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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:
- understand the oligopolistic market structure;
- appreciate the role of interdependence among the producers in deciding the output and price; and
- 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 ith^ firm is defined as the
Price and Output Determination-II
reaction of the jth^ firm, corresponding to a marginal adjustment in the ith^ firm’s strategy variable as perceived by the ith^ firm. For instance, if output were the strategic variable, then the CV of the ith^ firm would be given by (δqj/δqi)– 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 ith^ firm, as perceived by the ith^ 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
Each of the firms faces a linear market demand curve
Both sell identical products. In Counot’s model, the two are assumed to sell mineral water.
The cost functions are identical and the marginal cost (MC) of each firm is zero.
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 OP1, 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 OP1, 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.
Price and Output Determination-II
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 = (qi + qj) = total market demand and a, b > 0 Given the above assumptions, we can write the profit function of the ith^ firm as: Пi = pqi – C (qi ); where i = A, B = (a – bq) qi – C(qi ) = [a – b(qi + q (^) j ) ] qi – C(qi ) Each firm being a profit maximiser, we would differentiate Пi partially with respect to qi and set the derivatives equal to zero. Thus, δПi / δqi = a – 2bqi –b (q (^) j + qi δqj /δqi ) – δC/ δqi = 0. As in this model CV = 0, δqj/δqi = 0. Hence, we have, a – 2bqi –bqj = 0 (as δC/ δqi = 0, by assumption) From such an optimisation exercise we get: qi* = (a – bqj ) / 2b, qi * = R (^) i (qj ) where, qi * gives the profit maximising level of output of firm i(i, j =A, B; i ≠j)
D (^1)
D 2
P 1
E
C
C 1
O A 1 A B 1 B D (^1)
P (^2)
MR 2 A
MR 2 B MR 1 B
The equilibrium output levels of both the firms is obtained by solving the two^ Non-Collusive Oligopoly reaction function equations as:
qA * = a/3b, qB* = a/3b
For each firm, Ri represents the reaction function. Given the output level of the jth firm the reaction function shows the best response (i.e., qi *) of the i th firm, which maximises its profit. The reaction functions in this exposition will be downward sloping straight lines, as shown below in Figure 10.2 where SP is the reaction function of firm B and MN is the reaction function of firm A. For any level of output of firm B say qB^1 the level of output which would maximise firm A’s profit is given by qA^1 from A’s reaction function MN .We can similarly explain each point on the reaction curve of firm B.
Fig. 10.2: Cournot Equilibrium through Reaction Function
Equilibrium
Diagrammatically this is shown in Figure 10.2. The equilibrium is obtained where the two reaction functions intersect each other i.e., at point E corresponding to which we have the two equilibrium output levels as qA* and q (^) B*
Stability
For stability analysis, let us consider Figure 10. 3 where MN and SP represent the reaction functions of firms A and B respectively. To see if point E is a stable equilibrium, we would start from an arbitrary point and see if the inbuilt dynamics of the model would take us to point E. In the diagram, ON is the monopoly output of firm A and OS represents that of firm B. Suppose firm A enters the market first and produces the monopoly output ON. Next, firm B enters and assuming that firm A would continue producing at ON chooses to produce Ob from it’s reaction function SP. Therefore, the two firms end up at point 1 on B’s reaction function. As at point 1 firm A is off it’s reaction function, it would not be maximising it’s profit. Hence, A would not choose to be at 1. Assuming that firm B would continue producing at Ob, firm A would choose to be at point 2 on it’s reaction function whereby it would be producing Oa amount of output. From the point of view of firm B, point 2 is
M
qB
S
Km qB q^1 B
O
E
q*A (^) q^1 A N P qA
- What is a reaction function?^ Non-Collusive Oligopoly
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- Show using reaction functions that the Cournot equilibrium is a stable one?
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10.2.2 Bertrand Model of Duopoly
Bertrand model of duopoly examines the price setting behavior of a firm by taking price as is the strategic variable. The assumptions in it are the same as in Cournot model except that the firms have identical cost functions with constant marginal cost. To arrive at the equilibrium, let us consider two firms A and B.
Given any price P (^) B, set by firm B, firm A has 3 options:
To set a price P (^) A > P (^) B
To set a price P (^) A = P (^) B
To set a price P (^) A < P (^) B
In option 1, firm A looses the whole market. In 3, firm A captures the whole market and in 2, the market would be shared equally by them. Firm A will undercut B so long as MCA ≤ P (^) A .The same logic applies to firm B as well. This process will continue till P = MCA = MC (^) B
Diagrammatic Representation
In the following figures, we represent the process by which the firms reach equilibrium in the Bertrand model. The horizontal axis measures quantity of output produced by each firm along with cost (assumed to be identical). Along the vertical axis, we measure the price and marginal cost. In each of the figures, the horizontal line C represents the marginal cost (MC) of each firm.
When P (^) A is greater than P (^) B and both are greater than C as in Figure 10. 4, firm A will be loosing the market to firm B as they are selling homogeneous product. Therefore, profit of firm A will be less than that of B and firm A will undercut firm B. In the next period firm B will be loosing the market to firm A and therefore, would undercut firm A. This process will go on until both
Price and Output Determination-II
charge the same price, as shown in Figure 10. 5. In such a case, both of get an equal share of the market. However, it will not be an equilibrium situation because if any one of them reduces its price marginally, then it gets the full share of the market and earns more profit than when they were sharing the market equally. Finally, equilibrium will be attained when P (^) A = P (^) B = MC. This is shown in Figure 10. 6.
Figure 10. 6 gives a stable equilibrium because no one has any incentive to reduce or raise the price. If the former occurs, then there is the threat of loss because MC is greater than price. If the price is higher than the equilibrium price (P), then there is a threat of potential loss of customers. Therefore, if price is greater than or less than P it will have a tendency to move back to P*, which explains the stability of the equilibrium.
Fig. 10. 4 Fig. 10. 5 Fig. 10. 6
As in the Bertrand model P* = MC, the solution is a competitive one and therefore output will be produced at the competitive level (unlike in the Cournot model where output produced by each duopolist was equal to one- third of the total market demand).
Check Your Progress 2
- Describe how the firms under Bertrand model arrive at the equilibrium price.
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10.2.3 Edgeworth Model
To analyse the Edgeworth solution of oligopoly let us consider two profit maximising firms A and B, selling a homogeneous product and having identical cost function with marginal cost (MC) equals to zero. We also assume that they face a linear market demand curve. In the following figure, DA represents the market demand faced by firm A and DB represents that of firm B.
PA
PB C
PA =PB
C P*^ =C
Price and Output Determination-II
- Is there a stable equilibrium price in Edgeworth model? Why?
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10.2.4 Chamberlin’s Oligopoly Model
Chamberlin suggested that a stable equilibrium can be reached in an oligopolistic market if the firms charge monopoly price. This will be possible if the firms recognise their interdependence, unlike in the Cournot model where they act on the naïve assumption of rival marinating its previous period’s price or output level. In this model, the setting (i.e., assumption) is similar as that in Cournot’s except for the fact that the firms do recognise their mutual interdependence. Let us study the model on the basis of the following diagram. Let DQ be the linear market demand curve. Suppose firm A enters first in the market and sells OQ 1 units at the price OP 1 on the basis of (MR = MC), thereby reaping monopoly profit given by the area OQ 1 CP (^) 1. Let us now consider firm B’s entry into the market. Given that firm A produces OQ 1 , CQ becomes firm B’s relevant market demand curve. Therefore, the best B can do acting on the basis of (MR = MC) is to market Q 1 Q 2. As a result, price falls to OP 2 and the total profit accruing to both is given by the area OQ 2 FP 2.
Fig. 10. 8: Chamberlin Equilibrium
According to Chamberlin, firm A will survey the market situation after B’s^ Non-Collusive Oligopoly entry and will figure out that sharing the profit level OQ 1 CP 1 is the best for either of them. Therefore, firm A would reduce it’s output level from OQ 1 to OQ 3 and firm B would stick to the output level Q 1 Q 2 = Q 3 Q (^) 1. With this arrangement, the firms together produce OQ 1 and the price level is retained at OP1. Thus, we see that firm A produces OQ 3 = ½ OQ 1 and B Q 3 Q 1 = ½ OQ (^) 1. The total output is OQ 1 to be sold at a price OP 1 with firms A and B sharing the monopoly profit equally. Firms, in this kind of an agreement, produce more than in the Cournot case, where each one produces one-third of the total market demand.
Check Your Progress 4
- Describe briefly the method by which the firms arrive at the equilibrium price and output in Chamberlin’s model.
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- Is the output level in Chamberlin model higher than in Cournot? How much is it of the total market demand?
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10.2.5 Kinked Demand Curve: Sweezy Model
In an oligopolistic market situation, due to rivalry among the firms, any one lowering the price is interpreted by others as an attempt to eliminate their profit. Therefore, other firms also respond by cutting their prices as well. This chain of price cuts is called a price war. In the model by Sweezy, we would analyse what happens when the firms behave in the manner described above.
Each firm in an oligopolistic market faces two demand curves D 1 D 1 and D 2 D (^2) as shown in Figure 10.9. D 1 D 1 is the demand curve that a particular oligopolist faces on the assumption that others do not change their prices and D 2 D 2 has been drawn on the assumption that if one firm changes the price, then all others also change theirs.
The marginal revenue curve corresponding to the kinked demand curve is^ Non-Collusive Oligopoly shown in Figure 10. 10. MR 1 is the marginal revenue corresponding to D 1 D 1 and MR 2 is the marginal revenue curve corresponding to D 2 D (^) 2. To the right of Q 0 the demand curve is given by the segment of D 2 D 2 and hence the marginal revenue given by the corresponding segment of MR 2. At the quantity Q 0 there is a sudden drop in marginal revenue, from the point B to point C in Figure 10.10. The marginal revenue curve for the kinked demand curve in Figure 10.9 is given by the line EBCF in Figure 10.10.
See that there is some range within which changes in the firm’s marginal cost will not result in changes in price and quantity. This is shown in Figure 10. 11. Note that both for MC 1 and MC2, the price and quantity given by the equilibrium condition MC = MR are the same. Hence, the price is “sticky”, and the model is also known as the “sticky price model”.
Fig. 10. 11: MR and MC Curves with Sticky Price Kinked Demand Model
The kinked demand is derived on the assumption that price increase by one of the oligopolistic firm is not followed by others but price reductions.
Check Your Progress 5
- Which behaviour of the firms is central to Sweezy’s analysis?
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Price and Output Determination-II
- Describe the two different demand curves faced by a firm. Derive the MR curve diagrammatically.
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- Why is the model called the “sticky price model”?
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10.2.6 Stackelberg Model
In Stackelberg model, quantity is the strategic variable of the firms. We would analyse the model in terms of iso-profit curves. Let us assume that there are two profit maximising duopolistic firms A and B. An iso-profit curve shows the alternative combinations of output q (^) A (quantity of output of firm A) and q (^) B (quantity of output of firm B) that would yield the same profit. Let us suppose the following:
- Demand function faced by both the firms are p = p (q), where q = q (^) A+q (^) B , the aggregate output produced by both the firms.
- Cost function Ci = Ci (qi ) where i = A,B Derivation of Iso-profit Curve
Consider the profit function of firm A given by,
ПA = pq (^) A – C(q (^) A) and
take П 0 to be the desired level of profit. To arrive at the iso-profit curve of firm A choose those combinations of qA and qB, which would yield a profit level of П 0 to it;
i.e., П 0 = pq (^) A – C(qA) = p(q) q (^) A – C(qA) = p(qA+qB) qA – C(q (^) A )
To arrive at those combinations of qA and q (^) B, we totally differentiate П 0 , to get
dП 0 = pdq (^) A + qA dp – ∂C/∂qA dq (^) A
or, 0 = pdqA + qA dp/dq(dq (^) A + dqB ) – ∂C/∂qA dq (^) A
Price and Output Determination-II
reaction function of B. Given the iso-profit curves of A and the reaction function of B, firm A can find out from the tangency between the two. This is illustrated in the following diagram. In the diagram, point E gives the Stackelberg equilibrium. Firm B being on its reaction function would have no incentive to deviate from E. In addition, as firm A is maximising its profit, it has no incentive to deviate. Hence, this equilibrium is stable.
Fig.10. 13: Stackelberg Solution
Numerical Example Suppose the demand and cost conditions are the same as in above with firm 1 the leader and firm 2 the follower. In that case, derive the Stackelberg equilibrium quantity and profits of each firm.
P = 100 – 0.5(q 1 + q 2 ) and Ci = 5q (^) i where i = 1, П 2 = Pq (^) 2 – C 2 (q 2 ) = {100 – 0.5(q 1 + q 2 ) }q (^) 2 - 5q 2 = 95q (^) 2 - 0.5q 1 q 2 – 0.5q (^22) Differentiating partially with respect to q 1 and setting the derivatives equal to zero we get, δП 2 /δq 2 = 95 – 0.5q 1 – q 2 = 0 or, q (^) 2 = 95 – 0.5q 1 Firm A (leader) would now incorporate q 2 = 100 – 0.5q 1 into its profit function and proceed as above. П 1 = Pq (^) 1 – C 1 (q 1 ) = {100 – 0.5(q 1 + q 2 ) }q 1 – 5q 1 = 100q 1 – 0.5(q 1 + 95 – 0.5q 1 )q 1 – 5q 1
= 47.5q 1 – 0.25q 12
Therefore, δП 1 /δq 1 = 47.5 - 0.5q = 0
or, q (^) 1 = 95 and q 2 = 95 – 0.5*95 = 47.
Check Your Progress 6^ Non-Collusive Oligopoly
- What does the leader firm do in stackerberg this model?
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- Diagrammatically where does the equilibrium occur in Stackberg model?
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10.3 LET US SUM UP
In the non-collusive oligopolistic model, there is interdependence and rivalry among the firms. Depending on the conjectures made by the firms, we get different models.
The Cournot model deals with the case when the firms make conjecture that the rival would stick to the previous level of output. Here the firms deal with output changes. Finally, the firms together end up producing 2/3 of the total market demand.
In the Bertrand model, the case is similar to that of Cournot except that the firms compete in terms of price. Here they end up producing the competitive level of output.
In the Stackelberg model one firm acts as the leader and the other follower. A firm is a leader in the sense that it knows the reaction function of the follower. The leader maximises profit after incorporating the reaction function of the follower.
10.4 KEY WORDS
Conjectural Variation : CV of an i th^ firm is defined as the reaction of the jth firm, corresponding to a marginal adjustment in the ith^ firm’s strategy variable as perceived by the ith^ firm.
Mutual Interdependence: The action of one firm in the market affecting that of the other.
- Yes. It is ½ of the total market demand.^ Non-Collusive Oligopoly
Check Your Progress 5
In an oligopolistic market situation due to rivalry among the firms any firm’s lowering of price is interpreted by other firms as an attempt to eliminate their profit. Therefore, when a firm cuts price, other firms also respond by cutting theirs.
Do yourself after reading Section 10.2.5.
The Demand curve that a typical firm in this model face is a kinked one, so that the corresponding MR curve develops a discontinuous portion. If the MC curve is such that it passes through the discontinuous portion then we have a case when there is no change in price and quantity. Therefore, price becomes sticky in that range.
Check Your Progress 6
The leader firm incorporates the reaction function of the follower firm and thereafter maximises profit.
If A is the leader and B is the follower then the tangency between the lowest (one nearest to the origin) iso-profit curve of firm B and the reaction function of A gives the equilibrium.
10.7 EXERCISES
Check that if the reaction curves are reversed then the system would have an unstable equilibrium in Counot’s duopoly model.
Suppose the duopolists face the following demand and cost functions
P = 100 – 0.5(q 1 + q 2 ) and Ci = 5 q (^) i , where i = 1,
Derive the Cournot equilibrium quantities and price and profit (of both the firms).
Describe what happens in the Bertrand model when the identical cost structure assumption is relaxed.
If the demand function is given by P = 100 – 0.5(q 1 + q 2 ) and the cost functions of the two firms are C 1 = 5q 1 and C 2 = 0.5q^22 respectively then find the equilibrium quantity and price charged by the firms if (a) firm 1 is the leader (b) firm 2 is the leader.
In the Edgeworth model will an increase in demand raise price? Why?
In the kinked demand model of oligopoly at what price does the kink occur? How useful is the model in explaining pricing under oligopoly?
Discuss the welfare aspects of each model in terms of the consumers in the society.
[Hint: Assume that the welfare of the consumers in an economy depends on the amount of output being produced]
