1. Do the following for each of the profit-maximizing situations below:
        (1) Write the expression for the firm's MR and MC.
        (2) Set MR = MC and rearrange terms to solve for the firm's profit-maximizing output and profits.
        (3) Sketch the firm's (inverse) demand curve (P), its MR, MC and ATC curves, and illustrate the firm's profit-maximizing optimum. At the firm's optimal output level, label in your diagram the numerical values for Q, MC, P and ATC.
        (4) Indicate the area in your diagram which corresponds to the firm's profits (note: you may find it helpful to shade in TR and TC to net out your profits, but you don't have to do it this way).
Each of these cases comes from the "Exercises" in the Econ.102 computer exercise number 4, "Solving profit-maximization problems," so you can use the program to check your results. The parenthetical information for each case refers to the case's exercise number in the program.
    a. (Exercise 1): P = 60;   TC = 216 + 24Q - 6Q² + Q³
    b. (Exercise 2): P = 168 - 18Q;   TC = same as for a
    c. (Exercise 3): P = 120 - 9.6Q;   TC = 200 + 4Q + 2Q²
    d. (Exercise 4): P = 40 - 2Q;   TC = 12Q
    e. (Exercise 4B): P = 40 - 2Q;   TC = 48 + 12Q

    2. Use your calculations and/or diagrams from problem 1 to do each of the following:
    a. Case a: Check the second-order condition to be sure your profits are maximized.
    b. Case c: (1) What is the formula for your firm's AVC? (2) Sketch your AVC curve in your diagram.
    c. Case d: (1) What is the formula for your firm's ATC? (2) How do the rules about the relationship between marginal and average values account for the way the ATC curve behaves in this case?
    d. Case e: This case illustrates a natural monopoly, a situation in which the unit cost of output is lowest if just one firm provides all the output. Governments sometimes license such monopolists but charge them a franchise fee to operate. This franchise fee becomes part of the firm's fixed costs. (1) What is the size of the largest franchise fee a government could successfully impose in this case? (2) How did you decide? (3) In your diagram, show the consequences of the franchise fee.

    3. a. Write down the general expression for profits, then set up the first-order condition and rearrange terms to demonstrate that setting marginal profit = 0 is equivalent to setting MR = MC.
    b. For the optimum to be a maximum rather than a minimum, the second-order condition states that marginal profit must be falling at the optimum (in other words, the derivative of marginal profit with respect to output must be less than zero). If both the first- and second-order conditions are met, indicate whether each of the following would be positive, negative or zero:
    (1) marginal profit at the optimum;
    (2) marginal profit for output immediately above the optimum; and
    (3) marginal profit for output immediately below the optimum.

    4. Briefly explain in words why it is that MR = P for a perfectly competitive firm and MR < P for a monopolist.

    5. In the diagram to the right, indicate the output level(s) which correspond to each of the following:
    a. breakeven output level(s)
    b. the firm's profit-maximizing output level
    c. the firm's revenue-maximizing output level
    d. the socially desirable (efficient) output level

    6. For many of the total value functions used in economics, first derivatives yield some sort of marginal value function. Second derivatives in turn tell us how marginal value behaves as the independent variable changes. For each of the following, take the first derivative and then the second derivative to demonstrate the proposition asserted for each case:
    a. Consumption function: C = 100 + 0.6·DY. Demonstrate that the marginal propensity to consume (MPC) doesn't change as disposable income (DY) changes (in other words, show that the derivative of MPC with respect to DY = 0).
    b. Cost function: TC = 6 + 2Q + Q². Demonstrate that marginal cost (MC) for this total cost function is strictly increasing as output rises.
    c. Production function: Q = 2L^1/2. Demonstrate that this production function has strictly diminishing returns to labor. In other words, show that the marginal product of labor (MPL) is strictly decreasing as the quantity of labor rises.

    7. Consider the following short-run production function for a firm:
              Q = 2.4L² - .4L³
where Q=output and L=labor.
    a.. Suppose the firm wants to know the maximum amount of output it can produce in the short run.
      (1) Set up the first-order condition and solve for the quantity of labor which maximizes the firm's total output. Refer to this solution as Lt.
      (2) Check the second-order condition to be sure Lt maximizes instead of minimizes the firm's total output.
    b. Now consider the firm's marginal product of labor (MPL).
      (1) What is the formula for the firm's MPL given its production function?
      (2) Solve for the quantity of labor where MPL reaches its maximum. Refer to this solution as Lm.
      (3) Check the second-order condition to be sure Lm maximizes instead of minimizes MPL.
    c. Now consider the firm's average product of labor (APL).
      (1) What is the formula for the firm's APL given its production function?
      (2) Solve for the quantity of labor where APL reaches its maximum. Refer to this solution as La.
      (3) Check the second-order condition to be sure La maximizes instead of minimizes APL.
    d. Verify the following for this firm's production function :
      (1) At La, MPL = APL.
      (2) MPL > APL if and only if L < La.
 
    8. Consider the following demand curve for concert tickets in a small town: Qd = 400 - 40P + 20I,
where P=price per ticket and I=income per person.
    a. Calculate the partial derivative of quantity demanded with respect to (1) price and (2) income.
    b. Calculate the total differential for this demand curve. What happens to Qd whenever P rises by 5 and I rises by 10?
    c. Diagram demand curve 1 (D1) for an income level of 100 and demand curve 2 (D2) for an income level of 130.
    Suppose concert tickets are currently priced at 30 each.
    d. Calculate the quantity demanded for each demand curve.
    e. Calculate the elasticity of demand for each demand curve for your quantities from part d.
    f. Suppose the marginal cost to the concert promoter of allowing extra customers into the concert equals zero and the promoter wants to maximize profits. For each demand curve, would you recommend that the concert promoter raise the ticket price, lower it, leave it unchanged or is it impossible to tell?

    9. Consider the following demand curve for donuts:
            Qd = 60P^-2Pc^-0.1I^0.5
where P=price of donuts, Pc=price of coffee, and I=income.
    a. Based on what you know about power functions, what is the value of each of the following:
        (1) the own-price elasticity of demand for donuts;
        (2) the cross-price elasticity of demand for donuts with respect to the price of coffee; and
        (3) the income elasticity of demand for donuts.
    b. Make use of derivatives to support your answer to part a(1).
    c. Based on your elasticity values:
        (1) would you classify donuts and coffee as complements or substitutes?
        (2) would you classify donuts as a normal or inferior good?
Briefly explain both answers.
    d. If the price of donuts drops 10 percent, the price of coffee rises 50 percent and income rises 10 percent, what will be the approximate percentage effect on the quantity demanded of donuts?