______

Subject

Paper No and Title 5 : Advanced

Module No and Title 5 : Pricing in Imperfect Factor (Bilateral )

Module Tag ECO_P5_M5

ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)

______

TABLE OF CONTENTS

1. Learning Outcomes 2. Introduction 3. Mathematical Model of bilateral monopolist (upstream monopolist vs. downstream monopsonist) 4. Indeterminacy of and Employment 5. Summary

ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)

______

1. Learning Outcomes

After studying this module, you shall be able to

 Examine how factor is determined when monopoly exists in both demand and supply side of the factor market.  Learn the derivation static model and indeterminacy of solution.  Know how to derive dynamic model of price adjustment in bilateral monopoly

2. Introduction

Bilateral monopoly is said to exist in the market at which single buyer firm or monopsonist firm interact with labour union which acts as monopolist seller of factor input labour. Since, there exist two monopolists; one in demand side (firm) and other in supply side (labour union), the market is said to have bilateral monopoly. It is rarely occurred in the real world, but labour union of large manufacturing company in particular town (or one company town) or professional players’ union vs. management can be approximated as bilateral monopoly situation. In this market there are four possible situations observed in the determination of equilibrium price and quantity: (a) seller may play dominant role in the market and force the buyer to accept the price offered by the seller, (b) single buyer or monopsonist can dominate the market and compel the sellers to accept the price and quantity offered by him, (c) there may be collusion between buyer and seller(i.e., joint maximization) to determine optimum price and quantity and (d) the indeterminacy of price and output or due the noncooperation of buyer and seller. In this module we first represent simple mathematical model of joint profit maximization of upstream monopolist and downstream monopsonist under bilateral monopoly situation and then analyze the indeterminacy problem of factor pricing. Further we also analyze the dynamic model to solve the indeterminacy problem in determination of the product of upstream seller firm.

3. Mathematical Model of bilateral monopolist (upstream monopolist vs. downstream monopsonist

Suppose the seller (upstream monopolist) produces intermediate product 푥 and sells it to the downstream monopsonist buyer who employs it as an input to produce output 푦. The unit of the products 푥 and 푦 are 푚 and 푝 respectively. The profits of seller (Π푠)and buyer (Π푏) are given as: Π푠 = 푚. 푥 − 푐(푥) (1) ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)

______

Π푏 = 푝. 푓(푥) − 푚. 푥 (2) Where 푐(푥) shows the seller’s cost function and 푓(푥) represents buyer’s production function. In order to determine price and quantity in the bilateral monopoly situation we assume that seller purchases input in the perfectly competitive market and buyer firm also sales the product in the perfectly competitive market. We examine now the first two cases of bilateral monopoly in which either seller acts as a dominant role or buyer firm becomes dominant to determine the price. When seller firm is price setter, the price will be determined from the profit maximization process under pure monopoly of upstream seller firm. Similarly, if monopsonist buyer firm dominates, the price it obtains from its profit maximization will be market equilibrium price. But if no party dominates in the market and fails to recognize their interdependency in the determination of equilibrium price, the market mechanism breaks down and indeterminacy problem will arise. That is result of conventional bilateral monopoly theory. In order to avoid the indeterminacy in pricing decision Henderson & Quandt suggest the collusion between two parties and that can be accomplished through joint profit maximization and mutual bargaining for desirable price. The joint profit maximization takes place in the following way:

Π = Π푏 + Π푠

= [푝. 푓(푥) − 푚. 푥] + [푚. 푥 − 푐(푥)] (3)

= 푝. 푓(푥) − 푐(푥)

The first-order condition for joint profit maximization implies that

푑Π = 푝. 푓/(푥) − 푐/(푥) = 0 (4) 푑푥

Now the joint profit maximization provides the optimal output 푥∗ not the optimal price. The optimal price can be determined through the mutual bargaining between the buyer and seller. Henderson & Quandt in their model find that there is no unique price is obtained, instead there are upper and lower limits of the price. This is shown as

푐(푥∗) 푝.푓(푥∗) ≤ 푚 ≤ (5) 푥∗ 푥∗

In the static model there doesn’t exist any negotiated price and indeterminacy problem arises. In the dynamic model we can analyze how the bargaining process happens

ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)

______

between the upstream seller firm and downstream buyer firm. Devadoss(1998)1 suggests a dynamic model which captures the process of bargaining price negotiation and solves the indeterminacy problem. Devadoss assumes that each player (firm) in the market has some knowledge about the nominal profit of the other player. When buyer firm’s profit is higher than the seller firm’s profit, i.e. Π푏 > Π푠, the seller wants to enter in to the bargaining process to get higher price for his output 푥 for raising its profit. Similarly whenΠ푠 > Π푏, the monopsonist buyer firm wants to participate in bargaining to set a lower price so that it can increase its profit. Thus in this model there will be downward adjustment of price 푚 when Π푠 > Π푏, and upward adjustment of 푚 will tke place when Π푏 > Π푠. When Π푏 = Π푠, there will be no adjustment in price 푚 and the equilibrium price 푚∗ will be determined. According to Devadoss this dynamic adjustment process can be captured by the following equation:

푑푚 = 푚/(푡) = 훾[Π − Π ], 훾 > 0 (6) 푑푡 푏 푠

Where 훾 measures the speed of price adjustment or effectiveness of bargaining when price 푚 approaches towards the equilibrium price 푚∗. If the bargaining process is not taken place, the profit obtained by each party will be zero. Therefore, it would be rational for each party to enter into the bargaining process. Hence by substituting equations (1) and (2) in equation (6) we obtain

푑푚 = 푚/(푡) = 훾[{푝. 푓(푥) − 푚. 푥} − {푚. 푥 − 푐(푥)}] 푑푡 푑푚 = 푚/(푡) = 훾[푝. 푓(푥) − 2푚. 푥 + 푐(푥)] (7) 푑푡 Now by 푚/(푡) = 0 we can obtain the equilibrium price as 푝.푓(푥)+푐(푥) 푚∗ = (8) 2푥 The equilibrium price thus obtained as half of per unit revenue of buyer in addition with half of per unit cost of seller. The equilibrium price 푚∗ lies in the midpoint of the upper and lower bounds given in equation (5). At the equilibrium price profit of the buyer equals to the profit of the seller and that is given as 1 Π = Π = [푝. 푓(푥) − 푐(푥)] (9) 푏 푠 2 Thus at equilibrium, either party obtains the half of the joint profit (or monopoly profit). This implies that buyer and seller have equal bargaining power. The stability of the equilibrium price can be examined through phase diagram. This is shown in figure-3.1. 푑푚/(푡) The phase diagram is negatively sloped as < 0. For points above the horizontal 푑푚 axis, the price 푚 increases over time as 푚/(푡) > 0. By contrast, for any point below

1 Devadoss, Stephen, 1998, “A dynamic analysis of price determination under joint profit maximization in Bilateral monopoly”, American Association Annual Meeting. http://econpapers.repec.org/scripts/redir.pf?u=http%3A%2F%2Fpurl.umn.edu%2F20809;h=repec:ags:aae a98:20809 ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)

______

푚/(푡) < 0 the price 푚 will decrease over time. The arrowheads of the phase diagram shown in both upper and lower planes of the horizontal axis indicate that the phase line converges to the equilibrium price 푚∗. By solving the first order differential equation (7) we get the quantitative solution of dynamic stability. The solution of equation (7) is obtained as:

푝.푓(푥)+푐(푥) 푝.푓(푥)+푐(푥) 푚(푡) = [푚(0) − ] 푒−2훾푥푡 + 2푥 2푥 푚(푡) = [푚(0) − 푚∗]푒−2훾푥푡 + 푚∗ (10)

Here 푚(0) is initial price. As 훾 and 푥 are positive, 푒−2훾푥푡 approaches to zero with the increase in time 푡 and 푚(푡) tends to the equilibrium price 푚∗. The parameter 훾 measures the effectiveness of the bargaining or speed of price adjustment. For greater of 훾 the price will be dynamically adjusted at faster rate. The speed of price adjustment also depends on the magnitude of the upstream seller’s product 푥. The larger the quantity of 푥, the faster the equilibrium price approaches to 푚∗.

푚/(푡)

0 m 푚∗ 0

Figure-3.1

The above model can also be applied in the factor market by incorporating supplier of factor in place of upstream firm. For example, if we replace upstream firm by labour union, then its interaction with the downstream monopsonist buyer firm depicts the bilateral monopoly situation and we can obtain the indeterminacy of wage in the static model and stable equilibrium wage in dynamic model. Only the conventional result of wage indeterminacy under bilateral monopoly is analysed graphically in the next section.

ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)

______

4. Indeterminacy of and Employment

In bilateral monopoly situation, wage rate is set through the bargaining process of both the demand side and supply side monopolist agent in the factor market and equilibrium wage rate lies between upper and lower limits determined within this model. That is, we won’t get single equilibrium wage rate. In figure-4.1 the monopsonist firm’s is 퐷푓. This demand curve is 푀푅푃퐿 of factor input. For labour union this demand curve 퐷푓 represents the average revenue curve (i.e. 퐷푓 = 퐴푅푢). Since labour union acts as a monopolist, for obtaining equilibrium we will derive marginal revenue curve (푀푅푢) by using graphical technique. In figure-4.1 푀푅푢 curve is shown below the 퐷푓 or 퐴푅푢 curve. The 푆퐿 curve represents the supply curve of labour faced by the monopsonist firm. This curve shows average expenditure or of labour to the monopsonist firm. Now we draw corresponding marginal expenditure (푀퐸푓) curve. The 푆퐿 curve also reflects the of supplying labour to the monopolist labour union. Although we know that monopolist doesn’t have supply curve, here we can assume that the monopolist labour union behaves as if he were a perfectly competitive seller. Having considered the above cost revenue structure we can derive the equilibrium position of each monopolist agent in this model. The monopsonist firm maximizes its profit at point 퐹, where its marginal expense of labour (푀퐸푓) equals to the marginal revenue product of labour (푀푅푃퐿). Thus monopsonist wants to hire 퐿푓 units of labour at

ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)

______

wage rate equal to 푊푓. On the other hand, the monopolist labour union 푊

푀퐸푓

푊푢 퐹

푆퐿 = 퐴퐸푓 = 푀퐶푢

푊푓 퐷푓 = 푀푅푃퐿 = 퐴푅푢 푈 퐿 푂 퐿푢 퐿푓

푀푅푢 Figure-4.1

maximizes his gain at point 푈 by equating his marginal cost to his marginal revenue. The labour union desires to supply 퐿푢 units of labour at 푊푢 wage rate in order to fulfill his maximization of total wage gain. Thus, we obtain the upper limit of wage rate set by the labour union and lower limit of wage rate fixed by the monopsonist firm. The upper limit wage rate 푊푢 can be realized if and only if the monopsonist firm acts as a perfectly competitive firm. Similarly if labour union doesn’t exert monopoly power and behaves as a perfectly competitive seller, the lower limit wage rate 푊푓 can be realized. Thus, none of these monopolists can reach their wage and employment target and make the solutions of the bilateral monopoly market indeterminate. Hence, in this market the wage is settled through the bargaining process of these two monopoly agent. Since, only upper and lower limit of wage rates are determined in this market, the settlement of wage towards either of the upper and lower limits depends on the bargaining skill, power of the participants. For instance, if the government is pro labour or there is strong political support behind labour union, the bargaining power of union will be relatively stronger than the firm lobby. The wage will be set close to the upper limit of the wage rate. In contrast if state shows capitalist attitude, firm lobby will be more powerful in the bargaining process than union and wage rate will be mutually settled close to lower limit.

ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)

______

5. Summary

 Bilateral monopoly refers to the market situation in which single seller interacts with single buyer.  It arises in the factor market when there exist a single seller (monopolist labour union) and a single buyer (monopsonist firm).  In static model,there is an indeterminacy problem in the market solutions wage and employment. That is, no unique equilibrium wage and employment combination can be found in this market.  In the dynamic model the equilibrium price can be determined through negotiation.  In the bilateral monopoly situation only upper and lower limit of wage or employment can be determined. These are outcomes of two monopolists’ profit- maximizing behaviors.  The settlement of wage and employment depend on the bargaining power of each participant.

ECONOMICS PAPER No. 5: Advanced Microeconomics PAPER No. : TITLE MODULE No. 5: Pricing in Imperfect Factor Market MODULE No. : TITLE (Bilateral Monopoly)