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. 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. 2Y =-1,450 – 25P + 12. 5($4) + 0. 1($15,000) QD=100 – 25P When quantity is expressed as a function of price, the supply curve for Eye-de-ho Potatoes is: QS=-100 + 75P – 25PW – 12. 5PL + 10R =-100 + 75P – 25($4) – 12. 5($8) + 10(20) QS=-100 + 75P B.
The surplus or shortage can be calculated at each price level: | | | | | | | | | | | |Quantity |Quantity |Surplus (+) or | |Price |Supplied |Demanded |Shortage (-) | | | | | | |(1) |(2) |(3) |(4) = (2) – (3) | | | | | | |$1. 50: |QS = -100 + 75($1. 50) |QD = 100 – 25($1. 50) |-50 | | |= 12. 5 |= 62. | | | | | | | |$2. 00: |QS = -100 + 75($2) |QD = 100 – 25($2) |0 | | |= 50 |= 50 | | | | | | | |$2. 50: |QS = -100 + 75($2. 50) |QD = 100 – 25($2. 50) |+50 | | |= 87. 5 |= 37. | | C. The equilibrium price is found by setting the quantity demanded equal to the quantity supplied and solving for P: QD=QS 100 – 25P=-100 + 75P 100P=200 P=$2 To solve for Q, set: Demand: QD = 100 – 25($2)=50 (million bushels) Supply: QS = -100 + 75($2)=50 (million bushels) In equilibrium QD = QS=50 (million bushels). Ch 4 P4. 5Elasticity. The demand for personal computers can be characterized by the following point elasticities: price elasticity = -5, cross-price elasticity with software = -4, and income elasticity = 2. 5. Indicate whether each of the following statements is true or false, and explain your answer. A.
A price reduction for personal computers will increase both the number of units demanded and the total revenue of sellers. B. The cross-price elasticity indicates that a 5% reduction in the price of personal computers will cause a 20% increase in software demand. C. Demand for personal computers is price elastic and computers are cyclical normal goods. D. Falling software prices will increase revenues received by sellers of both computers and software. E. A 2% price reduction would be necessary to overcome the effects of a 1% decline in income. P4. 5SOLUTION A. True. A price reduction always increases units sold, given a downward sloping demand curve. The negative sign on the price elasticity indicates that this is indeed the case here.
The fact that price elasticity equals -5 indicates that demand is elastic with respect to price, and that a price reduction will increase total revenues. B. False. The cross-price elasticity indicates that a 5% decrease in the price of software programs will have the effect of increasing personal computer demand by 20%. C. True. Demand is price elastic (see part A). Since the income elasticity is positive, personal computers are a normal good. Moreover, since the income elasticity is greater than one, personal computer demand is also cyclical. D. False. Negative cross-price elasticity indicates that personal computers and software are compliments. Therefore, falling software prices will increase the demand for computers and resulting revenues for sellers.
However, there is no information concerning the price elasticity of demand for software, and therefore, one does not know the effect of falling software prices on software revenues. E. False. A 2% reduction in price will cause a 10% increase in the quantity of personal computers demanded. A 1% decline in income will cause a 2. 5% fall in demand. These changes will not be mutually offsetting. P4. 7Cross-Price Elasticity. The South Beach Cafe recently reduced appetizer prices from $12 to $10 for afternoon “early bird” customers and enjoyed a resulting increase in sales from 90 to 150 orders per day. Beverage sales also increased from 300 to 600 units per day. A. Calculate the arc price elasticity of demand for appetizers. B.
Calculate the arc cross-price elasticity of demand between beverage sales and appetizer prices. C. Holding all else equal, would you expect an additional appetizer price decrease to $8 to cause both appetizer and beverage revenues to rise? Explain. P4. 7SOLUTION A. [pic] B. [pic] C. Yes, the |EP| = 2. 75 > 1 calculated in part A implies an elastic demand for appetizers and that an additional price reduction will increase appetizer revenues. EPX = -3. 67 < 0 indicates that beverages and appetizers are complements. Therefore, a further decrease in appetizer prices will cause a continued growth in beverage unit sales and revenues. Alternatively, If P = a + bQ, then $12 = a + b(90) and $10 = a + b(150).
Solving for the demand curve gives P = $15 – $0. 033Q. At P = $12, total revenue is $1,080 (= $12 ? 90). If P = $10, total revenue is $1,500 (= $10 ? 150). At P = $8, total revenue is $1,680 (= $8 ? 210). In any case, to determine the profit effects of appetizer price changes it is necessary to consider revenue and cost implications of both appetizer and beverage sales. P4. 8Income Elasticity. Ironside Industries, Inc. , is a leading manufacturer of tufted carpeting under the Ironside brand. Demand for Ironside’s products is closely tied to the overall pace of building and remodeling activity and, therefore, is highly sensitive to changes in national income.
The carpet manufacturing industry is highly competitive, so Ironside’s demand is also very price-sensitive. During the past year, Ironside sold 30 million square yards (units) of carpeting at an average wholesale price of $15. 50 per unit. This year, household income is expected to ssurge from $55,500 to $58,500 per year in a booming economic recovery. A. Without any price change, Ironside’s marketing director expects current-year sales to soar to 50 million units because of rising income. Calculate the implied income arc elasticity of demand. B. Given the projected rise in income, the marketing director believes that a volume of 30 million units could be maintained despite an increase in price of $1 per unit.
On this basis, calculate the implied arc price elasticity of demand. C. Holding all else equal, would a further increase in price result in higher or lower total revenue? P4. 8SOLUTION A. [pic] B. Without a price increase, sales this year would total 50 million units. Therefore, it is appropriate to estimate the arc price elasticity from a before-price-increase base of 50 million units: [pic] C. Lower. Since carpet demand is in the elastic range, EP = -8, an increase (decrease) in price will result in lower (higher) total revenues. P4. 9Cross-Price Elasticity. B. B. Lean is a catalog retailer of a wide variety of sporting goods and recreational products.
Although the market response to the company’s spring catalog was generally good, sales of B. B. Lean’s $140 deluxe garment bag declined from 10,000 to 4,800 units. During this period, a competitor offered a whopping $52 off their regular $137 price on deluxe garment bags. A. Calculate the arc cross-price elasticity of demand for B. B. Lean’s deluxe garment bag. B. B. B. Lean’s deluxe garment bag sales recovered from 4,800 units to 6,000 units following a price reduction to $130 per unit. Calculate B. B. Lean’s arc price elasticity of demand for this product. C. Assuming the same arc price elasticity of demand calculated in Part B, determine the further price reduction necessary for B. B.
Lean to fully recover lost sales (i. e. , regain a volume of 10,000 units). P4. 9SOLUTION A. EPX=[pic] =[pic] =1. 5 (Substitutes) B. EP=[pic] =[pic] =-3 (Elastic) C. EP=[pic] -3=[pic] -3=[pic] -12P2 + $1,560=P2 + $130 13P2=$1,430 P2=$110 This implies a further price reduction of $20 because: ?P=$130 – $110 = $20 Ch 5 P5. 2Regression Analysis. Identify each of the following statements as true or false and explain why: A. A parameter is a population characteristic that is estimated by a coefficient derived from a sample of data. B. A one-tail t test is used to indicate whether the independent variables as a group explain a significant share of demand variation. C.
Given values for independent variables, the estimated demand relation can be used to derive a predicted value for demand. D. A two-tail t test is an appropriate means for testing direction (positive or negative) of the influences of independent variables. E. The coefficient of determination shows the share of total variation in demand that cannot be explained by the regression model. P5. 2SOLUTION A. True. A parameter is a population characteristic that is estimated by a coefficient derived from a sample of data. B. False. An F test is used to indicate whether or not the independent variables as a group explain a significant share of demand variation. C. True.
Given values for independent variables, the estimated demand relation can be used to derive a predicted (or fitted) value for demand. D. False. A one-tail t test is an appropriate means for tests of direction (positive or negative) or comparative magnitude concerning the influences of the independent variables on y. E. False. The coefficient of determination (R2) shows the share of total variation in demand that can be explained by the regression model. P5. 4Revenue vs. Profit Maximization. On weekends during summer months, Eric Cartman rents jet skis at the beach on an hourly basis. Last week, Cartman rented jet skis for 20 hours per day at a rate of $50 per hour.
This week, rentals fell to 15 hours per day when Cartman raised the price to $55 per hour. Using these two price-output combinations, the relevant linear demand and marginal revenue curves can be estimated as: P = $70 – $1Q and MR = $70 – $2Q A. Calculate the revenue-maximizing price-output combination. How much are these maximum revenues? If marginal cost is $30 per hour, calculate profits at this activity level assuming TC = MC ? Q. B. Calculate the profit-maximizing price-output combination along with revenues and profits at this activity level. P5. 4SOLUTION A. To find the revenue-maximizing price-output rental rate, set MR = 0, and solve for P. Because TR=P ? Q ($70 – $1Q)Q =$70Q – $1Q2 MR= ? TR/? Q MR=$70 – $2Q = 0 2Q=70 Q=35 At Q = 35, P=$70 – $1(35) =$35 Total revenue at a price of $35 is: TR=P ? Q =$35 ? 35 =$1,225 per day ?=TR – TC = $70Q – $1Q2 – $30Q = $70(35) – $1(352) – $30(35) =$175 per day (Note: ? 2TR/? Q2 < 0. This is a revenue-maximizing output level because total revenue is decreasing for output beyond Q > 35 hours. ) B. To find the profit-maximizing output level analytically, set MR = MC, or set M? = 0, and solve for Q. Because MR=MC $70 – $2Q=$30 2Q=40 Q=20 At Q = 20, P=$70 – $1(20) =$50 Total revenue at a price of $50 is: TR=P ? Q =$50 ? 20 =$1,000 per day ?=TR – TC = $70Q – $1Q2 – $30Q $70(20) – $1(202) – $30(20) =$400 per day (Note: ? 2? /? Q2 < 0. This is a profit maximum because total profit is falling for Q > 20. ) P5. 5Linear Demand Curve Estimation. Xerox Corporation develops, manufactures, and services document equipment and software solutions worldwide. Assume the company offered $75 off the $1,475 regular price on the Phaser 6360, a durable high-speed color copier, and Internet sales jumped from 700 to 800 units per week (see http://www. office. xerox. com). A. Estimate the color copier demand curve, assuming that it is linear. B. If marginal costs per unit are $650, calculate the profit-maximizing price/output combination. Remember: The marginal revenue curve has the same intercept as the demand curve, but has twice its negative slope (falls twice as fast). ] P5. 5SOLUTION A. When a linear demand curve is written as: P=a + bQ a is the intercept and b is the slope coefficient. From the data given previously, two points on this linear demand curve are identified. Given this information, it is possible to exactly identify the linear demand curve by solving the system of two equations with two unknowns, a and b: 1,475=a + b(700) minus 1,400=a + b(800) 75=-100 b b=-0. 75 By substitution, if b = -0. 75, then: 1,475=a + b(700) 1,475=a – 0. 75(700) 1,475=a – 525 a=2,000 Therefore, the color copier demand curve can be written: P=$2,000 – $0. 5Q B. To find the profit-maximizing output level, set MR = $650 = MC, and solve for Q. Because TR=P ? Q =($2,000 – $0. 75Q)Q =$2,000Q – $0. 75Q2 MR= ? TR/? Q = MC MR=$2,000 – $1. 5Q = $650 1. 5Q=1,350 Q=900 At Q = 900, the profit-maximizing price is P=$2,000 – $0. 75(900) =$1,325 (Note: ? 2? /? Q2 < 0. This is a profit-maximizing output level because MR > MC and total profit is falling for Q > 900. ) P5. 9Multiple Regression. Colorful Tile, Inc. , is a rapidly growing chain of ceramic tile outlets that caters to the do-it-yourself home remodeling market. In 2007, 33 stores were operated in small to medium-size metropolitan markets.
An in-house study of sales by these outlets revealed the following (standard errors in parentheses): Q=4 – 5P + 2A + 0. 2I + 0. 25HF (3) (1. 8) (0. 7) (0. 1) (0. 1) R2=93%, Standard Error of the Estimate = 6. Here, Q is tile sales (in thousands of cases), P is tile price (per case), A is advertising expenditures (in thousands of dollars), I is disposable income per household (in thousands of dollars), and HF is household formation (in hundreds). A. Fully evaluate and interpret these empirical results on an overall basis using R2, [pic], F-statistic and SEE information. B. Is quantity demanded sensitive to “own” price? C. Austin, Texas, was a typical market covered by this analysis.
During 2007 in the Austin market, price was $5, advertising was $30,000, income was an average $55,000 per household, and the number of household formations was 4,000. Calculate and interpret the relevant advertising point elasticity. D. Assume that the preceding model and data are relevant for the coming period. Estimate the probability that the Austin store will make a profit during 2008 if total costs are projected to be $300,000. P5. 9SOLUTION A. (i)Coefficient of determination = R2 = 93%, implying that 93% of demand variation is explained by the regression model. (ii)Corrected coefficient of determination = [pic] = R2 – (k – 1/n – k)(1 – R2) = 0. 93 – (4/28)(1 – 0. 93) = 0. 2, implying that 92% of demand variation is explained by the regression model when both coefficient number, k, and sample size, n, are controlled for. (iii)F statistic = (n – k/k – 1)(R2/1 – R2) = (28/4)(0. 93/0. 07) = 93 > F*4, 28, ? = 0. 01 = 4. 07 implying one can reject the null hypothesis H0: b1 = b2 = b3 = b4 = 0 and conclude with 99% confidence that the dependent variables as a group explain a significant share of demand variation. (iv)Standard error of the estimate = SEE = 6 implying that Q= [pic] ± 2. 048 ? 6 with 95% confidence. Q=[pic] ± 2. 763 ? 6 with 99% confidence. B. To determine whether quantity demanded depends upon “own” price, the question must be asked: is bP ? 0? If bP ? 0, then evidence exists that sales do indeed depend upon price.
For testing purposes, the null hypothesis one seeks to reject is the converse of the above question: H0: bP=0 (Two-tail test) where, |t|=[pic] Therefore, it is possible to reject H0: bP = 0 with 99% confidence and conclude that demand is sensitive to price. C. Because [pic]=4 – 5P + 2A + 0. 2I + 0. 25HF =4 – 5(5) + 2(30) + 0. 2(55) + 0. 25(40) =60(000) The point advertising elasticity is calculated as: ?A=[pic]? Q/? A ? A/Q =2 ? 30/60 =1 Because ? A = 1, a 1% increase in advertising will lead to commensurate percentage increase in demand. D. Pr = 0. 5 or 50%. To generate breakeven revenues of $300,000, Colorful Tile would have to sell Q = TR/P = TC/P = $300,000/$5 = 60,000 cases. From part C, [pic] = 60(000).
Because there is a 50/50 chance that actual sales will be above or below this level, there is a 50/50 chance that the Austin store will make a profit when TC = $300,000. Ch 6 P6. 3Sales Trend Analysis. Environmental Designs, Inc. , produces and installs energy-efficient window systems in commercial buildings. During the past ten years, sales revenue has increased from $25 million to $65 million. A. Calculate the company’s growth rate in sales using the constant growth model with annual compounding. B. Derive a five-year and a ten-year sales forecast. P6. 3SOLUTION A. St=S0(1 + g)t $65,000,000=$25,000,000(1 + g)10 2. 6=(1 + g)10 ln(2. 6)=10 ? ln(1 + g) 0. 956/10=ln(1 + g) e(0. 0956) – 1=g g=0. 100 or 10. 0% A. Five-Year Sales Forecast St=S0 (1 + g)t =$65,000,000 (1 + 0. 0)5 =$65,000,000 (1. 611) =$104,715,000 Ten-Year Sales Forecast St=S0 (1 + g)t =$65,000,000 (1 + 0. 10)10 =$65,000,000 (2. 594) =$168,610,000 P6. 4Cost Forecasting. Dorothy Gale, a quality-control supervisor for Wizard Products, Inc. , is concerned about unit labor cost increases for the assembly of electrical snap-action switches. Costs have increased from $80 to $100 per unit over the previous three years. Gale thinks that importing switches from foreign suppliers at a cost of $115. 90 per unit may soon be desirable. A. Calculate the company’s unit labor cost growth rate using the constant rate of change model with continuous compounding. B.
Forecast when unit labor costs will equal the current cost of importing. P6. 4SOLUTION A. Ct=C0egt $100=$80e3g 1. 25=e3g ln(1. 25)=3g g=0. 223/3 =0. 074 or 7. 4% B. Import Cost=C0egt $115. 90=$100e(0. 074)t 1. 159=e(0. 074)t ln(1. 159)=0. 074t t=0. 148/0. 074 =2 years P6. 7Cost Forecasting. Dr. Izobel Stevens is supervising physician at the Westbury HMO, a New York City-based medical facility serving the poor and indigent. Stevens is evaluating the cost effectiveness of a preventive maintenance program, and believes that monthly downtime on the packaging line caused by equipment breakdown is related to the hours spent each month on preventive maintenance. A.
Write an equation to predict next month’s downtime using the variables D = downtime, M = preventive maintenance, t = time, a0 = constant term, and a1 = regression slope coefficient. Assume that downtime in the forecast (next) month decreases by the same percentage as preventive maintenance increased during the month preceding the current one. B. If 40 hours were spent last month on preventive maintenance and this month’s downtime was 500 hours, what should downtime be next month if preventive maintenance this month is 50 hours? Use the equation developed in part A. P6. 7SOLUTION A. Dt+1=a0 + a1 M =Dt – ? D =Dt – [pic] B. Dt+1=500 – [pic] =375 hours of downtime P6. 5Unit Sales Forecasting.
Claire Littleton has discovered that the change in Product A demand in any given week is inversely proportional to the change in sales of Product B in the previous week. That is, if sales of B rose by X% last week, sales of A can be expected to fall by X% this week. A. Write the equation for next week’s sales of A, using the variables A = sales of Product A, B = sales of Product B, and t = time. Assume that there will be no shortages of either product. B. Last week, 100 units of A and 90 units of B were sold. Two weeks ago, 75 units of B were sold. What would you predict the sales of A to be this week? P6. 5SOLUTION A. At=At-1 + ? At-1 At=At-1 [pic] B. At=At-1 [pic] =100 [pic] =80.