Question

8.B.1A There are I firms in an industry. Each can try to convince Congress to give the industry a subsidy. Let h, denote the number of hours of effort put in by firm i, and let c,(hwi(h) where w, is a positive constant, be the cost of this effort to firm i. When the effort levels of the firms are (hi, . . . . h), the value of the subsidy that gets approved is αΣί hi + β(Π, A), where a and B are constants. Consider a game in which the firms decide simultaneously and independently how many hours they will each devote to this effort. Show that each firm has a strictly dominant strategy if and only if B0. What is firm i's strictly dominant strategy when this is so?

Answer 1

Product_1 = alpha (h_1 + h_2 + ellipsis + h_n) + beta (h_1. h_2 ellipsis h_n) - 2w_1 h_1 Best response of 1st person: partial differential product_1/ partial differential h_1 = alpha + beta (h_2 ellipsis h_n) - 2w_1 = 0 alpha + beta (h_2 ellipsis h_n) = 2w_1 Best response for i^th firm is: partial differential product_i/ partial differential h_i = alpha + beta product^i - 1_i = 1 h_i - 2w_i = 0 alpha + beta product^i - 1_i = 1 h_i = 2w_i If beta = 0 then, product_i = alpha sigma^n_i = 1 h_i - 2w_i h_i Best response in this case: partial differential product_i/ partial differential h_i = alpha - 2w_i = 0 alpha = 2w_i Intersection of Best Responses of every player is Nash equilibrium.