the number of rent-seekers and aggregate rent-seeking expenditures: an unpleasant result

The Number of Rent-Seekers and Aggregate Rent-Seeking Expenditures: An Unpleasant ResultAuthor(s): J. Atsu AmegashieSource: Public Choice, Vol. 99, No. 1/2 (1999), pp. 57-62Published by: SpringerStable URL: http://www.jstor.org/stable/30024508 .

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Public Choice 99: 57-62, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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The number of rent-seekers and aggregate rent-seeking expenditures: An unpleasant result *

J. ATSU AMEGASHIE 671 Highview Road, Pickering, Ontario, Canada LIV 4W2.

Accepted 21 May 1997

Abstract. I examine a rent-seeking contest in which the winner gets a minimum rent but also gets an additional rent which is an increasing function of his lobbying expenditure. I give real-world examples of such rent-seeking competitions. Contrary to the standard result in the rent-seeking literature, I obtain the perverse result that aggregate rent-seeking expenditures may be inversely related to the number of rent-seekers. However, I note that even if this result holds, the cost of administering rent-seeking competitions may imply that society is better off with fewer contenders than with an infinitely large number of contenders, although the optimal number may not be the smallest number.

1. Introduction

The literature on rent-seeking initiated by Tullock (1967), followed by Krueger (1974) and Posner (1975) has been extended in various ways by several authors. Most of these papers assume that the rent is exogenously fixed or that rent-seeking expenditures do not affect the size of the rent. Some exceptions are Aleexev and Leitzel (1996) who analyse a rent-seeking game which they refer to as rent-shrinking. In their model, n firms compete for a rent of size, say V. However, the rent which the winner gets is (V-EijjKj). That is, the winner's (i.e., the ith seeker's) rent shrinks by an amount equal to the total rent-seeking expenditures of the losing firms. Appelbaum and Katz (1987) analyse a game in which the size of the rent is determined by the rent-giver. Harstad (1995) examines a model in which rent-seekers are uncertain about the size of the rent and therefore invest resources to estimate the size of the rent. Thus, a seeker's estimate of the size of the rent depends on the resources invested. Finally, in Chung (1996), the winner's rent is an increasing function of aggregate rent-seeking expenditures.

* My thanks are due to Dan Usher for his help and encouragement. My thanks are also

due to an anonymous referee for his comments. I also thank Samuel Darku, Ed Kutsoati, Kwasi Ofori-Yeboah, John Spicer, Wisdom Tettey, and the Adinkrah Family for their immense support.



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In this note, I examine a rent-seeking game in which part of the size of a rent-yielding project or activity is fixed (i.e., independent of lobbying expenditures) and the other part is variable depending on the winner's amount of lobbying. My model differs from Chung (1996) because there is a fixed rent and the variable component is an increasing function of only the winner's (i.e., individual) rent-seeking expenditure instead of aggregate rent-seeking expenditure. The model has elements of what Lee (1985) refers to as fixed lobbying cost and marginal lobbying cost. In endnote 3, he writes: "By marginal lobbying cost we mean the increase in lobbying cost necessary to get a unit expansion in a government activity. It may be, of course, that the decision on a particular government project is made on an all-or-none basis. In this case there is no marginal lobbying cost, just a fixed lobbying cost. But it will often be the case that the size of a government project is variable between wide limits (i.e., a lower limit and an upper limit)1 with the decision on size depending on the amount of lobbying" (Parenthesis mine).

Examples of this rent-seeking game are the following: Firms competing for a monopoly franchise which has a minimum duration but the duration could be extended depending on the winner's amount of lobbying. Lobbying expenditure may include the collection of information (probably misleading) to convince the awarding authority of a lobbyist's competence. The length of the extension of the franchise depends on how impressed the awarding authority is. Similarly, most jobs have a minimum salary, but depending on one's qualifications, experience, and how the applicant performs at the interview, the salary may be increased accordingly. Hence applicants for such jobs will invest time and effort to ensure that they do well at the interview, embellish their CVs, and establish personal contacts with the interviewers. Fi-

nally, development agencies or research institutions who compete for funds to undertake development projects or research may receive a minimum amount when their proposals are successful, but this amount could be increased

depending how attractive the proposal is. Given that in the above examples, the activities yield a rent, it follows that

there is a fixed minimum rent, Vmin and an additional rent, R. It is important to note that I am not interested in why rent-givers some-

times design the competition in this way. Given this design of the rent-seeking competition, my aim is to show that a decrease (increase) in the number of rent-seekers may lead to an increase (decrease) in aggregate rent-seeking expenditures. I argue that this is a probable result. I also discuss some lim- itations of this result. To the best of my knowledge, there is no work in the

rent-seeking literature which gives this perverse result. In what follows, I assume that there are no rent-seeking expenditures when

there is only one rent-seeker. Thus, society is better off with one rent-seeker



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than with many rent-seekers. Assuming that the number of rent-seekers can only be reduced to a minimum of two, I shall show that society is not neces- sarily better off with two rent-seekers than with more than two rent-seekers. Without loss of generality but for the sake of exposition, I assume that all rent-seeking expenditures are socially wasteful.

2. The model

Consider n > 2 risk-neutral and identical agents competing in a rent-seeking game as described in Section 1. Let Ki, (i = 1... n), be the monetary equivalent of agent i's rent-seeking expenditure. Suppose that for Ki dollars of rent- seeking expenditures the winner gets a rent of R = bKi dollars in addition to the minimum rent of Vmin.2 Hence b may be referred to as the marginal lobbying return. Note that this is just the reverse of the marginal lobbying cost in Lee (1985). Therefore the higher is the marginal lobbying cost, the lower is the marginal lobbying return. I confine the analysis to a single stage one-off game of complete information. Using Tullock's (1980) probability function, the ith lobbyist maximises expected profits, 7ri, where

7i = Ki[Vmin + bKi]/(Ki + Ei]jKj) - Ki (1)

Note that (1) implies that the rent-giver is susceptible to rent-seeking expenditures in two ways: (i) rent-seeking expenditures influence his choice of the winner and (ii) they influence his decision to increase the winner's rent (e.g., extend the duration of a monopoly franchise).

The optimum for lobbyist i is given by 8rti/8Ki = 0. In a symmetric Cournot-Nash equilibrium, this gives

K* = (n - 1)Vmin/[n2 + (1 - 2n)b] (2)

For positive values of K*, given Vmin > 0 and n > 2, we require b < n2/(2n-1). Total rent-seeking expenditures, T*-nK*.

In this Cournot-Nash equilibrium, the expected profit for K* > 0, is given by

r* = (1 - b)Vmin/{n2 + (1 - 2n)b} (3)

For non-negative profits (i.e., r* > 0), we require 0 < b < 1. This condition also ensures that the second-order condition for a maximum is satisfied. Given T* = nK* and assuming continuity of n, gives 8T*/Sn = (Vmin/A2){(2n - 1)A - 2n(n - 1)(n - b)}, where A = n2 + (1 - 2n)b. ST*/8n < 0 if

bmin = n2/(2n2 - 2n + 1). Note that bmin < 1, given n > 2. Thus, if



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the marginal lobbying return is sufficiently high (i.e., the marginal lobbying cost is sufficiently low), a decrease in the number of rent-seekers will lead to an increase in aggregate rent-seeking expenditures. This is different from the standard result in the rent-seeking literature. However, as is the case in the rent-seeking literature 8K*/Sn < 0. To provide the intuition behind this perverse result (i.e., 8T*/8n < 0), consider the case of b = 0 (i.e., Tullock's (1980) model). Given b = 0, each agent's rent-seeking expenditure K^ = (n- 1)Vmin/n2 and T^ = nK^. This gives 8K^/8n < 0 and ST'/8n > 0. A decrease in n leads to an increase in each agent's rent-seeking expenditure because the equilibrium probability of victory (1/n) is higher. However, aggregate rent- seeking expenditures fall because the proportionate increase in K^ is less than the proportionate fall in n. Thus, K^ is inelastic with respect to n.3 In the case of b > 0, a fall in n also results in an increase in K*. However, if the benefit from an additional dollar spent in lobbying is sufficiently high (i.e., b > bmin), then the proportionate increase in K* will exceed the proportionate fall in n, which results in an increase in aggregate rent-seeking expenditures.

Note that bmin is decreasing in n. It follows that the maximum value of bmin (i.e., 0.8) occurs at the minimum value of n (i.e., 2). Thus, a sufficient but not necessary condition for this anomalous result to hold is 0.8 < b < 1.

By L'Hospital's rule, K* approaches zero and T* approaches Vmin as n

approaches infinity. These results are well-known for b = 0. Note however, that for a finite n, T* > Vmin if b > n/(2n-1). This condition will hold so long as b > bmin, since bmin > n/(2n-1). For example, given b = 1 (i.e., 7r* = 0), T* = nVmin/(n-1) = Vmin + bK* > Vmin.4 Thus, it may be welfare-improving if the number of rent-seekers is very large than if the number is small.5 This result will not hold when b = 0.

Given 82r*/8n < 0, it may be intuitively helpful to think of the lower (higher) expected profit when n increases (falls) as analogous to the lower (higher) profit that accrues to firms when the number of firms in an industry increases (falls). Suppose that the size of a firm is measured by the size of its

profits. Then an increase (decrease) in n is analogous to small (large) firms. Hence the above result suggests that when a small number of large firms are

lobbying for a monopoly prize, lobbying expenditures may be higher than the case of a very large number of small firms. Metaphorically, the saying that "when two elephants fight, it is the ground that suffers" may be applicable in this case; for when a thousand ants fight, the ground may not be so badly damaged.

If the rent is associated with a distortion(e.g., a monopoly), then society may be better off with many contenders than with few contenders, even if b < bmin (i.e., 8T*/Sn > 0). To see this, consider the example of the monopoly franchise. Suppose, given b > 0, that D = D(t) is the Harberger deadweight



61

loss when the monopoly franchise is extended for t years, where SD/St > 0.6 Evidently, the higher is the winner's lobbying expenditure K*, the higher is the number of years (t) by which the franchise is extended. Since D is increasing in t, t is increasing in K*, and K* is decreasing in n, it follows that SD/6n = (SK*/Sn)(St/IK *)(SD/St) < 0. Under these assumptions, the criterion for welfare maximisation, given a fixed irremovable rent, Vmin (i.e., a fixed distortion), is to choose n to minimise the loss function, L = D + T*. It follows that even if ST*/Sn > 0, welfare may be higher with an infinitely large number of contenders than with fewer contenders, if 8L/Sn < 0. However, if we take into account the cost of administering rent-seeking competitions (e.g., keeping a file for each rent-seeker, scheduling appointments, conducting interviews, carrying out inspections etc), it is very unlikely (if not impossible) that the optimal number of contenders will be infinitely large, although it may be a number greater than two. Denoting C = C(n) as the cost of administering the competition for n contenders, the loss function now becomes L = C + D + T*, where SC/8n > 0 and B2C/5n2 > 0. The optimal number of contenders n*, if it is an interior solution, must satisfy SL/Sn = 0 and 62L/8n2 > 0, where n* may be greater than two.

3. Conclusion

It is important to note that the perverse result obtained in this paper is less likely to hold when the marginal returns are declining rather than being constant. To see this, assume for simplicity, that there is no fixed rent (i.e., Vmin = 0). Suppose instead that all the rent, S, is a function of individual rent- seeking expenditures, such that S = c(Ki)a, where c > 0 and 0 < a < 1. Then the marginal return, SS/SKi > 0, is declining (i.e., 62S/SKi2 < 0). Using, as before [see (1)], Tullock's (1980) probability function with constant returns, it is easy to show that, in a symmetric Cournot-Nash equilibrium, total rent- seeking expenditures, T** = n[c(n+na-l)/n2]1/(1-l). It follows that ST**/Sn > 0, for a < 1/n. This is likely to be the case, given that for a non-negative equilibrium profit, we require 0 < a < 1/n.

Although the result (i.e., ST*/Sn < 0) contradicts the standard result in the rent-seeking literature, it could occur in practice. A priori, there is no reason why individual lobbying expenditures should always be inelastic with respect to changes in the number of lobbyists. The inelasticity of individual lobbying expenditures is a special case (i.e., b = 0) of our model. This paper shows that individual lobbying expenditures may be elastic, if the marginal return to individual rent-seeking expenditures is sufficiently high. This result may not be surprising if one considers the fact that in second-best situations (in this



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case a distortion, a rent, exists), we may obtain seemingly counter-intuitive results.7

Notes

1 I assume that the upper limit is sufficiently large.

2 Given this linear specification of the additional rent, we may interpret b as the proportion of the winner's lobbying expenditure which is refunded or recouped. In Aleexev and Leitzel (1996), the losing firms (not the winner) recoup a proportion of their rent-seeking expenditure.

3 It is easy to show that the absolute value of the elasticity of K^ with respect to n is (n- 2)/(n-1) < 1, given n > 2.

4 This implies that for any fixed rent V and finite n, there is full rent-dissipation (i.e., T* = V), if Vmin = (n-1)V/n and b = 1. V is the maximum rent that the winner can get. Note that in the Tullock rent-seeking model - with b = 0, Vmin= V, and constant returns to scale (r=1) - there is always under-dissipation of the rent, when n is finite.

5 It is obvious that this should be the case, given that ST*/8n < 0 when b > bmin and limnv4T* exists.

6 Of course, I am assuming that in the absence of the monopoly, the market would be competitive.

7 For a related but different discussion of rent-seeking and second-best situations, see Bhagwati (1982).

References

Alexeev, M. and Leitzel, J. (1996). Rent shrinking. Southern Economic Journal 62: 620-626. Appelbaum, E. and Katz, E. (1987). Seeking rents by setting rents: The political economy of

rent-seeking. Economic Journal 97: 685-699. Bhagwati, J. (1982). Directly unproductive profit-seeking (DUP) activities. Journal of Political

Economy 90: 988-1002. Chung, T-Y. (1996). Rent-seeking contest when the prize increases with aggregate efforts.

Public Choice 87: 55-66. Harstad, R. (1995). Privately informed seekers of an uncertain rent. Public Choice 83: 81-93.

Krueger, A.O. (1974). The political economy of the rent-seeking society. American Economic Review 64: 281-303.

Lee, D.R. (1985). Marginal lobbying cost and the optimal amount of rent-seeking. Public Choice 45: 207-213.

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

Tullock, G. (1967). The welfare cost of tariffs, monopolies, and theft. Western Economic Journal 5 (June): 224-232.

Tullock, G. (1980). Efficient rent-seeking. In J.M. Buchanan, R.D. Tollison and G. Tullock, Toward a theory of the rent-seeking society. Texas A&M University Press.



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