Question

1. Profit Maximization Problem. (Similar to Ch.1 Question 9 from the book) An analyst discovers the following about the demand for attendance at a local football team's games: P- 180-4A where P is the ticket price, measured in dollars, and A is their attendance measured in thousands of fans. Assume that marginal cost is $20, and total cost is 20A. 1. Suppose the goal of the analyst is to maximize profit for the team. Set up the profit maximization problem faced by the analyst

Answer 1

P = 180 - 4A [Demand function] TR = P times A TR = (180 - 4A)A TR = 180A - 4A^2 and TC = 20A (Given Profit = Total Revenue (TR) - Total cost (TC) pi = 180A - 4A^2 - 20A pi = 160A - 4A^2 equation (1) objective is to maximize pi = 160A - 4A^2 Differentiating equation (1) w. r. t A and set it equal to 0 partial differential pi/partial differential A = 160 - 8A = 0 160 - 8A = 0 160 = 8A A = 160/8 A* = 20 implies optimal value of A P = 180 - 4A P = 180 - 4(20) P = 180 - 80 P* = 100 rightarrow optimal value of P pi = 160A - 4A^2 pi = 160(20) - 4(20)^2 pi = 3200 - 1600 pi* = 1600 rightarrow Profit level associated with the optimal level of A* and P*