Problems attached. No graphs required, simple computations only.
1. UA = 5 + Sqrt(MA)
UB = Sqrt(MB)
MA + MB = 25
If the society has utilitarian social welfare functional then total utility is given by
SWF = UA + UB = 5 + Sqrt(MA) + Sqrt(MB)
(i) MA = 16 and MB = 9
Then SWF = 5 + Sqrt(16) + Sqrt(9) = 5 + 4 + 3 = 12
(ii) MA = 9 and MB = 16
Then SWF = 5 + Sqrt(9) + Sqrt(16) = 5 + 3 + 4 = 12
(b) To solve what distribution maximizes social welfare we have to consider the case where the marginal utility per dollar is the same. This condition maximizes total utility.
MU_x/P_x = MU_y/P_y
Since we do not have prices here, we just consider the case where the marginal utilities are the same. The marginal utility for A is 0.5 Sqrt(MA) and that of B is 0.5 Sqrt (MB). Utility is maximized when
0.5 Sqrt(MA) = 0.5 Sqrt (MB)
This is possible when MA = MB = 0 or MA = MB.
SWF = 5 + Sqrt(12.5) + Sqrt(12.5) = 12.071 > 12 as we obtained ...
Calculate the tax revenue generated for the government and tax burden for the buyers and sellers.