Question

Restate the utility-maximizing rule in algebraic terms.

Short answer

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.

Key concepts

Marginal Utility

When we talk about marginal utility, we refer to the extra satisfaction a consumer obtains from consuming an additional unit of a good. It's like adding one more scoop of ice cream to your dessert — how much more happiness does it bring you? Marginal utility is crucial in decision-making processes because consumers want to maximize the pleasure they derive from each expenditure.

In the context of utility maximization, marginal utility helps us understand the benefits of consuming one more unit of a commodity. It is expressed mathematically by taking the derivative of the utility function with respect to the quantity of the good consumed. For Goods X and Y, these are denoted as:

• \( MU_X = \frac{\partial U(X, Y)}{\partial X} \): This represents the marginal utility derived from an additional unit of Good X.
• \( MU_Y = \frac{\partial U(X, Y)}{\partial Y} \): This represents the marginal utility derived from an additional unit of Good Y.

Think of marginal utility as your guide to understanding how much extra satisfaction you gain and whether it is worth the extra cost.

Budget Constraint

A budget constraint represents the trade-offs consumers can make, given their financial resources. It illustrates the limit imposed on consumers by their income and the prices of goods and services they wish to purchase.

Imagine you're on a shopping spree with only a certain amount of money in your pocket. You have to choose between different kinds of goods — perhaps a new game and a pair of shoes. Your budget constraint will tell you how many of each you can afford, given their prices and your budget.

• Mathematically, the budget constraint is expressed as: \( I = P_x X + P_y Y \) where \( I \) is your income, \( P_x \) and \( P_y \) are the prices of Goods X and Y, and \( X \) and \( Y \) are the quantities consumed.

This constraint ensures that you don't spend more than your income, forcing you to make calculated choices to maximize satisfaction within your financial limits.

Utility Function

The utility function is a mathematical representation of a consumer's preference pattern. It helps relate different quantities of goods consumed to the level of satisfaction or happiness they provide.

Think of it as a recipe that combines different ingredients (quantities of goods) to produce a dish (level of utility). Each person's utility function is unique, reflecting personal tastes and preferences.

• Typically, it's expressed as \( U(X, Y) \) where \( X \) and \( Y \) represent the amounts of two different goods.

This function plays a central role in the utility maximization problem. It provides the foundation for determining how much of each good a consumer should purchase to achieve the highest possible utility.

The objective is always to find the combination of goods that maximizes the utility function, subject to the budget constraint, ensuring consumers get the best possible satisfaction given their limits.