Product Testing The total cost for a product-testing firm is C ( q ) = 60 + 10 q
ID: 2439719 • Letter: P
Question
Product Testing
The total cost for a product-testing firm is C(q) = 60 + 10q, where q is the number of products tested. The price of a product test equals the average cost per test, and each corporation in a region purchases one product test per year from a product-testing firm in the same city. All other inputs are ubiquitous (available at the same price at all locations). Suppose five corporations are initially distributed uniformly, with one corporation in each city (A, B, C, D, E).
a. Is the initial distribution a Nash equilibrium? Demonstrate with a suitable unilateral deviation.
b. Describe the Nash equilibrium in terms of (i) the distribution of firms and (ii) the price of a product test
Explanation / Answer
As per the given information, the price of a product test equals the average cost per test, and each corporation in a region purchases one product test per year from a product-testing firm in the same city the total cost for a product-testing firm is C(q) = 60 + 10q
a. When the total cost for a product-testing firm is C(q) = 60 + 10q the AC = C(q)/q = 60/q + 10 and MC = 10, thus in equilibrium P > MC for all firms. Therefore, we can say the initital distribution is a Nash equilibrium and a new unilateral deviation will lower payoff because if a corporation in a region A buys one product test per year from a product testing firm in the city B and similarly B corporation buys from city A leaving other things same the average cost of purchase and transporation in will be higher for A and B corporation which means the overall payoff will fall and unilateral deviation will be ineffective.
b. The nash equilibrium in terms of distribution of firms means the minimum distance or transporation cost for the corporations to buy from a product-testing firms in the same area. The nash equilibrium in terms of price of a product test means that the price turn out to be competitive price and fall equal to marginal cost for all the firms. The nash equilibrium in terms of price occur at P = MC therefore, it is not nash equilibrium in terms of price here.
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