Social Welfare: A social planner has decided to allocate income between two people so as to maximize the sum of utilities of persons 1 and 2 (W = U1 + U2). Each person’s utility function is given by U(Yi) = √Yi where Yi is person i’s income. The total income available in this two-person society is Y1 + Y2 = Y¯ .
(a) Determine the marginal rate of transformation of person 1’s income into person’s 2’s income, i.e., dY2/dY1. 2
(b) Determine the planner’s marginal rate of substitution between person 1’s income and person 2’s income, i.e. dY2/dY1 from the social welfare function.
(c) Find the socially optimal division of the total income Y¯ .
(d) Now suppose it is “more expensive” for the planner to give money to person 1 than to person 2. (Perhaps person 1 is forgetful and loses money, or perhaps person 1 is frequently robbed.) To capture this notion, let 2Y1 + Y2 = Y¯ . What is the new optimal allocation of Y¯ ?