Macroeconomics

1. Suppose that you have the following production function:
?? = 100??1/3??2/3
Where k represents the units of capital employed at production, L is the number of labor hours
employed for total production. The wage rate equals to $18 per hour. The cost of capital equals
$2,000 per unit.
a) What are the returns of scale for this production function? Prove it
b) You are currently employing 100 hours of labor. Calculate your total costs, average cost
and marginal costs of labor and capital if K=100 units.
c) Which input is more productive? Should you employ more labor or more capital?
d) Draw the production function, MPK and MPL for the different values of K and L.
2. Suppose that you are a firm that produces violins. You have a production technology to
produce violins that is
?? = ??1/2??1/2
Where k represents the units of capital employed and L is the labor hours employed to produce
the violins. Y is the value of the total production of violins. Assume that labor costs $20 per hour
and that rental price of capital is $400 per unit.
a) If the firm uses 500 units of capital. Calculate your optimal choice of labor to produce
$50,000 worth of violins.
b) What is the marginal cost of labor and the marginal cost of capital? Calculate the value if
the firm uses now 1000 units of capital and needs 1500 hours of labor.
3. Calculate the MPK and the MPL for these production functions:
?? = ??1/3??1/3 also prove what are the returns to scale that this production function presents.
Y=AK also give proof of the returns to scale
Y=K+2L give proof of the returns to scale
?? = ????2 give proof of the returns to scale
4. If a firm produces good called Y following this production function
?? = 400??1/2??1/2
Can you calculate the real wage the firm should pay for L=45 hours and knowing that the firm
uses 250 units o capital? What is the rental price? Draw the graphs to illustrate the equilibrium.