Information, Competition, and the Quality of Charities∗

Silvana Krasteva Huseyin Yildirim Department of Economics Department of Economics Texas A&M University Duke University Allen 3054 Box 90097 College Station, TX 77843 Durham, NC 27708 E-mail: [email protected] E-mail: [email protected]

April 16, 2015

Abstract We study a model of charity competition in which informed giving alone can explain quality heterogeneity across similar charities. It is this heterogeneity that also creates the demand for information. In equilibrium, too few donors pay to be informed; but interestingly, informed giving may increase with the cost of information. This is true if the charitable market is highly competitive or if giving is a relatively strong substitute to private consumption. Keywords: informed giving, quality of charity, competition, all-pay auctions. JEL Classiﬁcation: H00, H30, H50

1 Introduction

Up 30 percent from a decade ago, the number of public charities in the United States ex- ceeded one million in 2013 (Urban Institute, 2014).1 Charity Navigator rates 8000 of their largest by identifying nine categories of activity (e.g., animals) and 34 causes (e.g., zoos and aquariums, museums, and homeless services). Its rating reveals that the quality of charities for each cause varies signiﬁcantly, with about one-third failing industry standards.2

∗We thank seminar participants at various institutions for comments. All errors are ours. 1This figure is consistent with the fact that the Internal Revenue Service (IRS) approves more than 90 percent of all applications for a charity status (Reich et al. 2009). Unlike other nonprofits such as private founda- tions, public charities rely heavily on contributions from the general public, which consistently total about 1.5 percent of GDP in the U.S – $241.3 billion in 2013. 2For details, visit www.charitynavigator.org. Significant quality variation is also reported by other major charity evaluators including BBB Wise Giving Alliance, CharityWatch, and GuideStar. Not surprisingly, it is such variation that has facilitated recent empirical investigations of charity ratings (see Yoruk (forthcoming) and references therein).

1 The challenge for donors is therefore not finding a cause to support but choosing the charity that is most deserving. Despite the importance of informed giving, however, giving remains largely uninformed (Hope Consulting Report, 2010).3 In this paper, we propose a model of charity competition in which informed giving alone drives the vast heterogeneity in quality across similar charities and creates the endogenous demand for information in turn. Our model contains many potential charities that may begin fundraising for a given cause by incurring a setup cost – e.g., website design, staffing, and the IRS application – and committing to their quality of the public good or service.4 The quality entails such observable attributes as the living conditions of animals in zoos and aquariums; collections care and preservation in museums; and shelter maintanence and healthcare for the homeless. Each charity maximizes its net donations to be used toward the cause. Donors are assumed identical except for their information. An informed donor gives to the highest quality charity whereas an uninformed donor gives randomly to one. The latter implies that a charity may choose to “scam” the uninformed: begin fundraising but offer zero quality. Under free entry, this windfall profit is exhausted. A charity can therefore contribute to the cause only if it wins the informed donation. This incentive to win a lump-sum amount turns charity competition into an all-pay auction and leads to mixing over quality choices in equilibrium. In a symmetric equilibrium, we show that each charity mixes continuously over a positive interval of quality and has a mass point at zero. Such a strategy readily rationalizes the quality heterogeneity mentioned above and predicts a nontrivial probability of charity scams. We find that as the entry cost drops, more charities fundraise but interestingly, the increased competition (stochastically) decreases the highest quality offered and in turn lowers the expected welfare for all donors. The intuition is that charities are less willing to invest in a competition that they are less likely to prevail. And we show that this negative competition effect on quality dominates the positive scale effect associated with the

3Based on a nationwide survey of 4000 Americans with incomes over $80K, the 2010 Report found that while 85 percent of Americans say they care about nonprofit performance when they give, only 35 percent of donors research at least one of their donations each year; see www.hopeconsulting.us/moneyforgood. This finding parallels experimental evidence: whereas Eckel and Grossman (1996) document that individuals give generously when they are paired with recipients of preferred characteristics, Fong and Oberholzer-Gee (2011) observe that only one third of subjects are willing to pay for information about recipients. 4With nonrefundable donations, donors are unlikely to find any promised and/or uncertified quality cred- ible.

2 number of charities.5 Our analysis suggests distinct and necessary roles for uninformed and informed donors in the charitable market: the former entice entry while the latter induce quality provision. This implies that the cost of information can be neither too high nor too low for donors. As expected, too few donors pay to be informed in equilibrium, because each ignores the quality effect of his decision on others. Informed giving may, however, increase with a higher cost of information! This is true if the charitable market is highly competitive or if giving and private consumption are strong substitutes. In each case, decisions to get informed are strategic complements: the larger the number of the informed, the more one wants to become one. Our theoretical framework draws upon two influential papers on all-pay auctions: Var- ian (1980) and Che and Gale (2003). Varian considers a price competition with informed and uninformed consumers in order to explain equilibrium price dispersion. Unlike quality, price does not affect consumers’ reservation utility in his model. Che and Gale examine a research tournament where there is only one buyer, the procurer, whose value of inno- vation is endogenous to the winner’s effort and who decides informed. Siegel (2010) ably generalizes all-pay auctions with endogeneous valuations. Our paper relates to the few studies on charity competition.6 Rose-Ackerman (1982) shows that competitive fundraising can be “excessive” despite donors’ aversion to it. Cas- taneda et al. (2008) argue that such inefficiency may be reduced by nonprofits’ ability to contract on the use of donations while Aldashev et al. (2014) observe that it can be overcome by fundraising coordination, though such coordination is often difficult in this voluntary sector. In the same vein, Bilodeau and Slivinski (1997) find that rival charities may specialize in the provision of one public good or service in order to attract donations. With over one million charities, there are nevertheless many that provide similar – if not identical – services but differ in their quality of provision. In this sense, Scharf (2014) is closer to our work. Assuming an exogenous quality distribution, Scharf points out that competition can induce too much entry by low quality charities. We let quality choice be part of the competition and derive an endogenous distribution for it. Like her, we argue that increased entry into the charitable market decreases the (average) quality of charity.

5Clearly, if the quality distribution were exogenous, then the highest quality would only increase with the number of charities. 6For excellent surveys of the literature on charitable giving, see Andreoni and Payne (2013) and List (2011).

3 On the role of informed giving, our paper also relates to Vesterlund (2003), Andreoni (2006) and Krasteva and Yildirim (2013). Vesterlund shows that a large leadership gift can signal the (ﬁxed) quality of charity. Andreoni extends this argument by demonstrating that all else equal, it is the most wealthy who will lead. Krasteva and Yildirim explore a private value setting in which donors are uncertain about their private valuations of the charity and thus no signaling incentive exists. In all these papers, informed giving, on average, raises more funds and is therefore encouraged. In contrast, our analysis emphasizes that some uninformed giving is also necessary for the existence of a charitable market with costly entry. The rest of the paper is organized as follows. In the next section, we present the base model with exogenous donor information. In Section 3, we characterize equilibrium and perform comparative statics. In Section 4, we endogenize donor information. In the last two sections, we offer a robustness check for charity objective and then conclude. Proofs that do not appear in the text are relegated to the appendix.

2 Base model (with exogenous information)

A large number of ex ante identical charities simultaneously decide whether or not to enter the market to fundraise for a given cause such as zoos and aquariums, museums, and homeless services. Entry involves a setup cost k > 0 – e.g., designing a website, stafﬁng, and registering with the IRS. Observing entry decisions, active charities simultaneously

invest in their quality of the public good or service qi ∈ [0, ∞) at an additional cost qi. The quality is observable by donors upon inspection, perhaps with the help of third party evaluators, and it may reﬂect the living conditions of animals in zoos and the level of care for collections in museums, for instance. We assume that charity i maximizes its net reveneues Ri – donations minus costs – to be used toward the cause. If Ri ≤ 0, the charity is unable to contribute to the cause. On the supply side of the charitable market, there is a continuum of donors of mass M. Each donor picks a charity to support the cause. Donors are otherwise identical and come in two types: informed and uninformed. Informed donors, whose measure is I, ascertain the quality of charities prior to giving while uninformed donors, whose measure is U = M − I, give randomly to one. Each donor has a unit wealth and possesses Constant

4 Elasticity of Substitution (CES) preferences:7

ρ ρ u(xi, gi; q) = xi + qgi , ρ ∈ (0, 1) (1)

where xi ≥ 0 denotes private consumption and gi ≥ 0 denotes gift to the charity whose quality is q. The donor is assumed to receive no utility from giving if the charity of her 8 choice fails to support the cause, that is if Ri ≤ 0. We focus on symmetric (Nash) equilibrium with pure entry decisions.

2.1 Discussion

Our model is designed to highlight the role of donor information as the unique source of the quality heterogeneity. We therefore assume all charities and donors ex ante identical except for donor information, which is endogenized in Section 4. As is standard in the literature, we assume “warm-glow” charities that care only about their own provision of the public good. This is not critical, however. As long as charities care more about their own provision so that there is rivalry, the qualitative results will hold. From (1), it is evident that we also assume “warm-glow” givers (Andreoni 1990). This is a reasonable description of charitable behavior for a continuum of donors. As argued by Glazer and Konrad (1996), suppose Γ is the total provision of the public good by active charities in the market. Let ρ ρ the public good enter the utility in an additively separable way: xi + qgi + z(Γ). Since,

with a continuum of donors, dΓ/dgi = 0, such an altruistic motive has no effect on donor behavior.9 It is worth observing further that in the base model, charities value the level but not the quality of their provision. The latter is demanded by donors. This helps isolate the source of quality provision, but in Section 5, we show that our results are robust to a quality-adjusted provision by charities.

7This way of modeling preference for quality mirrors those of Vesterlund (2003) and Andreoni (2006). The CES form is assumed for exposition as the analysis readily generalizes to: u(xi, gi; q) = t(xi) + qw(gi) where both t and w are strictly increasing and strictly concave. 8Perhaps, the list of charities that support the cause such as open zoos, open museums and active homeless shelters is publicized. 9The fact that donations are driven purely by the warm-glow motive also obtains in a large ﬁnite economy (Ribar and Wilhelm 2002 and Yildirim 2014).

5 qr Figure 1: Optimal individual gift: g(q) = 1+qr

3 Equilibrium characterization

Maximizing utility in (1) subject to budget constraint, xi + gi = 1, donor i’s optimal gift is found to be qr g = g(q) ≡ (2) i 1 + qr 1 where r = 1−ρ ∈ (1, ∞) is the elasticity of substitution between private consumption and donation. Refer to Figure 1. It is readily veriﬁed that (a) g(0) = g0(0) = 0; (b) g0(q) > 0 for 1 00 00 r−1 r q > 0, and (c) g (q) > 0 for q < qc and g (q) < 0 for q > qc, where qc = r+1 . That is, giving is increasing in quality – at an increasing rate for its low levels and at a decreasing rate for its high levels. Since an informed donor learns the entire quality distribution of charities, he optimally

gives to the highest ranked, denoted by qmax. An uninformed donor, on the other hand,

can only conjecture the quality of his selected charity, denoted by qU. Using (2), informed and uninformed gifts can be respectively written:

gI = g(qmax) and gU = g(qU). (3)

Note that the behavior of informed donors turns the quality competition into an all-pay auction – e.g., Che and Gale (2003) and Siegel (2010). It is, therefore, unsurprising that

6 there will generically be no pure strategy equilibrium in quality choice. Before stating it formally, we develop some intuition behind equilibrium characterization. Suppose that entry cost, k, is sufﬁciently small so that at least two charities participate in the market in equilibrium, n∗ ≥ 2.10 If charity i sets quality q and ranks the highest, then it captures all informed donations as well as an equal share of the uninformed, resulting in net revenues: U R∗ (q) = Ig(q) + g(q∗ ) − q − k. (4) win n∗ U If charity i ranks lower, it loses informed donations but continues to receive the uninformed, generating net revenues: U R∗ (q) = g(q∗ ) − q − k. (5) lose n∗ U Clearly, charity i can always adopt a “scam” strategy: enter the market but choose zero quality.11 Since g(0) = 0, such a strategy can only target the uninformed and guarantee a U ∗ payoff: n∗ g(qU) − k. Ignoring integer problems, this payoff will be driven to zero under free entry: U g(q∗ ) − k = 0. (6) n∗ U Substituting for (6), (4) and (5) reduce to:

∗ ∗ Rwin(q) = Ig(q) − q and Rlose(q) = −q. (7) From (6) and (7), we see distinct roles for the two donor types in the charitable market: the uninformed entice charity entry while the informed engender competition in quality. It is, however, possible that the total informed donation is too small to recover the cost of ∗ quality – i.e., Rwin(q) ≤ 0 for all q – in which case no incentive to enter the market exists. To ∗ rule out such trivial cases, suppose that Rwin(q) > 0 for some q > 0. From (7), this implies that only the winning charity will supply the public good and that as with the informed, uninformed donors will also aim for the highest quality, whom they correctly select with 1 probability n∗ . The expected quality by the uninformed is therefore 1 q∗ = E[q∗ ], U n∗ emax 10A competitive charitable market is the case of most interest for us both because it is suggested by evidence and because, as we will see in the next section, a market with a single charity is not sustainable in equilibrium under endogenous information. 11We distinguish between a charity scam and failed fundraising. Although the donor receives no utility from giving in either case, in the latter, the charity invests in quality but ends up raising in- sufﬁcient funds to cover its costs. This distinction is also consistent with the popular view; see ¡http://www.consumer.ftc.gov/features/feature-0011-charity-scams¿.

7 where E[.] denotes the expectation operator. To complete the characterization, let F∗(q) denote the equilibrium distribution that rep- resents each fundraiser’s mixing over quality choices. It is determined by setting the expected net revenues to the free-entry payoff, 0:

∗ n∗−1 ∗ ∗ n∗−1 ∗ (F ) (q)Rwin(q) + [1 − (F ) (q)]Rlose(q) = 0, (8)

∗ where (F∗)n −1(q) is the probability that q is the highest quality. Simplifying (8), we obtain

1 1 q n∗−1 1 n∗−1 F∗(q) = = (q + q1−r) . (9) Ig(q) I

Note that F∗(0) > 0 (since r > 1), which suggests that there is a mass point at q = 0. Propositions 1 and 2 collect and formalize these observations.

Proposition 1 Suppose that there is a symmetric equilibrium in which n∗ ≥ 2. Then, in equilibrium,

(a) informed and uninformed gifts are respectively:

E[q∗ ] g∗ = g(q∗ ) and g∗ = g( emax ). (10) I max U n∗

(b) n∗ is uniquely determined by:

M − I E[q ] g( emax ) − k = 0. (11) n n

(c) Each charity continuously mixes over q ∈ [qL, qH] according to (9) and has a mass point at 1 r ∗ q = 0, where qL = (r − 1) and qH > qL is the largest root to Rwin(q) = 0.

Parts (a) and (b) are as explained above. In particular, indiscrimate giving by the uninformed attracts entry until it is exhausted. This means that each charity relies on informed donations for a positive net revenue. Refer to Figure 2. For a sufﬁciently low quality q ∈ (0, q), informed donations fall below the cost.12 As such, the charity will either choose zero quality and grab some uninformed donations or choose a high enough quality, q ≥ q, and have a chance to receive informed donations, too. This tradeoff generates the mass point at zero. Standard all-pay auction arguments indicate that another (interior)

12 ∗0 0 Note that Rwin(0) = g (0) − 1 = −1.

8 Figure 2: Winning and losing payoffs mass point is not possible; otherwise, a charity could discretely increase the probability of winning by slightly improving his quality. This “race to the top” also accounts for why the lower bound of quality distribution qL exceeds the intermediate break-even quality q as well as why the upper bound qH must obtain at the highest break-even quality. The equilibrium distribution F∗(q) is as derived in (9). Intuitively, the equilibrium distribution balances the probability of winning informed donations to the cost-to-donation ratio. Since, by deﬁnition, the distribution is strictly increasing in the quality level, so is the ratio. In this sense, the cost-to-donation ratio, often utilized by leading watchdogs such as Charity Navigator and CharityWatch, appears an unreliable measure for ranking charities (Steinberg, 1991; Gneezy et al. 2014).13 We emphasize that equilibrium heterogeneity in quality in our model is based solely on donor information – not on preference or income heterogeneity – which in turn creates demand for information; see Section 4. Proposition 1 characterizes equilibrium but does not prove its existence. Armed with ∗ ∗ ∗ M−I E[qemax] F (q), (11) implies that in equilibrium, n ≥ 2 if and only if k ≤ n∗ g( n∗ ). The right- hand side of this inequality – the share of uninformed gifts – is decreasing in n∗ and single-

13To be fair, Charity Navigator has recently expanded its rating methodology to include not only charities’ ﬁnancial health but also their accountability, transparency, and results reporting.

9 peaked in I, leading us to Proposition 2.

h ∗ i 1 M−I E[qemax] −1 Proposition 2 Let kL ≡ maxI∈[0,M] ∗ g( ∗ ) and suppose M > Ic ≡ r(r − 1) r . n n n∗=2 Then, for every k < kL, there are two cutoffs IL < IH such that there is a unique symmetric ∗ equilibrium with n ≥ 2 if and only if IL ≤ I ≤ IH, where Ic < IL and IH < M.

Proposition 2 indicates that the amount of informed giving cannot be too high or too low to sustain a competitive charitable market. Given the all-pay nature of quality choice, significant uninformed giving is needed to accommodate costly entry while at the same time, significant informed giving is also needed to encourage the costly provision of quality. The condition M > Ic merely ensures that a positive net revenue is feasible for at least ∗ the winning charity so trivial cases of no provision are avoided; formally, Rwin(q) > 0 for some q > 0 whenever I > Ic. In fact, I must sufficiently exceed the break-even level Ic so that at least two charities participate in the market. The uniqueness of equilibrium follows E[qemax] because the uninformed gift, g( n ), is decreasing in the number of fundraisers, n, resulting in a unique n∗ for a given entry cost k. The source of such diminished generosity by the uninformed is a negative competition effect on quality, which we elaborate on in the next proposition. Before stating it, however, it is useful to record donors’ expected payoffs. Note from (1) that a donor’s indirect utility conditional on charity quality q is:

v(q) = max (1 − g)ρ + qgρ, g∈[0,1] which, from (2), reduces to: r 1 v(q) = (1 + q ) r , (12) where v0(q) > 0 and v00(q) > 0. The equilibrium expected payoffs for informed and uninformed donors are therefore E[q∗ ] v∗(I) = E[v(q∗ )] and v∗ (I) = v( emax ). (13) I emax U n∗

Proposition 3 Suppose that IL < I < IH. Then, in equilibrium, as the entry cost k drops

(a) the number of charities n∗ increases and goes to inﬁnity as k → 0;

∗ (b) the expected quality of service, E[qemax], decreases and converges to 1 Z qH [q + (1 − r)q1−r]dq > 0 as k → 0, I qL

10 (c) each donor, informed or uninformed, receives a lower expected payoff.

Not surprisingly, as entry becomes less costly, more charities enter the market. The intensiﬁed competition, however, stochastically lowers the quality of service (i.e., the maximum quality) and thus the expected donor welfare. This is easily seen by inspecting the quality distribution in (9). Note that F∗(q) is increasing in n∗: charities are less willing to invest in quality in a competition (for the informed donors) that they are more likely to lose. While there is also a positive scale effect associated with the number of charities, the negative effect of competition dominates since the distribution for the maximum quality ∗ ∗ (F )n (q) is also increasing in n∗.14 In the most competitive market, the two effects bal- ( ∗ )n∗ ( ) → q ∗ → ance each other out: F q Ig(q) as n ∞, leaving the expected quality of service strictly positive in the limit economy. Part (c) obtains because all donors care about the quality of service, as implied by (13). Indeed, it is clear from part (a) of Proposition 1 that both informed and uninformed donors turn less generous in a more competitive market, with the uninformed gift approaching zero, as one would expect. An important corollary to Propositions 1-3 is that the equilibrium probability that all charities are scammers remains nonnegligible even in the most competitive market. For- mally, ∗ ∗ ∗ ∗ I (F )n (0) = (F )n (q ) → c as k → 0, L I 1 −1 15 where Ic ≡ r(r − 1) r as deﬁned in Proposition 2. Building on the insight from Proposi- tion 3, we now explore the impact of informed giving on market entry and donor welfare, both of which will be instrumental in endogenizing information in the next section.

Proposition 4 Suppose that IL ≤ I ≤ IH. In equilibrium,

(a) the number of charities n∗ is single-peaked in I,

∗ (b) the expected quality of service, E[qemax], is increasing in I,

∗ 14Fn (q) would obviously be decreasing in n∗ if F(q) were exogenous. 15As is evident from (9), this limit probability is positive if and only if there is a mass point at zero or F∗(0) > 0, which depends on the cost of quality and donor preferences. We have assumed a linear cost but ( ) c(0) > our point is more general. If the cost of quality is c q , a mass point at zero quality exists if and only if Ig(0) 0. For instance, if c(q) = qθ, θ > 0, this condition reduces to θ ≤ r.

11 Note from Proposition 1 that ﬁxing the number of charities, an increase in informed giving, I, raises the expected quality of service, both because it extends the quality com-

petition – i.e., ∂qH/∂I > 0 – and because it makes higher quality levels more likely to be chosen – i.e., ∂[1 − F∗(q)]/∂I > 0. Expecting a better service by the winning charity, the uninformed become more generous, which encourages further entry. Countervailing this incentive for entry, and the reason behind the non-monotonicity in part (a), is the fact that given the population size, more of informed giving means less of the uninformed, which discourages entry. This non-monotonicity is illustrated in Figure 3. To understand part (b), notice that the increased informed giving can adversely affect the expected quality of service only if it entices more charities into the market. This negative competition effect must, however, be limited in equilibrium: otherwise, by Proposition 1(b), lower expected quality of service along with more charities would depress uninformed giving and lead to fewer charities in equilibrium, yielding a contradiction. The same logic reveals that the ∗ E[qemax] expected quality of service by the uninformed, n∗ , is also increasing in I. From (13), this implies that the uninformed donor beneﬁts from informed giving. As stated in part (c), the informed donor also beneﬁts from informed giving by others because the indirect utility in (12) is increasing and convex in quality. Propositions 2 and 4 make clear the importance of donor information in the charitable market. We have, however, taken informed and uninformed giving as exogenous so far. A natural question then is: does it pay to be informed? We address this question next.

4 Endogenous information

Suppose that at the outset no donor is informed but each can get informed by paying a ﬁxed (utility) cost c > 0. This cost reﬂects time and energy spent in processing charities’ projects and/or their watchdog ratings. The value of being informed is simply the differ- ence between the expected informed and uninformed payoffs recorded in (13). Formally, given the mass of informed donors, I, the value of being informed for a representative donor is

∗ ∗ ∆(I) ≡ vI (I) − vU(I) E[q∗ ] = E[v(q∗ )] − v( emax ), (14) emax n∗

12 where n∗ = n∗(I). Since v(q) is convex, ∆(I) ≥ 0. The donor pays for information if and only if ∆(I) ≥ c. Therefore, under endogenous information, a triplet (I∗, n∗∗, F∗∗) constitutes a (Nash) equilibrium if:

• no charity has an incentive to enter or exit – i.e., n∗∗ = n∗(I∗),

∗∗ ∗ • no active charity has an incentive to deviate from symmetric mixing F = F |I=I∗ , and

• no uninformed donor has an incentive to be informed and no informed donor has an incentive to remain uninformed;16 in particular, the following must hold in equilibrium:

 ≥ ∗ =  c if I M   ∆(I∗) = c if 0 < I∗ < M (15)    ≤ c if I∗ = 0.

Our ﬁrst observation is that regardless of the information cost, there always exists a degenerate equilibrium in which the market fails.

Lemma 1 A degenerate equilibrium in which (I∗, n∗∗) = (0, 0) always exists.

∗ M ∗∗ Proof. Given I = 0, charity i in the market would receive payoff: Ri = n∗∗ gU − qi − k,

which is maximized at qi = 0. But then no uninformed donor would make a positive ∗∗ ∗∗ contribution – i.e., gU = 0, implying Ri = −k < 0 and in turn n = 0. Conversely, given n∗∗ = 0, staying uninformed is clearly a best response for each donor – i.e., I∗ = 0. Lemma 1 follows because as noted above, charities provide quality to win informed donors. In the absence of the latter, uninformed donors grow pessimistic about quality and are unwilling to contribute, which in turn deters entry. An important corollary to Lemma 1 is that for the charitable market to exist, there must be some informed giving in equilibrium, I∗ > 0. In fact, as indicated in Proposition 2, informed giving must be signiﬁcant, though not universal, to both accommodate entry

16Here we implicitly assume a pure information acquisition decision but given a continuum of donors, this I∗ is strategically equivalent to having each mix and acquire information with probability M .

13 and justify the cost of quality. Lemma 2 adds that for informed giving, at least two charities must be active.

∗ ∗∗ ∗ Lemma 2 If I > 0, then n ≥ 2 and IL ≤ I ≤ IH.

Proof. Suppose that I∗ > 0. Then n∗∗ ≥ 1 and ∆(I∗) ≥ c by Lemma 1 and (15), ∗∗ respectively. If n = 1, the sole charity would set qS > 0 that uniquely maximizes ∗∗ ∗ ∗ ∗∗ ∗∗ ∗∗ ∗ Rwin(q) = I g(q) + (M − I )gU − q − k. This implies that qmax = qU and that ∆(I ) = 0 from (14) – a contradiction. Hence, n∗∗ ≥ 2. Intuitively, for donors to be interested in informed giving, there must be quality uncertainty. But this can only arise under competition; otherwise, having a single-peaked net revenue (see Figure 2), a single charity would choose a perfectly predictable quality, leading all donors to stay uninformed instead.17 Lemma 2 implies that in a nondegenerate equilibrium, I∗ ∈ (0, M) and therefore solves

∆(I∗) = c. (16)

In order to determine when there is a solution to (16), let

∆min ≡ min ∆(I) and ∆max ≡ max ∆(I). (17) I∈[IL,IH ] I∈[IL,IH ]

These extreme values of information are well-deﬁned because ∆(I) is continuous. It is

readily veriﬁed that 0 < ∆min < ∆max < ∞. From here, Proposition 5 is immediate.

Proposition 5 A nondegenerate equilibrium exists if and only if the information cost is moderate:

∆min ≤ c ≤ ∆max.

Intuitively, if the information cost were too high, then donors would remain uninformed and without the informed, they would expect minimal quality of service and not give. If, on the other hand, the information cost were too low, charities would expect a much intense quality competition and with little uninformed giving, they would be unable to recoup their entry costs (however small they are). Proposition 5 thus suggests a limit to the beneﬁts of the freely available charity ratings. To understand this point further, we next study the effect of the information cost on informed giving and social welfare.

17This is reminiscent of why, in a pure strategy equilibrium of a sequential-move game, the follower may not pay to observe the leader’s action and thus engender a simultaneous-move outcome (e.g., Morgan and Vardy 2007).

14 Given I, we deﬁne social welfare to be the sum of expected donor payoffs:

∗ ∗ W(I) ≡ I[vI (I) − c] + (M − I)vU(I). (18)

Since, in equilibrium, informed donors ignore their positive externality on others, the following inefﬁciency is observed.

Lemma 3 In equilibrium, too few donors get informed.

0 ∗ ∗ ∗0 ∗ Proof. Differentiating (18) and evaluating at equilibrium, we have W (I ) = I vI (I ) + ∗ ∗0 ∗ (M − I )vU (I ) > 0 by (16) and Proposition 4(c). In equilibrium, using ∆(I∗) = c from (16), (18) reduces to:

∗ ∗ ∗ W(I ) = MvU(I ). (19)

Informed giving improves equilibrium welfare to the extent that it improves the unin- ∗ formed payoff, vU(I). Any additional beneﬁt is outweighed by its cost. Recall from Propo- sition 4(c) that the uninformed payoff is increasing in I. (19) therefore implies that a higher information cost will lower welfare if and only if it discourages informed giving. The latter depends criticially on whether ∆0(I∗) > 0 or ∆0(I∗) ≤ 0 – i.e., whether donors’ decisions to become informed are strategic complements or strategic substitutes. Under reasonable conditions, information acquisition decisions are strategic complements, leading us to:

Proposition 6 Suppose that k is sufﬁciently small or that r is sufﬁciently close to 1. Then, there exists some ∆b ∈ (∆min, ∆max) such that for ∆min ≤ c < ∆b, there is a unique nondegenerate equilibrium. Moreover, the mass of informed donors, I∗, and social welfare, W(I∗), are both increasing in the information cost, c.

Proposition 6 says that a higher information cost may actually encourage informed giving and raise total donor welfare in turn! Suppose that the setup cost k is negligible. Then we know from Proposition 3 that a large number of charities enter the market, which ∗ completely discourages uninformed giving. Formally, for k ≈ 0, we have vU(I) ≈ 1 and ∗ thus ∆(I) ≈ vI (I) − 1, which increases with I and indicates that information acquisition decisions are strategic complements. From (16), the mass of informed donors is found to be ∗ ∗−1 I ≈ vI (1 + c),

15 which is clearly increasing in c. The same comparative static also holds for a nonnegligible setup cost if donor preferences are (approximately) Cobb-Douglas, r ≈ 1. The intuition is similar but slightly more involved in this case due to the fact that uninformed giving is significant. Notice that for r ≈ 1, the indirect utility in (12) becomes v(q) ≈ 1 + q and reduces the value of information to: E[q∗ ] ∆(I) ≈ (1 + E[q∗ ]) − (1 + emax ) emax n∗ n∗ − 1 = E[q∗ ]. n∗ emax Clearly, the value of information is increasing in both the expected quality of service, ∗ ∗ E[qemax], and the number of fundraisers, n , because, unlike informed donors, the unin- 1 formed is able to pick the best fundraiser with probability n∗ . Under exogenous informa- ∗ tion, we know from Proposition 4 that E[qemax] is increasing in the size of informed donors, ∗ ∗ I. We also know that n = n (I) is single-peaked; so there is a threshold bI > IL such that ∗ for I ∈ (IL, bI), n (I) is also increasing in I, implying a higher value of information in this region, d∆(I)/dI > 0. From (16), it follows that I∗ rises with c for some intermediate cost levels, as stated in Proposition 6. This formal intuition is also illustrated in Figure 3(a) for r = 1.1, k = .05 and M = 10. Although we have been unable to prove this result for any r > 1, Figures 3(b) and 3(c) suggest that strategic complementarity of information acquisition decisions will be pro- nounced even more for higher r values. Intuitively, as implied by (2), under a greater substitutibility between private consumption and giving, donors grow more generous so long as the charity quality is sufficiently high –i.e., q > 1. This additional giving compen- sates for the diminished uninformed giving and extends the region in which charity entry, and therefore the value of information, rises with the size of the informed. Proposition 6 is important because, as alluded to in the Introduction, the sheer number of charities suggests little entry cost into the market for donations. Indeed, after natural disasters and national tragedies, donors are often warned of charity scams, raising money for victims.18 In such cases, Proposition 6 predicts that information cost must be significant – not negligible – to promote informed giving and raise total donor welfare in turn. Cost uncertainty and equilibrium robustness. In order to explain heterogeneity in donor information endogenously, we have assumed a homogenous information cost c.A

18http://www.hufﬁngtonpost.com/2013/05/06/boston-marathon-charity-scams n 3223366.html

16 (a) r = 1.1 and k = 0.05 (b) r = 2 and k = 0.05 (c) r = 10 and k = 0.05

Figure 3: Value of information and equilibrium entry similar result to Proposition 6, however, obtains under heterogeneous costs. Suppose that each donor draws his information cost c independently and privately from a continuous c.d.f. G(c; α) whose support is [cL, cH]. Assume Gα(.) < 0 so that a higher α means a ﬁrst-order stochastic increase in the information cost. As above, in equilibrium, a donor gets informed if and only if c ≤ ∆(I∗). Hence, the fraction of the ∗ I∗ ∗ informed is G(∆(I ); α), which must, by deﬁnition, equal to M whenever I ∈ (0, M). Formally, in a nondegenerate equilibrium, I∗ G(∆(I∗); α) − = 0. (20) M

Based on (17), let ∆max = ∆(Imax) and assume that 0 ≤ cL < cH ≤ ∆max. Since

∗ Ic Ic vI (Ic) = 0, we have ∆(Ic) = 0 and therefore G(∆(Ic); α) − M = − M < 0. Moreover, Imax G(∆max; α) − M > 0. By continuity of ∆(I), there exists a nondegenerate equilibrium in which I∗ ∈ (0, M). It is unique if 1 G0(.)∆0(I∗) − > 0. (21) M Note that ∆0(I∗) > 0 is necessary but not sufﬁcient for the uniqueness. For the latter, we must have ∆0(I∗) > G0(.)/M. In words, under heterogenous costs, a nondegenerate equi-

17 librium is unique if information acquisition decisions are strongly strategic complements. Suppose that (21) holds. Differentiating (20) with respect to α, we ﬁnd G (.) dI∗ d = − α > / α 0 0 ∗ 1 0. G (.)∆ (I ) − M Hence, if, as in Proposition 6, the nondegenerate equilibrium is unique, a stochastic increase in the information cost means more informed giving. To understand its impact on social welfare, we write expected sum of donor payoffs: ∗ Z ∆(I ) Z cH W(I∗) = M [v∗(I∗) − c]dG(c) + v∗ (I∗)dG(c) . I ∗ U cL ∆(I ) Integrating by parts yields Z ∆(I∗) ∗ ∗ ∗ W(I ) = M G(c)dc + MvU(I ). cL As a result, if, as in Proposition 6, the nondegenerate equilibrium is unique, a stochastic increase in the information cost also means a higher social welfare.

5 An extension: quality-adjusted provision

Up to now, we have assumed that charities aim to maximize the amount of public service,

Ri, and provide quality only to satisfy donors’ demand for it. Conceivably, charities may also have an intrinsic preference for quality. For instance, a charity may value not only how many homeless it shelters but also about how well it shelters them. Here we show that our results carry over. Suppose that charity i maximizes the following quality-adjusted provision:

Ri = φ(q)Ri, (22) where φ0(q) > 0 and φ(0) > 0. The latter rules out φ(0) = 0 to ensure that as in the base model, the charity cannot avoid entry cost by choosing q = 0. As before, if the collected funds turn out insufﬁcient to cover the cost, then no service can be provided, irrespective of quality. To intuitively see the equivalence of equilibrium with the base model, we recall the previous line of argument. Note that if charity i sets quality q and ranks the highest ∗ ∗ ∗ among n ≥ 2, then it receives net revenues: Rwin(q) = φ(q)Rwin(q). If it ranks lower, it ∗ ∗ receives net revenues: Rlose(q) = φ(q)Rlose(q). The indifference equation to determine the equilibrium quality distribution F∗ therefore becomes

∗ n∗−1 ∗ ∗ n∗−1 ∗ (F ) (q)φ(q)Rwin(q) + [1 − (F ) (q)]φ(q)Rlose(q) = 0, (23)

18 which, canceling out φ(q) > 0 from the left-hand side, reduces to (8). Moreover, since φ(0) > 0, it is clear that the free entry condition in (6) and thus (7) remain intact. Together, (23) reveals that F∗(q) is exactly as found in Proposition 1. Given this, it is immediate that the rest of the analysis, including endogenous information, continues to hold under this generalization.

6 Conclusion

Drawing upon the all-pay auction literature, especially Varian (1980) and Che and Gale (2003), this paper has offered a novel model of charity competition in which imperfect donor information alone can explain the quality heterogeneity across similar charities. We show that both significant uninformed and significant informed giving is necessary for the existence of the charitable market: the former entices costly entry whereas the latter induces quality provision by charities. As such, the information cost for donors cannot be too high or too low in this market. We also show that as the entry cost falls, more charities fundraise but the increased competition decreases the expected quality of the charitable service and donor welfare. Our findings indicate that the ease of entry into the charitable market, as suggested by the IRS approval rate (see Footnote 1), may adversely affect the quality of charitable service. Although our model is too stylized to determine the socially optimal number of charities, it does starkly imply that only two charities may generate enough quality competition for each charitable cause.19

A Appendix

Lemma A1 Let y1,...,yn be nonnegative iid random variables, with cdf Fb and pdf fb. Moreover, fb

is continuous in [yL, yH] and has a mass point at y = 0. Then

Z yH n−1 E[ymax] = ny fb(y)Fb (y)dy, yL

where ymax ≡ max{y1, ..., yn}.

19This is in line with Che and Gale’s (2003) prediction that the optimal number of contestants in a research tournament is two.

19 Fb(y)−Fb(yL) Proof. Deﬁne the conditional distribution: J(y) = , y ∈ [yL, yH]. Suppose 1−Fb(yL) exactly k out of n random variables have y > yL. Then,

k G(y, k) ≡ Pr{ymax ∈ (yL, y)} = J (y) and

g(y, k) = kh(y)Jk−1(y) ! !k−1 fb(y) Fb(y) − Fb(yL) = k 1 − Fb(yL) 1 − Fb(yL) k−1 fb(y) Fb(y) − Fb(yL) = k . k 1 − Fb(yL)

Note that n n k n−k E[ymax] = ∑ (1 − Fb(yL)) (Fb(yL)) E[ymax|k above yL] k=1 k R yH where E[ymax|k above yL] = yg(y, k)dy. Then yL

n Z yH k−1 n n−k E[ymax] = ∑ (Fb(yL)) ky fb(y) Fb(y) − Fb(yL) dy k=1 k yL ! Z yH n k−1 n n−k = y fb(y) ∑ (Fb(yL)) k Fb(y) − Fb(yL) dy. yL k=1 k

n n−1 Finally, since k(k) = n(k−1), we have n n k−1 n n − 1 k−1 (F(y ))n−kk F(y) − F(y ) = n (F(y ))n−k F(y) − F(y ) ∑ b L b b L ∑ − b L b b L k=1 k k=1 k 1 n−1 k n − 1 n−1−k = n ∑ (Fb(yL)) Fb(y) − Fb(yL) k=0 k = nFn−1(y) (by Binomial Theorem).

1 1 −1 Lemma A2. Let qL = (r − 1) r , Ic = r(r − 1) r and qH to be the largest root to Ig(q) − q =

0. Also let each fundraiser continuously mix over q ∈ [qL, qH] according to

1 q n−1 F(q) = , Ig(q) and have a mass point at q = 0 with F(0) = F(qL). Then, for I > Ic,

20 (a) limI→Ic qH = qL.

∂qH (b) qH < I and ∂I > 0.

∂E[qemax] (c) ∂I > 0.

∂E[qemax] 1 R qH 1−r (d) < 0 and limn→∞ E[qmax] = (q + (1 − r)q )dq > 0. ∂n e I qL

q r−1 r Proof. We can re-write eq. (7) as Rwin(q) = 1+qr Ω(q) where Ω(q) = q I − (1 + q ). We deﬁne qH to be the largest root that solves Rwin(q) = 0. Note that Ω(q) is maximized for − ∗ r−1 ∗ r−1 (r−1)r 1 r ∗ q = r I with Ω = Ω( r I) = rr I − 1. For I < Ic, Ω (q) < 0 for all q. Therefore, qH = 0 for I < Ic. For I ≥ Ic, Ω(q) ≥ 0 for some q > 0. Therefore, qH > 0 and solves 1 ∗ r ∗ Ω(qH) = 0. Note that limI→Ic q = (r − 1) = qL and limI→Ic Ω(q ) = 0, which implies

that limI→Ic Ω(q) < 0 for all q 6= qL. Therefore, by the deﬁnition of qH, limI→Ic qH = qL. r qH To prove part (b), note from eq. (7) that qH = r I < I. Differentiating Rwin(q) w.r.t. I 1+qH results in ∂q g(q ) H = H = ∂I rg(qH ) 1 − I r qH (1+qH ) g(qH) qH = r (because I = ) 1 − r g(qH) 1+qH r qH r r = r > 0 (because qH > qL = (r − 1)) 1 − r + qH .

To prove part (c), we ﬁrst integrate by parts E[q˜max|n, I] from Lemma A1:

Z qH n n E[qemax] = qH − qLF (qL) − F (q)dq. (A-1) qL

Differentiating with respect to I,

n Z qH n ∂E[qemax] ∂qH ∂F (qL) n ∂qH ∂F (q) = − qL − F (qH) − dq (A-2) ∂I ∂I ∂I ∂I qL ∂I n Z qH n ∂F (qL) ∂F (q) n = −qL − dq (because F (qH) = 1) ∂I qL ∂I > 0 (because ∂Fn(q)/∂I < 0).

n ∂E[qemax] ∂F (q) Finally, for part (d), ∂n < 0 follows from ∂n > 0. The limit of E[qemax] obtains by n( ) → q → observing that F q Ig(q) as n ∞.

21 Proof of Proposition 1. We begin with part (c). Suppose that n ≥ 2. As in standard all-pay auctions, no symmetric equilibrium in pure strategies exists in quality competition: given others’ quality choices, fundraiser i would have a strict incentive to slightly increase his and win all of informed donations. In equilibrium, let each fundraiser mix according to F(q) and S be its support. Clearly, due to free entry, the (common) expected payoff across quality choices in S is π = 0. Moreover, a standard argument from all-pay auctions reveals that F(q) cannot admit a mass point at q > 0; otherwise, fundraiser i could shift weight to q + ε and discretely increase his probability of winning informed donations while raising cost only by ε. From (4) and (5), the expected payoff of a fundraiser thus satisﬁes

U Fn−1(q)Ig(q) + g − k − q = 0 for all q ∈ S+, (A-3) n U where S+ ≡ {q ∈ S|q > 0}. Note that since a fundraiser could enter the market and choose q = 0, it must be that

U g − k ≤ 0. (A-4) n U Together with (A-4), (A-3) implies that F(q) > 0 for all q ∈ S+. Then q = 0 must be in S and have a probability mass. In particular, letting qL < qH be the lower and upper bounds + of S , we have F(0) = F(qL), as stated in part (c). Setting q = 0 in (A-3), (A-4) reduces to:

U g − k = 0, (A-5) n U as stated in part (b). Using (A-5), (A-3) reduces to:

Fn−1(q)Ig(q) − q = 0, from which the c.d.f. in part (c) obtains. Substituting for (2) and differentiating, we ﬁnd the p.d.f.: 1 1 − r + qr F0(q) = F(q) for q ∈ S+. n − 1 q(1 + qr) The continuous mixing over S+ requires that F0(q) > 0 in its interior, which in turn 1 1 requires that qL ≥ (r − 1) r . Suppose that qL > (r − 1) r . Then fundraiser i’s expected payoff from a deviation to q ∈ (0, qL) is

d n−1 πi (q, qL) = F(qL) Ig(q) − q.

22 Simple algebra shows

∂ r πd(q, q ) = − 1 ∂q i L 1 + qr q=qL L r < − 1 1 r 1 + (r − 1) r = 0.

d d This means that there exists some q ∈ (0, qL) such that πi (q, qL) > πi (qL, qL) = 0, contra- + 1 dicting qL being in S . Thus qL = (r − 1) r .

To complete the proof of part (a), note that Rlose(q) = −q by using (A-5). Therefore, only the highest quality fundraiser will provide charitable service. The informed donor as- certains the winner before giving while the uninformed donor picks him with probability 1 n . This explains their optimal gifts in part (a). To complete the proof of part (b), we substitute from (10) into (A-5) and write

M − I (E[q ]/n)r ( ) ≡ emax − = h n; I, r r k 0. (A-6) n 1 + (E[qemax]/n) From part (d) of Lemma A2,

∂h(n,.) < 0 and lim h(n,.) = −k. (A-7) ∂n n→∞ Therefore, a symmetric equilibrium with n ≥ 2 requires h(2; .) ≥ 0. Then, by (A-7) there is a unique n that solves h(n;.) = 0, proving part (b). Since n uniquely pins down F(q) and the gifts, there is also a unique symmetric equilibrium. Lemma A3. Let n ≥ 2. Then, h(n; I, r) is continuous and single-picked in I with h(n; I, r) =

−k for I ∈ [0, Ic] ∪ {M}. Proof. Note that h(n; I, r) is continuous in I as long as E[qemax] is continuous. By Lemma A2, E[qemax] is differentiable for n ≥ 2 and I > Ic, proving its continuity in this region. For I ≤ Ic, Rwin(q) ≤ 0 for all q > 0. Therefore, F(0) = 1 for I ≤ Ic, implying that E[qemax] = 0. Therefore, it follows that h(n; I, r) = −k for I ≤ Ic. Moreover, by Lemma A2, = [ ] = [ ] limI→Ic qH qL. Therefore, limI→Ic E qemax 0 establishing that E qemax is continuous in I at I = Ic. This establishes the continuity of h(n; I, r). The property h(n; M, r) = −k follows immediately from E[qemax] < ∞ (Lemma A2). r M−I qU To show that h(n; I, r) is single-picked, note that we can rewrite h(n; I, r) = r − n 1+qU k. Differentiating h(n; I, r) with respect to I to obtain

23 ∂h(n; I, r) 1 qr−1 ∂q = U ( − ) U − ( + r ) r 2 M I r qU 1 qU (A-8) ∂I n (1 + qU) ∂I

∂qU r Let Λ(n, I) = (M − I)r ∂I and Υ(n, I) = qU(1 + qU) and Φ(n, I) = Λ(n, I) − Υ(n, I). ∂h(n;I,r) sign ∂Fn(q) Then, ∂I = Φ(I, n) for I > Ic since qU > 0 for I > Ic. Given eq. (A-2) and ∂I = n F (q) ∂qU 1 n R qH n n R qH n − , = (qLF (qL) + F (q)dq). From eq. (A-1), qLF (qL) + F (q)dq = (n−1)I ∂I (n−1)I qL qL qH − nqU. Therefore, ∂q q − nq U = H U > 0 (A-9) ∂I (n − 1)I

1 r Consider I → Ic. From Lemma 2A, limI→Ic qH = qL = (r − 1) > 0 implying limI→Ic qU = 0. Therefore, limI→Ic Φ(n, I) = limI→Ic Λ(n, I) − Υ(n, I) > 0. Moreover, from eq. (A-8) it follows immediately that Φ(n, M) < 0. Therefore, by the continuity of

Λ(n, I) and Υ(n, I), Φ(I, n) = 0 has a solution with I ∈ (Ic, M). To show that h(n; I, r) is single-picked, it is sufﬁcient to show that this solution is unique. Suppose the contrary-Φ(n, I) does not yield a unique solution. Let I0 denote the lowest 0 0 ∂Φ(n,I0) ∂Λ(n,I0) value of I such that Φ(n, I ) = 0. Since Φ(n, I) > 0 for all I ∈ (Ic, I ), ∂I = ∂I − ∂Υ(n,I0) 0 ∂I < 0 so that h(n; I, r) reaches a local maximum at I . Note that ∂Λ(n, I) ∂q ∂2q = −r U + r(M − I) U (A-10) ∂I ∂I ∂2 I ∂Υ(n, I) ∂q = U [1 + (r + 1)qr ] > 0 (A-11) ∂I ∂I U 2 h i ∂ qU = 1 −( − ) ∂qU + ∂qH where differentiating eq. (A-9) yilds ∂2 I (n−1)I 2n 1 ∂I ∂I . By Lemma A2, r ∂qH qH ∂Φ(n,I) ∂qU = r > 0 and is continuous and differentiable since qU, and qH are ∂I 1−r+qH ∂I ∂I continuous and differentiable. ∂Φ(n,I) 0 0 0 If ∂I ≤ 0 for all I ∈ (I , M), then Φ(n, I) < 0 for all I ∈ (I , M) and I is the unique solution. Therefore, a necessary condition for the existence of another solution is ∂Φ(n,I00) 00 0 ∂Φ(n,I0) ∂Φ(n,I00) ∂I > 0 for some I ∈ (I , M). Then, ∂I < 0 and ∂I > 0 imply that there 000 000 000 2 000 2 000 000 ∈ ( 0 00) ∂Φ(n,I ) = ∂Λ(n,I ) − ∂Υ(n,I ) = ∂ Φ(n,I ) = ∂ Λ(n,I ) − exists I I , I such that ∂I ∂I ∂I 0 and ∂2 I ∂2 I 2 000 000 ∂ Υ(n,I ) > ∂Υ(n,I) > ∂Λ(n,I ) > ∂2 I 0. Since ∂I 0, it must be that ∂I 0, which from eq. (A-10) implies 2 ∂ qU that 2 > 0. ∂ I I=I000 ∂Λ(n,I) ∂Υ(n,I) Differentiating ∂I and ∂I given by eq. (A-10) and eq. (A-11) results in ∂2Λ(n, I) ∂2q ∂3q = −2r U + r(M − I) U ∂2 I ∂2 I ∂3 I

24 ∂2Υ(n, I) ∂2q ∂q 2 = U [1 + (r + 1)qr ] + (r + 1) U ∂2 I ∂2 I U ∂I 2 2 000 2 ∂ qU ∂ Υ(n,I ) ∂ qU Since 2 > 0, 2 > 0. Differentiating 2 w.r.t. I results in ∂ I I=I000 ∂ I ∂ I ∂3q 1 ∂2q 2n − 1 ∂2q 1 ∂2q U = − U − U + H ∂3 I I ∂2 I (n − 1)I ∂2 I (n − 1)I ∂2 I

2 r(r− )qr 2 3 ∂ qH 1 H ∂qH ∂ qU ∂ qU where 2 = − − + r < 0. Therefore, 2 > 0 , 3 < 0, and ∂ I 1 r qH ∂I ∂ I I=I000 ∂ I I=I000 2 000 2 000 2 000 2 000 ∂ Λ(n,I ) < ∂ Φ(n,I ) = ∂ Λ(n,I ) − ∂ Υ(n,I ) < ∂2 I 0. This, however, implies that ∂2 I ∂2 I ∂2 I 0 leading 00 0 ∂Φ(n,I00) to a contradiction. As a result, there is no I ∈ (I , M) such that ∂I > 0, contradicting the necessary condition for existence of another solution for Φ(n, I) = 0. This implies that I0 is a unique solution of Φ(n, I0) = 0, proving that h(n; I, r) is single-picked. ( ) Proof of Proposition 2. Since ∂h n;I,r < 0 for n ≥ 2 (by A-7), h(n; I, r) reaches a h ∂n i M−I E[qemax] maximum at I(2) = arg maxI g( ) = kL. By Lemma A3, h(n; I, r) is contin- 2 2 n=2 ∂h(2;I,r) ∂h(2;I,r) uous and single-picked. Therefore, ∂I > 0 for I < I(2) and ∂I < 0 for I > I(2). Consider ﬁrst the region I < I(2). Recall that h(2; I, r) < 0 for I ≤ Ic (Lemma A3) and ∂h(2;I,r) h(2; I(2), r) > 0 for every k ∈ (0, kL) (by the deﬁnition of kL). Therefore, ∂I > 0 for I < I(2) implies that there exists a unique IL ∈ (Ic, I(2)) such that h(2; IL, r) = 0 with

h(2; I, r) < 0 for I < IL and h(2; I, r) > 0 if I ∈ (IL, I(2)). Analogously, h(2; M, r) < 0 (by ∂h(2;I,r) ¯ Lemma 3A) and ∂I < 0 for I > I(2) implies that there exists a unique IH ∈ (I(2), M) such that h(2; IH, r) = 0 with h(2; I, r) > 0 for I ∈ [I(2), IH) and h(2; I, r) < 0 for I > IH. Since h(n; I, r) is strictly decreasing in n, h(n; I, r) ≤ h(2; I, r) < 0 for n ≥ 2 and I ∈/

[IL, IH]. Therefore, there is no equilibrium with n ≥ 2 for I ∈/ [IL, IH]. For I ∈ [IL, IH],

h(2; I, r) ≥ 0 with strict inequality for I ∈ (IL, IH). Therefore, for every I ∈ (IL, IH), there exists n∗(I) > 2 such that h(n∗(I); I, r) = 0, establishing the existence of a competitive equilibrium with n ≥ 2 for I ∈ [IL, IH]. Proof of Proposition 3. By deﬁnition h(n∗(I); I, r) = 0. Implicit differentiation results dn∗ = 1 < in dk ∂h(n;I,r)/∂n 0 (by A-7). Moreover, M − I (E[q∗ ]/n∗)r emax − = lim ∗ ∗ ∗ r k 0 (A-12) k→0 n 1 + (E[qemax]/n )

E[qemax] Note that n > 0 for all n < ∞ and I > Ic. Therefore, eq. (A-12) holds if and only ∗ if limk→0 n = ∞. Part (b) is an immediate consequence of Lemma 2A(b). To prove part dv∗ 1 ∗ − ∂q∗ ∗ U ∗r r −1 r 1 U ∂n (c), note that by eq. (13), dk = (1 + qU ) qU ∂n ∂k > 0, where the inequality follows ∗ ∂qU dn immediately from ∂n < 0 and dk < 0. Analogously, for the informed donor,

25 Z qH ∗ n∗ ∗ n∗−1 vI (I) = v(0)F (qL) + v(q)n f (q)F (q)dq (A-13) qL where v(q) is deﬁned by eq. (12). Integrating by parts, we obtain:

Z qH ∗ n∗ ∂v(q) n∗ vI (I) = v(qH) − [v(qL) − v(0)]F (qL) − F (q)dq (A-14) qL ∂q

∗ n ∂v(q) ∂vI (I) ∂F (q) where v(qL) − v(0) > 0 since ∂q > 0. ∂n < 0 follows from ∂n > 0. Therefore, ∗ ∗ dvI (I) ∂vI (I) ∂n dk = ∂n ∂k > 0. Proof of Proposition 4. To prove part (a), note that implicit differentiation of eq. (A-6), ∗ ∗ ∗ sign ∗ dn = − ∂h(n ;I,r)/∂I ∂h(n;I,r) < dn = ∂h(n ;I,r) ( ∗ ) results in dI ∂h(n∗;I,r)/∂n . Since ∂n 0, dI ∂I . By Lemma 3A, h n ; I, r is single-picked in I, which implies that n∗ is single-picked in I. ∗ ∗ ∗ ∗ ∗ dE[qemax] ∂E[qemax] ∂E[qemax] ∂n ∂E[qemax] To prove part (b), note that dI = ∂I + ∂n ∂I . By Lemma A2, ∂I > 0 ∗ ∗ ∗ ∗ ∂E[qemax] dE[qemax] ∂n ∂n and ∂n < 0. Therefore, dI > 0 for ∂I < 0. For ∂I > 0, we can re-write the equilibrium entry condition given by eq. (11) as

E [q∗ ] kn∗ g( emax ) = n∗ M − I Differentiating both sides with respect to I, results in

h ∗ i ∗ ∗ ∗ + ( − ) ∂n E [q ] d E[q ] k n M I ∂I g0( emax ) emax = > 0 (A-15) n∗ dI n∗ (M − I)2

∗ ∗ ∗ 0 ∂n d E[qemax] d E[qemax] Note that g (q) > 0 and ∂I > 0 imply that dI n∗ > 0. Moreover, dI n∗ = ∗ ∗ ∗ ∗ ∗ ∗ 1 dE[qemax] E[qemax] ∂n dE[qemax] E[qemax] ∂n n∗ dI − n∗ ∂I > 0, which implies that dI > n∗ ∂I > 0, completing the proof of part (b).

To prove part c), consider ﬁrst the uninformed donor. By eq. (13) vU is increasing in ∗ ∗ ∗ dqU d E[qemax] dvU qU. From the prove of part b), dI = dI n∗ > 0. Therefore, dI > 0. Next we consider the informed donor. By eq. (A-14),

∗ ∗ ∗ n Z qH n dvI (I) dF (qL) ∂v(q) dF (q) = −[v(qL) − v(0)] − dq (A-16) dI dI qL ∂q dI

∗ n∗ dvI (I) dF (q) Therefore, dI > 0 if dI < 0 for all q ∈ [qL, qH].

26 ∗ ∗ ∗ dFn (q) ∂Fn (q) ∂Fn (q) ∂n∗ = + = (A-17) dI ∂I ∂n∗ ∂I ∗ r ∗ n ∗ ∗ 1 + q 1 ∂n = − Fn (q) − Fn (q) ln = (n∗ − 1)I qr−1 I (n∗ − 1)2 ∂I ∗ ∗ r ∗ 1 ∗ (n − 1)n 1 + q ∂n = − Fn (q) + ln (n∗ − 1)2 I qr−1 I ∂I

∗ ∗ n ∗ ∂n dF (q) dvI (I) For < 0, < 0 for all q ∈ [qL, qH] and > 0. ∂I dI ∗ ∗ dI ∗ n n ∗ ∗ ∗ ∂n > dF (q) = dF (qH ) = − n < (n −1)n − For ∂I 0, limq→0 dI ∞ and dI (n∗−1)I 0. Moreover, I ∗ r ∗ n 1+q ∂n ( ) dF (q) > ln qr−1 I ∂I is increasing in q. Therefore, by continuity, there exists q˜ I such that dI ∗ dFn (q) 0 for q < q˜(I) and dI < 0 for q > q˜(I). Consider arbitrary I0 and I00 such that I00 > I0 and n∗(I00) > n∗(I0). Then by (A-14)

q00 ∗ ∗ ∗ ( 00 ) − ( 0 ) − R H ∂v(q) n ( | 00) = − ( 0 )[ − n ( 0 | 00)] + R qH ( ) n ( | 00) > Note that v qH v qH 0 ∂q F q I dq v qH 1 F qH I q0 v q dF q I qH H ∂v(q) 0 since ∂q > 0. Therefore,

n∗ 00 n∗ 0 ∗ 00 ∗ 0 If F (q|I ) < F (q|I ) for q = qL, then vI (I ) > vI (I ). n∗ 00 n∗ 0 0 Suppose instead that F (q|I ) > F (q|I ) for q < qˆ ∈ (qL, qH). Let ξ(q) = a + bq ∂ξ(qˆ) ∂v(qˆ) with the property ξ(qL) = v(qL) and ∂q = b = ∂q . These two equations completely determine the values of a and b. Moreover,

∗ E[ξ(q)|I] = a + bE[qemax] ∗ 00 0 Since E[qemax] is strictly increasing in I in equilibrium, E[ξ(q)|I ] > E[ξ(q)|I ].

27 ∗ 00 ∗ 0 00 0 We want to show that vI (I ) − vI (I ) > E[ξ(q)|I ] − E[ξ(q)|I ] > 0. This is equivalent ∗ 00 00 ∗ 0 0 to showing vI (I ) − E[ξ(q)|I ] > vI (I ) − E[ξ(q)|I ]. By (A-14),

∗ n∗ vI (I) − E[ξ(q)|I] = v(qH) − ξ(qH) + (v(0) − ξ(0)) F (qL|I) Z qH ∗ − v0(q) − ξ0(q) Fn (q|I)dq qL

where we used v(qL) = ξ(qL). Note that the strict convexing of v(q) and the linearly 0 0 0 0 of ξ(q) implies that v (q) < ξ (q) for q < qˆ. Moverover, v(qL) = ξ(qL) and v (q) < ξ (q) imply that v(0) − ξ(0) > 0. Then, we can write

Note that

00 Z qH ∗ [v(q00 ) − ξ(q00 )] − [v(q0 ) − ξ(q0 )] − v0(q) − ξ0(q) Fn (q|I00)dq = H H H H 0 qH 00 Z q n∗ 0 00 0 0 H n∗ 00 = −[1 − F (qH|I )][v(qH) − ξ(qH)] + 0 [v(q) − ξ(q)] dF (q|I ) > 0 qH

0 0 0 where the inequality follows from the fact that v (q) − ξ (q) > 0 for q > qˆ(< qH). n∗ 00 n∗ 0 n∗ 00 The term [v(0) − ξ(0)][F (qL|I ) − F (qL|I )] > 0 since [v(0) − ξ(0)] > 0 and F (qL|I ) − 0 ∗ ∗ ∗ n 0 R qH 0 0 n 00 n 0 F (qL|I ) > 0. Finally, the term − (v (q) − ξ (q)) [F (q|I ) − F (q|I )]dq > 0 since qL ∗ ∗ 1) for q < qˆ, v0(q) < ξ0(q) and Fn (q|I00) > Fn (q|I0); 2) for q > qˆ, v0(q) > ξ0(q) and ∗ ∗ Fn (q|I00) < Fn (q|I0). ∗ 00 ∗ 0 00 0 This proves that vI (I ) − vI (I ) > E[ξ(q)|I ] − E[ξ(q)|I ] > 0. Therefore, the equilib- ∗ rium informed payoff vI (I) is increasing in I. Proof of Proposition 5. A non-degenerate equilibrium satisﬁes eq. (16). Therefore, it sufﬁcies to establish that ∆(I) is continuous with ∆min > 0 and ∆max < ∞ where ∆min

and ∆max are deﬁned by eq. (17). ∆min > 0 follows immediately from the convexity of v(q) ∗ and the fact that n (I) ≥ 2 for I ∈ [IL, IH] (see proof of Proposition 2). Moreover, ∆max <

28 ∗ ∗ vI (M) < ∞. The continuity of ∆(I) follows from the fact that n (I) solving h(n; I, r) =

0 is continuous in I for I ∈ [IL, IH] due to the continuity of h(n; I, r) (see Lemma A3).

Therefore, for every c ∈ [∆min, ∆max], there exists I such that ∆(I) = c. Conversely, for

c ∈/ [∆min, ∆max], ∆(I) 6= c for any I ∈ [IL, IH]. Proof of Proposition 6. By eq. (16), a non-degenerate equilibrium satisﬁes ∆(I∗) = c. By implicit function ∗ dI∗ = 1 ∗ d∆(I ) > theorem, dc d∆(I∗)/dI . Therefore, I is increasing in c as long as ∂I 0. Note that

d∆(I) dv∗(I) dv∗ = I − U dI dI dI ∗ dvI (I) where dI is given by eq. (A-16) and

∗ ∗ dv 1 −1 − dq U = 1 + (q∗ )r r (q∗ )r 1 U (A-18) dI U U dI

Consider ﬁrst the case of r arbitrary close to 1. Then, limr→1 v(qL) = 1 + qL and ∂v(q) limr→1 ∂q = 1. Therefore, by eq. (A-16),

∗ ∗ ∗ n Z qH n dvI (I) dF (qL) dF (q) lim = lim −qL − dq r→1 dI r→1 dI qL dI dE[q∗ ] = lim emax r→1 dI

∗ ∗ ∗ dvU d E[qemax] n −1 ∗ while by eq. (A-18) limr→1 dI = limr→1 dI n∗ . Note that limr→1 ∆(I) = n∗ E[qemax] = ∆1(I). Therefore, d∆(I) d∆ (I) lim = 1 r→1 dI dI

d∆1(I) ∂∆1(I) First we show that there is exist a non-empty interval I ∈ [IL, eI] for which dI = ∂I + ∗ ∂∆1(I) ∂n (I) ∗ ∂∆1(I) ∂n ∂I > 0. Since E[qemax] is increasing in I (Lemma A2), ∂I > 0. Note that

∗ Z qH Z qH n − 1 ∗ n∗−1 q n∗ ∆1(I) = ∗ n q f (q|r = 1)F (q|r = 1)dq = F (q|r = 1)dq n qL qL (1 + q)

1 1 1 1+q n−1 where the second equality follows from the equilibrium pdf, f (q|r = 1) = n∗−1 1+q I . n∗ ∂F (q) ∂∆1(I) Since ∂n > 0 for q < qH, ∂n > 0. ∗ ∂n∗(I) 0 By Proposition 4, n (I) is single-picked with ∂I > 0 for some I < I < IH. Therefore, by the continuity of ∆1(I), there exists eI such that ∆1(I) is strictly increasing in I for I ≤ eI.

29 ∈ [ ( )] dI∗ = 1 > = As a result, for c ∆min, ∆ eI , there exist an equilibrium with dc d∆(I∗)/dI 0. If eI IH, this non-degenrate equilibrium is unique for all I ∈ [IL, IH] and ∆b1 = ∆(IH) = ∆max. Suppose instead that I < I . Let I = arg min ∆ (I). We want to show that e H b1 I∈(eI,IH ) 1 ∆1(bI1) = ∆b1 > ∆min, implying the uniqueness of the non-degenerate equilibrium for c ∈ ∗ ∗ ∗ [∆min, ∆b1). Recall that n (IL) = n (IH) = 2 (see proof of Proposition 2) and n is single 0 0 picked (Proposition 4) reaching a maximum at I < bI1. Therefore, there exists I ∈ [IL, I ) ∗ ∗ ∂∆1(I) such that n (I) = n (bI1). However, since ∂I > 0, ∆b1 = ∆1(bI1) > ∆1(I) ≥ ∆min.

Therefore, for c ∈ [∆min, ∆b1), the non-degenerate equilibrium is unique. By the continuity d∆(I) dI∗ of ∆(I) and dI in r , there is a unique non-degerate equilibrium with dc > 0 as long as r is sufﬁciently close to 1 and c ∈ [∆min, ∆b) where ∆b > ∆min. d∆(I) Consider next the case of k arbitrary close to 0. We want to show that limk→0 dI > 0 d∆(I) d∆(I) d∆(I) ∗ ∗ for all I ∈ [IL, IH]. By Proposition 3, limk→0 dI = limn →∞ dI . We establish limn →∞ dI > dv∗ dv∗ ∗ U ∗ I 0 by showing that limn →0 dI = 0 and limn →0 dI > 0. dv∗ r 1 −1 r−1 dq∗ ∗ U ∗ ∗ r ∗ U ∗ ∗ From eq. (A-18), limn →∞ dI = limn →∞ 1 + (qU) (qU) dI . Note that limn →∞ qU = dq∗ ∗ ∗ ∗ U 0 since limn →∞ E[qemax] < ∞ (Proposition 3). We next show that limn →∞ dI = 0. Note that dq∗ 1 ∂E[q∗ ] E[q∗ ] ∂n∗ U = emax − emax (A-19) dI n∗ ∂I n∗ ∂I

∗ eq. (A-2) n n ∂E[qemax] ∂F (qL) R qH ∂F (q) It is straightforward to verify that limn∗→∞ = limn∗→∞ −qL − dq < ∂I ∂I qL ∂I n r ∂F (q) n∗ n∗ 1+q ∗ − = ∗ − ( ) = < ∞ since limn →∞ ∂I limn →∞ (n∗−1)I F q qr−1 I ∞. From implicit differenti- ∗ ∗ dn∗ = − ∂h(n ;I,r)/∂I ∂h(n ;I,r) ation of eq. (A-6), dI ∂h(n∗;I,r)/∂n . ∂I is giveby by eq. (A-8) and

∂h(n; I, r) M − I qr−1 q (1 + qr ) ∂q = U − U U + r U ∂n n∗ r 2 n ∂n 1 + qU ∗ ∗ ∗ dqU qH −n qU Taking into account that dI = n∗−1 (eq. A-9)

∗ n∗r M−I ∗ ∗ ∗ r ∂n ∗ [qH − E[q ]] − E[q ](1 + (q ) ) = − n −1 I emax emax U r ∗ (A-20) ∂I ∗ ∗ ∗ ∂qU (M − I)[−qU(1 + qU ) + rn ∂n ] ∗ ∗ ∗ ∗ ∗ ∗ ∂qU qU 1 ∂E[qemax] ∂E[qemax] n ln F(qL) R qH n ln F(q) Note that = − ∗ + ∗ ∗ and by eq. (A-1) ∗ = qLF (qL) 2 + F (q) 2 dq. ∂n n n ∂n ∂n (n∗−1) qL (n∗−1) Therefore,

∗ ∗ M−I ∗ ∗ E[q ] ∂n r[q − lim ∗→ E[q ]] − lim ∗→ E[q ] emax = − I H n ∞ emax n ∞ emax lim∗ ∗ ∗ 2 ∗ n →∞ n ∂I (n ) ∂qU (M − I)[−1 + limn∗→∞ r ∗ ] E[qemax] ∂n

30 ∗ ∗ 2 ∗ ∗ ∂qU (n ) ∂qU By Proposition 3, limn∗→∞ E[qmax] < ∞. Substituting for , we obtain limn∗→∞ r ∗ = e ∂n E[qemax] ∂n E[q˜∗ ] ∗ dq∗ ∗ max ∂n ∗ U −1. This implies that limn →∞ n∗ ∂I ∈ (−∞, ∞). It immediately follows that limn →∞ dI = dv∗ (I) ∗ U 0 and limn →∞ dI = 0. ∗ ∗ dvI (I) dvI (I) It remains to show that is strictly increasing in the limit, i.e. limn∗→∞ > 0. By dI ∗ dI∗ dFn (q) dFn (q) eq. (A-16), it is sufficient to show that limn∗→∞ < 0 for all q ≥ qL where is give dI ∗ dI n∗ Fn (q) 1+qr ∗ − = − < by eq. (A-17) . Note that the first term in eq. (A-17) limn →∞ n∗−1 I qr−1 I2 0. ∗ 1+qr ∗ ∗ n 1 dn Therefore, it is sufficient to show that the second term limn →∞ −F (q) ln r−1 2 = q I (n∗−1) dI ∗ ∗ 1 dn 0. Given eq. (A-20), it is straightforward to verify that limn →∞ 2 = 0, implying (n∗−1) dI ∗ dFn (q) dv∗(I) ∗ ∗ I that limn →∞ dI < 0 for all q ≥ qL and limn →∞ dI > 0. d∆(I) d∆(I) ∗ It follows that limn →∞ dI > 0 for all I. By continuity of dI with respect to k, it follows that for sufficiently small k, the value of information is increasing in I for all

I ∈ [IL, IH]. Setting ∆b = ∆max obtains Proposition 6.

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