Working Paper No. -2-3 Can By-product Lobbying Firms Compete? Paul Pecorino
February 2 Can By-product Lobbying Firms Compete? Paul Pecorino* Department of Economics, Finance and Legal Studies Box 87224 University of Alabama Tuscaloosa, AL 35487-224 Phone: 25-348-379 E-Mail: ppecorin@cba.ua.edu Abstract Olson (965) has argued that one way large groups overcome the free-rider problem is through by-product lobbying. The by-product firm sells a private good to potential members of the interest group and finances lobbying with its profits. It has been argued that by-product lobbying firms cannot survive competition with for-profit firms, since this would compete away monopoly rents, leaving the firm unable to lobby. In a model of monopolistic competition, I show that the by-product firm can enter the market, and earn enough profits to exceed the noncooperative level of lobbying. This is true despite the free entry of for-profit firms. A model of Bertrand competition is also analyzed. This paper provide strong theoretical support for the argument that by-product lobbying firms can successfully compete against for-profit firms. *This research was supported by a grant from the University of Alabama's Culverhouse College of Commerce and Business Administration. I would like to thank Todd Sandler and Akram Temimi for providing helpful comments on the paper.
"The by-product theory invites an obvious objection: why should the association be able to charge more than the cost of the services which are appropriable as private goods? If an association seeks to add a charge for collective goods, a rival association which undertook no collective actions could undersell it and there are no barriers to entry into trade associations.. Introduction George Stigler (974: 36) In his classic work, The Logic of Collective Action, Mancur Olson introduces the byproduct theory of lobbying in order to explain how large groups are able to overcome the freerider problem. Under this theory, by-product firms are able to induce potential members of an interest group to join their lobbying organization by providing selective incentives. These selective incentives may be the provision of private benefits if the individual joins the group, or the imposition of private costs to individuals who fail to join the group. Profits from the provision of private benefits can be used for lobbying the government. Thus, lobbying is the byproduct of another activity, the provision of private goods and services. For example, in return for their dues, members of the American Association of Retired Persons (AARP) receive a magazine and group discounts. The AARP may then lobby the government with the profits it makes on the sale of memberships. While he does not use the term by-product lobbying, it should be noted that the discussion of this phenomenon by Moore (96: 3-6) predates Olson. One critique of the by-product theory is that such firms would not be able to survive competition from firms which do not lobby, because it would prevent by-product firms from earning any monopoly rents. This is evident in the opening quote from Stigler (974). In his book Collective Action, Todd Sandler (992: 6) notes this objection in the literature: "Some researchers have expressed doubts concerning the tying of private benefits to collective outputs as a mean of motivating successful collective action. In particular, these authors question whether the collective or public sector can compete against private sector firms that can provide the private benefits 'separately.' These critics recognize that if the collective has a monopoly over the private good provided, then their criticism does not hold. In fact, many private goods that are tied to collective provision problems (e.g., a journal given to members of a learned society or concert tickets given to supporters of a symphony) do, indeed, involve monopoly aspects. In
2 other instances, the joint production of multiple benefits may involve a technology of supply in which the private output may not be separated from the associated collective output." (footnote omitted) In this paper, I will show that by-product firms can compete with for-profit firms, even when the private good it sells can be sold separately from the public good of lobbying. While some element of monopoly power will prove to be important, I will show that by-product lobbying firms can compete under conditions of monopolistic competition, where there is free entry of for-profit firms into the industry. Thus, by-product firms can compete in a rather common market structure, and the monopoly power requirement is rather weak. Further, the byproduct firm is able to achieve a greater level of lobbying than would occur in the absence of such a firm, i.e., more lobbying will occur than in the noncooperative lobbying equilibrium. This analysis presumes that the private good is primarily consumed by members of the interest group in question. If, for a particular good, there is a large group of consumers who oppose the lobbying efforts of the interest group, then Stigler's critique will hold and the by-product firm will be unable to compete against for-profit firms in this market. I also analyze a model of Bertrand competition and find there are circumstances under which the by-product firm has a weakly dominant strategy to enter the private good market when it plays an entry game against a for-profit firm. In this case, there is a presumption that the byproduct firm will establish itself as a seller of the private good and the for-profit firm will stay out of the market. An important related work is Posnett and Sandler (986), who show that a charity in a perfectly competitive goods market can charge a premium for a private good which reflects the private valuation of charitable activity, a public good. Their work focuses mainly on the behavior of the donor and does not consider the types of market structures analyzed in this paper. Moe (98:233-244) contains an analytical treatment of an individual's decision to join an association, based on its provision of selective incentives and its lobbying activities. Morgan (997) is another important related work. He shows that lotteries may be an effective means of increasing voluntary provision of public goods. The fact that the lottery proceeds go to fund a public good
3 valued by consumers increases the willingness of consumers to pay for lottery tickets. Similarly, in this paper, the fact that consumers know that profits on the sale of the private good are used for lobbying by the by-product firm increases their willingness to pay for the private good. One major difference with Morgan (997) is that the role of competing firms is central in this paper. The analysis here is distinct from the related literature on impure public goods in which the technology of production (or preferences, i.e., the warm glow effect) leads a public good and a private good to be jointly supplied. For an overview of this literature, see the work of Sandler (992) and Cornes and Sandler (996). 2 By contrast, the private good in this paper may be provided separately from the public good of lobbying and lobbying may be provided separately from the private good. 2. The Model There are n identical consumers who are potential members of the interest group. While this ignores other consumers, this case applies when the private good sold by the by-product firm is demanded only by potential members of the interest group. For example, if the interest group consists of farmers, the private good may be inputs or information specific to farming. The aggregate quantity demanded of the private good is denoted Q, with the market inverse demand curve described by P(Q), where P'(Q)<. 2. Noncooperative and Cooperative Lobbying Equilibria Let F(L) describe the dollar value of the benefits received by each of the n consumers/interest group members when L dollars are devoted to the lobbying process, where F'(L) >, F''(L) <, F'()>, F'( )= and L. This benefit from lobbying is nonrival and nonexcludeable. The analysis here is partial equilibrium. We will ignore any income effects stemming from lobbying on the inverse demand curve introduced above. Similarly, there are no income effects on the demand for the policy put in place by lobbying. Formally this requires the assumption of quasi-linear preferences.
4 To understand what happens when the by-product firm does not enter the market for the private good, it is necessary to derive the first order conditions for lobbying contributions in a noncooperative game. We will also derive the conditions for the cooperative outcome which could be achieved if full cooperation were somehow possible. Let l i be the lobbying contribution n of individual i, with L = i = l i. Since the benefit of lobbying is measured in dollars, in the noncooperative equilibrium, each consumer will Max F( L) l, where l. The solution to this problem implies that for all n consumers, F '( L) =. () li i i Let the level of lobbying under the noncooperative equilibrium be L N and the associated individual payoff be F N. 3 The assumption that F'()> guarantees an interior solution. As is N N typical, in the absence of income effects, dl dn = df dn = (Cornes and Sandler, 996: 6-63). 4 Thus, the noncooperative level of lobbying is the level that a single individual finds it privately optimal to lobby for. Note that the analysis so far ignores the potential contributions of the person who is the manager of the by-product firm. This will be discussed further in section 3. Under the cooperative outcome, the joint welfare of the members of the lobbying group is maximized. This problem is to Max n F( L) L, and the first order condition implies L nf '( L) =. (2) From equation (2) we get the cooperative level of contributions L C and the individual payoff under full cooperation F C. Equation (2) and the restrictions on F(L) imply that both L C and F C are increasing in group size n. 2.2 The Demand Curve Faced by the By-product Firm The by-product firm is committed to using its profits to lobby on behalf of the members of the interest group (see the discussion in section 3). Consumers recognize that to the extent they pay a price above marginal cost, they are purchasing some lobbying services along with the private good. Further, it is assumed that consumers correctly value these lobbying services at the
5 margin. Because of this, the by-product firm faces a demand curve which reflects an increased willingness to pay on the part of consumers. In this subsection, we will derive the demand curve facing the by-product firm. For any given quantity on the horizontal axis (see Figure ), let P (Q) be the willingness to pay for the private good Q from the underlying demand curve, denoted D. Let P (Q)=P (Q)+k be the price that consumers are willing to pay for the private good cum lobbying, where k is used to denote the premium (relative to the demand curve D ) consumers are willing to pay. (While k is an endogenous variable, its functional dependencies will be suppressed throughout the paper.) If the firm charges a price of P (Q) and has a constant marginal cost of C, then the lobbying benefits to the consumer from the marginal purchase are ( df Π) dπ)( dπ dq) ( =F'(Π)(P - C), where Π=(P -C)Q=(P +k-c)q is the by-product firm's profits. 5 Thus, we have P = P + k = P + F'( Π)( P + k C). (3) The price the by-product firm can charge for a given quantity Q is a function of its profits, because these profits are used to provide lobbying benefits to the consumer. Rearrange (3) to get ( P + k C) Q) k = F'( Π) = F' P + k C. (4) The left-hand side of (4) is the percentage of the final mark up of price over marginal cost which is due to the lobbying benefit perceived by consumers. If, for example, the left-hand side of (4) equaled.3, then 3% of the total mark up of price over marginal cost reflects the lobbying benefit perceived by the consumer making the marginal purchase of good Q. By moving down the underlying demand curve D we can derive the demand curve D which faces the by-product lobbying firm. What can we say about the properties of demand curve D? (All references are to Figure. Note that while D is depicted as being linear in all diagrams, the results of the paper do not assume linearity of demand.) We assume that all fixed
6 costs represent sunk entry costs which the firm pays for in full before entering the industry. In this way the firm's profits for lobbying will simply be revenues minus variable costs. 6 First, D will generally either lie above D, or it will coincide with D. This reflects the increased willingness to pay by consumers for the private good cum lobbying relative to the private good alone. Also, we know that D must be downward sloping. Consider points g, h and j in the figure. A pair of points like g and h cannot lie on the same demand curve. At point h, the price P is the same as at g, but output is higher, and therefore profits are higher as well. The left hand side of (4) is greater since k is greater and P is smaller, and the right-hand side is lower (recall F''(Π) < ). Thus, if equation (4) holds at point g, it cannot hold at point h. By the same reasoning, it cannot hold at a point like j either. To further understand the nature of D is helpful to once again rewrite (3), this time as F' ( Π) k = ( P C). (5) F'( Π) Holding Π constant, k is increasing in the price-cost margin P -C. 7 The larger this margin, the greater the percentage of each dollar the consumer pays which goes to lobbying. Since the consumer values the lobbying, this increases the premium, the consumer is willing to pay above the price P. Starting from point a, profits are initially, because no units of good Q are sold. Moving down the demand curve, profits initially rise and k falls for two reasons. As Π rises, F'(Π) falls, and as we move down the underlying demand curve, P -C falls. Initially then we know the two demand curves are drawing closer together. Once we pass the point of profit maximization, however, further movements down the demand curve will lower profits and raise F'(Π). Since P -C continues to fall, it is not certain how the distance between the two curves changes past the point at which profits are maximized. If we are operating on the underlying demand curve at a point above e, we have P >C and from (4), we have k>. Taken together, this implies that the left-hand side of equation (4) is
7 a positive fraction. Recalling that F '( Π) =, under the noncooperative lobbying equilibrium, we can conclude that for all points to the left of e on D, the corresponding points on D will be associated with lobbying expenditures (which equal Π) above the noncooperative level. However, this analysis does not us allow to determine whether a particular point will be associated with a level of provision which is above or below the cooperative level derived from equation (2). Point e on the underlying demand curve and point f, the corresponding point on D are of particular importance. At point e, P =C (price equals marginal cost on the underlying demand curve), and from equation (3), we have = F '( Π). (6) This is the same condition, given in (), which is associated with the noncooperative equilibrium in lobbying. Thus, if the by-product firm operates at point f on D, the profits it obtains will just allow it to replicate the noncooperative level of lobbying. This is analogous to a result in Posnett and Sandler (986). The lobbying expenditure by the by-product firm must be greater than or equal to zero. At point b on demand curve D (or just below it), the by-product firm will earn negative operating profits. 8 If the firm has negative profits, it cannot lobby and the marginal purchase has no impact on lobbying. Since purchase of good Q is no longer buying any lobbying services, the willingness to pay by consumers is now given by the underlying demand curve D. This is indicated on the diagram by the vertical dashed line between points b and d. Note that for point e and points to the right of e, there is not a unique mapping between quantity and price. For example, at point e, Q N units will be demanded both at the price P N (this is discussed above) and at the price C from the demand curve D. If a price C is charged, profits per unit are zero and no lobbying takes place. As a result, consumers are placed back on the underlying demand curve D. We assume throughout that the by-product firm is a price setter interested in maximizing its profits; such firm would never choose to announce a price of C
8 rather than the corresponding price on D. A similar argument holds for all points between f and b on D. Note further, that in the analysis below, the by-product firm never chooses to operate at a point strictly to the right of point e. The analysis of D is summarized as Result. Result : The demand curve facing the by-product firm, D, has the following properties: i. The demand curve D is downward sloping. ii. The demand curve D either lies above the underlying demand curve D or it coincides with D. iii. At all prices P >C, the profits of the by-product firm allow it to exceed the noncooperative level of lobbying. iv. At P =C, the profits of the by-product firm allow it to achieve the noncooperative level of lobbying. Beyond the features discussed above, no further conclusion about the general shape of D should be drawn from the diagram in figure. In particular, D is generally not linear, and the results in the paper do not assume that it is linear. 2.3 A Numerical Example Consider the following numerical example which demonstrates how the relationship between P and P is established. For the purpose of this example only, I will assume that F' ( Π) is a constant. Let F '( Π) =.3, P =, and C = 7. From equation (5), we have k=$2.86 and therefore P = $2.86. The marginal consumer buys the private good for $2.86, and he values this good at $. In addition, $42.86 (=2.86-7) of profit from the sale goes to lobbying which the consumer values at $2.86 (=.3x42.86). This is turn equals the premium k. As long as consumers place some positive value on lobbying, the willingness to pay for the private good from the by-product firm will exceed the willingness to pay for the same good from a firm which retains for itself the profits from the sale.
9 3. The Bertrand Model I will develop the Bertrand model first. This is a rather stark model in many ways, but several of the results derived here will prove useful in the discussion of monopolistic competition in section 4. The utility of the by-product firm manager is identified with the number of dollars allocated to lobbying, whether these are contributed by herself or others. The objective of the manager of the by-product firm is to maximize the number of dollars allocated to lobbying. If the by-product firm enters the market for good Q, the manager will set prices to maximize "profits". This in turn will maximize lobbying expenditures and the utility of the manager. As noted earlier, the by-product lobbying firm has constant marginal costs C, and faces sunk capital costs S which must be incurred before the firm can enter the industry. We assume that the firm is able to pay S in cash at the time of entry, so that we may identify lobbying by the firm with the excess of revenues over variable costs. We will identify these operating profits simply as profits. There is one potential competitor which has an identical technology of production and produces a good which is a perfect substitute for the private good produced by the by-product firm. However, the other firm is for profit, and therefore does not undertake any lobbying on behalf of the interest group. This potential competitor has also raised the sunk cost of entry S and we will identify its profits with operating profits. If the firm does not enter, it earns a payout of S. The objective of the for-profit firm is to maximize profits. To assess the ability of the by-product firm to lobby in the face of competition or potential competition, we will analyze the firms' payoffs in a 2x2 game in which each firm has two strategies: enter the market or stay out. If both firms enter the market, they will engage in Bertrand competition. The structure of this game is as follows:. The firms simultaneously decide whether to enter the product market.
2. If the for-profit firm enters, it posts a price for the private good. If the by-product firm enters, it posts a price for the private good cum lobbying services. If both enter, these prices are posted simultaneously. 3. Consumers decide how much of the private good to buy from each firm at the posted prices. 4. The by-product firm lobbies the government. The payoffs for the firms in the entry game are displayed in the 2x2 by-matrices given in Tables and 2. If the by-product firm does not enter, the amount of lobbying is determined by a noncooperative game in which the interest group members and the firm manager simultaneously make lobbying contributions. Because the manager of the by-product firm acts to maximize lobbying expenditure, the noncooperative lobbying equilibrium must involve this manager contributing S to the lobbying effort, where S is the initial capital raised by the manager to meet the sunk costs of entry into the goods market. Recall that on their own, the members of the interest group would contribute L N, which is obtained from equation (). If N S < L, spending by the manager merely crowds out spending by other members of the interest group for due to the absence of income effects on demand for the policy produced by lobbying. The owner of the N N by-product firm enjoys the utility U = L. If N S > L, then other members of the interest group N N spend nothing on lobbying and the manager of the by-product firm enjoys utility U = S > L. Because the manager is, in effect, the highest demander for the policy favored by the interest group, she is subject to the 'exploitation of the great by the small' first pointed out by Olson (965). When N S < L, this 'exploitation' is limited by the funds available to the manager. For example, if equation () is satisfied for L N = and S = 8 under a noncooperative lobbying equilibrium, the by-product firm manager contributes 8, members of the interest group contribute 2 and total is spent on the lobbying effort. If S = 2 instead, then the firm manager contributes 2 and members of the interest group contribute nothing (since F'(2)<
if F'() = ). Thus the total lobbying effort is the 2 contributed by the by-product firm manager. If both firms stay out of the market, the for-profit firm earns a payout of S, while the byproduct firm manager gets utility U N. In Table, it is assumed that while in Table 2 it is assumed that N S > L, so she receives U N = S. N S < L N N, so she gets U = L, If the for-profit firm enters and the by-product firm stays out, then it earns the standard monopoly profits obtained by setting marginal revenue equal to marginal cost. Denote the associated price, output and profits, P M, Q M and Π, where Π S. The last inequality guarantees that the for-profit firm prefers to enter the market if it is the only firm to do so. The by-product firm will enjoy utility U N =L N or U N =S. If the by-product firm enters and the for-profit firm stays out, the by-product firm will also earn a monopoly profit all of which will be used to lobby on behalf of the interest group. When the by-product firm alone enters the market, its manager enjoys utility U M = Π and the for-profit firm earns a payout of S. The behavior of the by-product firm when it acts as a monopolist is summarized as Result 2. M M > M Result 2: When the by-product firm acts as a monopolist, it will i. Set output Q =, the same level as the for-profit firm when it acts as a M Q M monopolist. ii. Set price P > ; M P M iii. Earn profits Π >. M Π M A proof of (i) is provided in the appendix, but the intuition is as follows: In maximizing its profits, the by-product firm must assess how altering its output level affects profits deriving from sales based on the underlying demand curve D and profits deriving from the premium consumers are willing to pay due to the by-product lobbying. Denote this total premium kq. In choosing Q =, the by-product firm clearly maximizes profits with respect to the underlying M Q M
2 demand curve, but it turns out that this also maximizes kq. From (5), k is proportional to P -C. Thus, the total premium paid due to by-product lobbying is proportional to (P -C)Q. Since this is the profit function of the for-profit firm, it is maximized at revenue, this implies that at marginal costs being constant. Q M. Letting MR denote marginal Q M we have MR =MR =C. Note that this result does not depend Part (ii) of the result follows from part (i) and the fact that D lies above D. Part (iii) also immediately follows since the by-product firm produces the same level of output as the for-profit firm when it acts as a monopolist, but charges a higher price. Next, consider the outcome of the Bertrand game when both firms enter the market. This is depicted in Figure 2. The for-profit firm sets P =C, while the by-product firm sets price equal to the corresponding point on D. Denote this price P N since the profits which correspond to it support the noncooperative level of lobbying (Result, part iv). Importantly, all Q N units sold are sold by the by-product firm. If fewer units than this are sold by the by-product firm, its profits will drop and F'(Π) will rise above. If this were true, however, consumers would strictly prefer to buy from the by-product firm. Thus, when both firms enter, the byproduct firm charges P N, sells Q N and earns profits Π N, while the for-profit firm charges C and sells no units and earns zero profits. It is worth noting in general that when the by-product firm charges P and the forprofit firm charges the corresponding price P on the underlying demand curve D, consumers do not become indifferent between purchases from the two firms until the by-product firm sells all Q units indicated by the demand curve D. When Q < Q, consumers strictly prefer to buy the private good from the by-product firm. This follows since Π is increasing in Q and F''(Π)<. (See the discussion of this issue in Posnett and Sandler (986: 23).) If the by-product firm raises its price above P N, it will lose sales to the for-profit firm, but it will not lose all of them. Consumers have the option to buy from the for-profit firm at the price of C. The premium k must rise so that consumers at the margin are indifferent to buying from the two firms. If we let P B >P N be the price charged by the by-product firm, then we must have C + k = P = C + F' ( Π)( P C). The last equality follows from (3), where C has replaced P. B B
3 This equation implies that F'(Π)=. Thus, the by-product firm can charge an arbitrary price above P N and earn profits equal to the noncooperative level of lobbying L N, but no more. 9 This means that P B P N is a best response to a price of C by the for-profit firm. However, P B >P N is not consistent with equilibrium because C will not be a best response by the for-profit firm to such a price. (The for-profit firm could raise its price above C, retain some customers and earn positive profits.) If the for-profit firm charges any price above C, it will not attract any customers given the price P N set by the by-product firm. Since the for-profit firm will take a loss on all units sold if it charges less than C, the for-profit firm will not undercut a price of P N posted by the by-product firm. Thus a price of C by the for-profit firm is a best response to a price of P N posted by the byproduct firm. It is established in the appendix that no other pair of prices will constitute an equilibrium. When both firms enter, the utility of the by-product firm is Π N = L N, and the for-profit firm earns a payoff of. Result 3 follows from the payoffs we have derived which are displayed in Tables and 2. Result 3: i. If N S < L, the strategy "enter" weakly dominates the strategy "stay out" for the byproduct firm. Assuming that weakly dominated strategies are never played, in the unique equilibrium outcome of the game, the by-product firm enters the market and the for-profit firm stays out. ii. If N S > L, both pure strategy Nash equilibria remain possible outcomes of the game. Under one, the by-product firm enters and the for-profit firm does not, while under the second, the for-profit firm enters and the by-product firm does not. Part (i) may be verified from Table, where it is seen that the by-product firm has a weakly dominant strategy to enter the market. Part (ii) may be verified from Table 2, where it is seen that
4 neither firm has a dominant strategy to enter since this table covers the case S>L N. As a result, both firms are better off staying out, if the other firm has committed to entry first. Note under (ii) that the game also has a mixed strategy Nash equilibrium where each firm enters the market with some probability. If S>L N, then there is no presumption that the by-product firm will be able to establish itself as the lone entrant, though there is no presumption against the firm either. When S<L N, the by-product lobbying firm has a weakly dominant strategy to enter the market, and so there is a presumption that this firm will be able to establish itself as the lone entrant. This will allow it to lobby with profits which exceed the level of lobbying in the noncooperative equilibrium, L N. If the lobbying stakes per consumer are small, then the noncooperative level of lobbying will be small; recall that the noncooperative level of lobbying is the level a single individual (or firm) finds it privately optimal to lobby for. This suggests that it would be unlikely for the condition S<L N to be satisfied, when the stakes per person are small. On the other hand, if lobbying were on behalf of a concentrated industry with large firms, then the lobbying stakes per firm would be high and the condition is more likely to be satisfied. This suggests an advantage for a concentrated industry in having a by-product lobbying firm established on its behalf. When the by-product firm is the lone entrant in the market, an increase in the size of the interest group will shift out the demand curve facing this firm. As a result, equilibrium profits and lobbying will increase. Thus, we should observe a positive relationship between group size and lobbying. However, when stakes per person are small, the analysis in the paragraph above suggests that it is less likely that a by-product firm will be able to establish itself. If stakes tend to fall with group size, larger groups may be less likely to be represented by a by-product lobbying firm than small groups. The overall effects of an increase in group size on the extent of lobbying appear to be ambiguous. 4. Monopolistic Competition 4. The Model
5 The results of the Bertrand model are rather stark; because competition is so fierce when there is entry, in equilibrium only one firm enters the market. As an alternative, consider a model of monopolistic competition where firms produce differentiated goods within a commodity category which are imperfect substitutes for one another. Assume, as in Dixit and Stiglitz (977) and Krugman (979), that within the commodity category, all potential products enter into the utility function symmetrically. In particular, suppose that the subutility function V which describes preferences within the commodity group takes the form V = v( ), where v' > and v'' < (Krugman 979: 47). In the context of such a model, we will consider the behavior of one by-product lobbying firm interacting with for-profit firms, where there are a large number of for-profit firms which are potential entrants into the industry. The other assumptions of the model remain in tact. In particular, there are no income effects on demand caused by the policy put in place through lobbying. As before, there are sunk costs of entry S and constant marginal costs of production C which are common across all firms including the by-product firm. If two firms produce the exact same variety, they play a Bertrand game against one another. For two for-profit firms, this will result in marginal cost pricing under which neither firm will recover its fixed costs of entry. If the by-product firm and a for-profit firm produce the exact same variety, the outcome of the Bertrand game will be the same as the game analyzed in section 3. In the outcome of this game, the for-profit firm engages in marginal cost pricing, makes no sales and fails to recover its sunk costs of entry. Since there are a large (potentially infinite) number of varieties which may be produced, no firm entering the market will wish to produce the exact same variety as an existing firm. Thus in equilibrium, each firm, including the by-product firm, produces a distinct variety. Each firm takes the price posted by all other firms as given. Aside from by-product lobbying by the one firm, firms are otherwise symmetric with respect to cost structure and the way in which the commodity they produce enters consumer's utility functions. Thus, in i Q i
6 equilibrium, each for-profit firm faces an identical demand curve. Since we have a fixed cost and constant marginal cost, average cost is decreasing in output and approaches C asymptotically. All for profit firms set marginal revenue equal to marginal cost to determine the profit maximizing price and output, P MC and Q MC. In equilibrium, for-profit firms will enter until they earn zero profits at the point of profit maximization. This will occur at a point of tangency between the average cost curve and the demand curve D. This is depicted in Figure 3 at point a. At point a, for the for-profit firms, P equals average cost AC and exceeds marginal cost C. As before there will be a demand curve D facing the by-product firm which reflects consumer's willingness to pay for a particular product variety cum lobbying services. Since, by symmetry, the demand curve D which underlies D is identical to those faced by the for-profit firms, we know from Result 2 that the by-product firm will set Q = and P >. Thus MC Q MC MC P MC while the for-profit firms will merely recover their sunk costs, the operating profits of the byproduct firm will exceed its sunk costs. Further, since P MC > C, at the point where the byproduct firm produces, we know that its profits in equilibrium allow it to exceed the noncooperative level of lobbying which is achieved when the by-product firm stays out of the market (Result, (iii)). As a result, the by-product firm strictly prefers to enter the market rather than stay out. This analysis is summarized as Result 4. Result 4: In the model of monopolistic competition where (aside from by-product lobbying by one firm) firms are symmetric, in equilibrium, the by-product lobbying firm can earn profits which exceed its fixed costs of entry. In addition, the firm's profits allow it to exceed the noncooperative level of lobbying which results when the firm stays out of the market. As a result, the by-product manager strictly prefers to enter the market for the private good. So far the analysis has concentrated on a case where the by-product firm sells a good which is only of interest to members of the interest group. Suppose the by-product firm
7 attempted to sell a commodity which was of interest to consumers more generally, including those who were not members of the interest group. Suppose further that the second group of consumers placed a negative value on the lobbying efforts of the by-product firm because the policy put in place by such lobbying reduces their utility. The negative valuation placed on lobbying by these consumers would shift down their demand for the good sold by the by-product firm. If this second group of consumers were large, the market demand curve faced by the byproduct firm in the aggregate might be lower than the demand curve faced by for-profit firms. Since the for-profit firms just recover their sunk costs in equilibrium, in this case, the by product firm would not recover its sunk costs and its manager would strictly prefer to stay out of this market. This suggests that it may be important for by-product firm to sell goods which are generally only of interest to members of the interest group it represents. Presumably this is more important if the by-product firm represents a group, such as industrial polluters, which may be viewed unfavorably by the population at large. Stigler's (974) argument that by-product firms cannot compete should be valid when there are a large group of consumers of the good it sells, who oppose the lobbying efforts of the by-product firm. On the other hand, this issue should not be of importance for the provision of a pure public good, where no large group receives disutility from provision of the public good. 4.2. Comparative Statics In this section, we will analyze the comparative static effects of changes in group size and changes in the fixed costs of entry. 4.2. Changes in Group Size A central question of Olson's Logic of Collective Action is the relationship between the size of the interest group and its ability to provide a public good, in this case lobbying. As the group size n increases, the demand curve facing each firm shifts out. This raises profits and attracts new entry which pushes the demand curve down. The new entry also flattens the demand curve facing each firm. The new firms produce new varieties which are substitutes for the goods
8 produced by the existing firms. As the number of substitutes increases, each firm faces a more elastic (flatter) demand curve. In the new equilibrium, it must be the case that for-profit firms again produce at a tangency between their demand curve and the average cost curve to ensure zero profits. Thus, the effect of an increase in group size is to cause for-profit firms to move down the average cost curve. This is shown in Figure 4. When group size increases, the equilibrium moves from point a to point b on the average cost curve. The profit maximizing output of each firm rises as a result from Q MC to firm will also rise from ' Q MC. As established in Result 2, the profit maximizing output of the byproduct Q MC to ' Q MC. To determine the effect of a larger market (resulting from an increase in group size n) on the by-product firm, we merely need to move down the average cost curve to obtain price output pairs for the for-profit firms. The by-product firm will produce the same quantity as the for-profit firm, but will charge a premium k above the price these firms charge. This premium may be determined from equation (4). With a sunk cost S and constant marginal cost C, average cost AC = C + S/Q. Since P=AC in equilibrium, the price quantity pairs for the for-profit firms may be found from P + = C S /Q. (7) Substituting (7) into equation (4) and rearranging terms we get Q k S + Q k = F'( Q k + S), (8) where the term in parentheses on the right-hand side is the argument of the F' function. Profits for the by-product firm are Π = ( P C) Q = ( P + k C) Q. Use (7) to write this as
9 Π = S + kq. (9) An increase in Q represents a move down the average cost curve which would occur as a result of an increase in the size of the interest group. The relevant comparative static is dπ / dq. While noting that S is a constant, differentiate (8) with respect to Q to get dk / dq Q = k /. Differentiate (9) with respect to Q and substitute in this expression to find d Π / dq = k + Q ( dk / dq ) =. () As we move down the average cost curve, the price-cost margin falls, and this reduces the premium k that consumers are willing to pay over the price dictated by the underlying demand curve D. On the other hand, this smaller margin per unit is spread over a larger number of units. As equation () shows, these two effects exactly offset one another. As a result, the byproduct firm's profits and level of lobbying are independent of the size of the interest group. The analysis is summarized as Result 5. Result 5: The level of lobbying by the by-product firm is independent of the size of the interest group, holding sunk costs and stakes per member constant. The reason is that the profits of the by-product firm are independent of the firm's equilibrium level of output under monopolistic competition. Since the optimal level of lobbying (from the interest group's perspective) rises with group size, Result 5 implies that the gap between the actual and optimal levels of provision is increasing in group size. Result 5 will hold as long as stakes per interest group member (represented by the F function) are constant. When the stakes per person are smaller, the premium consumers are
2 willing to pay falls. If stakes per person tend to be smaller in larger groups, then there will be less lobbying in large groups, other things held equal. 4.2.2 Increases in the Fixed Cost of Entry Now consider the effect of sunk entry costs on the equilibrium profits of the by-product firm. From (), we see that for a given S, the product of kq is a constant. Thus, when we compute the comparative static with respect to S, we will treat the product kq as a single variable. Differentiate (8) with respect to S to get ( F'( Π) ''( Π) ) d ( kq ) / ds = ( F'( Π) + F''( Π)) F. () Differentiate (9) with respect to S and substitute from () to get ( F'( Π) F''( Π) ) > d Π / ds = + d( kq ) / ds =. (2) Note that F '( Π) F''( Π) > since F ''( Π) < and F '( Π) <. The last inequality holds because we are operating on a region of the demand curve D where P > C. In this region, the level of lobbying exceeds the noncooperative level (Result, (iii)) which implies F '( Π) < ). There is a sense in which industries with high sunk costs of entry are less competitive; holding Q constant, higher sunk costs imply a higher price-cost margin in equilibrium. The higher pricecost margin increases the premium k the by-product firm can charge for its product (equation (5)). At S=, we have costless entry and perfect competition. From (8) we get F'(Π)=; under conditions of perfect competition, the by-product firm is again only able to support the noncooperative level of lobbying with its profits (equation ()). It has been assumed that sunk costs are paid up front, but in equilibrium all firms recover these sunk costs which are then available for the by-product firm to lobby. Thus, it may be objected that the comparative static in (2) is being driven by this assumption. In other words, it
2 may be that the by-product firm lobbies more in equilibrium simply because we assume that is has raised a greater amount of capital at the start of the process to meet the higher sunk costs of entry. There are two ways to address this point. First, note from (2) that if F'(Π)+F''(Π)>, then the increase in profits is greater than the increase in sunk costs (dπ/ds>). Second, consider the alternative assumption that sunk costs are not paid up front, but rather financed out of sales revenues. Then profits become Π = ( P + k C) Q S = kq, (3) where use has been made of (7). Use (3) to rewrite (8) as Q k S + Q k = F'( Q k). (4) Differentiate (3) and (4) to get ( F'( Π) F''( Π) ) > d Π / ds = d( kq ) / ds = F'( Π) /. (5) Thus, the result that lobbying by the by-product firm is increasing in S is not sensitive to the assumption about how sunk costs are financed. 2 The analysis is summarized as Result 6. Result 6: The level of lobbying by the by-product firm is increasing in the level of the sunk entry costs S. 5. Conclusion
22 The results under Bertrand competition indicate that the by-product firm will have a dominant strategy to enter the market under some circumstances. However, the assumptions of the Bertrand model lead to very stark outcomes; there is so much competition when both firms enter the market, that in equilibrium only one firm enters. Under monopolistic competition, there may be a large number of competitors for the by-product firm, each producing a good which is an imperfect substitute for the good it produces. In this setting, there is a presumption that the by-product firm can successfully enter the industry and earn profits in excess of both its sunk costs and the noncooperative level of lobbying. These results strongly suggest that by-product firms can compete with for profit firms under conditions of free entry and still earn "rents" which allow the firm to lobby. Even under perfect competition (zero entry costs), the by-product firm can achieve the noncooperative level of lobbying, though we would not expect to see the emergence of byproduct firms if this were the best they could achieve. To achieve more requires some monopoly element, and this has long been recognized as crucial to by-product lobbying (Sandler, 992: 6). However, the monopolistic element inherent in monopolistic competition is rather weak since firms exists which produce goods in the same commodity group as the by-product firm; most people would not describe firms in such industries as monopolies. These results indicate that the market conditions required for by-product lobbying are not very special and that there is a theoretical presumption that such firms can survive in the marketplace. Further, this is true under a model of narrow rationality. If ideological warm glow effects exist, then the ability of byproduct firms to compete is further enhanced. 3 However, it may be important for by-product firms to sell goods which are only of interest to potential members of the interest group. Based on survey data, Moe (98: 2-28) shows that members of interest groups strongly value the selective incentives provided by the interest groups they belong to. In addition, the percentages of interest group members who believe their contributions have either a "big effect" or "some effect" on the groups political success or failure ranges from 62% to 7%. These results suggest that the specification of the model in this paper is plausible; interest group
23 members seem to value both the private goods they receive and the lobbying services purchased with their membership dollars. In the Bertrand model, if the by-product firm is the lone entrant, an increase in the size of the interest group n will shift out the demand curve which faces the firm. As a result, its monopoly profits and lobbying will both increase. In the model of monopolistic competition, an increase in n, holding the stakes per member and sunk costs of entry constant, will have no effect on equilibrium profits or lobbying by the by-product firm. Taken together, these results suggest that there is no presumption that larger groups have smaller provision of the public good of lobbying, though the latter result indicates that the underprovision relative to the optimum will increase with n. When the stakes per person are higher, then consumers are willing to pay a bigger premium k, other things equal. Thus, by-product firms which lobby on behalf of groups where the stakes per person are higher should attain a higher level of lobbying expenditure than firms which represent groups with lower stakes per person. To the extent that the stakes per person tend to be lower for large groups, these groups would tend to be disadvantaged in the lobbying process, even if lobbying were undertaken through by-product lobbying firms. Thus, the effects of group size on the provision of lobbying are mixed. If stakes per person are constant as group size grows, lobbying provision rises when the by-product firm is a monopoly and is constant under monopolistic competition. If stakes per person fall with group size, lobbying must fall under monopolistic competition, while the effects under monopoly will be ambiguous. In addition, in the Bertrand game, the by-product firm is more likely to have a dominant strategy to enter the goods market when the stakes per group member are high. 4 An analysis similar to the one presented in this paper may be relevant to an understanding of the economics of "green goods". 5 To the extent that consumers value the environmental benefits that certain products may provide (relative to other products), this will raise their demand for the product. Unlike the analysis in section 2.2, however, providing these benefits to
24 the consumer should raise the firm's cost of production. The ability of "green goods" to compete will depend on the magnitude of these two effects.
25 Appendix A.. Monopoly Outputs The purpose of this section of the appendix is to prove part (i) of Result 2 which states that the by-product firm acting as a monopolist will produce the same level of output as the for- profit firm acting as a monopolist ( Q = ). Before proceeding further, it will be useful to note from equation (3) that M Q M P =P +F'(Π)(P -C). (A) For the for-profit firm, profit maximization implies dπ dq = P dp + Q dq C =. (A2) When the by-product maximizes its profits, the first order condition requires dπ dq = P dp + Q dq C = P + F'( Π)( P dp C) + Q dq C =, (A3) where use have been made of (A). We will proceed by assuming the by-product firm is at its profit maximizing level of output and show that this implies that the for-profit firm is maximizing its profit at this same level of output. Formally, this requires showing that at the level of output which satisfies equation (A3), equation (A2) will also be satisfied. Taking the derivative of (A) with respect to Q we have dp dq dp dp dπ dp dp = + F' ( Π) + ( P C) F''( Π) = + F'( Π), (A4) dq dq dq dq dq where the term on the far right follows because profit maximization by the by-product firm implies d Π / dq =. Rearranging terms in (A4) we get dp dq dp = dq F'( Π) (A5)
26 From (A), F' ( P P ) = P C Π. Substitute this plus (A5) into (A3) and simplify to get dp P C P + ( P P ) + Q = C. dq P C This may be rewritten as dp dp P P P + Q C + ( P P ) + Q =. dq dq P C Further simplification reveals dp P C P + Q C =. (A6) dq P C Since the term in brackets is positive, for (A6) (and therefore (A3)) to hold, we must have dp P + Q C =. This implies that when (A3) holds, (A2) must also be satisfied. Thus, the dq same value of Q solves (A3) and (A2), which proves that Q =. M Q M Note that this proof does not require constant marginal costs of production. Let TC(Q) denote the total cost function. Then C can be replaced everywhere by dtc ( Q) dq, and the result will still hold. A.2. Uniqueness of the Bertrand Equilibrium The purpose of this section is to prove in the Bertrand game that when both firms enter, the unique equilibrium is for the by-product firm to charge P N and for the for-profit firm to charge C. The key to the proof is to show that if the for-profit firm charges a price by-product firm will optimally respond by charging a price less than or equal to P F > C P F, the corresponding point on the demand curve D. This is done by showing that the by-product firm will never choose a price, the P B > P F. As a result, the by-product firm will capture the entire market.