Chapter 13

John's Lawn Moving Service is a small business that acts as a price taker (i.e., $MR=P$ ). The prevailing market price of lawn mowing is $\mathrm{\$}20$ per acre. John's costs are given by \[ \text { total } \operatorname{cost}=.1 q^{2}+10 q+50 \] 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.

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

A firm faces a demand curve given by \[ q=100-2 P \] Marginal and average costs for the firm are constant at $\mathrm{\$}10$ per unit. a. What output level should the firm produce to maximize profits? What are profits at that output level? b. What output level should the firm produce to maximize revenues? What are profits at that output level? c. Suppose the firm wishes to maximize revenues subject to the constraint that it earn $\mathrm{\$}12$ in profits for each of the 64 machines it employs. What level of output should it produce? d. Graph your results.

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? The production function for a firm in the business of calculator assembly is given by \[ q=2 \sqrt{L} \] where $q$ is finished calculator output and $L$ represents hours of labor input. The firm is a price taker for both calculators (which sell for $P$ ) and workers (which can be hired at a wage rate of $w\text{ per hour })$ a. What is the supply function for assembled calculators $[q=f(P,w)]$ ? b. Explain both algebraically and graphically why this supply function is homogeneous of degree zero in $P$ and $w$ and why profits are homogeneous of degree one in these variables. c. Show explicitly how changes in $w$ shift the supply curve for this firm.

The production function for a firm in the business of calculator assembly is given by \[ q=2 \sqrt{L} \] where $q$ is finished calculator output and $L$ represents hours of labor input. The firm is a price taker for both calculators (which sell for $P$ ) and workers (which can be hired at a wage rate of $w\text{ per hour })$ a. What is the supply function for assembled calculators $[q=f(P,w)]$ ? b. Explain both algebraically and graphically why this supply function is homogeneous of degree zero in $P$ and $w$ and why profits are homogeneous of degree one in these variables. c. Show explicitly how changes in $w$ shift the supply curve for this firm.

The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for $\mathrm{\$}30$ per pound. In bad weather it sells for only $\mathrm{\$}20$ per pound. Caviar produced one week will not keep until the next week. A small caviar producer has a cost function given by \[ T C=\frac{1}{2} q^{2}+5 q+100 \] where $q$ is weekly caviar production. Production decisions must be made before the weather (and the price of caviar) is known, but it is known that good weather and bad weather each occur with a probability of 0.5 a. How much caviar should this firm produce if it wishes to maximize the expected value of its profits? b. Suppose the owner of this firm has a utility function of the form \[ \text { utility }=\sqrt{\pi} \] where $\pi$ is weekly profits. What is the expected utility associated with the output strategy defined in part (a)? c. Can this firm owner obtain a higher utility of profits by producing some output other than that specified in parts (a) and (b)? Explain. d. Suppose this firm could predict next week's price, but could not influence that price. What strategy would maximize expected profits in this case? What would expected profits be?

Suppose a firm engaged in the illegal copying of computer CDs has a daily short-run total cost function given by \[ S T C=q^{2}+25 \] a. If illegal computer CDs sell for $\mathrm{\$}20$, how many will the firm copy each day? What will its profits be? b. What is the firm's short-run producer surplus at $P=\mathrm{\$}20?$ c. Develop a general expression for this firm's producer surplus as a function of the price of illegal CDs.