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optimal level of output

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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 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.

a. how much labor should the firm employ? What are its resulting output and profit?

b. Over the next two years, labor costs are expected to be unchanged, but dress prices are expected to increase to $50. What effect 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 on the firm's output?

c. 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 this have on the firm's optimal output? Explain.

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

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
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

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

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The optimal level of output is determined.

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Optimal level of output/quadratic formula

Please can someone give some ideas, even if they don't know the final solution?

A firm receives a price of 120 for its output. Its total cost function is C=.02Q^3 +.4Q^2 - 5Q - 15.

a) Assuming the utility operates to maximize profits, what is this firm's profit maximizing output level?

I know we have to use the quadratic formula to solve for Q:

Q= [-b +/- (b2 - 4 a c).5]/2a

Where a,b,c are from a quadratic equation as follows: a Q2 + b Q + c

If we get two positive solutions, one of them will be a minimum, and one a maximum. We can find the maximum by looking at the sign of the second derivative, evaluated at the two solutions.

b) What are revenues at the optimal level of output?

c) what are costs at the optimal level of output?

d) What are profits at the optimal level?

e). Now suppose a tax of $15.00 is levied on the sale of the output (electricity), which we will assume lowers the price the utility receives to $105.00. (We will have more say subsequently about how a tax burden is distributed between the supply side and the demand side of the market). What is the utility's profit maximizing output level, and its level of revenue, cost, and profit level now?

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