A rational consumer maximizes utility subject to her budget constraint by choosing a consumption bundle where the

Question

BlackTom AI · Accepted Answer

At an interior optimum the indifference curve is tangent to the budget line, so the marginal rate of substitution (MRS) must equal the relative price ratio (Px/Py).

1) Relevant concepts:
- Budget constraint: Px x + Py y = I (income I, prices Px and Py).
- Marginal utility (MU): additional utility from one more unit of a good.
- Marginal rate of substitution (MRS): MUx/MUy, the rate at which a consumer will trade y for x while keeping utility constant (absolute slope of indifference curve).
- Utility maximization condition (interior solution): indifference curve tangent to budget line, so slopes equal.

2) Step-by-step reasoning using Lagrangian:
Set up L = U(x,y) + λ(I - Px x - Py y).
First-order conditions:
(a) ∂L/∂x = MUx - λ Px = 0 → MUx = λ Px.
(b) ∂L/∂y = MUy - λ Py = 0 → MUy = λ Py.
Divide (a) by (b): MUx/MUy = (λ Px)/(λ Py) = Px/Py.
Because MRS = MUx/MUy, we get MRS = Px/Py.

3) Intermediate algebraic steps shown above: MUx = λ Px and MUy = λ Py lead directly to MUx/MUy = Px/Py.

4) Why the correct answer is correct:
Equality MRS = Px/Py means the consumer equates the marginal rate at which she is willing to substitute goods (marginal benefit tradeoff) with the market rate of substitution (relative prices). At that point no small reallocation of spending raises utility while staying on the budget line.

5) Why other options are incorrect:
- "MRS is maximized": utility maximization requires MRS equals price ratio, not that MRS itself is maximized.
- "Ratios of all marginal utilities are equal": this would mean MUx = MUy (or MUx/MUy = 1), which is not generally true; the relevant ratio equals the price ratio, not 1.
- "Slope of the indifference curve exceeds the slope of the budget constraint": if the indifference curve’s slope exceeded the budget slope at the chosen bundle, you could reallocate to increase utility; optimum requires slopes equal, not one exceeding the other.

Conclusion: The consumer chooses the bundle where the marginal rate of substitution equals the relative price ratio, matching the provided correct answer.

A rational consumer maximizes utility subject to her budget constraint by choosing a consumption bundle where the单项选择题

类似问题

The figure shows Karim’s budget constraint, or feasibility frontier, on the same graph as Karim’s indifference curves. Based on this information, read the following statements and select the correct one(s).

The goal of utility maximization is to allocate your in order to maximize your .

You own a bond that pays $64 in interest annually. The face value is $1,000 and the current market price is $1,021.61. The bond matures in 11 years. What is the yield to maturity?

Which of the following statements is CORRECT?

It will cost $9,600 to acquire an ice cream cart that is expected to produce cash inflows of $3,600 a year for three years. After the three years, the cart is expected to be worthless. What is the payback period?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单