Question

Consider a 2-agent, 2-good economy with an aggregate endowment of 30 units of good 𝑋 and 10 units of good 𝑌. Agent 𝑖 has utility function
u_i (x_i , y_i ) = max {x_i , y_i }, 𝑖 = 1, 2. 
Select the choices below in which the specified allocation of the goods to the agents is Pareto optimal for this economy

• A. (𝑥₁, 𝑦₁, 𝑥₂, 𝑦₂ ) = (5, 5, 25, 5)
• B. (𝑥₁, 𝑦₁, 𝑥₂, 𝑦₂ ) = (10, 10, 20, 0)
• C. (𝑥₁, 𝑦₁, 𝑥₂, 𝑦₂ ) = (30, 0, 0, 10)
• D. (𝑥₁, 𝑦₁, 𝑥₂, 𝑦₂ ) = (0, 10, 30, 0)

Answer 1

The Correct Option is C, D

To determine which allocations of the goods are Pareto optimal, we need to consider the utility functions of the agents and the total endowments available in this economy.

Given: 

• Total endowment = 30 units of good X and 10 units of good Y.
• Utility function for each agent \( u_i(x_i, y_i) = \max \{x_i, y_i\} \), \( i = 1, 2 \).

An allocation is Pareto optimal if no reallocation can make one agent better off without making the other agent worse off. With the max utility function, each agent's utility is determined by their largest allocation of either good X or good Y.

We will evaluate each option for Pareto optimality:

1. Option 1: \((x_1, y_1, x_2, y_2) = (5, 5, 25, 5)\)
   • Agent 1's utility = \(\max(5, 5) = 5\)
   • Agent 2's utility = \(\max(25, 5) = 25\)
   • Possible improvement: Redistributing the goods can increase Agent 1's utility without decreasing Agent 2's utility significantly.
2. Option 2: \((x_1, y_1, x_2, y_2) = (10, 10, 20, 0)\)
   • Agent 1's utility = \(\max(10, 10) = 10\)
   • Agent 2's utility = \(\max(20, 0) = 20\)
   • Possible improvement: Reallocations could potentially increase one agent's utility without reducing the other’s.
3. Option 3: \((x_1, y_1, x_2, y_2) = (30, 0, 0, 10)\)
   • Agent 1's utility = \(\max(30, 0) = 30\)
   • Agent 2's utility = \(\max(0, 10) = 10\)
   • No reallocation possible that keeps one agent’s utility constant while increasing the other’s without making the first worse off.
   • This allocation is Pareto optimal.
4. Option 4: \((x_1, y_1, x_2, y_2) = (0, 10, 30, 0)\)
   • Agent 1's utility = \(\max(0, 10) = 10\)
   • Agent 2's utility = \(\max(30, 0) = 30\)
   • No reallocation possible that keeps one agent’s utility constant while increasing the other’s without making the first worse off.
   • This allocation is Pareto optimal.

Conclusion: The Pareto optimal allocations are Option 3: \((x_1, y_1, x_2, y_2) = (30, 0, 0, 10)\) and Option 4: \((x_1, y_1, x_2, y_2) = (0, 10, 30, 0)\), where both goods are allocated to one agent in such a way that no improvements can be made without worsening the other agent's situation.