Let M(U)a = z = 10 - x and M(U)b = z = 21 - 2y, where z is marginal utility per dollar measured in utils, x is the amount spent on product A, and y is the amount spent on product B.
Assume that the consumer has $10 to spend on A and B -- that is, x + y = 10.
How is the $10 best allocated between A and B?
How much utility will the marginal dollar yield?
The consumer maximizes his total utility when MUa = MUb
Let MUa = MUb
10 - x = 21 - 2y
2y = 11 + x ...
Given a consumer's utility functions and budget for two products, this solution shows how to calculate the consumer's utility-maximizing allocation between the two products in 87 words with calculations clearly shown.
Utility Maximization Exchange Economy
This question is about Walrasian equilibrium in an exchange economy with 2 goods and 2 consumers. Taxes are introduced in the question to solve for the equilibrium and allocation under Pareto theorem.
Consider an exchange economy with 2 goods and 2 consumers . Consumer 1's initial endowment is and consumer 2's endowment is . In each case , the first entry in the endowment vector denotes the initial endowment of good 1, and the second entry the initial endowment of good 2 . Both consumers have the same consumption set: , and the same utility function: Suppose that consumer 1's expenditure on good 1 is taxed at a rate of 50%, and that the revenue from this tax is paid as a lump-sum transfer to consumer 2.
Find the Walrasian equilibrium of this economy . Then find another allocation which Pareto-dominates the equilibrium allocation.