Negative Derivative of the Optimal Free Time Function Leads to Reduced Free Time

A new company creates a mobile app that allows people to pay someone to wait in line for them at a popular new restaurant. Despite the fact that both the person paying and the person being paid willingly agree to the exchange, the service is widely criticized as being unfair. Which of the following best explains the basis for this societal objection?

An economist observes that after a country-wide increase in the minimum wage, many low-wage workers who were previously working two jobs have started working only one, thereby increasing their leisure time. Let t* represent the optimal amount of free time and w represent the wage rate. Based on this observation, what can be inferred about the derivative of the optimal free time function with respect to the wage for this group of workers?

Labor Supply Decision Analysis

True or False: For an individual choosing between work and leisure, a positive value for the derivative of their optimal free time function with respect to their wage (dt*/dw > 0) indicates that the opportunity cost of an hour of free time is the dominant factor in their decision-making when wages rise.

Analyzing a Freelancer's Work-Leisure Choice

An individual's optimal amount of free time, denoted as t*, changes in response to a change in their wage, w. Match each mathematical expression for the rate of this change (the derivative) with the corresponding behavioral outcome.

Analyzing Worker Responses to Wage Increases

An empirical study finds that for a specific group of workers, a 10% increase in the hourly wage consistently leads to them choosing to work fewer hours and take more vacation time. This observed behavior indicates that for this group, the derivative of the optimal free time function with respect to the wage, $\frac{dt^*}{dw}$, has a ________ sign.

An employee receives a significant pay raise. In response, they decide to reduce their weekly work schedule to pursue a personal project. Arrange the following steps in the correct logical order to determine the sign of the derivative of this employee's optimal free time function with respect to their wage.

A government is considering a significant income tax cut, which would increase the net hourly wage for all workers. Advisor A predicts this will lead to people working more hours, thereby boosting economic output. Advisor B predicts that people will choose to work fewer hours, spending more time on leisure activities. Which of the following statements provides the strongest economic justification for Advisor B's prediction?

True or False: For an individual choosing between work and leisure, a positive value for the derivative of their optimal free time function with respect to their wage (dt*/dw > 0) indicates that the opportunity cost of an hour of free time is the dominant factor in their decision-making when wages rise.