# Consider inventions such as washing machines or self-propelled vacuum cleaners. Such inventions reduce the amount of time

## Question:

Consider inventions such as washing machines or self-propelled vacuum cleaners. Such inventions reduce the amount of time individuals have to spend on basic household chores
and thus in essence increase their leisure endowments.
A: Suppose that we wanted to determine the aggregate impact such labor saving technologies will have on a particular labor market in which the wage is w.
(a) Draw a graph with leisure on the horizontal axis and consumption on the vertical and assume an initially low level of leisure endowment for worker A. For the prevailing wage w, indicate this worker’s budget constraint and his optimal choice.
(b) On the same graph, illustrate the optimal choice for a second worker B who has the same leisure endowment and the same wage w but chooses to work more.
(c) Now suppose that a household-labor saving technology (such as an automatic vacuum cleaner) is invented and both workers experience the same increase in their leisure endowment. If leisure is quasilinear for both workers, will there be any impact on the labor market?
(d) Suppose instead that tastes for both workers are homothetic. Can you tell whether one of the workers will increase his labor supply by more than the other?
(e) How does your answer suggest that workers in an economy cannot generally be modeled as a single “representative worker” even if they all face the same wage?
B: Consider the problem of aggregating agents in an economy where we assume individuals have an exogenous income.
(a) In a footnote in this chapter, we stated that, when the indirect utility for individual m can be written as Vm (p1, p2, Im) = αm(p1, p2) + β(p1 , p2)Im , then demands can be written as in equation (15.2). Can you demonstrate that this is correct by using Roy’s Identity?
(b) Now consider the case of workers who choose between consumption (priced at 1) and leisure. Suppose they face the same wage w but different workers have different leisure endowments. Letting the two workers be superscripted by n and m, can you derive the form that the leisure demand equations lm (w, Lm) and ln (w, Ln) would have to take in order for redistributions of leisure endowments to not impact the overall amount of labor supplied by these workers (together) in the labor market?
(c) Can you re-write these in terms of labor supply equations ℓm (w, Lm) and ℓn (w, Ln)?
(d) Can you verify that these labor supply equations have the property that redistributions of leisure between the two workers do not affect overall labor supply?
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