chapter4 new

Presented by Suong Jian & Liu Yan, MGMT Panel , Guangdong University of Finance.

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Chapter 5 DEMAND ANALYSIS QUESTIONS & ANSWERS Q5.1 Is the economic demand for a product determined solely by its usefulness? Q5.1 ANSWER

No, two basic conditions must be met before economic demand is created. First, there must be value associated with acquiring and using the good or service. For individuals, this value is in terms of utility, well being, or satisfaction through consumption. For firms, this value is measured in terms of the profit created through resource employment. Second, there must be an ability to pay. Both individuals and firms must demonstrate an economic capability to acquire, or their wants will remain unfulfilled, and no economic demand will result.

Q5.2 John Baptiste Say, a French economist from the early 19th century, is credited with

stating, Asupply creates its own demand.@ Today, some economists appear to argue that Ademand creates its own supply.@ Explain why neither statement is precisely correct, and how demand and supply together create the market for goods and services.

Q5.2 ANSWER

In simplistic terms, supply is what is available, and demand is what is desirable. In the can-do world of the new millennium, some consumers have come to believe that everything is possible and that Ademand creates its own supply.@ Alas, that is not yet the case. While many companies continue to invent new and innovative products to meet consumer needs, not everything customers desire can be supplied. For example, there is no comprehensive cure for cancer. The forces of demand and supply work together to provide consumers what they want, within the realm of technological feasibility.

The point Say was trying to make in arguing Asupply creates its own demand@ was simply that there can be no general glut or oversupply of goods and services in free and competitive markets. While it is possible for temporary Agluts@ to arise for particular goods and services, these gluts will be balanced by temporary shortage or under supply in the markets for other goods and services. In a freely competitive market, permanent oversupply of all goods and services is not possible.

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On a historical note, it is perhaps worth mentioning that while the popular expression of Say's Law is Asupply creates its own demand,@ this quotation does not appear in Say's writings, nor in the writings of other prominent economists of his time. This Alaw@ was developed in the economics literature at a time when economists had begun to notice that the economic system could be subject to severe recessions. The nature of these crises, their causes, and relationship to the economic system became a topic of hot debate. A major figure on one side of the debate was Thomas Malthus, who argued that crises were a result of a Ageneral glut@ of goods and services. Because production could outrun people=s ability to purchase goods and services, Malthus argued that it was under consumption that led to recessions. On the other side of the debate were David Ricardo, James and John Stuart Mill, and Say who together argued that the under consumption thesis was wrong. Their ideas are today called Say's Law.

Q5.3 Why might customers be unwilling or unable to provide accurate demand

information? When might this prove helpful? Q5.3 ANSWER

Customers are often unable to supply accurate information about their demand for particular products given an inability to verbalize the method by which they select goods and services for consumption. Even when the decision process is understood and objective enough to provide relevant demand information, consumers might hesitate to supply such data because it could place them in a poorer bargaining position with suppliers. For example, a consumer might be reluctant to tell a retailer that he or she is willing to pay $3,000 for a sophisticated personal computer on the grounds that by withholding this information an even lower price might be obtained. Of course, providing suppliers with demand information can prove beneficial when such information improves the reliability or lowers the cost of supplied products. For example, projections of demand quantities are often provided to allow suppliers to develop the capacity necessary to meet demand, and better deal with demand fluctuations.

Q5.4 When is use of the arc elasticity concept valid as compared with the use of the point

elasticity concept? Q5.4 ANSWER

The arc versus the point elasticity concepts differ only in terms of the magnitude of the change in a given independent variable. Strictly speaking, the point elasticity

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concept is appropriate for measuring the responsiveness of a dependent Y variable to a change in an independent X variable at a given point on a function. The point elasticity concept is often used to measure the effect on Y of small, say 0-5%, changes in X. Use of the arc elasticity concept is appropriate when considering the effect of larger changes in X. The arc elasticity is an average elasticity over a range along a given function.

Q5.5 AWhen I go to the grocery store, I find cents-off coupons totally annoying. Why

can=t they just cut the price and do away with the clutter?@ Discuss this statement and explain why coupon promotions are an effective means of promotion for grocery retailers, and popular with many consumers.

Q5.5 ANSWER

At the grocery store, cents-off promotions remain an effective means of promotion. A 254 coupon turned in by 10% of all customers is said to have a 10% Aresponse rate.@ Retailers have found that a 254 coupon turned in by 10% of all customers has a much more beneficial effect on sales than an across-the-board price reduction of 2.54. By restricting price discounts to those customers conscientious enough to clip, retain, and turn in little bits of paper, retailers are able to target their price discounts to the most price-sensitive portion of the market. Price-conscious consumers love >em. Interestingly, many retailers have discovered that a substantial number of customers attracted by coupon and rebate promotions buy the product, but then fail to claim the coupon or rebate. For such customers, the idea of a price discount is just as good as a real price discount!

Q5.6 Market demand is influenced by price, the price of substitutes and complements,

product quality, advertising, income, and related factors. Explain why companies often find price changes to be the most important determinant of short-term changes in sales.

Q5.6 ANSWER

For consumer and industrial products, market demand is influenced by price, the price of substitutes and complements, product quality, advertising, income, and a host of related factors. Among such factors, companies often find price changes to be the most important determinant of short-term changes in market demand. Price changes are easily discovered, and both customers and competitors typically respond to them quickly. For example, when McDonald=s cuts the price of a jumbo order of fries, price-sensitive customers immediately recognize the change and the quantity

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demanded rises accordingly. On the other hand, when there is a slight change in the quality of potatoes used, or a modest change in cooking style, consumers and competitors may be slow to react. Similarly, while income is an important determinant of demand, changes in income affect all competitors, and the effect on any single competitor is thereby muted.

Q5.7 An estimated 80% increase in the retail price of cigarettes is necessary to cause a

30% drop in the number of cigarettes sold. Would such a price increase help or hurt tobacco industry profits? What would be the likely effect on industry profits if this price boost was simply caused by a $1.50 per pack increase in cigarette excise taxes?

Q5.7 ANSWER

The price elasticity of demand for cigarettes is highly inelastic if an 80% increase in retail prices would cause only a 30% drop in the number of cigarettes sold. An arc price elasticity for cigarettes on the order of EP = -0.375 (= -30%/80%) implies that tobacco industry revenues would rise sharply following a big price increase. Since the variable costs of producing and selling cigarettes would fall with a 30% drop in the number of cigarettes sold, a significant industry-wide increase in cigarette prices could cause industry profits to explode on the upside.

It is less certain what would happen to tobacco industry profits if the price increase described above were the simple result of an increase in excise taxes on each pack of cigarettes sold. For example, a new $1.50 tax on a pack of cigarettes would increase typical retail prices from $1.88 to $3.38 per pack, or by roughly 80%. If all new revenues went to the government in the form of taxes, then industry revenues and variable costs would both fall by 30% following a 30% drop in unit sales. Unless the tobacco industry cut fixed expenses, industry profits would fall in the long run. On the other hand, with such inelastic demand and government-sanctioned price increases, the tobacco industry might be expected to lobby hard for further profit-enhancing price increases in the post tax-increase era.

Q5.8 Is the price elasticity of demand typically greater if computed for an industry or for a

single firm in the industry? Why? Q5.8 ANSWER

The price elasticity of demand for a firm will typically be greater than that for the industry as a whole. Unless the firm is the industry, as in the case of monopoly, it will face a demand curve that is flatter than that faced by the industry as a whole. This stems from the fact that demand for the industry's product faces competition

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from goods that consumers view as alternatives as opposed to close substitutes. The firm, on the other hand, faces competition from substitute products produced by others in the industry, as well as competition from goods in general, for a place in the consumer's overall market basket.

Q5.9 Is the cross-price elasticity concept useful for identifying the boundaries of an

industry or market? Q5.9 ANSWER

Yes, the cross-price elasticity concept provides a practical means for identifying the boundaries of an industry or market. By definition, a direct relation between the price of one product and the demand for a second product holds for all substitutes. A price increase for a given product increases demand for substitutes; a price decrease for a given product will decrease demand for substitutes. This means that if the cross-price elasticity of demand is a very small negative number, say εPX = -5, a strong substitute good relation exists, and the products involved directly compete for a share of the consumer's dollar. If the cross-price elasticity of demand is a large negative number, say εPX = -0.05, a weak substitute good relation exists, and demand for the products involved is only slightly related. The cross-price elasticity is always positive for substitutes; the price of one good and the demand for the other always move in the same direction. Cross-price elasticity is negative for complements; price and quantity move in opposite directions for complementary goods and services. Finally, cross-price elasticity is zero, or nearly zero, for unrelated goods where variations in the price of one good have no effect on demand for the second.

The point to keep in mind is that the cross-price elasticity concept is an economic measure of the similarity or dissimilarity between products. If the cross-price elasticity of demand is near zero, then even products that are similar in physical terms cannot be regarded as substitutes. The demand for apparel, cosmetics, sporting goods, and a wide variety of services is such that many products that are genuinely comparable in physical terms cannot be regarded as economic substitutes. Similarly, identical products offered at different times or places are not economic substitutes. A Coca-Cola and a brat at the ball park are not offered at the same time and place as those sold at the grocery store, and price differentials reflect that fact.

Q5.10 ACollecting, analyzing and verifying elasticity of demand information is costly, and

that cost is an impediment to using these data help make better day-to-day pricing decisions.@ Discuss this statement.

Q5.10 ANSWER

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Collecting, analyzing and verifying elasticity of demand information is indeed costly, but that cost is scarcely an impediment to using these data to help business managers make better day-to-day pricing decisions. In case after case, evidence suggests that businesses are easily able to justify the expense involved with collecting and analyzing elasticity of demand information. Profit maximization requires a comparison between the marginal benefits and marginal costs of production, including the marginal benefits and marginal costs of producing correct market demand information. For example, cents-off coupons are a well-accepted means for probing the market demand curve at price-output combinations around the current market equilibrium. In grocery stores, information gleaned from electronic scanners is used to help adjust prices and price discounts for maximum effect. In car dealerships, detailed information is collected to determine the relative effectiveness of price rebates versus low-cost financing. In case after case, evidence emerges to suggest that manufacturers and retailers carefully collect, analyze and verify elasticity of demand information to accurately assess the market demand for their products.

SELF-TEST PROBLEMS & SOLUTIONS ST5.1 Elasticity Estimation. Distinctive Designs, Inc., imports and distributes dress and

sports watches. At the end of the company's fiscal year, brand manager J. Peterman has asked you to evaluate sales of the sports watch line using the following data:

Month

Number of

Sports Watches Sold

Sports Watch Advertising

Expenditures

Sports Watch Price, P

Dress Watch Price, PD

July

4,500

$10,000

26

50 August

5,500

10,000

24

50

September

4,500

9,200

24

50 October

3,500

9,200

24

46

November

5,000

9,750

25

50 December

15,000

9,750

20

50

January

5,000

8,350

25

50 February

4,000

7,850

25

50

March

5,500

9,500

25

55 April

6,000

8,500

24

51

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Month

Number of

Sports Watches Sold


Expenditures



May 4,000 8,500 26 51 June

5,000

8,500

26

57

In particular, Peterman has asked you to estimate relevant demand elasticities.

Remember that to estimate the required elasticities, you should consider months only when the other important factors considered in the preceding table have not changed. Also note that by restricting your analysis to consecutive months, changes in any additional factors not explicitly included in the analysis are less likely to affect estimated elasticities. Finally, the average arc elasticity of demand for each factor is simply the average of monthly elasticities calculated during the past year.

A. Indicate whether there was or was not a change in each respective independent

variable for each month pair during the past year.

Month-Pair


Expenditures, A



July-August

____________ ____________

____________

August-September

____________ ____________

____________

September-October

____________ ____________

____________

October-November

____________ ____________

____________

November-December

____________ ____________

____________

December-January

____________ ____________

____________

January-February

____________ ____________

____________

February-March

____________ ____________

____________

March-April

____________ ____________

____________

April-May

____________ ____________

____________

May-June

____________ ____________

____________

B. Calculate and interpret the average advertising arc elasticity of demand for

sports watches.

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C. Calculate and interpret the average arc price elasticity of demand for sports watches.

D. Calculate and interpret the average arc cross-price elasticity of demand

between sports and dress watches.

ST5.1 SOLUTION

A.

Month-Pair


Expenditures, A



July-August

No change

Change

No change August-September

Change

No change

No change

September-October

No change

No change

Change October-November

Change

Change

Change

November-December

No change

Change

No change December-January

Change

Change

No change

January-February

Change

No change

No change February-March

Change

No change

Change

March-April

Change

Change

Change April-May

No change

Change

No change

May-June

No change

No change

Change

B. In calculating the arc advertising elasticity of demand, only consider consecutive months when there was a change in advertising but no change in the prices of sports and dress watches:

August-September

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January-February

On average, EA = (2.4 + 3.6)/2 = 3 and demand will rise 3%, with a 1% increase in advertising. Thus, demand appears quite sensitive to advertising.

C. In calculating the arc price elasticity of demand, only consider consecutive months

when there was a change in the price of sports watches, but no change in advertising nor the price of dress watches:

July-August

November-December

April-May

2 1A

2 1

Q + A A = x E A + Q Q4,500 - 5,500 $9, 200 + $10,000 = x

$9, 200 - $10,000 4,500 + 5,500 = 2.4

ΔΔ

2 1A

2 1

Q + A A = x E A + Q Q4,000 - 5,000 $7,850 + $8,350 = x

$7,850 - $8,350 4,000 + 5,000 = 3.6

ΔΔ

2 1P

2 1

Q + P P = x E P + Q Q5,500 - 4,500 $24 + $26 = x

$24 - $26 5,500 + 4,500 = - 2.5

ΔΔ

2 1P

2 1

Q + P P = x E P + Q Q15,000 - 5,000 $20 + $25 = x

$20 - $25 15,000 + 5,000 = - 4.5

ΔΔ

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On average, EP = [(-2.5) + (-4.5) + (-5)]/3 = -4. A 1% increase (decrease) in price will lead to a 4% decrease (increase) in the quantity demanded. The demand for sports watches is, therefore, elastic with respect to price.

D. In calculating the arc cross-price elasticity of demand, we only consider consecutive

months when there was a change in the price of dress watches, but no change in advertising nor the price of sports watches:

September-October

May-June

On average, EPX = (3 + 2)/2 = 2.5. Since EPX > 0, sports and dress watches are substitutes.

ST5.2 Cross-Price Elasticity. Surgical Systems, Inc., makes a proprietary line of

disposable surgical stapling instruments. The company grew rapidly during the 1990s as surgical stapling procedures continued to gain wider hospital acceptance as an alternative to manual suturing. However, price competition in the medical supplies industry is growing rapidly in the increasingly price-conscious new millennium. During the past year, Surgical Systems sold 6 million units at a price of

2 1P

2 1

Q + P P = x E P + Q Q4,000 - 6,000 $26 + $24 = x

$26 - $24 4,000 + 6,000 = -5

ΔΔ

X2 X1PX

X 2 1

Q + P P = x E + Q QP3,500 - 4,500 $46 + $50 = x

$46 - $50 3,500 + 4,500 = 3

ΔΔ

X2 X1PX

X 2 1

Q + P P = x E + Q QP5,000 - 4,000 $57 + $51 = x

$57 - $51 5,000 + 4,000 = 2

ΔΔ

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$14.50, for total revenues of $87 million. During the current year, Surgical Systems' unit sales have fallen from 6 million units to 3.6 million units following a competitor price cut from $13.95 to $10.85 per unit.

A. Calculate the arc cross price elasticity of demand for Surgical Systems'

products.

B. Surgical Systems' director of marketing projects that unit sales will recover from 3.6 million units to 4.8 million units if Surgical Systems reduces its own price from $14.50 to $13.50 per unit. Calculate Surgical Systems' implied arc price elasticity of demand.

C. Assuming the same implied arc price elasticity of demand calculated in Part B,

determine the further price reduction necessary for Surgical Systems to fully recover lost sales (i.e., regain a volume of 6 million units).

ST5.2 SOLUTION

A. EPX = X 2 X1Y 2 Y1

X 2 X1 Y 2 Y1

- + Q Q P P x - + Q QP P

= 3,600,000 - 6,000,000 $10.85 + $13.95 x $10.85 - $13.95 3,600,000 + 6,000,000

= 2 (Substitutes)

B. EP = 2 12 1

2 1 2 1

- + Q Q P P x - + Q QP P

= 4,800,000 - 3,600,000 $13.50 + $14.50 x $13.50 - $14.50 4,800,000 + 3,600,000

= -4 (Elastic)

C. EP = 2 12 1

2 1 2 1

- + Q Q P P x - + Q QP P

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-4 = 2

2

6,000,000 - 4,800,000 + $13.50P x - $13.50 6,000,000 + 4,800,000P

-4 = 2

2

+ $13.50P9( - $13.50)P

-36P2 + $486 = P2 + $13.50

37P2 = $472.50

P2 = $12.77

This implies a further price reduction of 734 because:

ΔP = $12.77 - $13.50 = -$0.73.

PROBLEMS & SOLUTIONS

P5.1 Market Demand. Michael Kelso, a Wisconsin-based management consultant, has been asked to calculate and analyze market demand for a new video game that is to be marketed to retail (R) and wholesale (W) customers over the Internet.

Kelso=s client estimates fixed costs of $750,000 per year for the product, and that licencing fees and other marginal costs will be $20 per unit. The client has also provided Kelso with the following annual demand information:

PR = $62.50 - $0.0005QR Pw = $50 - $0.002QW

A. Express quantity as a function of price for both retail and wholesale customers. Add these quantities together to calculate the market demand curve. Graph the retail, wholesale, and market demand curves for prices ranging from $65 to $35 per unit.

B. Fill in the following table:

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Market Demand is Retail Plus Wholesale Demand

Price

Retail Demand

Wholesale Demand

Market Demand

Total

Revenue

Total Cost Total Profit

$65

$60

$55

$50

$45

$40

$35

C. What price-output combination will maximize total profits? P5.1 SOLUTION A. To find the market demand curve, it is necessary to express quantity as a function of

price for both retail and wholesale customers. Then, these quantities must be added together to calculate the market demand curve. For retail customers,

PR = $62.50 - $0.0005QR 0.0005QR = 62.50 - PR QR = 125,000 - 2,000PR

For wholesale customers,

PW = $50 - $0.002QW 0.002QW = 50 - PW QW = 25,000 - 500PW

For retail plus wholesale customers, the market demand curve can be expressed with quantity as a function of price as:

Q = QR + QW = 125,000- 2,000P + 25,000 -500P

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= 150,000 - 2,500P

For retail plus wholesale customers, the market demand curve can be expressed with price as a function of quantity as:

Q = 150,000 - 2,500P 2,500P = 150,000 - Q P = $60 - $0.0004Q

For graphic purposes, it is important to remember that the market demand curve is typically expressed with price as the dependent variable on the vertical Y axis. Quantity is usually shown as the independent variable on the horizontal X axis.

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B.

Market Demand is Retail Plus Wholesale Demand

Price

Retail Demand

Wholesale Demand

Market Demand

Total

Revenue

Total Cost Total Profit

$65

-

-

-

-

750,000

-750,000 $60

5,000

-

5,000

300,000

850,000

-550,000

$55 15,000

-

15,000

825,000

1,050,000

-225,000

$50 25,000

-

25,000

1,250,000

1,250,000

0

$45 35,000

2,500

37,500

1,687,500

1,500,000

187,500

$40 45,000

5,000

50,000

2,000,000

1,750,000

250,000

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$35 55,000 7,500 62,500 2,187,500 2,000,000 187,500 C. From the table, the profit-maximizing activity level can be seen as the $40 price and

50,000 unit activity level. Analytically, this can be found by first solving for MR and MC,

TR = P Η Q = ($60 - $0.0004Q)Q = $60Q - $0.0004Q2 MR = ΜTR/ΜQ = $60 - $0.0008Q TC = $750,000 + $20Q MC = ΜTC/ΜQ = $20

The profit-maximizing activity level occurs where MR = MC, and where Mπ = 0. MR = MC $60 - $0.0008Q = $20 0.0008Q = $40 Q = 50,000

At Q = 50,000, P = $60 - $0.0004(50,000) = $40 π = TR - TC = $60Q - $0.0004Q2 - ($750,000 + $20Q) = - $0.0004Q2 + $40Q - $750,000 = - $0.0004(50,000)2 + $40(50,000) - $750,000

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= $250,000

(Note: Μ2π/ΜQ2 < 0, and this is a profit maximum since profit is decreasing for Q > 50,000.)

P5.2 Elasticity. 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. P5.2 SOLUTION 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.

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D. False. A 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.

P5.3 Demand Curves. KRMY-TV is contemplating a T-shirt advertising promotion.

Monthly sales data from T-shirt shops marketing the AEye Watch KRMY-TV@ design indicate that

Q = 1,500 - 200P, where Q is T-shirt sales and P is price.

A. How many T-shirts could KRMY-TV sell at $4.50 each?

B. What price would KRMY-TV have to charge to sell 900 T-shirts?

C. At what price would T-shirt sales equal zero?

D. How many T-shirts could be given away?

E. Calculate the point price elasticity of demand at a price of $5.

P5.3 SOLUTION A. Q = 1,500 - 200P

= 1,500 - 200($4.50)

= 600 B. Q = 1,500 - 200P

900 = 1,500 - 200P

200P = 600

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P = $3 C. Q = 1,500 - 200P

0 = 1,500 - 200P

200P = 1,500

P = $7.50 D. Q = 1,500 - 200P

Q = 1,500 - 200(0)

Q = 1,500 E. The point price elasticity of demand at a price of $5 is calculated as follows:

εP = Q Px P Q

∂∂

= 5-200 x 500

= -2 (Elastic) P5.4 Optimal Pricing. In an effort to reduce excess end-of-the-model-year inventory,

Harrison Ford offered a 1% discount off the average price of 4WD Escape Limited SUVs sold during the month of August. Customer response was wildly enthusiastic, with unit sales rising by 50% over the previous month's level.

A. Calculate the point price elasticity of demand for Harrison Ford 4WD Escape

Limited SUVs sold during the month of August.

B. Calculate the profit-maximizing price per unit if Harrison Ford has an average wholesale (invoice) cost of $27,600 and incurs marginal selling costs of $330 per unit.

P5.4 SOLUTION

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A. εP = Percentage change in quantityPercentage change in price

= 0.500-0.01

= -50 (Elastic)

B. The profit-maximizing price can be found using the optimal price formula:

P =

P

MC1(1 + )ε

= $27,600 + $3301(1 + )

-50

= $28,500 P5.5 Cross-Price Elasticity. The South Beach Cafe recently reduced appetizer prices

from $10 to $6 for afternoon Aearly bird@ customers and enjoyed a resulting increase in sales from 60 to 180 orders per day. Beverage sales also increased from 30 to 150 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 $5 to cause both appetizer and beverage revenues to rise? Explain.

P5.5 SOLUTION A.

2 1P

2 1

Q + (180 - 60) ($6 + $10)P P = x = x = - 2E P + ($6 - $10) (180 + 60)Q QΔΔ

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B.

C. Yes, the |EP| = 2 > 1 calculated in part A implies an elastic demand for appetizers and that an additional price reduction will increase appetizer revenues. EPX = -2.67 < 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. To determine the profit effects of appetizer price changes it is necessary to consider revenue and cost implications of both appetizer and beverage sales.

P5.6 Income 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 expected to surge 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? P5.6 SOLUTION A.

X 2 X1PX

X 2 1

Q + (150 - 30) ($6 + $10)P P= x = x = - 2.67E + ($6 - $10) (150 + 30)Q QPΔΔ

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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:

C. Lower. Since carpet demand is in the elastic range, EP = -8, an increase (decrease) in price will result in lower (higher) total revenues.

P5.7 Cross-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).

P5.7 SOLUTION

A. EPX = X 2 X1Y 2 Y1

X 2 X1 Y 2 Y1

- + Q Q P P x - + Q QP P

2 1I

2 1

Q + I I = x E I + Q Q50 - 30 $58,500 + $55,500 = x

$58,500 - $55,500 50 + 30 = 9.5

ΔΔ

2 1P

2 1

Q + P P = x E P + Q Q30 - 50 $16.50 + $15.50 = x

$16.50 - $15.50 30 + 50 = -8 (Elastic)

ΔΔ

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= 4,800 - 10,000 $85 + $137 x $85 - $137 4,800 + 10,000

= 1.5 (Substitutes)

B. EP = 2 12 1

2 1 2 1

- + Q Q P P x - + Q QP P

= 6,000 - 4,800 $130 + $140 x $130 - $140 6,000 + 4,800

= -3 (Elastic)

C. EP = 2 12 1

2 1 2 1

- + Q Q P P x - + Q QP P

-3 = 2

2

10,000 - 6,000 + $130P x - $130 10,000 + 6,000P

-3 = 2

2

+ $130P4( - $130)P

-12P2 + $1,560 = P2 + $130

13P2 = $1,430

P2 = $110

This implies a further price reduction of $20 because:

ΔP = $130 - $110 = $20 P5.8 Advertising Elasticity. Enchantment Cosmetics, Inc., offers a line of cosmetic and

perfume products marketed through leading department stores. Product Manager Erica Kane recently raised the suggested retail price on a popular line of mascara products from $9 to $12 following increases in the costs of labor and materials. Unfortunately, sales dropped sharply from 16,200 to 9,000 units per month. In an effort to regain lost sales, Enchantment ran a coupon promotion featuring $5 off the

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new regular price. Coupon printing and distribution costs totaled $500 per month and represented a substantial increase over the typical advertising budget of $3,250 per month. Despite these added costs, the promotion was judged to be a success, as it proved to be highly popular with consumers. In the period prior to expiration, coupons were used on 40% of all purchases and monthly sales rose to 15,000 units.

A. Calculate the arc price elasticity implied by the initial response to the

Enchantment price increase.

B. Calculate the effective price reduction resulting from the coupon promotion.

C. In light of the price reduction associated with the coupon promotion and assuming no change in the price elasticity of demand, calculate Enchantment's arc advertising elasticity.

D. Why might the true arc advertising elasticity differ from that calculated in part

C? P5.8 SOLUTION

A. EP = 2 1

2 1

Q + P P x P + Q Q

ΔΔ

= 9,000 - 16, 200 $12 + $9 x $12 - $9 9,000 + 16, 200

= -2

B. The effective price reduction is $2 since 40% of sales are accompanied by a coupon: ΔP = -$5(0.4) or P2 = $12 - $5(0.4) = -$2 = $10 ΔP = $10 - $12 = -$2

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C. To calculate the arc advertising elasticity, the effect of the $2 price cut implicit in the coupon promotion must first be reflected. With just a price cut, the quantity demanded would rise to 13,000, because:

EP = 2 11

2 1 1

Q* - + Q P P x - Q* + QP P

-2 = Q* - 9,000 $10 + $12 x $10 - $12 Q* + 9,000

-2 = -11(Q* - 9,000)(Q* + 9,000)

-2(Q* + 9,000) = -11(Q* - 9,000) -2Q* - 18,000 = -11Q* + 99,000 9Q* = 117,000 Q* = 13,000

Then, the arc advertising elasticity can be calculated as:

EA = 2 12

2 1 2

- Q* + Q A A x - + Q*QA A

= 15,000 - 13,000 $3,750 + $3, 250 x $3,750 - $3, 250 15,000 + 13,000

= 1 D. It is important to recognize that a coupon promotion can involve more than just the

independent effects of a price cut plus an increase in advertising as is implied in Part C. Synergistic or interactive effects may increase advertising effectiveness when the promotion is accompanied by a price cut. Similarly, price reductions can have a much larger impact when advertised. In addition, a coupon is a price cut for only the most price sensitive (coupon-using) customers, and may spur sales by much more than a dollar equivalent across-the-board price cut.

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Synergy between advertising and the implicit price reduction that accompanies a coupon promotion can cause the estimate in Part C to overstate the true advertising elasticity. Similarly, this advertising elasticity will be overstated to the extent that targeted price cuts have a bigger influence on the quantity demanded than similar across-the-board price reductions, as seems likely.

P5.9 Profit Maximization. Rochester Instruments, Inc., operates in the highly competitive

electronics industry. Prices for its RII-X control switches are stable at $50 each. This means that P = MR = $50 in this market. Engineering estimates indicate that relevant total and marginal cost relations for the RII-X model are:

TC = $78,000 + $18Q + $0.002Q2 MC = ΜTC/ΜQ = $18 + $0.004Q

A. Set MR = MC and solve for Q to calculate the output level that will maximize RII-X profit. (Note: Setting MR = MC is equivalent to setting Mπ = 0, where total profit is π = TR - TC.)

B. Calculate this maximum profit.

P5.9 SOLUTION A. To find the profit-maximizing level of output, set MR = MC and solve for Q: MR = MC $50 = $18 + $0.004Q 0.004Q = 32 Q = 8,000

(Note: Μ2π/ΜQ2 < 0, and this is a profit maximum because profits are decreasing for Q > 8,000.)

B. The total revenue function for Rochester is: TR = P Η Q = $50Q

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Then, total profit is: π = TR - TC = $50Q - $78,000 - $18Q - $0.002Q2 = -$0.002Q2 + $32Q - $78,000 = -$0.002(8,0002) + $32(8,000) - $78,000 = $50,000 P5.10 Revenue Maximization. Desktop Publishing Software, Inc., develops and markets

software packages for business computers. Although sales have grown rapidly during recent years, the company's management fears that a recent onslaught of new competitors may severely retard future growth opportunities. Therefore, it believes that the time has come to "get big or get out."

The marketing and accounting departments have provided management with the following monthly demand and cost information:

P = $1,000 - $1Q TC = $50,000 + $100Q MR = ΜTR/ΜQ = $1,000 - $2Q MC = ΜTC/ΜQ = $100

A. Set MR = 0 and solve for Q to calculate monthly quantity, price, revenue, and profit at the short-run revenue-maximizing output level.

B. Set MR = MC and solve for Q to calculate these same values for the short-run

profit-maximizing level of output.

C. When would short-run revenue maximization lead to long-run profit maximization?

P5.10 SOLUTION A. To find the revenue-maximizing output level, set MR = 0 and solve for Q. Thus, MR = 1,000 - 2Q = 0 2Q = 1,000

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Q = 500

At Q = 500, P = $1,000 - $1(500) = $500 TR = P Η Q = $500(500) = $250,000 π = TR - TC = ($1,000 - $1Q)Q - $50,000 - $100Q = -$50,000 + $900Q - $1Q2 = -$50,000 + $900(500) - $1(5002) = $150,000

(Note: Μ2TR/ΜQ2 < 0, and this is a revenue maximum since revenue is decreasing for Q > 500.)

B. To find the profit-maximizing output level set MR = MC, or Mπ = 0, and solve for Q.

Since, MR = MC 1,000 - 2Q = 100 2Q = 900 Q = 450

At Q = 450,

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P = $1,000 - $1(450) = $550 TR = P Η Q = $550(450) = $247,500 π = -$50,000 + $900(450) - $1(4502) = $152,500

(Note: Μ2π/ΜQ2 < 0, and this is a profit maximum since profit is decreasing for Q > 450.)

C. In pursuing a short-run revenue rather than a profit-maximizing strategy, Desktop

Publishing can expect to gain a number of important advantages, including: enhanced product awareness among consumers, increased customer loyalty, potential economies of scale in marketing and promotion, and possible limitations in competitor entry and growth. To be consistent with long-run profit maximization, these advantages of short-run revenue maximization must be at least worth the sacrifice of $2,500 per outlet in monthly profits.

CASE STUDY FOR CHAPTER 5 Optimal Level of Advertising for CSI, Inc. The concept of multivariate optimization is important in managerial economics because many demand and supply relations involve more than two variables. In demand analysis, the concept is particularly important in markets where firms face the difficult question of how to set both prices and advertising at profit-maximizing levels. In demand analysis, it is often typical to consider the quantity sold as a function of the price of the product itself, the price of other goods, advertising, income, and other factors. In cost analysis, cost is determined by output, input prices, the nature of technology, and so on. As a result, the process of multivariate optimization is often employed in the process of optimization.

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To further explore the concepts of multivariate optimization and the optimal level of advertising, consider a hypothetical multivariate product demand function for CSI, Inc., where the demand for product Q is determined by the price charged, P, and the level of advertising, A: Q = 5,000 - 10P + 40A + PA - 0.8A2 - 0.5P2 When analyzing multivariate relations such as these, one is interested in the marginal effect of each independent variable on the quantity sold, the dependent variable. Optimization requires an analysis of how a change in each independent variable affects the dependent variable, holding constant the effect of all other independent variables. The partial derivative concept is used in this type of marginal analysis.

In light of the fact that the CSI demand function includes two independent variables, the price of the product itself and advertising, it is possible to examine two partial derivatives: the partial of Q with respect to price, or ΜQ/ΜP, and the partial of Q with respect to advertising expenditures, or ΜQ/ΜA.

In determining partial derivatives, all variables except the one with respect to which the derivative is being taken remain unchanged. In this instance, A is treated as a constant when the partial derivative of Q with respect to P is analyzed; P is treated as a constant when the partial derivative of Q with respect to A is evaluated. Therefore, the partial derivative of Q with respect to P is:

QP

∂∂

= 0 - 10 + 0 + A - 0 - P

= -10 + A - P The partial with respect to A is:

QA

∂∂

= 0 - 0 + 40 + P - 1.6A - 0

= 40 + P - 1.6A The maximization or minimization of multivariate functions is similar to that for single variable functions. All first-order partial derivatives must equal zero. Thus, maximization of the function Q = f(P,A) requires:

and

To maximize the value of theCSI, demand function, each partial must equal zero:

Q = 0,P

∂∂

Q = 0.A

∂∂

Demand Analysis


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and

Solving these two equations simultaneously yields the optimal price-output-advertising combination. Because -10 + A - P = 0, P = A -10. Substituting this value for P into 40 + P - 1.6A = 0, gives 40 + (A - 10) - 1.6A = 0, which implies that 0.6A = 30 and A = 50(00) or $5,000. Given this value, P = A - 10 = 50 - 10 = $40.

Inserting these numbers for P and A into the CSI demand function results in a value for Q of 5,800. Therefore, the maximum value of Q is 5,800 reflects an optimal price of $40 and optimal advertising of $5,000.

The process of simultaneously determining optimal levels of price and advertising can be visualized by referring to Figure 2.A1, a three-dimensional graph of the CSI demand function. For positive values of P and A, this demand function maps out a surface with a peak at point X*. At the peak, the surface of the figure is level. Alternatively stated, a plane that is tangent to the surface at point X* is parallel to the PA plane. This means that the slope of the figure with respect to either P or A is zero; as is required for locating the maximum of a multivariate function.

Unfortunately, on the basis of Figure 2.A1 it is not possible to conclusively determine whether point X* locates an optimal point for price and advertising that will result in maximum profits or an inflection point that indicates only a local maximum for profits. On the basis of Figure 2.A1, it does not appear that point X* is a local or global point for minimum profits, but in the absence of further analysis, the process described above can lead to mistakes in identifying minimums versus maximums, and vice versa. Absent a check of second-order conditions, the possibility of misidentifying inflection points and points of maxima and minima is always present.

One attractive use of computer spreadsheet analysis is to create simple numerical examples that can be used to conclusively show the change in sales, profits and other variables that occur as one moves beyond points such as point X* identified for CSI Inc. A. Set up a table or spreadsheet for CSI, that illustrates the relationships among quantity

(Q), price (P), the optimal level of advertising (A), the advertising-sales ratio (A/S), and sales revenue (S). In this spreadsheet, use the relations developed in the case study to define appropriate values for each of these items. Importantly,

Q = 5,000 - 10P + 40A + PA- 0.8A2 - 0.5P2 A = $25 +$0.625P

Q = -10 + A - P = 0,P

∂∂

Q = 40 + P - 1.6A = 0.A

∂∂

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A/S = (100ΗA)/S S = PΗQ

Establish a range for P from 0 to $125 in increments of $5 (i.e., $0, $5, $10, ..., $125). To test the sensitivity of all other variables to extreme bounds for the price variable, also set price equal to $1,000, $2,500, $10,000.

B. Based on the CSI table or spreadsheet, determine the price-advertising combination that

will maximize the number of units sold. C. Give an analytical explanation of the negative quantity and sales revenue levels

observed at very high price-advertising combinations. Do these negative values have an economic interpretation as well?

CASE STUDY SOLUTION A. The table or spreadsheet for CSI's unit sales quantity (Q), price (P), optimal advertising

level (A), advertising intensity (A/S), and sales revenue (S) is:

Quantity Price Advertising Ad/Sales Sales Revenue 5,500 $0 $25.000 ----- $0.00 5,570 5 28.125 10.10% 27,851.56 5,631 10 31.250 5.55% 56,312.50 5,683 15 34.375 4.03% 85,242.19 5,725 20 37.500 3.28% 114,500.00 5,758 25 40.625 2.82% 143,945.31 5,781 30 43.750 2.52% 173,437.50 5,795 35 46.875 2.31% 202,835.94 5,800 40 50.000 2.16% 232,000.00 5,795 45 53.125 2.04% 260,789.06 5,781 50 56.250 1.95% 289,062.50 5,758 55 59.375 1.87% 316,679.69 5,725 60 62.500 1.82% 343,500.00 5,683 65 65.625 1.78% 369,382.81 5,631 70 68.750 1.74% 394,187.50 5,570 75 71.875 1.72% 417,773.44 5,500 80 75.000 1.70% 440,000.00 5,420 85 78.125 1.70% 460,726.56 5,331 90 81.250 1.69% 479,812.50 5,233 95 84.375 1.70% 497,117.19

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5,125 100 87.500 1.71% 512,500.00 B. Based on the CSI table or spreadsheet, the optimal price-advertising combination that

will maximize the number of units sold is the analytical solution P = $40 and A = $5,000. Notice that quantity rises until this point is reached, but then falls as both price and advertising climb beyond this point.

C. The negative quantities and sales revenues observed at very high price-advertising

combinations have a simple analytical interpretation. The quadratic expressions for both price and advertising are negative in the CSI demand function. At very high levels for price and advertising, the importance of these negative quadratic terms will outweigh positive first-order effects. Therefore, at very high levels for price and advertising, both quantity and sales revenue turn negative. However, negative values for quantity and sales revenue have no economic interpretation. Remember that price-quantity-advertising relations are clearly sensitive to the price and advertising levels considered. The nonlinear relation described in the CSI demand function would undoubtedly change dramatically with dramatic changes in the price-advertising levels considered.

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