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Suppose that a fully automated production process can produce output according to the production function Q = 10K – 0.5K2 where Q = output and K = capital input. Assume further that the expression for the marginal product of capital, MPK, is given as MPK = 10 – K. In this production process capital costs $90 to rent (MFC = $90) and the output sells for $30 per unit (and therefore P = MR = $30).

a. Based on this information, what is the expression for the marginal revenue product of capital, MRPK in this production process? 

b. Given that the production function has a single variable input, what is the optimal quantity of capital the firm should rent (K*) given the existing rental cost of capital and price per unit of output?

c. Based on the optimal rental of capital, how much output (Q) would this company produce per production period? Total Output (Q) = ____________________.

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Related questions

The long-run production function for a rm's product is given by q = f(K; L) = 5 K L. The price of capital is $10 and the price of labor is $15.
a. Suppose the rm wishes to produce output of 500. List 5 combinations of capital and labor that the rm can transform into 500 output.
b. For each of your 5 combinations from part a, give the cost of using that combination of capital and labor. Which is the lowest?
c. For your lowest cost combination from part b, calculate the marginal product of capital (MPK) and the marginal product of labor (MPL).
d. For your answer in parts b-c, is your marginal product per dollar equal across the two inputs.

2. A firm plans to produce 1,000 units per day of good X. The firm's production engineer finds two technically efficient processes (i.e., input combinations of labor and capital) to produce 1,000 units per day:

Process 1 Process 2

Labor 10       8

Capital 20    25

a. If the production function for the existing technology is Q = f(L, K), where Q is the maximum possible output, L is the amount of labor used, and K is the amount of capital used, then f(10, 20) =__________ and f(8, 25) = ____________.

b. If the firm must pay $200 per day for a unit of labor and $100 per day for a unit of capital, which process is economically efficient?

c. If the firm must pay $250 per day for a unit of labor and $75 per day for a unit of capital, which process is economically efficient?

d. No matter what input prices prevail, as long as the firm employs either process 1 or process 2 it will be technically efficient. Evaluate this statement.

You manage a plant that mass-produces engines by teams of workers using assembly

machines. The technology is summarized by the production function

q = 5 KL

where q is the number of engines per week, K is the number of assembly machines, and L

is the number of labor teams. Each assembly machine rents for r = $10,000 per week, and

each team costs w = $5000 per week. Your plant has a fixed installation of 5assembly machines as part of its design.

In the SHORT RUN:

For any given level of output q, write the expression for the number of Labor teams required.

b. What is the cost function for your plant � namely, how much would it cost to produce q engines? (Hint: TC=TFC+TVC, TFC=rK, TVC=wL)

c. What are average and marginal costs for producing q engines? How do average costs vary with output?

d. How many Labor teams are required to produce 250 engines? What is the average cost per engine?

In the LONG RUN:

e. You are asked to make recommendations for the design of a new production facility. What capital/labor (K/L) ratio should the new plant accommodate if it wants to minimize the total cost of producing at any level of output q? MPL=5K, MPK=5L.

f. For output q=500, how many L and K should you use in order to minimize cost?

g. What is the minimum cost of production for q=500?

h. What is the minimum cost of product for any output q? (Note: write TC as a function of q).

i. If w is increased to $10,000 per week, how does this change the capital/labor (K/L) ratio if you want to minimize the total cost of production?

j. Given the new wage rate, how many L and K should you use in order to minimize cost for output q=500?

k. Is the expansion path a straight or a curved line? Is it upward or downward sloping? Why?

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