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Homework answers / question archive / Production function, Cost Function, MC, AC You manage a plant the mass produces engines by teams of workers using assembly machines

Production function, Cost Function, MC, AC You manage a plant the mass produces engines by teams of workers using assembly machines

Economics

Production function, Cost Function, MC, AC

You manage a plant the mass produces engines by teams of workers using assembly machines. The technology is summarized by the production:

q = 5KL

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 = 5,000$ per week. Engine costs are given by the cost of labor teams and machines, plus 2,000$ per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its design at the current moment (short run).

a) What is the short run production function?
b) What is the total cost function for your plant to produce q engines (in terms of q only)?
c) What are average and marginal costs for producing q engines in the short run?
d) How many teams are required to produce 250 engines? What is the average cost per engine at this output?

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a) What is the short run production function?

In a short run production function either K or L is fixed. Since you are told that your capital if fixed, your short run production function in this case is q = 25L.

b) What is the total cost function for your plant to produce q engines (in terms of q only)?

The total cost function is the sum of the costs of labor, capital, and raw materials:
TC = rK +wL + 2000q
Solving our short term cost function for L, we have L =25/q
Using this and values we're given, we have
TC(q) = 10000 (5) +5000(25/q) +2000q = 50000 + 2,200 q

c) What are average and marginal costs for producing q engines in the short run?

Average cost is total cost divided by quantity produced so
AC = TC(q)/q = 50,000 + 2,200q /q
The slope of the total cost function equals marginal cost. So given TC(q) =50000 + 2,200q,
MC=2200. The marginal cost is the change in total cost which results from producing one additional unit.

d) How many teams are required to produce 250 engines? What is the average cost per engine at this output?

Solving our short term cost function for L, we have L =25/q or L =10. This is the number of teams required. Average costs are then
AC = 50,000 +2,200(250)/250 = 2400

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