Optimal rent-seeking group.

Lee, Sanghack

Introduction

Following the seminal works of Posner (6) and Tullock (8,9) many authors have extended the theory of rent seeking in several directions. |For a survey of the literature, see Tollison (7) and Brooks and Heijdra (2).~ The received literature focuses mainly on the extent of social wastes due to rent-seeking competition under a variety of assumptions on the number of rent-seekers, the possibility of rent-avoidance, uncertainty, etc.

In most of the literature rent seekers are assumed to behave non-cooperatively against each other. That is, the possibility of partial or full cooperation between rent seekers is excluded from consideration. However, it is often observed that rent seekers form a group to share the prize (rent) and costs of rent seeking. For instance, as Appelbaum and Katz (1, p. 175) notes, "|c~ar manufacturers may fight jointly against tariffs on steel imports. . .". Tollison (7) also acknowledges the possibility that consumers form a group to fight against the monopoly rent. The possibility of group formation by rent seekers indeed raises several interesting issues, aside from its effect on social welfare, such as those on the internal organization of groups and the free-rider problem associated with it. However, to the author's knowledge, no attempt has been made to characterize the rent-seeking competition under the possibility of cooperation between rent seekers. The purpose of this note is to examine the implication of that possibility on rent-seeking competition and on social welfare.

The next section sets out the basic model and characterizes equilibrium rent-seeking under the possibility of group formation. It employs the argument the structure of which is similar to that of the club theory pioneered by Buchanan (3). Rational individuals are shown either to form a group to fight for the rent or not, depending upon costs and benefits of group formation. The conventional case in which rent seekers compete independently for the rent can also arise as a special case. In the normative respect it is found that group formation by rent seekers reduces social wastes resulting from rent-seeking competition.

The Analysis

There are N identical individuals in the society, each competing for the rent in the amount of L, where N |is greater than or equal to~ 2. They are allowed to form groups prior to competition for the rent. Rent-seeking competition among groups is considered as a two-stage procedure. In the first stage groups are formed. In the second stage those groups are engaged in competition for the rent. The rent obtained by a particular group and costs of group formation are equally shared with by members of the group. Each member decides his or her contribution to the rent-seeking activities of the group, based upon a Nash conjecture about the contributions of other groups and the other members of his or her own group. The paper derives the optimality condition for rent-seeking groups under which each member's expected value of rent-seeking is maximized. The solution is obtained by a backward induction.

Let j denote the number of groups competing for the rent. Each group has the same number of members, m. Then, disregarding the integer problem, m = N/j. The i-th member of the k-th group contributes |x.sub.ik~ amount to the rent-seeking activities of the k-th group, for i = 1,..., m and k = 1,... J.

Define |Mathematical Expression Omitted~ for k = 1,..., j.

The variable Xk represents the k-th group's total expenditure on rent-seeking activities.

As in the literature that follows Tullock (9), the probability of the k-th group's winning the rent L, Pk, is specified as:(1)

|Mathematical Expression Omitted~

Each member of the winning group receives L/m amount of the rent. Following the literature individuals are assumed to be risk-neutral. The i-th member of the k-th group thus solves the problem:

Max (|P.sub.k~)(L/m) - xik (2)

The first order condition is:

(|Delta~|P.sub.k~/|Delta~X|i.sub.k~) (L/m) - 1 = (L/m) |R|(|X.sub.k~).sup.R - 1~(|Sigma~ |(|X.sub.i~).sup.R~) - R|(|X.sub.k~).sup.2R - 1~~/(|Sigma~|(|X.sub.i~).sup.R~).sup.2~ - 1 = 0. (3)

Utilizing the symmetry condition that X1 = . . . = |X.sub.j~, the first order condition is simplified to obtain:

R(j - 1)L = m|(j).sup.2~|X.sub.k~. (4)

The second order condition, which is assumed to be satisfied, can also be simplified as

R|j(R - 1) - 2R~(j - 1) |is less than~ 0.

When j = 1, the weak form of the second order condition is satisfied. From the first order condition one finds that

|X.sub.k~ = X = R(j - 1)L/m|j.sup.2~,

and

x|i.sub.k~ = x = |X.sub.k~/m = R(j - 1)L/|N.sup.2~.

Without group formation, each rent-seeker may allocate R(N - 1)L/|N.sup.2~ amount to rent seeking. Note that x |is less than~ R(N - 1)L/|N.sup.2~ if j |is less than~ n. This means that each rent seeker allocates less amount of resources to rent-seeking activities with group formation than without group formation.

Expected gain of rent-seeking for each member, E, is given as

E = (|P.sub.k~)(L/m) - x = L/N - R(j - 1)L/|N.sup.2~. (5)

Note that E is a decreasing function of j. A decrease in j means an increase in the size of each group. Hence, the expected gain of rent-seeking is maximized when all rent-seekers form a grand coalition. However, some costs may be associated with group formation. The optimal group formation problem should take those costs into account.

Optimal group formation

Group formation in the first stage is now considered. Various kinds of costs such as information-transmission costs and staff costs are associated with group formation. This note does not examine details of those costs. Instead, those costs are assumed to be represented by a cost function c(j) that denotes per-member cost of group formation when each group has N/j number of members. In equilibrium, of course, j denotes the number of groups in the society. Now the optimality condition for group formation is obtained by solving the problem as follows:(2)

Max |L/N - R(j - 1)L/|N.sup.2~~ - c(j) (6)

where 1 |is less than or equal to~ j |is less than or equal to~ N.

There is no a priori restriction on c(j). Let j* denote the solution to the above problem. If c|prime~ (j) |is greater than or equal to~ 0, then it easily follows that j* = 1. This case is, however, of little interest since there no longer exists rent-seeking competition. In the following discussion it is assumed that c|prime~ |is less than or equal to~ 0.

This means that per-member cost of group formation decreases as the size of each group decreases. The first order condition to (6) is given as

- RL/|N.sup.2~ - c|prime~ (j) = 0. (7)

If c|double prime~ |is greater than~ 0, then the second order condition is satisfied and an interior solution can be obtained.(3) With an interior solution a comparative statics analysis can be done by totally differentiating (7), to derive |Delta~j*/|Delta~R |is less than~ 0, |Delta~j*/|Delta~L |is less than~ 0 and |Delta~j*/|Delta~N |is greater than~ 0.

When c|double prime~ |is less than~ 0, the second order condition is violated. In such a case, a boundary solution is obtained, i.e., j* = 1 or j* = N.

The effect of group formation on social welfare is now examined in the second-best situation. Suppose that for some reason rent-seeking competition is unavoidable. Then, should group formation by rent seekers be allowed or not? To answer this question, rent-seeking expenditures with group formation are compared with those without group formation. Note that each rent seeker spends R(j* - 1)L/|N.sup.2~ amount on rent seeking plus c(j*) amount on group formation. From definition of j* it follows that

R(j* - 1)(L/|N.sup.2~ + c(j*) |is less than~ R(N - 1)L/|N.sup.2~ + c(N) if j* |is less than~ N.

Hence allowing for group formation reduces social wastes resulting from rent-seeking competition.

Notes

1. A different form of a rent-seeking success function is proposed by Hirshleifer (4)

2. Note the structural similarity between this argument and that of the club theory. However, this model is different from the club theory in several respects. First, the rent has no public-good characteristics. In addition, group members decide their contribution to rent-seeking activity of the group based upon a non-cooperative Nash conjecture.

3. Notice that c|double prime~ |is greater than~ 0 is the necessary but not sufficient condition for an interior solution.

References

Appelbaum, E. and Katz, E., "Transfer seeking and avoidance: On the full social costs of rent seeking," Public Choice 48. (1986): pp. 175-181.

Brooks, M. A. and Heijdra, B. J., "An exploration of rent seeking," Economic Record 65. (1989) pp. 32-50.

Buchanan, J. M., "An Economic theory of clubs," Economica 32. (1965) pp. 1-14.

Hirshleifer, J., "Conflict and rent-seeking success functions: Ratio vs. difference models of relative success", Public Choice 63. (1989) pp. 101-112.

Lee, S., "Inter-Group competition for a pure private rents," mimeo. (1991). Kookmin University.

Posner, R. A., "The social cost of monopoly and regulation," Journal of Political Economy 83. (1975) pp. 807-827.

Tollison, R. D., "Rent-seeking: A survey," Kyklos 35. (1982) pp. 525-602.

Tullock, G., "The welfare costs of tariffs, monopolies and theft," Western Economic Journal 5. (1967) pp. 224-232.

Tullock, G., "Efficient rent seeking," In J. M. Buchanan, R. D. Tollison and G. Tullock (Eds.), Towards a theory of rent-seeking society College Station, Tx.: Texas A & M University Press. (1980).