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Collective rent seeking with endogenous group sizes
European Journal of Political Economy Vol. 13 Ž1997. 121–130
Collective rent seeking with endogenous group sizes Kyung Hwan Baik a , Sanghack Lee a b
b,)
Department of Economics, Sung Kyun Kwan UniÕersity, Seoul 110-745, South Korea Department of International Trade, Kookmin UniÕersity, Seoul 136-702, South Korea
Abstract We examine a contest in which two groups compete to win larger shares of a private-good rent, and where group size is endogenized by allowing inter-group mobility of group members. We also relax the restriction on the range of parameter values of the sharing rules adopted by Nitzan Ž‘Rent-seeking with non-identical sharing rules’, Public Choice 71, 43–50, 1991, and ‘Collective rent dissipation’, Economic Journal 101, 1522– 1534, 1991. and Lee Ž‘Endogenous sharing rules in collective-group rent-seeking’, Public Choice 85, 31–44, 1995.. We find that the two groups tend to be of equal size, and that the optimal sharing rules place great emphasis on relative outlays. The rent is ‘substantially’ dissipated in a collective contest with endogenous group sizes and sharing rules. JEL classification: D72 Keywords: Collective rent seeking; Endogenous sharing rule; Inter-group mobility
1. Introduction The theory of rent seeking, initiated by seminal work of Tullock Ž1967, 1980., Krueger Ž1974., and Posner Ž1975., has been extended in many directions. 1 While encompassing a variety of scenarios on rent seeking, most papers in the literature )
Corresponding author. Tel.: Žq82. 2-910.4546; fax: Žq82. 2-910.4519. The literature on rent seeking has been surveyed by Tollison Ž1982., Brooks and Heijdra Ž1989., and Nitzan Ž1994.. 1
0176-2680r97r$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved. PII S 0 1 7 6 - 2 6 8 0 Ž 9 6 . 0 0 0 4 1 - 9
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K.H. Baik, S. Lee r European Journal of Political Economy 13 (1997) 121–130
have focused on rent-seeking competition between individuals. However, rents are often sought by groups of individuals. Reflecting this, Nitzan Ž1991a,b., Baik Ž1994., Baik and Shogren Ž1995., and Lee Ž1995. have recently analyzed collective rent seeking. The purpose of this paper is to extend the theory of collective rent seeking in two respects. First, we endogenize group size by allowing inter-group mobility of group members. Second, we relax the restriction on the range of parameter values of the sharing rules adopted in Nitzan Ž1991a,b. and Lee Ž1995.. The paper is organized as follows. Section 2 sets out the basic model without inter-group mobility. As in Lee Ž1995., collective rent seeking is modelled as a two-stage game in which intra-group sharing rules and individual outlays are determined in the first and the second stage, respectively. Characteristics of the subgame-perfect equilibrium are derived for the cases with or without the restriction on the parameter value of the intra-group sharing rule. Section 3 endogenizes group sizes by allowing inter-group mobility of group members. We find that the two groups tend to be of equal size. We offer our conclusions in Section 4.
2. Collective rent seeking without inter-group mobility Combining Nitzan Ž1991a,b. with Long and Vousden Ž1987., we consider a contest in which two groups compete to win larger shares of a divisible private-good rent. The rent is worth S. Each group consists of a fixed number of identical risk-neutral individuals, denoted by n i , for i s 1, 2. Without loss of generality, we assume that n1 G n 2 . The size of total population N, Ž N s n1 q n 2 ., is assumed to exceed 2. Individual k of group i contributes X k i to rent-seeking activity of his own group. Group i expends X i , Ž X i s Ý k X k i ., on rent-seeking. Total rent-seeking outlay of the two groups is denoted by X Ž X s X 1 q X 2 .. A negative value of X k i or X i is allowed, provided that X ) 0. That is, an individual or a group in the aggregate can take away resources contributed by other individuals or group, rather than expending resources on rent-seeking, as long as total rent-seeking expenditure X is positive. Group i’s share of the rent, P i , is given by the ratio of its rent-seeking outlay X i to the total rent-seeking outlay X 2 :
P i s X irX .
Ž 1.
If X i - 0, then P i - 0, since X ) 0. In such a case, group i in the aggregate takes away resources contributed by the other group, and pays a penalty in the amount 2 The functional form of P i is identical to the logit-form probability-of-winning function employed in Tullock Ž1980., Ursprung Ž1990., Nitzan Ž1991a,b., Baik Ž1994., and Baik and Shogren Ž1995., the difference being the possibility that P i can now have a negative value or exceed 1.
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123
of yP i S, when the rent is actually distributed. On the other hand, the other group, which has expended resources, receives both the rent and the penalty paid by group i. Of course, the size of the rent is not affected, since P 1 q P 2 s 1. Each group distributes its share of the rent among members according to its prespecified intra-group sharing rule. Let f k i denote the proportion of the share accruing to individual k of group i. Each group’s share of the rent, P i S, is fully distributed so that Ý k f k i s 1, for i s 1, 2. The payoff of rent-seeking activity for individual k of group i is given by Vk i s P i Sf k i y X k i .
Ž 2.
The proportion f k i is assumed to be determined by his own outlay and by outlays of the other members of group i. Following Nitzan Ž1991a,b., this paper focuses on a family of distribution rules given as f k i s Ž 1 y a i . X k irX i q a irn i .
Ž 3.
From Eqs. Ž2. and Ž3., we find that the amount Ž1 y a i . P i S is distributed in proportion to relative outlay, and that the amount a i P i S is equally shared with by members of group i. Nitzan Ž1991a,b. and Lee Ž1995. have adopted the restriction that a i should belong to the closed unit interval w0,1x. However, there seems to be no a priori reason for such restriction. The parameter a i may have a negative value or exceed 1. 3 A negative value of a i implies that group i’s sharing rule places great emphasis on member’s relative outlay, X k irX i . In this case, group i collects Žya i . P i Srn i from each of its members, and then distributes Ž1 y a i . P i S among members in proportion to relative outlay. On the other hand, when a i ) 1, voluntary contribution is discouraged. In this case group i collects Ž a i y 1. P i S from its members according to relative outlay, and then distributes a i P i S equally among its members. Note that the distribution rule given in Eq. Ž3. is compatible with any value of a i , since Ý k f k i s 1 for any value of a i . We model the collective rent-seeking contest as a two-stage game, drawing on Lee Ž1995.. We assume that the parameters a 1 and a 2 are determined prior to individual decisions on the extent of voluntary contribution. In the first stage the representative of each group chooses the value of the sharing-rule parameter which maximizes the group’s aggregate payoffs, respectively. 4 In so doing, the actions of members in the second stage are suitably taken into account. In the second stage all the members of the groups choose their outlays simultaneously and independently. All of the above is common knowledge to the individuals. We employ a subgame-perfect equilibrium as the solution concept. 3
Baik and Shogren Ž1995. have also noted this possibility. Maximizing aggregate payoffs of the group is equivalent to maximizing individual payoff in a symmetric equilibrium in which each one obtains the same payoff. Thus, each member will agree to the sharing rule proposed by the representative. 4
K.H. Baik, S. Lee r European Journal of Political Economy 13 (1997) 121–130
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To obtain the solution, we firstly analyze the second-stage actions of individuals. We then examine the first-stage interaction between the groups. Substituting Eqs. Ž1. and Ž3. into Eq. Ž2., we obtain Vk i s P i Sf k i y X k i s S Ž 1 y a i . X k irX q S a i X irn i X y X k i .
Ž 4.
Individual k of group i solves the problem given as max Vk i s S Ž 1 y a i . X k irX q S a i X irn i X y X k i .
Ž 5.
Xk i
For individual k of group i, the first-order condition for the payoff maximization is 5
E Vk irE X k i s S Ž 1 y a i . Ž X y X k i . rX 2 q S a i Ž X y X i . rn i X 2 y 1 s 0. Ž 6 . The N equations in Ž6. determine the Nash equilibrium values of X k i for given values of a 1 and a 2 , under a mild restriction that X be positive. Summation of Eq. Ž6. over the relevant range of k and simplification gives, for i s 1, 2, X irX s a i q Ž 1 y a i . n i y Ž n i XrS . .
Ž 7.
Simultaneous solution of Eq. Ž7., for i s 1,2, gives X 1 Ž a 1 , a 2 . s Ž SrN 2 . Q n1 n 2 Ž a 2 y a 1 . q n 2 a 1 y n1 a 2 q n1 ,
Ž 8.
X 2 Ž a 1 , a 2 . s Ž SrN 2 . Q n1 n 2 Ž a 1 y a 2 . q n1 a 2 y n 2 a 1 q n 2 ,
Ž 9.
and X Ž a 1 , a 2 . s X 1 Ž a 1 , a 2 . q X 2 Ž a 1 , a 2 . s Ž SrN . Q,
Ž 10 .
where Q s w N y n1 a 1 y n 2 a 2 q a 1 q a 2 y 1x. To assure a positive value of X, Q should also be positive. We suitably restrict our attention to the set of a 1 and a 2 which guarantees positive value of Q. This set encompasses a wide range of a 1 and a 2 . For example, when both a 1 and a 2 belong to the closed unit interval w0, 1x, as in Nitzan Ž1991a,b. and Lee Ž1995., Q is positive, and thus an equilibrium always exists. If a i - 0, for i s 1 and 2, positive value of Q is assured as well. We now analyze the first-stage interaction between the groups. The representative of each group solves, respectively: maxV1 Ž a 1 , a 2 . s ÝVk1 Ž a 1 , a 2 . s S P 1 Ž a 1 , a 2 . y X 1 Ž a 1 , a 2 . a1
Ž 11 .
k
and maxV2 Ž a 1 , a 2 . s ÝVj2 Ž a 2 , a 2 . s S P 2 Ž a 1 , a 2 . y X 2 Ž a 1 , a 2 . . a2
5
j
The second-order condition is satisfied.
Ž 12 .
K.H. Baik, S. Lee r European Journal of Political Economy 13 (1997) 121–130
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The first-order conditions for Eqs. Ž11. and Ž12. are calculated as
E V1rEa 1 s Ž n1 y 1 . S Ž n1 y n 2 . Ž Ž n 2 y 1 . a 2 q 1 . y 2 n 2 Ž n1 y 1 . a 1 rN 2 s 0,
Ž 13 .
and
E V2rEa 2 s Ž n 2 y 1 . S Ž n 2 y n1 . Ž Ž n1 y 1 . a 1 q 1 . y 2 n1 Ž n 2 y 1 . a 2 rN 2 s 0,
Ž 14 .
respectively. The second-order conditions are satisfied. In the case without the restriction simultaneous solution of Eqs. Ž13. and Ž14. gives the set of the equilibrium sharing-rule parameters Ž a 1) , a 2) .. In the case with the restriction, the set of the equilibrium sharing-rule parameters can be obtained through application of the Kuhn–Tucker conditions. We firstly consider the case with the restriction. In this case, from Eq. Ž10., X is always positive. Thus, an equilibrium exists. The existence issue raised by Nitzan Ž1991a. does not apply to this case. Following the same procedure as in Lee Ž1995. we can obtain the set of the subgame-perfect equilibrium sharing-rule parameters. In the case with no restriction on values of a i , simultaneous solution of Eqs. Ž13. and Ž14. gives the set of the subgame-perfect equilibrium sharing-rule parameters Ž a 1) , a 2) .. In this case, of course, we focus on the set of Ž a 1 , a 2 . which assures a positive value of Q. It is easy to see that the solution is unique and stable if n 2 ) 1. Lemma 1 summarizes the results on equilibrium intra-group sharing rules. Lemma 1. Ž1. If a i belongs to the closed interÕal w0, 1x , then the set of the subgame-perfect equilibrium intra-group sharing rules is giÕen as: Ža. Ž a 1) , a 2) . s ŽŽ n1 y n 2 .rw2Ž n1 n 2 y n 2 .x, 0. , if n 2 ) 1 , and Žb. Ž a 1) , a 2) . s Ž1r2, r . where r g w0, 1x , if n 2 s 1. Ž2. With no restriction on Õalues of a 1 and a 2 , the set of the subgame-perfect equilibrium intra-group sharing rules is: Ža. Ž a 1) , a 2) . s ŽŽ n1 y n 2 .rŽ n1 y 1. N, Ž n 2 y n1 .rŽ n 2 y 1. N . if n 2 ) 1 , and Žb. Ž a 1) , a 2) . s Ž1r2, r . where r denotes an arbitrary real number, if n 2 s 1. The case without the restriction calls for some explanations. Note that group 2 distributes more than the share in proportion to relative outlay to overcome the size disadvantage, if n 2 ) 1. On the other hand, group 1 can spare some of the share to distribute on an egalitarian basis, while its sharing rule places more emphasis on relative outlay than in the case with the restriction as well. Substituting Ž a 1) , a 2) . into Eqs. Ž8. and Ž9., we can find the equilibrium rent-seeking expenditure of each group. The extent of rent-dissipation and the payoff to each member can be easily calculated as well. Let t denote the rent-dissipation rate, i.e., t s X )rS. Also, define Õi as the payoff to a member of
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group i, for i s 1, 2. The characteristics of the subgame-perfect equilibrium with or without the restriction on a i are given by Proposition 1. Ž1. When a i belongs to the closed interÕal w0, 1x , the subgameperfect equilibrium is characterized as: Ža. P 1) s P 2) s 1r2, Žb. t s Ž2 n 2 y 1.r2 n 2 , and Žc. Õ 1 s Sr4n1 n 2 and Õ 2 s Sr4w n 2 x 2 . Ž2. With no restriction on Õalues of a 1 and a 2 , the subgame-perfect equilibrium has the following characteristics: Ža. If n 2 ) 1, P 1) rP 2) s n 2rn1 , t s Ž N y 1.rN, Õ1 s n 2 Srn1 N 2 , and Õ 2 s n1 Srn 2 N 2 . Žb. If n 2 s 1, P 1) s P 2) s 1r2, t s 1r2, Õ 1 s Sr4n1 and Õ 2 s Sr4. In the case with the restriction, the two groups always share the rent equally. Note that the rent-dissipation rate t is determined by the size of group 2. The larger group 2 is, the higher is the rent-dissipation rate. Each member obtains a positive payoff. Note also that the payoff to each member of group 2 is not affected by the size of group 1. In the case without the restriction, group 2 takes the larger share of the rent, if n 2 ) 1. This is because the larger group, group 1, suffers from the negative size effect due to free-riding. However, the rent-dissipation rate t is independent of the distribution of population across groups. Notice that it is identical to the result obtained by Tullock Ž1980. in the N-person rent-seeking contest. The social cost associated with collective rent seeking can be inferred by observing the value of the rent and the size of total population. However, the payoff to each member is affected by his or her membership. When n 2 s 1, collective rent seeking exhibits characteristics of a two-person rent-seeking contest in both cases with or without the restriction. From Proposition 1, we find that the restriction on values of 1 and 2 lowers the extent of rent dissipation, unless n 2 s 1. The restriction has adverse effect on the rent-seeking efforts of the smaller group, which would otherwise have placed more emphasis on relative effort. When n 2 s 1, the restriction has no effect on rent-seeking efforts of the only member of group 2, and thus has no effect on rent-seeking efforts of group 1, either.
3. Endogenous group sizes Inter-group mobility of group members is now introduced into the model. We add one more stage to the game specified in the previous section. 6 In the first stage a chance to move to the other group is given to every member of each group. 6 A similar argument has been used in Baik and Shogren Ž1995.. Our model is also related to the literature on provision of local public goods and the theory of clubs. See, for example, Rubinfeld Ž1987..
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Each member takes all the other members’ membership decisions as given and makes his own membership decision. No cost is incurred with movement of members across groups. In the second stage, each group adopts its own sharing rule. All the members choose their outlays simultaneously and independently in the third stage. From Proposition 1, the payoff to each member of group i, Õi , can be written as a function of n1 and n 2 , respectively. If Õ1Ž n1 , n 2 . - Õ 2 Ž n1 y 1, n 2 q 1., a member of group 1 has an incentive to move to group 2. On the other hand, a member in group 2 will move to group 1, if Õ 1Ž n1 q 1, n 2 y 1. ) Õ 2 Ž n1 , n 2 .. When equality holds, an individual is indifferent between staying in his group and moving to the other group. No one can increase his payoff by moving to the other group if the following inequalities are jointly satisfied: Õ1 Ž n1 , n 2 . G Õ 2 Ž n1 y 1, n 2 q 1 . ,
Ž 15 .
Õ1 Ž n1 q 1, n 2 y 1 . F Õ 2 Ž n1 , n 2 . .
Ž 16 .
and
The weak inequality Ž15. implies that a member of group 1 cannot increase his payoff by moving to group 2. Similarly, the weak inequality Ž16. means that a member in group 2 cannot augment his payoff by moving to group 1, either. Denote n i satisfying both Eq. Ž15. and Eq. Ž16. by n )i , for i s 1, 2. We now derive the equilibrium group sizes, n1) and n )2 , in the case when a i belongs to w0, 1x. From Proposition 1, we find that Õ 1Ž n1 q 1, n 2 y 1. - Õ 2 Ž n1 , n 2 ., if n1 ) n 2 and n 2 G 2. Thus, no one in group 2 will move to group 1, if n 2 G 2. If n1 s n 2 and n 2 G 2, a member of each group has an incentive to move to the other group. Straightforward calculation also shows that Õ 1Ž n1 , n 2 . - Õ 2 Ž n1 y 1, n 2 q 1. if n1 G Ž n 2 q 3.. That is, a member of group 1 has an incentive to move to group 2 if group 1 is larger than group 2 by three members or more. When n 2 s 1, Õ 2 s Sr4. In this case we need to consider the possibility of the formation of a grand coalition by movement of the only member of group 2 to group 1. In the grand coalition each individual receives SrN. It is easy to show that SrN - Sr4 if N ) 4. In such a case, the only member of group 2 has no incentive to move to group 1. Thus, a grand coalition is not feasible if N ) 4. If N s 4, the only member of group 2 is indifferent between remaining in group 2 and forming a grand coalition. When N s 3, the grand coalition is formed. Proposition 3 reports the results on equilibrium group sizes. Proposition 2. If a i belongs to w0, 1x , then the equilibrium group sizes Ž n1) , n )2 . are: Ž1. If N s 3, a grand coalition is formed. Ž2. If N s 4, Ž n1) , n )2 . s Ž3, 1. , or a grand coalition is formed. Ž3. If N G 5 and N is an odd number, Ž n1) , n )2 . s ŽŽ N q 1.r2, Ž N y 1.r2.. Ž4. If N G 5 and N is an eÕen number, Ž n1) , n )2 . s ŽŽ Nr2. q 1, Ž Nr2. y 1..
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For a large population, the two groups are of almost the same size, the difference being only one or two. Each group adopts an intra-group sharing rule which places great emphasis on relative effort Žsee Lemma 1.. Thus, the rent is almost fully dissipated. Inter-group mobility is now incorporated in the collective rent-seeking contest with no restriction on a 1 and a 2 . Let us first examine the incentives of members of group 1 to move to group 2. Utilizing Proposition 1, we find that Õ 1Ž n1 , n 2 . Õ 2 Ž n1 y 1, n 2 q 1., if n1 ) Ž n 2 q 1. and n 2 G 2. When n 2 s 1, also from Proposition 1, it follows that Õ1Ž n1 , 1. - Õ 2 Ž n1 y 1, 2. if N G 6. In these cases, a member of group 1 will move to group 2. However, Õ 1Ž n1 , 1. ) Õ 2 Ž n1 y 1, 2. if N - 6. Then, members of group 1 have no incentive to move to group 2. The incentives of members of group 2 to move to group 1 are now analyzed. If n 2 G 3, Õ 1Ž n1 q 1,n 2 y 1. - Õ 2 Ž n1 , n 2 .. Hence, a member of group 2 has no incentive to move to group 1. When n 2 s 2, each member of group 2 obtains Ž N y 2. Sr2 N 2 amount of payoff. On moving to group 1, the migrant’s payoff becomes Sr4Ž N y 1.. If N G 6, Ž N y 2. Sr2 N 2 ) Sr4Ž N y 1.. Thus, no one will move from group 2 to group 1. For N s 4, 5, however, a member in group 2 will move to group 1. 7 When n 2 s 1, a grand coalition may be formed, if the only member of group 2 moves to group 1. In the grand coalition each member receives SrN. If N G 5, the only member of group 2 obtains more by staying in group 2 than by joining a grand coalition. If N s 4, the only member of group 2 receives the same amount of payoff Sr4 both when he joins the grand coalition and when he remains in group 2. If N s 3, the member of group 2 will agree to form a grand coalition since Sr3 ) Sr4. Combining these results, we obtain
Proposition 3. With no restriction on a 1 and a 2 , the equilibrium group sizes Ž n1) , n )2 . are: Ž1. If N s 3, a grand coalition is formed. Ž2. If N s 4 , Ž n1) , n )2 . s Ž3, 1. , or a grand coalition is formed. Ž3. If N s 5, Ž n1) , n )2 . s Ž4, 1.. Ž4. If N G 6 and N is an eÕen number, Ž n1) , n )2 . s Ž Nr2, Nr2.. Ž5. If N G 6 and N is an odd number, then Ž n1) , n )2 . s ŽŽ N q 1.r2, Ž N y 1.r2..
For a large and even N, in equilibrium, the two groups are of the same size. Both groups adopt the sharing rule based solely on relative effort. Individuals also obtain the same payoff regardless of membership. For a large and odd N, the two groups are of almost the same size, the difference being a single member. The sharing rules of the two groups are very close to that based on relative effort. Notice that, in both cases, the rent is substantially dissipated.
7
Since n1 G n 2 and n 2 s 2, we have N G 4.
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4. Concluding remarks We have examined a contest in which two groups compete to win larger shares of a rent. Group size has been endogenized by allowing inter-group mobility of group members. We have also relaxed the restriction on the range of parameter values of the sharing rules adopted in Nitzan Ž1991a,b. and Lee Ž1995.. We have found that the two groups tend to be of equal size, and that the optimal sharing rules place great emphasis on relative outlay. The rent is substantially dissipated in a collective contest with endogenous group sizes and sharing rules. Our analysis in this paper is restricted to a bipartisan contest in which each individual belongs to one of two groups. An interesting extension would examine a rent-seeking contest in which there are more than two groups with or without individuals outside the groups. In such an extended model, one could endogenize the number of groups and their size distribution.
Acknowledgements We are grateful to three anonymous referees and seminar participants at Kookmin University for their helpful comments and suggestions. The usual disclaimer applies.
References Baik, K.H., 1994, Winner-help-loser group formation in rent-seeking contests, Economics and Politics 6, 147–162. Baik, K.H. and J.F. Shogren, 1995, Competitive-share group formation in rent-seeking contests, Public Choice 83, 113–126. Brooks, M.A. and B.J. Heijdra, 1989, An exploration of rent seeking, Economic Record 65, 32–50. Krueger, A.O., 1974, The political economy of the rent-seeking society, American Economic Review 64, 291–303. Lee, S., 1995, Endogenous sharing rules in collective-group rent-seeking, Public Choice 85, 31–44. Long, N.V. and N. Vousden, 1987, Risk-averse rent seeking with shared rents, Economic Journal 97, 971–985. Nitzan, S., 1991a, Rent-seeking with non-identical sharing rules, Public Choice 71, 43–50. Nitzan, S., 1991b, Collective rent dissipation, Economic Journal 101, 1522–1534. Nitzan, S., 1994, Modelling rent-seeking contests, European Journal of Political Economy 10, 41–60. Posner, R.A., 1975, The social costs of monopoly and regulation, Journal of Political Economy 83, 807–827. Rubinfeld, D.L., 1987, The economics of the local public sector, In: A.J. Auerbach and M. Feldstein, eds., Handbook of public economics, Vol. II ŽNorth-Holland, Amsterdam. 571–645. Tollison, R.D., 1982, Rent seeking: A survey, Kyklos 35, 575–602. Tullock, G., 1967, The welfare costs of tariffs, monopolies, and theft, Western Economic Journal 5, 224–232.
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Tullock, G., 1980, Efficient rent seeking, In: J.M. Buchanan, R.D. Tollison and G. Tullock, eds., Toward a theory of rent-seeking society ŽTexas A&M University Press, College Station, TX. 97–112. Ursprung, H.W., 1990, Public goods, rent dissipation, and candidate competition, Economics and Politics 2, 115–132.
Collective rent seeking with endogenous group sizes Kyung Hwan Baik a , Sanghack Lee a b
b,)
Department of Economics, Sung Kyun Kwan UniÕersity, Seoul 110-745, South Korea Department of International Trade, Kookmin UniÕersity, Seoul 136-702, South Korea
Abstract We examine a contest in which two groups compete to win larger shares of a private-good rent, and where group size is endogenized by allowing inter-group mobility of group members. We also relax the restriction on the range of parameter values of the sharing rules adopted by Nitzan Ž‘Rent-seeking with non-identical sharing rules’, Public Choice 71, 43–50, 1991, and ‘Collective rent dissipation’, Economic Journal 101, 1522– 1534, 1991. and Lee Ž‘Endogenous sharing rules in collective-group rent-seeking’, Public Choice 85, 31–44, 1995.. We find that the two groups tend to be of equal size, and that the optimal sharing rules place great emphasis on relative outlays. The rent is ‘substantially’ dissipated in a collective contest with endogenous group sizes and sharing rules. JEL classification: D72 Keywords: Collective rent seeking; Endogenous sharing rule; Inter-group mobility
1. Introduction The theory of rent seeking, initiated by seminal work of Tullock Ž1967, 1980., Krueger Ž1974., and Posner Ž1975., has been extended in many directions. 1 While encompassing a variety of scenarios on rent seeking, most papers in the literature )
Corresponding author. Tel.: Žq82. 2-910.4546; fax: Žq82. 2-910.4519. The literature on rent seeking has been surveyed by Tollison Ž1982., Brooks and Heijdra Ž1989., and Nitzan Ž1994.. 1
0176-2680r97r$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved. PII S 0 1 7 6 - 2 6 8 0 Ž 9 6 . 0 0 0 4 1 - 9
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K.H. Baik, S. Lee r European Journal of Political Economy 13 (1997) 121–130
have focused on rent-seeking competition between individuals. However, rents are often sought by groups of individuals. Reflecting this, Nitzan Ž1991a,b., Baik Ž1994., Baik and Shogren Ž1995., and Lee Ž1995. have recently analyzed collective rent seeking. The purpose of this paper is to extend the theory of collective rent seeking in two respects. First, we endogenize group size by allowing inter-group mobility of group members. Second, we relax the restriction on the range of parameter values of the sharing rules adopted in Nitzan Ž1991a,b. and Lee Ž1995.. The paper is organized as follows. Section 2 sets out the basic model without inter-group mobility. As in Lee Ž1995., collective rent seeking is modelled as a two-stage game in which intra-group sharing rules and individual outlays are determined in the first and the second stage, respectively. Characteristics of the subgame-perfect equilibrium are derived for the cases with or without the restriction on the parameter value of the intra-group sharing rule. Section 3 endogenizes group sizes by allowing inter-group mobility of group members. We find that the two groups tend to be of equal size. We offer our conclusions in Section 4.
2. Collective rent seeking without inter-group mobility Combining Nitzan Ž1991a,b. with Long and Vousden Ž1987., we consider a contest in which two groups compete to win larger shares of a divisible private-good rent. The rent is worth S. Each group consists of a fixed number of identical risk-neutral individuals, denoted by n i , for i s 1, 2. Without loss of generality, we assume that n1 G n 2 . The size of total population N, Ž N s n1 q n 2 ., is assumed to exceed 2. Individual k of group i contributes X k i to rent-seeking activity of his own group. Group i expends X i , Ž X i s Ý k X k i ., on rent-seeking. Total rent-seeking outlay of the two groups is denoted by X Ž X s X 1 q X 2 .. A negative value of X k i or X i is allowed, provided that X ) 0. That is, an individual or a group in the aggregate can take away resources contributed by other individuals or group, rather than expending resources on rent-seeking, as long as total rent-seeking expenditure X is positive. Group i’s share of the rent, P i , is given by the ratio of its rent-seeking outlay X i to the total rent-seeking outlay X 2 :
P i s X irX .
Ž 1.
If X i - 0, then P i - 0, since X ) 0. In such a case, group i in the aggregate takes away resources contributed by the other group, and pays a penalty in the amount 2 The functional form of P i is identical to the logit-form probability-of-winning function employed in Tullock Ž1980., Ursprung Ž1990., Nitzan Ž1991a,b., Baik Ž1994., and Baik and Shogren Ž1995., the difference being the possibility that P i can now have a negative value or exceed 1.
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of yP i S, when the rent is actually distributed. On the other hand, the other group, which has expended resources, receives both the rent and the penalty paid by group i. Of course, the size of the rent is not affected, since P 1 q P 2 s 1. Each group distributes its share of the rent among members according to its prespecified intra-group sharing rule. Let f k i denote the proportion of the share accruing to individual k of group i. Each group’s share of the rent, P i S, is fully distributed so that Ý k f k i s 1, for i s 1, 2. The payoff of rent-seeking activity for individual k of group i is given by Vk i s P i Sf k i y X k i .
Ž 2.
The proportion f k i is assumed to be determined by his own outlay and by outlays of the other members of group i. Following Nitzan Ž1991a,b., this paper focuses on a family of distribution rules given as f k i s Ž 1 y a i . X k irX i q a irn i .
Ž 3.
From Eqs. Ž2. and Ž3., we find that the amount Ž1 y a i . P i S is distributed in proportion to relative outlay, and that the amount a i P i S is equally shared with by members of group i. Nitzan Ž1991a,b. and Lee Ž1995. have adopted the restriction that a i should belong to the closed unit interval w0,1x. However, there seems to be no a priori reason for such restriction. The parameter a i may have a negative value or exceed 1. 3 A negative value of a i implies that group i’s sharing rule places great emphasis on member’s relative outlay, X k irX i . In this case, group i collects Žya i . P i Srn i from each of its members, and then distributes Ž1 y a i . P i S among members in proportion to relative outlay. On the other hand, when a i ) 1, voluntary contribution is discouraged. In this case group i collects Ž a i y 1. P i S from its members according to relative outlay, and then distributes a i P i S equally among its members. Note that the distribution rule given in Eq. Ž3. is compatible with any value of a i , since Ý k f k i s 1 for any value of a i . We model the collective rent-seeking contest as a two-stage game, drawing on Lee Ž1995.. We assume that the parameters a 1 and a 2 are determined prior to individual decisions on the extent of voluntary contribution. In the first stage the representative of each group chooses the value of the sharing-rule parameter which maximizes the group’s aggregate payoffs, respectively. 4 In so doing, the actions of members in the second stage are suitably taken into account. In the second stage all the members of the groups choose their outlays simultaneously and independently. All of the above is common knowledge to the individuals. We employ a subgame-perfect equilibrium as the solution concept. 3
Baik and Shogren Ž1995. have also noted this possibility. Maximizing aggregate payoffs of the group is equivalent to maximizing individual payoff in a symmetric equilibrium in which each one obtains the same payoff. Thus, each member will agree to the sharing rule proposed by the representative. 4
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To obtain the solution, we firstly analyze the second-stage actions of individuals. We then examine the first-stage interaction between the groups. Substituting Eqs. Ž1. and Ž3. into Eq. Ž2., we obtain Vk i s P i Sf k i y X k i s S Ž 1 y a i . X k irX q S a i X irn i X y X k i .
Ž 4.
Individual k of group i solves the problem given as max Vk i s S Ž 1 y a i . X k irX q S a i X irn i X y X k i .
Ž 5.
Xk i
For individual k of group i, the first-order condition for the payoff maximization is 5
E Vk irE X k i s S Ž 1 y a i . Ž X y X k i . rX 2 q S a i Ž X y X i . rn i X 2 y 1 s 0. Ž 6 . The N equations in Ž6. determine the Nash equilibrium values of X k i for given values of a 1 and a 2 , under a mild restriction that X be positive. Summation of Eq. Ž6. over the relevant range of k and simplification gives, for i s 1, 2, X irX s a i q Ž 1 y a i . n i y Ž n i XrS . .
Ž 7.
Simultaneous solution of Eq. Ž7., for i s 1,2, gives X 1 Ž a 1 , a 2 . s Ž SrN 2 . Q n1 n 2 Ž a 2 y a 1 . q n 2 a 1 y n1 a 2 q n1 ,
Ž 8.
X 2 Ž a 1 , a 2 . s Ž SrN 2 . Q n1 n 2 Ž a 1 y a 2 . q n1 a 2 y n 2 a 1 q n 2 ,
Ž 9.
and X Ž a 1 , a 2 . s X 1 Ž a 1 , a 2 . q X 2 Ž a 1 , a 2 . s Ž SrN . Q,
Ž 10 .
where Q s w N y n1 a 1 y n 2 a 2 q a 1 q a 2 y 1x. To assure a positive value of X, Q should also be positive. We suitably restrict our attention to the set of a 1 and a 2 which guarantees positive value of Q. This set encompasses a wide range of a 1 and a 2 . For example, when both a 1 and a 2 belong to the closed unit interval w0, 1x, as in Nitzan Ž1991a,b. and Lee Ž1995., Q is positive, and thus an equilibrium always exists. If a i - 0, for i s 1 and 2, positive value of Q is assured as well. We now analyze the first-stage interaction between the groups. The representative of each group solves, respectively: maxV1 Ž a 1 , a 2 . s ÝVk1 Ž a 1 , a 2 . s S P 1 Ž a 1 , a 2 . y X 1 Ž a 1 , a 2 . a1
Ž 11 .
k
and maxV2 Ž a 1 , a 2 . s ÝVj2 Ž a 2 , a 2 . s S P 2 Ž a 1 , a 2 . y X 2 Ž a 1 , a 2 . . a2
5
j
The second-order condition is satisfied.
Ž 12 .
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The first-order conditions for Eqs. Ž11. and Ž12. are calculated as
E V1rEa 1 s Ž n1 y 1 . S Ž n1 y n 2 . Ž Ž n 2 y 1 . a 2 q 1 . y 2 n 2 Ž n1 y 1 . a 1 rN 2 s 0,
Ž 13 .
and
E V2rEa 2 s Ž n 2 y 1 . S Ž n 2 y n1 . Ž Ž n1 y 1 . a 1 q 1 . y 2 n1 Ž n 2 y 1 . a 2 rN 2 s 0,
Ž 14 .
respectively. The second-order conditions are satisfied. In the case without the restriction simultaneous solution of Eqs. Ž13. and Ž14. gives the set of the equilibrium sharing-rule parameters Ž a 1) , a 2) .. In the case with the restriction, the set of the equilibrium sharing-rule parameters can be obtained through application of the Kuhn–Tucker conditions. We firstly consider the case with the restriction. In this case, from Eq. Ž10., X is always positive. Thus, an equilibrium exists. The existence issue raised by Nitzan Ž1991a. does not apply to this case. Following the same procedure as in Lee Ž1995. we can obtain the set of the subgame-perfect equilibrium sharing-rule parameters. In the case with no restriction on values of a i , simultaneous solution of Eqs. Ž13. and Ž14. gives the set of the subgame-perfect equilibrium sharing-rule parameters Ž a 1) , a 2) .. In this case, of course, we focus on the set of Ž a 1 , a 2 . which assures a positive value of Q. It is easy to see that the solution is unique and stable if n 2 ) 1. Lemma 1 summarizes the results on equilibrium intra-group sharing rules. Lemma 1. Ž1. If a i belongs to the closed interÕal w0, 1x , then the set of the subgame-perfect equilibrium intra-group sharing rules is giÕen as: Ža. Ž a 1) , a 2) . s ŽŽ n1 y n 2 .rw2Ž n1 n 2 y n 2 .x, 0. , if n 2 ) 1 , and Žb. Ž a 1) , a 2) . s Ž1r2, r . where r g w0, 1x , if n 2 s 1. Ž2. With no restriction on Õalues of a 1 and a 2 , the set of the subgame-perfect equilibrium intra-group sharing rules is: Ža. Ž a 1) , a 2) . s ŽŽ n1 y n 2 .rŽ n1 y 1. N, Ž n 2 y n1 .rŽ n 2 y 1. N . if n 2 ) 1 , and Žb. Ž a 1) , a 2) . s Ž1r2, r . where r denotes an arbitrary real number, if n 2 s 1. The case without the restriction calls for some explanations. Note that group 2 distributes more than the share in proportion to relative outlay to overcome the size disadvantage, if n 2 ) 1. On the other hand, group 1 can spare some of the share to distribute on an egalitarian basis, while its sharing rule places more emphasis on relative outlay than in the case with the restriction as well. Substituting Ž a 1) , a 2) . into Eqs. Ž8. and Ž9., we can find the equilibrium rent-seeking expenditure of each group. The extent of rent-dissipation and the payoff to each member can be easily calculated as well. Let t denote the rent-dissipation rate, i.e., t s X )rS. Also, define Õi as the payoff to a member of
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group i, for i s 1, 2. The characteristics of the subgame-perfect equilibrium with or without the restriction on a i are given by Proposition 1. Ž1. When a i belongs to the closed interÕal w0, 1x , the subgameperfect equilibrium is characterized as: Ža. P 1) s P 2) s 1r2, Žb. t s Ž2 n 2 y 1.r2 n 2 , and Žc. Õ 1 s Sr4n1 n 2 and Õ 2 s Sr4w n 2 x 2 . Ž2. With no restriction on Õalues of a 1 and a 2 , the subgame-perfect equilibrium has the following characteristics: Ža. If n 2 ) 1, P 1) rP 2) s n 2rn1 , t s Ž N y 1.rN, Õ1 s n 2 Srn1 N 2 , and Õ 2 s n1 Srn 2 N 2 . Žb. If n 2 s 1, P 1) s P 2) s 1r2, t s 1r2, Õ 1 s Sr4n1 and Õ 2 s Sr4. In the case with the restriction, the two groups always share the rent equally. Note that the rent-dissipation rate t is determined by the size of group 2. The larger group 2 is, the higher is the rent-dissipation rate. Each member obtains a positive payoff. Note also that the payoff to each member of group 2 is not affected by the size of group 1. In the case without the restriction, group 2 takes the larger share of the rent, if n 2 ) 1. This is because the larger group, group 1, suffers from the negative size effect due to free-riding. However, the rent-dissipation rate t is independent of the distribution of population across groups. Notice that it is identical to the result obtained by Tullock Ž1980. in the N-person rent-seeking contest. The social cost associated with collective rent seeking can be inferred by observing the value of the rent and the size of total population. However, the payoff to each member is affected by his or her membership. When n 2 s 1, collective rent seeking exhibits characteristics of a two-person rent-seeking contest in both cases with or without the restriction. From Proposition 1, we find that the restriction on values of 1 and 2 lowers the extent of rent dissipation, unless n 2 s 1. The restriction has adverse effect on the rent-seeking efforts of the smaller group, which would otherwise have placed more emphasis on relative effort. When n 2 s 1, the restriction has no effect on rent-seeking efforts of the only member of group 2, and thus has no effect on rent-seeking efforts of group 1, either.
3. Endogenous group sizes Inter-group mobility of group members is now introduced into the model. We add one more stage to the game specified in the previous section. 6 In the first stage a chance to move to the other group is given to every member of each group. 6 A similar argument has been used in Baik and Shogren Ž1995.. Our model is also related to the literature on provision of local public goods and the theory of clubs. See, for example, Rubinfeld Ž1987..
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Each member takes all the other members’ membership decisions as given and makes his own membership decision. No cost is incurred with movement of members across groups. In the second stage, each group adopts its own sharing rule. All the members choose their outlays simultaneously and independently in the third stage. From Proposition 1, the payoff to each member of group i, Õi , can be written as a function of n1 and n 2 , respectively. If Õ1Ž n1 , n 2 . - Õ 2 Ž n1 y 1, n 2 q 1., a member of group 1 has an incentive to move to group 2. On the other hand, a member in group 2 will move to group 1, if Õ 1Ž n1 q 1, n 2 y 1. ) Õ 2 Ž n1 , n 2 .. When equality holds, an individual is indifferent between staying in his group and moving to the other group. No one can increase his payoff by moving to the other group if the following inequalities are jointly satisfied: Õ1 Ž n1 , n 2 . G Õ 2 Ž n1 y 1, n 2 q 1 . ,
Ž 15 .
Õ1 Ž n1 q 1, n 2 y 1 . F Õ 2 Ž n1 , n 2 . .
Ž 16 .
and
The weak inequality Ž15. implies that a member of group 1 cannot increase his payoff by moving to group 2. Similarly, the weak inequality Ž16. means that a member in group 2 cannot augment his payoff by moving to group 1, either. Denote n i satisfying both Eq. Ž15. and Eq. Ž16. by n )i , for i s 1, 2. We now derive the equilibrium group sizes, n1) and n )2 , in the case when a i belongs to w0, 1x. From Proposition 1, we find that Õ 1Ž n1 q 1, n 2 y 1. - Õ 2 Ž n1 , n 2 ., if n1 ) n 2 and n 2 G 2. Thus, no one in group 2 will move to group 1, if n 2 G 2. If n1 s n 2 and n 2 G 2, a member of each group has an incentive to move to the other group. Straightforward calculation also shows that Õ 1Ž n1 , n 2 . - Õ 2 Ž n1 y 1, n 2 q 1. if n1 G Ž n 2 q 3.. That is, a member of group 1 has an incentive to move to group 2 if group 1 is larger than group 2 by three members or more. When n 2 s 1, Õ 2 s Sr4. In this case we need to consider the possibility of the formation of a grand coalition by movement of the only member of group 2 to group 1. In the grand coalition each individual receives SrN. It is easy to show that SrN - Sr4 if N ) 4. In such a case, the only member of group 2 has no incentive to move to group 1. Thus, a grand coalition is not feasible if N ) 4. If N s 4, the only member of group 2 is indifferent between remaining in group 2 and forming a grand coalition. When N s 3, the grand coalition is formed. Proposition 3 reports the results on equilibrium group sizes. Proposition 2. If a i belongs to w0, 1x , then the equilibrium group sizes Ž n1) , n )2 . are: Ž1. If N s 3, a grand coalition is formed. Ž2. If N s 4, Ž n1) , n )2 . s Ž3, 1. , or a grand coalition is formed. Ž3. If N G 5 and N is an odd number, Ž n1) , n )2 . s ŽŽ N q 1.r2, Ž N y 1.r2.. Ž4. If N G 5 and N is an eÕen number, Ž n1) , n )2 . s ŽŽ Nr2. q 1, Ž Nr2. y 1..
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For a large population, the two groups are of almost the same size, the difference being only one or two. Each group adopts an intra-group sharing rule which places great emphasis on relative effort Žsee Lemma 1.. Thus, the rent is almost fully dissipated. Inter-group mobility is now incorporated in the collective rent-seeking contest with no restriction on a 1 and a 2 . Let us first examine the incentives of members of group 1 to move to group 2. Utilizing Proposition 1, we find that Õ 1Ž n1 , n 2 . Õ 2 Ž n1 y 1, n 2 q 1., if n1 ) Ž n 2 q 1. and n 2 G 2. When n 2 s 1, also from Proposition 1, it follows that Õ1Ž n1 , 1. - Õ 2 Ž n1 y 1, 2. if N G 6. In these cases, a member of group 1 will move to group 2. However, Õ 1Ž n1 , 1. ) Õ 2 Ž n1 y 1, 2. if N - 6. Then, members of group 1 have no incentive to move to group 2. The incentives of members of group 2 to move to group 1 are now analyzed. If n 2 G 3, Õ 1Ž n1 q 1,n 2 y 1. - Õ 2 Ž n1 , n 2 .. Hence, a member of group 2 has no incentive to move to group 1. When n 2 s 2, each member of group 2 obtains Ž N y 2. Sr2 N 2 amount of payoff. On moving to group 1, the migrant’s payoff becomes Sr4Ž N y 1.. If N G 6, Ž N y 2. Sr2 N 2 ) Sr4Ž N y 1.. Thus, no one will move from group 2 to group 1. For N s 4, 5, however, a member in group 2 will move to group 1. 7 When n 2 s 1, a grand coalition may be formed, if the only member of group 2 moves to group 1. In the grand coalition each member receives SrN. If N G 5, the only member of group 2 obtains more by staying in group 2 than by joining a grand coalition. If N s 4, the only member of group 2 receives the same amount of payoff Sr4 both when he joins the grand coalition and when he remains in group 2. If N s 3, the member of group 2 will agree to form a grand coalition since Sr3 ) Sr4. Combining these results, we obtain
Proposition 3. With no restriction on a 1 and a 2 , the equilibrium group sizes Ž n1) , n )2 . are: Ž1. If N s 3, a grand coalition is formed. Ž2. If N s 4 , Ž n1) , n )2 . s Ž3, 1. , or a grand coalition is formed. Ž3. If N s 5, Ž n1) , n )2 . s Ž4, 1.. Ž4. If N G 6 and N is an eÕen number, Ž n1) , n )2 . s Ž Nr2, Nr2.. Ž5. If N G 6 and N is an odd number, then Ž n1) , n )2 . s ŽŽ N q 1.r2, Ž N y 1.r2..
For a large and even N, in equilibrium, the two groups are of the same size. Both groups adopt the sharing rule based solely on relative effort. Individuals also obtain the same payoff regardless of membership. For a large and odd N, the two groups are of almost the same size, the difference being a single member. The sharing rules of the two groups are very close to that based on relative effort. Notice that, in both cases, the rent is substantially dissipated.
7
Since n1 G n 2 and n 2 s 2, we have N G 4.
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4. Concluding remarks We have examined a contest in which two groups compete to win larger shares of a rent. Group size has been endogenized by allowing inter-group mobility of group members. We have also relaxed the restriction on the range of parameter values of the sharing rules adopted in Nitzan Ž1991a,b. and Lee Ž1995.. We have found that the two groups tend to be of equal size, and that the optimal sharing rules place great emphasis on relative outlay. The rent is substantially dissipated in a collective contest with endogenous group sizes and sharing rules. Our analysis in this paper is restricted to a bipartisan contest in which each individual belongs to one of two groups. An interesting extension would examine a rent-seeking contest in which there are more than two groups with or without individuals outside the groups. In such an extended model, one could endogenize the number of groups and their size distribution.
Acknowledgements We are grateful to three anonymous referees and seminar participants at Kookmin University for their helpful comments and suggestions. The usual disclaimer applies.
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