Point C as the Hypothetical Choice in Figure E3.4

Simplification of the Utility Equivalence Equation in Figure E3.4

An individual's daily well-being is modeled by the expression w × t², where w is their hourly wage and t is the hours of free time they choose. Initially, their wage is $20 per hour. After a promotion, their wage increases to $25 per hour, and they are observed to choose 16 hours of free time per day. An economist wants to determine the level of well-being at the new wage and then find the hypothetical choice of free time (t_H) that would provide this same level of well-being if the individual were still facing the original $20 wage. Which equation correctly represents the relationship needed to find t_H?

Formulating a Utility Equivalence Equation

Calculating Hypothetical Free Time for Utility Equivalence

An individual's satisfaction from their daily choices is modeled by the expression w × t², where w is their hourly wage and t is the hours of free time they enjoy. This individual's wage increases from $50/hour to $72/hour. At the new, higher wage, they are observed to choose 10 hours of free time. To find the hypothetical amount of free time (t_H) that would provide this same level of satisfaction if they were still earning their original wage, the following equation must be solved: 50 × t_H² = ____ × 10².

An analyst is studying a person whose daily satisfaction is represented by the formula w × t², where w is the hourly wage and t is hours of free time. The person's wage increases from $49/hour to $64/hour, and at the new wage, they choose 7 hours of free time. The analyst states that to find the hypothetical amount of free time (t_H) that would have provided this same level of satisfaction at the original wage, one must solve the equation: $64 × t_H² = $49 × 7². Is the analyst's statement correct?

An economist is studying how a wage change affects an individual's choice of free time. The individual's daily satisfaction is modeled by the expression U = w × t², where w is the hourly wage and t is hours of free time. The wage increases from w_initial to w_final, and the individual is now observed choosing t_final hours of free time. The economist sets up the following equation to solve for a hypothetical amount of free time, t_hypothetical:

w_initial × (t_hypothetical)² = w_final × (t_final)²

What does the value of t_hypothetical represent in this context?

An individual's daily satisfaction is modeled by the expression U = w × t², where U is the satisfaction level, w is the hourly wage, and t is hours of free time. This person's wage increases from $36/hour to $64/hour. At the new, higher wage, they choose to have 9 hours of free time. To analyze this change, an economist sets up the following equation to find a hypothetical choice of free time (t_H): $36 × t_H² = $64 × 9². Match each mathematical expression from the equation to the concept it represents.

An economist wants to analyze the impact of a wage increase on an individual's choices. The individual's daily satisfaction is modeled by the expression U = w × t², where w is the hourly wage and t is hours of free time. The wage increases from w_initial to w_final, and at the new wage, the individual chooses t_final hours of free time. The economist needs to find the hypothetical amount of free time (t_hypothetical) that would provide the same level of satisfaction as the final choice, but if the individual were still facing the w_initial. Arrange the following steps in the correct logical order to solve for t_hypothetical.

Evaluating an Economic Interpretation

An economist is analyzing an individual's response to a wage change, where daily satisfaction is modeled by the expression U = w × t² (w = hourly wage, t = hours of free time). After the individual's wage increased to $72/hour, they were observed to choose 10 hours of free time. The economist calculated that to achieve this same level of satisfaction at the original wage, the individual would have needed 12 hours of free time. Based on this analysis, what was the individual's original hourly wage?

An individual's satisfaction from their daily choices is modeled by the expression w × t², where w is their hourly wage and t is the hours of free time they enjoy. This individual's wage increases from $50/hour to $72/hour. At the new, higher wage, they are observed to choose 10 hours of free time. To find the hypothetical amount of free time (t_H) that would provide this same level of satisfaction if they were still earning their original wage, the following equation must be solved: 50 × t_H² = ____ × 10².