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.
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A
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B
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C
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D
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E
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F
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G
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H
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I
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1
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2
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OPTIMAL INPUTS
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3
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Output
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136.0
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4
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Price
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10.0
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5
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Labor
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20.0
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Capital
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50.0
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6
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MPL
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1.600
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MPK
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1.000
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MR
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10.00
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7
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Revenue
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1360.0
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8
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MRPL
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16.0
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MRPK
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10.0
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9
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MCL
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10.0
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MCK
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20.0
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Cost
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1200.0
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10
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Ave Cost
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8.8
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11
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12
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Profit
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160.0
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13
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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 your
spreadsheet'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?