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Rival charities
Journal of Public Economics 66 (1997) 449–467
Rival charities a, b Marc Bilodeau *, Al Slivinski a
´ ´ , Universite´ de Sherbrooke, and Department of Economics, IUPUI, Departement d’ Economique Cavanaugh Hall 517, Indianapolis, IN 46202 -5140, USA b Department of Economics, University of Western Ontario, London, Ontario, Canada Received 1 June 1996; received in revised form 1 February 1997; accepted 8 April 1997
Abstract The paper develops a model in which a number of charities (or other nonprofit firms) provide various bundles of public goods or services through private donations. The motivation for individuals to found and operate such firms is that it allows them to influence the mix of public goods. It is their decisions regarding the allocation of donations across uses that matter in the end. Donors to these firms take into account the allocation decisions that will be made by the organizations to which they contribute. We find a propensity for such organizations to specialize in the provision of services, and further find that diversification by such firms diminishes the equilibrium level of contributions they will collect. We demonstrate the possibility that a commitment by a monopoly charity to an allocation rule that is, ex-post, privately sub-optimal can eliminate this effect, and may therefore be advantageous, ex-ante. The allocation rule which accomplishes this involves honouring donor designations of their contributions to specific uses. This is a policy that is frequently adopted by local chapters of the United Way. 1997 Elsevier Science S.A. Keywords: Nonprofit enterprises; Public goods; Voluntary contributions; Spatial competition JEL classification: H41; L31
1. Introduction This paper is an attempt to understand rivalry among ‘‘donative’’ nonprofit firms, which we define as firms that provide public goods using voluntary *Corresponding author. E-mail: [email protected] 0047-2727 / 97 / $17.00 1997 Elsevier Science S.A. All rights reserved. PII S0047-2727( 97 )00046-7
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donations. While ‘‘commercial’’ proprietary and nonprofit firms producing private goods compete for paying customers, these nonprofit firms compete for donations toward the provision of the particular public goods they supply. The type of competition we focus on occurs when firms can attempt to differentiate themselves by offering to provide public goods that have particular characteristics. For example, communities often include several nonprofit organizations that provide a variety of in-kind assistance to the indigent, shelters for battered spouses or runaway teenagers, or support alternative kinds of medical research. Private post-secondary educational institutions in the U.S. differ considerably in the nature of the education they provide, and are partly funded through private contributions. The towns of London, Ontario and Sherbrooke, Quebec are each home to a number of youth hockey leagues, each of them offering different programs and each soliciting private contributions to aid their operations. In this paper we present a simple model in which the following decisions are analyzed: At the first, entry stage, individuals decide whether or not to found a nonprofit firm. Doing so is costly, but entitles one to collect voluntary donations from others, which may then be used to provide some mix of public goods. Next, at the contributions stage, all individuals choose whether and how much to contribute to the various nonprofit firms that are operating. Finally, at the allocation stage, the entrepreneurs determine how they will allocate the funds collected to the provision of various public goods. To keep the analysis tractable, we analyze a model in which there are only two public goods. The model predicts an inherent propensity for donative nonprofit organizations to specialize. This explains why, for example, it is more common to see several separate organizations raising funds for medical research into particular diseases, rather than a single organization raising funds for research into a whole range of diseases. Specialization is useful to donors because they can then control how their contributions are used. The fact that a diversified charity may allocate donations differently than would the donors themselves affects the amounts that they are willing to contribute. The model predicts that, in a well-defined sense, the level of provision of both public goods will be higher if there are two rival specialized firms rather than a single diversified firm. An institution like the United Way can be regarded as a diversified nonprofit firm. Bilodeau (1992) shows that donations to a ‘united charity’ like the United Way can be rationalized without reference to any informational or fund-raising advantages, but the model in that paper does not incorporate the incentives of those who operate the organization. We show below that a diversified charity like the United Way can arise as an equilibrium outcome in the richer model presented here even though it may end up providing less of both goods than would two specialized organizations. It has become increasingly common for local United Way chapters to allow donors to designate how their contributions are to be used. To consider the implications of this, we allow the firm to commit to honour such donor
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designations. We show that such a policy benefits donors, and may also be in the best interests of the organization. Allowing donor designations is a way to reintroduce the benefits of inter-firm competition in situations in which there is a monopoly charity. However, such a policy may be ineffective if a significant fraction of contributors fail to earmark their contributions. This is because a pool of undesignated donations gives the organization a means to offset the effects of any donor designations, and doing so is in the organizations interest once it has been given the donations. This result is a mirror-image of Becker’s (Becker, 1974) Rotten Kid Theorem, in which an altruistic household head guarantees an efficient outcome by reallocating income among household members. We go on to show, however, that by also committing to allocate undesignated funds according to a fixed rule, the monopoly charity can overcome this problem, and regain the mutual benefits that arise from donor designation or specialization by firms.
2. The model Consider a population of n individuals, with individual i having private wealth w i which can be used to buy private consumption, x i , or to provide the public goods Z1 and Z2 . Assume that individual preferences can be represented by utility functions u i (x i ,Z1 ,Z2 ).1 We assume utility functions are increasing in each good, strictly quasi-concave, and that all goods are normal. We further assume that preferences over the public good bundle (Z1 ,Z2 ) are separable from x i , and are homothetic. Utility functions can therefore be written as u i (x i , g i (Z1 ,Z2 ) with g i homogeneous of degree one. Homotheticity is not necessary for any of the results below, but allows us to characterize each individual i by the constant share of expenditures s i 5 Z2 /(Z1 1 Z2 ) which i would like to see allocated to good 2, and thus simplifies the exposition.2 We assume finally that the public goods can be produced at a constant cost of 1 per unit, and that there are no economies of joint production. The environment is one of complete information and the only providers of public goods are nonprofit firms. The model used is a three-stage game with observed actions. 1 The different Zi can be thought of as different characteristics that can be embodied in a public good (e.g., abortion and / or religious counselling at a shelter for pregnant teenagers), or simply as different public goods (e.g., a shelter for pregnant teenagers and a shelter for runaway kids). We also abstract from the fact that some public facilities may confer private benefits to the direct recipients of the services. The specification used here implies that individuals care only about the total quantity of each public good or characteristic. In particular, they are indifferent about whether a given total quantity of a public good is provided by one or many firms, and about whether any firm provides more than one public good. 2 It is sufficient that the income-expansion paths of different individuals in (Z1 ,Z2 ) space not intersect, so that individuals can still be characterized by their desired relative level of expenditure on the two public goods.
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The first, entry stage, produces a set of nonprofit firms. Each firm is operated by some nonprofit entrepreneur who is also a member of the community. The entry of any individual into nonprofit entrepreneurship entails a cost, c, which is the cost of founding a firm to collect donations from the rest of the population.3 At the contributions stage, all individuals (including those who chose previously to become entrepreneurs) simultaneously make non-negative contributions of their private wealth to the k firms founded at Stage I.4 We let d i 5 (d 1i , . . . ,d ki ) be individual i’s vector of contributions to these firms, where d hi is his contribution to k the firm founded by entrepreneur h. Then D i 5 o h51 d hi is the amount contributed n i by i, and D 5 o i 51 D is the total contributed by everyone. b h 5 o in51 d hi is the total (i.e., the budget) received by firm h and b 5 (b 1 , . . . , b k ) is the vector of i j donations received by the k firms. It will also be useful to denote b 2 h 5 o j ±i d h as 2i 2i the amount received by firm h from all donors other than i, and b 5 (b 1 , . . . , i b2 k ) as the amounts collected by each firm from donors other than i. At the final allocation stage, all entrepreneurs simultaneously allocate the contributions they have received at Stage II (i.e., the amounts b h ) to the various public goods. That is, entrepreneur h chooses a vector z h 5 (z h1 ,z h2 ) such that z h1 1 z h2 5 b h . The fact that these are nonprofit firms implies that a non-distribution constraint (NDC) exists, and thus it cannot be that z 1h 1 z 2h , b h . None of the donations received by firm h, including any contribution the entrepreneur may have made from her own wealth, can be used to augment her private consumption, x h . In our model this is the important sense in which these are nonprofit firms, since if this constraint were not present, entrepreneurs could choose to devote less than b h to the provision of public goods. Zj 5 o kh51 z hj will denote the total quantity of public good j provided and 2h Z j 5 o i ±h z ij will be the quantity of good j provided by all firms except h. To reiterate, the order of play in this one-shot game is as follows: Stage I: Individuals simultaneously choose whether or not to become entrepreneurs. Stage II: Individuals simultaneously make private donations to the firms that arose in Stage I. Stage III: Entrepreneurs simultaneously choose allocations of the donations 3
See Bilodeau and Slivinski (1994) and Bilodeau and Slivinski (1996) for models in which an individual rationally founds a nonprofit firm despite the fact that it is privately costly for her to do so. 4 One could have the entrepreneurs make any personal contributions before the general public, as in Varian (1994), or both before and after, as in Bilodeau and Slivinski (1994). The specification used here is simpler, and allows us to focus on the effect of the entrepreneurs’ ability to allocate donations received across various uses.
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received in Stage II to the two public goods. These allocations must satisfy the NDC. The game then ends, and individuals receive their payoffs. The payoff function of individual i is u i (w i 2 D i 2 E i c, g i (Z1 ,Z2 )), where E i is 1 if i chooses to become an entrepreneur, and is 0 if not, and the Zj are derived from the choices of d ih , z hj as defined above.
3. The allocation stage We seek to characterize the subgame perfect equilibria of this game and therefore we start by deriving equilibrium behaviour at the last stage, when k entrepreneurs have already founded nonprofit firms, and have received donations in the amounts (b 1 ,b 2 , . . . ,b k ). Without loss of generality we label them according to the share, s h , of funds they would prefer to devote to good 2, so that s h , s h11 . 2h Given b h , then, and any allocation of funds Z 2 h 5 (Z 2h 1 ,Z 2 ) by the other firms, h h h h h 2h h 2h entrepreneur h chooses z 5 (z 1 ,z 2 ) to maximize g (z 1 1 Z 1 ,z 2 1 Z 2 ) subject to h h 5 z 1 1 z 2 5 b h . The form of this best-response is most easily understood via a diagram like Fig. 1. The curve z h ( ? ) emanating from the origin is entrepreneur h’s income expansion path, defined as z h (D) 5 z h * (D,0). That is, z h ( ? ) expresses how individual h would allocate total donations D across the two public goods, were he the only one with funds to allocate, (i.e., if Z 2 h were 0).6 The allocations of all other firms determine point Z 2 h . Entrepreneur h can then allocate the funds b h to obtain an allocation anywhere on the line segment labelled ab.7 If z h (D) does not intersect F h (b h ,Z 2 h ), the entrepreneur’s best-response will be to allocate nothing to the good which she feels is already relatively overprovided by other firms. This is the case for good 2 in Fig. 1. Firms that provide only one public good will be said to be specialized. The only situation in which an entrepreneur would allocate positive amounts to the provision of both goods is when z h (D) . Z 2 h . Then entrepreneur h would, if alone, allocate total donations D to provide more of both public goods than are being provided by her rivals’ current allocations. In this case z h (D) intersects 5 The separability assumption implies that the entrepreneur’s view of the optimal allocation of b h is independent of his private consumption, and hence does not depend on his wealth, or on the amount he may have donated on his own. 6 The homotheticity assumption implies that z h (D)5((12s h )D,s h D) for some s h between 0 and 1. Fig. 1 is drawn for an entrepreneur with non-homothetic preferences, as none of the analysis of this stage depends on this assumption. 7 This is formally defined as the set
H
O Z 5O Z 2
F h (b h ,Z 2h ) 5 (Z1 ,Z2 )u
j 51
2
j
j 51
2h j
J
1 b h , Zj $ Z 2h . j , j 5 1,2
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Fig. 1. A specialized firm.
F h (b h ,Z 2 h ), as in Fig. 2 and entrepreneur h’s allocation will be z h * (b h ,Z 2 h ) 5 z h (D) 2 Z 2 h . If in equilibrium there is a firm h such that z h (D) . Z 2 h , this firm will be called a dominant firm, because the resulting quantities of the two public goods will be as
Fig. 2. A diversified firm.
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if that firm were a monopoly allocating the entire D on it’s own.8 Further, any h reallocation of funds by other firms that leaves each Z 2 less than z hj (D) can and j will be offset by firm h so as to maintain the outcome Z 5 z h (D). A firm is more likely to be dominant the more donations it has collected, because then F h (b h ,Z 2 h ) spans a larger fraction of the line Z1 1 Z2 5 D. However, what is required for firm h to be dominant is that, given total donations of D, z jh (D) . Z j2 h for j 5 1,2. For any set of entrepreneurs, and any non-negative amounts b h collected by them, the allocation stage has a unique Nash equilibrium outcome in pure strategies.9 We denote the resulting public goods bundle as Z * (b), and denote the equilibrium share of good 2 as r (b) 5 Z 2* (b) /(Z *1 (b) 1 Z *2 (b)). To lighten the notation, the dependence of these variables on b will be suppressed when it will result in no confusion. Given any set of entrepreneurs with distinct s h , in equilibrium at most one of them can be dominant, and the equilibrium share of total donations allocated to good 2 will then be the one preferred by that entrepreneur. Further, in any equilibrium, all entrepreneurs whose s h is less than r will provide only good 1, while only good 2 is provided by all entrepreneurs whose s h is greater than r. The equilibrium allocations for any set of k entrepreneurs and any contributions b 5 hb 1 , . . . ,b k j by the public, have the following properties: • There will be at most one dominant firm. That is, r (b) 5 s h for at most one entrepreneur h, and this entrepreneur will allocate his funds so that z h * (b h ,Z 2 h) 5 z h (D) 2 Z 2h , so Z * (b) 5 z h * (D,0). • If there is no dominant firm, then there must be entrepreneurs i, j such that s i , r (b) , s j . • In any equilibrium, s 1 # r (b) # s k . • Any entrepreneur h for whom s h , r (b) provides z h1 5 b h (specializes in providing good 1), and any for whom s h . r (b) provides z h2 5 b h (specializes in providing good 2). A noteworthy aspect of this allocation stage is the tendency for the nonprofit firms to specialize, even when entrepreneurs value the provision of both public
8 Note that the definition of a dominant firm does not require that the firm has collected more than half the total donations. Both z h (D) and Z 2 h are vectors of provision of the two goods, not scalar sums of money. z h (D) is the bundle that entrepreneur h would provide in the hypothetical situation in which his firm was a monopoly and had collected total amount D, while Z 2 h is the bundle actually provided by all other firms in equilibrium. 9 This last stage game has a finite set of players choosing allocations z h from a compact and convex set, and payoff functions g h (o kh 51 z h1 , o kh 51 z h2 ) which are continuous and strictly concave in all z hj , so that a Nash Equilibrium in pure strategies exists (see Fan (1952) or Debreu (1952)). The proof of uniqueness in the two goods case is identical to that found in Bilodeau (1994) p. 55.
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goods. This result from this two-good model is an extreme manifestation of a tendency that would still be present with m goods, as it follows from each entrepreneur’s desire to modify the overall package of public goods to be closer to that which she most prefers. To summarize:
• If there is a dominant firm, then the equilibrium outcome is the same as if this firm was a monopoly, regardless of how many competitors it has. This is true in the sense that the final allocation of donations across goods would be no different if the dominant firm received all the donations. All other firms will specialize in providing only one public good. • If there is no dominant firm, then all firms specialize in producing only one public good and at least one firm specializes in each good.
4. The contributions stage At this stage, k entrepreneurs have entered and founded nonprofit firms. Each individual i chooses a vector of contributions d i 5 (d i1 , . . . ,d ik ) to the k firms in existence, and it may be that many individuals contribute nothing to some or all firms. This stage differs from the allocation stage in two basic ways. First, the total amount donated is chosen by each individual at this stage, while at the following stage the NDC implied that the total amount to be allocated by each firm was fixed. Entrepreneurs could not allocate less to public good provision than they had received, and had no desire to allocate more. Second, individual donors do not contribute directly to the provision of particular goods, but must contribute to firms whose entrepreneurs then allocate their contributions as they wish. Thus, donors are responding to the contributions of others while also taking into account the way in which firms will later allocate the donations they receive. Therefore, individual i’s best-response to any b 2 i will depend on the equilibrium outcome of the last stage, Z * (b 2 i 1 d i ), because his equilibrium payoff is u i (w i 2 D i 2 E i c, g i [Z 1* (b 2 i 1 d i ), Z 2* (b 2i 1 d i )]). In what follows, we simplify notation by writing Z * (b 2 i ) 5 Z 2 i for the mix of public goods that would be provided in equilibrium from the donations of all contributors other than i. A donor’s best-response to any Z 2 i will depend on the set of firms that have entered. However for any entry configuration, if firms behave optimally at the next stage, a donor at this stage faces one of only two qualitatively different possibilities: either there is a dominant firm or not. Suppose first that a single firm has entered. Then trivially this firm is dominant and individuals have a simple choice: to contribute to it or not at all. If they contribute, their donations buy a composite public good in which the two goods
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are present in the ratio chosen by the firm. Thus, if the entrepreneur prefers the public good mix s h [ ]0,1[, an individual contributor with s i ± s h would prefer that donations be allocated in a different ratio. However, so long as he values at least one of the Zj he will place some value on the provision of the two goods in the ratio Z2 /Z1 5 s h /(1 2 s h ) chosen by the firm and may, therefore, be willing to contribute to this firm. The outcome is then essentially the same as a game where individuals contribute simultaneously to a single public good, as in Bergstrom et al. (1986). If two or more firms have entered, then two possibilities can arise: either all firms are specialized, or one of them is dominant. If one firm is dominant, it does not matter how many firms have entered, as the outcome must be as if this firm is a monopoly. If some firm m is dominant, the other firms will allocate all their funds to one good, with firms 1 through m 2 1 providing only good 1 and firms m 1 1 through k providing only good 2. It may seem that this implies that every donor for whom s i , s m also contributes only to the first m 2 1 firms, in an effort to increase the provision of good 1. However, it does not matter at the margin to which firm any donor contributes. While any dollar contributed to firm m 1 1, say, will be allocated by that firm solely to the provision of good 2, this will also result in Firm m altering it’s own allocation in the next stage so as to keep the proportion of good 2 at s m . Thus, at the margin, all donations to any firm will result in an increase in provision of a composite public good which contains goods 1 and 2 in the proportions 1 2 s m and s m , respectively. It follows that the targeting of donations to firms by individuals is not unique if in equilibrium one firm is dominant. Note then that a dominant firm, if one exists, plays a role much like the benevolent head of the household in the ‘‘Rotten Kid Theorem’’ in Becker (1974). There, a household head who has sufficient income of his own and cares about the consumption of all family members will neutralize actions of other family members which would otherwise alter the distribution of consumption within the family from that which the head views as optimal.10 The other possibility is that all firms are specialized. Then the equilibrium mix of public goods, r, must differ from the s h of all entrepreneurs, otherwise one of them would be dominant. If s 1 , r , s 2 , for example, then firms 2, . . . ,k allocate all their funds to good 2, while Firm 1 allocates everything to good 1. All donors for whom s i , r contribute only to firm 1, and all those with s i . r contribute only to firms 2, . . . ,k (the division of any given amount between these firms being 10 The economic literature contains other models of similar flavour, in which some players are essentially rendered impotent by other players’ ability to completely offset anything they do. In Barro (1974)’s model of government debt with altruistic bequests, the bequestors can completely offset changes in government debt, rendering changes in government policy meaningless. Warr (1983) and Bernheim (1986)’s neutrality results in an economy with a privately provided public good are also in the same vein: if the government is limited to small enough income redistributions, its efforts to alter the income distribution would be completely offset by changes in donations.
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a matter of unanimous indifference). Only an individual with s i 5 r (if any such individuals exist) might contribute to Firm 1 and to one or more of the others. In such an equilibrium firms are acting as perfect agents for donors in that they can be relied on to use all donations in a particular way-providing one of the public goods. In equilibrium, each donor is then trying to move the equilibrium mix closer to the one he prefers by contributing to a specialized firm whose entrepreneur feels the same way. To summarize this discussion, for any entry configuration, only two types of equilibrium continuations are possible: • All firms specialize and individuals contribute to the firm(s) which provide the one good they feel is relatively underprovided by other firms. We will refer to this outcome as a ‘specialized firms outcome’. • One firm is dominant and it does not matter at the margin to which firm anyone contributes. This outcome will be referred to as a ‘‘dominant firm outcome’’. Both of these outcomes could arise as equilibria for a given entry configuration, and in general there could be more than one equilibrium outcome of each type. Proposition 1 in the appendix shows that if preferences are additively separable then there can be at most one equilibrium outcome of the first type (in which all firms specialize). On the other hand, given any set of entering firms, there may be an outcome in which any one of those firms is dominant.11 It is possible to compare the behaviour of individual donors when all firms are specialized (which gives them the choice to contribute to the good they like the most) to their behaviour when there is a dominant firm (which constrains them to contribute toward a composite good). In general, the response of each individual will differ, some contributing more, others less, depending on the proportions favoured by the dominant firm. Also, since the proportions of both goods may not be the same in each case, a dominant firm might provide more of one good but less of the other. To establish a benchmark for comparison, suppose that with specialized firms the unique equilibrium bundle is Z b and that the share of funds devoted to good 2 is s b . For such an equilibrium to exist, no individual i for whom s i 5 s b , if any exist, can have founded a firm. Now suppose that such an individual does exist and has founded a firm. At the equilibrium where his firm is dominant, the share of good 2 would also be s b . Let Z m be the bundle provided in this case. The following proposition, stated formally and proved in the appendix, shows that, 11 Conditions for the existence of a dominant firm are established in Bilodeau (1992). For any firm h, if everyone contributes only to h, then unless some contributor is willing and able to deviate by contributing more to one of the other specialized firms than firm h is allocating to the same good, this is an equilibrium. Any firm whose s h is not too close to 0 or 1 (i.e., whose preferences are not too extreme) can therefore be dominant.
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under a mild preference assumption, more of both goods are provided in the specialized equilibrium. Proposition. If all individuals view the two public goods as substitutes, then Z bj . Z mj , for j 5 1,2. An easy way to see the intuition behind this is to suppose that all those individuals for whom s i , s b in fact get no utility from Z2 , and those for whom s i . s b get none from Z1 . Call these groups 1 and 2, respectively. Then a dominant firm essentially ‘‘taxes’’ their contributions by devoting a fraction of every dollar they contribute to a good they care nothing about, which raises the unit price of contributing a dollar to the provision of their favourite good. The firm also cross-subsidizes the good they care about by allocating to it a fraction of the contributions of the individuals who do not care about it. In equilibrium, this leads each group of individuals to reduce their own contributions below what they would be when two specialized firms are present. As for individuals for whom s i 5 s b , they see no difference in the two situations, except that the other groups contribute less to the dominant firm. Although they respond by increasing their own contributions, they do so by less than a dollar for each dollar reduction in the group 1 and 2 donations. In the general case where individuals care about both goods things are not so simple, but the assumption that individuals view the two public goods as substitutes is sufficient to maintain the result. If the dominant firm prefers a proportion other than s b , it is generally ambiguous whether total contributions to it will be higher or lower than when there are two specialized firms. In Fig. 3, for the values of s defined by each point on the curve labelled v (s), the corresponding vector Z 5 v (s) indicates the public good bundle that would emerge from the existence of a monopoly firm whose founder preferred that particular value of s. From the above proposition, we know that v (s b ) , Z b . It is immediate that v (.) is continuous, however, so that, as Fig. 3 illustrates, more of both goods will be provided by specialized firms than by a dominant firm whose s h is anywhere between s9 and s0. Since there will generally be underprovision of the public goods even in the specialized equilibrium, it follows that (neglecting the entry cost to the entrepreneurs) specialization represents a Pareto-improvement over any of the monopoly equilibria involving entrepreneurs with s h in this range.
5. Entry decisions Many different equilibrium continuations can arise in any subgame following the entry of more than one firm, and this significantly complicates the analysis of entry decisions. Virtually any entry configuration can be sustained as an equilibrium, including configurations in which there are redundant firms collecting no
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Fig. 3. Specialization vs. diversified monopoly.
money. However, all of the equilibrium outcome possibilities can be summarized in two cases:
1. Diversified ‘‘monopoly’’. Whenever one firm is dominant, the outcome is as if that firm were in fact a monopoly. 2. Rival specialized firms. There can be more than one firm specializing in each good, but the outcome is invariant to the number of firms (again neglecting the entry costs). Given this, we do not attempt to characterize equilibrium entry decisions in greater detail. We merely note that adding more firms to the specialized outcome requires the payment of additional entry costs, without affecting the outcome, unless one of the entrants becomes dominant. If that happens, however, there is no reason for the other firms to remain, and if it does not, there is no reason to enter. Similarly, there is no reason for other firms to remain active if some firm is dominant, and no reason to enter against a dominant firm unless the result is specialization.
6. Commitments In subgame perfect equilibria, the entrepreneurs must, in the last stage, allocate received donations in accordance with their own preferences. In this section we
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analyze the possibility that entrepreneurs are able to commit to rules for utilizing donations – even if it is not optimal for the entrepreneur to adhere to the rule after donations have been made. If a commitment to some allocation rule results in an entrepreneur’s firm receiving greater donations, then such a commitment may pay even if the rule implies a less desirable allocation of those donations. There are an infinity of possible rules an entrepreneur might commit to, and attempting to determine an optimal commitment is beyond the scope of this paper. We restrict ourselves to a particular strategy that corresponds to observed behaviour in some nonprofit organizations: a commitment to honour donor instructions regarding the allocation of their donations. We saw above that when there is a diversified monopoly, donors contribute less than they would to rival specialized firms, if the allocation of donations is sufficiently similar in both cases. If a monopoly can find a way to present donors with the same choice opportunities that they would have when faced with specialized firms, more of both public goods can then be provided. One policy to consider involves donors having the option of designating to which uses their contributions should be allocated, together with a commitment to honour these designations. If enough donors 12 take advantage of this option, the outcome will be just as if two rival specialized firms were operating. However, if enough donations are not earmarked, the designation option becomes meaningless because the monopoly will be able to allocate these undesignated funds so as to offset the effect of any designations. So if the monopoly retains discretionary use of undesignated funds, either of these outcomes could obtain in equilibrium even when the monopoly commits to honouring donor designations. Nonprofit managers then must face the fact that the failure to earmark by a significant set of donors can render the designation option meaningless, and prevent the firm from achieving an outcome which everyone may prefer. We noted above the apparent similarity between the dominant-firm equilibrium and Becker’s ‘‘Rotten Kid Theorem’’. However, the result of the household head’s ability to reallocate income within the household is the attainment of an efficient outcome while here the presence of a dominant firm with sufficient non-earmarked funds to allocate can result in a Pareto-inferior allocation of resources to public goods. This is because Becker’s patriarch is altruistic, in that his utility depends positively on the utilities of all household members, while our dominant firm entrepreneur’s utility depends only on her own consumption and the mix of public goods. The difficulty caused by non-designated donations can be circumvented by a further commitment to a fixed rule for allocating those contributions. That is, the organization can specify the proportion of its undesignated revenues that will be 12 An exact formulation of what ‘‘large enough’’ means is given in Bilodeau (1992). Essentially, individuals must earmark more money in aggregate to each good than the monopoly would from the undesignated funds it has.
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allocated to each good. Almost any such commitment will cause everyone to earmark because it prevents the firm from using undesignated funds to offset individual designations, while also implying an allocation of those funds that individuals do not find optimal. A commitment to honour donor designations, coupled with a commitment to allocate non-designated funds according to a fixed rule would therefore always yield a preferred outcome if the entrepreneur’s s m is close to the s b that results from specialization. In practice, perhaps due to decision-making costs (like the cost of gathering information about each member charity of a United Way organization) it is likely that some individuals will fail to earmark. The monopoly may also receive undesignated funds from government grants, fees for service, sales, or investment income. If so, the rule the monopoly commits to for using undesignated funds will then affect the outcome. By choosing the rule appropriately, the entrepreneur can influence the final allocation for the better. This is illustrated in Fig. 4. The monopolist prefers that the fraction s m of each dollar be allocated to good 2 and has D u in undesignated revenues. Designated donations are Z¯ 1 and Z¯ 2 . If she does not commit to a rule, fewer contributions will be received and the outcome will be Z m . By committing to allocate the fraction s c of all undesignated donations to good 2, the firm receives more donations, and achieves outcome Z b which she prefers. What is crucial is that undesignated funds are committed to being spent in a particular way before individuals start contributing, and not allocated after they have done so. It has recently become more common for local United Way Chapters, which can surely be viewed as diversified charities, to allow this sort of donor designation.
Fig. 4. Donor designations.
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However, those chapters which do allow donors to designate vary in the way they treat these designations, and in the way they allocate undesignated funds. In practice, when United Way chapters do allow for donor designation of contributions (and not all do so) they do it in one of two ways. One is to count the designated funds as part of the ‘‘regular allocation’’ of funds to that use. That is, the United Way chapter lays out an ex-ante allocation policy, and any funds designated to a particular use are simply subtracted from the amount of nonearmarked funds that will go to it. In effect this leaves the United Way in a position to use the undesignated funds to offset individual designations at the margin and therefore is not useful. Unless enough individuals designate their contributions that more is designated to each particular use than the United Way would allocate to it otherwise, all designations are meaningless. The other method used is to have a separate allocation policy that applies only to non-designated funds. If this policy specifies a particular fraction of non-designated funds going to each use, this is a commitment to a particular s for non-designated funds only. Since these policies are typically set by ‘‘citizen review boards’’, this can be seen as a means of committing to some s. This second method should, according to our model, yield higher amounts of total contributions. This suggests that data on the way in which various chapters deal with designations, and the amount of designated donations they receive, could allow for an empirical test of this prediction.
7. Conclusion The model developed in this paper generates a number of results regarding the behaviour of nonprofit firms competing for voluntary contributions. We showed that in the presence of competitors, firms will tend to differentiate themselves by specializing, rather than imitating their competitors. Competition between nonprofit firms may also be desirable in the sense that if public goods are underprovided, competing nonprofit firms will elicit larger voluntary contributions from the public than will a monopoly firm. We also showed that commitments to allocation rules may be in the nonprofit entrepreneur’s interest. A monopoly firm may wish to commit to honour donor designation about how their contribution should be spent in order to increase the level of total contributions. For such a commitment to be successful however, the firm must also commit to a fixed rule for allocating any undesignated funds.
8. Appendix Proposition 1. Consider a voluntary provision game in which n individuals simultaneously contribute voluntarily to the provision of 2 public goods. If
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individual preferences can be represented by additively separable utility functions which are strictly concave in each of x i , Z1 and Z2 , then this game has a unique equilibrium outcome. Proof. The separability assumption implies that utility functions have the form u i 5 fi (x i ) 1 gi (Z1 ) 1 h i (Z2 ). Suppose there is an equilibrium in which the outcome is (Z1 ,Z2 ), and the sets of contributors to each good are C1 and C2 . For all i in C1 we have g 9i (Z1 ) 5 f 9i (x i ) $ h i9 (Z2 ), and for all i in C2 we have g 9i (Z1 ) # f 9i (x i ) 5 h 9i (Z2 ). Suppose by way of contradiction that there is another equilibrium in which the outcome is (Zˆ 1 ,Zˆ 2 ), and assume without loss of generality that Zˆ 1 . Z1 . Then Zˆ 1 . Z1 ⇒ g 9i (Zˆ 1 ) , g 9i (Z1 ) by strict concavity. ;i [ ⁄ C2 , g i9 (Z1 ) # f 9i (x i ), so g 9i (Zˆ 1 ) , f 9i (x i ). If any such i contributed to good 1 in the second equilibrium, he would choose xˆ i such that g 9i (Zˆ 1 ) 5 f 9i (xˆ i ). So f i9 (xˆ i ) , f i9 (x i ) and hence xˆ i . x i . And if he were to contribute nothing to good 1, he would also choose xˆ i $ x i . Therefore xˆ i $ x i ;i [ ⁄ C2 and since x i 1 d 1i 5 w i then they must all be contributing no more to good 1 in the second equilibrium. Since Zˆ 1 . Z1 by assumption, it must then be that some i [ C2 contributes more to good 1 in the second equilibrium. Therefore the set C2 > Cˆ 1 is not empty. However, ;i [ C2 > Cˆ 1 , g 9i (Z1 ) # f 9i (x i ) 5 h 9i (Z2 ), and f 9i (xˆ i ) 5 g 9i (Zˆ 1 ) $ h 9i (Zˆ 2 ). But since Zˆ 1 . Z1 , then g i9 (Zˆ 1 ) , g i9 (Z1 ). So f 9i (x i ) 5 h 9i (Z2 ) $ g 9i (Z1 ) . g 9i (Zˆ 1 ) 5 f 9i (xˆ i ) $ h 9i (Zˆ 2 ). This implies that Zˆ 2 .Z2 and xˆ i . x i ;i [C2 >Cˆ 1 . Now consider all i [C2 \Cˆ 1 . If any are in Cˆ 2 then f i9 (xˆ i )5h i9 (Zˆ 2 ) and since ˆZ2 .Z2 then h i9 (Zˆ 2 ),h 9i (Z2 ). So for any such i, f 9i (xˆ i )5h i9 (Zˆ 2 ),h 9i (Z2 )5f 9i (x i ). Therefore it must be that xˆ i .x i . And if some are not in Cˆ 2 then they are not contributing at all and for them also xˆ i 5w i .x i . But we cannot have xˆ i $x i ;i at the same time as Zˆ 1 .Z1 and Zˆ 2 .Z2 without violating the budget constraints. i i 2i i i Proposition 2. Let x i1 (Z 2i and 1 ,t,Z 2 )5argmax x i hu (x ,Z 1 1t(w 2 x ),Z 2 ), i 2i 2i i i i 2i i 2i h 1 (Z 1 ,t,Z2 )5Z 1 1t(w 2 x ). Define x 2 (Z 2 ,t,Z1 ) and h 2 (Z 2 ,t,Z1 ) symmetrically. i b m If ≠h |j / ≠Zj #0 for all i, then Z j .Z j , for j51,2. b
m
b
Proof. Since the two goods are provided in the same ratio s in both Z and Z , the only other possibility is that Z b ,Z m so suppose by way of contradiction that this is the case. Let T 1 be the set of individuals i for whom s i ,s b , let T 2 be the set of all i for whom s i .s b , and let T 0 be the set of those for whom s i 5s b . If any i [T 1 chooses d im 50, then d im #d ib trivially. Suppose that d im .0 for
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some i [T 1 . Dropping the i superscripts for the remainder of the proof, define x 1 (Y1 ,t,Z2 ) and h 1 (Y1 ,t,Z2 ) as above 13 , letting Y1 5Z 12i . 21 Now, define G(x,t,Z2 )5h 1 (x 21 1 (xut,Z 2 ),t,Z 2 ), where x 1 (?ut,Z 2 ) is the inverse of the x 1 function in Y1 . G(x,t,Z2 )5Z1 is then just the function that describes the income expansion path showing the combination of x i and Z1 that i will choose as his ‘‘income’’ varies, when the price of Z1 is t, and the level of public good 2 is fixed at Z2 . Normality implies that the income expansion path is upward sloping so G(.) is increasing in x, and so long as w.x, G is also increasing in t. Let f ;x 21 to ease the notation, and noting that f(x 1 (A,t,Z2 )ut,Z2 ); A, it follows 1 ≠f ≠x 1 ≠f ≠f ≠f ≠x 1 ] 1 ] ; 0, so that ] 5 2 ] ] that ] ≠x ≠Z 2 ≠Z 2 ≠Z 2 ≠x ≠Z 2 . ≠h 1 ≠h 1 ≠f ≠h 1 ≠h 1 ≠G ] ] ] ] ] From the definition of G it follows that ] ≠Z 2 5 ≠Z 2 1 ≠ A ≠Z 2 5 ≠Z 2 2 ( ≠ A / ≠x 1 ≠x 1 ≠x ≠f 1 ] ) ] , where the last equality also uses the fact that ] 5 1 / ] ≠ A ≠Z 2 ≠x ≠A . ≠h 1 ≠x 1 ≠h 1 ≠G ] ] ] Since ] ≠Z 2 and ≠Z 2 cannot have the same sign, it follows that ≠Z 2 # 0⇔ ≠Z 2 # 0, ≠G and thus the assumption in the Proposition implies that ] ≠Z 2 # 0. Now, in the specialized equilibrium, an individual from T 1 donates only to Z1 , b b b b b and therefore his private consumption, x , and Z 1 satisfy Z 1 5G(x ,1,Z 2 ). For the same individual, in the dominant firm equilibrium his private consumption, x m must be the solution to max x hu(x, (12s)(w2x)1Y1 , s(w2x)1Y2 )j, and so x m must satisfy: u x (x m ,Zm ) 5 (1 2 s)u Z 1 (x m ,Zm ) 1 su Z 2 (x m ,Zm ) with the subscripts on u denoting partial derivatives, and Z m1 5(12s)(w 2 x m)1Y1 , and Z m2 5s(w 2 x m)1Y2 . Further, since he is in T 1 , it must be that u mZ 1 . u mZ 2 where these are the derivatives evaluated at (x m ,Z m). However, if we now let t m ;(12s)1su Zm2 /u Zm1 , 1, and y m ;Z 1m 2t m (w2x m), then it can be shown that if he faces the problem: max x hu(x, y m 1t m (w2x),Z m2 )j, then the solution is in fact x m . It follows that x m 5x 1 ( y m ,t m ,Z 2m ) and Z 1m 5h 1 ( y m ,t m ,Z 2m ), so that if the hypothesis is true, then G(x m ,t m ,Z m2 )5Z m1 .Z b1 5G(x b ,1,Z b2 ). Then t m ,1 and Z 2m .Z 2b , together with the above results on the derivatives of G imply that x m .x b if x m ,w, and thus no individual in T 1 donates more in the dominant firm equilibrium. A symmetric result holds for all those in T 2 . Now all i [T 0 can be viewed as consumers of two goods x i and vi 5g i (Z1 ,Z2 ). i i In the dominant firm case, Z5(12s b ,s b )D so letting r 5g (12s b ,s b ), we can write u i (x i ,g i (Z1 ,Z2 )) 5 u i (w i 2 d i ,(D2i 1 d i )r i ). 13
These are just individual i’s demands for private consumption and one public good, given that the provision of the other public good is fixed at some level, donations by others to the one public good are ~ some Z 2i j , and the ‘‘price’’ of the one public good is 1 / t. Note further that for each i, then, a change in Z1 , say, cannot move x i2 and h i2 in the same direction.
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Therefore it must be that if d mi .0, then m m Dm 5 argmax D hu i (w i 2 (D 2 D 2i ),Dr i )j ; Hi ( r i ,D 2i ) m
i
m
m
m
Therefore d i $Hi ( r ,D 2 i )2D 2 i (with equality if d i .0). 2i 2i 2i 2i Let a1 (Z )5(Z 2 /s b )2(Z 1 1Z 2 ) and a2 (Z 2 i )5s b (Z 12i 1Z 22i )1Z 22i . Note 2i 2i 2i 2i that a1 (Z )5 a2 (Z )50 if Z 2 /(Z 1 1Z 2i 2 )5s b and that at most one of a1 (.) or a2 (.) can be positive. Define g (Z 2i ,di )5(g1 (Z 2 i ,di ), g2 (Z 2 i ,di )) as follows:
g (Z 2i ,di )
2i 5 (Z 2i 1 1 d i ,Z 2 ) 2i
if 0 # d i # a1 (Z 2i )
2i
2i
if 0 # d i # a2 (Z )
5 (Z 1 ,Z 2 1 d i ) 2i
2i
5 (Z 1 1 Z 2 1 d i )(1 2 s b ,s b )
otherwise
If there are two specialized firms then the payoff to any i [T 0 is u i (w i 2 d i , g i (g (Z 2 i ,d i ))) if he allocates any total d i optimally for any Z 2 i . Further, 2i g i (g (Z 2i ,d i )) # (Z 2i 1 1 Z 2 1 d i )ri
(with equality iff d i $maxha1 (Z 2 i ), a2 (Z 2 i )j). In particular, in equilibrium g i (g (Z 2ib ,d bi )) 5 (Z 2ib 1 Z 2ib 1 d bi )r i 1 2 b m b m also. So D b 5Hi ( r i ,D 2 i ). But D .D by assumption, so if d i .0, we have m b D m 5 Hi ( r i ,D 2i ) . D b $ Hi ( r i ,D 2i )
and therefore m m b b d mi 5 Hi ( r i ,D 2i ) 2 D 2i , Hi ( ri ,D 2i ) 2 D 2i # d ib m
m
b
m
b
and if d i 50 then d i #d i trivially. So d i #d i for all i [T 0 , also. Therefore if Z m $Z b it must be that d im #d ib for all i, which is a contradiction.
Acknowledgements We thank David Austen-Smith, Rich Steinberg, Pete Streufert and an anomymous referee for helpful suggestions, and gratefully acknowledge financial support from the Aspen Institute’s Nonprofit Sector Research Fund and the Social Sciences and Humanities Research Council of Canada. Part of the work on this paper was done while the authors were visitors at the Indiana University Centre on Philanthropy. We thank the Centre and the IUPUI Department of Economics for their financial and other support during these visits.
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References Barro, R.J., 1974. Are government bonds net wealth?. Journal of Political Economy 82, 1095–1117. Becker, G.S., 1974. A theory of social interactions. Journal of Political Economy 82 (6), 1063–1093. Bergstrom, T., Blume, L., Varian, H., 1986. On the private provision of public goods. Journal of Public Economics 29, 25–49. Bernheim, D., 1986. On the voluntary and involuntary provision of public goods. American Economic Review 75, 789–793. Bilodeau, M., 1992. Voluntary contributions to united charities. Journal of Public Economics 48, 119–133. Bilodeau, M., 1994. Tax-earmarking and separate school financing. Journal of Public Economics 54, 51–63. Bilodeau, M., Slivinski, A., 1994. Rational nonprofit entrepreneurship. Working Paper 93-2, Universite´ de Sherbrooke. Bilodeau, M., Slivinski, A., 1996. Volunteering nonprofit entrepreneurial services. Journal of Economic Behavior and Organization 31, 117–127. Debreu, G., 1952. A social equilibrium existence theorem. In: Proceedings of the National Academy of Sciences, USA, pp. 386–393. Fan, K., 1952. Some properties of convex sets related to fixed-point theorems. Math. Ann. pp. 519–537. Varian, H., 1994. Sequential provision of public goods. Journal of Public Economics 53, 165–186. Warr, P.G., 1983. The private provision of a public good is independent of the distribution of income. Economics Letters 13, 207–211.
Rival charities a, b Marc Bilodeau *, Al Slivinski a
´ ´ , Universite´ de Sherbrooke, and Department of Economics, IUPUI, Departement d’ Economique Cavanaugh Hall 517, Indianapolis, IN 46202 -5140, USA b Department of Economics, University of Western Ontario, London, Ontario, Canada Received 1 June 1996; received in revised form 1 February 1997; accepted 8 April 1997
Abstract The paper develops a model in which a number of charities (or other nonprofit firms) provide various bundles of public goods or services through private donations. The motivation for individuals to found and operate such firms is that it allows them to influence the mix of public goods. It is their decisions regarding the allocation of donations across uses that matter in the end. Donors to these firms take into account the allocation decisions that will be made by the organizations to which they contribute. We find a propensity for such organizations to specialize in the provision of services, and further find that diversification by such firms diminishes the equilibrium level of contributions they will collect. We demonstrate the possibility that a commitment by a monopoly charity to an allocation rule that is, ex-post, privately sub-optimal can eliminate this effect, and may therefore be advantageous, ex-ante. The allocation rule which accomplishes this involves honouring donor designations of their contributions to specific uses. This is a policy that is frequently adopted by local chapters of the United Way. 1997 Elsevier Science S.A. Keywords: Nonprofit enterprises; Public goods; Voluntary contributions; Spatial competition JEL classification: H41; L31
1. Introduction This paper is an attempt to understand rivalry among ‘‘donative’’ nonprofit firms, which we define as firms that provide public goods using voluntary *Corresponding author. E-mail: [email protected] 0047-2727 / 97 / $17.00 1997 Elsevier Science S.A. All rights reserved. PII S0047-2727( 97 )00046-7
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donations. While ‘‘commercial’’ proprietary and nonprofit firms producing private goods compete for paying customers, these nonprofit firms compete for donations toward the provision of the particular public goods they supply. The type of competition we focus on occurs when firms can attempt to differentiate themselves by offering to provide public goods that have particular characteristics. For example, communities often include several nonprofit organizations that provide a variety of in-kind assistance to the indigent, shelters for battered spouses or runaway teenagers, or support alternative kinds of medical research. Private post-secondary educational institutions in the U.S. differ considerably in the nature of the education they provide, and are partly funded through private contributions. The towns of London, Ontario and Sherbrooke, Quebec are each home to a number of youth hockey leagues, each of them offering different programs and each soliciting private contributions to aid their operations. In this paper we present a simple model in which the following decisions are analyzed: At the first, entry stage, individuals decide whether or not to found a nonprofit firm. Doing so is costly, but entitles one to collect voluntary donations from others, which may then be used to provide some mix of public goods. Next, at the contributions stage, all individuals choose whether and how much to contribute to the various nonprofit firms that are operating. Finally, at the allocation stage, the entrepreneurs determine how they will allocate the funds collected to the provision of various public goods. To keep the analysis tractable, we analyze a model in which there are only two public goods. The model predicts an inherent propensity for donative nonprofit organizations to specialize. This explains why, for example, it is more common to see several separate organizations raising funds for medical research into particular diseases, rather than a single organization raising funds for research into a whole range of diseases. Specialization is useful to donors because they can then control how their contributions are used. The fact that a diversified charity may allocate donations differently than would the donors themselves affects the amounts that they are willing to contribute. The model predicts that, in a well-defined sense, the level of provision of both public goods will be higher if there are two rival specialized firms rather than a single diversified firm. An institution like the United Way can be regarded as a diversified nonprofit firm. Bilodeau (1992) shows that donations to a ‘united charity’ like the United Way can be rationalized without reference to any informational or fund-raising advantages, but the model in that paper does not incorporate the incentives of those who operate the organization. We show below that a diversified charity like the United Way can arise as an equilibrium outcome in the richer model presented here even though it may end up providing less of both goods than would two specialized organizations. It has become increasingly common for local United Way chapters to allow donors to designate how their contributions are to be used. To consider the implications of this, we allow the firm to commit to honour such donor
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designations. We show that such a policy benefits donors, and may also be in the best interests of the organization. Allowing donor designations is a way to reintroduce the benefits of inter-firm competition in situations in which there is a monopoly charity. However, such a policy may be ineffective if a significant fraction of contributors fail to earmark their contributions. This is because a pool of undesignated donations gives the organization a means to offset the effects of any donor designations, and doing so is in the organizations interest once it has been given the donations. This result is a mirror-image of Becker’s (Becker, 1974) Rotten Kid Theorem, in which an altruistic household head guarantees an efficient outcome by reallocating income among household members. We go on to show, however, that by also committing to allocate undesignated funds according to a fixed rule, the monopoly charity can overcome this problem, and regain the mutual benefits that arise from donor designation or specialization by firms.
2. The model Consider a population of n individuals, with individual i having private wealth w i which can be used to buy private consumption, x i , or to provide the public goods Z1 and Z2 . Assume that individual preferences can be represented by utility functions u i (x i ,Z1 ,Z2 ).1 We assume utility functions are increasing in each good, strictly quasi-concave, and that all goods are normal. We further assume that preferences over the public good bundle (Z1 ,Z2 ) are separable from x i , and are homothetic. Utility functions can therefore be written as u i (x i , g i (Z1 ,Z2 ) with g i homogeneous of degree one. Homotheticity is not necessary for any of the results below, but allows us to characterize each individual i by the constant share of expenditures s i 5 Z2 /(Z1 1 Z2 ) which i would like to see allocated to good 2, and thus simplifies the exposition.2 We assume finally that the public goods can be produced at a constant cost of 1 per unit, and that there are no economies of joint production. The environment is one of complete information and the only providers of public goods are nonprofit firms. The model used is a three-stage game with observed actions. 1 The different Zi can be thought of as different characteristics that can be embodied in a public good (e.g., abortion and / or religious counselling at a shelter for pregnant teenagers), or simply as different public goods (e.g., a shelter for pregnant teenagers and a shelter for runaway kids). We also abstract from the fact that some public facilities may confer private benefits to the direct recipients of the services. The specification used here implies that individuals care only about the total quantity of each public good or characteristic. In particular, they are indifferent about whether a given total quantity of a public good is provided by one or many firms, and about whether any firm provides more than one public good. 2 It is sufficient that the income-expansion paths of different individuals in (Z1 ,Z2 ) space not intersect, so that individuals can still be characterized by their desired relative level of expenditure on the two public goods.
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The first, entry stage, produces a set of nonprofit firms. Each firm is operated by some nonprofit entrepreneur who is also a member of the community. The entry of any individual into nonprofit entrepreneurship entails a cost, c, which is the cost of founding a firm to collect donations from the rest of the population.3 At the contributions stage, all individuals (including those who chose previously to become entrepreneurs) simultaneously make non-negative contributions of their private wealth to the k firms founded at Stage I.4 We let d i 5 (d 1i , . . . ,d ki ) be individual i’s vector of contributions to these firms, where d hi is his contribution to k the firm founded by entrepreneur h. Then D i 5 o h51 d hi is the amount contributed n i by i, and D 5 o i 51 D is the total contributed by everyone. b h 5 o in51 d hi is the total (i.e., the budget) received by firm h and b 5 (b 1 , . . . , b k ) is the vector of i j donations received by the k firms. It will also be useful to denote b 2 h 5 o j ±i d h as 2i 2i the amount received by firm h from all donors other than i, and b 5 (b 1 , . . . , i b2 k ) as the amounts collected by each firm from donors other than i. At the final allocation stage, all entrepreneurs simultaneously allocate the contributions they have received at Stage II (i.e., the amounts b h ) to the various public goods. That is, entrepreneur h chooses a vector z h 5 (z h1 ,z h2 ) such that z h1 1 z h2 5 b h . The fact that these are nonprofit firms implies that a non-distribution constraint (NDC) exists, and thus it cannot be that z 1h 1 z 2h , b h . None of the donations received by firm h, including any contribution the entrepreneur may have made from her own wealth, can be used to augment her private consumption, x h . In our model this is the important sense in which these are nonprofit firms, since if this constraint were not present, entrepreneurs could choose to devote less than b h to the provision of public goods. Zj 5 o kh51 z hj will denote the total quantity of public good j provided and 2h Z j 5 o i ±h z ij will be the quantity of good j provided by all firms except h. To reiterate, the order of play in this one-shot game is as follows: Stage I: Individuals simultaneously choose whether or not to become entrepreneurs. Stage II: Individuals simultaneously make private donations to the firms that arose in Stage I. Stage III: Entrepreneurs simultaneously choose allocations of the donations 3
See Bilodeau and Slivinski (1994) and Bilodeau and Slivinski (1996) for models in which an individual rationally founds a nonprofit firm despite the fact that it is privately costly for her to do so. 4 One could have the entrepreneurs make any personal contributions before the general public, as in Varian (1994), or both before and after, as in Bilodeau and Slivinski (1994). The specification used here is simpler, and allows us to focus on the effect of the entrepreneurs’ ability to allocate donations received across various uses.
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received in Stage II to the two public goods. These allocations must satisfy the NDC. The game then ends, and individuals receive their payoffs. The payoff function of individual i is u i (w i 2 D i 2 E i c, g i (Z1 ,Z2 )), where E i is 1 if i chooses to become an entrepreneur, and is 0 if not, and the Zj are derived from the choices of d ih , z hj as defined above.
3. The allocation stage We seek to characterize the subgame perfect equilibria of this game and therefore we start by deriving equilibrium behaviour at the last stage, when k entrepreneurs have already founded nonprofit firms, and have received donations in the amounts (b 1 ,b 2 , . . . ,b k ). Without loss of generality we label them according to the share, s h , of funds they would prefer to devote to good 2, so that s h , s h11 . 2h Given b h , then, and any allocation of funds Z 2 h 5 (Z 2h 1 ,Z 2 ) by the other firms, h h h h h 2h h 2h entrepreneur h chooses z 5 (z 1 ,z 2 ) to maximize g (z 1 1 Z 1 ,z 2 1 Z 2 ) subject to h h 5 z 1 1 z 2 5 b h . The form of this best-response is most easily understood via a diagram like Fig. 1. The curve z h ( ? ) emanating from the origin is entrepreneur h’s income expansion path, defined as z h (D) 5 z h * (D,0). That is, z h ( ? ) expresses how individual h would allocate total donations D across the two public goods, were he the only one with funds to allocate, (i.e., if Z 2 h were 0).6 The allocations of all other firms determine point Z 2 h . Entrepreneur h can then allocate the funds b h to obtain an allocation anywhere on the line segment labelled ab.7 If z h (D) does not intersect F h (b h ,Z 2 h ), the entrepreneur’s best-response will be to allocate nothing to the good which she feels is already relatively overprovided by other firms. This is the case for good 2 in Fig. 1. Firms that provide only one public good will be said to be specialized. The only situation in which an entrepreneur would allocate positive amounts to the provision of both goods is when z h (D) . Z 2 h . Then entrepreneur h would, if alone, allocate total donations D to provide more of both public goods than are being provided by her rivals’ current allocations. In this case z h (D) intersects 5 The separability assumption implies that the entrepreneur’s view of the optimal allocation of b h is independent of his private consumption, and hence does not depend on his wealth, or on the amount he may have donated on his own. 6 The homotheticity assumption implies that z h (D)5((12s h )D,s h D) for some s h between 0 and 1. Fig. 1 is drawn for an entrepreneur with non-homothetic preferences, as none of the analysis of this stage depends on this assumption. 7 This is formally defined as the set
H
O Z 5O Z 2
F h (b h ,Z 2h ) 5 (Z1 ,Z2 )u
j 51
2
j
j 51
2h j
J
1 b h , Zj $ Z 2h . j , j 5 1,2
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Fig. 1. A specialized firm.
F h (b h ,Z 2 h ), as in Fig. 2 and entrepreneur h’s allocation will be z h * (b h ,Z 2 h ) 5 z h (D) 2 Z 2 h . If in equilibrium there is a firm h such that z h (D) . Z 2 h , this firm will be called a dominant firm, because the resulting quantities of the two public goods will be as
Fig. 2. A diversified firm.
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if that firm were a monopoly allocating the entire D on it’s own.8 Further, any h reallocation of funds by other firms that leaves each Z 2 less than z hj (D) can and j will be offset by firm h so as to maintain the outcome Z 5 z h (D). A firm is more likely to be dominant the more donations it has collected, because then F h (b h ,Z 2 h ) spans a larger fraction of the line Z1 1 Z2 5 D. However, what is required for firm h to be dominant is that, given total donations of D, z jh (D) . Z j2 h for j 5 1,2. For any set of entrepreneurs, and any non-negative amounts b h collected by them, the allocation stage has a unique Nash equilibrium outcome in pure strategies.9 We denote the resulting public goods bundle as Z * (b), and denote the equilibrium share of good 2 as r (b) 5 Z 2* (b) /(Z *1 (b) 1 Z *2 (b)). To lighten the notation, the dependence of these variables on b will be suppressed when it will result in no confusion. Given any set of entrepreneurs with distinct s h , in equilibrium at most one of them can be dominant, and the equilibrium share of total donations allocated to good 2 will then be the one preferred by that entrepreneur. Further, in any equilibrium, all entrepreneurs whose s h is less than r will provide only good 1, while only good 2 is provided by all entrepreneurs whose s h is greater than r. The equilibrium allocations for any set of k entrepreneurs and any contributions b 5 hb 1 , . . . ,b k j by the public, have the following properties: • There will be at most one dominant firm. That is, r (b) 5 s h for at most one entrepreneur h, and this entrepreneur will allocate his funds so that z h * (b h ,Z 2 h) 5 z h (D) 2 Z 2h , so Z * (b) 5 z h * (D,0). • If there is no dominant firm, then there must be entrepreneurs i, j such that s i , r (b) , s j . • In any equilibrium, s 1 # r (b) # s k . • Any entrepreneur h for whom s h , r (b) provides z h1 5 b h (specializes in providing good 1), and any for whom s h . r (b) provides z h2 5 b h (specializes in providing good 2). A noteworthy aspect of this allocation stage is the tendency for the nonprofit firms to specialize, even when entrepreneurs value the provision of both public
8 Note that the definition of a dominant firm does not require that the firm has collected more than half the total donations. Both z h (D) and Z 2 h are vectors of provision of the two goods, not scalar sums of money. z h (D) is the bundle that entrepreneur h would provide in the hypothetical situation in which his firm was a monopoly and had collected total amount D, while Z 2 h is the bundle actually provided by all other firms in equilibrium. 9 This last stage game has a finite set of players choosing allocations z h from a compact and convex set, and payoff functions g h (o kh 51 z h1 , o kh 51 z h2 ) which are continuous and strictly concave in all z hj , so that a Nash Equilibrium in pure strategies exists (see Fan (1952) or Debreu (1952)). The proof of uniqueness in the two goods case is identical to that found in Bilodeau (1994) p. 55.
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goods. This result from this two-good model is an extreme manifestation of a tendency that would still be present with m goods, as it follows from each entrepreneur’s desire to modify the overall package of public goods to be closer to that which she most prefers. To summarize:
• If there is a dominant firm, then the equilibrium outcome is the same as if this firm was a monopoly, regardless of how many competitors it has. This is true in the sense that the final allocation of donations across goods would be no different if the dominant firm received all the donations. All other firms will specialize in providing only one public good. • If there is no dominant firm, then all firms specialize in producing only one public good and at least one firm specializes in each good.
4. The contributions stage At this stage, k entrepreneurs have entered and founded nonprofit firms. Each individual i chooses a vector of contributions d i 5 (d i1 , . . . ,d ik ) to the k firms in existence, and it may be that many individuals contribute nothing to some or all firms. This stage differs from the allocation stage in two basic ways. First, the total amount donated is chosen by each individual at this stage, while at the following stage the NDC implied that the total amount to be allocated by each firm was fixed. Entrepreneurs could not allocate less to public good provision than they had received, and had no desire to allocate more. Second, individual donors do not contribute directly to the provision of particular goods, but must contribute to firms whose entrepreneurs then allocate their contributions as they wish. Thus, donors are responding to the contributions of others while also taking into account the way in which firms will later allocate the donations they receive. Therefore, individual i’s best-response to any b 2 i will depend on the equilibrium outcome of the last stage, Z * (b 2 i 1 d i ), because his equilibrium payoff is u i (w i 2 D i 2 E i c, g i [Z 1* (b 2 i 1 d i ), Z 2* (b 2i 1 d i )]). In what follows, we simplify notation by writing Z * (b 2 i ) 5 Z 2 i for the mix of public goods that would be provided in equilibrium from the donations of all contributors other than i. A donor’s best-response to any Z 2 i will depend on the set of firms that have entered. However for any entry configuration, if firms behave optimally at the next stage, a donor at this stage faces one of only two qualitatively different possibilities: either there is a dominant firm or not. Suppose first that a single firm has entered. Then trivially this firm is dominant and individuals have a simple choice: to contribute to it or not at all. If they contribute, their donations buy a composite public good in which the two goods
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are present in the ratio chosen by the firm. Thus, if the entrepreneur prefers the public good mix s h [ ]0,1[, an individual contributor with s i ± s h would prefer that donations be allocated in a different ratio. However, so long as he values at least one of the Zj he will place some value on the provision of the two goods in the ratio Z2 /Z1 5 s h /(1 2 s h ) chosen by the firm and may, therefore, be willing to contribute to this firm. The outcome is then essentially the same as a game where individuals contribute simultaneously to a single public good, as in Bergstrom et al. (1986). If two or more firms have entered, then two possibilities can arise: either all firms are specialized, or one of them is dominant. If one firm is dominant, it does not matter how many firms have entered, as the outcome must be as if this firm is a monopoly. If some firm m is dominant, the other firms will allocate all their funds to one good, with firms 1 through m 2 1 providing only good 1 and firms m 1 1 through k providing only good 2. It may seem that this implies that every donor for whom s i , s m also contributes only to the first m 2 1 firms, in an effort to increase the provision of good 1. However, it does not matter at the margin to which firm any donor contributes. While any dollar contributed to firm m 1 1, say, will be allocated by that firm solely to the provision of good 2, this will also result in Firm m altering it’s own allocation in the next stage so as to keep the proportion of good 2 at s m . Thus, at the margin, all donations to any firm will result in an increase in provision of a composite public good which contains goods 1 and 2 in the proportions 1 2 s m and s m , respectively. It follows that the targeting of donations to firms by individuals is not unique if in equilibrium one firm is dominant. Note then that a dominant firm, if one exists, plays a role much like the benevolent head of the household in the ‘‘Rotten Kid Theorem’’ in Becker (1974). There, a household head who has sufficient income of his own and cares about the consumption of all family members will neutralize actions of other family members which would otherwise alter the distribution of consumption within the family from that which the head views as optimal.10 The other possibility is that all firms are specialized. Then the equilibrium mix of public goods, r, must differ from the s h of all entrepreneurs, otherwise one of them would be dominant. If s 1 , r , s 2 , for example, then firms 2, . . . ,k allocate all their funds to good 2, while Firm 1 allocates everything to good 1. All donors for whom s i , r contribute only to firm 1, and all those with s i . r contribute only to firms 2, . . . ,k (the division of any given amount between these firms being 10 The economic literature contains other models of similar flavour, in which some players are essentially rendered impotent by other players’ ability to completely offset anything they do. In Barro (1974)’s model of government debt with altruistic bequests, the bequestors can completely offset changes in government debt, rendering changes in government policy meaningless. Warr (1983) and Bernheim (1986)’s neutrality results in an economy with a privately provided public good are also in the same vein: if the government is limited to small enough income redistributions, its efforts to alter the income distribution would be completely offset by changes in donations.
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a matter of unanimous indifference). Only an individual with s i 5 r (if any such individuals exist) might contribute to Firm 1 and to one or more of the others. In such an equilibrium firms are acting as perfect agents for donors in that they can be relied on to use all donations in a particular way-providing one of the public goods. In equilibrium, each donor is then trying to move the equilibrium mix closer to the one he prefers by contributing to a specialized firm whose entrepreneur feels the same way. To summarize this discussion, for any entry configuration, only two types of equilibrium continuations are possible: • All firms specialize and individuals contribute to the firm(s) which provide the one good they feel is relatively underprovided by other firms. We will refer to this outcome as a ‘specialized firms outcome’. • One firm is dominant and it does not matter at the margin to which firm anyone contributes. This outcome will be referred to as a ‘‘dominant firm outcome’’. Both of these outcomes could arise as equilibria for a given entry configuration, and in general there could be more than one equilibrium outcome of each type. Proposition 1 in the appendix shows that if preferences are additively separable then there can be at most one equilibrium outcome of the first type (in which all firms specialize). On the other hand, given any set of entering firms, there may be an outcome in which any one of those firms is dominant.11 It is possible to compare the behaviour of individual donors when all firms are specialized (which gives them the choice to contribute to the good they like the most) to their behaviour when there is a dominant firm (which constrains them to contribute toward a composite good). In general, the response of each individual will differ, some contributing more, others less, depending on the proportions favoured by the dominant firm. Also, since the proportions of both goods may not be the same in each case, a dominant firm might provide more of one good but less of the other. To establish a benchmark for comparison, suppose that with specialized firms the unique equilibrium bundle is Z b and that the share of funds devoted to good 2 is s b . For such an equilibrium to exist, no individual i for whom s i 5 s b , if any exist, can have founded a firm. Now suppose that such an individual does exist and has founded a firm. At the equilibrium where his firm is dominant, the share of good 2 would also be s b . Let Z m be the bundle provided in this case. The following proposition, stated formally and proved in the appendix, shows that, 11 Conditions for the existence of a dominant firm are established in Bilodeau (1992). For any firm h, if everyone contributes only to h, then unless some contributor is willing and able to deviate by contributing more to one of the other specialized firms than firm h is allocating to the same good, this is an equilibrium. Any firm whose s h is not too close to 0 or 1 (i.e., whose preferences are not too extreme) can therefore be dominant.
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under a mild preference assumption, more of both goods are provided in the specialized equilibrium. Proposition. If all individuals view the two public goods as substitutes, then Z bj . Z mj , for j 5 1,2. An easy way to see the intuition behind this is to suppose that all those individuals for whom s i , s b in fact get no utility from Z2 , and those for whom s i . s b get none from Z1 . Call these groups 1 and 2, respectively. Then a dominant firm essentially ‘‘taxes’’ their contributions by devoting a fraction of every dollar they contribute to a good they care nothing about, which raises the unit price of contributing a dollar to the provision of their favourite good. The firm also cross-subsidizes the good they care about by allocating to it a fraction of the contributions of the individuals who do not care about it. In equilibrium, this leads each group of individuals to reduce their own contributions below what they would be when two specialized firms are present. As for individuals for whom s i 5 s b , they see no difference in the two situations, except that the other groups contribute less to the dominant firm. Although they respond by increasing their own contributions, they do so by less than a dollar for each dollar reduction in the group 1 and 2 donations. In the general case where individuals care about both goods things are not so simple, but the assumption that individuals view the two public goods as substitutes is sufficient to maintain the result. If the dominant firm prefers a proportion other than s b , it is generally ambiguous whether total contributions to it will be higher or lower than when there are two specialized firms. In Fig. 3, for the values of s defined by each point on the curve labelled v (s), the corresponding vector Z 5 v (s) indicates the public good bundle that would emerge from the existence of a monopoly firm whose founder preferred that particular value of s. From the above proposition, we know that v (s b ) , Z b . It is immediate that v (.) is continuous, however, so that, as Fig. 3 illustrates, more of both goods will be provided by specialized firms than by a dominant firm whose s h is anywhere between s9 and s0. Since there will generally be underprovision of the public goods even in the specialized equilibrium, it follows that (neglecting the entry cost to the entrepreneurs) specialization represents a Pareto-improvement over any of the monopoly equilibria involving entrepreneurs with s h in this range.
5. Entry decisions Many different equilibrium continuations can arise in any subgame following the entry of more than one firm, and this significantly complicates the analysis of entry decisions. Virtually any entry configuration can be sustained as an equilibrium, including configurations in which there are redundant firms collecting no
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Fig. 3. Specialization vs. diversified monopoly.
money. However, all of the equilibrium outcome possibilities can be summarized in two cases:
1. Diversified ‘‘monopoly’’. Whenever one firm is dominant, the outcome is as if that firm were in fact a monopoly. 2. Rival specialized firms. There can be more than one firm specializing in each good, but the outcome is invariant to the number of firms (again neglecting the entry costs). Given this, we do not attempt to characterize equilibrium entry decisions in greater detail. We merely note that adding more firms to the specialized outcome requires the payment of additional entry costs, without affecting the outcome, unless one of the entrants becomes dominant. If that happens, however, there is no reason for the other firms to remain, and if it does not, there is no reason to enter. Similarly, there is no reason for other firms to remain active if some firm is dominant, and no reason to enter against a dominant firm unless the result is specialization.
6. Commitments In subgame perfect equilibria, the entrepreneurs must, in the last stage, allocate received donations in accordance with their own preferences. In this section we
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analyze the possibility that entrepreneurs are able to commit to rules for utilizing donations – even if it is not optimal for the entrepreneur to adhere to the rule after donations have been made. If a commitment to some allocation rule results in an entrepreneur’s firm receiving greater donations, then such a commitment may pay even if the rule implies a less desirable allocation of those donations. There are an infinity of possible rules an entrepreneur might commit to, and attempting to determine an optimal commitment is beyond the scope of this paper. We restrict ourselves to a particular strategy that corresponds to observed behaviour in some nonprofit organizations: a commitment to honour donor instructions regarding the allocation of their donations. We saw above that when there is a diversified monopoly, donors contribute less than they would to rival specialized firms, if the allocation of donations is sufficiently similar in both cases. If a monopoly can find a way to present donors with the same choice opportunities that they would have when faced with specialized firms, more of both public goods can then be provided. One policy to consider involves donors having the option of designating to which uses their contributions should be allocated, together with a commitment to honour these designations. If enough donors 12 take advantage of this option, the outcome will be just as if two rival specialized firms were operating. However, if enough donations are not earmarked, the designation option becomes meaningless because the monopoly will be able to allocate these undesignated funds so as to offset the effect of any designations. So if the monopoly retains discretionary use of undesignated funds, either of these outcomes could obtain in equilibrium even when the monopoly commits to honouring donor designations. Nonprofit managers then must face the fact that the failure to earmark by a significant set of donors can render the designation option meaningless, and prevent the firm from achieving an outcome which everyone may prefer. We noted above the apparent similarity between the dominant-firm equilibrium and Becker’s ‘‘Rotten Kid Theorem’’. However, the result of the household head’s ability to reallocate income within the household is the attainment of an efficient outcome while here the presence of a dominant firm with sufficient non-earmarked funds to allocate can result in a Pareto-inferior allocation of resources to public goods. This is because Becker’s patriarch is altruistic, in that his utility depends positively on the utilities of all household members, while our dominant firm entrepreneur’s utility depends only on her own consumption and the mix of public goods. The difficulty caused by non-designated donations can be circumvented by a further commitment to a fixed rule for allocating those contributions. That is, the organization can specify the proportion of its undesignated revenues that will be 12 An exact formulation of what ‘‘large enough’’ means is given in Bilodeau (1992). Essentially, individuals must earmark more money in aggregate to each good than the monopoly would from the undesignated funds it has.
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allocated to each good. Almost any such commitment will cause everyone to earmark because it prevents the firm from using undesignated funds to offset individual designations, while also implying an allocation of those funds that individuals do not find optimal. A commitment to honour donor designations, coupled with a commitment to allocate non-designated funds according to a fixed rule would therefore always yield a preferred outcome if the entrepreneur’s s m is close to the s b that results from specialization. In practice, perhaps due to decision-making costs (like the cost of gathering information about each member charity of a United Way organization) it is likely that some individuals will fail to earmark. The monopoly may also receive undesignated funds from government grants, fees for service, sales, or investment income. If so, the rule the monopoly commits to for using undesignated funds will then affect the outcome. By choosing the rule appropriately, the entrepreneur can influence the final allocation for the better. This is illustrated in Fig. 4. The monopolist prefers that the fraction s m of each dollar be allocated to good 2 and has D u in undesignated revenues. Designated donations are Z¯ 1 and Z¯ 2 . If she does not commit to a rule, fewer contributions will be received and the outcome will be Z m . By committing to allocate the fraction s c of all undesignated donations to good 2, the firm receives more donations, and achieves outcome Z b which she prefers. What is crucial is that undesignated funds are committed to being spent in a particular way before individuals start contributing, and not allocated after they have done so. It has recently become more common for local United Way Chapters, which can surely be viewed as diversified charities, to allow this sort of donor designation.
Fig. 4. Donor designations.
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However, those chapters which do allow donors to designate vary in the way they treat these designations, and in the way they allocate undesignated funds. In practice, when United Way chapters do allow for donor designation of contributions (and not all do so) they do it in one of two ways. One is to count the designated funds as part of the ‘‘regular allocation’’ of funds to that use. That is, the United Way chapter lays out an ex-ante allocation policy, and any funds designated to a particular use are simply subtracted from the amount of nonearmarked funds that will go to it. In effect this leaves the United Way in a position to use the undesignated funds to offset individual designations at the margin and therefore is not useful. Unless enough individuals designate their contributions that more is designated to each particular use than the United Way would allocate to it otherwise, all designations are meaningless. The other method used is to have a separate allocation policy that applies only to non-designated funds. If this policy specifies a particular fraction of non-designated funds going to each use, this is a commitment to a particular s for non-designated funds only. Since these policies are typically set by ‘‘citizen review boards’’, this can be seen as a means of committing to some s. This second method should, according to our model, yield higher amounts of total contributions. This suggests that data on the way in which various chapters deal with designations, and the amount of designated donations they receive, could allow for an empirical test of this prediction.
7. Conclusion The model developed in this paper generates a number of results regarding the behaviour of nonprofit firms competing for voluntary contributions. We showed that in the presence of competitors, firms will tend to differentiate themselves by specializing, rather than imitating their competitors. Competition between nonprofit firms may also be desirable in the sense that if public goods are underprovided, competing nonprofit firms will elicit larger voluntary contributions from the public than will a monopoly firm. We also showed that commitments to allocation rules may be in the nonprofit entrepreneur’s interest. A monopoly firm may wish to commit to honour donor designation about how their contribution should be spent in order to increase the level of total contributions. For such a commitment to be successful however, the firm must also commit to a fixed rule for allocating any undesignated funds.
8. Appendix Proposition 1. Consider a voluntary provision game in which n individuals simultaneously contribute voluntarily to the provision of 2 public goods. If
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individual preferences can be represented by additively separable utility functions which are strictly concave in each of x i , Z1 and Z2 , then this game has a unique equilibrium outcome. Proof. The separability assumption implies that utility functions have the form u i 5 fi (x i ) 1 gi (Z1 ) 1 h i (Z2 ). Suppose there is an equilibrium in which the outcome is (Z1 ,Z2 ), and the sets of contributors to each good are C1 and C2 . For all i in C1 we have g 9i (Z1 ) 5 f 9i (x i ) $ h i9 (Z2 ), and for all i in C2 we have g 9i (Z1 ) # f 9i (x i ) 5 h 9i (Z2 ). Suppose by way of contradiction that there is another equilibrium in which the outcome is (Zˆ 1 ,Zˆ 2 ), and assume without loss of generality that Zˆ 1 . Z1 . Then Zˆ 1 . Z1 ⇒ g 9i (Zˆ 1 ) , g 9i (Z1 ) by strict concavity. ;i [ ⁄ C2 , g i9 (Z1 ) # f 9i (x i ), so g 9i (Zˆ 1 ) , f 9i (x i ). If any such i contributed to good 1 in the second equilibrium, he would choose xˆ i such that g 9i (Zˆ 1 ) 5 f 9i (xˆ i ). So f i9 (xˆ i ) , f i9 (x i ) and hence xˆ i . x i . And if he were to contribute nothing to good 1, he would also choose xˆ i $ x i . Therefore xˆ i $ x i ;i [ ⁄ C2 and since x i 1 d 1i 5 w i then they must all be contributing no more to good 1 in the second equilibrium. Since Zˆ 1 . Z1 by assumption, it must then be that some i [ C2 contributes more to good 1 in the second equilibrium. Therefore the set C2 > Cˆ 1 is not empty. However, ;i [ C2 > Cˆ 1 , g 9i (Z1 ) # f 9i (x i ) 5 h 9i (Z2 ), and f 9i (xˆ i ) 5 g 9i (Zˆ 1 ) $ h 9i (Zˆ 2 ). But since Zˆ 1 . Z1 , then g i9 (Zˆ 1 ) , g i9 (Z1 ). So f 9i (x i ) 5 h 9i (Z2 ) $ g 9i (Z1 ) . g 9i (Zˆ 1 ) 5 f 9i (xˆ i ) $ h 9i (Zˆ 2 ). This implies that Zˆ 2 .Z2 and xˆ i . x i ;i [C2 >Cˆ 1 . Now consider all i [C2 \Cˆ 1 . If any are in Cˆ 2 then f i9 (xˆ i )5h i9 (Zˆ 2 ) and since ˆZ2 .Z2 then h i9 (Zˆ 2 ),h 9i (Z2 ). So for any such i, f 9i (xˆ i )5h i9 (Zˆ 2 ),h 9i (Z2 )5f 9i (x i ). Therefore it must be that xˆ i .x i . And if some are not in Cˆ 2 then they are not contributing at all and for them also xˆ i 5w i .x i . But we cannot have xˆ i $x i ;i at the same time as Zˆ 1 .Z1 and Zˆ 2 .Z2 without violating the budget constraints. i i 2i i i Proposition 2. Let x i1 (Z 2i and 1 ,t,Z 2 )5argmax x i hu (x ,Z 1 1t(w 2 x ),Z 2 ), i 2i 2i i i i 2i i 2i h 1 (Z 1 ,t,Z2 )5Z 1 1t(w 2 x ). Define x 2 (Z 2 ,t,Z1 ) and h 2 (Z 2 ,t,Z1 ) symmetrically. i b m If ≠h |j / ≠Zj #0 for all i, then Z j .Z j , for j51,2. b
m
b
Proof. Since the two goods are provided in the same ratio s in both Z and Z , the only other possibility is that Z b ,Z m so suppose by way of contradiction that this is the case. Let T 1 be the set of individuals i for whom s i ,s b , let T 2 be the set of all i for whom s i .s b , and let T 0 be the set of those for whom s i 5s b . If any i [T 1 chooses d im 50, then d im #d ib trivially. Suppose that d im .0 for
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some i [T 1 . Dropping the i superscripts for the remainder of the proof, define x 1 (Y1 ,t,Z2 ) and h 1 (Y1 ,t,Z2 ) as above 13 , letting Y1 5Z 12i . 21 Now, define G(x,t,Z2 )5h 1 (x 21 1 (xut,Z 2 ),t,Z 2 ), where x 1 (?ut,Z 2 ) is the inverse of the x 1 function in Y1 . G(x,t,Z2 )5Z1 is then just the function that describes the income expansion path showing the combination of x i and Z1 that i will choose as his ‘‘income’’ varies, when the price of Z1 is t, and the level of public good 2 is fixed at Z2 . Normality implies that the income expansion path is upward sloping so G(.) is increasing in x, and so long as w.x, G is also increasing in t. Let f ;x 21 to ease the notation, and noting that f(x 1 (A,t,Z2 )ut,Z2 ); A, it follows 1 ≠f ≠x 1 ≠f ≠f ≠f ≠x 1 ] 1 ] ; 0, so that ] 5 2 ] ] that ] ≠x ≠Z 2 ≠Z 2 ≠Z 2 ≠x ≠Z 2 . ≠h 1 ≠h 1 ≠f ≠h 1 ≠h 1 ≠G ] ] ] ] ] From the definition of G it follows that ] ≠Z 2 5 ≠Z 2 1 ≠ A ≠Z 2 5 ≠Z 2 2 ( ≠ A / ≠x 1 ≠x 1 ≠x ≠f 1 ] ) ] , where the last equality also uses the fact that ] 5 1 / ] ≠ A ≠Z 2 ≠x ≠A . ≠h 1 ≠x 1 ≠h 1 ≠G ] ] ] Since ] ≠Z 2 and ≠Z 2 cannot have the same sign, it follows that ≠Z 2 # 0⇔ ≠Z 2 # 0, ≠G and thus the assumption in the Proposition implies that ] ≠Z 2 # 0. Now, in the specialized equilibrium, an individual from T 1 donates only to Z1 , b b b b b and therefore his private consumption, x , and Z 1 satisfy Z 1 5G(x ,1,Z 2 ). For the same individual, in the dominant firm equilibrium his private consumption, x m must be the solution to max x hu(x, (12s)(w2x)1Y1 , s(w2x)1Y2 )j, and so x m must satisfy: u x (x m ,Zm ) 5 (1 2 s)u Z 1 (x m ,Zm ) 1 su Z 2 (x m ,Zm ) with the subscripts on u denoting partial derivatives, and Z m1 5(12s)(w 2 x m)1Y1 , and Z m2 5s(w 2 x m)1Y2 . Further, since he is in T 1 , it must be that u mZ 1 . u mZ 2 where these are the derivatives evaluated at (x m ,Z m). However, if we now let t m ;(12s)1su Zm2 /u Zm1 , 1, and y m ;Z 1m 2t m (w2x m), then it can be shown that if he faces the problem: max x hu(x, y m 1t m (w2x),Z m2 )j, then the solution is in fact x m . It follows that x m 5x 1 ( y m ,t m ,Z 2m ) and Z 1m 5h 1 ( y m ,t m ,Z 2m ), so that if the hypothesis is true, then G(x m ,t m ,Z m2 )5Z m1 .Z b1 5G(x b ,1,Z b2 ). Then t m ,1 and Z 2m .Z 2b , together with the above results on the derivatives of G imply that x m .x b if x m ,w, and thus no individual in T 1 donates more in the dominant firm equilibrium. A symmetric result holds for all those in T 2 . Now all i [T 0 can be viewed as consumers of two goods x i and vi 5g i (Z1 ,Z2 ). i i In the dominant firm case, Z5(12s b ,s b )D so letting r 5g (12s b ,s b ), we can write u i (x i ,g i (Z1 ,Z2 )) 5 u i (w i 2 d i ,(D2i 1 d i )r i ). 13
These are just individual i’s demands for private consumption and one public good, given that the provision of the other public good is fixed at some level, donations by others to the one public good are ~ some Z 2i j , and the ‘‘price’’ of the one public good is 1 / t. Note further that for each i, then, a change in Z1 , say, cannot move x i2 and h i2 in the same direction.
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Therefore it must be that if d mi .0, then m m Dm 5 argmax D hu i (w i 2 (D 2 D 2i ),Dr i )j ; Hi ( r i ,D 2i ) m
i
m
m
m
Therefore d i $Hi ( r ,D 2 i )2D 2 i (with equality if d i .0). 2i 2i 2i 2i Let a1 (Z )5(Z 2 /s b )2(Z 1 1Z 2 ) and a2 (Z 2 i )5s b (Z 12i 1Z 22i )1Z 22i . Note 2i 2i 2i 2i that a1 (Z )5 a2 (Z )50 if Z 2 /(Z 1 1Z 2i 2 )5s b and that at most one of a1 (.) or a2 (.) can be positive. Define g (Z 2i ,di )5(g1 (Z 2 i ,di ), g2 (Z 2 i ,di )) as follows:
g (Z 2i ,di )
2i 5 (Z 2i 1 1 d i ,Z 2 ) 2i
if 0 # d i # a1 (Z 2i )
2i
2i
if 0 # d i # a2 (Z )
5 (Z 1 ,Z 2 1 d i ) 2i
2i
5 (Z 1 1 Z 2 1 d i )(1 2 s b ,s b )
otherwise
If there are two specialized firms then the payoff to any i [T 0 is u i (w i 2 d i , g i (g (Z 2 i ,d i ))) if he allocates any total d i optimally for any Z 2 i . Further, 2i g i (g (Z 2i ,d i )) # (Z 2i 1 1 Z 2 1 d i )ri
(with equality iff d i $maxha1 (Z 2 i ), a2 (Z 2 i )j). In particular, in equilibrium g i (g (Z 2ib ,d bi )) 5 (Z 2ib 1 Z 2ib 1 d bi )r i 1 2 b m b m also. So D b 5Hi ( r i ,D 2 i ). But D .D by assumption, so if d i .0, we have m b D m 5 Hi ( r i ,D 2i ) . D b $ Hi ( r i ,D 2i )
and therefore m m b b d mi 5 Hi ( r i ,D 2i ) 2 D 2i , Hi ( ri ,D 2i ) 2 D 2i # d ib m
m
b
m
b
and if d i 50 then d i #d i trivially. So d i #d i for all i [T 0 , also. Therefore if Z m $Z b it must be that d im #d ib for all i, which is a contradiction.
Acknowledgements We thank David Austen-Smith, Rich Steinberg, Pete Streufert and an anomymous referee for helpful suggestions, and gratefully acknowledge financial support from the Aspen Institute’s Nonprofit Sector Research Fund and the Social Sciences and Humanities Research Council of Canada. Part of the work on this paper was done while the authors were visitors at the Indiana University Centre on Philanthropy. We thank the Centre and the IUPUI Department of Economics for their financial and other support during these visits.
M. Bilodeau, A. Slivinski / Journal of Public Economics 66 (1997) 449 – 467
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