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# The profit maximization

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1. John's Lawn Mowing Service is a small business that acts as a price taker (MR = P). The prevailing market price of lawn mowing is \$20 per acre. John's costs are given by

total cost = .1q^2 + 10q + 50

where q = the number of acres John chooses to cut a day

a. How many acres should John choose to cut in order to maximize profit?
b. Calculate John's maximum daily profit
c. Graph these results and label John's supply curve

2. Would a lump-sum profits tax affect the profit-maximizing quantity of output? How about a tax assessed on each unit of output?

3. Universal Widget produces high-quality widgets at its plant in Gulch, Nevada for sale throughout the world. The cost function for total widget production (q) is given by

total cost = .25q^2

Widgets are demanded only in Australia (where the demand curve is given by q = 100 - 2P) and Lapland (where the demand curve is given by q = 100 - 4P). If Universal Widget can control the quantities supplied to each market, how many should it sell in each location in order to maximize total profits? What price will be charged in each location?

4. Young's theorem can be used in combination with the envelope theorem to derive some useful results

a. Show that dl(P,v,w)/dv = dk(P,v,w)/dw. Interpret this result using substitution and output effects

b. Use the result from part (a) to show how a unit tax on labor would be expected to affect capital input

c. Show that dq/dw = -dl/dP. What is the interpretation of the result?

d. Use the result from park (c) to discuss how a unit tax on labor input would affect quantity supplied