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# Production Function of a Dressmaking Firm: How much labor should the firm employ and what is the resulting output and profit?

Making dresses is a labor-intensive process. Indeed, the production function of a dressmaking firm is well described by the equation Q=L-L^2/800, where Q denotes the number of dresses per week and L is the the number of labor hours per week. The firm's additional cost of hiring an extra hour of labor is about \$20 per hour (wage plus fringe benefits). The firm faces the fixed selling price P=\$40.

How much labor should the firm employ? What is its resulting output and profit?

Over the next two years, labor costs are expected to be unchanged, but the dress prices are expected to increase to \$50. What affects will this have on the firm's optimal output? Explain. Suppose instead that inflation is expected to increase the firm's labor cost and output price by identical (percentage) amounts. What effect would this have the firm's output?

Finally, suppose once again that MCL=\$20 and P=\$50 but that labor productivity (i.e. output per labor hour) is expected to increase by 25 percent over the next five years. What effect would have this on the firm's optimal output? Explain.

#### Solution Preview

How much labor should the firm employ? What is its resulting output and profit?

Q=L-L^2/800
Marginal Product of labor=MP=dQ/dL=1-L/400
Marginal Revenue of labor=MP*Price=(1-L/400)*40=40-L/10
Marginal Cost of labor=20 per hour
For optimal level of output, Put Marginal Cost equal to Marginal Revenue of labor
40-L/10=20
L/10=20
L=200 hours
Total output at L=200 hours=200-200^2/800=150 units
Total revenue=Price*optimal output=40*150=\$6000
Total cost=L*wage rate=200*20=\$4000
Profit=6000-4000=\$2000

Over the next two years, labor costs are expected to ...

#### Solution Summary

This complete solution provides step-by-step calculations to find optimal level of input, resulting output and profits in various situations.

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