# what is the optimal allocation for this economy

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.

• Allocation in option D does not exhaust all of $y$. Is it $\left(\left(\frac{1}{4}, 0\right), \left(\frac{3}{4}, 1\right)\right)$ ?
– Amit
May 31 at 19:14

## 1 Answer

Allocation $$((x_A, y_A), (x_B, y_B)) = \left(\left(\frac{1}{4}, 0\right), \left(\frac{3}{4}, 1\right)\right)$$ is Pareto superior to $$\left(\left(\frac{1}{4}, \frac{1}{2}\right), \left(\frac{3}{4}, \frac{1}{2}\right)\right)$$, so option A is ruled out.

Allocation $$\left(\left(\frac{1}{2}, 0\right), \left(\frac{1}{2}, 1\right)\right)$$ is Pareto superior to $$\left(\left(\frac{1}{2}, \frac{1}{4}\right), \left(\frac{1}{2}, \frac{3}{4}\right)\right)$$, so option B is ruled out.

Allocation $$\left(\left(\frac{1}{2}, 0\right), \left(\frac{1}{2}, 1\right)\right)$$ is Pareto superior to $$\left(\left(\frac{1}{4}, 1\right), \left(\frac{3}{4}, 0\right)\right)$$, so option C is ruled out.

So, allocations in options A, B and C are not Pareto efficient.

• i was trying to solve this through edgeworth box, could you please see to it, i have just added that. May 31 at 20:45
• Shape of the IC looks correct to me. If you want to answer this MCQ using Edgeworth box, then draw ICs of the two individuals passing through the allocation in option D and check that it is efficient in the usual way.
– Amit
May 31 at 23:35
• Also note that not all boundaries of the Edgeworth box are efficient. First argue that if a feasible allocation is efficient then it must satisfy $y_A=0$ or $y_A=1$. Note that the converse is not true (see option C).
– Amit
May 31 at 23:43