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Sam has preferences for consumption goods (C) and time spent on leisure (L). The utility function is u(C,L) = CL. The household also has a home production technology summarized by a production function. The production function produces consumption goods with labor (l) according to f(l) = 100*sqrt(l). Time spent on leisure and labor adds up to 24, L + l = 24.
a) How much time will Sam spend in leisure? How many units of consumption good will he produce?
b) Using the production function from part a), what is the marginal product of labor at the amount of labor in part a)?
c) Now suppose that there is a competitive firm with the production function f(l) = 100*sqrt(l). Wage (w) equal to marginal product of labor from part b) and the price of output is 1. What is the profit maximizing demand for labor and amount of output? What is the profit?
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