Andy purchases only two goods, apples (a) and kumquats (k) ....

Andy purchases only two goods, apples (a) and kumquats $(k) .$ He has an income of $\$ 40$ and can buy apples at $\$ 2$ per pound and kumquats at $\$ 4$ per pound. His utility function is $U(a, k)=3 a+5 k$ That is, his (constant) marginal utility for apples is 3 and his marginal utility for kumquats is $5 .$ What bundle of apples and kumquats should he purchase to maximize his utility? Why? A

Key Concepts

Corner Solutions in Linear Utility Maximization

In cases where the utility function is linear, the consumer’s optimization often leads to a corner solution, where all available income is spent on a single good. This occurs because the marginal utility per dollar is constant for each good, and when one good offers a higher rate than the other, there is no benefit to diversifying the expenditure; instead, the consumer maximizes utility by allocating all resources to the good with the highest marginal utility per dollar.

Marginal Utility per Dollar

Marginal utility per dollar measures the additional utility obtained per unit of currency spent on a good. It is calculated by dividing the marginal utility of a good by its price, and it assists consumers in making efficient spending decisions by identifying which good provides the most utility for each dollar spent.

Marginal Utility

Marginal utility is the additional satisfaction or benefit gained from consuming one more unit of a good. In decision-making, comparing the marginal utilities of different goods helps determine how a consumer allocates their budget, particularly when the marginal utilities are constant as in linear utility functions.

Utility Maximization

Utility maximization is the process of choosing the combination of goods that provides the highest level of satisfaction or utility, subject to the consumer’s budget constraint. This principle underpins consumer decision-making, where the goal is to achieve the greatest possible happiness or benefit from limited resources.

Budget Constraint

The budget constraint represents all combinations of goods that a consumer can purchase given their limited income and the prices of the goods. It is a fundamental concept in microeconomics that shows the trade-offs available to a consumer because spending more on one good implies spending less on another.

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