PS2 Economics

PART II – Long Questions
5. An island nation has two residents, Teddy and Frankie. Both
Teddy and Frankie like to use boats so there is demand for a coast
guard. Teddy loves to take his boat far offshore so his utility over
private goods x and total coast guard services g is UTeddy = 2 log(x) +
10 log(2g). Frankie keeps his boat close to shore so his utility is UFrankie = 4 log(x) + 5 log(g). The total amount of coast guard services provided is g = gf + gt. Teddy and Frankie both have an income of 100, and the price of both the private good and coast guard service is 1. They are limited, for the purposes of this problem, to providing between 0 and 100 units of coast guard service.
a) How much coast guard service is hired if the government does not
intervene? How much is provided by Teddy? By Frankie?
b) What is the socially optimal amount of coast guard service? If your
answer differs from (a), why? (Hint: using brute force will not work. Instead of equalizing the sum of MRS to MRT, solve for the government’s problem, i.e. solve for the social welfare function).
c) Suppose the government is not happy with the private equilibrium,
and it decides to provide 10 units of coast guard service. It taxes Teddy and Frankie equally to pay for the new service. What is the new total amount of coast guard service? How does your answer compare to (a)? Have we achieved the social optimum? Why or why not?
d) How can the government achieve the social optimum?
6. Gruber, question 5.16. It is the same in the 2nd and 4th editions – if you have the 3rd edition, please let me know if it is different (it asks about commuters and highways).