So far, we have assumed that the two firms in the duopoly produce an identical product. We also assumed an (inverse) industry demand curve given by P = 100 – Y = 100 – (y1 + y2), and zero costs. N0w let the two firms produce somewhat different goods, and define s as a similarity index of the two firms’ products, where s ranges from 1 (identical products) to 0 (completely different products, so neither has any effect on the market for the other). We now assume that P1 = 100 – y1 – s y2 P2 = 100 – s y1 – y2. Assume quantity competition, meaning that each firm observes the quantity sold by the other firm and reacts to it, as in the Cournot model.
a. Assume s = ½ . Find and graph the firms’ reaction curves, and find the Nash-Cournot solution (i.e. the quantity produced by each firm, the price charged by each firm, and the profit earned by each firm). b. What happens to the price, quantity, and profit of each firm as s approaches zero? c. What happens to the price, quantity, and profit of each firm as s approaches 1?
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The two firms manufacture items 1 and 2 respectively. The similarity index of the products is given by s. The price firm 1 gets for its products is given by P1 = 100 – y1 – s*y2 and the price firm 2 gets for its products is P2 = 100 – s*y1 – y2
Each of the firms would like to increase the number of units bought from it.
If s = 1/2:
The profit made by firm 1 is given by
P1*y1 = y1*(100 - y1 - 0.5*y2)
The derivative of this taking y2 constant is 100 - 2*y1 - 0.5*y2
Equating the derivative to 0 gives 100 - 2*y1 - 0.5*y2 = 0 or y1 = 50 - 0.25*y2
Similarly the firm 2 also has a similar expression to maximize profits.
y2 = 50 - 0.25*y1
Equating the 2 gives y1 = y2
Each of firms should produce 40 units and the price at which they are sold is $40. They make a profit of $1600
As s approaches 0, each firm makes 50 products and they are sold at $50 to give a profit of $2500.
As s approaches 1, each firm makes 100/3 products and they are sold at $33.33 to give a profit $1111
It should be noticed that the total profits decrease as the similarity of the products increases.
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