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Chapter 22: Problem 5

A family with two adult members seeks to maximize a utility function of the form \\[ U\left(C, H_{1}, H_{2}\right) \\] where \(C\) is family consumption and \(H_{1}\) and \(H_{2}\) are hours of leisure of each family member. Choices are constrained by \\[ C=w_{1}\left(24-H_{1}\right)+w_{2}\left(24-H_{2}\right)+N \\] where \(w_{1}\) and \(w_{2}\) are the wages of each family member and \(N\) is nonlabor income. a. Without attempting a mathematical presentation, use the notions of substitution and income effects to discuss the likely signs of the cross- substitution effects \(\partial H_{1} / \partial w_{2}\) and \(\partial H_{2} / \partial w_{1}\) b. Suppose that one family member (say, individual 1 ) can work in the home, thereby converting leisure hours into consumption according to the function \\[ C_{1}=f\left(H_{1}\right) \\] where \(f^{\prime}>0, f^{\prime \prime}<0 .\) How might this additional option affect the optimal division of work among family members?

Short Answer

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Question: Discuss the likely signs of cross-substitution effects between family members when considering consumption and leisure, and explain how the presence of a home work function might affect the optimal division of work among family members. Answer: The cross-substitution effects between family members can have either positive or negative signs. A negative sign indicates that when one family member's wage increases and they work more, the other family member compensates by working less and enjoying more leisure. A positive sign implies that both family members decide to work more to capitalize on higher income possibilities. When considering a home work function, the optimal division of work among family members depends on the efficiency of the home work function relative to market wages and the preferences of both family members. The presence of a home work function might lead one family member to work more at home and enjoy more leisure, while the other family member might decide to work fewer hours as the desired consumption is achieved through home work.

Step by step solution

01

Explain the Substitution and Income Effects

The substitution effect is the change in consumption and leisure due to a change in wages, with the goal of keeping the utility constant. In this case, if the wage of one family member increases, they may work more (reducing leisure) since the opportunity cost of leisure has gone up. The income effect is the change in consumption and leisure caused by a change in income, holding the wage constant. If a family member's wage increases, the overall income increases, which can result in consuming more goods and leisure.
02

Contrast Cross-Substitution Effects

The cross-substitution effects relate to how the change in one family member's wage will impact the other family member's leisure (\(\partial H_{1}/\partial w_{2}\) and \(\partial H_{2}/\partial w_{1}\)). Since leisure has value, an increase in one member's wage can potentially change the other member's leisure preferences. For example, if member 1's wage increases, they may work more, reducing their leisure. This can lead to member 2 compensating for the lost income by working less and enjoying more leisure, in which case the sign of the cross-substitution effect would be negative. Alternatively, both members might decide to work more as a result of higher income possibilities, leading to a positive cross-substitution effect. #Part B#
03

Introduce the Home Work Function

Given that individual 1 can work at home, we can incorporate a function for at-home consumption as \(C_{1} = f(H_{1})\), where \(f^{\prime} > 0\) and \(f^{\prime \prime} < 0\). This means as individual 1 devotes more hours to home work, their consumption increases at a decreasing rate.
04

Analyze the Impact on Optimal Division of Work

The presence of the home work function adds a new element to the decision-making process for the family. If the home work function is more efficient (higher returns as compared to market wage rates), individual 1 may choose to work at home rather than market work, leading to more consumption without reducing leisure to the same extent as market work. This will depend on how well the returns from at-home work compare to the wages earned from formal work. Individual 2's decision may also be influenced by the home work function. If individual 1's home work contribution helps to achieve the desired level of consumption, individual 2 might decide to work fewer hours and spend more time on leisure, depending on their preferences. Thus, the optimal division of work among family members will depend on the efficiency of the home work function relative to market wages rates, and the preferences of both family members.

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Most popular questions from this chapter

Using the concept of the opportunity cost of time, discuss the following: a. Which persons might you expect to pay the higher fares to fly the faster Concorde to Europe? b. Which individuals do you expect would be more likely to stand in long lines and even camp out overnight to purchase tickets to a sporting event? c. Are greens fees a larger fraction of the total cost of a golf game for a prospering physician or a peanut vendor? d. How would the degree of traffic congestion affect which individuals drive to work and which take mass transit?

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