Suppose that an invention has just resulted in everyone being able to cut their sleep requirement by 10 hours per week—thus providing an increase in their weekly leisure endowment.
A. For each of the cases below, can you tell whether a worker will work more or less?
(a) The worker’s tastes over consumption and leisure are quasilinear in leisure.
(b) The worker’s tastes over consumption and leisure are homothetic.
(c) Leisure is a luxury good.
(d) Leisure is a necessity.
(e) The worker’s tastes over consumption and leisure are quasilinear in consumption.
(f) Do any of your answers have anything to do with how substitutable consumption and leisure are? Why or why not?
B. Suppose that a worker’s tastes for consumption c and leisure ℓ can be represented by the utility function u(c,ℓ) = cαℓ(1−α) .
(a)Write down the worker’s constrained optimization problem and the Lagrange function used to solve it, using w to denote the wage and L to denote the leisure endowment.
(b) Solve the problem to determine leisure consumption as a function of w, α and L. Will an increase in L result in more or less Leisure consumption?
(c) Can you determine whether an increase in leisure will cause the worker to work more?
(d) Repeat the above parts using the utility function u(c,ℓ) = c +αlnℓ instead.
(e) Can you show that, if tastes can be represented by the CES utility function u(c,ℓ) = (αc−ρ(1− α)ℓ−ρ)−1/ρ, the worker will choose to consume more leisure as well as work more when there is an increase in the leisure endowment L? (Warning: The algebra gets a little messy. You can occasionally check your answers by substituting ρ = 0 and checking that this matches what you know to be true for the Cobb-Douglas function u(c,ℓ) = c0.5ℓ0.5.)