Josephine is buying candy and gum. If she is maximizing her utility for the combination of candy and gum she buys, which of the following must hold?

Question

BlackTom AI · Accepted Answer

Equalizing marginal utility per dollar spent across goods is the condition for utility maximization under a budget constraint, so MU_c/P_c = MU_g/P_g.

1) Relevant knowledge points and concepts
- Utility maximization: a consumer chooses quantities of goods (candy C and gum G) to maximize utility U(C,G) subject to a budget constraint.
- Marginal utility (MU): additional utility from one more unit of a good.
- First-order conditions (Lagrange multiplier method) yield the interior optimality condition that marginal utility per dollar spent must be equal across goods.
- Assumes an interior solution (positive amounts of both goods) and diminishing marginal utility.

2) Complete reasoning process (step by step)
Set budget: P_c C + P_g G = M (income). Form Lagrangian:
L = U(C,G) + λ(M − P_c C − P_g G).
First-order conditions:
∂L/∂C = MU_C − λ P_c = 0  => MU_C = λ P_c. 
∂L/∂G = MU_G − λ P_g = 0  => MU_G = λ P_g.
∂L/∂λ = M − P_c C − P_g G = 0 (budget holds).

3) Intermediate algebraic steps
From MU_C = λ P_c and MU_G = λ P_g, divide the two equations:
(MU_C) / (P_c) = λ = (MU_G) / (P_g),
hence MU_C / P_c = MU_G / P_g.

4) Why the correct answer is correct
This condition equates the extra utility per dollar from spending on candy to the extra utility per dollar from spending on gum. If they differed, the consumer could reallocate spending toward the higher MU per dollar good and increase total utility until equality holds.

5) Why other options are incorrect
- Total utility × price equalization: irrelevant — total utility is not the marginal decision metric and multiplying by price has no economic meaning for marginal tradeoffs.
- Marginal utility × price equalization: MU_c·P_c = MU_g·P_g is algebraically different and not derived from the FOCs.
- Total utility/price equality: uses total rather than marginal values; optimization depends on marginal changes.
- MU/Total utility equality: MU/Total is not the correct criterion; decisions are based on MU per dollar, not MU relative to total utility.

6) Conclusion
Therefore the required condition is MU_c/price_c = MU_g/price_g, which matches the provided correct answer.

Josephine is buying candy and gum. If she is maximizing her utility for the combination of candy and gum she buys, which of the following must hold?单项选择题

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