A two-person two commodity economy has social endowment of x = 1 unit of food and y = 1 unit of wine. Agents preferences are increasing in own consumption but decreasing in wine consumption of the other person. Preferences of agents A and B are as follows, $$u_A(x_A, y_A, y_B) = x_A[1 + max(y_A − y_B, 0)],$$ $$u_B(x_B, y_B, y_A) = x_B[1 + max(y_B − y_A, 0)],$$ where A consumes $x_A$ and $y_A$ units of x and y respectively, similarly B’s consumption is $x_B$ and $y_B$. Which of the following is a Pareto optimum allocation.
$A. (1/4,1/2) \ (3/4,1/2)$
$B. (1/2,1/4) \ (1/2,3/4)$
$C. (1/4,1) \ (3/4,0)$
$D. (1/4,0) \ (3/4,1)$
Answer is supposed to be the option D
My approach: if we take out the utilities associated with the given allocations then we'll get that Option B and C are providing the same utilities and for Agent B, option D is providing the best satisfaction.
the answer must be a boundary solution according to me
i absolutely do not understand how to proceed after that. can someone please guide me? thanks a lot.