Thoreau 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√ l. Time spent on leisure and labor adds up to 24, L + l = 24.

a) How much time will Thoreau 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√ l. The wage (w) is equal to the 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?

d) Let π be the profit from part c. Prices are also the same as in part c). Now suppose that Thoreau has decided to rejoin civilization. He now owns a firm with the production function f(l) = 100√ l and receives its profits. He can also provides some of his time to labor markets. His budget constraint is C + wL = 24w + π. How much leisure time and consumption goods will he provide.

e) Now consider an economy that consist of 10 consumers that are identical to Thoreau who each own one firm like Thoreau's firm. Prices are the same as in part c). Show that both the labor market and the consumption good market clear at these prices. That is the sum of demand for leisure time from the consumers and demand for labor time from the firms equals the total hours which is 240. And the supply and demand for consumption goods are equal.