Rice, Salmon Or Sushi? Political competition for supply of a regulated input.

Kahane, Leo H.

The early work by Olson (1965) and Tullock (1967) on the political competition for economic rents has fostered a growing body of literature devoted to the study of pressure group influence on public policy. For example rent-seeking behavior, with a focus on the degree of rent dissipation, has been examined by a variety of researchers including Bhagwati (1987); Hillman and Samet (1987); Varian (1989). Others, including Becker (1983); Findlay and Wellisz (1982, 1983); Nitzan (1991), have emphasized the nature of competition among pressure groups for political influence. This paper continues along this latter line of research by developing a simple, game-theoretic model which describes how rival groups politically compete for the supply of a government regulated input. Of particular interest in this study is the effects of the degree of internal lobby group organization on a group's ability to effectively compete for the regulated input. It should be acknowledged that given the hypothetical nature of the model d escribed below, (and the lack of relevant data to test the model), its primary function is pedagogical. That is, the goal of this paper is to provide an interesting example of how models of endogenous policy formation can applied to a variety of interesting policy issues.

As a means of motivating the model described below, an example of the on-going struggle for government controlled water in California will be used. The six years of drought in California from the mid-1980s to 1992 left water reserves at their lowest level since 1977. This scarcity of water intensified the struggle between competing users of government-controlled water supplies. (1) For example rice and salmon production depend heavily on the availability of water; for any given level of water, the larger the proportion of water rice growers receive, the less salmon farmers receive. This resulting conflict between rice growers and salmon farmers gives rise to an interesting model for policy formation. Notwithstanding the fact that these two products are quite diverse and do not compete in the goods market, rice growers and salmon farmers do compete in a political market for water rights. If the government (which has control over water rights) is responsive to political lobby pressure, then water rights may be endogenous to the lobby activities of groups that stand to gain or lose from a particular water distribution policy. In such a framework, an interesting question arises as to the effects on the lobbying process of differing degrees of lobby group organization or coordination. That is, if say rice growers come together and determine their optimal level of lobby expenditure (given some level of lobby expenditure by salmon farmers), how does this level of expenditure compare to the case when individual rice growers do not come together but act alone to lobby the government? The model presented below captures the effects of different degrees of lobby group organization by developing a simple n-firm Cournot model with endogenous policy formation.

The remainder of this paper is organized into three sections. The first section develops the above mentioned model of endogenous policy formation. An equilibrium involving optimal quantities of output and lobby contributions for individual group members is described. The next section examines the effects on optimal lobby contributions of different conjectures about internal lobby group organization. The last section contains concluding remarks.

1. The Model

To examine the importance of internal lobby group organization, suppose there is a continuous policy measure, m, that is being considered by local political leaders. We can assume that there are two groups of producers, rice growers and salmon farmers, in the locality which do not compete in the goods market but are both affected by the measure. In particular, if we let m be the amount of government controlled water allotted to rice production, then rice growers favor a larger m because it reduces the marginal cost of rice production. On the other hand, salmon farmers prefer a smaller m since less water for rice farmers means more water for salmon farming which will decrease their marginal cost of production. Assume that both the rice growing sector and salmon farming sector are n-firm Cournot oligopolies producing a homogeneous good (internally) and that within each group members are identical in size and have the same production technology. It is assumed internally that each group acts collectively on the political front but competitively in the goods market. The key element for this analysis then will be how groups fare under different conjectures regarding internal lobby group organization.

It is assumed that both groups are able to contribute lobbying funds in an effort to influence the outcome of the decision by local politicians on the level of m chosen. (2) Suppose that each group believes that the level of m chosen by the government is determined by the following policy formation function: (3)

M = m(L, L')

where: L = [summation over (n/i=1)] [L.sub.i] L' = [summation over (n'/,j=1)] [L.sub.j]

and [partial]m/[partial]L > 0, [[partial].sup.2]m/[partial][L.sup.2] <, [partial]m/[partial]L' < 0, [[partial].sup.2]m/[partial][L'.sup.2] < 0 (1.1)

Where L and L' are the total contributions by rice growers and salmon farmers, respectively, to the lobbying effort.

Let the rice grower's market demand curve have the following simple linear form:

P(Q) = a - bQ

with: Q = [summation over (n/i=1)][q.sub.i] (1.2)

where Q represents the industry output and [q.sub.i] the ith individual producer's output. For simplicity we will assume that there are no fixed costs of production. Given the above industry demand function, we can now write the profit function of an individual rice grower as:

[[pi].sub.i] = P(Q)[q.sub.i] - [kq.sub.i] - [L.sub.i] (1.3)

where k is the marginal cost to production and [L.sub.i] is the ith individual producer's contribution to the lobbying effort. As was described above, the policy measure m is assumed to reduce the marginal cost of production for rice farmers. To this end let marginal cost for rice farmers be determined by the following simple function:

k = (c - m) (1.4)

where c is constant with respect to output.

We can now use equation (1.1) and (1.4) to rewrite the profit function in (1.3) as:

[[pi].sub.i] = (a - bQ)[q.sub.i] - (c - m)[q.sub.i] - [L.sub.i] (1.5)

In order to maximize equation (1.5) with respect to [q.sub.i] and [L.sub.i] it is necessary to make an assumption on how individuals view the behavior of the salmon farmers' group when members of the rice growers' group choose output quantities and lobby expenditures. (4) To this end, a Nash assumption is adopted meaning that individuals in the rice growers group take the level of contributions by salmon growers as given when choosing their optimal contribution level.

In maximizing (1.5) it will be assumed that individual rice growers first choose a level of contributions [L.sub.i]. Given this level of contributions, the rice growers then choose output levels, [q.sub.i]. Under this assumption, we can solve the model first for optimal output levels. Doing so yields the following:

[q.sup.*.sub.i] = (a - c + m)/b(n + 1) (1.6)

Equation (1.6) resembles the familiar n-firm Cournot solution for optimal quantities, but is augmented by the effect of m on the marginal cost of production. (5) Having solved for optimal output levels for firms, the result shown in equation (1.6) can now be substituted into the profit function in (1.5) giving the reduced profit function below:

[[pi].sub.i] = [(a - c + m).sup.2] / b[(n+1).sup.2] - [L.sub.i] (1.7)

The reduced form profit function shown in equation (1.7) would now be maximized with respect to contribution levels [L.sub.i]. Before carrying out this step however, more information is required. In particular, individuals in the rice growers' group must form a conjecture regarding how members within their group behave. That is, the policy on water distribution represented by m has a collective good character in that all rice farmers gain from a higher m whether or not they contribute to the lobbying effort and thus a free rider problem emerges. As such, how the rice growers organize themselves becomes an important issue. Before dealing with this issue we can, however, make several observations regarding the profit function shown in (1.7). The first part on the right hand side of (1.7) represents the gross profits for any individual given [q.sub.i] and contribution level [L.sub.i]. The second part, [L.sub.i], is simply the cost of that lobby contribution level which is assumed to be linear in [L.sub.i]. (6) T aken together the right hand side then gives profits net of lobby expenditures. Note that given the assumptions on the policy formation function given in (1.1), gross profits will be a monotonically increasing, concave function in [L.sub.i]. (7) The graph shown in Figure 1 gives an example of these functions for a given level of contributions by salmon farmers. The gap between the gross profit curve and the cost of contribution line represents net profits. The profit maximizing level of contributions will be the point at which the gap between the two functions is the widest or, in other words, where the marginal gross profit is equal to the marginal cost of contributing. Assuming symmetry across groups we can also determine salmon farmers' optimal output levels and lobby contributions, for a given level of contributions by rice growers.

The graph for rice growers was drawn for a given level of contributions by salmon farmers. If the level of contributions by salmon farmers changes, then this would give rise to a new gross profit function and equilibrium contribution level for rice growers. Thus the optimum contribution level for rice growers, [L.sub.i], is a function of total contributions of salmon farmers, which in turn is a function of individual salmon farmers' contributions, [L'.sub.i]. And conversely, optimal [L'.sub.i] is a function of [L.sub.i]. In other words, we have two reaction functions represented as follows:

[L.sub.i] = R([L'.sub.i])

and

[L'.sub.i] = R'([L.sub.i]) (1.8)

Finding equilibrium individual contribution levels for both groups involves solving both the above reaction functions simultaneously for [L.sub.i] and [L'.sub.i]. (8) In order to ensure an equilibrium exists, it is assumed that the slopes of the reaction functions are positive and less than one with respect to their arguments. (9) That is, in the case for rice growers, if an individual salmon farmer increases their contributions by 1 unit, an individual rice grower is assumed to increase their contributions but by less than 1 unit. (10) Examples of reaction functions satisfying the assumptions listed above are shown in Figure 2.

The intersection of two reaction functions gives equilibrium contribution levels for individual rice growers and salmon farmers. Notice also that, given the specification of the policy formation function in (1.1), there are "iso-policy" curves such as [m.sup.1] - [m.sup.1] in Figure 2 which represent the same policy outcome ([m.sup.1] for various combinations of [L.sub.i] and [L.sub.i]'.

2. Optimal Lobby Contributions and Lobby Group Coordination

The model above describes an equilibrium in lobby group contributions. As was mentioned in the introduction of this paper, one of the interesting issues relating to lobby group confrontation is the effects of group organization or coordination. This section analyzes the effects on optimal lobby group contributions and profits of individual rice growers and salmon farmers of different conjectures over internal lobby group organization.

The degree of internal lobby group coordination can be captured in the above model by looking more closely at equation (1.1), the policy formation function. Consider individual rice growers who choose their contribution [L.sub.i] so as to maximize profits represented by equation (1.7). When taking the partial derivative of the gross profit with respect to [L.sub.i] we obtain the following result:

[partial][[pi].sub.gross]/[partial][L.sub.i] = 2(a - c + m)/b[(n + 1).sup.2] [partial]m/[partial]L [partial]L/[partial][L.sub.i] (2.1)

The final factor on the right hand side of equation (2.1), the partial derivative of total lobby group contributions with respect to individual contributions, represents the individual's behavioral conjecture for internal lobby group organization. That is, it represents what the individual contributor believes will be the response from fellow group members to an increase in their contribution level. We can consider two extreme cases for this conjecture. If the individual holds a Nash conjecture, meaning that when they consider optimal contribution levels it is believed that other group members will not change their contribution levels, then the final term on equation (2.1) will be:

[partial]L/[partial][L.sub.i] = 1 (2.2)

We can compare this conjecture to one where lobby groups are perfectly coordinated, by perhaps a planner, and determine optimal contributions for the group as a whole which is then divided across all individual members. Under this scenario, the final term for equation (2.1) would be:

[partial]L/[partial][L.sub.i] = n (2.3)

where n represents the number of individuals in the group. The qualitative effects of these two conjectures in terms of optimal contributions for individuals can be determined by referring back to equation (2.1) and noting that the slope of the gross profit function is increasing in the conjecture for group organization. That is, the slope of gross profit function will be steeper under perfect lobby group coordination than it would be under the Nash conjecture. Thus, as Figure 3 shows, for any given level of contributions by salmon farmers, optimal level of individual contributions and profits for rice growers are greater under the case of perfect lobby group coordination (denoted by P) than it would be under the Nash case (denoted by N). This result supports Mancur Olson's (1965) well-known finding that as a group eliminates free riding it provides itself with more of a collective good (i.e. m via greater lobby contributions) than it would otherwise. Furthermore it is easily seen that, all else equal, the di vergence of equilibrium individual lobby contributions between the Nash and perfectly organized case increases with n, which is also noted by Olson (1965).

Maintaining the assumption for the slopes of the reaction functions made earlier, and starting from the case where both groups are organized internally under the Nash conjecture the result of rice growers moving from the Nash conjecture to that of a perfectly coordinated lobby effort would be to shift up the reaction function for rice growers. This is shown in Figure 4 (where the subscripts on [L.sub.i] and [L.sub.i] denote the conjecture for the rice growers and salmon framers, respectively). The new equilibrium contribution levels for both groups results in greater contributions by both groups and a new, higher equilibrium iso-policy [m.sup.2] - [m.sup.2], (where [m.sup.2] > [m.sup.1]).

If we assume that the two groups are perfectly symmetric (11) an interesting possibility emerges. Consider the case where both rice growers and salmon farmers are initially organized under the Nash conjecture. Suppose then that each group decides to adopt the perfectly planned conjecture. The result then is a shift upward of the individual rice grower's reaction function and a similar shift rightward of individual salmon farmer's reaction function. Given then the assumption that each group enters the policy formation symmetrically, the resulting policy m will be the same as when both groups were organized under the Nash conjecture. This case is shown in Figure 5 where the intersection of the two reaction functions moves along the same iso-policy curve. The result of this last exercise is that individuals in both groups are unambiguously worse off with greater internal lobby group coordination. This is because gross profits have not changed for any individual since the same level of m prevails in both cases, b ut each individual member of both groups has expended more on lobbying. An analogy of this last result can be found in a game of "tug-of-war." That is, consider two identical but opposing groups pulling on different ends of a rope. The position of a flag, which is fastened to the middle of the rope, relative to a point on the ground could denote the policy measure in. On any one team, individual members may free ride on the strength of their teammates and hence not pull as hard as they can, (this would be the Nash case). Thus both teams would pull weakly on the rope and a particular m results. Each team may then acquire a volunteer coach who is able to spot, and eliminate free riding. As a consequence, each team ends up pulling harder on the rope and, assuming symmetry, the flag remains in the same position as before when each team pulled weakly. Obviously, in this example all individuals are worse off with a coach than without since all individuals pull harder, but gain no ground. (12)

Finally, it is interesting to consider how groups would choose to organize themselves if they had the opportunity to meet and decide to adopt a perfectly organized conjecture or not. Under the assumptions made for this model, and given the discussion for Figure 3 above, each group member's profit is perceived to increase with greater coordination in lobbying, given the behavior of the other group. Thus starting from the case where both groups are internally organized under the Nash conjecture, each group member believes they can increase their profits by becoming perfectly coordinated. (13) As such, each group separately decides to become perfectly organized and as they do so all individuals are made worse off. In other words, in the case where the two opposing groups are identical, we get a Prisoner's dilemma result when each group can choose whether or not to perfectly coordinate internal lobby activity. Finally, it is interesting to note that from the policy makers' perspective, if lobbying is in the form of campaign contributions they would prefer the case where both groups are perfectly organized since under this scenario lobby contributions are the greatest.

3. Conclusion

Tullock's (1967) article inspired a great deal of work on economic models with endogenous policy. (14) There has been very little said, however, in terms of the effects of free riding in models where policy is determined endogenously by the confrontation of opposing lobby groups and it is this issue that was the focus of this paper. (15)

The example used in this paper concerning the conflict over water rights in California, to be sure, is more complicated than as they are described above. Indeed, battles over government controlled water from the Klamath river have involved cattle ranchers, farmers and salmon growers from both California and Oregon, (who share the Klamath river waters), as well as the Karuk Indian tribe which depends heavily on the salmon harvest for their survival. (16) Although the model developed above is much less complex than this case, it nonetheless presented some interesting results. It was shown that as a group increased its degree of internal organization or coordination, it decreased the degree of free riding by group members and the equilibrium level of contributions by the group rose. Another interesting finding above was that as two identical groups intensify their lobby battle against each other by eliminating free riding in lobby contributions, each can be made worse off. This result thus expands on Tullock's (1967) argument that when considering the welfare effects of various economic policies we must also include in that analysis the deadweight losses due to unproductive lobbying expenditure. (17)

Possible extensions of the above model are numerous. For example, one can imagine a more complicated policy formation function where the government has a more active role in the determination of policy. In addition, in the above analysis two polar cases for internal lobby group organization were analyzed: the Nash conjecture and the perfectly coordinated case. The conjecture for lobby group organization, however, could also fall anywhere between those two polar cases, and the degree of lobby group organization itself could in fact be endogenously determined.

Another extension would be to consider the above model in the context of a repeated game. In this case the analysis may quickly become complicated. Repeating the game may affect the equilibrium within each group as well as across groups and would depend on whether the game is repeated a finite number of times or if it is repeated infinitely. (18) One possibility could be that by infinitely repeating the game a cooperative outcome could prevail which may entail less expenditure on lobbying.

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Notes

(1.) For example, rainfall during the 1994-95 rainy season totaled nearly 150 percent of the average season rainfall. Nonetheless, many water users in California were not expected to receive their full allotment. The reason for the continued shortage was simply that there were more users than the present water delivery system could handle, (San Francisco Chronicle, March, 1995: p. 1, 17.)

(2.) Under this assumption the local government behaves in a manner Bhagwati (1987, p. 3) describes as a "clearing house" where political officials make decisions on policy solely on the basis of net lobby group pressure.

(3.) This policy formation function is similar to Findlay and Wellisz's (1982) tariff formation function which they use in a model with endogenous tariffs.

(4.) Note that the level of one group's contributions enters into the other's objective function through m.

(5.) See, for example, Tirole (1989, p. 220). Note that given the assumption that all individuals are identical, the solution of equation (1.6) will be the same for all rice growers.

(6.) The units of L are defined such that the cost of each unit is equal to 1.

(7.) It is assumed that the second derivative of m in (1.1) with respect to L is sufficiently negative so that this result obtains. If this condition was not met, optimal lobby contributions could be infinite.

(8.) Note that given the assumption of symmetry across groups and that individuals within a group are identical, we can characterize lobby group behavior by analyzing the behavior of individual members of each group.

(9.) It is also assumed that in the absence of contributions by one group there will be a positive optimal level of contributions by the other group.

(10.) This assumption, of course, need not be the case. In fact, negatively sloped reaction functions are possible and would lead to slightly different results.

(11.) "Perfect symmetry" meaning that the two groups are identical, in terms of size and individual reduced profit functions (i.e. n = n' and [pi] = [pi]'), yet opposing. It is also assumed that the marginal effect of a unit of lobby expenditure is identical, yet opposing. That is,

[partial]m/[partial]L = - [partial]m/[partial]L'

(12.) This of course assumes that individuals do not value the larger muscles they obtain from pulling harder on the rope.

(13.) Recall the assumption that each group member adopts a Nash assumption with regard to the behavior of the members of the rival group.

(14.) See Hillman (1989), for example, for a summary of international trade models with endogenous commercial policy.

(15.) Findlay and Wellisz (1982, 1983) develop a model where the action of two opposing lobby groups determine commercial policy, but the free rider problem is assumed away.

(16.) "Battle for water could be fatal to Klamath salmon," San Francisco Chronicle, September 8, 1992, p. A13.

(17.) Not addressed in this paper is the possibility of private property rights to water. It is clear that if private property rights to water were well defined, enforceable and transferable, water would be allocated in the most efficient manner since owners of water rights could auction off water to the highest bidder and in doing so allocate water to its highest valued use. In the model developed above, the government could be viewed as auctioning off water rights through the policy formation function given in equation (1.1). However, if it is assumed, as it is here, that lobby expenditures aimed at inducing government policy are not directly productive, then the private market auction would be more efficient. See Anderson and Leal (1991) for a more detailed discussion on privatizing water rights.

(18.) If the game is repeated a finite number of times, under given conjectures for within group organization, then the sub-game perfect equilibrium across the groups would be the Nash outcome played for each play of the game, (see Gibbons, 1992, p. 84).

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Leo H. Kahane *

* California State University, Hayward, CA 94542, (510) 885-3369, (510) 885-2923 (fax), [email protected]