Reference no: EM132414058

This problem concerns production externality. Life has got tougher in Gotham City and Mr. Batman has lost all of his businesses to bad guys, except for ?rm 1, which produces apples. Though less rich than before, Mr. Batman is still a cheerful man and he sometimes sings while producing apples (but he is better at ?ghting than at singing if I may add). Singing helps with the production to some extent because it reminds Mr. Batman of good old days but it becomes costly if he sings too much. Let a be the amount of apples and s the amount of singing. The cost function for Mr. Batman (?rm 1) is given by c1(a,s) = a2 + s2 -2s for all a,s ≥ 0. Let pa > 0 be the price of an apple. Tony Stark (yes, that Tony Stark again) has moved to Gotham City recently and acquired ?rm 2, which produces bananas. Mr. Stark and Mr. Batman are neighbors. So when producing bananas, Mr. Stark can hear Mr. Batman singing, which makes him happy and thus reduces the production cost. Let b be the amount of bananas. The cost function for Mr. Stark (?rm 2) is given by c2(b,s) = 2b2 -s. Let pb > 0 be the price of a banana. Now we search for the market outcome arising from the self-interested behavior of the two ?rms and compare it to a social optimum.

(a) Set up and solve the pro?t maximization problem for Mr. Batman. Find the optimal choice (a∗,s∗).

(b) Set up and solve the pro?t maximization problem for Mr. Stark. Find the optimal choice b∗.

(c) The choice (a∗,s∗,b∗) may not be socially optimal. To ?nd a socially optimal choice, suppose that the two ?rms are merged. Set up and solve the pro?t maximization problem for the merged ?rm. Find the optimal choice (ˆ a, ˆ s,ˆ b).

(d) Compare the choices (a∗,s∗,b∗) and (ˆ a, ˆ s,ˆ b). How does the externality of singing a?ect the optimal choice of s? Explain using the concepts of private bene?t and social bene?t.