University of Pittsburgh



University of Pittsburgh

The Joseph M. Katz Graduate School of Business

Economic Analysis for Managerial Decisions-Firms and Markets

PROBLEM SET

Supply and Demand Analysis

1. Consider the following demand and supply curves: Qd = 5,800 – 6P and Qs = 4P – 120.

a. Graph the supply and demand curves.

b. What are the equilibrium quantity and price, and consumer and producer surplus?

c. What are the excess supply, cost of government subsidy, dead weight loss, and consumer and producer surplus if a price floor of $600 is placed on this good?

d. What are excess demand, the full economic price, dead weight loss, black market gains and consumer and producer surplus if a price ceiling of $500 were placed on this good?

e. What is the quantity exchanged, buyer and producer price, tax revenue, dead weight loss, consumer and producer surplus if excise tax of $60 is imposed on the equilibrium? What are the values of the above variables if a 10% sales tax is used instead?

Tariffs and Quotas

2. Assume US demand for and supply of steel are Qd = 184 - 20P and Qs = 124 + 4P.

a. Find the competitive market price and output.

b. Assuming free trade and $2 world price, calculate US demand, supply and imports.

c. Calculate the elements of lost consumer surplus if government imposes a $0.25 tariff.

d. Calculate the elements of lost consumer surplus if government imposes a quota of 6 tons.

e. If demand increases to Qd = 200 - 20P calculate the elements of lost consumer surplus first with a $0.25 tariff and then a 6 tons quota.

Relationship between Sales Revenue and Demand Characteristics

3. Complete the following table representing data for the market demand for a good.

| |Quantity |Average |Total |Marginal |Arc Price Elasticity |

|Price |Demanded |Revenue |Revenue |Revenue | |

|$50 |1 |_______ |_______ | _______ | |

|$40 |2 |_______ |_______ |_______ |__________ |

|$30 |3 |_______ |_______ |_______ |__________ |

|$20 |4 |_______ |_______ |_______ |__________ |

|$13 |5 |_______ |_______ |_______ |__________ |

|$8 |6 |_______ |_______ |_______ |__________ |

4. Given that the demand equation is P = 81 - 9Q.

a. What is the equation for MR?

b. At what output is MR=0?

c. At what output is TR maximized?

d. Determine the price elasticity of demand at the output where TR is maximum.

5. What is the price elasticity at a price of $1 and then at a price of $4 if demand is Q = 20 - 3P?

6. What is the price elasticity at a price of $4 and then at a price of $3, for Q = 16 + 9P - 2P2.

7. Determine equations for TR, MR and price elasticity at Q = 10, for P = 1,000 + 3Q - 4Q2,

8. Given that the relationship between product A and product B is QA = 80PB – 0.5PB2 where QA are units of product A demanded by consumers each day and PB is selling price of product B.

a. Determine the cross-elasticity for the two products when the price of product B=$10.

b. Are A and B complements, substitutes, or independent, and how "strong" is the relationship?

9. Given the current price levels for steel, would you expect that the higher price elasticity of demand for the USX Corporation or for the steel industry as a whole? Why?

10. A minor league baseball team is determining its pricing policy. Past data suggest the following elasticities: ticket price = -0.6, refreshment price = -0.2, and local population = 0.7.

a. If the local population increases from 60,000 to 61,500, what will be the impact on ticket sales? What if refreshment prices rise 40 percent on average?

b. Currently a typical fan buys an average ticket of $5 and $4 of refreshments at the game. Management is thinking of raising the average ticket price to $5.50. Compute the percentage changes in tickets sold, ticket revenue, and total revenue from admissions and refreshment sales. (If it helps, assume that 5,000 tickets are current sold are per game.)

11. The demand for cars is less elastic than the demand for stereos because a $50 lower price does not affect the sales of cars nearly as much as sales of stereos. Is this statement correct?

Production Theory, Technology and Optimal Choice of Inputs

12. For the production function Q = 12X, where Q and X are units of output and variable input:

a. Determine the equations for MP and AP.

b. Determine APfi when 10 units of variable and 5 units of fixed input are combined.

c. Graph the production function and the corresponding MP and AP functions.

d. How would you describe the important properties of this production function?

13. Short-run production function is estimated as Q = 72X + 15X2 - X3.

a. Determine the equations for MP and AP.

b. What is MP when 7 units of variable input are employed?

c. How much does output rise when the usage of variable input increases from 7 to 8 units?

d. At what usage of variable input is the diminishing marginal productivity encountered?

e. What are the maximum output and number of units of variable input required for it?

f. Graph Q, MP and AP functions. Where does output increase at an increasing and at a decreasing rate? Where does the diminishing average productivity of variable input start?

14. Determine the optimal L and C combination for Q = 140L + 160C – 5L2 – 2C2, when

PL = $12, PC = $24, TC = $732. Now assume PL = $5, PC = $10 and TC = $180 for Q = 6LC.

15. Consider the production function Q = 10L – 0.5L2 + 24K – K2, for L and K in the 0-10 range. Does each input have diminishing marginal productivity? Are returns to scale decreasing?

16. A 200-pound steer can be sustained on these combinations of grass and grain diet:

Pounds of Grass: 50 Pounds of Grain: 80

56. 70

60. 65

68. 60

80. 54

88 52

a. Plot the isoquant corresponding to the inputs necessary to sustain a 200-pound steer.

b. The cost of grass and grain is $0.10 and $0.07 per pound. Rancher prefers a feed mix of 68 pounds of grass and 60 pounds of grain. Is this a least--cost mix? If not, what is?

c. The rancher believes there are constant returns to scale in fattening cattle. At current feed prices, what input quantities should he choose to raise the steer's weight to 250 pounds?

17. Which of these production functions has increasing, constant or decreasing returns to scale?

a. Q = 15 + 0.5KL + 30L

b. Q = 0.01K3 + 4KL + L2K + 0.0001L3

c. Q = 25K1/2L1/3

d. Q = 0.001K + 50,000L

e. Q = AK1-αL2α, α > 0

The Theory of Costs

18. Complete the table assuming that units of fixed and variable input cost $10 and $20 each.

| | | | | | | | |

|Units of Fixed |Units of |Units of Output |Marginal Product|Average Product |TFC |TVC |TC |

|Input |Variable Input | |of Variable |of Variable | | | |

| | | |Input |Input | | | |

| | | | | | | | |

|0 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

|1 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

|2 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

|3 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

|4 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

|5 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

|6 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

|7 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

|8 |_____ |_____ |_____ |_____ |_____ |_____ |_____ |

22. What does U shaped short-run average cost curve imply about the marginal productivity of the variable input? What does flat long-run average cost curve imply about returns to scale? What does this mean for the firm's long-run marginal cost?

23. One of plants with these cost TCA = 80 + 2QA + 0.5QA2 and TCB = 50 + QB2 will be build.

a. Which plant to build for 8 units of output? What output would justify building plant A?

b. Suppose the firm already has built both plants. If a total planned output is sufficiently large, the firm should use both facilities. Explain why. Suppose planned production is 22 units. How should the firm divide production between the plants to minimize total cost? (Hint: Confirm that the firm should divide production to ensure that [pic]

24. Assuming that L is the only variable input and that it has fixed price, complete this table.

| | | | | | | | |

|L |Q |APL |MPL |SMC |AVC |AFC |STC |

| | | | | | | | |

|0 |0 |– |– |– |– |0 | |

| | | | | | | | |

|2 |10 | |5.0 |40.00 |40.00 | | |

| | | | | | | | |

|4 | |7 | |22.22 |28.57 | | |

| | | | | | | | |

|6 |48 | |10.0 | | |14 | |

| | | | | | | | |

|8 |56 |7 | |50.00 |28.57 | | |

| | | | | | | | |

|10 | |6 | | | | | |

| | | | | | | | |

|12 |63 | |1.5 |133.33 |38.10 | | |

The Perfectly Competitive Industry

25. a. TR is the sum of the MR for each unit sold. True or false?

b. TC is the sum of the MC for each unit produced. True or false?

c. T( is the sum of the M( for each unit produced and sold. True or false?

26. Typical perfectly competitive firm with TC = 200 + 25Q - 6Q2 + (1/3)Q3 faces price of $70. Determine the profit-maximizing output and the amount of its short-run profits or losses.

27. Use the per unit cost and revenue curves to graph the short-run profit-maximizing price and output for a perfectly competitive firm whose production function has decreasing average productivity of the variable input over its entire output. (Hint: Determine the shape of SAC and MC curves for a production with diminishing marginal productivity of the variable input and then locate the price and output at which MR = MC). Indicate the firm's total profits.

28. Perfectly competitive firm faces price of $40. Firm's total cost is [pic]

a. Determine profit maximizing output. Write firm's supply curve in terms of price P.

b. Would increase in the fixed cost from 100 to 144 affect the output? Firm’s supply curve?

c. How the increase in fixed costs affect long-run equilibrium price and the number of firms?

The Model of Monopoly

29. A television station is considering selling promotional videos. Supplier A will charge the station a set-up charge of $1,200 plus $2 for each cassette. Supplier B has no set-up charge but will charge $4 per cassette. Estimated demand for the cassettes is Q = 1,600 – 200P.

a. How many cassettes, from which supplier should be ordered if video is to be given away?

b. How many cassettes to order from which supplier and how much to charge public to maximize profit from sales of the cassettes. (Hint: Using MR = MC rule compare profits if cassettes are ordered from supplier A and then from supplier B).

30. Burger Queen (BQ) receives 20% of its franchise revenue. Franchise’s cost per Slopper (ingredients from BQ, labor, etc.) is $0.8. The weekly demand for Sloppers is P = 3 – Q/800.

a. What quantity and price would BQ set? How much do BQ and franchise receive?

b. What quantity and price will franchise set? Compare with part a. what two parties receive?

c. Will profit sharing remove the conflict? What will be the price and quantity of Sloppers? (Does the exact split of the profit affect your answer?) What is the parties' total profit?

d. What are disadvantages of profit sharing relative to more common revenue sharing?

31. a. What is the max profit for a monopolist with demand P = 20 – 0.5Q and TC = 30 + 5Q.

b. What are the price, output and profit if cost per unit of output increases by $3 due to higher input price? Is raising the price of output by $3 optimal?

c. What are the price, output and profits if property taxes increase by $20?

d. What are the price, output and profits if tax on profit increase by $20?

e. What are the price, output and profits if sales tax increase 20%?

32. Soda crackers are produced in three plants with these MC. Demand schedule is as follows:

| | | | | |

|Daily Output |MC of Plant 1 |MC of Plant 2 |MC of Plant 3 |P of Crackers |

| | | | | |

|0 | | |$0.10 |$0.50 |

|1 |$0.14 |$0.13 |$0.13 |$0.48 |

|2 |$0.16 |$0.14 |$0.16 |$0.46 |

|3 |$0.18 |$0.15 |$0.18 |$0.44 |

|4 |$0.20 |$0.16 |Capacity |$0.42 |

|5 |Capacity |Capacity | |$0.40 |

|6 | | | |$0.38 |

|7 | | | |$0.36 |

|8 | | | |$0.34 |

|9 | | | |$0.32 |

|10 | | | |$0.30 |

|11 | | | |$0.28 |

|12 | | | |$0.26 |

What is the most profitable price, output and allocation of output among the firm's three plants.

Monopolistic Competition and Oligopoly

33. Firm manufactures and sells chairs in monopolistically competitive market. The demand is estimated as P = 1,625 – 6Q. The firm costs are TC = 25,000 + 25Q – 6Q2 + (1/3)Q3.

a. Determine the firm's short-run profit-maximizing price and output rate.

b. How much profit will the firm earn at this price and output rate?

c. Calculate and explain the impact of 10% higher TFC on firm's price, output and profits.

34. European air travel market is highly regulated: severely restricted entry and fares are set by regulators. Partly as a result, European fares exceed US for comparable routes. Demand for a given European route and cost per passenger are Q = 1,500 – 3P and $200.

a. Find the profit maximizing fare and number of trips if European air travel market is an implicit cartel of airlines that charge monopoly fares under the shield of regulation.

b. Find the fare and number of trips if market is deregulated into perfectly competitive one.

35. OPEC’s short run (5 years) demand and average cost are Q = 52.5 – 1.25P and $10 per barrel.

a. What is OPEC's optimal production? What is the prevailing price of oil at this level?

b. Many experts contend that maximizing short-run profit is counterproductive in the long run: high prices induce buyers to conserve energy and/or seek other supplies. If the oil price is $20 or, the above demand remains unchanged. Price above $20, curtails long-run demand (over a second 5 years) to Q = 60 – 2P. Find output and price that maximize total profit over the next decade? (Assume all values are present values.)

Game Theory

36. Firm A and B are solely interested in maximizing TR from market share in 2 separate markets worth $30 and $18 million in revenue. Firm A can allocate 3 salespeople (3-0, 2-1, 1-2, and 0-3) and firm B 2 (2-0, 1-1, 0-2.). Firm's revenue is proportional to the number of salespeople assigned to market. For example, if firm A puts 2 and firm B 1 salesperson in market I, A's revenue from this market is [2/(2+1)]$30 = $20 million and B's revenue is the remaining $10 million. The firms split a market equally if neither assigns a salesperson to it.

a. Compute the payoff table. Is this a constant-sum game?

b. Does either firm have dominant (or dominated) strategies? What outcome is predicted?

37. Strict enforcement of laws (speeding, drunk driving, etc.) lowers the rate of auto accidents but is very costly. Table below lists the payoffs of a typical motorist and a town government.

| | | |

| |Town Enforces Law |Town Doesn't Enforce Law |

| | | |

|Motorist Obeys Law |0,-15 |0,0 |

| | | |

|Motorist Disobeys Law |-20,-20 |5,-10 |

a. What is the town's optimal strategy? What is the typical motorist's behavior in response?

b. What if the town could commit and motorists believe that strict enforcement would be used? Would the town wish to do so?

38. From 1989 to 1991 the Trump Shuttle and the Pan Am Shuttle battled for market share on the Boston/New York and Washington DC/New York routes. In addition to service quality and dependability (claimed or real), the airlines competed over price via periodic fare changes. Per seat profit for each airline under different combinations of one-way fares are:

| | |

| |Pan Am Shuttle Fares |

|Trump Shuttle Fares | |

| | | | |

| |$139 |$119 |$99 |

| | | | |

|$139 |$34, $38 |$15, $42 |$6, $32 |

|$119 |$42, $20 |$22, $22 |$10, $25 |

|$99 |$35, $7 |$27, $9 |$18, $16 |

a. What fares should the airlines set if fares are select independently and for ever?

b. What pattern of fares would you predict if the airlines will set fares over the next 18 months and in any month, each airline is free to change its fare if it wishes?

Pricing Strategies

39. Domestic and international plastic wax demand are PD = 100 – 5QD and PF = 60 – 5QF. Short-run production function for plastic wax is Q = 10X. Price of variable input X is $200. Calculate price and output levels that will maximize profits from the sale of plastic wax.

40. A garage operator has identified two distinct market segments: short-term and all-day parkers with respective demand curves of PS = 3 – QS/200 and PC = 2 – QC/200. The garage owner is considering charging different per-hour prices for short-term and all-day parking. Garage capacity is 600 cars, and the cost of adding extra cars (up to this limit) is negligible.

a. Given these facts, what is the owner's appropriate objective? How can he ensure that members of each market segment effectively pay a different hourly price?

b. What are the price and number of cars for each type of parkers? Will the garage be full? What are the price and total number of parkers if discrimination is not feasible?

c. Answer the questions in part b assuming the garage has a capacity of only 400 cars.

41. Demand for wine is Qw = 12,000 – 100Pw. Wine vinegar, co-produced at constant 1-to-1 ratio with wine, has demand Qv = 6,000 – 200Pv. Winery’s cost is TC = 100,000 + 4.16Q + 0.006Q2. How many cases of wine and vinegar should be produced and sold at what price?

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