Question

Consider a 2-agent, 2-good exchange economy where agent 𝑖 has utility function 𝑢𝑖 (𝑥𝑖 , 𝑦𝑖 ) = max{𝑥𝑖 , 𝑦𝑖 }, 𝑖 = 1,2. The initial endowments of goods 𝑋 and 𝑌 that the agents have are (𝑥̅̅1̅, 𝑦̅̅1̅, 𝑥̅̅2̅, 𝑦̅̅2̅) = (25, 5, 5, 5). Then select the CORRECT choice below where the price vector (𝑝𝑥, 𝑝𝑦) specified is part of a competitive equilibrium.

• A. (𝑝_𝑥, 𝑝_𝑦) = (2,1)
• B. (𝑝_𝑥, 𝑝_𝑦) = (2,2)
• C. (𝑝_𝑥, 𝑝_𝑦) = (1,2)
• D. (𝑝_𝑥, 𝑝_𝑦) = (4,2)

Answer 1

The Correct Option is B

To determine the competitive equilibrium in the given 2-agent, 2-good exchange economy, we need to analyze the utility functions and the initial endowments of both agents.

Each agent has the utility function \(u_i(x_i, y_i) = \max\{x_i, y_i\}\), which suggests that each agent derives utility based on the maximum of the two goods they possess. This means that each agent wants to balance the quantities of goods \(X\) and \(Y\) they have to maximize the lower of the two, assuming they can exchange at given prices. 

The initial endowments are:

• Agent 1: \((x_1, y_1) = (25, 5)\)
• Agent 2: \((x_2, y_2) = (5, 5)\)

Let's analyze each price vector option to see which one results in an equilibrium.

1. Price Vector: \((p_x, p_y) = (2, 1)\)
   • At these prices, good X is more expensive than good Y. Agent 1, who has more of good X initially, will try to exchange some X for Y to balance his quantities. Agent 2 would need a balancing opportunity as well, but given the prices, they would not be willing to pay a higher price per unit for X than Y for equal worth, so this is unlikely to be an equilibrium.
2. Price Vector: \((p_x, p_y) = (2, 2)\)
   • With equal prices, both goods are valued the same in the market. Agent 1 and Agent 2 can trade units of X for Y 1-to-1, perfectly allowing them to equalize their bundles according to each other's initial endowments and trading needs. Here, they reach an equilibrium where neither wants to deviate as the exchange can help balance their satisfaction derived from the number of goods.
3. Price Vector: \((p_x, p_y) = (1, 2)\)
   • Here, good Y is priced higher than X. Similar reasoning from option 1 applies here with the roles of X and Y switched. Agent 2 has no incentive to trade 1-to-1 for X being cheaper.
4. Price Vector: \((p_x, p_y) = (4, 2)\)
   • Good X is twice the price of Y, leading to disproportionate valuation, causing disequilibrium. Agent 1 would have surplus X, and trade viable but unfavorable for Agent 2, unbalancing desires for a balanced utility maximization.

Therefore, the correct price vector for a competitive equilibrium is \((p_x, p_y) = (2, 2)\) because it allows agents to exchange goods without any apparent utility loss, balancing their utilities.