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

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