Group Size, Collective Action and Complementarities in Efforts

Transcription

1 Group Size, Collective Action and Complementarities in Efforts Guillaume Cheibossian Université de Montpellier (CEE-M) & Toulouse School of Economics (TSE) Romain Fayat Université de Montpellier (CEE-M) Postal adress: Faculté d Economie Site Richter Avenue Raymond Dugrand CS Montpellier, France Abstract: We revisit the group size paradox in a model where two groups of different size compete for a prize exhibiting a varying degree of rivalry and where group effort is given by a CES function of individual efforts. We then show that the larger group has a higher probability of success than the smaller group only for an intermediate range of the degree of complementarity of individual efforts, and that this range is decreasing with the degree of rivalry of the prize. We also show that the members of the larger group have a higher payoff for an even smaller interval of the degree of complementarity. Keywords: group size paradox; group contest; complementarity; (impure) public good JEL classification: D72;D74 (Corresponding author). Adress guillaume.cheibossian@umontpellier.fr Declarations of interest: none. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

2 Introduction According to Olson (965) larger groups are less effective than smaller groups because they face a greater free-rider or collective action problem. Yet, even though individuals contribute less in larger groups, it does not necessarily imply that larger groups produce lower levels of collective action since by definition a larger number of fellow members can contribute to group action. Esteban and Ray (200) develop a model of group contest for a prize that can have mixed public-private characteristics, and show that if the individual cost of contributing to group action is sufficiently convex relative to the degree of rivalry of the prize, then larger groups are more successful than smaller groups. The reason is that the higher level of individual effort contributed in smaller groups is not sufficient to counterbalance the lower number of contributors. In this paper, we also consider a contest between two groups of different size for a prize exhibiting a certain degree of rivalry, and where group members incur constant, or increasing, marginal cost of contributing to collective action. The difference is that the effective level of group action is given by a CES function of individual efforts, instead of being given by a summation technology implying perfect substitutability between individual contributions. An important result is that the larger group has a greater probability of success than the smaller group only for intermediary values of the degree of complementarity, while the reverse holds when individual efforts tend to be more substitutable or more complementary. The range of these intermediary values is increasing with the degree of the cost convexity and decreasing with the degree of rivalry of the prize. We also show that the members of the larger group have a higher payoff for an even smaller interval of the degree of complementarity than that for which the larger group has a greater probability of success. Few recent papers analyze contests between groups where group members efforts are not perfect substitutes. Lee (202) considers weaest-lin contests for group-specific public prizes, while Chowdhury and al. (203) tae the other extreme by considering best-shot group contests. More closely related to the present analysis, Kolmar and Rommeswinel (203) assume that group effort is given by a CES function of group members efforts and consider that group members are heterogeneous in their valuation of the pure public prize. As a result of this heterogeneity, groups with higher complementarity perform worse than similar groups with lower complementarity. Yet, they assume that individuals face linear costs and that groups compete for a pure public good. In the present paper, all agents have the same valuation of the prize and thus we focus primarily on the collective action problem as a function of group size when the prize is an impure public good. Furthermore, we focus both on probabilities of success and on individual welfare. 2 The model Assume that two groups A and B with n A and n B identical members respectively, compete for a impure public prize Y. We denote by e (e, e 2,...e n ) the vector of individual efforts in group for = A, B. Group effort depends on group members efforts according to a CES function, that is n F (e ) = = e σ i σ for = A, B and σ {(, 0) (0, ]}, () where σ measures the degree of complementarity between individual efforts. The elasticity of substitution is /( σ). For σ =, we have perfect substitutability between individual efforts and Eq. () becomes the standard summation technology, i.e. F (e ) = e i. For a survey of the literature on the group-size paradox, see Pecorino (205). 2

3 For σ, we have perfect complementarity and in the limit we have F (e ) = Min{e i } (referred as to the weaest-lin function). Finally, if σ < 0 and if there exists such that e i = 0, F (e ) is not well defined. We will therefore tae the limit of F (e ) as e i 0, which means F (e ) = 0 in that case. The winning probability of group, for = A, B and, is given by the following contest-success function, i.e. p (e, e ) = F (e ) F (e ) + F (e ) if (F (e ) + F (e )) (0, 0), 2 otherwise. All individuals in the two groups have the same valuation of the prize. The individual welfare of member i in group is thus (2) π i (e i, e, e ) = p (e, e ) Y n β c(e i ). (3) β (0, ) measures the degree of rivalry of the prize Y. For β = 0, the prize is a pure public good while for β =, it is fully private and divisible among group members. The greater β, the larger is the rivalry of the prize. We will also assume that the cost of individual effort is isoleastic, i.e., c(e i ) = γ eγ i for i =, 2,..., n, = A, B and γ. (4) We have the following Proposition. 2 Proposition : The group contest game admits a Nash equilibrium in pure strategies. In equilibrium, the level of effort of member i in group, for = A, B and, is given by the following first-order condition where F (e ) / e i = F (e ) e i F (e ) F (e ) + F (e )] 2 Y n β eσ i] σ e σ i. e γ i 0, (5) We now focus on the symmetric interior equilibrium such that all members in a group exert the same level of effort that is e i = e for any i =, 2,..., n and = A, B. In this case, we have F (e ) = n /σ e and F (e ) / e ei i =e = n ( σ)/σ. First-order conditions (5) can then be rewritten as n ( σ)/σ n /σ e F (e ) + F (e )] 2 Y n β = e γ. (6) Thus, from these two equilibrium conditions, we can obtain e = ( n 2 The proof of Proposition is in the Appendix n ) (+β)/γ e. (7) 3

4 In a symmetric equilibrium, the winning probability of group for = A, B and is thus given by ( p e, e ) n θ = n θ + nθ We then have the following Proposition. where θ = σ + β. (8) γ Proposition 2: The larger group has a larger probability of success than the smaller group for σ (0, min {γ/( + β), }], while the reverse holds for σ (, 0) if γ + β or σ (, 0) (γ/( + β), ] if γ < + β. Using (6) and (7), the equilibrium level of individual effort in the symmetric equilibrium is given, for = A, B and, by e = nθ n θ n θ + nθ ] 2 Y n β γ. (9) We have that e > e if n+β > n +β. This is the illustration of the free-rider or collective action problem: the members of the larger group produce lower levels of efforts than the members of the smaller group. Finally, substituting (8) and (9) into (3), individual welfare in the symmetric equilibrium is given, for = A, B and, by ( ) ] π ( e, e ) n θ β γ n θ + nθ n n θ = ] 2 Y. (0) γ n θ + nθ 3 The Group Size Paradox We now establish two corollaries that allow us to expose the group size effect, on the one hand, and the complementarity effect on the other hand. Corollary : Assuming perfect substitutability between group members efforts, i.e. σ =, then the larger group has a higher probability of success than the smaller group if and only if γ + β. Let define the group size effect as the impact of the size of the group on the arithmetic sum of contributions of the larger group (that is for σ = ). The group size effect is neutral if the prize is a pure public good, i.e. β = 0, and if the costs of contributing to collective action are linear, i.e. γ =. This corresponds to Katz and al. (2000) s result. The reason is that the larger number of contributors ust counter-balances the lower individual contribution in the larger group due to the free rider problem. When the prize exhibits some degree of rivalry, i.e. β 0, and if γ =, individual returns depend negatively on the size of the group reducing even more individual contributions in the larger group. In this case, the group size effect is negative, which corresponds to the Olson s group size paradox. Finally, if the cost function is sufficiently convex, and in particular if γ 2, then the group size effect is positive even though the prize is fully divisible (i.e. β = ). Indeed, the higher level of individual contribution in the smaller group is not sufficient to compensate 4

5 its inherent disadvantage of having a smaller number of contributors. This corresponds to Esteban and Ray (200) s analysis. Now, let consider that σ, but γ =, and that the prize is either a pure public or a divisible good. Corollary 2: Let γ =. (i) If the prize is a pure public good, i.e. β = 0, then the larger group has a higher probability of success than the smaller group for σ (0, ], while the reverse holds for σ (, 0). (ii) If the prize is fully private and divisible among group members, i.e. β =, then the larger group has a higher probability of success than the smaller group for σ (0, /2], while the reverse holds for σ (, 0) (/2, ]. We define the complementarity effect as the impact of σ on the probability of success of the larger group when the group size effect is neutralized, i.e. when β = 0. This effect is positive for σ (0, ], but negative for σ (, 0). 3 Recall that the lower σ, the greater is the complementarity between group members contributions. Thus, in the case of a pure public prize, individuals contributions in the larger group must not be too much complementary for the larger group to be more successful. However, when the prize is private, i.e. β =, the larger group has greater probability of success only for intermediary values of σ. For σ < 0, the group size effect and the complementarity effect are both negative, and thus the group size paradox holds. But, when σ (0, ] the complementarity effect is positive and compensates the negative group size effect only for σ (/2, ]. Thus, in the case a fully divisible prize, the group size paradox is reversed when group members efforts are not too, but sufficiently, complementary. Now, for γ, β 0 and σ, the probability of success of the larger group is always lower than that of the smaller group if σ < 0, independently of β and of γ. When σ > 0, the group size paradox is reversed only for σ (0, γ/( + β)] provided that γ < + β, as shown by Proposition 2. Thus, the higher the cost convexity and the lower the degree of rivalry of the prize, the larger the size of the interval for which the larger group has a greater probability of success. 4 Individual Welfare We now determine whether individuals are better-off in the larger (say group ) or in the smaller group (say group ). For the sae of simplicity, we assume γ = and that the prize is either a pure public good, i.e. β = 0, or a fully divisible prize, i.e. β =. From (0) we can write the difference between the welfare of a member of the larger group and that of a member of the smaller group as: Π Π = n θ β n θ + nθ n n θ ] n θ β ] n θ + nθ n nθ ] 2 Y. () n θ + nθ Recall that θ = (/σ) ( + β) and let consider that Ψ = n θ + nθ n n θ and Ψ = n θ + nθ n nθ. We can see that Ψ > Ψ for σ (0, ], while the reverse holds for σ (, 0], independently of β. Indeed Ψ > Ψ can be reduced to n θ+ > n θ+, where θ + = σ β. And the sign of θ + depends only on the sign of σ. Furthermore, for β = 0 the sign of θ depends also only on σ. And thus, n θ > nθ and Ψ > Ψ for σ (0, ], while the reverse holds for σ (, 0]. Consequently, the sign of Π Π depends only on the sign of σ when β = 0. 3 Point (i) of Corollary 2 for a public prize corresponds to a specific example given by Kolmar and Rommeswinel (203) in their appendix when they consider homogenous groups. 5

6 Now, for β =, we still have Ψ > Ψ when σ > 0, while the reverse holds for σ < 0. Meanwhile, we have that n θ < n θ, and thus Π Π < 0 for σ < 0, where θ = σ 3. However, when σ > 0, we have that nθ > n θ only for σ (0, /3]. Thus, for σ (/3, ] we have that n θ < n θ and Ψ > Ψ. Therefore, we rewrite equation (2) as: ( ) ( )] n 2θ n 2θ + (n n ) θ n n n 2 n 2 Π Π = ] 2 Y. (2) n θ + nθ ( ) ( ) The expression n n n 2 n 2 is always negative for n > n and the expression n 2θ n 2θ is also negative for σ (, 0] (0.4, ] since 2θ = (2/σ) 5 when β =. Therefore, we have: Proposition 3: (i) When the prize is a pure public good, i.e. β = 0, then a member of the larger group gets higher welfare than a member of the smaller group for σ (0, ], while the reverse holds for σ (, 0). (ii) When the prize is a fully divisible good, i.e. β =, a member of the larger group gets a higher welfare than a member of the smaller group for σ (0, /3], while the reverse holds for σ (, 0) (0.4/]. However, the result is ambiguous for σ (/3, 0.4]. When the prize is a pure public good, i.e. β = 0, obviously, the interval of values of σ for which a member of the larger group gets a higher or lower welfare than a member of the smaller group coincides to that for which the larger group has a higher or lower probability of success. When the prize is perfectly divisible, i.e. when β =, the group size impacts directly and negatively the individual return of the prize. Consequently, the size of the interval of values of σ for which a member of the larger group gets a higher welfare is smaller than that for which the larger group has a higher probability of success. Due to the ambiguity of the result for σ (/3, 0.4], we present in the following some numerical examples. We set the value of the fully divisible prize at Y = 00. Table : Numerical examples for σ (/3, 0.4] σ n n Π Π Table shows that the sign of Π Π depends on both σ and on the asymmetry in group size, but also on the total number of players involved in the contest. For example, whether σ = 0.34 or σ = 0.37 and for the same ratio of group size n /n = 2, the sign of Π Π changes as the total numbers of players increases from (4 + 2) = 6 to ( ) = 50. 6

7 5 Conclusion Our main conclusion is that Olson (965) s thesis of the lower efficiency in collective action of larger groups remains broadly valid in a quite general setup. Indeed, we show that the members of the larger group have a greater probability of success and a higher level individual of welfare than their competitors in the smaller group only for intermediary values of the degree of complementarity in efforts, depending on the degree of rivalry of the prize. As a result, the question as to why group size is an asset in real-world politics remains open. 6 Appendix From the first-order condition, the first derivative of π i (e i, e, e ) with respect to e i can be written as follows p (.)( p (.)) F (e ) F (e ) e i Y n β e γ i, (A) where p (.) stands for p (e, e ). Now let Ψ (.) p (.)( p (.))/F (e ). This derivative can thus be re-written as Ψ (.) F (e ) e i Y n β e γ i. (A2) We now characterize the second derivative of π i (e i, e, e ) with respect to e i. We have We have Ψ (.) e i = We also have ] Ψ (.) F (e ) + Ψ (.) 2 F (e ) e i e i e 2 i Y n β (γ ) e γ 2 i. (A3) { p (.) F (e )] 2 ( 2p (.))F (e ) p (.)( p (.)) F (e } ). (A4) e i e i Substituting into (A4), we thus have which can be rewritten as p (.) = p (.)( p (.)) e i F (e ) F (e ) e i. (A5) Ψ (.) e i = 2p (.) p (.)( p (.)) F (e )] 2 F (e ) e i, (A6) Ψ (.) e i = 2p (.) Ψ (.) F (e ). (A7) F (e ) e i Substituting into (A3), we have 2p (.) Ψ ( ) ] (.) F (e ) 2 + Ψ (.) 2 F (e ) Y F (e ) e i e 2 i n β (γ ) e γ 2 i. (A8) 7

8 We also have F (e ) / e i = i] eσ σ e σ i and hence 2 F (e ) e 2 i = ( ) ( σ ) eσ i σ 2 σe σ i ( i + (σ ) e σ eσ i ) σ e σ 2 i. (A9) Simplifying we have that 2 F (e ) e 2 i ( = eσ i ) σ 2 e σ 2 i Simplifying again, it can be rewritten as 2 F (e ) e 2 i ( σ) e σ i + (σ ) ] eσ i. (A0) ( ) α = (σ ) eσ σ 2 ( ) i i eσ e σ 2 i. (A) which is always negative since σ. It follows that the second derivative of π i (e i, e, e ) with respect to e i given by (A3) is always negative. Therefore, the game admits a Nash equilibrium in pure strategies References ] Chowdhury, S.M., Lee, D., and R. Sheremeta, 203, Top Guns May Not Fire: Best- Shot Group Contests with Group-Specific Public Good Prizes, Journal of Economic Behavior and Organization 92, ] Esteban, J., and D. Ray, 200, Collective Action and the Group Size Paradox, American Political Science Review 95, ] Katz, E., Nitzan S. and J. Rosenberg, 990, Rent-Seeing for Pure Public Goods, Public Choice 65, ] Kolmar, M., and H. Rommeswinel, 203, Contests with Group-Specific Public Goods and Complementarities in Efforts, Journal of Economic Behavior and Organization 89, ] Lee, D., 202, Weaest-Lin Contests with Group-Specific Public Good Prizes, European Journal of Political Economy 28, ] Olson, M., 965, The Logic of Collective Action, Cambridge, MA: Harvard University Press. 7] Pecorino, P., 205, Olson s Logic Collective Action at Fifty, Public Choice 62(3-4),