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Duopoly and residual demand

September 29, 2011

The model of a monopoly firm I made had a demand function of Q = 500 – P , no fixed cost and a constant marginal cost of 150.

Monopoly model

The firm was producing output of 175, selling at a price of 325. It was making profits of 30625 and had a Lerner Index of 0.538.

Now what would happen if another firm entered the market? First consider this being an identical firm, again with no fixed cost and a constant marginal cost of 150, and assume that the two firms could not differentiate their product in any way, they were producing identical products. The market has now gone from monopoly to duopoly.

Call the first firm (the model firm) firm A and the new rival firm B.

Now that there is another firm in the market, firm A does not have a monopoly on the whole market demand curve of Q = 500 – P. Instead it faces a residual demand curve. The residual demand curve is the market demand curve minus the quantity supplied by other firms, we can write this D^r (P) = D (P) - S^o (P). The P in brackets indicates that the quantities are functions of price, like the original demand curve.

So in this model, firm A now faces a demand curve ofQ_A = 500 - P - Q_B. This is fairly logical. When there was no other firm supplying the market, it faced a demand curve of Q = 500 – P, but if another firm comes in and supplies 100 to the market, then firm A faces a demand curve of 400 – P.

Each firm’s residual demand curve depends on the output decision of the other – or if they do not know the output decision of the other, the amount they expect the other to produce. Each firm will need to have a good estimate of what its residual demand curve will be in order to choose the correct quantity of output to produce, as the firm will want to produce the quantity where MR = MC.

Consider what would happen if firm B entered the market unknown to firm A, and produced 175 just like firm A did. Now with both firms producing 175, the total amount of output supplied to the market is 350, so the market price falls to 500-350 = 150.

On this graph the duopoly market price, Pd, is 150, which is just the same as each firm’s marginal cost. The quantity each firm is producing (QA and QB) is 175, and the duopoly market quantity, Qd, is 350. As each firm is selling at a price equal to its marginal cost this is like a competitive market, each firm is breaking even, neither firm is making a profit. This is clearly not an optimal choice for either firm.

The key concept to duopoly strategic behaviour (which can be extended to strategic behaviour with any number of firms) is that the output decision of each firm depends on the output decision it expects from its rival firm.

If both firms know that there is another firm in the market, then they need a framework for thinking about what is the optimal output to produce bearing in mind the decision of the other. This is where best response curves come in.

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