An economy has $72, to divide between Hamlet and Ophelia.
Hamlet has the utility function uH = UH(mH) = 3mH, where mH denotes his quantity of money.
Ophelia has the utility function uO = UO(mO) = 42 sq rt mO, where mO denotes her quantity of money.
(a) What are the Pareto efficient allocations of the $72?
(b) Calculate the allocation of money that maximizes utilitarian social welfare. In that allocation, what is the maximized value of social welfare? How much money does Hamlet receive? How much money does Ophelia receive?
A different social welfare function is the “Rawlsian” function W(uH,uO) = min(uH,uO), meaning that welfare equals whichever of the utility values is smaller.
(c) Calculate the allocation of money that maximizes Rawlsian social welfare. Explain how maximizing the Rawlsian social welfare does or does not achieve equality in this economy.
(d) Does (selfish) Hamlet prefer the allocation that maximizes utilitarian social welfare or the allocation that maximizes Rawlsian social welfare?
(e) Now suppose that Hamlet’s utility function is scaled up to UH(mH) = 7mH. Repeat parts (b) and (c) for this situation. (Everything is the same except for Hamlet’s utility function.)
(f) Given Hamlet’s new utility function, does he prefer the allocation that maximizes utilitarian social welfare or the allocation that maximizes Rawlsian social welfare?
(g) Compare your answers to (d) and (f). Can you provide intuition for Hamlet’s preferences in these two situations?