# P2. number of tickets sold (quantity) can be

### P2. number of tickets sold (quantity) can be

P2. 6Price and Total Revenue. The Portland Sea Dogs, the AA affiliate of the Boston Red Sox major league baseball team, have enjoyed a surge in popularity.

During a recent home stand, suppose the club offered \$5 off the \$12 regular price of reserved seats, and sales spurted from 3,200 to 5,200 tickets per game. A. Derive the function that describes the price/output relation with price expressed as a function of quantity (tickets sold). Also express tickets sold as a function of price.

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B. Use the information derived in part A to calculate total revenues at prices in \$1 increments from \$5 to \$15 per ticket.What is the revenue-maximizing ticket price? If variable costs are negligible, is this amount also the profit-maximizing ticket price? P2. 6SOLUTION A. When a linear demand curve is written as: P=a + bQ a is the intercept and b is the slope coefficient.

Because 3,200 seats were sold at a regular price of \$12 per game, and 5,200 seats were sold at the discount price of \$7, two points on the firm’s linear demand curve are identified. Given this information, it is possible to identify the linear demand curve by solving the system of two equations with two unknowns, a and b: 2=a + b(3,200) minus 7=a + b(5,200) 5=-2,000 b b=-0. 0025 By substitution, if b = -0. 0025, then: 12=a + b(3,200) 12=a – 0. 0025(3,200) 12=a – 8 a=20 With price expressed as a function of quantity, the reserved seat demand curve can be written: P=\$20 – \$0.

0025Q Similarly, the number of tickets sold (quantity) can be expressed as a function of price: P=\$20 – \$0. 0025Q 0. 0025Q=\$20 – P Q=8,000 – 400P This simple linear characterization of the firm’s demand curve can be used to profitably guide production, pricing and promotion decisions.

B.The Portland Sea Dogs could use the estimated linear market demand curve to estimate the quantity demanded during the same marketing period for ticket prices in the range from \$5 to \$15 per ticket, using \$1 increments: |Price |Quantity |TR=P? Q | |\$5 | | 30,000 | | |6,000 | | |6 | | 33,600 | | |5,600 | | |7 | | 36,400 | | |5,200 | | |8 | | 38,400 | | |4,800 | | 9 | | 39,600 | | |4,400 | | |10 | | 40,000 | | |4,000 | | |11 | | 39,600 | | |3,600 | | |12 | | 38,400 | | |3,200 | | |13 | | 36,400 | | |2,800 | | |14 | | 33,600 | | |2,400 | | |15 | | 30,000 | | |2,000 | |From the table, the revenue-maximizing ticket price is \$10. This is also the profit-maximizing ticket price if variable costs and, hence, marginal costs are negligible. The pricing promotion resulted in declining revenues, and the \$7 price results in an activity level that is above the revenue-maximizing output.

Because the marginal cost of fan attendance cannot be less than zero, the profit-maximizing price cannot be less than the revenue-maximizing price of \$10. P2. 7Profit Maximization: Equations.

21st Century Insurance offers mail-order automobile insurance to preferred-risk drivers in the Los Angeles area.The company is the low-cost provider of insurance in this market but doesn’t believe its annual premium of \$1,500 can be raised for competitive reasons. Rates are expected to remain stable during coming periods; hence, P = MR = \$1,500.

Total and marginal cost relations for the company are as follows: TC=\$41,000,000 + \$500Q + \$0. 005Q2 MC=? TC/? Q = \$500 + \$0. 01Q A.

Calculate the profit-maximizing activity level. B. Calculate the company’s optimal profit, and optimal profit as a percentage of sales revenue (profit margin).

P2. 7SOLUTION A.Set MR = MC and solve for Q to find the profit-maximizing activity level: MR=MC \$1,500=\$500 + \$0. 01Q 0.

01Q=\$1,000 Q=100,000 This is a profit maximum because profits are decreasing for Q > 100,000. B. The total revenue function for 21st Century Insurance is: TR=P ? Q = \$1,500Q Then, total profit is ?=TR – TC =\$1,500Q – \$41,000,000 – \$500Q – \$0. 005Q2 =1,500(100,000) – 41,000,000 – 500(100,000) – 0. 005(100,0002) =\$9,000,000 TR=\$1,500(100,000) =\$150,000,000 or \$150 million Profit Margin=? /TR =\$9,000,000/\$150,000,000 =0. 06 or 6 percent P2.

9Average Cost Minimization. Giant Screen TV, Inc. is a Miami-based importer and distributor of 60-inch screen HDTVs for residential and commercial customers.

Revenue and cost relations are as follows: TR=\$1,800Q – \$0. 006Q2 MR=? TR/? Q = \$1,800 – \$0. 012Q TC=\$12,100,000 + \$800Q + \$0. 004Q2 MC=? TC/? Q = \$800 + \$0. 008Q A. Calculate output, marginal cost, average cost, price, and profit at the average cost-minimizing activity level.

B. Calculate these values at the profit-maximizing activity level. C.

Compare and discuss your answers to parts A and B. P2. 9SOLUTION A. To find the average cost-minimizing level of output, set MC = AC and solve or Q. Because, AC=TC/Q =(\$12,100,000 + \$800Q + \$0. 004Q2)/Q =\$12,100,000/Q + \$800 + \$0.

004Q Therefore, MC=AC \$800 + \$0. 008Q=\$12,100,000/Q + \$800 + \$0. 004Q 0.

004Q=12,100,000/Q Q2=12,100,000/0. 004 Q=(12,100,000/0. 004)1/2 =55,000 And, MC=\$800 + \$0. 008(55,000) =\$1,240 AC=\$12,100,000/(55,000) + \$800 + \$0. 004(55,000) =\$1,240 P=TR/Q =(\$1,800Q – \$0.

006Q2)/Q =\$1,800 – \$0. 006Q =\$1,800 – \$0. 006(55,000) =\$1,470 ?=P ? Q – TC =\$1,470(55,000) – \$12,100,000 – \$800(55,000) – \$0. 004(55,0002) =\$12,650,000 This is an average-cost minimum because average cost is rising for Q > 55,000.B.

To find the profit-maximizing level of output, set MR = MC and solve for Q (this is also where M? = 0): MR=MC \$1,800 – \$0. 012Q=\$800 + \$0. 008Q 0. 02Q=1,000 Q=50,000 And MC=\$800 + \$0.

008(50,000) =\$1,200 AC=\$12,100,000/(50,000) + \$800 + \$0. 004(50,000) =\$1,242 P=\$1,800 – \$0. 006(50,000) =\$1,500 ?=TR – TC =\$1,800Q – \$0. 006Q2 – \$12,100,000 – \$800Q – \$0. 004Q2 =-\$0.

01Q2 + \$1,000Q – \$12,100,000 =-\$0. 01(50,0002) + \$1,000(50,000) – \$12,100,000 =\$12,900,000 This is a profit maximum because profit is falling for Q > 50,000. C. Average cost is minimized when MC = AC = \$1,240.Given P = \$1,470, a \$230 profit per unit of output is earned when Q = 55,000. Total profit ? = \$12. 65 million.

Profit is maximized when Q = 50,000 since MR = MC = \$1,200 at that activity level. Since MC = \$1,200 < AC = \$1,242, average cost is falling. Given P = \$1,500 and AC = \$1,242, a \$258 profit per unit of output is earned when Q = 50,000. Total profit ? = \$12.

9 million. Total profit is higher at the Q = 50,000 activity level because the modest \$2 (= \$1,242 – \$1,240) decline in average cost is more than offset by the \$30 (= \$1,500 – \$1,470) price cut necessary to expand sales from Q = 50,000 to Q = 55,000 units.P3.

3Demand Analysis. The demand for housing is often described as being highly cyclical and very sensitive to housing prices and interest rates. Given these characteristics, describe the effect of each of the following in terms of whether it would increase or decrease the quantity demanded or the demand for housing. Moreover, when price is expressed as a function of quantity, indicate whether the effect of each of the following is an upward or downward movement along a given demand curve or involves an outward or inward shift in the relevant demand curve for housing.Explain your answers.

A. An increase in housing prices B. A fall in interest rates C. A rise in interest rates D. A severe economic recession E.

A robust economic expansion P3. 3SOLUTION A. An increase in housing prices will decrease the quantity demanded and involve an upward movement along the housing demand curve.

B. A fall in interest rates will increase the demand for housing and cause an outward shift of the housing demand curve. C. A rise in interest rates will decrease the demand for housing and cause an inward shift of the housing demand curve. D.

A severe economic recession (fall in income) will decrease the demand for housing and result in an inward shift of the housing demand curve. E. A robust economic expansion (rise in income) will increase the demand for housing and result in an outward shift of the housing demand curve.

Ch 3 Problems P3. 3SOLUTION A. An increase in housing prices will decrease the quantity demanded and involve an upward movement along the housing demand curve. B.

A fall in interest rates will increase the demand for housing and cause an outward shift of the housing demand curve.C. A rise in interest rates will decrease the demand for housing and cause an inward shift of the housing demand curve. D. A severe economic recession (fall in income) will decrease the demand for housing and result in an inward shift of the housing demand curve. E.

A robust economic expansion (rise in income) will increase the demand for housing and result in an outward shift of the housing demand curve. P3. 5Demand Function.

The Creative Publishing Company (CPC) is a coupon book publisher with markets in several southeastern states.CPC coupon books are sold directly to the public, sold through religious and other charitable organizations, or given away as promotional items. Operating experience during the past year suggests the following demand function for CPC’s coupon books: Q = 5,000 – 4,000P + 0. 02Pop + 0. 25I + 1.

5A, where Q is quantity, P is price (\$), Pop is population, I is disposable income per household (\$), and A is advertising expenditures (\$). A. Determine the demand faced by CPC in a typical market in which P = \$10, Pop = 1,000,000 persons, I = \$60,000, and A = \$10,000. B.Calculate the level of demand if CPC increases annual advertising expenditures from \$10,000 to \$15,000. C.

Calculate the demand curves faced by CPC in parts A and B. P3. 5SOLUTION A. The demand faced by CPC in a typical market in which P = \$10, Pop = 1,000,000 persons, I = \$60,000, and A = \$10,000 is: Q=5,000 – 4,000P + 0.

02Pop + 0. 25I + 1. 5A =5,000 – 4,000(10) + 0.

02(1,000,000) + 0. 25(60,000) + 1. 5(10,000) =15,000 B.

If advertising rises from \$10,000 to \$15,000, CPC demand rises to: Q=5,000 – 4,000P + 0. 02Pop + 0. 25I + 1.

5A =5,000 – 4,000(10) + 0. 02(1,000,000) + 0. 25(60,000) + 1. (15,000) =22,500 C. The effect of an increase in advertising from \$10,000 to \$15,000 is to shift the demand curve upward following a 7,500 unit increase in the intercept term.

If advertising is \$10,000, the CPC demand curve is: Q=5,000 – 4,000P + 0. 02(1,000,000) + 0. 25(60,000) + 1. 5(10,000) =55,000 – 4,000P Then, price as a function of quantity is: Q=55,000 – 4,000P 4,000P=55,000 – Q P=\$13. 75 – \$0.

00025Q If advertising is \$15,000, the CPC demand curve is Q=5,000 – 4,000P + 0. 02(1,000,000) + 0. 25(60,000) + 1.

5(15,000) =62,500 – 4,000P Then, price as a function of quantity is:Q=62,500 – 4,000P 4,000P=62,500 – Q P=\$15. 625 – \$0. 00025Q P3. 6Demand Curves. The Eastern Shuttle, Inc. , is a regional airline providing shuttle service between New York and Washington, D. C.

An analysis of the monthly demand for service has revealed the following demand relation: Q = 26,000 – 500P – 250POG + 200IB – 5,000S, where Q is quantity measured by the number of passengers per month, P is price (\$), POG is a regional price index for other consumer goods (1967 = 1. 00), IB is an index of business activity, and S, a binary or dummy variable, equals 1 in summer months and 0 otherwise.A. Determine the demand curve facing the airline during the winter month of January if POG = 4 and IB = 250.

B. Determine the demand curve facing the airline, quantity demanded, and total revenues during the summer month of July if P = \$100 and all other price-related and business activity variables are as specified previously. P3. 6SOLUTION A.

The demand curve facing Eastern during the winter month of January can be calculated by substituting the appropriate value for each respective variable into the firm’s demand function: Q=26,000 – 500P – 250POG + 200IB – 5,000S 26,000 – 500P – 250(4) + 200(250) – 5,000(0) Q=75,000 – 500P With price expressed as a function of quantity, the firm demand curve can be written: Q=75,000 – 500P 500P=75,000 – Q P=\$150 – \$0. 002Q B. During the summer month of July, the variable S = 1. Therefore, assuming that price-related values remain as before, the firm demand curve is: Q=26,000 – 500P – 250(4) + 200(250) – 5,000(1) =70,000 – 500P The quantity demanded during July is: Q=26,000 – 500(100) – 250(4) + 200(250) – 5,000(1) =20,000 passengers Total July revenue for the company is: TR=P ? Q \$100(20,000) =\$2,000,000 P3.

10Market Equilibrium. Eye-de-ho Potatoes is a product of the Coeur d’Alene Growers’ Association. Producers in the area are able to switch back and forth between potato and wheat production depending on market conditions. Similarly, consumers tend to regard potatoes and wheat (bread and bakery products) as substitutes.

As a result, the demand and supply of Eye-de-ho Potatoes are highly sensitive to changes in both potato and wheat prices. Demand and supply functions for Eye-de-ho Potatoes are as follows: QD=-1,450 – 25P + 12. 5PW + 0. Y,(Demand) QS=-100 + 75P – 25PW – 12.

5PL + 10R,(Supply) where P is the average wholesale price of Eye-de-ho Potatoes (\$ per bushel), PW is the average wholesale price of wheat (\$ per bushel), Y is income (GDP in \$ billions), PL is the average price of unskilled labor (\$ per hour), and R is the average annual rainfall (in inches). Both QD and QS are in millions of bushels of potatoes. A. When quantity is expressed as a function of price, what are the Eye-de-ho Potatoes demand and supply curves if PW = \$4, Y = \$15,000 billion, PL = \$8, and R = 20 inches?B. Calculate the surplus or shortage of Eye-de-ho Potatoes when P = \$1. 50, \$2, and \$2. 50.

C. Calculate the market equilibrium price/output combination. P3.

10SOLUTION A. When quantity is expressed as a function of price, the demand curve for Eye-de-ho Potatoes is: QD=-1,450 – 25P + 12. 5PW + 0. 