In this problem, we’ll compare the total-surplus-maximizing advertising level and the equilibrium advertising level in the Grossman and Shapiro model (which we discussed in Wednesday’s class). The utility of a consumer that purchases a unit of the good from a firm that is x units away and purchases at a price p is 1 − p − tx.

a. In terms of X1 and X2, what fraction of consumers receive ads from both firms? Feel free to assume X1 = X2. What is the average transportation cost for consumers that receive ads from both firms?

b. In terms of X1 and X2, what fraction of consumers receive ads from only one firm? Feel free to assume X1 = X2. What is the average transportation cost for consumers that receive ads from only one firm?

c. Using your answer to parts (a) and (b), compute the total surplus, making sure to subtract the advertising cost, κ X1 (squared)/2 + κ X2(squared)/2 , paid by the two firms. Feel free to assume X1 = X2.

d. Find the total-surplus-maximzing X∗.

e. For κ = 1/2 , find the t for which the competitive equilibrium X matches the socially optimal level. For high/low values of t, is there too little or too much advertising? Hint: It would be sufficient to check how X∗ and Xequilibrium compare when t = 1/16 , t = 1/4 , and t = 1.