I only need help with D and E Thanks Sam has preferences for consumption goods (
ID: 1198942 • Letter: I
Question
I only need help with D and E
Thanks
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 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 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 be the profit from part c. Prices are also the same as in part c). Now suppose that Sam 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 + . How much leisure time and consumption goods will he provide.
e) Now consider an economy that consist of 10 consumers that are identical to Sam who each own one firm like Sam'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.
Here is the solution to A-C
The production function is given as
The producer at profit maximizing level equate marginal revenue product with the cost of employing one more unit of labor, that is the wage rate. That is at equilibrium
This is the demand for labor.
Now the supply of labor is determined through the utility function.
The utility function of the consumer is given as
The FOC of utility maximization is
At equililbrium
Explanation / Answer
50/ l +50/ l. .(24-(C/100)2 = 24(50/ l).+100 2
100 2+100 2 (24-(C/100)2 = 24 (100 2) +100 2
C2 2/100
Hence he will spend his time like this in leisure and labour.
e. If there are 10 identical custmers, they will spend it identically. and their demand and supply curves will be similar.