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A firm's production function is well described by the equation Q = 2L - .01L2 + 3K - .02K2. Input prices are $10 per labor hour and $20 per machine hour, and the firm sells its output at a fixed price of $10 per unit.
a. In the short run, the firm has an installed capacity of K = 50 machine hours per day, and this capacity cannot be varied. Create a spreadsheet (based on the example below) to model this production setting. Determine the firm's profit-maximizing employment of labor. Use the spreadsheet to probe the solution by hand before using your spreadsheet's optimizer. Confirm that MRPL = MCL.
A
B
C
D
E
F
G
H
I
1
2
OPTIMAL INPUTS
3
Output
136.0
4
Price
10.0
5
Labor
20.0
Capital
50.0
6
MPL
1.600
MPK
1.000
MR
10.00
7
Revenue
1360.0
8
MRPL
16.0
MRPK
9
MCL
MCK
Cost
1200.0
10
Ave Cost
8.8
11
12
Profit
160.0
13
b. In the long run, the firm seeks to produce the output found in part (a) by adjusting its use of both labor and capital. Use yourspreadsheet's optimizer to find the least-cost input amounts. (Hint: Be sure to include the appropriate output constraint for cell I3.)
c. Suppose the firm were to downsize in the long run, cutting its use of both inputs by 50 percent (relative to part b). How much output would it now be able to produce? Comment on the nature of returns to scale in production. Has the firm's profitability improved? Is it currently achieving least-cost production?
Suppose that the capital investment of Alternative 1 is known with certainty. By how much would the estimate of capital investment for Alternative 2 have to vary so that the initial decision based on these data would be reversed? The annual MARR is 1..
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