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   5 UNIT 10 NON-COLLUSIVE OLIGOPOLY Structure 10.0 Objectives 10.1 Introduction 10.2 Non-Collusive Oligopoly 10.2.1 Cournot Model of Duopoly 10.2.2 Bertrand Model of Duopoly 10.2.3 Edgeworth Model 10.2.4 Chamberlin’s Oligopoly Model 10.2.5 Kinked Demand Curve: Sweezy Model 10.2.6 Stackelberg Model 10.3 Let Us Sum Up 10.4 Key Words 10.5 Some Useful Books 10.6 Answer or Hints to Check Your Progress 10.7 Exercises 10.0 OBJECTIVES   After going through this unit, you will be able to: ã understand the oligopolistic market structure; ã appreciate the role of interdependence among the producers in deciding the output and price; and ã examine the important models developed for analysing oligopolistic  behaviour. 10.1 INTRODUCTION An oligopolistic market is characterised by the existence of a small number of firms who have the market power in the sense that they can affect the market  price by changing their output level. In such a market, the firms may produce identical or differentiated products. The distinguishing feature in it is strategic interdependence among the firms with regard to price and output decisions. 10.2 NON-COLLUSIVE OLIGOPOLY Oligopoly can be of two types: non-collusive and collusive. In the non-collusive oligopoly, there is rivalry among the firms due to the interdependence. On the other hand, in collusive oligopoly the rival firms enter into a collusion to maximise joint profit by reducing the uncertainty due to rivalry. Under non-collusive oligopoly each firm develops an expectation about what the other firms are is likely to do. This brings us to an important concept of “Conjectural Variation” (CV) of a firm. CV of i th  firm is defined as the    Price and Output Determination-II 6 reaction of the j th  firm, corresponding to a marginal adjustment in the i th  firm’s strategy variable as perceived by the i th  firm. For instance, if output were the strategic variable, then the CV of the i th  firm would be given by ( δ q  j / δ q i )– the amount of change in the output level that would be brought about by the jth firm for an additional change in the output level of the i th  firm, as perceived by the i th  firm. Depending on CV, we can have different models under oligopoly. For instance, in the Cournot Duopoly model, CV of each firm is zero because each of the duopolists assumes that the other would stick to its previous  period’s output level. In the Stackelberg model, there is a leader and a follower. Here the leader knows what the follower is likely to do; hence, the CV of the leader is positive. In the following sections, we would see how equilibrium is arrived at in the important models of non-collusive oligopoly—Cournot model of duopoly, Bertrand model, Stackelberg model, Edgeworth, Chamberlin and the Kinked Demand curve analysis of Sweezy. To do this we would make use of the concept of reaction functions (RF). A reaction function of a firm gives the best response of the firm, given the decision taken by the rival firm. 10.2.1 Cournot Model of Duopoly The model by Augustin Cournot deals with two profit maximising firms. Let the two firms be A and B. Assumptions 1) Each of the firms faces a linear market demand curve 2) Both sell identical products. In Counot’s model, the two are assumed to sell mineral water. 3) The cost functions are identical and the marginal cost (MC) of each firm is zero. 4) Each firm assumes that the other would continue to produce the same output as in the last period.  Diagrammatic Representation To arrive at the Cournot solution, let us assume that firm A is the first to  produce and sell in the market. Let D 1  D 1  be the linear market demand curve, as shown in Figure 10.1.The marginal cost =0 for both the firms. In the figure, this corresponds to the horizontal axis. Firm A being a profit maximiser, equates MR with MC and arrives at the output level OA (= ½ OD 1 ) and price OP 1 Suppose now firm B enters the market. As firm A is already selling OA amount of output, firm B would cater to OD 1 minus OA amount of the market demand, assuming that firm A will continue producing OA. Therefore, the  portion of the market demand relevant to firm B is CD 1 .This is so because B cannot sell anything at a price higher than OP 1,  as firm A is already present in the market and they are selling the same product.   Hence the only other option open to firm B is to sell at a price lower than OP 1,  whereby the market demand curve for B shrinks to CD 1 . Firm B being a profit maximiser, produces output AB (= ½ A D 1 ) where MR  B = MC = 0.   7 Non-Collusive Oligopoly The output level supplied in the market after firm B’s entrance is OA + AB (= ½ OD 1  + ¼ OD 1 ) = ¾ OD 1 . As the output level goes up, the price in the market goes down to, say OP 2.   Next, firm A assumes firm B to continue producing AB and therefore the market demand that A can cater to is OD 1  minus AB. In the diagram the market demand curve relevant to A is D 2  B . Once again setting MR  A = MC = 0, firm A will produce OA 1 (= ½ OB= ½ * ¾ OD 1  =3/8 OD 1 ) .  Firm B now is assuming that firm A will continue producing OA 1 and has ED 1  as the relevant market demand curve. Setting MR  B = MC =   0 firm B would  produce A 1 B 1 = ½ A 1  D 1 = ½ *(1- 3/8 ) OD 1 = 5/16 OD 1. The total supply in the market would be OA 1  + A 1 B 1 = 3/8 OD 1 + 5/16 OD 1  = 11/16 OD 1 . As the market supply goes up the price comes down to say, OP 3. Thus, we see that the output of firm A goes down whereas that of firm B goes up. This process continues until each one of the firms produces 1/3 OD 1. To see how, we derive the equilibrium output levels of each firm in the following. Let the total market demand be x units of output .  Output levels of firm A: Period 1:   x/2 Period 2: ½(1-1/4) x = 3x/8 = x/2 – x/8 Period 3: ½(1- 5/16) x = 11x/32 = x/2 – x/8 – x/32 Period 4: ½(1 – 21/64) x = 43x/128 = x/2 – x/8 – x/32 –x/128 In the nth period, the output of firm A is = x/2 – x/8 – x/32 –x/128 - ……. = x/2 – [1/8 + 1/8 * (¼) + 1/8*(1/4) 2  + ……… ] * x = x/2 –1/8 * [ 1/(1- 1/4)]*x = x/3 Output levels of firm B: Period 2: ½(1/2)x = x/4 Period 3: ½(1- 3/8)x = 5x/16 = x/4 + x/16 Period 4: ½(1 – 11/32) x = 21x/64 = x/4 + x/16 + x/64 Period 5: ½(1 – 43/128) x = 85x/256 = x/4 + x/16 + x/64+x/256 In the n th  period, the output of firm B is = x/4 + x/16 + x/64+x/256 +... = [(¼)/1- ¼] * x = x/3    Price and Output Determination-II 8 Fig. 10. 1: Demand Analysis of Cournot Equilibrium Reaction Function Approach The reaction function approach is a useful tool in analysing oligopolistic markets. With this approach, it becomes easier to analyse the equilibrium condition of the different oligopolistic models. We would apply it to the Cournot duopoly model in the following. In his duopoly model, Cournot makes a very naïve assumption that the firms think their rivals would stick to their past periods output level. Therefore, the conjectural variation (CV) of both the duopolists is equal to zero. Retaining the same assumptions that both the duopolists i) face linear market demand curve, ii) maximise profit and iii) have MC = 0.We can write the model as follows: Let the demand function be p = a - bq, where q = (q i  + q  j ) = total market demand and a, b > 0 Given the above assumptions, we can write the profit function of the i th  firm as: П i  = pq i  – C (q i ); where i = A, B = (a – bq) q i  – C(q i ) = [a – b(q i  + q  j ) ] q i  – C(q i ) Each firm being a profit maximiser, we would differentiate П i  partially with respect to q i  and set the derivatives equal to zero. Thus, δП i / δ q i  = a – 2bq i  –b (q  j + q i δ q  j / δ q i ) – δ C/ δ q i  = 0. As in this model CV = 0, δ qj/ δ q i  = 0. Hence, we have, a – 2bq i  –bq  j = 0 (as δ C/ δ q i  = 0, by assumption) From such an optimisation exercise we get: q i * = (a – bq  j )   / 2b, q i * = R  i (q  j ) where, q i * gives the profit maximising level of output of firm i(i, j =A, B; i ≠  j) D 1 D 2 P 1 ECC 1 D 1 BB 1 AA 1 OP 2 M    R 2   A   M    R 2   B    M    R 1   B   
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