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Market Demand Analysis and Estimation
After reading this chapter, you should be able to:
• Identify the relationship between the demand curve and the demand function and between the demand curve and the total and marginal revenue curves.
• Discuss the relationship between price elasticity of demand and the change in total rev- enue for a price reduction or increase.
• Explain the concepts and usefulness for managerial decision making of income elasticity, cross-price elasticity, advertising elasticity, and other elasticities of demand.
• Describe how primary data required for the estimation of demand functions and curves might be collected using marketing research methods.
• Explain how regression analysis can be utilized to estimate the demand function using secondary data and how these estimates can be used to derive the demand curve and various elasticity of demand measure.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Introduction
The firm’s market demand for a particular product is the aggregation (or horizon-tal summation) of the demand curves of individual consumers for that product.1 Managers need to understand the underlying determinants of market demand and, in particular, how responsive it is likely to be to changes in the firm’s controllable variables (such as the four Ps of marketing) and to changes in uncontrollable variables (such as changes in consumer incomes or the strategic actions of rival firms), since the firm’s revenues depend on the market demand for its products. The sensitivity of quantity demanded to a change in an underlying determinant variable is known as an elasticity of demand. Elasticities of demand convey important information to managers about the impact on market demand (and hence on the firm’s revenues) due to changes in control- lable and uncontrollable variables. In this chapter we will investigate several elasticities of demand that are of interest to practicing managers.
In the second half of this chapter, we concern ourselves with the estimation of market demand for the firm. Managers need to estimate the volume of demand in the current and future periods so that they can plan effectively for hiring and training employees, order- ing raw materials, expanding physical plant and equipment, introducing new products, replacing obsolete products, and so on. Estimation of market demand involves gathering data and interpreting that data to provide a numerical estimate of demand in the cur- rent and future time periods. We consider the gathering of data via interviews, surveys, and market experiments, as well as the use of regression analysis to estimate the respon- siveness of quantity demanded to changes in the firm’s controllable and uncontrollable variables.
4.1 The Demand Function and the Inverse Demand Curve
To clarify terms, note that the demand function refers to the relationship that exists between the quantity demanded of a particular product and all the determinants of that demand that we discussed in detail in the preceding chapter. The demand curve, on the other hand, refers to the relationship that exists between the quantity demanded of a particular product and the price of that product, with all other determinants held constant. The demand curve is thus a subset of the demand function where ceteris pari- bus applies to all determinants except price. As noted in Chapter 3, when we simply say “demand for a product,” we will generally mean the demand curve for that product. So that if we say “a change in demand” or “demand has changed,” we will mean that the demand curve for that product has shifted. To reiterate, changes in price cause move- ments along the demand curve, while changes in all other determining variables cause a shift of the demand curve.
1. For example, hundreds of consumers might buy 0, 1, 2, 3 or more units each at the price of $10 per unit, and in aggregate the market demand might be, say, 680 units at price $10. At a lower price, for example, $9, these consumers might increase their quantity demanded by one or a few units each, such that market demand is, say, 920 units.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Controllable and Uncontrollable Variables That Affect the Demand Curve In Chapter 3 we considered the main determinants of consumer demand. Those that are controllable by the firm are the four Ps of marketing, namely price, product design (or quality), promotion, and the place of sale (distribution system). We saw that changes in price cause a movement along the consumer’s demand curve, while changes in the other controllable variables (and in uncontrollable variables) will cause a shift of the consumer’s demand curve. Since the market demand curve is the horizontal summation of these indi- vidual consumer demand curves, we expect the market demand curve to shift in the same direction (as individual demand curves) when these “demand shifters” change. The direc- tion of the shift for each of the other three Ps is summarized in Table 4.1.
Table 4.1: Controllable shift variables for the firm’s demand curve
Controllable shift variable Demand curve will shift outward for:
Demand curve will shift inward for:
Promotion and advertising Promotional campaigns that motivate consumers to buy the product for the first time, or to buy more of it
Reductions in promotional activity or unsuccessful promotion that upsets people and turns them against the product
Product design or quality Product design changes that are perceived as enhancements by the market
Perceived reductions in product design or quality aspects
Place of sale (distribution) Changes to the distribution system that makes purchasing more convenient for the customer
Changes to the distribution system that reduce convenience or accessibility
In addition, a variety of uncontrollable variables affect the firm’s demand. These uncontrol- lable variables can be discussed under three main headings, namely (a) actions by related- product firms; (b) consumer variables, and (c) changes in the business environment.
Price and Nonprice Actions of Related-Product Firms
As we saw in Chapter 3, related products are those that are either substitutes or comple- ments for the focal firm’s product. Producers of substitute products are in direct competi- tion with the focal firm—their actions that increase demand for their products will simul- taneously reduce the demand for the focal firm’s product. For example, if Ford reduces the price of its line of compact sedans it will sell more of these, but General Motors, Toyota, and other rivals will sell fewer compact sedans after consumers adjust their purchases to maximize their utility. The impact on quantity demanded due to a substitute product’s price reduction may, in some market situations,2 lead to a retaliatory price reduction by
2. Price wars are likely to happen in oligopoly markets (where there are relatively few competing firms, such as the automobile market), which we examine in Chapter 7.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
the focal firm (and by other competitors), perhaps followed then by another price reduc- tion by the firm that initiated the price cutting. We call this price competition, and if the price cutting continues it might degenerate into a price war. In extreme cases prices may be driven down to below costs and all firms might lose money. Since this is a predictable outcome, firms are usually sensible enough to avoid being drawn into a price war and usually try to increase their demand by adjusting one of the remaining three controllable variables.
Adjusting the nonprice controllable variables (i.e., product design, promotion, and place of sale) is called nonprice competition and this is the most common form of competitive rivalry in most mar- kets. Whereas price competition is reactive and immediate, nonprice competition is proactive and delayed—it takes time and talent to design increased quality into the product, to develop and execute an effective promotional campaign, or to set up new distribution channels. Since it takes time to retaliate to nonprice competition, and since the success of such retaliation is not at all assured (in prospect), firms that initiate non- price competition usually benefit from the change they have initiated for a considerable period of time and thereby earn financial payback on the investment they have made in changing product design, promotion, or other nonprice variable.
Producers of complementary products should also be expected to adjust their controllable vari- ables in their own best interests, but in this case what is good for the complementary firm is also good for the focal firm. As we saw in Chapter 3, actions by complementary firms that increase the demand for their products will also increase the demand for the focal firm’s products, and oppo- sitely, actions that reduce demand for the comple-
ment will also reduce demand for the focal firm’s product. For example, if international airfares to France were reduced (increasing the quantity demanded of air travel to France) the demand curve for French hotel accommodation would shift to the right, increasing the quantity demanded at any price. Conversely, if the price of French hotels went up significantly, for example due to the reduced value of the U.S. dollar in terms of the Euro, the demand curve for air travel from the United States to France would shift to the left, causing a reduced quantity demanded at the current airfare levels.
©Photodisc/Thinkstock
Since consumers adjust their purchases to maximize their utility, when a substitute firm reduces its price, it will sell more, and the focal firm will sell less, leading to price competition.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Consumer Variables: Incomes, Tastes, and Expectations
As we saw in the preceding chapter, increases in consumers’ incomes will cause increased quantity demanded for superior3 products (by definition), while increases in consumer income will cause reduced quantity for inferior products (by definition). Thus, if the focal firm’s product is an inferior product, then demand for this product will move oppositely to changes in consumer incomes. When consumers’ incomes are generally rising, as in a period of macroeconomic expansion, the firm’s demand should be expected to fall as its customers switch to a superior substitute, and conversely when consumers’ incomes are falling, as in a recession, the demand for an inferior good should be expected to increase. Macroeconomic expansions and recessions, also known as the business cycle, are likely to cause changes in the incomes of many (but not all) consumers, but note that consumers might at any time receive salary increases or bonuses, or conversely work fewer hours or lose their jobs and thus suffer reduced incomes, independent of the general trend of mac- roeconomic conditions.4
Consumers’ tastes may change and cause the quantity demanded of the firm’s product to increase when tastes change in favor of that product, or oppositely cause the quantity demanded to fall when tastes change against the firm’s product. As we learned in Chap- ter 3, consumer tastes relate to attributes of the product which consumers see as benefits offered by the product. Their “tastes” are really a euphemism for the utility they expect to gain from the consumption of the product, and underlying that is the utility they expect to gain from the attributes of the product. Changes in tastes are usually prompted by new information, such as a medical report showing that carotene is related to the prevention of cancer, or that saturated fats are related to heart disease. Similarly, if the firm engages in “green” practices to save the natural environment, its brand name may be viewed more favorably by at least some consumers and it should expect increased market demand as a result. Conversely, if a firm pollutes the environment or practices discrimination in its workforce, it should expect a negative change in tastes for the firm’s products for at least some of its customers, and hence, there will be a leftward shift of its demand curve.5
Consumer expectations underlie their purchasing behavior. For example, if a person expects to continue earning a salary, he or she might take on a large mortgage for a new house, or take a loan to buy an expensive luxury car, because he or she expects to be able to afford the
3. Some texts use the term normal products to describe what we are calling superior products here. Later in this chapter we will see that superior (also known as normal) products can be divided into necessities, which increase by a lesser proportion than does income, and luxuries, that increase by a greater proportion than does income.
4. The macroeconomic system tends to follow a cyclical pattern of faster growth followed by slower (or negative) growth of gross national product (GNP), followed by faster growth again. These cycles of economic expansion followed by recession are known as business cycles. Thus, we say that the demand for superior products is procyclical, meaning that it is generally in synchroniza- tion with the business cycle, whereas the demand for inferior products is anticyclical, meaning that it generally moves oppositely to the changes in GNP.
5. Note that the firm’s brand name is an attribute of the product because it connotes information about the firm’s attitudes to business ethics, product quality, the natural environment, social equity and justice, and so on. A brand name is effectively a stock of knowledge held by the con- sumer, so that new information that enhances the brand is likely to result in increased demand for the firm’s product.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
loan payments into the future. If these expectations change, for example due to a global finan- cial crisis, he or she might want to proceed more cautiously and subsequently defer or cancel such purchases. Consumers will also form expectations about future price and quality levels and should be expected to defer purchases into a future period if they expect prices to fall sub- stantially or quality to improve substantially. For example, if a firm announces that next month it will put its products on sale, it should expect to experience reduced demand in the current month because at least some consumers will defer their pur-
chase of these products until next month. On the other side of the coin, expectations of quality reductions or price increases in future periods might arise and cause the consumer to accelerate purchases of some products rather than buy lesser quality or at higher prices in later periods. The same applies for expectations of limited availability (shortages) of preferred products in the future—consumers will tend to stock up in advance rather than be unable to buy the product in a future period.
Changes in the Business Environment
In this section we will discuss changes that could happen in the firm’s external business environment and consequently affect demand for its product (see Table 4.2). Actions by governments may lead to changes in demand for the firm’s product. Governments pass new laws and regulations banning some products, legitimizing others, and mandating consumption of still others, such as seat belts in cars. Governments take action to discour- age consumption of cigarettes, alcohol, and drugs. A government might prohibit or place temporary restrictions on trade with particular nations, or, oppositely, open trade rela- tions with nations that were previously closed. As an example of government regulation, increasing social concern over global warming has led various governments to implement “carbon taxes” that will have a detrimental impact on the demand for products that have a relatively large carbon footprint as consumers switch to suppliers that have smaller car- bon footprints. Similarly, new taxes in some countries on fat in food will reduce demand for foods with relatively high-fat content.
©iStockphoto/Thinkstock
Purchasing behavior is shaped by consumer expectations. A homeowner might take on a larger mortgage because he or she expects to afford the future loan payments.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Growth in population will typi- cally mean more demand for a firm’s product, but demand for particular products is likely to be related to changes in the structure of total population. Demographic change refers to variables such as age, ethnicity, gender, geographic distribu- tion, and employment type that change over time and are likely to affect the demand for prod- ucts that are consumed more (or less) by a particular age, ethnic, gender, or regional group. For example, changes in the rela- tive size of an age cohort (i.e., people of the same age, or in the same five-year age bracket, such as 25–29 years old) occur due to changes in birth, death, immigration and emigration rates, such that a particular age cohort might be growing or shrinking in size. Thus, the market for things that a particu- lar cohort consumes is likely to move in the same direction as the size of that age cohort. Where a firm’s product is targeting a relatively narrow age group (e.g., training wheels for bicycles typically used by kids 3–5 years old), changes in the size of this cohort (due to ear- lier changes in the birth rate) can be expected to significantly affect sales from year to year.
A third factor that impacts the external business environment for the firm is weather conditions. Severe snowstorms might paralyze the transportation system and cause elec- tric power failures, meaning that sales will be lost as consumers stay at home or allocate their limited funds to blankets, snowplows, or household repairs. At the other extreme, heat waves cause a re-allocation of consumer expenditures toward air conditioners and electricity to drive the air conditioners at the expense of other expenditures. A recurring weather phenomenon with major economic impact is the cycle of El Niño and La Niña weather events that are due to the changing temperature of Pacific Ocean currents. In summary, products for which consumption patterns are weather-related should expect to have increases or decreases in demand due to changing weather conditions. Finally, natural disasters, most prominently earthquakes and tsunamis, but also forest fires and floods, are low probability but high impact external events that cause massive infrastruc- ture disruption and loss of life and property. They reduce demand for many firms as con- sumers delay or forego consumption due to diverting their limited income towards other products that are necessary to rectify the damage caused by the natural disaster.
©iStockphoto/Thinkstock
When severe snowstorms occur, sales often decline due to limited transportation and consumers choosing to stay at home or allocate their funds elsewhere.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Table 4.2: Uncontrollable shift variables for the firm’s demand curve
Shift variable Demand curve will shift outward for:
Demand curve will shift inward for:
Prices of substitutes Price increases for substitutes Price reductions for substitutes
Prices of complements Price reductions for complements
Price increases for complements
Nonprice competition by substitutes (rivals)
Changes in nonprice controllable variables that shift the rival’s demand curve inward
Changes in nonprice controllable variables that shift the rival’s demand curve outward
Nonprice competition by complements
Changes in nonprice variables that shift the complement’s demand curve outward
Changes in nonprice variables that shift the complement’s demand curve inward
Consumer incomes Increases in incomes (for superior goods) OR decreases in income (for inferior goods)
Decreases in incomes (for superior goods) OR increases in income (for inferior goods)
Consumer tastes Changes of consumer tastes in favor of the focal firm’s product (or its attributes)
Changes of consumer tastes away from the focal firm’s product (or its attributes).
Consumer expectations Changes in expectations that cause consumers to buy now rather than in a future period
Changes in expectations that cause consumers to postpone purchases into a future period
Actions by governments Changes to laws or regulations that encourage consumption of the product
Changes to laws or regulations that discourage consumption of the product
Demographic changes Increases in the age and gender cohorts that buy the focal firm’s product
Decreases in the age and gender cohorts that buy the focal firm’s product
Weather conditions Changes in weather patterns or conditions that cause more to be demanded
Changes in weather patterns or conditions that cause more to be demanded
Natural disasters Events causing damage that requires purchase of products necessary to cope with the damage caused by the event
Events that cause consumption of a product to be impossible or inappropriate to consume
The Form of the Demand Function We are now ready to consider the business implications of the demand function, which we shall express in mathematical forms as shown in equation 4-1. You may be alarmed to think we are about to embark on a mathematical discussion; but fear not, the symbols are used as a shorthand way of identifying the variables and will serve to facilitate our
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
discussion, which in turn will facilitate your understanding of the important issues. Let us express the demand function6 in symbols as:
QDx 5 a 1 1Px 1 2Py 1 3Ax 1 4Ay 1 5GNI (4-1)
Where QDx represents the quantity demanded of product X (the dependent variable); a(alpha) represents the part of QDx that is unexplained by the independent variables listed;
1 2 5 (the betas) show the impact of a one unit change in each independent variables (PX and PY; AX and AY) on QDx;
PX and PY represent the prices of products X and Y, respectively;
AX and AY represent the advertising expenditures for product X and Y, respectively; and GNI represents the level of Gross National Income.
Demand estimation techniques (to be introduced later in this chapter) allow us to esti- mate the values for a and the various s, such that, if we know the current values of the independent variables (those on the right hand side), we can predict the value of the dependent variable, in this case QDx. For now, let us suppose that we have collected data on the independent variables shown in equation 4-1 and have conducted multiple regres- sion analysis to find the following values for a and the various s for a particular product:
QDx 5 5,030 2 3,806.2(Px) 1 1,458.5(PY) 1 256.6(AX) 2 32.3(AY) 1 0.18(GNI) (4-2)
Suppose we want to predict demand (QDX) for the current month using this demand func- tion estimated from recently collected data. First, we will substitute into this equation the current values for the independent variables—suppose these are PX 5 $8; PY 5 $6; AX 5 $168 (thousands); AY 5 $182 (thousands), and GNI 5 $12,875 (billions).
7 Multiplying each of these variables by the appropriate coefficient, and summing the results, we would find QDx 5 22,879 units, and this would be our prediction for quantity demanded in the current month.
6. Note that only some of the controllable and uncontrollable determinants of market demand are listed in this particular demand function. We suppose that either data could not be collected on the other independent variables, or that data collected on the other potential determinants revealed that these variables had insignificant impact on QDx for this product. Also note that the linear (i.e., additive) demand function depicted by equation 4-1 is just one form of the demand function—the actual form is an empirical issue, demand could be a nonlinear function of some variables, for example, it depends on what the data reveals when we estimate the demand func- tion. We use a linear demand function here for simplicity of exposition.
7. We are using gross national income (a macroeconomic concept) here as a proxy measure of the income levels of consumers of product X, assuming that their incomes would rise or fall in synchronization with GNI. This is an oversimplification, of course, but we note that GNI data is readily available and allows the manager to avoid the search costs of gaining actual data on changes in consumers’ incomes, which may be only slightly more accurate.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Since a demand curve shows the impact on quantity demanded for a change in PX with all other determinants held constant, we can isolate the impact of price on quantity demanded by amalgamating the impact of the other variables into a single quantity, which we shall call AOV (for “All Other Variables”), and express the demand curve in the form QDX 5 AOV 1 PX as follows:
QDx 5 53,328.7 2 3,806.2(PX) (4-3)
Note that this expression for the demand curve depends on all other variables remaining constant, and we require this so we can isolate the impact on quantity demanded of price, alone.
The Inverse Demand Curve Economists, following the convention set by the classical economist Alfred Marshall, tra- ditionally draw the demand curve with price (the independent variable) on the vertical axis and quantity (the dependent variable) on the horizontal axis, which is opposite to the way mathematicians like to draw graphs—they would place the dependent variable on the vertical (Y) axis and the independent variable on the horizontal (X) axis. The econo- mist’s convention makes it easier to compare prices and costs in later chapters, however. To express equation 4-3 in terms of PX we will add 3,806.2PX to both sides, take QDX from both sides, and divide both sides by 3,806.2 to find:
PX 5 53,328.7/3,806.2 2 1/3,806.2(QDx) (4-4)
This simplifies to PX 5 14.011 2 0.000263(QDx). This very small coefficient to QDx is hard to comprehend, so for convenience we shall now express QDx in units of one thousand, and rewrite this as:
PX 5 14.011 2 0.263QDx (4-5)
The demand for product X is now expressed in the form of an inverse demand curve of the generic form:
Px 5 a 1 bQDx (4-6)
where a 5 AOV/2 and b 5 1/. If we now plot this inverse demand curve on a graph with Px on the vertical axis, it will intercept the vertical axis at 14.011 (since Px 5 14.011 when QDx 5 0). Further, from equation 4-3 above, we would find the horizontal intercept by noting that QDx 5 53,328.7 when Px 5 0. These intercept values of the inverse demand curve serve to locate the demand curve at the correct height within the graph, as shown in Figure 4.1, so that this demand curve will provide useful results within the relevant range of prices, that is, for prices around the current price level. Note that, henceforth, we shall sim- ply refer to it as the demand curve rather than repeatedly saying the inverse demand curve.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Figure 4.1: The inverse demand curve
Horizontal intercept at 53.3287
The relevant range of prices
Vertical intercept at Px = 14.011Px
QDX/t (thousands)
15
10
10 20 30 40 50 60
5
0
Total Revenue and Marginal Revenue In Figure 4.1 you can see the full range of prices and quantities demanded for the market demand curve. The manager will want to know what happens to total revenue (TR) when price is changed and whether TR will rise or fall when price is increased (or reduced). There is a simple relationship between the demand curve, the total revenue curve and the marginal revenue curve, which we will now demonstrate. In Table 4.3 we show several levels of price ranging from $14.011 (when QDx 5 0) to zero, (when QDx 5 53.328). Note that total revenue is equal to price times the quantity demanded, that is:
TR 5 PxQDx (4-7)
In Table 4.3 you will note that TR starts from zero (when nothing is sold), rises to a maxi- mum when price is $7, and then falls all the way to zero again as price is reduced further. Notice that TR is maximized at $186,797 halfway down the demand curve (where Px 5 $7, which is half of the vertical intercept value and QDx 5 26.685, which is half of the horizon- tal intercept value). You can see that TR rises in a smooth curve to its maximum value and then falls in a smooth curve, as shown in Figure 4.2.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Table 4.3: Price times quantity demanded equals total revenue
Price Quantity demanded Total revenue
$14.011 0 $0
$12 7,654 $91,848
$10 15,267 $152,670
$8 22,879 $183,032
$7 26,685 $186,795
$6 30,492 $182,952
$4 38,104 $152,416
$2 45,716 $91,432
$0 53,329 $0
Now we consider marginal revenue (MR), which is defined as the change in total revenue for a one-unit change in quantity demanded. MR is important because it indicates “where the firm is” on the TR curve and the firm surely does not want to be on that part of the TR curve where total revenue is falling. Given the definition of MR above, we can express it in terms of TR and quantity demanded, as follows:
MR 5 TR/QDx (4-8)
where the symbol (uppercase delta) signifies a discrete change in each of the variables indicated. You will quickly appreciate that MR is a measure of the slope of the TR curve, since TR represents the rise of the TR curve and QDx represents the run of that curve. Clearly the slope (and hence MR) is positive but falling as the TR curve rises at a decreas- ing rate (becoming progressively less steep) until MR 5 0 when TR is maximized, and thereafter MR is negative and takes increasingly larger negative values as TR falls toward zero. In Figure 4.2 the MR curve is shown as starting at the vertical intercept and falling exactly twice as fast as the demand curve (since it cuts the horizontal axis at half the quan- tity demanded, i.e., about 26,685 compared to 53,328). The MR curve starts at about $14, fractionally below the intercept value (which is 14.011) where the first unit is demanded, because the first unit sold caused TR to increase from zero to about $14. Thus, the MR curve can be represented as having the same vertical intercept as the demand curve and twice the slope of the demand curve8 as follows:
MRx 5 a 1 2bQDx (4-9)
8. This is because TR is a quadratic expression in Q, which we can specify by recalling from equa- tion 4-6 that Px 5 a 1 bQDx and substituting for Px into equation 4-7 we find TR 5 aQDX 1 bQDx
2. Marginal revenue (MR) is the derivative of the TR expression, so MR 5 a 12bQDX.
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CHAPTER 4Section 4.1 The Demand Function and the Inverse Demand Curve
Figure 4.2: Demand, total revenue and marginal revenue curves
ε > |1| . Demand is “elastic”
ε < |1|. Demand is “inelastic”
QDX
QDX
PX
TR
TR
MR
D
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CHAPTER 4Section 4.2 Elasticities of Demand
What we learn from all this is that there is a happy medium in terms of the price to be set by the firm. If price is set “too high” on the demand curve, TR could be increased by reducing price, and oppositely, if price is set “too low” on the demand curve, TR could be increased by increasing price. But note that TR is not profit—we have not yet considered the costs of production, so we are not yet ready to say exactly where on the demand curve the price should be set to maximize profits. We can readily see that the lower half of the demand curve is a bad idea. Marginal revenue would be negative, so TR would be increased by moving to the upper half of the demand curve. But, the profit-maximizing price level will depend on the level of costs, as we shall see in Chapter 7.
4.2 Elasticities of Demand
Managers will be interested in the elasticity of demand with respect to each of the determining variables because these measure the relative responsiveness of quantity demanded to a small change in a particular determining variable. Because many independent variables operate to determine quantity demanded, there are many elasticity measures that the manager will be interested in. We shall look at the main ones here, but you will see that an elasticity value can easily be calculated for any variable that significantly influences demand.9
Price Elasticity of Demand Price elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price. That is:
5 %DQDx %DPx
(4-10)
where (the Greek letter epsilon) is the conventional symbol for price elasticity and (the Greek letter [capital] delta) represents a discrete change in the relevant variable. Expand- ing this out, we can equivalently say:
5 DQDx/QDx # 100/1
DPx/Px # 100/1 (4-11)
Cancelling the 100/1 terms and rearranging we find:
5 DQDx DPx
# Px
QDx (4-12)
9. Elasticities of demand are deduced from the values found in the demand function. Thus, although we have for convenience expressed QDx in thousands in the demand curve, we will continue to express QDx in single units in the calculations of the various elasticities. Alternatively, we could divide each of the coefficients by 1000 (e.g., 1 which was found to be 3,806.2 would be 3.8062, and so on, for the other coefficients.
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CHAPTER 4Section 4.2 Elasticities of Demand
Now note that the first term in this expression (QDx/Px) is equal to 1 from the demand function, so:
51? Px
QDx (4-13)
Note that 1 shows the responsiveness of QDx to a small change in Px. Since 1 5 23,806.2 in our earlier example, we can say that a $1 increase in the price of product X will cause a decrease in quantity demanded of 3,806.2 units. But note that elasticities of demand are measures of relative responsiveness, so the 1 coefficient is weighted by the ratio of price to quantity demanded to find the price elasticity value.10 Evaluating equation 4-14 for 1 5 23,806.2; Px 5 $8, and QDx 5 22,879 we find 5 21.331. So, price elasticity is equal to the coefficient to price from the demand function weighted by the ratio of price to quantity demanded. Since the ratio of Px to QDx varies from infinity (when QDx 5 0) to zero (when Px 5 0) price elasticity must vary from very high negative numbers to very low negative numbers as we move down the demand curve. We can see this in Table 4.4, using the data from the previous example.
Table 4.4: Price elasticity at various price levels
Px QDx Px/QDx β1 ε
$14.011 0 ∞ 23,806.2 ∞
$12 7,654 1.568 23,806.2 25.967
$10 15,267 0.655 23,806.2 22.493
$8 22,879 0.350 23,806.2 21.331
$7 26,685 0.262 23,806.2 21.000
$6 30,492 0.197 23,806.2 20.749
$4 38,104 0.105 23,806.2 20.400
$2 45,716 0.044 23,806.2 20.167
$0 53,329 $0 23,806.2 0
Note that the price elasticity values are negative because 1 is negative (due to the inverse relationship between price and quantity demanded). Also note that the price elasticity rises from high negative numbers to smaller negative numbers as we move down the demand curve, and that 5 21 at the midpoint of the demand curve. Talking about numerically larger negative values can be confusing. A better way of communicating
10. This is an important difference between the jargons of marketing and economics. When market- ers speak of “price responsiveness,” they usually mean the 1 value. Economists use the term “price elasticity” to show the relative responsiveness of QDx to Px and they use value to convey information about the change in total revenue that is associated with a price change, as we will see.
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about price elasticity is to talk about it in absolute terms, ignoring the negative sign. Thus, economists say the price elasticity gets higher (in absolute terms,) as we move up the demand curve. For example, we say that price elasticity at price $12 is higher than it is at price $10, and so on. By convention we say demand is price elastic above the midpoint of the demand curve (for |1|, i.e., for values of greater than one in absolute terms) and conversely is price inelastic below the midpoint of the demand curve (for |1|, i.e., for values of less than one in absolute terms).11
We can now summarize the relationships between price, price elasticity, total revenue, and marginal revenue that are apparent in Figure 4.2. If price elasticity is greater than 1 (in absolute terms) then the current price must be on the upper half of the demand curve; MR must be positive; and TR will decrease for a price increase (or increase for a price decrease). Oppositely, if 1 (in absolute terms) then the current price must be on the lower half of the demand curve; MR must be negative; and TR will increase for a price increase (or decrease for a price reduction). Thus, price elasticity is a number that conveys a lot of useful information to the managerial decision maker.
Income Elasticity of Demand Income elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in income (that caused the change in demand). That is:
5 %DQDx %DGNI
(4-14)
where (the Greek letter theta) is the conventional symbol for income elasticity. Follow- ing the earlier example we are using GNI (gross national income) to represent the income level. Following the same procedure as before we can “cut to the chase” and simply say that:
5 5 # GNI QDx
(4-15)
where 5 is the coefficient to GNI from the demand function estimated earlier as equation 4-1. Evaluating equation 4-15, using the earlier values of 5 5 0.18; GNI 5 12,875, and QDx 5 22,879, we find 5 0.1013. Thus, according to our data, we expect that the responsive- ness of demand to a one-unit change (i.e., one billion) in GNI will be about 0.18 units of product X (i.e., the value 5), but that the relative responsiveness (i.e., the income elasticity) is 0.1013. The income elasticity implies that quantity demanded would increase by only about one tenth of 1% for a 1% increase in GNI (or 1.013% for a 10% increase in GNI), which is to say that quantity demanded is virtually unresponsive to changes in GNI. As
11. Note that firms do not want to have their prices fall on the lower half of the demand curve, since MR is negative and TR would be increased by raising price until it was higher than the midpoint of the demand curve. As we shall see in Chapter 7, only firms operating in oligopo- lies (few competing firms) who expect heavy sales reductions if they raise prices (because they expect their rivals to not follow their price increases) have to worry about that.
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with price elasticity, economists use the income elasticity value () to provide information over and above the responsiveness value, as we shall soon see.
From our earlier discussion of superior and inferior goods we know that product X must be a superior good since its quantity demanded increases when income increases. Indeed we could have concluded that simply by looking at the positive sign of the 5 coefficient to income in the demand function. If on the other hand, 5 had been a negative number, we would know that the product was an inferior good. Note that some consumers may indeed consider product X to be an inferior good but on balance the income elasticity shows it is a superior good, since the overall effect of an income change on quantity demanded is slightly positive, and this overall effect is what the manager will want to know.
Luxury and Necessity Goods
We can make a further distinction within superior goods based on the size of the income elasticity. If is greater than 1, we say that the product is a lux- ury good. Values of 1 mean that demand for the product increases by a larger percentage than the percentage increase in consumer incomes. For example, suppose your income is currently $100,000 and you buy caviar four times a year, spending say $400 a year on 20 ounces of caviar. Suppose your income rises by 10% to $110,000 a year, and you then increase your caviar consump- tion to six times (30 ounces total) a year. The per- centage increase in your demand for caviar is 50% and the percentage increase in your income is 10%, so 5 5. As we suspected of course, this makes caviar a luxury good for you, but not necessarily for everyone, since everybody’s tastes and income levels are different. Extraordinarily rich people may already be buying as much caviar as they can eat and may respond to an increase in their incomes by not buying more, and most people on lower incomes will not be buying any caviar at all. Again, the manager will look at the aggregate change in market demand when incomes change rather than the variation across consumers within his or her market.12
12. Of course all caviars are not the same. There are quality differences that reside in the different attributes and attribute ratios for the various brands of caviar. So a brand manager pursuing a niche marketing strategy will be very interested in the tastes and preferences of consumers in the targeted niche market, and it is the niche market’s demand, rather than the total market demand for all caviars, that is important to the manager.
©iStockphoto/Thinkstock
Although caviar is a luxury good for many, this is not the case for everyone since everybody’s tastes and income levels are different.
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Conversely, if is positive but less than one we say that the product is a necessity good. In our earlier example product X would be classified as a necessity good, since 5 0.1013. For necessity goods, demand certainly increases when income rises, but its relative respon- siveness is less than one. Examples of necessity goods include all basic foodstuffs, ordinary clothing, basic housing and transportation services, and other basic things that consumers need to buy to stay warm and stay well as they go about their normal lives whether work- ing, unemployed, or retired. Luxury goods are those substitute products that one aspires to but cannot afford while on relatively low incomes. For example, as income rises people might want to replace a low-status brand handbag with a high-status brand handbag, or replace cheap Scotch whisky with single-malt Scotch whisky. Notice that the low-status handbag and the cheap Scotch are inferior goods for these people in these examples, since their quantity demanded for these products decreases as their incomes rise. But also note that these products are not necessarily inferior goods at the aggregate market level—for very poor people they might be luxury goods that can only be purchased as incomes rise from very low levels. Managers will want to predict changes in the market demand for their products when incomes change, and while this will include some people increasing and some people reducing their quantity demanded, they will want to be able to predict that the net effect will go one way (superior goods) or the other (inferior goods).
Cross-Price Elasticity of Demand Cross-price elasticity of demand is defined as the percentage change in quantity demanded for product X divided by the percentage change in the price of a related product Y. That is:
5 %DQDx %DPY
(4-16)
where (the Greek letter eta) is the conventional symbol for cross-price elasticity of demand. Following the same procedure as before we can say that:
5 2 # PY
QDx (4-17)
where 2 is the coefficient to PY in the demand function shown earlier as equation 4-1. Evaluating equation 4-17 for the values of 2 5 1,458.5, PY 5 6, and QDx 5 22,879, we find 5 0.382. Thus, according to our data, we expect that the responsiveness of QDx to a one- unit increase in PY (i.e., from $6 to $7), will be 1,485.5 additional units of X, but that the relative responsiveness (i.e., the cross-price elasticity) will be 0.382. This implies that a 10% increase in the price of product Y would cause the quantity demanded for X to increase by about 3.82%, or conversely a 10% price reduction for product Y would cause the demand for product X to decline by about 3.82%. Thus product X and product Y are substitutes for each other—increases in the price of one would cause some consumers to switch from that product towards the other in order to maximize their utility.
To summarize, the sign of the coefficient to the price of a related product in the demand function (2 in this example) and the sign of the cross-price elasticity measure () indi- cates whether that product is a substitute or a complement for the focal product X. If the
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coefficient and hence the cross-price elasticity had shown a negative sign, we would know that product Y is a complement for product X, that is, X and Y are complementary in con- sumption, meaning they are typically consumed together. The size of the coefficient will show the responsiveness of QDx to a one unit change in PY, and the size of the cross-price elasticity measure will imply the percentage change in QDx for a one-percentage change in PY.
Other Elasticities of Demand Because the estimated demand function we introduced earlier provides data on the responsiveness of demand for product X to changes in advertising for product X, we can calculate the elasticity of demand with respect to the firm’s advertising expenditures. Again, the elasticity measure will be the percentage change in QDX divided by the per- centage change in advertising, but is more quickly calculated by weighting the coefficient to the advertising variable in the demand function (3) by the ratio of the advertising expenditure (Ax) over the dependent variable (QDx). Evaluating for 3 5 256.6, Ax 5 168 (in thousands) and QDx 5 22,879 we find the advertising elasticity to be 1.884.
Thus, the responsiveness of quantity demanded to advertising expenditures is expected to be 256.6 additional units of product X for every additional $1,000 of advertising expendi- ture. The relative responsiveness of demand to advertising is that a 1% increase in advertis- ing would lead to a 1.884% increase in quantity demanded. But whether the firm should do any more advertising depends on how much each additional unit of product X would contribute to the firm’s expenses or contribution margin.13 Obviously it is only worth spending another $1,000 on adverting if the extra sales (256.6 units) contribute at least $1,000 (i.e., about $4 each per unit) to overhead and profit. The information gained from the estimated demand curve, plus information gained from cost estimation (see Chapter 6) will help the manager make the decision whether to change the advertising level or not. We shall also consider the firm’s advertising decision in Chapter 11.
Cross-Advertising Elasticity
The estimated demand function also provided data on the impact of firm Y’s advertising on the quantity demanded of product X. The cross-advertising elasticity will be the coef- ficient (4) to the AY variable in the demand function weighted by the ratio of the inde- pendent variable (AY) over the dependent variable (QDx). Evaluating for 4 5 232.3, AY 5 182 (in thousands) and QDx 5 22,879 we find the cross-advertising elasticity to be 20.257. Thus, the responsiveness of quantity demanded of X to increased advertising expenditures by Y is expected to be 232.3 units of product X for every additional $1,000 spent on adver- tising by firm Y, and the relative responsiveness of demand for X to advertising by Y is that a 10% increase in advertising by Y would lead to a 2.57% decrease in the quantity demanded of X. Thus, the firm’s demand is not very sensitive to changes in the advertis- ing of this rival firm. But managers at firm X need to know how vulnerable they are to
13. The contribution margin is the difference between price and variable cost per unit and represents the contribution that each unit of product X makes toward the firm’s overhead (fixed) costs and profits after the incremental variable costs have been covered. We will spend more time talking about the contribution margin in the next two chapters.
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CHAPTER 4Section 4.3 Estimating Market Demand From Primary Data
increases in advertising (or promotional efforts more generally) by related-product firms, so they can decide on their own advertising policy. Should they spend more on advertis- ing to win these customers back? Would it be worth it? That depends, again, on the con- tribution margin for product X. If it is thousands of dollars per unit of X (such as it might be for heavy machinery, MBA programs, or sales of luxury cars) then firm X may well find it more profitable to increase its own advertising, or reduce their price, to increase market share. On the other hand, if the contribution margin is relatively small it will not be worth either adjustment, given the costs of changing prices or mounting an advertising campaign. We defer resolution of this decision problem to Chapter 11.
Quality Elasticity of Demand
Finally, we will briefly consider the firm’s quality elasticity of demand. Quality elasticity is defined as the percentage change in quantity demanded of product X divided by the percentage change in the quality of product X. We argued in Chapter 3 that a perceived increase in quality would cause an outward shift in the demand curve for individual con- sumers, and that this, in aggregate, would cause an outward shift in the market demand curve. The manager will want to know “how far will the demand curve shift if I increase product quality?” Keep in mind that increased product quality will typically cause per unit costs to be higher, so again the manager will want to know “will an increase in qual- ity lead to an increase in my profits?” In a simple example, suppose that increased quality causes variable costs to rise by $1 per unit, but the market will buy the same or greater volume at a price that is $1.25 higher than before, you can see that the firm will be more profitable as a result. In Chapter 11 we will look at more complex examples where there are diminishing returns to advertising effectiveness and also where variable costs increase at higher levels of production.14
4.3 Estimating Market Demand From Primary Data
In the remainder of this chapter, we will discuss methods that are used to obtain data relating to the firm’s demand function so we can derive the coefficients that indicate the responsiveness of quantity demanded to each of the independent variables that collectively determine the firm’s demand. We begin by discussing direct methods of demand estimation whereby primary data is collected from actual or potential buyers via interviews, surveys, and market experiments. Indirect methods of demand estimation involve the statistical examination of data previously collected for official government sta- tistics, or found in reports researched and written for other purposes (so-called secondary data). We will discuss the use of regression analysis as a tool to quantify the dependence of quantity demanded on the independent variables in the demand function. Special atten- tion is given to the interpretation of the regression results and the avoidance of six major pitfalls of regression analysis.
14. To include product quality as an independent variable in an estimated demand function, the firm would need to have reliable data that reflects product quality. Assuming efficiency in pro- duction (see Chapter 6), and other things remaining the same, such as the output rate, wages, and other input costs, the cost of goods sold per unit might be a sufficiently reliable indicator of product quality.
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CHAPTER 4Section 4.3 Estimating Market Demand From Primary Data
Interviews, Focus Groups, and Surveys The most direct method of demand estimation is to simply ask buyers or potential buyers about their probable reactions to changes in price or other determining variables. Interviews usually follow a questionnaire to ensure that respondents provide answers to specific questions. Question- naire design is an art form and should not be treated lightly—unless the questions are asked in words that the respondent fully understands, the results might not be reliable. Focus groups are less structured and are useful for finding new information about what consumers want in terms of product design or their collective response to planned price changes, for example. Usually the researcher will let conversation range freely within the focus group, letting people’s ideas and opinions emerge, and occasionally redirect- ing discussion back to issues of concern to the researcher. Surveys utilize structured question- naires and are administered either by mail, by telephone, by email, or via online survey tools such as SurveyMonkey.
Let us work through a simple example of a survey to find the demand curve for a new product. Sup- pose a firm plans to introduce a new product and wants to know how much would be demanded at various price levels. From the population of potential buyers a random sample of 500 people is drawn, perhaps by contacting every 10th name on a list of probable buyers. Sup- pose we then contact these people after dividing them into five groups of 100 each. For each group we would describe the new product and its usefulness in the same way, but would state a different price for each group, as shown in Table 4.5. We would then ask them whether they would buy the product at the price stated, and list the number of posi- tive responses in the table as shown.15
Table 4.5: Survey results for various suggested prices
Sample Group 1 2 3 4 5
Price stated $6 $4 $5 $7 $8
Quantity demanded 198 305 262 155 97
©Digital Vision/Thinkstock
Focus groups are useful for uncovering new information about buyers’ probable reactions to changes in price and other determining variables.
15. Rather than a Yes/No answer, we might ask the respondent to choose a point on a 1–5 scale where 1 5 not at all likely to buy; 2 5 somewhat likely to buy; 3 5 not sure; 4 5 quite likely to buy; and 5 5 very likely to buy. We would then record in a table the number of people who chose either 4 or 5.
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These price2quantity coordinates would then be plotted, as shown in Figure 4.3, and a line of best fit would be drawn freehand across the data points. A line of best fit sum- marizes the apparent relationship between the two variables. We should not expect all (or any) of the observations to lie exactly on the line of best fit. There is likely to be random variations among the five groups that cause them to demand a little more, or a little less, than another group would have at the same price. We position the line of best fit such that we think it minimizes the gap between the data points and the line. Then we note the intercept point on the price axis and the slope of the line of best fit. The intercept appears to be close to $10 and the slope appears to be roughly 21/50 5 20.02. Thus, our estimate of the sample’s demand curve for this new product is Px 5 10 2 0.02QDx.
Figure 4.3: Freehand line of best fit to price-quantity data pairs
Sample Quantity Demanded / t
Line of best fit: PX =10 – 0.02QDX (sample) or
PX=10 – 0.001Q DX (population)
Market Quantity Demanded / t0 2000 3000 4000 5000 6000
10
9
8
7
6
5
4
3
2
1
98 155 198 262 305
D
Price ($)
But note that these results relate to a sample of only 500 people. To estimate the demand curve for the entire market we would multiply the slope term by the sampling proportion. Suppose 5% (or one in 20 people) of the population was sampled, so we need to multiply
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the slope term by 1/20 to arrive at an estimate of the market demand curve, which would be PX 5 10 2 0.001QDX. From this, we can easily determine that the horizontal intercept of the estimated demand curve occurs at about 10,000 units. Thus, marginal revenue would cut the horizontal axis at about 5,000 units, so that the total revenue curve would rise to a maximum of about ($5 3 5,000 units 5) $25,000 (per period) and would then fall if price were to be reduced below $5.
Potential Problems With Interview and Survey Data
Survey methods may not lead to reliable results if any one of the seven following problems exists. First, we may not have a random sample of our target con- sumers. Methods exist to ensure that samples are sufficiently random, and tests can be made to ensure that sample selection bias or nonresponse bias are not likely to be present in the data (Hair, Black, Babin, & Anderson, 2010). A second problem is that answers reported to you directly (often called espoused data) by the survey respondent may be unreliable. The presence of the interviewer, or even the fact that someone later will see the answers given, may cause the respondents to be less than fully frank with their answers, which is known as interviewer bias. For example, people may overstate their income in order to seem more successful than they really are. A third problem is social desirability bias, that is, if the researcher asks a personal question, the respondents might understate or over- state the true amount because they would be embarrassed to reveal the truth. Questions about politics, religion, income, and lifestyle are likely to induce a social desirability bias. A fourth problem is self-serving bias, where interviewees have the incentive to not reveal the exact truth since it might adversely affect future outcomes for them. For example, if a firm were to ask how much more would consumers pay for a 20% improvement in qual- ity, respondents would have an incentive to say an amount smaller than what they really would be prepared to pay. A fifth problem is disinterested or busy participants who give random answers or rush through ticking boxes without really considering the questions asked. Even if the answers given are completely truthful, a sixth problem is the best-of- intentions problem whereby the respondents say what they believe they will do (e.g., I will buy the product at a price of $7) but do not subsequently actually buy it due to the inter- vention of other issues, such as losing a job or discovering another product that is a better value proposition. Finally, the responses may be unreliable if the questions are confusing, misinterpreted, or unknowable. New products, when described briefly for the first time, may not easily be imagined by consumers as part of their lifestyle or work environment. For
©iStockphoto/Thinkstock
Hiring a marketing research firm to design a professional questionnaire is money well spent for managers contemplating large investments in new products, repositioning of product prices or qualities, and product line extensions.
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example, IBM vastly underestimated the demand for its personal computers in the early 1980s after surveying hundreds of business executives who saw the desktop computer simply as a fancy typewriter for their secretaries.
To combat these problems, it is important to design the questionnaire carefully, which is best done by experts. Marketing research firms, or freelance consultants specializing in survey design, can be utilized to design questionnaires that will minimize these problems; for example, rather than asking questions about price directly, the question might ask about “value” (recall that value equals quality over price) that consumers perceive in a series of price–quality combinations. Money spent on professional questionnaire design will be money well spent for managers contemplating large investments in new products, repositioning of product prices or qualities, product line extensions, and so on (Hair et al., 2010).16
Simulated Market Situations Marketers often conduct simulated market situations, also known as consumer clinics, where they construct an artificial shopping environment and observe the choices of cus- tomers, while varying the prices of some products, the shelf placement of products, and point-of-purchase information about product quality, for different groups of shoppers. Participants are usually attracted to these experiments by the promise of free products; for example, they may be given $100 of “monopoly money” to spend as they wish within an hour. Researchers monitor hidden cameras to observe shopping behavior and tally up the purchases at the cash register on the way out. If the participants are randomly drawn from the population of potential customers for the focal products, market researchers may conclude that the entire market would react to price changes and changes in other product attributes in the same way. The quantitative results of a simulated market experiment (e.g., how many units of X were purchased by each sample of 100 shoppers) can be analyzed in the same manner as above for the survey data, with an estimated line of best fit scaled up to reflect the ratio of sample size to the population of potential customers for the focal product. Similarly, for different qualities of the same product, higher shelf placement, and
16. Market researchers often use “conjoint analysis” to provide data that is revealed by choices among alternatives, rather than simply espoused. Respondents consider several product attri- butes conjointly (i.e., together, in the context of each other). For example, the manager might want to know whether consumers value higher product quality and increased purchasing con- venience enough to warrant charging a higher price. In the experiment, the level of price, qual- ity, and convenience of purchasing are varied across multiple scenarios. For example, scenario 1 might be stated as high price, high quality, and high convenience (i.e., high-high-high), while scenario 2 might be stated as low-high-high; scenario 3 as high-high-low; and so on. Respon- dents are asked to rate the attractiveness of each scenario on a 1–7 scale ranging from 1 5 highly unattractive to 7 5 highly attractive. The conjoint method assigns values to the high setting for each of the three variables to find revealed attitudes to high price, high quality, and high con- venience. These will reveal whether the disutility of the higher price is offset by the increased utility associated with the higher quality and higher convenience. See Hair, J.F., Black, W.C., Babin, B.J., and Anderson, R.E. Multivariate Data Analysis, Pearson, 2010.
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end-of-aisle placement, the researchers could multiply the sample groups’ responses to estimate the overall market’s response to changes in these variables.17
Market Experiments Market experiments involve real people in real markets spending their own money on the products they probably really do want. The firm will select a specific city or region that is representative of other cities or regions or perhaps is representative of the entire nation (e.g., San Diego is said to be quite representative of southern California). The firm then introduces a new price, or new quality (i.e., changed attributes) of its product, or new promotional campaign into all stores in this test market (via its established distribution systems), and observes the impact on quantity demanded in that city or region. The firm then predicts a similar result will occur in other cities or regions that are similar to the test market, perhaps proceeding to a “national rollout” to the entire market. Such experiments are obviously very large scale and will require a large investment that may be lost if the experiment is not successful or conducted well. On the other hand, such market experi- ments limit the potential loss to only some fraction of what it might be if the firm had gone directly to a national rollout. Thus, market experiments can be used to validate managers’ decisions about changes in controlled variables in a limited context before they embark on full-scale implementation of the changes. As we saw in Chapter 2, managers will want to integrate such risk considerations into their decision making.
Direct Marketing Experiments
Direct marketing occurs when the producer of the product sells directly to the consumer of the product rather than utilizing conventional distribution channels that involve whole- salers and retailers as intermediaries between the producer and the consumer. An exam- ple of direct marketing is Internet sales direct from the manufacturer to the consumer, and many retailers also sell products via their Internet websites. However, many people still read newspapers, magazines, and letters sent to them via “snail mail,” and these remain effective vehicles for direct marketing experiments. Researchers send different price and quality offers to different potential customers, who then decide whether to buy the prod- uct at the price and quality they have been offered. Researchers expect to find a negative price effect and a positive quality effect, of course, but it is a measure of the extent of those effects (i.e., the responsiveness of demand to the changes made) that they seek. For example, a firm might want to test three different price points and two different quality points before choosing one price–quality combination to commit to full-scale production. It would send mail to, call, or email all six possible combinations as an offer (only one
17. Again, such results are not always reliable, for three main reasons: First, the samples may not be a random sample of the overall market, perhaps under-representing consumers who (a) are working and cannot participate during the day, or (b) that have higher incomes and don’t consider $100 worth of free products worth the trouble, or (c) that value their privacy highly. Second, people may spend other people’s money differently than how they spend their own, and might decide to buy luxury goods in the market simulation that they would not buy from their own income. A third problem is that these market simulations are very expensive. Hun- dreds of shoppers take away thousands of dollars worth of products. It is expensive to set up initially and to monitor the “store.”
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offer to each person) to individuals on its mail- ing, telephone, or email list, or social network. By making different offers to different potential customers and tallying up the actual sales made for each combination, the firm’s managers would estimate the price effect and the quality effect and decide which combination to proceed with.
Internet websites allow firms to compile a list of potential customers by encouraging people to ask for further information or a price quote, and con- sumers must supply their email address to receive an answer. Since those customers are typically not connected to each other, or are not likely to com- municate with each other effectively, they are not likely to know that the firm is making multiple offers and will typically make up their own mind on the basis of the offer made to them. Similarly, special-interest magazines and national news- papers that are printed in two or more locations to save freight and postage cost, or are printed in two or more print runs, can contain different advertisements for the same product being read by different potential customers, with subsequent sales data providing information on the price effect, quality effect, impact of packaging or pro- motional changes, and so on.
4.4 Regression Analysis of the Demand Function
Regression analysis calculates a mathematical equation that best summarizes the relationship existing between two or more variables. Bivariate regression analysis, also known as correlation analysis, calculates the relationship between two vari- ables, such as price and quantity demanded, and provides an equation that specifies the intercept and slope terms of the line of best fit to the data. In the simple two-variable case, it is quite easy to sketch in a freehand line of best fit for a small number of observations as we did in Figure 4.3. But imagine if you had 100 or more data points; it would take a long time to carefully plot them on a graph before you even got to the point of sketching in the line of best fit, and so we rely on computers to quickly and accurately produce the parameters of the line of best fit. Multiple regression analysis, that is, with multiple inde- pendent variables, effectively provides the responsiveness coefficients () of the relation- ships between quantity demanded and each of the independent variables in the demand function, so that the impact on demand of all the independent variables can be found simultaneously, with all other independent variables effectively held constant. The Excel
©Jerry Arcieri/Corbis
Direct marketing occurs when the producer of the product sells directly to the consumer. This can be accomplished through Internet sales, newspapers advertisements, or direct mailings sent in the mail.
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CHAPTER 4Section 4.4 Regression Analysis of the Demand Function
spreadsheet package (with a statistical add-in module) can accomplish bivariate or mul- tiple regression analysis in milliseconds once you have arranged the data in columns. More powerful statistical software packages, such SPSS or STATA, are generally used by academics and other researchers, but these rather expensive programs are less likely to be provided in a business firm, or, if you are self-employed you would have to buy one of those packages separately. For our purposes here, Excel is more than enough.18
To illustrate how a computer regression program finds the line of best fit, we shall start with a simple two-variable (bivariate correlation) case, where Y depends on X. In Figure 4.4, we show several Y,X data points plotted on a graph with the line of best fit shown as Y 5 A 1 X. The ordinary least squares method positions the line of best fit such that it minimizes the sum of the squared deviations of the observations from the line—the devia- tions are squared to avoid positive deviations offsetting negative variations and to more heavily weight the larger deviations. A computer regression program calculates the sum of the squared deviations in a few milliseconds, and effectively compares different loca- tions of the line of best fit (or the multivariate analog) and selects the one that minimizes the sum of these squares. Essentially, the mathematical procedure passes the line through the point representing the mean values of the Y and X observations (which we call Y-bar and X-bar), and then pivots that line around this mean–mean point to find the slope (and thus the intercept) that minimizes the sum of the squared deviations from the line. A simi- lar process is involved for multiple regression analysis. The mathematical procedure can be imagined to pivot the lines of best fit between the dependent variable Y and each of the multiple independent variables simultaneously until it finds the values for the various coefficients that minimize the sum of the squares of the deviations (which are also known as the residuals or error terms).
18. Computer programs have turned the old fashioned way of computing the regression equation by hand into an unnecessary waste of time, and also eliminate calculation errors. Accordingly, students in managerial economics courses do not need to learn the complex equations that allow the regression parameters to be calculated by hand. We can rely on computers and con- centrate on ensuring that the input data quality is high.
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CHAPTER 4Section 4.4 Regression Analysis of the Demand Function
Figure 4.4: Ordinary least-squares method of fitting the line of best fit
Deviation of the observation (Y1) from the predicted value shown by the line (Y1′)
Y1
Y
′
1
Y= α+βX
Y
X1 X
Statistics Provided by Regression Software Before we use Excel to find the coefficients for a demand function, we need to consider some statistics that are provided by regression analysis and that allow us to judge how well the line of best fit actually fits the data, and how reliable are predictions that are made based on the data.
The Coefficient of Determination (R2)
The coefficient of determination indicates the explained variance of Y, that is, the propor- tion of the variation in Y (from its mean value) that is explained by the variance in X (or multiple X variables) from its (their) mean value(s). In effect the coefficient of determina- tion, commonly called R2, tells us how well the regression equation fits the data.19 The value of R2 will be between 0 and 1; for example, an R2 of 0.98 would indicate an amaz- ingly good fit to the data, such as we saw in Figure 4.3. By contrast, the R2 for the data shown in Figure 4.4 is probably only about 0.6, indicating that only 60% of the variance
19. Note that a “good fit” does not confirm causality, however, since other factors might be driv- ing this, such as misspecification of constructs, poor measurement of variables, endogeneity of variables, and so on.
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CHAPTER 4Section 4.4 Regression Analysis of the Demand Function
in Y is explained by variance in X. That means there is 40% of the variance that is unex- plained by the X. This unexplained variance will be largely attributed to missing vari- ables, that is, other determinants of Y for which data has not been collected or that were not entered into the regression analysis. A second cause of unexplained variance could be measurement error in the data. We have to be sure we are measuring the variables correctly or we will introduce variance into the analysis.20 For example, the use of GNI in the earlier example as a proxy for the incomes of the consumers in the target market is a fairly impre- cise measure that almost certainly contains measurement error, since GNI includes busi- ness profits as well as household incomes and not all profits are paid out as dividends. To minimize measurement error we need to seek the most precise measure we can get for each variable and only use proxy variables when we have no better data available.
The Standard Error of Estimate
The standard error of estimate (Se) is a measure of the dispersion of the data points from the line of best fit. Using this statistic we can calculate confidence intervals around the predicted value of Y given a set of values for the independent variables. A confidence interval is a range of values that we expect the actual value of Y to fall within for every value of X within a particular statistical confidence level. We commonly use the 95% con- fidence interval to indicate the range, between a lower estimate and an upper estimate of the Y value (associated with any X value), within which we expect the actual value of Y to fall 95% of the time.
Assuming that the error terms are normally distributed around the line of best fit, we can use the properties of the normal distribution to calculate the upper and lower points of the 95% confidence interval. As we saw in Chapter 2, a normal distribution will be a symmetric bell-shaped distribution of data points around the mean of those data points, and the height of the bell will be such that 68% of the observations will lie within plus or minus one standard deviation from the mean; and 95% of the observations will lie between plus or minus two standard deviations from the mean; and 99.7% of the observations will lie between plus or minus three standard deviations from the mean. Note that a standard deviation is the square root of the variance of the observations around the mean of a single variable. The standard error of estimate is the analog when there are two or more vari- ables—it measures the square root of the variance of the Y values around the predicted value given the observed values of the independent variables that cause that variance in Y. So, the 95% confidence interval around the predicted value of Y will be given by the predicted value of Y plus or minus two standard errors of estimate. Put another way, there is only a 5% chance of the predicted value of Y falling outside that confidence interval for any selected value of X.
The Standard Error of the Coefficient
The standard error of the coefficient (S) is a measure of the accuracy of the calculated value of the coefficient generated by the regression analysis. For multiple regression there is a S value for each one of the independent (X) variables. If S is relatively small, we can be more confident that the estimated value of is close to the true value of , and
20. Unexplained variance could also be due to the pure randomness. Note that the same consumer may end up with different decisions on the same product with the same price on different days.
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CHAPTER 4Section 4.4 Regression Analysis of the Demand Function
conversely, if it is relatively large we can have less confidence that the estimated is close to the true value. (The true value of the coefficients could be found if we were to survey the entire population of customers rather than just a sample). In short, S is the standard deviation of the mean observed for each independent variable from the sample. Again, using the properties of a normal distribution, we can establish confidence intervals around the estimated value of , and be confident at the 95% confidence level that the responsive- ness of the dependent variable to changes of an independent variable will lie within the range described by the relevant plus or minus two standard errors of the coefficient.
Using Excel to Conduct Regression Analysis There are many statistics add-in modules that work with the Microsoft Excel spreadsheet program that will make you quite capable of conducting basic regression analyses.21 Let us now work through how to set up and conduct regression analysis using Excel. You would first open a new worksheet in Excel. Supposing you have 100 observations, leave the top row empty and enter the numbers 1–100 down the first column (in cells A2, A3, A4, and so on down to cell A101). This allows us to refer to specific observations by their identifying observation number in case we later find there are serious outliers that need to be removed.22 Then label the columns across the top of the spreadsheet with the variable names, starting with the dependent variable in cell B1, and the names of the independent variables in cells C1, D1, E1, and so on. Enter the data for the dependent variable observations all the way down the second column (cells B2–B101), and then enter the independent variables corre- sponding to each observation down each of the columns to the right, as shown in Table 4.6.
Table 4.6: Setting up an Excel spreadsheet to conduct multiple regression analysis
Obs’n Sales of X
Price of X
Price of Y
Advert X
Advert Y
Income Avg. temp
Exch. rate
1 9,637 6.99 6.99 128.4 96.3 3,100.2 48.4 1.0434
2 10,815 6.49 6.99 143.1 98.1 3,327.1 52.4 1.0889
3 12,886 5.99 5.99 165.9 120.8 3,654.7 64.8 1.1054
4 9,847 6.49 5.99 132.5 105.6 3,229.3 58.8 1.0226
etc. etc. etc. etc. etc. etc. etc. etc. etc.
21. One such “statistics add-in” is Statpro, developed to accompany a popular Statistics textbook by Albright et al. This add-in is free to download and will allow you to conduct multiple regres- sions using Excel spreadsheets. Use Google to find the Statpro website and follow the instruc- tions to download it into your Excel program.
22. An outlier is an observation that has a value for one or more variables that is way outside what appears to be the normal range of deviations from the line of best fit (i.e., the regression equation). It is important not to delete observations that include large variations that might reasonably be correctly measured but simply include a large random error term—that would amount to falsify- ing the data. As a simple rule of thumb, and mindful of the properties of a normal distribution, you might decide to delete observations that are more than three standard deviations from the mean value of the variable, assuming an approximately normal distribution of the data.
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CHAPTER 4Section 4.4 Regression Analysis of the Demand Function
Leave a spare row below the last observation, and calculate the mean and standard devia- tion for the dependent variable by entering 5avg(B2-B101) and 5stdev(B2-B101) in cells B103 and B104, respectively. Then enter 5min(B2-B101) and 5max(B2-B101) in cells B105 and B106, respectively to find the smallest and largest values for the dependent variable. To drag these formulas over to the right for the other data columns, highlight the cells B103–106, locate the small square in the bottom right hand corner of cell B106 and drag it to the right, out to the last column of data. This should immediately produce the means, standard deviations, minima and maxima of the independent variables. We want to take a quick look at these statistics of the data to give us a feel for the data and to identify any outliers. Looking down each column and comparing the data points with the above statis- tics might also reveal obvious data entry errors that we can fix before moving on.
To conduct the regression analysis we first move the cursor to a location below the data so that the results will be posted there. We then go to the “Statistics” menu and pull down the “Regression” tab to identify the algorithm we need. Clicking on that algorithm will place it in the chosen cell, and we then click on the column heading for the dependent variable (or highlight the data down that column), then in sequence identify which of the indepen- dent variables are to be entered into the regression analysis. After all have been entered, close the bracket in the algorithm and press “Enter” and the calculations will proceed. A table showing the results will appear in the chosen area of the spreadsheet. This will include the value of the a and the various statistics, as well as the R2, Se , and S statistics.
We then observe and interpret these statistics. The a statistic represents that part of the dependent variable that is due to all other variables that are not included in the regres- sion equation. Each of the statistics shows the responsiveness of the independent vari- able to the relevant independent variable, and the signs (positive or negative) should be as hypothesized; that is, negative for the price effect; positive for the other three Ps (if entered into the equation)23; either positive or negative for the income effect; positive for the price of substitutes; negative for the price of complements; negative for the nonprice strategic variables of substitutes; positive for the nonprice strategic variables of comple- ments; positive for supporting business environment variables (such as population growth) and negative for damaging business environment variables (such as bad weather or new legislation).
From this data we are able to calculate the inverse demand curve (P 5 a 1 bQDx) by first calculating the compressed form of the demand function (QDx 5 AOV 1 1Px) where AOV includes the impact of all other variables except Px, as detailed earlier in this chapter. The intercept term in the demand curve expression is calculated as a 5 2AOV/1 and the slope term of the demand curve is b 5 1/1. The marginal revenue curve is then MR 5 a 1 2bQDx and the total revenue function is TR 5 aQ 1 bQ
2. We can also calculate the elasticity
23. If the for any of the other three Ps (product design, promotion, or place of sale) is negative, this would indicate that that variable has been taken too far, beyond the optimum, and is hav- ing a negative impact on quantity demanded, so should be reduced if the firm wishes to maxi- mize its profits.
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CHAPTER 4Summary
value for each independent variable as the responsiveness coefficient () times the ratio of the relevant independent variable and the value for QDx, such as 5 1. Px /QDx.
Pitfalls of Regression Analysis There is an old expression “garbage in, garbage out” that certainly relates to regression analysis. If your data is garbage, your regression results will be too! I will briefly mention here six common pitfalls of regression analysis that may compromise the accuracy of your results. Specification errors relate to the hypothesized functional form of the regression equation. If you set up the equation in linear form, but in reality the dependent vari- able has a curvilinear relationship with any independent variable, or there are impor- tant missing variables, the output statistics of the regression analysis must be inaccurate. We have also previously mentioned measurement errors whereby the data collected is not an accurate measure of the variable that you want to measure. Next, if there is a simultaneous equation relationship, such as Y depends on X and simultaneously X depends on Z, we cannot expect a single equation to be a true reflection of a multiple- equation relationship. Multicollinearity occurs when there is a significant correlation between two or more of the supposedly independent variables—this violates a basic assumption of regression analysis that the independent variables are independent of each other. Heteroscedasticity occurs when the variance of the error terms depends on the magnitude of the independent variables and is therefore not as assumed by the math- ematics of regression analysis. Finally autocorrelation (also known as serial correlation) might occur in time–series data and indicates that the error terms are serially related, that is, get progressively larger or smaller as time increases.
Any one of these problems will seriously compromise your regression results. Fortunately, most of these problems can be fixed by respecifying the functional form of the equation; by collecting better data; by using a structural equation model (with multiple equations); by eliminating one of the colinear independent variables; or by plotting and observing the error terms and subsequently respecifying the regression model. But these “fixes” would take us beyond a managerial economics course into a quantitative methods course or a research method textbook.24 Our purpose here was to show that basic regression analysis can be conducted easily using the readily available Excel spreadsheet package (with a sta- tistical module added in) to provide a first level estimate of the demand function and the responsiveness of quantity demanded to each of the independent variables for managerial decision-making purposes.
Summary In this chapter we began by reviewing the impact of various factors on the firm’s demand function, and stated the demand function as a linear equation that showed quantity demanded as a function of the independent variables that collectively determine market demand. Market demand is influenced by the firm’s controllable variables (price, pro- motion, product design, and place of sale) and by other variables that are uncontrolla- ble by the firm (the 4 Ps of other firms, consumer incomes, consumer tastes, consumer
24. For example, Hair et al., 2010, Multivariate Data Analysis text, cited earlier.
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CHAPTER 4 Summary
expectations, actions by governments, demographic changes, weather events, and natural disasters). We converted the demand function to an inverse demand curve of the form P 5 a 1 bQDx, which states the price in terms of a vertical intercept term (a) and a slope term (b). We then derived the relationships between the demand curve, the total revenue curve and the marginal revenue curve. The marginal revenue (MR) curve has the same intercept value but twice the slope of the demand curve. The total revenue (TR) curve has an inverted-U shape, increasing up to the point where MR falls to zero, and then falling. Next we introduced price elasticity of demand (denoted by the Greek letter epsilon, ) and related this to the above curves—price elasticity is a summary measure that indi- cates what will happen to marginal and total revenue when price is either increased or decreased. We say that demand is price elastic when price elasticity measure is greater than one in absolute terms, that is, |1|, and that demand is price inelastic when |1|. We know that TR will be rising (for price reductions) when demand is price elastic and falling (for price reductions) when demand is inelastic.
Similarly, other elasticities were introduced as summary measures of the relative respon- siveness of quantity demanded to changes in variables that affect the quantity demanded. In each case the responsiveness of quantity demanded is shown by the sign and size of the relevant coefficient in the demand function, and the elasticity (relative responsiveness) measure is the coefficient weighted by the ratio of the relevant independent variable to the quantity demanded. Managers need to be aware of these elasticities because they each communicate relevant information to the decision maker. Income elasticity of demand (denoted by theta, ) is a measure of the relatively responsiveness of quantity demanded to changes in consumers’ incomes. Normal or superior goods have positive income elastic- ity ( 0) while inferior goods have negative income elasticity ( 0). Superior goods can be further categorized as luxury goods if income elasticity exceeds one ( 1), necessity goods if income elasticity value lies between zero and one (0 1). Cross-price elasticity of demand was denoted where (the Greek letter eta) was positive for price changes of substitute goods and negative for price changes of complementary goods. Cross-advertising elasticity of demand was similarly positive for substitute goods and negative for comple- mentary goods. We also considered the firm’s own advertising elasticity of demand and quality elasticity of demand, noting that when these are positive it means that additional advertising or quality will lead to additional quantity demanded, but where negative it means that the market has responded badly to the firm’s change in that controllable variable.
In the second half of the chapter we introduced methods for estimating demand functions and demand curves. We considered primary data collection methods such as interviews, surveys, and market experiments and sketched in the line of best fit to indicate the respon- siveness of quantity demanded to changes in price or whichever other determining vari- able the manager may be interested in. We then considered multiple regression analysis to compute the dependency of quantity demanded on each of the independent variables simultaneously. Regression analysis generates statistics that provide a measure () of the responsiveness of quantity demanded to changes in each of the independent variables included in the regression equation, as well as the coefficient of determination (R2) that indicates how well the regression equation fits the data. The standard error of estimate (Se) indicates how confident we can be in the prediction made for the magnitude of quan- tity demanded, with a range of plus or minus 2 times the Se indicating the 95% confi- dence interval around that prediction. The standard error of the coefficient (S) similarly
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CHAPTER 4Questions for Review and Discussion
indicates how confident we can be that the coefficients derived from the sample data reflect the true relationship between the variables that would be found in the population as a whole. We concluded by briefly considering six common pitfalls of regression analy- sis and emphasizing the need to call in the experts when conducting regression analysis in a business situation when the stakes are high and the cost of making a mistake is likely to be higher than the cost of the consultant who could design and implement a study that would gain sufficiently accurate data.
In the next two chapters we turn our attention to the cost side of the profit equation (where profit equals revenues minus costs), and again we will utilize regression analysis to estimate cost functions, so that the manager will have the data required to make profit- maximizing price and other strategic decisions, and that in turn is the subject matter for the following four chapters of this book.
Questions for Review and Discussion
1. What independent variables do you think should enter the demand function for tick- ets to the home games of a National Football team?
2. Assign the value “relatively high” or “relatively low” to the price elasticity of market demand for each of the following, and explain why you chose one or the other value.
a. Coca-Cola b. Dr Pepper c. Artificial limbs d. Levi’s jeans e. Cure for AIDS
3. If you knew that, for a particular product, the current price is $45, current quantity demanded is 250 (thousand units per month), and the price elasticity of demand is equal to 21.5, explain how you would find the expression for the demand curve (in the form P 5 a 1 bQ).
4. Given your knowledge of the elasticity concept, define the rainfall elasticity of umbrellas. What possible usefulness could such a concept have?
5. Why would you expect the market demand for luxury goods, such as jewelry, to be more volatile in periods of fluctuating incomes, as compared to items such as milk and bread?
6. List 10 questions you would ask people in a survey designed to estimate the demand function for a particular brand of toothpaste.
7. Suppose you had annual data on the price and quantity demanded of newsprint over the past 20 years and plotted these (or conducted bivariate regression analysis) to find a line of best fit. Why would this be an unreliable estimate of the demand curve? What other data would you need to make a more reliable estimate of the demand curve for newsprint?
8. Summarize the issues you would need to check before concluding that the results of a regression analysis were a reliable basis for estimating the demand function.
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CHAPTER 4Decision Problems
Decision Problems
1. The demand for Fritz Reinhart premium beer in a particular city has been estimated to be:
Qx 5 37,986.5 2 4,476.9Px 1 2,994.2Py 1 668.2Ax 2 849.7Ay Where Qx is quantity demanded (per month) and Px is the price of Fritz Reinhart beer (in six-packs); Py is the price of the main competitor beer, Urquhart Pilsner; Ax is the advertising expenditure for Reinhart (in thousands of dollars per month) and Ay is the advertising expenditure of Urquhart (thousands of dollars per month). The current values of the independent variables are Px 5 9.95; Py 5 8.95; Ax 5 36; and Ay 5 22.
a. Calculate the price elasticity of demand for Fritz Reinhart beer, and comment on the extent to which demand and total revenue would change if the price of this beer (per six-pack) were to be raised by a dollar.
b. Calculate the cross-price elasticity of demand and speculate how a price decrease to $8.50 for Urquhart beer would impact the quantity demanded of Reinhart beer.
c. Would it be profit maximizing for Fritz Reinhart to increase its advertising expenditures by another $1,000 per month? (Assume the cost of production is constant at $4 per six-pack).
d. What can Fritz Reinhart do to reduce the negative impact of Urquhart’s advertis- ing on its quantity demanded?
2. The demand function for Crispie Chips has been estimated as follows:
Qx 5 227.6887 2 37.73585Px 1 44.1177Py 1 0.2315Ax Where Qx represents thousands of packets of chips; Px is the price per packet; Py is
the average price per packet of the many other brands of similar chips; and Ax repre- sents thousands of dollars spent advertising Crispie Chips. The current values of the independent variables are Ax 5216.0; Px 5 0.85; Py 5 0.79.
a. Calculate the price elasticity of demand for Crispie Chips and comment on its value.
b. Derive an expression for the (inverse) demand curve for Crispie Chips and sketch this on a piece of paper.
c. Suppose the cost of producing Crispie Chips is constant at $0.19 per packet. Should they reduce price and produce more (assuming they want to maximize profit)?
d. Should Crispie Chips spend more on advertising? e. What assumptions have you made regarding the reliability of the data and the
accuracy of the estimated demand function?
3. Jose Hermanos Tequila (JHT) has conducted an experiment in six different liquor stores that are spread around the suburbs of a major southwestern city, with approxi- mately similar customers for each store. JHT set different prices in each of the stores for its Hermanos Gold product, as follows, in each case with its product promi- nently displayed between the two other major tequila brands. Fortunately for JHT’s
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CHAPTER 4Decision Problems
experiment there were no changes in the prices or promotion of any other liquor products during the week of the experiment.
Store Price ($) Quantity demanded
A B C D E F
19.10 15.70 16.50 21.50 12.90 13.90
17 24 21 10 32 28
a. Sketch the line of best fit representing the demand curve for Hermanos Gold in the typical liquor store in that city.
b. What price promises to maximize total revenue (TR) for the JHT company? c. What is the price elasticity of demand at that price?
4. Axton Auto Accessories (AAA) has manufactured a new a product—a high-mounted rear brake light to replace the one that typically sits on the back window shelf of a passenger car. AAA’s product offers several new features, however, such as the abil- ity to scroll messages across the light-emitting diode (LED) screen. These prerecorded messages are designed to facilitate communication between vehicles, with displays such as “Sorry!”; “Thank You!”; “Please let me in!”; and “This car for sale.” The memory of the device can also be loaded (via your computer) with new messages such as your phone number, advertisements for products, and support for sports teams or political parties. This device is about a foot wide, and these messages only scroll across the screen when the vehicle is accelerating. The display reverts to bright red when the vehicle is braking, but also offers another innovation by displaying the word “CAUTION” when the driver lifts his or her foot off the accelerator. Trial mar- keting of the product at different prices for a month in 10 stores of a major auto parts retail company has returned the following data:
Retail store Retail price ($) Quantity demanded
1 2 3 4 5 6 7 8 9
10
33.90 25.00 49.98 27.98 37.75 45.90 23.98 31.00 25.50 29.98
2,208 2,682 2,061 2,526 2,158 1,732 2,877 2,312 2,606 2,488
a. Load this price and quantity data into columns in an Excel spreadsheet and con- duct bivariate regression analysis to determine the demand function in the form Qx 5 AOV 1 Px.
b. Convert the demand function to an inverse demand curve expressed in the form Px 5 a 1 bQx.
c. At what price level is total revenue (TR) maximized?
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CHAPTER 4Decision Problems
d. Assuming that the cost of production is constant at $8 and that the auto parts store marks up the wholesale price (its cost) by 100% to arrive at the retail price for its customers, what wholesale price should AAA set to maximize its profit?
5. The consulting firm that you work for has been hired by the U.S. Government to provide an independent analysis of the demand-side effects of a contemplated increase in the tax on gasoline. It provide you with a data set relating to the period 1962–1987, which contains valuable historic lessons relating to the impact of volatile pump prices due to the supply restrictions imposed by the Organization of Petro- leum Exporting Countries (OPEC) and the Corporate Average Fuel Economy (CAFE) regulations that required car manufacturers to increase the fuel efficiency of the cars they sold, while at the same time Real Disposable Income (RDI) per capita was rising, the number of passenger cars (NPC) almost doubled, and inflation was pushing up the Consumer Price Index (CPI).
Year QDx Px NPC MPG RDI CPI
1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1084 1985 1986 1987
43,771 45,246 47,567 50,273 53,312 55,110 58,524 62,448 65,784 69,514 73,463 78,011 74,217 76,457 78,847 80,677 83,233 80,233 73,375 71,718 72,848 73,156 71,180 69,450 71,404 70,984
20.36 20.11 19.98 20.70 21.57 22.55 22.93 23.85 24.55 25.20 24.46 26.88 40.41 45.44 47.44 50.70 53.09 74.33 104.73 112.75 102.65 95.36 91.46 89.64 63.63 66.33
66,638 69,842 72,969 76,634 80,106 82,367 85,793 89,156 92,095 96,144 100,658 106,119 109,823 111,679 115,170 118,711 121,717 125,750 127,448 129,123 129,500 131,723 133,751 137,308 140,693 142,209
14.37 14.26 14.25 14.15 14.10 14.05 13.91 13.75 13.70 13.73 13.67 13.29 13.65 13.74 13.93 14.15 14.26 14.49 15.32 15.68 16.36 16.81 17.80 18.28 18.35 19.29
6,271 6,378 6,727 7,027 7,280 7,513 7,728 7,891 8,134 8,322 8,562 9,042 8,867 8,944 9,175 9,381 9,735 9,829 9,722 9,769 9,725 9,930 10,419 10,662 10,947 10,976
90.6 91.7 92.9 94.5 97.2 100.0 104.2 109.8 116.3 121.3 125.3 133.1 147.7 161.2 170.5 181.5 195.4 217.4 246.8 272.4 289.1 298.4 311.1 322.2 328.4 340.4
Where: Qx is the gasoline consumption by passenger cars (in millions of gallons); Px is the retail (pump) price of gasoline, in cents per gallon; NPC is the number of regis- tered passenger cars (in thousands); MPG is the national average of miles travelled per gallon of gasoline; RDI is Real Disposable Income per capita (in 1982 dollars); and CPI is the Consumer Price Index (base year 1967).
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CHAPTER 4Key Terms
This data illustrates some very interesting issues that were happening over that tumul- tuous period of our history. You will note that the pump price of gasoline more than doubled five-fold from the mid-1960s to the mid-1970s, and then doubled again in the early 1980s, due to the OPEC crisis. The number of passenger cars climbed relentlessly with the love affair with “muscle cars” despite the increasing pump price of gasoline, and indeed outpacing the increases in real disposable income per capita. The aver- age MPG climbed only slowly as manufacturers increased the fuel efficiency of new cars and consumers slowly traded up to the more efficient new cars and retired their older vehicles. The changes in CPI show that the rate of inflation was generally much greater than the rate of increase of pump prices as the increased production and trans- portation costs, due to rising fuel prices, pervaded the entire economy, pushing up the prices of food and other household items that drive the CPI.
a. Reconcile the fact that while the quantity demanded of gasoline and pump prices both rise over this period generally, they are inversely related along a demand curve.
b. Conduct a multiple regression analysis to explain the quantity demanded of gas- oline in terms of the other data provided. (Enter this data into an Excel spread- sheet and use the Excel regression tool, if loaded, or alternatively, download an add-in regression program such as Statpro to find the regression statistics).
c. What proportion of the variance in Qx is explained by these other variables? What missing variables might account for the remainder of the variance in the quantity demanded of gasoline?
d. Use the regression equation to predict the quantity demanded of gasoline in 1988 for the values Px 5 68.5; NPC 5 145,885; MPG 5 20.36; RDI 5 11,192; and CPI 5 354.6.
e. What is the 95% confidence interval for your prediction?
Key Terms
absolute terms An expression meaning that we ignore the negative sign in front of a number, such that we say, for example, that 25 is larger than 24, typically used when dealing with price elasticity values, which are always negative.
autocorrelation The dependence of data values in the current period on their value in the preceding period. Autocorrelation violates an assumption of regression analy- sis that the dependent variable is deter- mined by the independent variables alone, not also by its own prior value.
business cycle The pattern of growth of the macroeconomy, typically with alternat- ing periods of expansion and recession (or at least slower growth rates) with associ- ated fluctuations in macroeconomic vari- ables such as interest rates, inflation rates, and unemployment rates.
coefficient of determination A statistic produced by regression analysis that indi- cates what proportion of the total variance in the dependent variable (Y) is explained by variations in the independent variables (Xs).
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confidence interval A range of values around the value of the dependent vari- able (Y) that is predicted (by the regres- sion equation) for particular values of the independent variables (Xs), within which range we can be confident that the actual value of Y subsequently observed will fall (for example) 95% of the time.
consumer clinics Artificial shopping environments created by marketers to conduct simulated market situations. Marketers observe the choices of custom- ers, while varying the prices of some products, the shelf placement of products, and point-of-purchase information about product quality, for different groups of shoppers.
contribution margin The excess of price per unit over average variable costs per unit, which contributes towards the firm’s fixed costs and profit.
cross-advertising elasticity The relative responsiveness of one firm’s quantity demanded to changes in another firm’s advertising expenditures. It can be calcu- lated as the percentage change in quantity demanded divided by the percentage change in the other firm’s advertising expenditures.
demand function The functional rela- tionship that exists between the quantity demanded of a particular product (depen- dent variable) and all determinants of that demand (the independent variables).
demographic change Changes over time in variables such as age cohort size, ethnicity proportions, gender balance, geographic distribution, and employment type. These changes are likely to affect the demand for products that are consumed more (or less) by a particular age, ethnic, gender, regional, or employment group.
dependent variable A variable (such as quantity demanded in different time peri- ods) that is dependent on the concurrent values of independent variables (such as price, advertising, consumer incomes, and so on).
direct marketing Directly selling goods and services to consumers through direct marketing channels (such as the Internet, mailing campaigns, and telemarketing) rather than indirectly via wholesale and retail firms (who interface directly with the consumer).
direct methods Demand estimation whereby primary data is collected from actual or potential buyers via interviews, surveys, and market experiments.
elasticity of demand The relative responsiveness of quantity demanded to a change in one of the independent variables that help to determine quantity demanded. For example, price elasticity of demand is measured by the percentage change in quantity demanded divided by the percentage change in the price level.
estimation of market demand A process by which the volume of demand in the current and future periods is estimated. This process involves gathering and interpreting data to provide a numerical estimate of demand in the current and future time periods.
heteroscedasticity The circumstance where the error terms associated with a regression equation vary in a system- atic manner relative to the magnitude of an independent variable, rather than occurring randomly as assumed by the mathematics of the regression equation. It introduces unreliability into the standard error of estimate and the coefficient of determination.
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independent variables Variables included on the right-hand side of a regression equation because they are expected to exert a determining influence on the dependent variable.
indirect methods Demand estimation involving the statistical examination of data previously collected for official government statistics, or found in reports researched and written for other purposes (so-called secondary data).
inverse demand curve Expresses price as a function of quantity demanded for product X, in the form Px 5 a 1 bQDx. This form is derived from the demand func- tion which expresses quantity demanded as a function of the various independent variables, including price.
line of best fit A line best summarizing the apparent relationship between two variables. We should not expect any of the observations to lie exactly on the line of best fit as there are likely to be random variations within the data.
luxury good Items for which the quan- tity demanded increases more than pro- portionately when consumer incomes increase. For example, income rises 5% and demand for caviar increases 10%.
marginal revenue (MR) A measure of how much the total revenue changes when one more unit is sold.
market demand The aggregate sum of the quantity demanded by all consumers of a particular product within a period of time.
missing variables Other determinants (Xs) of the dependent variable (Y) for which data has not been collected or that were not entered into the regression analysis.
multicollinearity The circumstance where the independent variables in a regression equation are in fact not inde- pendent of each other but instead are significantly correlated with each other. This causes the regression statistics to be unreliable.
necessity good A product that has an income elasticity value that is positive but less than one, meaning that when incomes rise, quantity demanded of that prod- uct rises but by a lesser percentage than income has risen.
nonprice competition Competition among rival firms that does not involve cutting price but rather involves the other three Ps of marketing, namely product design, promotion, and place of sales.
ordinary least squares method A process that positions the line of the best fit so that it minimizes the sum of the squared deviations of the observations from the line. The deviations are squared to avoid positive deviations offsetting negative variations and to more heavily weight the larger deviations.
price competition A situation among two or more rival firms, where each sets its price in an attempt to maximize its profits.
price elasticity The relative responsive- ness of quantity demanded to changes in the price level, for a particular product. It allows an estimate of by what propor- tion demand is likely to change when an item’s price is increased or decreased by a particular proportion.
quality elasticity The relative respon- siveness of quantity demand to a change in the quality of a particular product. It is measured as the percentage change in quantity demanded of product X divided by the percentage change in the quality of product X.
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regression analysis A method of analysis that determines the statistical relationship between a particular dependent variable and the independent variables that are expected to determine the value of that dependent variable.
simultaneous equation relation- ship Exists when a dependent variable Y depends on X but simultaneously X depends on Z. Thus, a single regression equation cannot reliably be estimated for the dependence of Y on X or Z.
specification error Occurs when the hypothesized functional form of the regression equation does not reflect the true relationship between the variables. Linear regression equations are most com- monly used, but sometimes there will be a nonlinear relationship (e.g., a quadratic function) between the dependent vari- able and at least one of the independent variables.
standard error of estimate A measure of the dispersion of the data points from the line of best fit. Using this statistic we can calculate confidence intervals around the predicted value of Y given a set of values for the independent (X) variables.
standard error of the coefficient A measure, for each of the independent variables, of the accuracy of the calculated value of the coefficient generated by the regression analysis.
total revenue (TR) The combined sum of the revenues collected from all the buyers during a particular period of time. It is equal to the price per unit multiplied by the number of units sold.
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