THE Cournot Model - summary PDF

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THE COURNOT MODEL 1. OLIGOPOLY MODELS There are 3 main oligopoly models that will be considered:   

The Cournot model The Bertrand model The Stackelberg model

The first two models involve static games, while the third involves a dynamic game. Although both the Cournot and Bertrand models were developed independently, are based on different assumptions, and lead to different conclusions, there are some common features of the models that can be discussed at this stage before examining each model in detail. These features are easier to discuss in a two-firm framework, meaning a duopoly. Both models consider the situation where each firm considers the other firm’s strategy in determining its own demand function. The other firm’s strategy is considered to relate to either the output or price variable. Thus these demand functions can be expressed as reaction or response curves which show one firm’s strategy, given the other firm’s strategy. The equilibrium point is where the two response curves intersect, meaning that the two firms’ strategies coincide. To understand what all this is about we now have to consider each model separately.

2. NATURE AND ASSUMPTIONS OF COURNOT MODEL The Cournot model, originally developed in 1835, initially considered a market in which there were only two firms, A and B. In more general terms we can say that the Cournot model is based on the following assumptions: 1. There are few firms in the market and many buyers. 2. The firms produce homogeneous products; therefore each firm has to charge the same market price (the model can be extended to cover differentiated products). 3. Competition is in the form of output, meaning that each firm determines its level of output based on its estimate of the level of output of the other firm. Each firm believes that its own output strategy does not affect the strategy of its rival(s). 4. Barriers to entry exist. 5. Each firm aims to maximise profit, and assumes that the other firms do the same. An essential difference between this situation and the ones considered until now is that strategies are continuous in the Cournot model. This allows a more mathematical approach to analysis. The situation can be illustrated by using the example relating to the cartel in the previous chapter. In that case the market demand was given by P = 400 – 2Q and each firm had constant marginal costs of £40 and no fixed costs. We saw that the monopoly price and

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output were £220 and 90 units, while price and output in perfect competition were £40 and 180 units.

3. ANALYSIS The analytical procedure can be viewed as involving the following steps. Step 1 - Transform the market demand into a demand function that relates to the outputs of each of the two firms. Thus we have: P = 400 – 2(QA + QB) P = 400 – 2QA - 2QB

(3.1)

Step 2 - Derive the profit functions for each firm, which are functions of the outputs of both firms. Bearing in mind that there are no fixed costs and therefore marginal cost and average cost are equal, the profit function for firm A is as follows: ΠA = (400 – 2QA - 2QB)QA – 40QA = 400QA –2QA2 - 2QBQA – 40QA ΠA = 360QA - 2QA2 - 2QBQA

(3.2)

Step 3 - Derive the optimal output for firm A as a function of the output of firm B, by differentiating the profit function with respect to QA and setting the partial derivative equal to zero: ∂ΠA = 360 - 4QA - 2QB = 0 ∂QA 4QA = 360 – 2QB QA = 90 – 0.5QB

(3.3)

Strictly speaking, the value of QB in this equation is not known with certainty by firm A, but is an estimate. Equation 10.3 is known as the best response function or response curve of firm A. It shows how much firm A will put on the market for any amount that it estimates firm B will put on the market. The second and third steps above can then be repeated for firm B, to derive firm B’s response curve. Because of the symmetry involved, it can be easily seen that the profit function for firm B is given by: ΠB = 360QB - 2QB 2 - 2QBQA

(3.4)

And the response curve for firm B is given by: QB = 90 – 0.5QA

(3.5)

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This shows how much firm B will put on the market for any amount that it estimates firm A will put on the market. The situation can now be represented graphically, as shown in Figure 10.1. Step 4 - Solve the equations for the best response functions simultaneously to derive the Cournot equilibrium. The properties of this equilibrium will be discussed shortly. QA = 90 – 0.5QB QB = 90 – 0.5QA QA = 90 – 0.5( 90 – 0.5QA) QA = 90 – 45 + 0.25QA 0.75QA = 45 QA = 60 QB = 90 – 0.5(60) = 60

FIGURE 1

COURNOT RESPONSE CURVES

QB 180 90 QB* = 60 QA* = 60

90

180 QA

The market price can now be determined: RA + 60) P = 400 – 2(60 P = £160 We can now compare this situation with the situations discussed in the previous chapter relating to perfect competition, monopoly and cartels. This is shown in Table 1.

TABLE 1

COMPARISON OF PERFECT COMPETITION, MONOPOLY, AND COURNOT DUOPOLY RB

Market structure

Price (£)

Perfect Competition Monopoly (or cartel) Cournot Duopoly

40

Output in the industry 180

220

90

16,200

160

120

14,400

3

Profit in the industry (£) 0

Obviously, a Cournot duopoly does not make as much profit as a cartel involving the two producers; by colluding, the two firms could restrict output to increase profit. The reason for this is that, when one firm increases output, this reduces the price for the market as a whole. However, the firm does not consider the effect of the reduced revenue of the other firm, called the revenue destruction effect, since it is only concerned with maximising its own profit. Thus it expands its output more aggressively than a cartel would, since the cartel is concerned with the profit of the industry as a whole. This Cournot equilibrium is also called the Cournot-Nash equilibrium (CNE), since it satisfies the conditions stated in the last subsection regarding the nature of a Nash equilibrium. The CNE represents the situation where the strategies of the two firms ‘match’, and there will be no tendency for the firms to change their outputs; at any other pair of outputs there will be a tendency for the firms to change them, since the other firm is not producing what they estimated. The Cournot equilibrium is therefore a comparative static equilibrium. This can be illustrated in the following example of the adjustment process. Let us assume that both firms start by producing 80 units. Neither firm will be happy with their output, since they are producing more than they want, given the other firm’s actual output. Say that firm A is the first to adjust its output. It will now produce 90 – 0.5(80) = 50 units. Firm B will react to this by producing 65 units; firm A will then produce 57.5 units; firm B will then produce 61.25 units and so on, with the outputs converging on 60 units for each firm. It should be noted that the Cournot-Nash equilibrium has stronger properties than other Nash equilibria. If strictly dominated strategies are eliminated, as shown in the above example, only one strategy profile remains for rational players, the Cournot-Nash equilibrium. It can therefore be concluded that this represents a unique iterated strictly dominant strategy equilibrium of the Cournot game. All the preceding analysis has been based on a two-firm industry. As the number of firms in the industry increases, the market price is reduced and market output increases. The reason for this is related to the revenue destruction effect described earlier. With more firms in the industry, any increase in output by one firm has a smaller effect on the market price, and on its own profit, but the effect on the combined revenues of all the other firms increases in comparison to the effect on its own profit.

PROBLEM How is the equilibrium situation affected if there are 3 similar firms in the industry?

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