Chapter 15: Problem 555

Restate the utility-maximizing rule in algebraic terms.

Short Answer

Expert verified

The utility-maximizing rule in algebraic terms can be represented as \(\frac{MU_X}{Px} = \frac{MU_Y}{Py}\), where \(MU_X\) and \(MU_Y\) are the marginal utilities of Good X and Good Y, and \(Px\) and \(Py\) are their respective prices. This rule states that the ratio of the marginal utility of each good to its price should be equal for a consumer to maximize their utility while allocating their income between different goods.

Step by step solution

Conceptualise the Consumer Problem

Consider a consumer who has a fixed income I and wants to consume two goods, Good X and Good Y. The consumer aims to maximize their utility derived from the consumption of these goods. Let the prices of Goods X and Y be Px and Py, respectively. The objective is to allocate their income in such a way that maximizes the utility.

Set Up the Budget Constraint and Utility Function

The consumer's budget constraint equation is given by the total spending on both goods, equal to their total income: \[ I = PxX + PyY \] Let the utility function be denoted by: \[ U(X, Y) \] The consumer wants to maximize this utility function, given the budget constraint.

Define the Marginal Utility of Each Good

The marginal utility refers to the additional satisfaction or utility gained by a consumer with the consumption of an additional unit of a good. The marginal utility of Good X and Good Y can be represented as partial derivatives of the utility function with respect to the quantity consumed: \[ MU_X = \frac{\partial U(X, Y)}{\partial X}\] \[ MU_Y = \frac{\partial U(X, Y)}{\partial Y}\]

Apply Utility Maximization Rule to Marginal Utilities

According to the utility-maximizing rule, the ratio of the marginal utility of each good to its price should be equal. Mathematically, that can be represented as: \[ \frac{MU_X}{Px} = \frac{MU_Y}{Py} \] This equation represents the utility-maximizing rule in algebraic terms. It shows the optimal allocation of income between Goods X and Y to maximize a consumer's utility.

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