Question

An economy consists of two people. Person 1 has a utility function U₁ = ln(x) + y and an initial endowment of (x, y) (12,8). Person 2 has a utility function U₂ = 2ln(x) + y and an initial endowment of (x, y) = (12, 16). = = (a) Using x as the numeraire, find the equilibrium price and quantities for each person. (b) Now consider the same utility functions but with different initial endowments: (xi, yi): (4,0) and (x, y) = (20, 10). As usual, let p = 1; find the equilibrium price and quantities.¹ (c) For your answer in part b, draw the Edgeworth box. Label the initial allocation, the equilibrium allocation, and the budget line derived from the initial allocation and equilibrium prices.

Answer 1

The question asks for equilibrium prices and quantities for two agents with specific utility functions and endowments, an understanding of total utility and budget constraints, and requires plotting an Edgeworth box diagram to illustrate allocation and prices.

Step-by-step explanation:

The students question relates to finding the equilibrium price and quantities for two consumers with given utility functions and different initial endowments in an economy. To solve this, we need to understand the concept of total utility which is the total satisfaction a consumer derives from consuming a certain amount of goods and services, and how diminishing marginal utility affects consumer choices.

Additionally, we need to consider the budget constraint or budget line which represents the combinations of goods and services that a consumer can afford given their income. In such a scenario, individuals will aim to maximize their utility subject to their budget constraints, which leads to an equilibrium where the market clears, meaning that the total amount of goods supplied is equal to the total amount demanded by these two individuals.

Understanding how consumer preferences change with income effects and substitution effects in response to price changes is also crucial. Higher incomes typically lead to purchasing more normal goods, whereas for inferior goods the opposite might happen. The Edgeworth box, a tool used in microeconomics, can be utilized to illustrate the initial and equilibrium allocation of resources corresponding to the preferences and endowments of the two people in the economy.