Yuen, a travelling salesman for snake oil, can produce the stuff at a marginal cost of 1. There are 100 potential customers in Vernon, each of whom has the following demand function for snake oil: P = 2 - y. The trouble is that none of the customers have any idea that Yuen is in town. The solution to his problem is, of course, advertising. The local radio station has explained to Yuen that their research shows that the probability that his message gets through to any one of the customers is [1 - 1/(1+A)], where A is the amount he spends on radio advertising.
a. What is Yuen's profit function?
Total demand:
Qd= y*100 = (2-P)*100=200-100P
P= (200-Qd)/100 = 2-.01Qd
revenues= Qd*P = [2 -.01Qd] *Qd
costs = TVC + ad costs = 1*Qd + A
b. What are the profit maximizing values of P and A?