Rent-seeking firms, consumer groups, and the social costs of monopoly.

Kyung Hwan Baik

I. INTRODUCTION

When a positive monopoly rent is secured under government protection, firms compete with one another to win the monopoly by expending resources. The opportunity costs of such rent-seeking activities - the values of goods and services that could be produced with resources expended on such rent-seeking activities - are part of the social costs of monopoly. Many economists have examined the social costs of monopoly, defining them as the sum of the deadweight loss and the firms' rent-seeking costs. Examples include Tullock [1967], Krueger [1974], Posner [1975], Rogerson [1982], and Fisher [1985].(1)

Recently, several economists, such as Wenders [1987], Ellingsen [1991], and Schmidt [1992], have pointed out that consumers do not always allow government protection of a monopoly. Rather, consumers often seek government protection of themselves - they try to defend their consumer surplus against a monopoly by spending resources. For example, consumers often lobby politicians for government regulation that requires a natural monopolist to set its price below the unregulated profit-maximizing level. The opportunity costs of such consumer-surplus-defending (henceforth, "CS-defending") activities are also part of the social costs of monopoly. An interesting question then becomes whether consumers' CS-defending activities increase or decrease the social costs of monopoly, as compared with those incurred in the case of inactive consumers. Wenders [1987] argues that consumers' CS-defending activities lead to higher social costs of monopoly. On the other hand, Ellingsen [1991], and Schmidt [1992] show that consumers' participation to defend their consumer surplus lowers the social costs of monopoly.

This paper also addresses the question of whether consumers' CS-defending activities increase or decrease the social costs of monopoly, but differs from previous research by analyzing a model in which the government decides first whether or not to regulate a monopoly and then decides which firm to be the monopolist.(2) We compare the social costs of monopoly in the case of active consumers with those incurred in the case of inactive consumers. We demonstrate that, given just one rent-seeking firm, consumers' CS-defending activities generally increase the social costs of monopoly, but given two or more rent-seeking firms, such activities generally reduce the social costs of monopoly.

Section II sets up the basic model. We consider a rent-seeking and CS-defending (henceforth, RCS) contest with one consumer group. It has two stages. In the first stage, firms and the consumer group compete by expending outlays to win their prizes.(3) The prize for each firm is an unregulated monopoly and that for the consumer group is a regulated monopoly. Each firm and the consumer group choose their outlays simultaneously and independently. In the second stage, after knowing the government's decision on the form of the monopoly, the firms compete against each other to win the monopoly. They expend their outlays simultaneously and independently. At the end of the second stage, the government chooses the winning firm.

Ellingsen [1991] argues that his sequential-move framework is more realistic than his simultaneous-move framework, since in most cases consumers compete against existing monopolists, rather than potential ones, over the forms of monopolies. In his sequential-move framework, potential monopolists first compete to win the monopoly, and then the winner and consumers compete over the form of the monopoly. However, consider a situation in which the government periodically reassigns a monopoly. In this situation, consumers compete against potential monopolists including the incumbent monopolist over the form of the monopoly, and the firms periodically compete to be the monopolist. The sequential-move game in this paper is more appropriate for this situation than the sequential-move game of Ellingsen [1991].

Section III solves for the subgame-perfect equilibria of the RCS contest. In so doing, we obtain and utilize an interesting observation that, given identical firms, to obtain the first-stage equilibrium outlay of the firms and that of the consumer group, we only need to solve a reduced game in which any one firm and the consumer group compete to win their first-stage prizes. This observation is due to the fact that the prize for each firm in the first-stage competition is a group-specific public good: If a firm wins its prize, an unregulated monopoly, all the firms enjoy being candidates for the unregulated monopoly.

In Section IV, we first compare the firms' expected outlay in the RCS contest with the firms' outlay in the rent-seeking contest. By the rent-seeking contest, we mean the contest in which there is no consumer group engaging in CS-defending activities and only the firms compete against each other to be the unregulated monopolist. We find that, given just one rent-seeking firm, the firm's outlay in the rent-seeking contest is less than that in the RCS contest, but given two or more rent-seeking firms, the firms' outlay in the rent-seeking contest is greater than the firms' expected outlay in the RCS contest. Next, we compare the expected total outlay in the RCS contest with the total outlay in the rent-seeking contest. We find that, given just one rent-seeking firm, the total outlay in the rent-seeking contest is less than that in the RCS contest, but given two or more rent-seeking firms, it is greater than the expected total outlay in the RCS contest. Finally, we compare the social costs of monopoly in the RCS contest with those in the rent-seeking contest. We find that, given just one rent-seeking firm, the social costs of monopoly in the rent-seeking contest are in general less than those in the RCS contest, but given two or more rent-seeking firms, the social costs of monopoly in the rent-seeking contest are in general greater than those in the RCS contest.

Section V extends the basic model and considers an RCS contest with multiple consumer groups. We show that the results obtained in Section IV still hold in this case. Section VI provides our conclusions.

II. THE BASIC MODEL

There are n identical firms which try to win a monopoly - for example, the monopoly illustrated in Figure 1. Consider a situation in which the government first decides whether to regulate the monopoly and then decides which firm to be the monopolist. In the case of active consumers, two kinds of conflicts arise. First, conflicts between firms and consumers arise over the form of the monopoly. If the government does not regulate the monopoly, the monopoly price is [P.sub.m] and the monopoly profits are T. If the government regulates the monopoly and sets a price of [P.sub.r], then the monopoly profits are zero, and consumer surplus increases by T + H, as compared with the unregulated monopoly.(4) Hence, the firms - the potential monopolists - lobby for the unregulated monopoly while consumers lobby for the regulated monopoly. Second, conflicts among the firms arise over the winner of the monopoly. After the government decides whether to regulate the monopoly, the firms lobby to win the monopoly.

We formally consider the following two-stage RCS contest. In the first stage, the n firms and one consumer group compete by expending outlays to win their prizes. The prize for each firm is the unregulated monopoly and that for the consumer group is the regulated monopoly. In the second stage, after knowing the government's decision on the form of the monopoly, the firms compete against each other to win the monopoly. At the end of the second stage, the government chooses the winning firm.

First, consider the first stage. The consumer group is treated as a single player. Each firm and the consumer group are risk neutral. The prize for each firm is a group-specific public good - that is, if a firm wins its prize, the unregulated monopoly, then all the firms enjoy being candidates for the unregulated monopoly. The firms have the same valuation for their prize: their second-stage equilibrium expected payoffs resulting when the unregulated monopoly is chosen in the first stage. Let v represent the consumer group's valuation for its prize, where 0 [less than] v [less than or equal to] T + H. Note that, if the consumer group is associated with all the consumers in the market, then v = T + H. Let [x.sub.il] represent the irreversible rent-seeking outlay expended by firm i in the first stage and let [X.sub.1] represent the outlay expended by all the firms in the first stage:

[X.sub.1] = [summation of] [X.sub.kl] where k = 1 to n. Let y represent the irreversible CS-defending outlay expended by the consumer group. Let P([X.sub.1], y) be the probability that the firms win their prize, the unregulated monopoly, given [X.sub.1] and y. Following the rent-seeking literature (e.g., Tullock [1980]), we assume that the probability-of-winning function for the firms takes a logit form:

(1) P([X.sub.1], y) = [Beta][X.sub.1] / ([Beta][X.sub.1] + y) for [X.sub.1] + y [greater than] 0

1/2 for [X.sub.1] + y = 0,

where [Beta] [greater than or equal to] 1.(5) The parameter [Beta] represents the firms' lobbying ability relative to the consumer group. A value of [Beta] greater than one implies that the firms have more ability than the consumer group: if both the firms and the consumer group expend the same positive outlay, the firms' probability of winning is greater than one half. Let [Mathematical Expression Omitted] denote firm i's expected payoff in the first stage. Then firm i's payoff function in the first stage is given by

(2) [Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is firm i's second-stage equilibrium expected payoff resulting when the unregulated monopoly is chosen in the first stage (see Lemma 1). The expected payoff of the consumer group is

(3) [[Pi].sup.C] = v[1 - P([X.sub.1], y)] - y.

Although the firms have the same goal of winning their prize, the unregulated monopoly, each firm chooses its outlay independently. We assume that the firms and the consumer group choose their outlays simultaneously.

Next, consider the second stage. Let [x.sub.i2] represent the irreversible outlay expended by firm i in the second stage and let [X.sub.2] represent the outlay expended by all the firms in the

second stage: [X.sub.2] = [summation of] [x.sub.k2] where k = 1 to n. Let [p.sub.i]([x.sub.2]) be the probability that firm i wins the monopoly when the outlays of the firms are [x.sub.2] [equivalent to] ([x.sub.12], ..., [x.sub.n2]). The probability-of-winning function for firm i is given by

[p.sub.i]([x.sub.2]) = [x.sub.i2]/[X.sub.2] for [X.sub.2] [greater than] 0

1/n for [X.sub.2] = 0.

This implies that the winning firm is determined by the second-stage outlays only. Let [[Pi].sub.i] denote firm i's expected payoff in the second stage. Then firm i's payoff function in the second stage is

[[Pi].sub.i] = V[p.sub.i]([x.sub.2]) - [x.sub.i2],

where V represents profits from the monopoly. We have V = T for the unregulated monopoly and V = 0 for the regulated monopoly. We assume that the firms choose their outlays simultaneously and independently.

We assume that all of the above is common knowledge. We employ subgame-perfect equilibrium as the solution concept.

III. SUBGAME-PERFECT EQUILIBRIA

To obtain a subgame-perfect equilibrium, we work backward. Consider the second stage. Either the regulated monopoly or the unregulated monopoly has been chosen in the first stage. If the regulated monopoly has been chosen, then the Nash equilibrium of the second stage is trivial. Realizing that the profits from the regulated monopoly are zero, each firm expends zero outlays.

To obtain a Nash equilibrium of the second stage in the case where the unregulated monopoly has been chosen, we begin by deriving firm i's reaction function. Firm i's reaction function shows its best response to every possible combination of outlays that the other firms might choose. Given a positive outlay of the other firms, the best response of firm i is obtained from the first-order condition for maximizing its expected payoff: [[Pi].sub.i] = T[p.sub.i]([x.sub.2]) - [x.sub.i2]. Firm i's expected payoff [[Pi].sub.i] is strictly concave in its own outlay [x.sub.i2], and thus the second-order condition is satisfied. Firm i's reaction function is then

[r.sub.i]([X.sub.-i2]) = [-square root of [TX.sub.-i2] - [X.sub.-i2]] for 0 [less than] [X.sub.-i2] [less than or equal to] T

0 for [X.sub.-i2] [greater than or equal to] T,

where [X.sub.-i2] represents an outlay of the other firms in the second stage and [r.sub.i] ([X.sub.-i2]) is firm i's best response to [X.sub.-i2]. Note that when the other firms expend zero outlays, firm i's best response is to expend an infinitesimally small outlay.(6) Since a Nash equilibrium is an n-tuple vector of outlays, one for each firm, at which each firm's outlay is the best response to its opponents' outlays, it satisfies all the reaction functions. It is straightforward to see that there is a unique Nash equilibrium and the Nash equilibrium is symmetric.

Lemma 1 shows each firm's outlay, the outlay expended by all the firms, and each firm's expected payoff, at the Nash equilibrium of the second stage.

LEMMA 1. For the unregulated monopoly, at the Nash equilibrium of the second stage, firm i's outlay is (n - 1)T/[n.sup.2], the outlay expended by all the firms is [Mathematical Expression Omitted], and firm i's expected payoff is [Mathematical Expression Omitted]. For the regulated monopoly, firm i's outlay and expected payoff are zero.

A Group-Specific Public-Good Prize

We turn now to the first stage of the full game. The firms lobby for the unregulated monopoly while the consumer group lobbies for the regulated monopoly. Each firm chooses its outlay independently and the firms and the consumer group choose their outlays simultaneously. We show that, to obtain the first-stage outlay of the firms and that of the consumer group in the subgame-perfect equilibria of the full game, we only need to solve a reduced game in which any one firm and the consumer group compete to win their first-stage prizes. This observation is due to the fact that the prize for each firm is a group-specific public good.

We derive a firms-specific equilibrium - an n-tuple vector of outlays, one for each firm, at which each firm's outlay is the best response to the other firms' outlays - given an outlay of the consumer group.(7) It follows from expression (2) and Lemma 1 that firm i's payoff function in the first stage is [Mathematical Expression Omitted]. Given a positive outlay of the consumer group, y, firm i's best response to an outlay of the other firms is obtained from the first-order condition for maximizing its expected payoff [Mathematical Expression Omitted]. Firm i's expected payoff [Mathematical Expression Omitted] is strictly concave in its own outlay [x.sub.i1] and thus the second-order condition is satisfied. Firm i's reaction function is then

[Mathematical Expression Omitted]

[Mathematical Expression Omitted],

where [X.sub.-i1] represents an outlay of the other firms in the first stage and [Mathematical Expression Omitted] is firm i's best response to [X.sub.-i1], given y. Utilizing the fact that these n reaction functions drawn in the n-dimensional space coincide, we obtain Lemma 2.

LEMMA 2. Given an outlay of the consumer group, y, a firms-specific equilibrium is an n-tuple vector of outlays ([x[prime].sub.11], ..., [x[prime].sub.n1]) which satisfies 0 [less than or equal to] [x[prime].sub.k1] [less than or equal to] [-square root of Ty]/n [-square root of [Beta]] - y/[Beta] for k = 1, ..., n

and [summation of] [x[prime].sub.kl] where k = 1 to n = [-square root of Ty]/n [-square root of [Beta]] - y/[Beta].

Lemma 2 implies the following. First, there are multiple firms-specific equilibria given y. Second, there are firms-specific equilibria with free riders - firms expending zero outlays. Third, the outlay of the firms is constant across the firms-specific equilibria. Finally, the outlay of the firms at the firms-specific equilibria, [-square root of Ty]/n [-square root of [Beta]] - y/[Beta], is equal to firm i's best response to y when firm i is the only firm competing against the consumer group in the first stage.

It follows immediately from Lemma 2 that the full game has multiple subgame-perfect equilibria and the outlay of the firms and that of the consumer group are constant across the subgame-perfect equilibria. Lemma 3 also follows from Lemma 2.

LEMMA 3. The first-stage outlay of the firms and that of the consumer group in the subgame-perfect equilibria of the full game are equal to those resulting when only one firm and the consumer group are allowed to choose their outlays in the first stage.

Lemma 3 implies that, to obtain the first-stage equilibrium outlay of the firms and that of the consumer group, we only need to solve a reduced game in which any one firm and the consumer group compete to win their first-stage prizes.

A Reduced Game and the First-Stage Equilibrium Outlays

To obtain the first-stage equilibrium outlay of the firms and that of the consumer group, we solve a reduced game in which firm 1 and the consumer group compete to win their first-stage prizes. Firm 1's and the consumer group's payoff functions are then

[Mathematical Expression Omitted]

and

[[Pi].sup.C] = v[1 - P([x.sub.11], y)] - y.

We first derive the reaction function of firm 1. Firm 1's reaction function shows its best response to every possible outlay that the consumer group might choose. Given a positive outlay of the consumer group, y, the best response of firm 1 is obtained from the first-order condition for maximizing its expected payoff, [Mathematical Expression Omitted]. Firm l's expected payoff [Mathematical Expression Omitted] is strictly concave in its own effort level [x.sub.11], and therefore the second-order condition is satisfied. Firm 1's reaction function is then

[Mathematical Expression Omitted]

0 for y [greater than or equal to] [Beta]T/[n.sup.2],

where [Mathematical Expression Omitted] is firm 1's best response to y.

Similarly, the consumer group's reaction function shows its best response to every possible outlay that firm 1 might choose. Given a positive outlay of firm 1, [x.sub.11], the best response of the consumer group is obtained from the first-order condition for maximizing its expected payoff, [[Pi].sup.C]. The consumer group's expected payoff [[Pi].sup.C] is strictly concave in its own effort level y and therefore the second-order condition is satisfied. The consumer group's reaction function is then

[R.sup.C]([x.sub.11] = [-square root of [Beta]v[x.sub.11]] = [Beta][x.sub.11] for 0 [less than] [x.sub.11] [less than or equal to] v/[Beta]

0 for [x.sub.11] [greater than or equal to] v/[Beta],

where [R.sup.C]([x.sub.11]) is the consumer group's best response to [x.sub.11].

Let [Mathematical Expression Omitted] be a Nash equilibrium of the reduced game. Then it satisfies the two reaction functions above. The reduced game has a unique Nash equilibrium which is reported in Lemma 4.

LEMMA 4. The Nash equilibrium of the reduced game is [Mathematical Expression Omitted], [Beta][n.sup.2][v.sup.2]T/[([Beta]T + [n.sup.2]v).sup.2]).

Let [Mathematical Expression Omitted] and [y.sup.*] be the first-stage outlay of the firms and that of the consumer group in the subgame-perfect equilibria of the full game. Since they are equal to firm 1's and the consumer group's outlays in the Nash equilibrium of the reduced game (see Lemma 3), using Lemma 4, we obtain Lemma 5.

LEMMA 5. The first-stage outlay of the firms and that of the consumer group in the subgame-perfect equilibria of the full game are [Mathematical Expression Omitted] and [y.sup.*] = [Beta][n.sup.2][v.sup.2]T/[([Beta]T + [n.sup.2]v).sup.2]. Therefore, the first-stage equilibrium total outlay is [Mathematical Expression Omitted].

The first-stage outlay of the firms and that of the consumer group in the subgame-perfect equilibria depend on the consumer group's valuation for its prize v, the profits from the unregulated monopoly T, the number of firms n, and the ability parameter [Beta].

IV. EQUILIBRIUM OUTLAYS AND THE SOCIAL COSTS OF MONOPOLY

We first compare the firms' expected outlay in the RCS contest with the firms' outlay in the rent-seeking contest.(8) In the rent-seeking contest, there is no consumer group engaging in CS-defending activities and only the n firms compete against each other to be the unregulated monopolist making the profits of T. Hence, the firms' outlay in the rent-seeking contest, F[O.sup.*], is equal to [Mathematical Expression Omitted] in the RCS contest (see Lemma 1). Next, we derive the firms' expected outlay in the RCS contest. If the firms lose in the first stage, or equivalently, if the regulated monopoly is chosen in the first stage, then the firms' outlay is just their first-stage outlay, [Mathematical Expression Omitted], since the firms expend zero outlays in the second stage (see Lemma 1). If the firms win in the first stage, or equivalently, if the unregulated monopoly is chosen in the first stage, then the firms' outlay is the sum of their first-stage and second-stage outlays, [Mathematical Expression Omitted], since the firms expend positive outlays to win the unregulated monopoly in the second stage. Given the first-stage outlays of the firms and the consumer group, using function (1), we obtain the probability that the firms win in the first stage: [Mathematical Expression Omitted]. Therefore, the expected outlay of the firms in the RCS contest is [Mathematical Expression Omitted]. Using Lemma 1, we have EF[O.sup.*] = [Beta]v[T.sup.2]/[([Beta]T + [n.sup.2]v).sup.2] + (n - 1)[Beta][T.sup.2]/n([Beta]T + [n.sup.2]v). Comparing EF[O.sup.*] with the firms' outlay in the rent-seeking contest, F[O.sup.*] we obtain Proposition 1.

PROPOSITION 1. If there is just one firm, then the firm's outlay in the rent-seeking contest is less than that in the RCS contest. If there is more than one firm, then the firms' outlay in the rent-seeking contest is greater than the firms' expected outlay in the RCS contest.

Proposition 1 holds regardless of the values of [Beta] and v.(9) If there is just one firm, the firm expends zero outlays when there is no consumer group engaging in CS-defending activities, but positive outlays when confronting the consumer group. Given n [greater than or equal to] 2, since the RCS contest has additional competition, the first-stage competition, compared with the rent-seeking contest, it is tempting to conclude that the firms' expected outlay in the RCS contest exceeds the firms' outlay in the rent-seeking contest. In the RCS contest, however, the consumer group's CS-defending activities in the first stage reduce the probability that the firms expend positive outlays in the second stage. Therefore, we obtain the opposite.

Next, we compare the expected total outlay in the RCS contest with the total outlay in the rent-seeking contest. We begin by examining the expected total outlay in the RCS contest. If the consumer group wins its prize in the first stage, the total outlay is only the sum of the first-stage firms' and consumer group's outlays: [Mathematical Expression Omitted]. If the firms win their prize in the first stage, then the total outlay is the sum of the first-stage outlays and the second-stage firms' outlay: [Mathematical Expression Omitted]. The expected total outlay in the RCS contest is then [Mathematical Expression Omitted]. The total outlay in the rent-seeking contest, T[O.sup.*], is equal to [Mathematical Expression Omitted] in the RCS contest (see Lemma 1). Comparing ET[O.sup.*] with T[O.sup.*], we obtain Lemmas 6 and 7.

LEMMA 6. If

(4) [Mathematical Expression Omitted],

then [Mathematical Expression Omitted].

LEMMA 7. If there is just one firm, then T[O.sup.*] [less than] ET[O.sup.*] regardless of the values of [Beta] and v. In the case where there is more than one firm, (a) if 1 [less than or equal to] [Beta] [less than] [n.sup.3](n - 1)(T + H)/[[n.sup.2]H + (n + 1)T] then T[O.sup.*] [greater than] ET[O.sup.*] regardless of the value of v, and (b) if [Beta] [greater than or equal to] [n.sup.3] (n-1) (T + H)/[[n.sup.2]H + (n + 1)T], then [Mathematical Expression Omitted] if [Mathematical Expression Omitted].

Proof. In the case where there is just one firm, the left-hand side of expression (4) is zero and its right-hand side is positive. Therefore, we have T[O.sup.*] [less than] ET[O.sup.*].

Consider the case where there is more than one firm. Expression (4) can be rewritten as

(5) [Mathematical Expression Omitted].

Given n [greater than or equal to] 2, the first term of expression (5) is positive. It is easy to see that for [Beta] [less than or equal to] n(n- 1), the second term of expression (5) is nonnegative. Then, for [Beta] [less than or equal to] n(n- 1), the left-hand side of expression (5) is positive, and therefore it follows from Lemma 6 that T[O.sup.*] [greater than] ET[O.sup.*], regardless of the value of v. Next, consider the case where [Beta] [greater than] n(n- 1) and thus the second term of expression (5) is negative. Expression (5) can be rewritten as [Mathematical Expression Omitted], where G [equivalent to] [Beta]T([n.sup.2] - n - 1)/[n.sup.2][[Beta] - n(n - 1)]. Then, using Lemma 6, we have: if [Mathematical Expression Omitted], then [Mathematical Expression Omitted]. This and the assumption that v [less than or equal to] T + H, yield the following. If G [greater than] T + H, or equivalently, if n(n - 1) [less than] [Beta] [less than] [n.sup.3](n - 1)(T + H)/[[n.sup.2]H + (n + 1)T], then G is always greater than v and therefore T[O.sup.*] [greater than] ET[O.sup.*] regardless of the value of v. If G [less than or equal to] T + H, or equivalently, if [Beta] [greater than or equal to] [n.sup.3](n- 1)(T + H)/[[n.sup.2]H + (n + 1)T], then [Mathematical Expression Omitted] if [Mathematical Expression Omitted].

In the case where there is more than one firm, [n.sup.3](n - 1)(T + H)/[[n.sup.2]H + (n + 1)T] is greater than n(n - 1). Hence, believing that an RCS contest with [Beta] [greater than] n(n - 1) rarely arises, we say that part (a) of Lemma 7 covers most cases occurring when there is more than one firm.

Proposition 2 then follows immediately from Lemma 7.

PROPOSITION 2. If there is just one firm, then the total outlay in the rent-seeking contest is less than in the RCS contest. If there is more than one firm, then the total outlay by firms in the rent-seeking contest is greater than the expected total outlay by firms and consumers in the RCS contest.

The first part of Proposition 2 holds regardless of the values of [Beta] and v. The second part holds regardless of the value of v, unless the firms have "much" more ability than the consumer group. The intuitive explanations for Proposition 2 are similar to those for Proposition 1 and therefore are omitted.

Finally, we compare the social costs of monopoly in the RCS contest with those in the rent-seeking contest. The social costs of monopoly in the rent-seeking contest are the sum of the deadweight loss and the firms' rent-seeking outlay. The firms' rent-seeking outlay is [Mathematical Expression Omitted] (see Lemma 1). Therefore, the social costs of monopoly in the rent-seeking contest are [C.sup.S] = H + [Delta] + (n - 1)T/n, where [Delta] represents a difference in welfare between the average cost pricing regulation and the marginal cost pricing regulation.(10) Next, we compute the social costs of monopoly in the RCS contest. From Lemma 5, we know that the firms and the consumer group expend [Mathematical Expression Omitted] and [y.sup.*] = [Beta][n.sup.2][v.sup.2] T/[([Beta]T + [n.sup.2]v).sup.2] in the first stage. If the consumer group wins in the first stage, then the social costs of monopoly are [Mathematical Expression Omitted]. If the firms win in the first stage, then the social costs of monopoly are increased by the sum of the deadweight loss H and the second-stage firms' outlay [Mathematical Expression Omitted], and thus they are equal to [Mathematical Expression Omitted]. Therefore, the social costs of monopoly in the RCS contest are

[Mathematical Expression Omitted].

Comparing [C.sup.S] with [C.sup.D], we obtain Lemma 8.

LEMMA 8. If

(6) [Mathematical Expression Omitted],

then [Mathematical Expression Omitted].

Lemma 9 compares the social costs of monopoly in the RCS contest with those in the rent-seeking contest, when there is just one firm.

LEMMA 9. In the case where there is just one firm, (a) if H [less than] T, then [C.sup.S] [less than] [C.sup.D] regardless of the values of [Beta] and v, (b) if H = T, then [C.sup.S] [less than] [C.sup.D] regardless of the value of v when [Beta] [greater than] 1, and [C.sup.S] = [C.sup.D] regardless of the value of v when [Beta] = 1, and (c) if H [greater than] T, then [C.sup.S] [greater than] [C.sup.D] regardless of the value of [Beta] when 0 [less than] v [less than or equal to] H - T, and [Mathematical Expression Omitted] if [Mathematical Expression Omitted], when H - T [less than] v [less than or equal to] T + H.

Proof. Given n = 1, Lemma 8 is simplified to: If

(7) [Mathematical Expression Omitted],

then [Mathematical Expression Omitted].

Parts (a) and (b) follow immediately from the fact that the right-hand side of expression (7) is increasing in [Beta] and the fact that, for [Beta] = 1, the right-hand side of expression (7) is T.

To prove part (c), note that the right-hand side of expression (7) is increasing in [Beta] but its limit is (T + v) as [Beta] approaches plus infinity. Therefore, if 0 [less than] v [less than or equal to] H - T, then the left-hand side of expression (7) is greater than its right-hand side and thus [C.sup.S] [greater than] [C.sup.D] regardless of the value of [Beta]. If H - T [less than] v [less than or equal to] T + H, we can rewrite expression (7) as [Mathematical Expression Omitted].

Suppose that H [less than] [Beta]T(T + v)/([Beta]T + v). Then

Proposition 3 follows immediately from Lemma 8 or 9.

PROPOSITION 3. If there is just one firm, then the social costs of monopoly in the rent-seeking contest are less than those in the RCS contest - that is, consumers' CS-defending activities (or lobbying) increase the social costs of monopoly.

When there is just one firm, the social costs of monopoly in the rent-seeking contest are "small," since there is no rent-seeking outlay. On the other hand, the social costs of monopoly in the RCS contest are "large." In the RCS contest, the firm realizes that, if it wins its prize in the first stage, it secures the positive monopoly profits without any effort in the second stage - in other words, the firm's valuation for its prize in the first stage is high - and thus it tries hard. Therefore, we obtain Proposition 3.

However, even in this one-firm case, consumers' CS-defending activities (or lobbying) may reduce the social costs of monopoly. This occurs, for example, when the monopoly profit T is less than the deadweight loss H and the consumer group's valuation v is less than the difference between H and T (see Lemma 9).

Lemma 10 compares the social costs of monopoly in the RCS contest with those in the rent-seeking contest, when there is more than one firm.

LEMMA 10. In the case where there is more than one firm, (a) if 1[less than or equal to] [Beta] [less than] [n.sup.3](T + H) [nH + (n - 1)T]/(n + 1)[T.sup.2], then [C.sup.S][greater than][C.sup.D] regardless of the value of v, and (b) if [Beta][greater than or equal to][n.sup.3](T + H)[nH + (n-1)7]/(n + 1)[T.sup.2], then [Mathematical Expression Omitted] if [Mathematical Expression Omitted].

Proof. Expression (6) can be rewritten as

(8) [Mathematical Expression Omitted].

Given n [greater than or equal to] 2, the first term of expression (8) is positive. It is easy to see that for [Beta] [less than or equal to] [[Beta].sub.1], the second term of expression (8) is nonnegative, where [[Beta].sub.1] [equivalent to] n(n - 1) + [n.sup.2]H/T. Then, for [Beta] [less than or equal to] [[Beta].sub.1], the left-hand side of expression (8) is positive and therefore it follows from Lemma 8 that [C.sup.S] [greater than] [C.sup.D] regardless of the value of v. Next, consider the case where [Beta] [greater than] [[Beta].sub.1] and thus the second term of expression (8) is negative. Expression (8) can be rewritten as [Mathematical Expression Omitted], where K [equivalent to] [Beta]T[[n.sup.2]H + ([n.sup.2] - n - 1)7]/[n.sup.2] [[Beta]T- [n.sup.2] H - n (n- 1)T]. Then, using Lemma 8, we have: If [Mathematical Expression Omitted], then [Mathematical Expression Omitted]. This and the assumption that v [less than or equal to] T + H, yield the following. If K [greater than] T + H, or equivalently, if [[Beta].sub.1] [less than] [Beta] [[Beta].sub.2], where [[Beta].sub.2] [equivalent to] [n.sup.3] (T + H) [nH + (n - 1)T]/(n+1)[T.sup.2], then K is always greater than v and therefore [C.sup.S] [greater than] [C.sup.D] regardless of the value of v. If K [less than or equal to] T + H, or equivalently, if [Beta] [greater than or equal to] [[Beta].sub.2], then [Mathematical Expression Omitted] if [Mathematical Expression Omitted].

Note that [n.sup.3](T + H) [nH +(n-1)T]/(n+1)[T.sup.2] is greater than n(n - 1) + [n.sup.2] H/T for n [greater than or equal to] 2. Hence, believing that an RCS contest with [Beta] [greater than] n(n - 1) + [n.sup.2]H/T rarely arises, we say that part (a) of Lemma 10 covers most cases occurring when there is more than one firm.

Suppose that [n.sup.2]H + n(n - 1)T [greater than] [Beta]T(T + [n.sup.2]v)/([Beta]T + [n.sup.2]v). Then Proposition 4 follows immediately from Lemma 8 or 10.

PROPOSITION 4. If there is more than one firm, then the social costs of monopoly in the rent-seeking contest are greater than those in the RCS contest - that is, consumers' CS-defending activities (or lobbying) reduce the social costs of monopoly.

Consumers' CS-defending activities (or lobbying) have two opposing effects on the social costs of monopoly, as compared with the rent-seeking contest. On one hand, they increase the social costs of monopoly by inducing the first-stage firms' and consumer group's outlays. On the other hand, they reduce the social costs of monopoly by decreasing the probability that the deadweight loss H is incurred and the firms expend positive outlays in the second stage. Note that in the rent-seeking contest this probability is unity. In general the second effect dominates the first one, and thus we obtain Proposition 4. However, Lemma 10 shows that the social costs of monopoly in the rent-seeking contest may be less than those in the RCS contest - that is, consumers' CS-defending activities (or lobbying) may increase the social costs of monopoly. This happens when the consumer group's valuation is very high and the firms have "much" more ability than the consumer group. Intuitive explanations follow. When the consumers value their prize very high, they try hard. Furthermore, when the consumer group is "much" less effective than the firms, the consumers try harder. On the other hand, when the consumer group is "much" less effective than the firms, the probability that the consumer group wins in the first stage is low, and therefore the probability that the deadweight loss H is incurred and the firms expend positive outlays in the second stage is high. Therefore, consumers' CS-defending activities (or lobbying) increase the social costs of monopoly.

Using the probability-of-winning function used in perfectly discriminating contests, Ellingsen [1991] shows, both in his simultaneous-move framework and in his sequential move framework, that consumers' CS-defending activities lower the social costs of monopoly. Using the simplest logit-form probability-of-winning function, he shows in both frameworks that consumers' CS-defending activities tend to reduce the social costs of monopoly. Comparing the latter with the results in Lemma 10, we find that the conditions in Lemma 10 for consumers' CS-defending activities to reduce the social costs of monopoly are considerably weaker. The reasons are twofold. First, consider the stage in which the firm(s) and the consumer group compete over the form of the monopoly - the first stage in our game and the second stage in the sequential-move game of Ellingsen [1991]. In our game, each firm's valuation for its first-stage prize is much less than the monopoly profits while in his game the winning firm's valuation for its second-stage prize is the monopoly profits. Hence, less outlays are expended in our game (see Baik [1994]). Also, the probability that the consumer group wins is higher and thus the probability that the deadweight loss H is incurred is lower, in our game. Second, consider the stage in which the firms compete against each other to win the monopoly-the second stage in our game and the first stage in his game. In our game, the firms expend positive outlays only when they win in the first stage, but in his game the firms always expend positive outlays.

V. MULTIPLE CONSUMER GROUPS

We have so far assumed that there is a single consumer group in the RCS contest. Do we obtain results different from those in Section IV if there are multiple consumer groups in the RCS contest? Surprisingly, the answer is no. We prove this by showing that the first-stage equilibrium outlay of the firms and that of the consumer groups are exactly equal to [Mathematical Expression Omitted] and [y.sup.*], respectively, in Lemma 5. (In this case, v represents the highest of the consumer groups' valuations for their prize.)

We consider the following modified two-stage RCS contest. In the first stage, the n firms and m consumer groups compete by expending outlays to win their prizes, where m [greater than or equal to] 2. In the second stage, after knowing the government's decision on the form of the monopoly, the firms compete against each other to win the monopoly.

Consider the first stage. Each consumer group is treated as a single player. Each firm and each consumer group are risk-neutral. The prize for each firm is a group-specific public good. The prize for each consumer group is also a group-specific public good - that is, if a consumer group wins its prize, the regulated monopoly, then all the consumers enjoy the lower price. The consumer groups' valuations for their prize may differ.(11) Let [v.sub.i] represent consumer group i's valuation for its prize. Without loss of generality, we assume that 0 [less than] [v.sub.m] [less than or equal to] [v.sub.m-1] [less than or equal to] . . . [less than or equal to] [v.sub.1] [less than or equal to] T + H. Let [y.sub.j] represent the irreversible CS-defending outlay expended by consumer group j and let Y represent the outlay expended by all the consumer

groups: Y = [summation of] [y.sub.k] where k = 1 to m. Let P([X.sub.1], Y) be the probability that the firms win their prize, the unregulated monopoly, given [X.sub.1] and Y. We assume that the probability-of-winning function for the firms is

P([X.sub.1, Y) = [Beta][X.sub.1]/([Beta][X.sub.1] + Y) for [X.sub.1] + Y [greater than] 0

1/2 for[X.sub.1] + Y = 0,

where [Beta] [greater than or equal to] 1. The parameter [Beta] represents the firms' lobbying ability relative to the consumer groups. Firm i's expected payoff in the first stage is given by [Mathematical Expression Omitted], where [Mathematical Expression Omitted], is firm i's second-stage equilibrium expected payoff resulting when the unregulated monopoly is chosen in the first stage (see Lemma 1). Consumer group j's expected payoff is [Mathematical Expression Omitted]. Each firm and each consumer group choose their outlays independently and simultaneously.

The second stage is the same as that of the original RCS contest.

To obtain a subgame-perfect equilibrium, we work backward. Since the second stage is the same as that of the original RCS contest, we use Lemma 1 intact. Consider then the first stage. We begin by noting that Lemma 2 holds when we replace y with Y. Next, we derive a consumers-specific equilibrium - an m-tuple vector of outlays, one for each consumer group, at which each consumer group's outlay is the best response to the other consumer groups' outlays - given an outlay of the firms. Given a positive outlay of the firms, [X.sub.1], the best response of consumer group j to an outlay of the other consumer groups is obtained from the first-order condition for maximizing its expected payoff [Mathematical Expression Omitted]. Consumer group j's expected payoff [Mathematical Expression Omitted] is strictly concave in its own outlay [y.sub.j] and thus the second-order condition is satisfied. Consumer group j's reaction function is then

[Mathematical Expression Omitted]

for 0 [less than or equal to] [Y.sub.-j] [less than or equal to] [-square root of [Beta][v.sub.j][X.sub.1]] - [Beta][X.sub.1]

0

for [Y.sub.-j] [greater than or equal to] [-square root of [Beta][v.sub.j][X.sub.]1] - [Beta][X.sub.1],

where [Y.sub.-j] represents an outlay of the other consumer groups and [Mathematical Expression Omitted] is consumer group j's best response to [Y.sub.-j]' given [X.sub.1]. Using these m reaction functions, we obtain Lemma 11.

LEMMA 11. Given an outlay of the firms, [X.sub.1], a consumers-specific equilibrium is an m-tuple vector of outlays (y[prime].sub.1], ..., [y[prime].sub.m]) such that (a) if [v.sub.1] [greater than] [v.sub.2], then [y[prime].sub.1] = [-square root of [Beta][v.sub.1][X.sub.1]] - [Beta][X.sub.1] and y[prime]h = 0 for h = 2, ..., m, (b) if [v.sub.1] = [v.sub.s] [greater than] [v.sub.s+1] for some s, where 2 [less than or equal to] s [less than or equal to] m - 1, then 0 [less than or equal to] [y[prime].sub.k] [less than or equal to] [-square root of[Beta][v.sub.1][X.sub.1]] -[Beta][X.sub.1] for k = 1, ..., s, [summation of] [y[prime].sub.k] where k = 1 to s = [-square root of[Beta][v.sub.1][X.sub.1]] -[Beta][X.sub.1], and [y[prime].sub.h] = 0 for h = s + 1, ..., m, and (c) if [v.sub.1] = [v.sub.m], then 0 [less than or equal to] [y[prime].sub.k] [less than or equal to] [-square root of [Beta][v.sub.1][X.sub.1]] - [Beta][X.sub.1] for k = 1, ..., m, and [summation of] [y[prime].sub.k] where k = 1 to m = [-square root of[Beta][v.sub.1][X.sub.1]] - [Beta][X.sub.1].

The formal proof of Lemma 11 is omitted (see Baik [1993], for a proof in a more general setting). Figure 2 illustrates the case where there are only two consumer groups, 1 and 2, and [v.sub.1] [greater than] [v.sub.2] holds. The two reaction curves are parallel to each other and have a slope of-1. The intersection of the reaction curves - and thus the consumers-specific equilibrium ([y[prime].sub.1], [y[prime].sub.2]) - occurs at point A on the horizontal axis. Lemma 11 implies the following. First, if one and only one consumer group has the highest valuation, there is a unique consumers-specific equilibrium. If more than one consumer group has the highest valuation, there are multiple consumers-specific equilibria. Second, a consumer group expending a positive outlay is one of the consumer groups with the highest valuation. There are consumers-specific equilibria, however, at which some of the consumer groups with the highest valuation expend zero outlays. A consumer group whose valuation for the prize is less than some other group's expends zero outlays and therefore is a free rider. Third, the outlay of the consumer groups is constant across the consumers-specific equilibria. Finally, the outlay of the consumer groups at the consumers-specific equilibria, [-square root of[v.sub.1][[Beta][X.sub.1]] - [Beta][X.sub.1], is equal to consumer group 1's best response to [X.sub.1] when consumer group 1 is the only consumer group competing against the firms in the first stage.

Now it follows immediately from Lemmas 2 (with y replaced by Y) and 11 that the full game has multiple subgame-perfect equilibria and the outlay of the firms and that of the consumer groups are constant across the subgame-perfect equilibria. Lemma 12 also follows from the lemmas.

LEMMA 12. The first-stage outlay of the firms and that of the consumer groups in the subgame-perfect equilibria of the full game are equal to those resulting when only one firm and consumer group 1 are allowed to choose their outlays in the first stage.

Let [Mathematical Expression Omitted] and [Y.sup.**] be the first-stage outlay of the firms and that of the consumer groups in the subgame-perfect equilibria of the full game. According to Lemma 12, we can obtain them by solving a reduced game in which firm 1 and consumer group 1 compete to win their first-stage prizes. And they are equal to firm l's and consumer group 1's outlays, respectively, at the Nash equilibrium of the reduced game. Since the reduced game is virtually the same as that in Section III, we obtain its Nash equilibrium by replacing v with [v.sub.1] in Lemma 4: firm 1 expends [Beta][v.sub.1][T.sup.2]/[([Beta]T + [n.sup.2][v.sub.1]).sup.2], and consumer group 1 expends [Mathematical Expression Omitted]. Therefore, the first-stage equilibrium outlay of the firms and that of the consumer groups are [Mathematical Expression Omitted] and [Mathematical Expression Omitted].(12) Note that they are equal to the first-stage equilibrium outlay of the firms and that of the consumer group, respectively, in the original RCS contest. More precisely, if we let v = [v.sub.1], then [Mathematical Expression Omitted] and [Mathematical Expression Omitted] (see Lemma 5). This implies that the results in Section IV still hold when there are multiple consumer groups in the RCS contest[middle dot] The only change to make is to replace v in Section IV with the highest of the consumer groups' valuations [v.sub.1].

We obtain this interesting result because the prize for each firm and that for each consumer group in the first-stage competition are group-specific public goods, each player's marginal cost is constant, and all the players have the same marginal cost.

VI. CONCLUSIONS

This paper has compared the social costs of monopoly in the RCS contest with those in the rent-seeking contest. The RCS contest is a two-stage game in which firms and consumer groups first compete to win their prizes, an unregulated monopoly for each firm and a regulated monopoly for each consumer group, and then after knowing the government's decision on the form of the monopoly, the firms compete against each other to win the monopoly. The rent-seeking contest is a game in which there is no consumer group engaging in CS-defending activities and only the firms compete against each other to be the unregulated monopolist. We have found that, given just one rent-seeking firm, the social costs of monopoly in the rent-seeking contest are in general less than those in the RCS contest, but given two or more rent-seeking firms, the social costs of monopoly in the rent-seeking contest are in general greater than those in the RCS contest. This result implies that lobbying by consumers generally reduces the social costs of monopoly, compared with those incurred in the case where there is no lobbying by consumers.

I am grateful to Jacques Cremer, Laura Baldwin, Scott Masten, Richard Milam, Tim Perri, Gary Shelley, Gyu Ho Wang, two anonymous referees, and seminar participants at Appalachian State University, Sung Kyun Kwan University, and Kyung Hee University for their helpful comments and suggestions. This research was supported by a 1993 summer research grant of the Department of Economics at Appalachian State University. Earlier versions of this paper were presented at the Sixty-Fourth Annual Conference of the Southern Economic Association, Orlando, Fla., November 1994 and the 1995 Annual Conference of the Korean Econometric Society, Seoul, Korea, December 1995.

Baik: Associate Professor, Department of Economics, Sung Kyun Kwan University, Seoul 110-745, South Korea, Phone 82-2-760-0432, Fax 82-2-744-5717 E-mail khbaik@yurim.skku.ac.kr

1. Tullock [1967] emphasizes that resources expended in lobbying for tariffs and monopolies should be included in the welfare costs of tariffs and monopolies. He also discusses the social costs of theft and defines them as the sum of the efforts invested in the activity of theft, private protection against theft, and the public investment in police protection. Krueger [1974] considers a model of competitive rent seeking in which rents are created by quantitative restrictions upon international trade. She shows, among other things, that the welfare cost of quantitative restrictions includes the cost of rent-seeking activities which equals the value of the rents. Posner [1975], assuming that competition for a monopoly transforms expected monopoly profits into social costs, shows that public regulation is probably a larger source of social costs than private monopoly. Rogerson [1982] shows that some monopoly rents are not transformed into social costs if firms are inframarginal in the competition for the rents. According to the paper, firms can be inframarginal when they have lower fixed organization costs and/or possess incumbency advantages. Fisher [1985], commenting on Posner [1975], says that there are some circumstances in which monopoly rents are transformed into the social costs of monopoly.

2. We assume that if the government regulates a monopoly, it does so by means of price control. Rasmusen and Zupan [1991] address an interesting question: which regulatory policies are preferred by an incumbent monopolist and the government? They consider four types of policies: subsidies, demand stimulation, price/quality controls, and entry barriers. Employing a model in which a monopolist lobbies for government protection against potential entrants, they show that the monopolist and policymakers may prefer entry barriers.

3. This first-stage competition is competition between groups: group of firms and group of consumers. Becker [1983; 1985] studies models in which pressure groups compete for political favors.

4. The government may set the price determined by the intersection of the marginal cost curve and the demand curve, and subsidize the monopoly by an amount equal to its fixed costs. For future use, let us call this regulation the marginal cost pricing regulation and the regulation in the text the average cost pricing regulation. Our analysis holds for both types of regulation. Note, however, that H associated with the marginal cost pricing regulation may be greater than that associated with the average cost pricing regulation.

5. Consider a contest in which n players compete with one another to win a prize. Let [x.sub.i] represent player i's effort level and let [p.sub.i] represent the probability that player i wins the prize. Then a general logit-form probability-of-winning function for player i is given by [p.sub.i]([x.sub.1], ..., [x.sub.n]) = [h.sub.i]([x.sub.i])/[[h.sub.1]([x.sub.1]) + ... + [h.sub.n]([x.sub.n])], where [h.sub.1] through [h.sub.n] are increasing functions. Logit-form functions are used in Tullock [1980], Rogerson [1982], Dixit [1987], Hillman and Riley [1989], Ellingsen [1991], Nitzan [1991], Balk and Shogren [1992], Balk [1994], and Baik and Kim [1997]. Tullock [1980] uses [Mathematical Expression Omitted], where r [greater than] 0.

Another form of probability-of-winning functions is found in perfectly discriminating contests (see Hillman and Riley [1989] and Ellingsen [1991]). In such contests, the player who bids the highest bid wins the prize. More precisely, the probability-of-winning function for player i is given by: [p.sub.i]([x.sub.1], ... [x.sub.n]) = 1/h if player i is one of h players expending the largest effort, where 1 [less than or equal to] h [less than or equal to] n, and [p.sub.i]([x.sub.1], ..., [x.sub.n]) = 0 if [x.sub.i] [less than] [x.sub.k] for some k. Note that this function is the limit of the function in Tullock [1980] as r approaches to plus infinity.

One may prefer logit-form functions to the function used in perfectly discriminating contests in that logit-form functions assume that the player expending the largest effort may not win the prize and player i's probability of winning is increasing in his own effort.

Many economists study contests in which players' abilities and/or their valuations differ. Examples include Rogerson [1982], Dixit [1987], Hillman and Riley [1989], Ellingsen [1991], Baik and Shogren [1992], Baik [1993; 1994], and Baik and Kim [1997].

6. Throughout the paper, whenever we derive a reaction function, we need a remark similar to this. We will omit it for concise exposition.

7. The firms play a game of the noncooperative provision of a public good (see Bergstrom, Blume, and Varian [1986]).

8. Outlays in this section are all equilibrium ones.

9. Fabella [1995] considers a situation in which sellers in a competitive market lobby to be the monopolist, while consumers lobby to maintain the competitive market. He sets up a model in which the sellers and the consumer coalition choose their lobbying expenditures simultaneously. He obtains results quite similar to Propositions 1 and 3 in this paper, and a result opposite to Proposition 2.

10. If the marginal cost pricing regulation does not involve administration costs, it is the social optimum. It maximizes welfare, which is defined as the sum of consumer surplus and the monopoly profits (or losses). In Figure 1, the difference in welfare between the average cost pricing and marginal cost pricing regulation is L. Note that, if fixed costs of a monopoly are zero, then the two types of regulation are the same and therefore [Delta] = 0.

11. A consumer group associated with more consumers may put a higher value on the prize. For example, a consumer group associated with state residents may put a higher value on the prize than a consumer group associated with county residents.

12. The first-stage equilibrium outlay of the firms and that of the consumer groups do not depend on the number of consumer groups, the distribution of valuations across the consumer groups, or the sum of the consumer groups' valuations.

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