Charley Citrus has to leave town quickly and cannot take his car with him. Since he does not have time to seek another buyer, he must either sell it to his neighbor Harriet Hardnose or simply destroy it. Charley and Harriet both know that the car is worth $500 to Harriet. There is no time for negotiation. Harriet must make one offer for the car and Charley will decide either to sell it to her at that price or destroy it. Harriet has no sympathy for Charley and his dilemma, but she does know that Charley has a temper and she fears that if she offers too little, Charley might get angry and destroy the car. In particular, she believes that the probability that Charley will sell her the car is x/500 for any x ≤ $500. Harriet realizes that if she can buy the car for $x, her profit will be $500 − x, but if he destroys the car, her profit is zero.
(a) Write an expression for Harriet’s expected profit as a function of x. _____________ What price should she offer Charley in order to maximize her expected profit? _________
(b) Write an expression for Harriet’s expected profit as a function of x. _________ What price should she offer in order to maximize her expected profit? __________
(c) Suppose that Harriet thinks that Charley is sure to sell the car if she offers more than $200 and that the probability that Charley will sell her the car is x/200 for x ≤ 200. What price should she offer in order to maximize her expected profit? _________