Assume that Alegra and Blake are the only two people in an economy and total available income is fixed at $100. Alegra's utility is given by

U_A = (y_A 20)^2/3 where y_A is Alegra's income and Blake's utility is U_B = (y_B 10)^2/3 where y_B is Blake's income.

(a) In this case, there is only one condition that a Pareto efficient allocation of income must satisfy. What is it?

(b) What distribution of income maximizes social welfare when the social welfare function is W₁ = U_A + U_B ?

(c) What distribution of income maximizes social welfare when the social welfare function is W₂ = 6U_A + 3U_B ?

(d) What distribution of income maximizes social welfare when the social welfare function is W₃ = min(U_A,U_B)?