treatment or special benefits, in terms of their private consumption, from the noncontributing consumers.

9. Versions of this specification of the utility function are often used in models of quality selection. See recent work by Gabszewicz, Shaked, Sutton, and Thisse [1986]. For simplicity, it is assumed that utility is not affected by the number of members, their identity, or their rate of utilization of the club good. The qualitative results of the equilibrium solution, reported below, can be derived using a general form utility function u(x,z).

10. Extending the analysis to continuous income distributions implies that N, the total number of consumers in the club, is inversely related to the price p for a given level of quality Z. This N will be determined by the lowest income level (I) for which (I - p)Z [greater than or equal to] AI. Such an extension adds little insight to the main issue investigated in this paper at very high cost of complicating the exposition. Therefore, it is assumed that N is determined exogenously and, under the single-price regime, all consumers will join the club.

11. One could substitute the word "quantity" for "quality" of the club good.

12. The constraint being that the total cost of production is equally shared by all members who are paying the single uniform market price p.

13. When the income diversity is "too large," the socially optimal quality of the club-good and the pricing of a single price p paid by all members cannot be maintained simultaneously. As becomes clear below, the existence of voluntary contributions can correct such situations and generate a Preto-optimal quality of the club good which is otherwise not feasible. Under these circumstances, dividing the group into "a more" homogeneous single-priced club can be socially preferred. Similar phenomenon are discussed in McGuire [1991, 1405]. Unlike McGuire, the main concern here is with the financing incentives of the quality of the club good.

14. This case will also be referred to as the "baseline case" as it represents the no-contribution equilibrium solution that low-income consumers can secure.

15. In order to avoid the feasibility constraint noted above, it is implicitly assumed that the price [p.sub.mn] preferred by the high-income club members is lower than the reservation price of the low-income consumers [p.sup.-I]. Otherwise, low-income consumers will not participate.

16. Cornes and Sandler [1984; 1985] and others consider a contribution game where the market price of the nonexcludable good is fixed at the zero level.

17. I assume that the price and the quality of the club good are higher than the majority-rule equilibrium. Therefore, low-income consumers have no incentive to contribute, and thus only a "small" part of the population will contribute.

18. This objective function is not endogenous to this model. Possible objective functions, such as maximizing the utility of the poor or of the rich, were discussed above. Other potential objectives, such as maximizing the quality or the size of the club good or subsidizing the price of the club good, could be analyzed in a very similar fashion.

19. It is often alleged that the potential contributors are brought together in order to increase the amount of contributions. The reasons quoted for this are (1) the crowd (herd) effect of giving, and/or (2) the tie-in sale effect which is believed to be the case when one organization supports the production of a seemingly unrelated club good produced by another organization.

20. Useem and Kutner [1986] report that in corporate giving the degree of intercorporate communication and information exchange is very high. They report among their findings that "peer company comparisons were a major factor in setting the contribution dollar level." It should be recognized that as the number of donors (h) increases, the likelihood of free riding and the degree to which the aggregate contribution (c) falls short of a Pareto-efficient amount both increase. See McGuire [1974b].

21. This represents the typical crowd (bandwagon) effect of giving. In this model, the crowd effect is a result of "levering" the externality of the cooperative game, where it is assumed that each contributor is contributing the same amount.

22. This should not be confused with the free-rider effect. In this case, as N increases, the number of low-income members (l) increases (holding h unchanged). The revenue from membership fees (Np) as well as the quality of the club good produced (Z) increases. Thus the incentive to contribute decreases (no congestion).

23. See Sugden [1985] for a discussion regarding consistent conjectures.

24. It is sometimes observed that charitable contributions are rejected by the recipient. These are cases where the utility of the recipients is reduced below their alternative no-contribution situation.

25. Note that the main characteristics of the results reported here are independent of the number of contributors. When h = 1, there is only one contributor and he will internalize the externalities such that his contribution will lead to the socially optimal provision of the club good. As a single contributor, he might bargain differently for the level of k compared to a larger group of contributors.

26. The problem faced by the low-income consumers resembles the problem of choosing the lowest amount of benefit per dollar of contribution which is acceptable to the contributors.

27. This mechanism of a progressive income tax might not work with selfish consumers when they have the same income but differ in their taste for the club good.

28. Lump-sum subsidies, like most price-related subsidies given to the club good, will not support the Pareto efficient quality.

29. The model developed here can be extended to explicitly include the phenomenon where government funding crowds out private contributions to the production of the public good.

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