Answer to Question #127846 in Microeconomics for Wondater muluneh

Question

The utility function U(X, Y) = X + Y satisfies all but one of the axioms of preference discussed in class. Which one is not satisfied and explain why.
Can you explain why taking a monotonic transformation of a utility function doesn’t change the marginal rate of substitution?
If a consumer has a utility function u(x1, x2) =x1x24, what fraction of her income will she spend on good 2?
Suppose that the original budget constraint is p1x1+p2x2=m.
What is the budget constraint if we tax the consumption of good 1 at a rate of t quantity tax?

Answer 1

The utility function U(X, Y) = X + Y satisfies all but one of the axioms of preference discussed in class. Which one is not satisfied and explain why.

Preferences exhibit the following axioms: completeness, transitivity and continuity, so a utility function exists. Indifference curves are strictly downward sloping. However, the utility function U(X, Y) = X + Y does not satisfy this fact that and it requires finding the points x and y on the same indifference curve such that yi ≥ xi for all i, and yi > xi for some i. Doing this however contradicts monotonicity, which posits that the agent strictly prefers y to x.

Can you explain why taking a monotonic transformation of a utility function doesn’t change the marginal rate of substitution?

Monotonic transformation effected through addition of a constant to the utility equation eventually disappears since the constants disappear in the differentiation process and as a result, there is no change in change in the marginal rate of substitution.

If a consumer has a utility function u(x₁, x₂) =x1x2^4, what fraction of her income will she spend on good 2?

The utility function exhibits a cobb-douglas utility function and thus the fraction of income spent on good 2 will be equal to 4/(1+4)= 4/5.

Suppose that the original budget constraint is p₁x₁+p₂x₂=m. What is the budget constraint if we tax the consumption of good 1 at a rate of t quantity tax?

Introduction of tax shifts the budget constraint from the original p₁x₁+p₂x₂=m to (p₁+t)x₁+(p₂+t)x₂=m