Production managers for the Cosmic Paper Co

1. Production managers for the Cosmic Paper Co., estimate that their production process is currently characterized by the following short-run production function:
Q = 72L + 15L2 – L3
Where, Q = tons of paper products per production period and L = units of variable input employed per production period.
a. Determine the equations for MP and AP of variable input L.
b. At what rate of usage of variable input is the point of diminishing marginal returns encountered?
c. What rate of usage of variable input is required to reach the maximum output level?

2. a.

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b.

1

You manage PowerGuns Co., a Connecticut based firm that mass produces power guns by teams of workers using assembly machines.

PowerGuns Co production function is given by:
Q = 5 KL
where Q is the number of guns produced per week, K is the number of assembly machines used in production, and L is the number of workers employed.
The weekly wage rate (w) is $5,000 per worker and the weekly rental rate (r)for an assembly machine is $10,000. Guns costs are given by the cost of labor and machines, plus $2,000 per gun for raw materials. Power Guns Co has a fixed installation of 5 assembly machines as part of its design. What is the law of diminishing returns? Does PowerGuns Co’s production function display this characteristic? Explain in full. Be sure to provide convincing mathematical evidence to support your conclusion. What is the total cost function for PowerGuns Co — namely, how much would it cost to produce Q guns?

c. What are average and marginal costs for producing Q guns? average costs vary with output?

d. How many workers are required to produce 250 guns? What is the average cost per gun?

e. You are asked to make recommendations for the design of a new PowerGuns Co production facility. What capital/labor (K/L) ratio should the new plant accommodate if it wants to minimize the total cost of producing any level of output Q?

3. 1. Suppose the production of air beds is given by the following production function:

Q = (L½ + K½)2.

[The marginal products for this production function are MPL = (L½ + K½)L?½

and MPK = (L½+ K½)K?½. ]

Suppose that the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital for an air bed manufacturer that wants to produce 121,000 airbeds.