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 of two goods at the optimal choice is known as the Marginal Rate of Substitution (MRS), which can be expressed as \(MRS = \frac{MU_1}{MU_2} = \frac{P_1}{P_2}\). This rule implies the consumer's willingness to trade off between the two goods is equal to the market's rate of exchange (price ratio). If this rule does not hold, it suggests that either the consumer values one good more or less than the market does, and can increase their utility by reallocating their budget and adjusting their consumption of the goods. The choice cannot be utility-maximizing if the rule does not hold because the consumer has not yet reached their optimal balance between the two goods' marginal utility and prices.

Step-by-step solution

Define Marginal Utility

Marginal Utility (MU) is the additional satisfaction gained from consuming an extra unit of a good or service. For each good, we denote marginal utility as \(MU_1\) and \(MU_2\) for Good 1 and Good 2, respectively.

Define Marginal Rate of Substitution (MRS)

Marginal Rate of Substitution (MRS) is the rate at which a consumer is willing to trade off one good for another while maintaining the same level of utility or satisfaction. In other words, it is the slope of the indifference curve, which represents the consumer's preferences. Mathematically, MRS can be expressed as the ratio of the marginal utilities: \[MRS = \frac{MU_1}{MU_2}\]

Define the Optimal Choice Rule

At the optimal choice, a consumer will allocate their budget to maximize their utility. According to the optimal choice rule, the consumer will equalize the MRS with the ratio of prices of the two goods. The rule can be expressed mathematically as: \[MRS = \frac{MU_1}{MU_2} = \frac{P_1}{P_2}\]

Explain why the choice is not utility-maximizing if the rule does not hold

If the optimal choice rule does not hold, it means that \(\frac{MU_1}{MU_2} ≠ \frac{P_1}{P_2}\), which implies that the consumer's willingness to trade off between the two goods is not equal to the market's rate of exchange (price ratio). This leads to two possibilities: 1. If \(\frac{MU_1}{MU_2} > \frac{P_1}{P_2}\), the consumer values Good 1 more than the market does, so they can increase their utility by purchasing more of Good 1 and exchanging it for Good 2 in the market. 2. If \(\frac{MU_1}{MU_2} < \frac{P_1}{P_2}\), the consumer values Good 2 more than the market does, so they can increase their utility by purchasing more of Good 2 and exchanging it for Good 1 in the market. In either case, by reallocating their budget and adjusting the consumption of Good 1 and Good 2, the consumer can achieve higher utility, implying that the initial choice was not utility-maximizing. The consumer's optimum choice will be reached when the MRS is equal to the ratio of the prices of the two goods.