Pre-negotiation commitment and internalization in public good provision through bilateral negotiations

Journal of Public Economics 175 (2019) 84–93

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Journal of Public Economics journal homepage: www.elsevier.com/locate/jpube

Pre-negotiation commitment and internalization in public good provision through bilateral negotiations☆ Noriaki Matsushima a, Ryusuke Shinohara b,⁎ a b

Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan Department of Economics, Hosei University, 4342 Aihara-machi, Machida, Tokyo 194-0298, Japan

a r t i c l e

i n f o

Article history: Received 31 August 2017 Received in revised form 4 March 2019 Accepted 27 March 2019 Available online 4 June 2019 JEL classification: C78 D42 D62 H41 H44

a b s t r a c t A profit-maximizing public good supplier endogenously determines the level of the public good and simultaneously negotiates with beneficiaries of the good one by one. A pre-negotiation commitment on the production level of the public good by the supplier enhances the internalization of beneficiaries' preferences. With the commitment, the supplier produces the public good at an efficient level in equilibrium if and only if its bargaining power is sufficiently weak. In addition, the public good is produced excessively as a result of the commitment when the supplier's bargaining power is sufficiently strong. © 2019 Elsevier B.V. All rights reserved.

Keywords: Public good Simultaneous bilateral bargaining Supplier bargaining power Nash bargaining solution

1. Introduction We consider a game in which a profit-maximizing public good supplier endogenously determines the commitment level of the public good, and then simultaneously negotiates with beneficiaries of the good one by one. We explain applicable situations of the game below.

☆ This paper extends a previously circulated unpublished paper “The efficiency of monopolistic provision of public goods through simultaneous bilateral bargaining.” We would like to thank the Co-Editor, Kai A. Konrad, and the three anonymous referees for their constructive and helpful comments. We are also grateful to Naoki Watanabe, Wataru Kobayashi, Emmanuelle Taugourdeau, Takuro Yamashita, and Shintaro Miura for their useful discussions and suggestions. In addition, we thank the participants of the 12th Meeting of the Society for Social Choice and Welfare, the 68th European Meeting of the Econometric Society, PET 15, 71st Annual Congress of the International Institute of Public Finance, the meeting of the Japanese Economic Association, and the meeting of the Japan Institution of Public Finance. We gratefully acknowledge the financial support from the KAKENHI Grant-in-Aid for Scientific Research (S) No. JP15H05728, (A) No. JP17H00984, (B) Nos. JP15H03349, JP18H00847, (C) Nos. JP24530248, JP15K03361, JP18K01519, and Young Scientists (B) No. JP24730165, RISTEX (JST), and the program of the Joint Usage/Research Center for “Behavioral Economics” at ISER, Osaka University. The usual disclaimer applies. ⁎ Corresponding author. E-mail addresses: [email protected] (N. Matsushima), [email protected] (R. Shinohara).

https://doi.org/10.1016/j.jpubeco.2019.03.009 0047-2727/© 2019 Elsevier B.V. All rights reserved.

In the agricultural industry, a marketing board of farm products (e.g., beef, milk, oranges) provides generic advertising campaigns to enhance demand for farm products. Because the universal benefits for farmers have a public good nature, funds to launch these generic campaigns are necessary based on the assessments of farmers, governmental support, and so on. In the United States, however, mandatory collections of promotional funds (e.g., mandatory promotional programs) have been controversial, inducing litigation on generic advertising programs since the Agricultural Marketing Agreement Act of 1937 (Crespi, 2003; Messer et al., 2007).1 Given the controversy over these programs, agricultural economists have considered several nonmandatory mechanisms to overcome the legal problem (e.g., Messer et al., 2005, 2007), because generic advertisements in product categories are beneficial to those producers (e.g., Rickard et al., 2011). Concretely, by using experimental methods, they have considered the effectiveness of the provision point mechanism with money-back guarantee rule (PPM-MBG rule) inspired by Bagnoli and Lipman (1989), who show that a public good is provided efficiently in an equilibrium 1 The mandatory promotional programs for agricultural products common in the United States are supported through referendum processes by involved farmers, although there is a non-negligible amount of litigation on some of these programs (Crespi, 2003; Messer et al., 2007).

N. Matsushima, R. Shinohara / Journal of Public Economics 175 (2019) 84–93

of voluntary contribution games if the level of the public good is exogenously restricted to either zero or a positive value.2 Given the need to propose a useful mechanism to overcome the problem of generic advertising campaigns, we show the possibility that the level of a public good can be endogenously determined at the Paretoefficient level even when a profit-maximizing representative, who negotiates with public good consumers one by one, determines it. This possibility implies that a group of farmers can ask a profit-maximizing advertising agency to be the representative to promote their agricultural products through generic advertising campaigns. We believe that the presented framework can also be applied to generic advertising by (i) tourism marketing organizations, which represent firms including hoteliers and gift shops in those regions, and (ii) franchisors, which represent the promotions of their company names on behalf of their franchisees which just borrow the franchisors' names for their sales. We consider a three-stage game in which a profit-maximizing public good supplier endogenously determines the commitment level of the public good; it simultaneously and bilaterally negotiates with n consumers of the good concerning monetary transfers to the supplier through Nash bargaining3; finally, the supplier determines whether to provide the public good under the endogenous level of the public good and predetermined transfers, and then incurs the supply cost if and only if it provides the public good. Our results show that the efficiency of the negotiation outcome changes drastically depending on whether the supplier commits to the level of the public good before negotiations. First, as a benchmark, we examine the case in which the supplier does not commit to the public good level before negotiations with consumers (Section 3). In this case, the supplier and each consumer negotiate the joint production level of the public good as well as transfers from the consumer to the supplier. We prove that the realized public good level through the negotiation is far from the efficient level (Proposition 1), because of a kind of the free rider problem on public goods. Second, we examine the case in which the supplier commits to the public good level before negotiations (Section 4). In this case, the level of the public good is separated from the negotiation, and the supplier and each consumer negotiate transfers only. We prove that under mild conditions, the public good is produced efficiently in equilibrium if and only if the supplier's bargaining power is sufficiently weak (Theorem 1). Under such conditions, each consumer anticipates that his/her transfer is necessary to provide the public good because the amount of transfer from each consumer to the supplier in the negotiations is small. Hence, the pre-commitment achieves the internalization of consumers' preferences so as to efficiently provide the public good. In addition, we point out the possibility that the pre-commitment leads to an excessive provision of the public good when the supplier's bargaining power is strong. Under these conditions, at the efficient level of the public good, not all consumers' transfers are necessary, implying that some consumers can free ride. To impede such free riding, the supplier sets an excessively high level of the public good to induce all consumers to contribute. The pre-commitment in our model restricts the possible levels of the public good to either zero or a pre-committed level. Because of this restriction, the public good in our model is converted into a discrete public good, which has been widely studied since Palfrey and Rosenthal (1984) and Bagnoli and Lipman (1989). This ex post discreteness of the public good improves the efficiency of public good provision. On top of the well-known results in the literature, our contribution is thus to examine whether the provision level of the public good is Pareto-efficient if a profit-maximizing agent endogenously determines the level from positive real numbers. In addition, the provision level is exogenously given in most earlier studies, including in Palfrey and Rosenthal (1984) and Bagnoli and 2 Krishnamurthy (2001) also experimentally investigates the effectiveness of the PPMMBG rule in a marketing context. 3 In the industrial organization context, several studies employ the simultaneous bilateral bargaining model (Chipty and Snyder, 1999; Raskovich, 2003; Marshall and Merlo, 2004; Matsushima and Shinohara, 2014; Collard-Wexler et al., 2019).

85

Lipman (1989). One exception is the working paper by Morelli and Vesterlund (2000), who also consider the endogenous choice of the provision point when a fundraiser aims to maximize (i) the total contribution and (ii) welfare. These are different from our model in which the supplier is motivated by profit maximization. We show that the public good can be efficiently supplied even by a profit-maximizing supplier and that the efficient level is not necessarily chosen endogenously. We briefly review several studies. As in our study, Ray and Vohra (1997, 2001) and Dixit and Olson (2000) examine public good provision through bargaining under complete information, and show that the internalization of beneficiaries' preferences fails even through bargaining and that the provision of public goods is inefficient, except in specific cases.4 Our Proposition 1 shares the negative results of these studies. Fortunately, in our model, the pre-commitment on the level of the public good can improve the efficiency of providing the good.5 In a different context, by incorporating incomplete information on agents' costs when providing public goods into the model of Hoel (1991), Konrad and Thum (2014) show that a unilateral pre-negotiation commitment by an agent on the level of the public good reduces the likelihood of achieving efficient outcomes.6 In addition, similar to the incomplete contract literature (Grossman and Hart, 1986), Beccherle and Tirole (2011) show that multiple agents manipulate their bargaining positions through prenegotiation commitments which substantially reduce the efficiency of the negotiation outcomes, although our model of public good provision differs from theirs. The remainder of this paper is organized as follows. Section 2 presents the preliminaries. Section 3 analyzes the case of no pre-commitment and Section 4 analyzes the case of pre-commitment. Section 5 discusses the extension of the basic model. Section 6 concludes the paper. 2. The model Consider an economy with a pure public good and a private good (money), with a supplier of the public good and n ≥ 2 consumers of the public good.7 Only the supplier has the technology to provide the public good, but without any compensation from consumers, it has no incentive to provide it. On the contrary, consumers enjoy the public good, but cannot produce the public good by themselves. Formally, the set of players is denoted by {s, 1, … , n}, where s is the supplier and i (i = 1, … , n) is a consumer. Let N be the set of consumers. The level of the public good is typically denoted by g ≥ 0, and consumer i's transfer to the supplier is denoted by Ti. The cost function of the public good is denoted by c : ℝ+ → ℝ+, such that c(0) = 0, and c is an increasing, convex, and twice continuously differentiable function. When the supplier provides g units of the public good and receives payment Ti from each i ∈ N, its payoff is ∑i∈NTi − c(g) (a profit). Each consumer i receives payoff v(g) − Ti, where v : ℝ+ → ℝ+ is a benefit function from the public good, such that v(0) = 0 and v is an increasing, concave, and twice continuously differentiable function. In this model, each consumer has the same benefit function, v. Definition 1. For each m ∈ {1, … , n}, g(m) is defined as the socially optimal level of the public good in the group with the supplier and m consumers such that g ðmÞ∈ arg max mvðg Þ  cðg Þ: g≥0

4 Ellingsen and Paltseva (2012) show that even when providing excludable public goods, bargaining may not achieve allocative efficiency through negotiation. 5 Several researchers have examined how the restriction of feasible bargaining outcomes affects the efficiency of negotiations (MacKenzie and Ohndorf, 2013; Ellingsen and Paltseva, 2016), although the ways in which such outcomes are restricted are different from those in our study. 6 Hoffmann et al. (2015) test Konrad and Thum's (2014) results experimentally. 7 The public good is always produced efficiently in all subsequent negotiation models if n = 1.

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Furthermore, we impose the following conditions on v(g) and c(g). Assumption 1. (1.1) lim v0 ðgÞ ¼ ∞, lim v0 ðgÞ ¼ 0, lim c0 ðgÞ ¼ 0, and g→0

g→∞

g→0

lim c0 ðgÞ ¼ ∞. (1.2) For all g N 0, c′(g)/c(g) N v′(g)/v(g).

g→∞

Remark 1. Condition (1.1) guarantees interior solutions of the model; that is, g(m) N 0 for each m ∈ {1, … ,n}.8 Condition (1.2) describes the relation between the ratio of the marginal benefit of the public good to the gross benefit and the ratio of the marginal cost to the overall cost. Condition (1.2) implies that c(g)/v(g) is increasing in g, that is, the cost per value is increasing in the level of the public good, which sounds reasonable.9 We model the bargaining process between the supplier and consumers in Sections 3 and 4 based on simultaneous and bilateral bargaining, which is often used in the context of buyer–supplier relationships (e.g., Horn and Wolinsky, 1988; Marshall and Merlo, 2004). The question we address is how the commitment to the production level of the public good before negotiations affects the efficiency of the provision of the good. To answer this question, we consider the case in which the supplier does not pre-commit to the level of the public good (Section 3) and then the case in which the supplier unilaterally commits to the level of the public good before negotiations (Section 4).

We solve this model in a backward manner. In the second stage, given (gj, Tj)j∈N, the supplier approves if ∑j∈NTj N c(∑j∈N gj), is indifferent between approval and disapproval if ∑j∈NTj = c(∑j∈N gj), and otherwise disapproves. Henceforth, we assume that when ∑j∈NTj = c (∑j∈N gj), the supplier disapproves.11 In the first stage, the supplier and consumer i ∈ N negotiate over (gi,Ti) and maximize the Nash product function, given (gj,Tj)j≠i. The supplier and consumer i's surpluses from the bilateral negotiation take different forms depending on the outcomes of the other bilateral negotiations. First, consider the case of ∑j≠iTj ≤ c(∑j≠i gj). Then, based on the second-stage equilibrium, if this bargaining breaks down, the supplier does not approve (gj, Tj)j≠i in the next stage. Hence, the payoffs to the supplier and consumer i are zero. If this bargaining reaches an agreement (gi, Ti) and the supplier approves (gj, Tj)j∈N in the next stage, then the supplier and consumer i receive the payoffs Ti + ∑j≠iTj − c(gi + ∑j≠i gj) and v(gi + ∑j≠i gj) − Ti, respectively. Therefore, the supplier and consumer i decide (gi, Ti) to maximize the Nash product function β ln (Ti + ∑j≠iTj − c(gi + ∑j≠i gj)) + (1 − β) ln (v(gi + ∑j≠i gj) − Ti). Thus, from the first-order condition, given (gj, Tj)j≠i, the bargaining between the supplier and consumer i achieves (gi∗, Ti∗) such that 0 1 0 1 X X
 g i ¼ arg max v@g i þ g j A  c@g i þ g j A and gi ≥ 0

j≠i

3. No pre-commitment on the level of the public good In this section, we examine the benchmark case in which the supplier does not commit to the production level of the public good before negotiations. We provide a two-stage model of negotiation based on simultaneous and bilateral bargaining. In the first stage, the supplier and each consumer i ∈ N bilaterally negotiate the joint production level of the public good gi ≥ 0 and the transfer to the supplier from consumer i Ti(≥0). Hence, in this stage, the supplier faces n independent bilateral negotiations. We assume that every bilateral negotiation is simultaneous and Nash bargaining. We further assume that the outcome of each negotiation is given by the (asymmetric) Nash bargaining solution, in the belief that the bargaining outcomes with the other parties are determined in the same way. Hence, the joint surplus of one bilateral bargaining process is maximized through the choice of the public good level, and the maximized joint surplus is divided between the consumer and the supplier in the proportion of 1 − β to β, where β ∈ [0,1] represents the supplier's bargaining power. We assume that, in equilibrium, the negotiators in each bilateral negotiation have consistent beliefs about the other negotiations: the outcome of each negotiation is predicted correctly by the others. The second stage is an approval stage by the supplier, in which it decides whether to execute the first-stage outcome (gj,Tj)j∈N. If the supplier approves (gj,Tj)j∈N, it provides ∑j∈N gj units of the public good in exchange for transfers ∑j∈NTj, and the supplier's and consumer i's payoffs are ∑j∈NTj − c(∑j∈N gj) and v(∑j∈N gj) − Ti, respectively. If the supplier disapproves, it annuls (gj,Tj)j∈N, and the payoff to every player is zero.10 Our model resembles those of Chipty and Snyder (1999) and Raskovich (2003). However, in their models, free riding is impossible. See the condition “vi(0,q−i) = 0” in Raskovich (2003, p. 410). 8 Our main results hold whenever g(m) is interior for each m ∈ {1, … ,n}. Thus, we can obtain these results if we assume lim v0 ðgÞ ¼ ∞, lim v0 ðgÞ ¼ 0, and c′(g) is finite, instead of g→∞ g→0 assuming Condition (1.1). 9 The conditions in Assumption 1 are satisfied by reasonable parametric benefit and cost ^pital's rule and (1.1), we obtain functions (see Section B in the online appendix). From l'Ho

limimg→0 10

cðg Þ ¼ 0: vðgÞ

(1.3)

See Section 4.3 for a discussion on simultaneous bargaining and the supplier’s approval. Further, we do not explicitly model consumers' approval of (gj,Tj)j∈N because the participation constraint always holds for all consumers because the Nash bargaining solution guarantees them non-negative payoffs. Thus, consumers will approve the second-stage bargaining outcome.

0 T i

v@g i

¼

þ

0 0

X

1 g j A−ð1−βÞ

j≠i

 @v@g i

þ

ð1Þ

j≠i

X

1

0

g j A−c@g i

þ

X

j≠i

1 g jA þ

X

j≠i

1 T j A:

ð2Þ

j≠i

From Eq. (2),

T i

þ

X

0 T j −c@g i

þ

j≠i

2 0

¼

β4v@g i

þ

X

X

1 g jA

j≠i 1

0

g j A−c@g i

þ

X

j≠i

1 g jA þ

j≠i

X

3 T j 5:

j≠i

Thus, Ti∗ + ∑j≠iTj N c(gi∗ + ∑j≠i gj) if and only if 0 v@g i

þ

X j≠i

1

0

g j A−c@g i

þ

X j≠i

1 g jA þ

X

T j N0:

ð3Þ

j≠i

Because the left-hand side of Eq. (3) is the joint surplus of the supplier and consumer i, the supplier approves ((gi∗, Ti∗), (gj, Tj)j≠i) if and only if the joint surplus is positive. If the supplier approves ((gi∗, Ti∗), (gj, Tj)j≠i), consumer i's payoff v(gi∗ + ∑j≠i gj) − Ti∗ = (1 − β)(v(gi∗ + ∑j≠i gj) − c(gi∗ + ∑j≠i gj) + ∑j≠iTj) is non-negative. By contrast, if the joint surplus is negative (i.e., Eq. (3) does not hold), the bargaining does not reach an agreement; that is, (gi∗, Ti∗) = (0,0). Second, consider the case of ∑j≠iTj N c(∑j≠i gj). Based on the secondstage equilibrium, even if this bilateral bargaining breaks down, the supplier approves the outcome. Hence, in the case of disagreement, the supplier and consumer i receive ∑j≠iTj − c(∑j≠i gj) and v(∑j≠i gj), respectively. If this bargaining reaches agreement (gi, Ti) and the supplier approves (gj, Tj)j∈N in the next stage, then the supplier and consumer i receive Ti + ∑j≠iTj − c(gi + ∑j≠i gj) and v(gi + ∑j≠i gj) − Ti, respectively. As in the previous case, (gi∗, Ti∗) maximizes the Nash product function β ln (Ti − c(gi + ∑j≠i gj) + c(∑j≠i gj)) + (1 − β) ln (v(gi + 11

This assumption does not change the subsequent result.

N. Matsushima, R. Shinohara / Journal of Public Economics 175 (2019) 84–93

∑j≠i gj) − Ti − v(∑j≠i gj)); that is,

4. Pre-commitment on the level of the public good

0 1 0 1 X X
 @ A @ g j  c gi þ g j A and g i ¼ arg max v g i þ gi ≥ 0

0 T i

¼ v@g i þ

j≠i

 @v@g i þ

j≠i

0 1 X g j A−v@ g jA

j≠i

−ð1−β Þ 0 0

ð4Þ

1

X

j≠i

X

1

0

g j A−c@g i þ

j≠i

X

1 0 0 1 0 111 X X g j A−@v@ g j A−c@ g j AAA:

j≠i

j≠i

j≠i

ð5Þ The maximized joint surplus of this negotiation is v(gi∗ + ∑j≠i gj) − c (gi∗ + ∑j≠i gj) − (v(∑j≠i gj) − c(∑j≠i gj)), which is non-negative from Eq. (4). From Eq. (5), T i þ

X

0 T j −c@

j≠i

X

1

j∈N

0 −c@

ð ÞÞ

0

g j A ¼ β v@gi þ

X j≠i

1 g jA

X

1

0

g j A−c@gi þ

j≠i

X

1 0 1 X g j A−ðv@ g jA

j≠i

j≠i

0 1 X X þ T j −c@ g j AN0; j≠i

j≠i

implying that the supplier's payoff is positive and so it always approves ((gi∗, Ti∗),(gj, Tj)j≠i). From Eqs. (1) and (4), in equilibrium, the supplier and every consumer i anticipate (gj∗, Tj∗)j≠i and maximize v(gi + ∑j≠i gj∗) − c(gi + ∑j≠i gj∗) through the choice of gi. Thus, we find that fg i g ¼ arg max v gi ≥ 0

ðg i þ ∑ j≠i g j Þ  cðg i þ ∑ j≠i g j Þ for each i ∈ N, leading to ∑ j∈N g j ¼ gð1Þ ¼ arg max vðgÞ  cðgÞ. By using this argument, we show the existence g≥0

and properties of the equilibria.12 Proposition 1. (gj∗, Tj∗)j∈N is an equilibrium only if ∑j∈N gj∗ = g(1). (gj∗, Tj∗)j∈N is an equilibrium if (gi∗, Ti∗) = (g(1), βv(g(1)) + (1 − β)c(g (1))) for some i ∈ N and (gj∗, Tj∗) = (0, 0) for each j ∈ N\{i}. In addition, there is a joint production equilibrium by M (M ⊆ N) such that 8      < g ð1Þ ; βvðg ð1ÞÞ þ ð1−βÞcðg ð1ÞÞ for each i∈M; gi ; T i ¼ m β þ ð1−βÞm : ð0; 0Þ for each i∈NnM; in which ∣M∣ ≡ m ≥ 2, if and only if β = 0 and c is linear. If M = N in Proposition 1, the equilibrium is symmetric although the necessary and sufficient condition is quite limited. The first statement in Proposition 1 shows that the equilibrium level of the public good is unique even if multiple equilibria exist. From Proposition 1, in each bilateral negotiation, the total surplus nv (g)-c(g) is never maximized because no bilateral negotiation internalizes the consumers' benefits from the other bilateral negotiations. Proposition 1 resembles the well-known result of the underprovision of the public good through voluntary contribution (e.g., Bergstrom et al., 1986) in that no player takes other players' benefits into account. In our model, because of the externalities among bilateral negotiations, some consumers may be able to enjoy the public good at no cost. If g (1) units of the public good are provided by some consumer, then the negotiations between the supplier and other consumers generate no surplus because g(1) maximizes the surplus of each bargaining pair. Thus, some consumers become free riders. 12

87

A detailed proof of Proposition 1 is available in Section C of the online appendix.

In this section, the supplier commits to the level of the public good before negotiations. In the first stage, the supplier decides the commitment level of the public good g ≥ 0, which is realized if it approves supplying the commitment level in the third stage. That is, in the first stage, the supplier does not provide the public good at this level or incur the cost. In the second stage, the supplier and each consumer i ∈ N bilaterally and simultaneously negotiate over the division of the joint surplus, thus determining Ti ≥ 0. The equilibrium outcome is determined in a similar way to the model in the previous section. In the third stage, the supplier decides whether to approve project (g, (Tj)j∈N). If the supplier approves the project, then it provides g units of the public good conditionally on receiving Ti from each consumer i. As a result, the supplier's payoff is ∑i∈NTi − c(g) and each consumer i's payoff is v(g) − Ti. Otherwise, no public good is provided and no money is transferred. Then, the supplier's and each consumer's payoffs are zero. 4.1. The second and third stages The analysis of the third stage is the same as before. Given (g,(Ti)i∈N), the supplier approves it if and only if ∑i∈NTi N c(g). In the second stage, who contributes to the project and how much money each contributor transfers to the supplier are determined. Who contributes to the project depends on who is pivotal to the project, as defined below. Definition 2. Let g ≥ 0 be the commitment level of the public good. Let i ∈ N. Let Tj be the transfer from consumer j ∈ N\{i}. Given (Tj)j≠i, consumer i is pivotal to the approval of project (g,(Tk)k∈N) if ∑j≠iTj ≤ c(g) b Ti + ∑j≠iTj. Because ∑j≠iTj ≤ c(g), transfers from a pivotal consumer i are necessary for the supplier's approval. If consumer i agrees to pay Ti, then Ti + ∑j≠iTj = ∑j∈NTj N c(g) holds and hence the supplier approves in the next stage. Then, the supplier's and consumer i's payoffs are ∑j∈NTj − c(g) N 0 and v(g) − Ti, respectively. We apply the bargaining outcome in the case of ∑j≠iTj ≤ c(∑j≠i gj) in Section 3 to that in this section. By replacing gi∗ + ∑j≠i gj in Eq. (2) with g, we obtain consumer i's transfer as follows: 0 1 X T i ¼ vðg Þ−ð1−βÞ@vðg Þ−cðg Þ þ T jA ¼ βvðgÞ þ ð1−βÞcðg Þ−ð1−βÞ

j≠i X

T j:

ð6Þ

j≠i

By arranging Eq. (6), we find that the supplier's payoff is ∑j∈NTj − c (g) = β(v(g) + ∑j≠iTj − c(g)). Hence, ∑j∈NTj N c(g) if and only if v(g) + ∑j≠iTj − c(g) N 0, and the joint surplus of the supplier and consumer i is positive. If consumer i is not pivotal to the project, then (i) Ti + ∑j≠iTj ≤ c(g) or (ii) ∑j≠iTj N c(g) is satisfied. In case (i), the joint surplus of the bargaining is zero because the supplier disapproves. Case (ii) means that the supplier approves the project in the third stage irrespective of whether the bargaining with consumer i succeeds. Hence, the supplier's net surplus from this bargaining is Ti + ∑j≠iTj − c(g) − (∑j≠iTj − c(g)) = Ti and consumer i's is v(g) − Ti − v(g) = − Ti; the joint surplus of this bargaining is thus zero. In any case, the joint surplus of the bargaining with the nonpivotal consumer i is zero, which implies Ti = 0; thus, consumer i becomes a free rider. Finally, we examine how many consumers become pivotal and how much money the pivotal consumers transfer to the supplier. Suppose that M is a set of pivotal consumers such that ∣M∣ = m (1 ≤ m ≤ n). We assume that the supplier randomly chooses m consumers out of n consumers when m b n.13 13 Note that because the consumers are identical, the way to select contributors does not affect the main results of this study.

88

N. Matsushima, R. Shinohara / Journal of Public Economics 175 (2019) 84–93

Because condition (6) holds for all i ∈ M and Ti = 0 for all i ∈ N\M, we obtain the transfers and the supplier's payoff in equilibrium as follows: Lemma 1. For each j ∈ M,

T mj ¼

Definition 3. g–m for each m ∈ {1, … , n} andg m for each m ∈ {2, … , n} are – public good levels that satisfy

βvðg Þ þ ð1−βÞcðg Þ ðN0 if gN0Þ β þ ð1−βÞm

ð7Þ

and for each j ∈ N\M, Tm j = 0. The payoff to the supplier is

πm S ðg Þ ≡

X

T mj −cðg Þ ¼

j∈M

βðmvðg Þ−cðg ÞÞ ; β þ ð1−βÞm

where the superscript “m” of consumers.

Tm j

and

contributors as in Eq. (9) and hence it can indirectly choose the number of contributors.14 To investigate which second-stage equilibrium maximizes the supplier's payoff, we start by restating Lemma (2.b).

πm S

ð8Þ

refers to the number of pivotal

Henceforth, we call pivotal consumers contributors and non-pivotal consumers free riders. The total joint surplus of the bargaining sessions with m contributors is mv(g)-c(g) and the supplier's share of the surplus is β/(β + (1 − β)m). Because β + (1 − β)m is the sum of the bargaining powers of the supplier and m contributors, the surplus is distributed in proportion to the bargaining power. Lemma 2 shows that the equilibrium number of contributors is determined according to the commitment level of the public good in the first stage. Lemma 2. (2.a) For any given g ≥ 0 such that nv(g) ≤ c(g), the equilibrium number of contributors at g is zero. (2.b) For any given g N 0 such that nv(g) N c(g), m is the equilibrium number of contributors at g if and only if cðg Þ cðg Þ bm≤ þ 1: vðg Þ βvðg Þ

ð9Þ

At least one integer m exists that satisfies Eq. (9). Proof. (2.a) If g satisfies nv(g) ≤ c(g), there is no surplus in any bilateral bargaining. Hence, no consumer pays a positive fee; the number of contributors is zero. (2.b) From Definition 2, m is the equilibrium number of contributors m m if and only if ∑j∈M\{i}Tm j ≤ c(g) b Ti + ∑j∈M\{i}Tj for each i ∈ M. From Eq. (7), these inequalities hold if and only if (m − 1)βv(g) ≤ c(g) b mv (g), implying Eq. (9). Clearly, at least one integer m exists that satisfies c(g)/v(g) b m ≤ (c(g)/v(g)) + 1 ≤ (c(g)/βv(g)) + 1. ■ First, note that Eq. (9) may include multiple integers. Second, note that if nv(g) N c(g), then the condition in Definition 2 restricts the number of contributors, which means that the supplier does not necessarily receive positive transfers from all consumers. We summarize the second-stage equilibria as follows. The second-stage equilibria. If g is the first-stage level of the public good, the second-stage equilibrium outcome is (M, (Tm j )j∈N), where M is the set of contributors such that m = ∣M∣ is determined according to Lemma 2 and (Tm j )j∈N conforms to Lemma 1. We consider a refinement of the second-stage equilibria because multiple second-stage equilibria may support different equilibrium numbers of contributors for some commitment levels of the public good. For the backward induction analysis, we focus on the equilibrium that maximizes the payoff to the supplier among the second-stage equilibria in every second-stage subgame. This equilibrium refinement seems reasonable because the supplier's choice of g before negotiations determines the number of

 c gm cðg–m Þ m ¼ –m and m ¼ – þ 1: vðg Þ βv g m –

ð10Þ

Here, g 1 ≡ 0, from (1.3) (see footnote 9). – Because c(g)/v(g) is monotonically increasing in g (Remark 1), g–m can be defined uniquely for each m ∈ {1, … , n}, and g m can be defined – uniquely for each m ∈ {2, … , n}. Lemma 3 shows the properties of g m – and g–m . Lemma 3. (3.a) For any given g N 0 such that nv(g) N c(g), m is the equilibrium number of contributors at g if and only if g m ≤gbg–m . –

1.

(3.b) For each m ∈ {2, … ,n}, g m bg–m−1 if β b 1, and g m ¼ g–m−1 if β = – –

Proof. (3.a) is a restatement of Lemma (2.b). We now show (3.b). By Definition 3, cðg–m−1 Þ=vðg–m−1 Þ ¼ m−1 and cðg m Þ=ðβvðg m ÞÞ ¼ m−1 . – – Under these conditions, we obtain cðg m Þ=vðg m Þ ¼ βðm−1Þ≤m−1 ¼ cð – – g–m−1 Þ=vðg–m−1 Þ . Because c(g)/v(g) is increasing in g, we obtain g m ≤ – g–m−1 . ■ Lemma (3.a) shows that g m and g–m are the lower and upper bounds – of the commitment level of the public good, respectively, where m is the equilibrium number of contributors in the second stage. In addition, Lemma (3.b) shows the conditions under which the second-stage subgame has multiple equilibria that support different numbers of contributors. From Lemma (3.b), for m ∈ {2, … ,n}, the range of m − 1 contributors (g m−1 ≤gbg–m−1 ) and the range of m contributors (g m ≤gbg–m ) – – overlap if β b 1. Thus, if β b 1 and the supplier chooses g between g m – and g–m−1 in the first stage, multiple second-stage equilibria exist that support the existence of m − 1 contributors and that of m contributors. Lemma 4 clarifies which second-stage equilibrium maximizes the supplier's payoff. m+1 Lemma 4. For each g N 0 and each m ∈ {1, … ,n − 1}, πm (g), S (g) ≤ πS which is satisfied with equality if and only if β = 0.

Proof. From Eq. (8), ðgÞ−πm πmþ1 S ðg Þ ¼ S

βðβvðg Þ þ ð1−βÞcðg ÞÞ ≥0; ðβ þ ð1−βÞðm þ 1ÞÞðβ þ ð1−βÞmÞ

which is satisfied with equality if and only if β = 0. ■ From Lemma 4, for each g N 0, πm S (g) is non-decreasing in m. Hence, in the second stage, after the supplier chooses g such that nv(g) N c(g), the second-stage equilibrium that maximizes the supplier's payoff within the set of the second-stage equilibria is the equilibrium at which the number of contributors is the maximal integer among m 14 Moreover, if the supplier can choose among alternative sequential negotiations paths, our assumption that it will choose the path that maximizes its payoff is obvious. We show that our main result still holds under sequential bargaining (see Section 5.1). We thank an anonymous referee for pointing out the justification.

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that satisfies Eq. (9). By Lemmas 3 and 4, we obtain the refined secondstage equilibrium: The refinement of the second-stage equilibria. If g is the commitment level of the public good, the second-stage equilibrium outcome is (M, (Tm(g) )j∈N), where ∣M ∣ = m(g) and j 8 0 if g ¼ g 1 ð¼ 0Þ; > > > – > > 1 2 > > > < 1 if g∈h–g ;–g ; mðg Þ ¼ k if g∈ g k ; g kþ1 ðk∈f2; …; n−1gÞ; > > h– – > > > > n if g∈ g n ; g–n ; > > – : 0 if g ≥g–n :

ð11Þ

Finally, in Lemmas 5 and 6, we obtain monotonic properties of the supplier's payoff function with respect to the number of contributors, which generate a driving force for our main result (Theorem 1) with Lemma 4 (see the proofs of the lemmas in Appendix A). Lemma 5. For each m ∈ {1, … ,n − 1},

m πm Þbπmþ1 ðg mþ1 Þ. S ðg S – –

m+1 (g(m + 1)). Lemma 6. For each m ∈ {1, … ,n − 1}, πm S (g(m)) b πS

4.2. The first stage To express the threshold value of β in Theorem 1, we impose the following definition by using Eq. (10).

89

some increase of the public good level, increases the supplier's payoff. Therefore, the supplier desires to receive transfers from all consumers. Because the public good is pure, the supplier must set the level of the public good sufficiently high to receive transfers from all consumers; if the level of the public good is low, so that the cost of the public good is compensated by the transfers from fewer than n consumers, then free riders exist. However, there is another problem: the supplier may choose an excessively high level of the public good. If the supplier's bargaining power is strong, the transfers from contributors are large. Then, there may be the case in which the cost of g(n) is compensated by fewer than n consumers. In this case, the supplier sets the commitment level of the public good over g(n) to eliminate free riders. Therefore, the supplier provides the public good efficiently if and only if its bargaining power is sufficiently weak, such that the cost of g(n) is compensated by all consumers, but not by fewer than n consumers. Nevertheless, the precommitment of the public good level is a useful device for supplying a large amount of the public good. Remark 2. Condition 0 ≤ β ≤ β(g(n),n) in Theorem 1 implies that in the second-stage equilibrium given g = g(n), all consumers are contributors, but does not directly show that g(n) maximizes the supplier's payoff. In the proof, first, for each m ∈ {1, … , n}, we seek the level of the public good that maximizes the supplier's payoff under the constraint that m is the number of contributors. By adopting this method, we find n constrained maximizers, which are candidates for a global maximizer. g(n) is shown to be the constrained maximizer when m = n. Second, by comparing the supplier's payoffs at the constrained maximizers, we show that g(n) is the global maximizer.

Definition 4. For each g ≥ 0 and m ∈ (1, n], β(g, m) ∈ ℝ+ is defined as βðg; mÞ ≡

cðg Þ 1  : vðg Þ m−1

4.3. Discussion ð12Þ

Note from Eq. (10) that β(g, m) ∈ ℝ+ is the value of bargaining power at which g is the lower bound over which m contributors exist; that is, g ¼ g m if β = β(g, m). – In Theorem 1, we state that the necessary and sufficient condition for the efficient provision of the public good is that all n consumers are contributors when the commitment level is g(n). By Eq. (9), the condition is n≤

cðg ðnÞÞ cðg ðnÞÞ 1 þ 1 or β ≤  ¼ βðg ðnÞ; nÞ: βvðg ðnÞÞ vðg ðnÞÞ n−1

By Definition 4, gðnÞ ¼ g n if β = β(g(n),n). Because c(g)/v(g) is in– creasing in g, g n bgðnÞ if and only if β is smaller than β(g(n),n). This im– plies that g(n) is on ½g n ; g–n Þ , which is the range of g inducing n – consumers to contribute, if β ≤ β(g(n), n) (see Eq. (11)). Theorem 1. Under Assumption 1, there is an equilibrium at which the supplier produces the efficient level of the public good (g(n)) if and only if β satisfies 0 ≤ β ≤ β(g(n), n). The proof of Theorem 1 is in Appendix A. As Theorem 1 shows, β(g (n), n) marks the threshold value of the supplier's bargaining power, below which the supplier produces the public good efficiently. In an equilibrium at which the supplier's bargaining power is strong, the supplier provides the public good excessively, although it receives transfers from all consumers.15 We can confirm from Lemma 4 that for any level of the public good, the more consumers from which the supplier receives transfers, the higher the supplier's payoff is. We can also confirm from Lemmas 5 and 6 that an increase in the number of contributors, accompanied with 15

See the analysis when β N β(g(n),n) in Section A in the online appendix.

4.3.1. Credibility of the pre-negotiation commitment The supplier can credibly commit to the level of the public good if its payoff with the commitment is larger than that without it. From the analysis in Section 4.2, the supplier's equilibrium payoff with the commitment is πnS (g(n)) if β ≤ β(g(n), n), otherwise, πnS ðg n Þ.16 As we see in – Section 3, the supplier's equilibrium payoff without the commitment is π1S (g(1)) = β(v(g(1)) − c(g(1))) because only one contributor generally exists in equilibrium if β N 0. Proposition 2 shows that, for the supplier, the outcome with the commitment is preferable to that without the commitment (the proof is in Appendix A): Proposition 2. π1S (g(1)) b πnS (g(n)) if β ≤ β(g(n),n), otherwise, π1S ðgð1ÞÞb πnS ðg n Þ. – The supplier provides a higher level of the public good with the commitment. This increases transfers from consumers as well as the cost of the public good. Proposition 2 suggests that the commitment to the level of the public good secures larger transfers from consumers, which outweighs the cost increase, and hence, the supplier has an incentive to decide the level of the public good before negotiations. 4.3.2. On the supplier's approval stage As explained in the Introduction, some agricultural economists emphasize the usefulness of the PPM-MBG rule in the context of generic advertising (e.g., Messer et al., 2007). The money-back guarantee rule is consistent with the supplier's equilibrium approval decision in our model. We briefly discuss modified models without the supplier's approval stage; that is, the supplier may have to provide a public good even if the sum of contributions falls short of its cost. This possibility may require some outside resource to fill a deficit of public good provision. However, the existence of the outside resource may affect the bargaining outcomes. First, we consider the no-commitment scenario. Let i ∈ N and (gj, Tj)j≠i. Because the outside resource always fills the deficit, ∑j≠i gj 16

See Section A in the online appendix for the case of β N β(g(n),n).

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can be produced even without consumer i's transfer. Thus, (gi, Ti) maximizes β ln (Ti − c(gi + ∑j≠i gj) + c(∑j≠i gj)) + (1 − β) ln (v(gi + ∑j≠i gj) − v(∑j≠i gj) − Ti). From the similar analysis in Section 3, we can show that g(1) is the unique equilibrium level of the public good. This is the same as the first statement in Proposition 1. Second, we consider the commitment scenario. Because any pre-committed level g can be provided by the existence of the outside resource, no bilateral bargaining generates a positive surplus and hence Ti = 0 for each i ∈ N. As the supplier is fully compensated by the outside resource, any g leads to zero profit. This is different from Theorem 1. From this result, we conclude that the pre-commitment on the level of the public good should be done with the post-bargaining supplier's approval. 4.3.3. On simultaneous bilateral bargaining A non-cooperative foundation for the asymmetric Nash bargaining solution has been presented by several studies. As Binmore et al. (1986) show, the subgame perfect equilibrium in Rubinstein's bargaining model, with a risk of breakdown, approximates the Nash bargaining solution. Hence, we can approximately interpret that in the bargaining stage of our model, each pair of the supplier and a consumer plays Binmore et al.'s (1986) game, anticipating the other sessions' outcomes. That is, while negotiating, the supplier and a consumer in a bilateral negotiation do not know whether the others' negotiations are continuing or what agreements they might reach even if they are completed. Participants in a bilateral bargaining learn the outcomes of the other negotiations only after their bargaining finishes. This information assumption is justified because the simultaneous Nash bargaining solution implicitly assumes that the supplier delegates n agents to negotiate with consumers and each pair of an agent and a consumer simultaneously negotiates.17 5. Extensions The formal analyses in this section are available in Section D of the online appendix. 5.1. A sequential bargaining model Sequential bargaining seems to be an alternative bargaining procedure in our situation. We briefly discuss how the main result changes if we consider a model in which the supplier bilaterally, but sequentially negotiates with each consumer.18 Formally, we replace the secondstage bargaining game of the model in Section 4 with the following sequential bilateral bargaining game: the supplier first negotiates with consumer 1, then negotiates with consumer 2 after the bilateral bargaining with consumer 1, …, and finally, bilaterally negotiates with consumer n after the bilateral bargaining sessions with the other consumers. Each bilateral bargaining session is assumed to be Nash bargaining. The supplier has only one bilateral negotiation with each of the n consumers. We can show that the equilibrium transfer attained through this sequential bargaining is the same as that in Lemma 1. Hence, we obtain a result similar to Theorem 1. 5.2. Take-it-or-leave-it offers and other belief assumptions Our model resembles the situation in which a monopolistic input supplier negotiates with competing downstream firms (e.g., McAfee and Schwartz, 1994). Many studies in this line of research assume that the supplier makes take-it-or-leave-it offers publicly or privately to each downstream firm. A public offer is observable by not only the firm that receives it but also the other firms. A private offer is observable only by the 17 The strategic delegation problem reported by several studies such as Segendorff (1998) and Besley and Coate (2003) does not matter in our model because each agent is assumed to behave in the best interest of the supplier rather than pursuing his/her own objective. 18 A sequential bargaining protocol based on the Nash bargaining solution is analyzed by Valencia-Toledo and Vidal-Puga (2017).

firm that receives it, and the other firms form beliefs about the private offer. In existing studies, three sorts of belief formation with regard to the other offers (symmetric, passive, and wary beliefs) have been examined.19 These belief formations are different from the consistent belief assumed in the simultaneous Nash bargaining solution. Hence, it is worth clarifying how different belief formations affect the equilibrium outcomes. By examining the take-it-or-leave-it offer game with and without pre-negotiation commitment on a public good level, we show that the difference in belief formation does not influence our main result. Without the commitment, the public good is never provided efficiently irrespective of whether offers are public or private. By contrast, with the commitment, the supplier provides the public good efficiently if and only if gn ≤gðnÞbgn, which is the same as the condition in Theorem – 1 when β = 1. 5.3. Commitment on transfers Because one might wonder whether the level of transfers is an alternative commitment device for efficiency, we briefly discuss the effect of commitment on transfers. Our analysis is based on an extension of the benchmark model in Section 3. First, in each bargaining, the level of transfers from a consumer to the supplier is decided by the supplier or the consumer before bilateral negotiations. Second, in each bargaining, the supplier and each consumer i negotiate only on gi, the level of the public good that is jointly produced. Whether the commitment on transfers achieves efficiency depends on who decides the transfer level and how strong the supplier is. When the supplier decides the transfer level from all consumers, it provides the public good efficiently in equilibrium if its bargaining power is sufficiently weak. In this equilibrium, the transfers are set so that all consumers are non-pivotal. This contrasts with the result when the supplier commits to g (i.e., all consumers are pivotal; see Section 4). When consumers decide the level of transfers, whether efficiency is achieved depends on the number of consumers who commit to transfers. Our analysis shows that commitment by one consumer never achieves the efficient provision of the public good in equilibrium, while commitment by all consumers achieves efficiency in equilibrium if the supplier's bargaining power is sufficiently weak. 5.4. Heterogeneous consumers We prove that it is partially correct that the supplier produces the public good inefficiently if its bargaining power is sufficiently strong, even when consumers are heterogeneous. We consider the case in which there are two consumers (N = {1,2}) that have different benefit functions and bargaining power over the supplier. The payoff to consumer i ∈ N is vi (g) − Ti, where vi : ℝ+ → ℝ+ is consumer i's benefit function from the public good that satisfies vi(0) = 0, vi 0 N0, vi 00 ≤0, and is twice continuously differentiable. Let βi ∈ [0,1] be the supplier's bargaining power with consumer i ∈ N. The cost function c(g) is the same as before. The timing of the game is the same as that for the basic model in Section 4. In this extended model, the Pareto-efficient allocation is not achieved in equilibrium if the supplier's bargaining power over some consumers (but not every consumer) is sufficiently strong. In this sense, the main result in the case of identical consumers remains partially true. 6. Concluding remarks We examine public good provision through simultaneous bilateral negotiations between a public good supplier, which endogenously determines the level of the public good before those negotiations, and the consumers of the good. Our question is whether the pre-negotiation 19

See the definition of these belief conditions in McAfee and Schwartz (1994).

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commitment for the public good plays an effective role in achieving its efficient provision. First, we show that under simultaneous bilateral negotiations, the efficient provision of the public good is never accomplished without the pre-commitment (Proposition 1). Second, we show that if the supplier has an opportunity to pre-commit to the production level of the public good, then under some conditions, it produces the public good efficiently in equilibrium if and only if its bargaining power is sufficiently weak (Theorem 1). We further clarify the relation between the effectiveness of the precommitment and the supplier's bargaining strength. This finding provides some positive and negative aspects of the commitment before negotiations: while the pre-commitment on the level of the public good plays a role in enhancing the internalization of beneficiaries' preferences, the public good is not necessarily provided efficiently, even under this enhanced internalization.

91

When the threshold level of the public good is endogenously chosen by the profit-maximizing supplier, fund-raising through bargaining, rather than through voluntary contribution, may provide the supplier a strong incentive to provide the public good. This is because the supplier's profit is zero in any equilibrium of the voluntary contribution game after the threshold level of the public good is set at any positive level (e.g., Bagnoli and Lipman, 1989). On the contrary, whenever β N 0, the supplier obtains a positive profit in some equilibrium of the simultaneous bargaining game under the threshold level g such that nv(g) N c(g). Theorem 1 suggests that a positive but very small profit to the supplier will be a strong incentive for it to provide the public good efficiently. Thus, when we consider the generic advertising of agricultural products as the comprehensive process of setting the provision point and fund-raising, fund-raising through bargaining rather than through voluntary contribution may be helpful to provide a higher public good.

Appendix A. Appendix Proof of Lemma 5 First, 0 ¼ π1S ðg 1 Þbπ 2S ðg 2 Þ from 0 ¼ g 1 bg 2 bg–2. Hereafter, we focus on m ∈ {2, … ,n − 1}. For each g N 0, define a real number μ(g) such that μ(g) = c – – – – (g)/(βv(g)) + 1 (see Eq. (10)). For instance, when g ¼ g m , μ(g) = m. We obtain that – μ 0 ðg Þ ¼

vðg Þc0 ðg Þ−cðg Þv0 ðg Þ βðvðg ÞÞ2

N0;

ð13Þ

according to Assumption 1. From Eq. (8), differentiating πμ(g) (g) with regard to g yields S μ ðg Þ

dπ S ðg Þ β½μ 0 ðg Þðβvðg Þ þ ð1−βÞcðg ÞÞ−ðμ ðg Þv0 ðg Þ−c0 ðg ÞÞðβ þ ð1−βÞμ ðg ÞÞ ¼ dg ðβ þ ð1−βÞμ ðg ÞÞ2 0

¼

v ðg Þðβðvðg Þ  cðg ÞÞ þ cðg ÞÞ2 βðvðg ÞÞ2 ðβ þ ð1  βÞμ ðg ÞÞ2

N0;

ð14Þ

in which the second line is obtained by substituting μ(g) = c(g)/(βv(g)) + 1 and Eq. (13) into the first line. Thus, πμ(g) (g) is increasing in g. Because S μðg m Þ μðg mþ1 Þ m Þ ¼ π S – ðg m ÞbπS – ðg mþ1 Þ ¼ πmþ1 ðg mþ1 Þ for each m ∈ {2, … , n − 1}. ■ g m bg mþ1 from Eq. (10), then πm S ðg S – – – – – – Proof of Lemma 6 m+1 (g(m)). We also easily find the following two properties: From Lemma 4, πm S (g(m)) ≤ πS m 1. From Definition 1 and Eq. (8), for fixed m, g = g(m) maximizes πm S (g). This implies that πS (g) is increasing in g ∈ [0,g(m)) and decreasing in g ∈ [g (m),∞). 2. g(m) b g(m + 1) for each m ∈ {1, … , n − 1}.

From the observations, πm+1 (g(m)) b πm+1 (g(m + 1)). ■ S S Preliminaries for proof of Theorem 1 Before proving Theorem 1, we remember that: g(n) is the efficient level of the public good because g(n) maximizes nv(g)-c(g). We summarize how g(m) and g m (or g mþ1 ) are related according to the values of m and β as in Lemma 7. – – Lemma 7. Let β ∈ (0, 1] and m ∈ (1, n]. Then, g mþ1 ≤g ðmÞif β ≤βðg ðmÞ; m þ 1Þ; –

ð15Þ

g m ≤gðmÞbg mþ1 if βðg ðmÞ; m þ 1Þbβ ≤βðg ðmÞ; mÞ; and – –

ð16Þ

g ðmÞbg m if βNβðg ðmÞ; mÞ: –

ð17Þ

Proof. From Eqs. (10) and (12),  c gm cðg ðmÞÞ  – ; ¼ β ≤βðg ðmÞ; mÞ ¼ v ð g ð mÞÞðm−1Þ m v g ðm−1Þ –

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that is, β ≤ β(g(m),m) if and only if g m ≤gðmÞ from Remark 1. Similarly, from Eqs. (10) and (12), –  c g mþ1 cðg ðmÞÞ  – ¼ β ≤βðg ðmÞ; m þ 1Þ ¼ ; v ðg ðmÞÞm mþ1 m v g – that is, β ≤ β(g(m),m + 1) if and only if g mþ1 ≤gðmÞ. In addition, we immediately find from Eq. (12) that β(g(m),m + 1) b β(g(m), m), and then, we – obtain Eqs. (15)–(17). ■ m mþ1 Þ, for each m and β. Lemma 8 shows what level of the public good g maximizes πm S (g) under the constraint g∈½g ; g – –

Lemma 8. Let m ∈ {1, … ,n} and β ∈ (0, 1]. m mþ1 Þ. Thus, there is no maximizer when 8a. If β ≤ β(g(m), m + 1), then g mþ1 ≤gðmÞ (see Eq. (15)); hence, πm S (g) is increasing within the interval ½g ; g – – – restricting to this interval.

8b. If β(g(m),m + 1) b β ≤ β(g(m), m), then gðmÞ∈½g m ; g mþ1 Þ (see Eq. (16)); hence, within the interval ½g m ; g mþ1 Þ, πm S (g) is maximized at g = g(m). – – – – m 8c. If β(g(m), m) b β, then gðmÞbg m (see Eq. (17)); hence, within the interval ½g m ; g mþ1 Þ, πm S (g) is maximized at g ¼ g . – – – –

Proof of Theorem 1 (⇐) Suppose that 0 ≤ β ≤ β(g(n), n). If β = 0, then the supplier's payoff is zero irrespective of its choice of g. Thus, g(n) is one of the optimal levels for the supplier. Hereafter, we restrict our focus to the case in which 0 b β ≤ β(g(n), n). Given that the number of contributors is m(g) in Eq. (11), we investigate the public good level that maximizes the supplier's payoff. From 0 b β ≤ β(g(n),n) and Eq. (16), we obtain g n ≤g ðnÞbg n : –

ð18Þ

We partition the set {1, … , n − 1} (the number of contributors is fewer than n) into three subsets as follows: n o
N 1 ≡ m∈f1; …; n−1gj g mþ1 ≤g ðmÞ ; N 2 ≡ m∈f1; …; n−1gj g –

m

≤g ðmÞbg

mþ1



; and N 3 ≡ fm∈f1; …; n−1gj g ðmÞbg

m

g:

We examine which level of the public good maximizes the supplier's payoff for each m in the three subsets. First, consider the numbers of conm mþ1 Þ. As πm tributors in N 1. Let m∈N 1. m is the number of contributors if and only if g∈½g m ; g mþ1 Þ. Because g mþ1 ≤gðmÞ, πm S (g) is increasing in ½g ; g S (g) is – – – – – m mþ1 Þ and continuous at every g, we can define π S ðg –

m lim mþ1 g↑g π S ðg Þ¼

–

Thus,

πm S

sup g∈½g m ;g mþ1 Þ

    β mv g mþ1 −c g mþ1 – – mþ1 : ¼ g – β þ ð1−βÞm m mþ1 mþ1 πm Þ. From Lemma 4, πm Þbπ nS ðg mþ1 Þ. Because πnS (g) is increasing if g b g(n) and g mþ1 ≤g n ≤gðnÞ, we further obS ðgÞ ¼ π S ðg S ðg – – – – –

– – tain πnS ðg mþ1 Þ≤π nS ðgðnÞÞ. Therefore, –

 g mþ1 bπ nS ðg ðnÞÞ for each m∈N 1 : πm S –

ð19Þ

Second, we consider the numbers of contributors in N 2. Let m∈N 2. Because g m ≤gðmÞbg mþ1 , πm S (g) is maximized at g = g(m). Then, from Lemma 6, – – n πm S ðg ðmÞÞbπ S ðg ðnÞÞ for each m∈N 2 :

ð20Þ

Third, consider the numbers of contributors in N 3. Let m∈N 3. Given that the number of contributors is m, the supplier's payoff πm S (g) is maximized m at g ¼ g m because gðmÞbg m . From Lemma 4, π m ÞbπnS ðg m Þ. Because πnS (g) is increasing if g b g(n) and g m bg n ≤gðnÞbg–n , we further obtain πnS ðg m Þb S ðg – – – – – – – πnS ðgðnÞÞ. Therefore,  g m bπnS ðg ðnÞÞ for each m∈N 3 : πm S –

ð21Þ

Finally, we prove that the supplier's payoff is maximized at g = g(n). Given that the number of contributors is n, the suppliers' payoff is maximized at g = g(n) from Eq. (18). Then, Eqs. (19)–(21) imply that when the number of contributors belongs to N 1 ∪N 2 ∪N 3, the supplier's payoff is less than πnS (g(n)). Therefore, the supplier chooses g(n) in equilibrium. (⇒) On the contrary, suppose that β N β(g(n),n) and an equilibrium exists at which the supplier chooses g(n). Suppose that on the path of this equilibrium, the supplier chooses g(n) in the first stage, the number of contributors is μ ∈ {1, … , n} in the second stage, and the project is approved

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in the third stage. The supplier obtains πμS(g(n)) in this equilibrium. The supplier chooses a level of the public good in the set ½g μ ; g μ  when μ is the – number of contributors. At the same time, from Eq. (17), we obtain gðnÞbg n from β N β(g(n),n). Thus, in this equilibrium, μ b n holds. – We need to consider two possibilities:g μ bgðnÞ and g μ ¼ gðnÞ. Suppose first that g μ bgðnÞ. We obtain g(μ) b g(n) because μ b n. Then, because πμS(g) is – – – decreasing in g if g N g(μ), if the supplier sets a level of the public good a bit lower than g(n) in the interval½g μ ; g μ , its payoff increases (the supplier can – μ μ choose such a level because of g μ bgðnÞ). Second, suppose that g μ ¼ gðnÞ. From Lemma 5 and μ b n, πS ðg μ Þ ¼ πS ðgðnÞÞbπnS ðg n Þ. In any case, the supplier – – – – does not choose g(n) in equilibrium, which is a contradiction. ■ Remark 3. The proof of the necessity above does not depend on the equilibrium selection in the second and third stages presented in Section 4. In addition to the equilibrium presented in Section 4, in the third stage, there is an equilibrium at which the supplier chooses execution if and only if ∑j∈NTj ≥ c(g). Depending on which third-stage equilibrium, the condition under which μ consumers transfer to the supplier is gðnÞ∈½g μ ; g μ Þ or gðnÞ∈ð – g μ ; g μ ; that is, in any equilibrium, if u consumers transfer to the supplier, then gðnÞ∈½g μ ; g μ . In the proof, we consider this interval. – –

Proof of Proposition 2 From Lemma 6, π1S (g(1)) b πnS (g(n)) if β ≤ β(g(n),n). If β N β(g(n),n), then define N 2 similarly to the proof of Theorem 1 and let m ∗ be the maximal ∗ ∗ m  n n integer in N 2 . From Lemma 6, π1S (g(1)) ≤ πm S (g(m )) and from Claim A.1 in Section A in the online appendix, π S ðgðm ÞÞbπ S ðg Þ. ■ –

Appendix B. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.jpubeco.2019.03.009.

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