Question

What is the rule relating the ratio of marginal utility to prices of two goods at the optimal choice? Explain why, if this rule does not hold, the choice cannot be utility-maximizing.

Short answer

The rule relating the ratio of marginal utility to prices at the optimal choice is \(\frac{MU_1}{P_1} = \frac{MU_2}{P_2}\), which ensures that the marginal utility per dollar spent on each good is the same. This rule is necessary for utility maximization because if it does not hold, the consumer could increase their total utility by reallocating their expenditure between the two goods while staying within their budget constraint.

Step-by-step solution

Understand the Concept of Marginal Utility

Marginal utility refers to the additional satisfaction or utility gained from consuming an additional unit of a good or service. Marginal utility plays a crucial role in determining the optimal choice of goods by consumers, as it helps to find the combination of goods that maximizes the total utility within the budget constraint.

Define the Optimal Choice

Optimal choice refers to the combination of goods and services that maximizes total utility for the consumer while staying within their budget constraint. It means that the consumer allocates their income in such a way that the utility of the last dollar spent on each good is equal to the utility of the last dollar spent on any other good.

Discuss the Budget Constraint

The budget constraint represents the combinations of two goods that the consumer can afford given the prices of the goods and the consumer's income. Mathematically, the budget constraint can be expressed as \(P_1Q_1 + P_2Q_2 \le I\), where \(P_1\), \(P_2\) are the prices of good 1 and good 2, respectively, \(Q_1\) and \(Q_2\) are the quantities of good 1 and good 2, respectively, and I is the consumer's income.

Define the Marginal Rate of Substitution (MRS)

The marginal rate of substitution (MRS) refers to the rate at which a consumer is willing to trade one good for another while keeping the utility constant. The MRS depends on the consumer's preferences for the two goods and is equal to the negative slope of an indifference curve (a curve that represents combinations of goods that provide the consumer with equal utility). Mathematically, MRS is defined as the ratio of marginal utilities: MRS = \(MU_1/MU_2\), where \(MU_1\) and \(MU_2\) are the marginal utilities of good 1 and good 2, respectively.

State the Rule for the Optimal Choice

The rule relating the ratio of marginal utility to prices at the optimal choice can be derived from the condition that the consumer allocates income in such a way that the marginal utility per dollar spent on each good is the same. Mathematically, it can be expressed as \(\frac{MU_1}{P_1} = \frac{MU_2}{P_2}\) or \(MU_1/P_1 = MU_2/P_2\).

Explain Why the Rule is Necessary for Utility Maximization

If the rule does not hold, i.e., \(\frac{MU_1}{P_1} \ne \frac{MU_2}{P_2}\), then the consumer could increase their utility by reallocating their expenditure between the two goods. Suppose \(\frac{MU_1}{P_1} > \frac{MU_2}{P_2}\). In this case, the consumer would gain more utility by spending another dollar on good 1 than by spending it on good 2. By reallocating their expenditure from good 2 to good 1, the consumer could increase their total utility while still staying within their budget constraint. In conclusion, the rule relating the ratio of marginal utility to prices at the optimal choice is \(\frac{MU_1}{P_1} = \frac{MU_2}{P_2}\), and if this rule does not hold, the choice cannot be utility-maximizing.