Solving for the Second Optimal Variable via Back-Substitution

A student is determining their optimal amount of consumption (c) and free time (t). Their choice must satisfy two conditions simultaneously:

1. Optimality Condition: c / t = 15
2. Budget Constraint: c = 15(24 - t)

To find the value of the single variable 't', the budget constraint must be substituted into the optimality condition. Which of the following equations correctly shows the result of this specific substitution?

An individual's optimal choice between consumption (c) and free time (t) is determined by a system of two related equations. To find the optimal value for free time (t), the system must be reduced to a single equation and solved. Given the initial system below, arrange the subsequent equations into the correct logical sequence that solves for t.

Initial System:

1. c = 10t
2. c + 5t = 120

Solving for a Choice Variable via Substitution

Guiding a Decision-Making Process

When solving a consumer's choice problem, the standard procedure is to substitute the budget constraint into the optimality condition to solve for one variable. An alternative procedure of isolating a variable in the optimality condition and substituting it into the budget constraint would lead to an incorrect final optimal choice.

To find a single optimal variable, like free time (t), in a choice problem, an expression for the other variable, consumption (c), is taken from the budget constraint and substituted into the optimality condition. Match each system of equations (representing an optimality condition and a budget constraint) to the correct resulting equation that contains only the variable 't'.

The Logic of Substitution in Consumer Choice

Solving for an Optimal Variable with an Unarranged Constraint

A person is solving for their optimal mix of daily consumption (c) and free time (t). They correctly combine their optimality condition (c = 20t) and their budget constraint (c = 10(24 - t)) into a single new equation: 10(24 - t) = 20t. What is the primary analytical advantage of deriving this new equation?