Upstream mergers, downstream mergers, and unionized oligopoly.

Chang, Shu-hua

I. Introduction

In a vertical structure, an integrated industry makes more profits than a non-integrated industry, and the consumer price is lower in the case of the former. The objective of vertical integration is to avoid the double price distortion that occurs when each firm adds its own price-cost margin at each stage of production. However, an upstream monopoly can impose a two-part tariff by choosing a franchise fee in such a way that all of the profit is extracted from the downstream firm. By setting the wholesale price equal to its marginal cost, the downstream firm will produce a quantity that is optimal for the upstream monopoly. Based on this observation, Hart and Tirole (1990) argue that double marginalization would offer no motive for integration when a two-part tariff is allowed. Bonanno and Vickers (1988) use a simple duopoly model to show the advantage that a manufacturer will have if it sells its product through an independent retailer (vertical separation) rather than directly to consumers (vertical integration). Vertical separation is profitable insofar as it induces more friendly behavior from the rival manufacturer. Fumagalli and Motta (2001) consider an industry characterized by secret vertical contracts, and examine a benchmark case where there are two vertical chains, in which two upstream manufacturers sell to two downstream retailers, thereby demonstrating that a downstream merger is more welfare detrimental than an upstream merger.

All of the above arguments are based on the assumption that the cost functions of both firms are independent of the specific contractual form. If wages are bargained over at the firm level, then this assumption may not be tenable. Labor market structures may have important effects on imperfectly competitive rivalries between firms. This development calls for more research on the analysis of vertical contracts under the wage bargaining model. Grandner (2000b) analyzes the effects of certain specific institutional wage bargaining arrangements at the firm level on the incentive that firms may have to integrate vertically.

In this paper we build on the previous literature and especially on the paper by Fumagalli and Motta (2001) to further analyze wage bargaining that has important effects on imperfectly competitive rivalries between firms. To do so, we first construct a pre-merger two-vertical-chains case where two upstream firms supply two downstream firms. The manufacturers choose the wholesale prices with which they will supply retailers while downstream competition takes place in terms of quantities. It is also assumed that retailers achieve Nash equilibrium in terms of quantities. We analyze two situations: one is where the manufacturer can charge a franchise fee to the retailer so as to fully extract the retailer's surplus', and the other is where the retailer can charge the manufacturer a franchise fee so as to fully extract the manufacturer's surplus. We show that the wage is jointly determined by the union and the upstream firm through bargaining and will have different impacts on the pre-merger and after-merger cases in the event of an upstream merger or a downstream merger.

In the next section, we describe the model and analyze the case where there are unobservable contracts (i.e. contracts agreed between a retailer and its supplier that cannot be seen by the other agents in the economy), and that are based on the assumption that retailers compete in terms of quantities. In Sections III and IV we analyze the situation where the firm imposes a two-part tariff and chooses the optimal franchise fee, and where the downstream firm transforms the intermediate product into the final one on a one-for-one basis and at zero marginal cost. We confine our attention to the duopoly case where the upstream wage is negotiated through union-manufacturer bargaining and discuss the different effects that the bargaining power of the union has on the upstream and downstream mergers, respectively. In Section V we concentrate on the union's welfare analysis, and the different effects that the bargaining power of the union has on the upstream and downstream mergers, respectively. Finally, in Section VI we present our discussion and conclusions.

II. The Model:

The model is characterized as follows: (a) the product market is imperfect, organized as a duopoly, and the upstream firms produce homogeneous substitutes; (b) downstream competition occurs in terms of quantities; (c) in each upstream firm one distinct union is active; (d) the manufacturers have the same demand functions, production functions, unions' utilities, and bargaining power; (2) and (e) it is assumed that contracts stipulated by a producer and a downstream firm remain unobserved by the rival upstream and downstream firms. We will discuss an industry in which there are two manufacturers or upstream firms, each of which stipulates an exclusive and secret contract with a retailer or downstream firm. We will analyze the equilibrium outputs, prices and profits arising both before and after a merger occurring either in the upstream or the downstream sectors.

All firms are assumed to exhibit constant returns to scale. We also assume that each upstream firm charges a wholesale price in excess of the unit production cost as well as choosing the optimal franchise fee, and that the downstream firms transform the intermediate product into the final good on a one-for-one basis and at zero marginal cost. We confine our attention to the duopoly case where the upstream wage is negotiated through union-manufacturer bargaining and discuss the bargaining power of the union in terms of its having different effects on an upstream merger and a downstream merger.

As noted previously, there are two upstream firms manufacturing a homogeneous product with output levels [q.sub.i] and [q.sub.j], and thus an aggregate output of Q = [q.sub.i] + [q.sub.j]. The market price associated with this output (the inverse demand function) is taken to be P(Q) [equivalent to] P([q.sub.i] + [q.sub.j]). The downstream producers compete in terms of quantities, and the upstream firms exhibit constant returns to scale technologies. To simplify the analysis, each retailer faces a linear inverse demand function: (3)

P = k - a([q.sub.i] + [q.sub.j]). (1)

The production technology of the upstream firm i is

[q.sub.i] = [L.sub.i], (2)

where [L.sub.i] is employment in firm i and is the only variable factor. In the next section, we will compare the equilibrium outcome of an upstream merger with that of a downstream merger where there are secret contracts.

III. Upstream merger

1. The pre-merger case

In the first stage we use a right-to-manage model where the upstream wage is negotiated through union-manufacturer bargaining. In the second stage each manufacturer [U.sub.i] stipulates an exclusive and secret contract with a retaileror a downstream firm [D.sub.i] while simultaneously charging each retailer a wholesale price in excess of the unit production cost and choosing the optimal franchise fee. In the third stage, each retailer chooses its profit maximizing quantity and price. The problem is solved backwards by starting with the third stage.

Suppose that each retailer transforms the intermediate product into the final one on a one-for-one basis and at zero marginal cost. The profit of each retailer can be expressed as:

[[pi].sup.d.sub.i] = (P(Q) - [P.sup.u.sub.i])[q.sub.i] - [F.sub.i], (3)

where [[pi].sup.d.sub.i] is the profit of retailer i, [P.sup.u.sub.i] is the wholesale price charged by upstream firm i, and [F.sub.i] is the franchise fee a downstream firm pays to an upstream firm.

From the first-order condition, we can derive a linear reaction function with a negative slope:

[q.sub.i] = k - a[q.sub.j] - [P.sup.u.sub.i]/2a. (4)

Equation (4) defines the retailer's best reply function [q.sub.i] = [q.sub.i]([q.sub.j], [P.sup.u.sub.i]). It should be noted that, since contracts are unobservable, the best reply does not depend on the wholesale price established by the other upstream producer. In other words, the upstream producer expects that only the output of its own downstream firm will respond to changes in its wholesale price. Therefore, the game is played as if q = ([q.sub.i], [q.sub.j]) and P = ([P.sup.u.sub.i], [P.sup.u.sub.j]) are determined simultaneously and not sequentially.

The technology of an upstream firm is assumed to be one unit of output and is produced by one unit of labor. For a given [P.sup.u.sub.j], the optimal wholesale price [P.sup.u.sub.i] is determined by the following maximization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [[pi].sup.d.sub.i] is the profit of upstream firm i. Solving the above equation yields the solution:

[P.sup.u.sub.i] - [w.sub.i]. (6)

The optimal wholesale prices are decided by the outcomes of the bargaining wage. When upstream wages increase, upstream firms will raise their wholesale prices.

By combining equation (4) with equation (5), we have the Cournot equilibrium,

[q.sup.c.sub.i] = (k - [w.sub.i])/3a, P = (k + 2[w.sub.i])/3.

The next step is to formulate the wage bargaining. Unions are interested in their rents, L(w - [bar.w]), where [bar.w] is the exogenously given reservation wage, while firms are interested in their profits. In the efficient bargaining model, (4) both the wage and employment levels are bargained over simultaneously. However, there is considerable evidence indicating that unions and firms do not bargain simultaneously over wages and employment. Booth (1995) notes that the wage is jointly determined by the union and the firm through bargaining, while the employer retains unilateral control over employment. Thus, we adopt a more plausible right-to-manage union model to derive the bargaining wages of the upstream industry under the Cournot duopoly competing case:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where the superscript "c" expresses the Cournot duopoly competing case, [a.sup.c] is the union's bargaining power, and 1 - [a.sup.c] is the bargaining power of the upstream firm.

If the wages are jointly determined by the unions and the firms through bargaining, then the solution is given by: (5)

[w.sup.c.sub.i] = [w.sup.c.sub.j] = [a.sup.c]/2 (k - [bar.w]) + [bar.w]. (8)

Upstream wages depend on the exogenously given reservation wage [bar.w], the parameters of the demand function k, and the union's bargaining power [a.sup.c]. In the symmetric equilibrium,

[q.sup.c.sub.i] = [q.sup.c.sub.j] = (1 - [a.sup.c]/2)(k - [bar.w])/3a, (9)

[[pi].sup.c.sub.i] [[pi].sup.c.sub.j] = [(1 - [a.sup.c]/2).sup.2][(k - [bar.w]).sup.2]/9a. (10)

Equations (8), (9) and (10) indicate that, when an increase in [a.sup.c] causes the wage cost to rise, the upstream firm will set a higher wholesale price and this will result in a higher retail price. Therefore, an increase in the wage bargaining power of the union will depress outputs and profits. This result leads us to establish the following proposition:

Proposition 1. A stronger union will give rise to higher equilibrium wages, the upstream firms will set higher wholesale prices, and retail prices will increase, but the outputs and the profits of upstream firms will decrease under the Cournot duopoly competing case.

2. After an upstream merger

Consider the case where the industry is characterized by an upstream monopolist and two downstream firms. The timing of the game is unchanged with the upstream firm offering each downstream firm a two-part tariff contract. For the upstream monopolist the wage bargaining may be described as it was before in terms of being characterized by the generalized Nash bargaining solution. The profit of the upstream monopolist [[pi].sup.u] is given by:

[[pi].sup.u] = (P([Q.sup.m]) - w)[Q.sup.m], (11)

where the superscript "m" indicates the existence of a monopoly whenever an upstream merger or a downstream merger takes place. The profit maximizing output is [Q.sup.m], and therefore the monopoly offers [D.sub.i] and [D.sub.j], respectively, the same output:

[Q.sup.m]/2 = [q.sup.m.sub.i] = [q.sup.m.sub.j] = (1 - [a.sup.c]/2)(k - [bar.w]/4a, (12)

where the superscript "u" indicates an upstream merger case. As previously stated, we can calculate the upstream wage and the retail price in a regime where there is an upstream merger as follows, respectively:

[w.sup.u.sub.i] = [w.sup.u.sub.i] = [a.sup.u]/2 (k - w) + [[bar.w].sup.6] (13)

[p.sup.u] = [a.sup.u]/4 (k - [bar.w]) + 1/2 (k + [bar.w]) (14)

Thus,

[[pi].sup.u] = [(1 - [a.sup.u]/2).sup.2] [(k - [bar.w]).sup.2]/4a (15)

The profit of the upstream monopolist decreases when increases.

Given that contracts are not observable, the monopolist can still fully exert its monopoly power. Since the monopolist can maximize its profit by offering each downstream firm a two-part tariff contract, this is sufficient to induce each retailer to buy [Q.sup.m]/2, with each retailer believing that the contract is credible. If [D.sub.i] accepts the offer, then the monopolist has no incentive to change the offer to [D.sub.j]. The intuition is that when the upstream monopolist offers output [Q.sup.m]/2 to [D.sub.i] and offers more than output [Q.sup.m]/2 to [D.sub.j], then an increase in total output will depress the retail price, which will in turn reduce the profit of the upstream monopolist.

Therefore, when contracts are unobserved, an upstream monopolist does not have a commitment problem and is then able to fully exert its monopoly power. Moreover, the wage bargaining power is stronger after an upstream merger. The upstream monopolist's profit in an upstream merger case is more than the sum of the upstream duopolistic firms' profits in the upstream pre-merger.

In this case, we obtain the following proposition: (7)

Proposition 2. When contracts are unobserved, an upstream monopolist does not have a commitment problem and is then able to fully exert its monopoly power.

IV. Downstream merger

In this section we compare the equilibrium arising before and after a merger that occurs downstream.

1. The pre-merger case

In this case, we assume that the downstream firm possesses the franchise right, and that it can use a franchise fee to extract all of the upstream firms' profit ([F.sup.i] = ([P.sup.u.sub.i] - [w.sub.i])[q.sub.i]). The decision process is as follows. In the first stage, the wage is negotiated through union-manufacturer bargaining. In the second stage, each retailer simultaneously offers each upstream firm a two-part tariff contract, and each [D.sub.i] orders a quantity of intermediate product [q.sub.i] and receives a franchise fee [F.sup.i]. In the third stage, [D.sub.i] and [D.sub.j] transform the intermediate product into the final one, compete in terms of quantities, and choose their respective profit maximizing outputs. The profit of each retailer can be expressed as:

[[pi].sup.d.sub.i] = (P(Q) - [P.sup.u.sub.i])[q.sub.i] - [F.sub.i]. (16)

From the first-order condition, we can derive a linear reaction function [q.sub.i] = [q.sub.i] ([q.sub.j], [P.sup.u.sub.i]), which is the same as equation (4). Since contracts are unobservable, the upstream firms' maximization problem is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

For any [P.sup.u.sub.j] charged by the rival manufacturer, the first-order condition for profit maximization is that the best response of producer [U.sub.i] is to set the wholesale price equal to the marginal cost. Therefore, a downstream firm is indifferent between stipulating an exclusive contract with an upstream firm and integrating vertically.

In equilibrium, each manufacturer chooses [P.sup.u.sub.i] = [w.sub.i], where the symmetric Nash equilibrium quantities, profits, and wage are, respectively, [q.sup.c.sub.i]= [q.sup.c.sub.j], [[pi].sup.c.sub.i]= [[pi].sup.c.sub.j] and [w.sub.i] = [w.sub.j], which are exactly the same solutions as in the case of duopolistic vertical chains where the upstream firm can use the franchise fee [F.sub.i] to extract all of the downstream firms' profits. When either the retailer or the upstream firm possesses the right to franchise, then each vertical structure will act in order to maximize its total profits. Thus, there is no impact on production, and only the distribution of profits within the vertical chain is affected. Based on the previous analysis, we can establish the following proposition:

Proposition 3. An upstream firm or a downstream firm, regardless of whether it possesses the right to franchise, will exert no impact on production or total profit, and will only affect the distribution of profits within the vertical chain.

2. After a downstream merger

Consider the case where the industry is characterized by two upstream firms serving a downstream producer. We assume that the downstream monopolist possesses the right to franchise, and can therefore charge the manufacturers a franchise fee so as to fully extract the manufacturers' surplus. The downstream monopolist can also force [U.sub.i] and [U.sub.j] to sell their input at the marginal cost [w.sub.i]. This implies that the downstream monopolist will choose to sell the optimal quantity that maximizes its aggregate profit:

[[pi].sup.d] = (P([Q.sup.m]) - [w.sub.i])[Q.sup.m]. (18)

The profit maximizing output is [Q.sub.m], and therefore [D.sub.i] and [D.sub.j] offer the same outputs [Q.sup.m]/2, respectively:

[q.sup.m.sub.i] = [q.sup.m.sub.j] = (1 - [a.sup.d]/2)(k - [bar.w])/4a (19)

where the superscript "d" indicates a downstream merger case. Therefore, we can calculate the bargaining wages, the retail price and the profit of the downstream monopolist in the regime where there is a downstream merger as follows:

[w.sup.d.sub.i] = [w.sup.d.sub.j] = [a.sup.d]/2 (k - [bar.w]) + [bar.w], (20)

[p.sup.d] = [a.sup.d]/4 (k + [bar.w]) + 1/2 (k + [bar.w]), (21)

[[pi].sup.d] = [(1 - [a.sup.d]/2).sup.d] [(k - [bar.w]).sup.2]/4a (22)

The profit of the downstream monopolist is decreasing when [a.sup.d] is increasing. We next compare equation (14) with equation (20) as follows:

[[pi].sup.u] = [(1 - [a.sup.u]/2).sup.2] [(k - [bar.w]).sup.2]/4a >/< [[pi].sup.d] = [(1 - [a.sup.u]/2).sup.2] [(k - [bar.w]).sup.2]/4a; if [a.sup.u] >/< [a.sup.d]. (23)

Equation (21) indicates that the profit of the upstream monopolist in the regime where there is an upstream merger is more than, equal to or less than the profit of the downstream monopolist in the regime where there is a downstream merger. It is dependent on the wage bargaining power of the union in the different regime. We also compare equation (13) with equation (19), which respectively express the retail price in the upstream and downstream merger cases, as follows:

[p.sup.u] = [a.sup.u]/4 (k - [bar.w] + 1/2 (k + [bar.w]) </> [p.sup.d] = [a.sup.d]/4 (k - [bar.w] + 1/2 (k + [bar.w]); if [a.sup.u] </> [a.sup.d]. (24)

When we consider the wage to be jointly determined by the union and the firm through bargaining that determines the upstream firm's marginal cost, the retail price in the upstream merger case is lower, equal to or higher than the retail price in the downstream merger case. It is also dependent on the wage bargaining power of the union in the different cases. Therefore, (1) if the wage bargaining power of the upstream monopolist in the regime where there is an upstream merger is stronger than the wage bargaining power of the upstream duopoly firms in the regime where there is a downstream merger ([a.sup.u] < [a.sup.d]), then an upstream merger reveals that the retail price will be lower and the profit of the monopolist higher than in the case of a downstream merger. (2) If the wage bargaining power of the upstream monopolist in the regime where there is an upstream merger is equal to the wage bargaining power of the upstream duopoly firms in the regime where there is a downstream merger ([a.sup.u] = [a.sup.d]), then an upstream merger and a downstream merger reveal that the retail price and the profit of the monopolist will be the same. (3) If the wage bargaining power of the upstream monopolist in the regime where there is an upstream merger is weaker than the wage bargaining power of the upstream duopoly firms in the regime where there is a downstream merger ([a.sup.u] > [a.sup.d]), in contrast to a downstream merger, a merger involving upstream firms will result in a higher retail price and will lower the profit of the monopolist. The stronger union will benefit from this rent extraction that enhances the retail price and the profit of the monopolist will be reduced. Therefore, the union plays a key role in this economy.

V. The Union's Welfare

Following the utilitarian approach developed by McDonald and Solow (1981) and Oswald (1982), the union's objective is to maximize the sum of its members' utilities, U, i.e.:

U = Q(w - [bar.w]) (25)

First, from equations (8) and (9) we can obtain the union's welfare function in a pre-merger case as follows:

[U.sup.c] = [a.sup.c] (1 - [a.sup.c]/2)[(k - [bar.w]).sup.2]/3a. (26)

Next, from equations (12) and (13) we can obtain the union's welfare function in an upstream merger case as follows:

[U.sup.u] = [a.sup.u] (1 - [a.sup.u]/2)[(k - [bar.w]).sup.2]/4a. (27)

Similarly, from equations (19) and (20) we can obtain the union's welfare function in a downstream merger case as follows:

[U.sup.d] = [a.sup.d] (1 - [a.sup.d]/2)[(k - [bar.w]).sup.2]/4a. (28)

If the bargaining power of the union is the same in all three scenarios, i.e. two separate vertical chains, a merger between the two upstream firms, and a merger between the two downstream firms, we can compare equations (26) and (27) with equation (28) to conclude that the union's welfare will decrease after a merger. The intuition is that the total output (i.e. total employment) will decrease regardless of whether an upstream merger or a downstream merger takes place. Because either an upstream monopolist or a downstream monopolist is able to fully exert its monopoly power, which will decrease the total output and enhance the price, this will result in the monopolist obtaining more profit. If the bargaining power of the union in a merger between the two upstream firms is different from that of a merger between the two downstream firms, it will determine the different degree of the welfare detrimental effect under different regimes.

Then, we differentiate [U.sup.i] in equations (26), (27) and equation (28) with respect to [a.sup.i] as follows:

[partial derivative][U.sup.i]/[partial derivative][a.sup.i] [greater than or equal to] 0, (29)

where the superscript "i" refers to the pre-merger, upstream merger and downstream merger cases, respectively. Equation (29) indicates that the higher relative bargaining power of the union will result in a higher union's welfare.

Consequently, the results obtained above may be summarized in the following proposition:

Proposition 4. Regardless of whether an upstream merger or a downstream merger takes place, both will reduce a union's welfare. Moreover, the wage bargaining power of the union under different regimes will determine the degree of the welfare detrimental effect. (8)

VI. Concluding Remarks

Wage bargaining is found to have a substantial impact on imperfectly competitive rivalries between firms. This development calls for more research on the analysis of vertical contracts under the wage bargaining model. We have assumed that the unions and the upstream firms bargain over the wage and settle on the (generalized) Nash bargaining solution, but the firm is free to set whatever employment level it wishes. This paper shows that if the upstream firm can use a franchise fee to extract the retailers' profits, although contracts are unobserved, the upstream monopolist does not have a commitment problem and will then be able to fully exert its monopoly power to obtain more profit. We show in a pre-merger case that when either the retailer or the upstream firm possesses the right to franchise, then each vertical structure will act in order to maximize its total profits. Thus, there is no impact on production, and only the distribution of profits within the vertical chain is affected. Moreover, a downstream firm is indifferent between stipulating an exclusive contract with an upstream firm and integrating vertically. Regardless of whether an upstream merger or a downstream merger takes place, both will reduce a union's welfare. Furthermore, the wage bargaining power of the union under different regimes will determine the degree of the welfare detrimental effects.

Before ending this paper, one further point should be made. A franchise fee is a simple and powerful instrument in this environment. However, in more complex environments a franchise fee can also have its drawbacks. First, when the retailer is risk-averse and the retail cost or the final demand is random, the retailer bears too much risk, because it claims all the residual profits. Second, supposing that at the contracting date the retailer possesses private information regarding the retail cost or the final demand that the manufacturer does not have, because the retailer's profit is not known to the manufacturer, it is difficult for him to charge a franchise fee to the retailer so as to appropriate the retailer's profit.

References

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Notes

(1.) Fumagalli and Motta (2001) analyze two alternative mergers and show that a downstream merger (which gives the downstream monopolist all the bargaining power) is more welfare detrimental than an upstream merger (which gives the bargaining power to the upstream monopolist).

(2.) The definition of bargaining power is different from that of Fumagalli and Motta (2001). In this paper, bargaining power is defined as that which exists between the union and the firms that decide the equilibrium wage and employment.

(3.) See Grandner, (2000a) for a detailed discussion.

(4.) Several surveys exist on this topic. See Oswald (1985) and Farber (1986) for detailed discussions.

(5.) We have the profit function of the upstream firm [[pi].sup.u.sub.i] = (P(Q) - [w.sub.i])[q.sub.i] = (k - [(w.sub.i).sup.2]/9a, thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], after taking log of the objective function, the objective function becomes:

[a.sup.c] ln (k - [w.sub.i])/3a ([w.sub.i] - [bar.w]) + (1 - [a.sup.c]) ln [(k - [w.sub.i]).sup.2]/9a.

From the first-order condition, we can derive the bargained wages.

(6.) We have the profit function of the upstream firm [[pi].sup.u] = (P(Q) - [w.sub.i])Q = [(k - [w.sub.i]).sup.2]/4a, thus, max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the first-order condition, we can derive the bargained wages.

(7.) Fumagalli and Motta (2001) believe that an upstream monopolist that offers unobservable contracts will suffer from a lack of commitment power.

(8.) Fumagalli and Motta (2001) showed that downstream mergers are more likely to give rise to welfare detrimental effects than upstream mergers.

Shu-hua Chang, Department of Accounting, National Taichung Institute of Technology, Taiwan. I am grateful to an anonymous referee of this journal for excellent guidance in revising the paper. Needless to say, any remaining deficiencies are the author's responsibility.