Finding an Optimum for a Single-Variable Function using First and Second-Order Conditions

An individual's preferences for consumption (c) and free time (t) are represented by the utility function u(t, c) = t * c. The individual has 24 hours available per day, which can be divided between work and free time. They earn a wage of $20 per hour and spend all their income on consumption. By using the substitution method to incorporate the time and income constraint, what is the resulting utility function expressed solely in terms of free time (t)?

Logic of the Substitution Method in Constrained Choice

An individual aims to find the optimal consumption bundle that maximizes their utility, subject to a budget constraint. To achieve this, they use the substitution method to convert the two-variable optimization problem into a single-variable problem. Arrange the following steps in the correct logical order to execute this method.

Analyzing a Flawed Optimization Setup

When using the substitution method to solve a constrained choice problem, the solution that maximizes the transformed single-variable utility function must be checked separately to ensure it also satisfies the original constraint equation.

Evaluating the Substitution Method in Constrained Choice

To solve a constrained choice problem, a utility function of two variables, u(t, c), can be transformed into a function of a single variable by substituting the constraint into the utility function. For each combination of a utility function and a constraint provided, match it to the correctly substituted utility function that is expressed solely in terms of free time, t.

An individual's utility is derived from consumption (c) and free time (t), represented by the function u(t, c) = 4t + c. The individual's choices are limited by a constraint, which can be expressed as c = 10(24 - t). To find the optimal choice, the first step is to substitute the constraint into the utility function. This transforms the utility function into an expression of a single variable, t. The transformed utility function is u(t) = _________.

Setting Up a Constrained Choice Problem for Substitution

Interpreting a Transformed Utility Function