Cross elasticity of demand

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In economics, the cross (or cross-price) elasticity of demand measures the effect of changes in the price of one good on the quantity demanded of another good. This reflects the fact that the quantity demanded of good is dependent on not only its own price (price elasticity of demand) but also the price of other "related" good.

The cross elasticity of demand is calculated as the ratio between the percentage change of the quantity demanded for a good and the percentage change in the price of another good, ceteris paribus:^[1]

{\text{Cross elasticity of demand}}={\frac {\%{\text{ change in quantity demanded of good A}}}{\%{\text{ change in price of good B}}}}

The sign of the cross elasticity indicates the relationship between two goods. A negative cross elasticity denotes two products that are complements, while a positive cross elasticity denotes two products are substitutes. If products A and B are complements, an increase in the price of B leads to a decrease in the quantity demanded for A, as A is used in conjunction with B.^[2] Equivalently, if the price of product B decreases, the demand curve for product A shifts to the right reflecting an increase in A's demand, resulting in a negative value for the cross elasticity of demand. If A and B are substitutes, an increase in the price of B will increase the market demand for A, as customers would easily replace B with A,^[3] like McDonald's and Domino's Pizza.

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So far, we've been focused on the elasticity of demand for only one good. We've thought about how changes in the price of that good affect changes in its quantity. Now what we're going to explore is how we can go across goods. So we're going to talk about the cross elasticity of demand. And there's multiple different scenarios we could think about, but it's really thinking about how a price change in one good might affect the quantity demanded in another good. And to see an example of this, think about two airlines-- two competing airlines-- maybe it's the same exact route going at the exact same time, maybe between New York and London. So airline one, right over here-- airline two, very competitive, price right over here is $1,000 for a round trip. Quantity demanded is 200 tickets, let's say, in a given week. Airline two, price is $1,000 for the round trip, and the quantity demanded is 200 tickets as well. Now let's think about what will happen. What will happen if airline one raises its price from $1,000 to $1,100? In fact, we could even do something less dramatic than that, to $1,050-- so a relatively small increase in price. And remember, when we think about the percentage price increase, when we're thinking about elasticities in general, we don't just say, OK, $50 on top of $1,000, that's a 5% price increase. That's what we would do in everyday thinking. If you said you went from $1,000 to $1,050, you would say that's a $50 increase on a base of $1,000 or that is a 5% increase. But when you think about elasticities, because we want to have the same percent change between-- if you go from $1,000 to $1,050, or if you go from $1,050 down to 1,000-- we actually use the average point as a base. So the percent change in this scenario-- let me write it right over here. So our percent change-- and I'll write it in quotes, because it's a little bit different than what you do in traditional mathematics when you think about percent changes-- is you had a 50 change in price. Your price went up by 50, and on our base we will use 1,025, which is the average of 1,000 and 1,050. And so that gives us a change of 50 divided by 1,025 is equal to, let's say, roughly 4.9%. So this is approximately 4.9%, we'll say, "increase" in price, although we're going to put that increase in quotes, because we're using it on the average. And we do that so that if we said it was 1,050 to 1,000, it would still be a 4.9% decrease using this same idea-- using the midpoint as the base. Now, when that happens-- Everyone today, they use these travel sites where you can compare prices-- If these really are the exact same route, going from the exact same airport to the exact same other airport in London, leaving at the exact same time, everyone is going to gravitate to this one now, because it's only $1,000-- even just to save $50. Why would they ride on this airline? So this quantity demand is going to go to 0. And this quantity demanded is going to go to 400. And we're not going to think about the actual capacity of the planes and all that. We're going to have a very simple model here. So what was the percent change in quantity for airline two right over here? Well, once again, our change in quantity is 200, not 400. We went from 200 to 400. So we gained 200. And our base, we want to use the average of 200 and 400, which is 300. And so this is approximately 67%. So we have, all of a sudden, our cross elasticity of demand for airline two's tickets, relative to a1's price. And we get the percent change in the quantity demanded for a2's tickets, which is 67% over the percent change, not in a2's price change, but in a1's price change. That's why we call it cross elasticity. We're going from one good to another. So let's just say, for simplicity, roughly 5%. And so you do the math. So if you have 67% divided by 5%, you get to roughly 13.4. So this is approximately 13.4. So you have a very high cross elasticity of demand. In fact, if you even increase this, maybe by $5, you might have had the same effect. And so you would have had a very large number here. And that situation right here, for this cross elasticity of demand-- it's because these things are near perfect substitutes. The way that we set up this problem, we said, well, people don't care which one they take. They're just going to go for the cheapest one. And so when you have near substitutes, or nearly perfect substitutes, for each other, like this example right here, the cross elasticity of demand approaches infinity. It gets higher and higher and higher. In theory, if these are really, really, really identical, even if you raise this a penny, people will say, well, why would I waste a penny? I would just use airline two. And so this number would be even lower right over here. And so this thing might approach infinity. And notice this was a positive. When we just did regular price elasticity of demand, the only way that you would increase quantity for a traditional goods was by lowering price. But here, we raise price on a substitute competitive product, and we raise the demand for airline two's product, which actually made a lot of sense. So it wasn't a negative relationship. It's actually a positive value right over here. But you could have things in other-- you could have that negative relationship using cross elasticity of demand. This is an example of a substitute. We could think about the example of a complement. So what if we're talking about e-books? So let's say I have some type of an e-book, and the current quantity demanded in a given week is 1,000. And let's say that the price of an e-reader that you would need for my e-book is $100. But let's say that price of the e-reader goes down from $100 to $80. So you had a $20 decrease in price. Well, what's going to happen to my e-book, assuming its price does not change? Well, then the quantity demanded for my e-book will go up. So let's say the quantity demanded for my e-book goes up by 100, because more people are going to be able to afford this, or they're going to have money left over when they buy this to buy more e-books. And so I don't even know what the price for my e-book is, but at a given price point, the quantity demanded will go up. And so this goes to 1,100. And so I'll leave it to you to calculate this price elasticity of demand. But you will see that you will actually get a negative value, like we're used to seeing for regular price elasticity of demand. And when you do calculate it, remember, you want to do your percent price change in e-book quantity over percent change in e-reader price. And the other thing you have to remember, you don't just take negative 20 over 100. You take negative 20 over the average of these two, when you're thinking of it in the elasticity context. So this right over here-- actually, maybe we'll just work it through. Pause it, and try to do it yourself. So this value right over here is negative 20 over 90-- the average of those two-- and this value right over here is going to be plus 100 over the average of these two. So the average of those two is 1,050. And so this is 100 divided by 1,050, which gets you to about 0.95. So about 9 and 1/2% change in quantity demanded for my book. And then this denominator right here is negative 20 divided by 90. So you get a drop of 22%. And so if you divide the numerator by the denominator, you get 0.952 divided by negative 0.22222-- I'll just put couple of 2's there-- and you get a negative 0.43. So this is equal to negative 0.43. And this makes sense. If you lower the price of an e-reader-- this complement product, a product that goes along with my e-book-- it increases the demand. So just like you get with price elasticity of demand, you get a negative value over here. And what about completely two unrelated products? So let's say that I have basketballs, and the price of basketballs goes from, let's say, $20 to $30. What's going to happen to my e-book? Well, my e-book's not going to change. It's going to stay at $1,000. So my percent change in the quantity demanded of my e-book is going to be 0 in this example. So we're going to have 0, when we want to do this cross elasticity of demand, over my percent change in basketballs, which would be 30 over 25. So whatever that is-- 30 over 25 would be 10 over 25-- which is a 40% increase. So that would be 0 over 40%, which equals 0. So for unrelated products, products where the price of change in one of them does not affect the quantity demanded in the other, it makes complete sense that you have a 0 cross elasticity of demand. If they're complements, you would have a negative cross elasticity of demand. And if they're substitutes, you would have a positive one. And the closer the substitutes they are, the more positive your cross elasticity of demand is going to be.

History

The concept of "price elasticity of demand" originated by Alfred Marshall predicted relative changes between price and quantity. In the Cellophane case, Professor Stocking believed that a change in the price of one product will induce a price change of its rivalry in the same direction, so he firstly regarded that movement of two prices in the same direction explicitly reflects a high cross-price elasticity.^[4] However, during 1924–1940, du Pont cellophane prices moved independently from its perceived competitors' (waxed paper, vegetable parchment, etc) price; independent price movements reflect noncompetitive pricing between cellophane and its rival products.^[5] Thus, Professor Stocking's emphasis on the same movement of prices was too rigid, as the price of cellophane changed induced by three factors:

Change in demand due to price change of rival products
The production function of the cellophane
The slope and position of the cost curves of rival products.

In other words, the competitive relationship between two goods (cross-price elasticity) can not be simply concluded by price change, as price change arises from both cost and demand factors. Furthermore, instead of a high positive or low positive elasticity concluded by observing respective price change, cross-elasticity of demand should be either positive or negative to represent if there is a complementary or substitutive relationship between two goods.

Calculation and interpretation

Cross elasticity of demand of product B with respect to product A (η_BA):

\eta _{BA}={\frac {\Delta Q_{B}/Q_{B}}{\Delta P_{A}/P_{A}}}={\frac {P_{A}}{Q_{B}}}\cdot {\frac {\Delta Q_{B}}{\Delta P_{A}}}

$\eta _{BA}>0$ implies two goods are substitutes. Consumers purchase more B when the price of A increases. Example: the cross elasticity of demand of butter with respect to margarine is 0.81, so 1% increase in the price of margarine will increase the demand for butter by 0.81%.

$\eta _{BA}<0$ implies two goods are complements. Consumers purchase less B when the price of A increases. Example: the cross elasticity of demand of entertainment with respect to food is −0.72, so 1% increase in the price of food will decrease the demand for entertainment by 0.72%.

$\eta _{BA}=0$ implies two goods are independent (a price change of good A is unrelated to demand change of good B), so changes in the price of product A have no effect on the demand for Product B. Example: bread and cloths .

Interpretation of cross elasticity of demand^[6]
If the sign of cross elasticity of demand is...	the elasticity range	the goods are
negative	−∞	perfect complements
negative	$(-\infty，0)$	highly or somewhat complements
0	0	unrelated goods (neither complements or substitutes)
positive	$(0, +\infty)$	somewhat or highly substitutes
positive	+∞	perfect substitutes

The higher the positive cross elasticity of demand, the more substitutable two products are; thus, the more competition between them.^[7] Similarly, the lower the negative cross elasticity of demand, the more complementary two goods are. In general, monopolies usually possess a low-positive cross elasticity of demand with respect to their competitors.^[8]

Degree of response

The higher the positive cross elasticity of demand, the more substitutable two products are; thus, the more competition between them.^[9] Similarly, the lower the negative cross elasticity of demand, the more complementary two goods are. In general, monopolies usually possess a low-positive cross elasticity of demand with respect to their competitors.^[10]

Elastic demand

If the absolute value of the cross elasticity of demand is greater than 1, the cross elasticity of demand is elastic, this means that a change in price of good A results in a more than proportionate change in quantity demanded for good B. In other words, a change in price of good A has a relatively high impact on the change in quantity demanded for good B.

|XED|>1

Inelastic demand

If the absolute value of the cross elasticity of demand between 1 and 0, the cross elasticity of demand is inelastic, this means that a change in price of good A results in a less than proportionate change in quantity demanded for good B. In other words, a change in price of good A has a relatively small impact on the change in quantity demanded for good B.

0<|XED|<1

Unitary demand

If the value of the cross elasticity of demand is 1, the cross elasticity of demand is unitary, this means that a change in price of good A results in an exactly proportionate change in quantity demanded for good B.

XED=1

Results for main types of goods

For two goods, fuel and new cars (consists of fuel consumption), are complements; that is, one is used with the other. In these cases the cross elasticity of demand will be negative, as shown by the decrease in demand for cars when the price for fuel will rise. In the case of perfect substitutes, the cross elasticity of demand is equal to positive infinity (at the point when both goods can be consumed). Where the two goods are independent, or, as described in consumer theory, if a good is independent in demand then the demand of that good is independent of the quantity consumed of all other goods available to the consumer, the cross elasticity of demand will be zero i.e. if the price of one good changes, there will be no change in demand for the other good.

When goods are substitutable, the diversion ratio, which quantifies how much of the displaced demand for product j switches to product i, is measured by the ratio of the cross-elasticity to the own-elasticity multiplied by the ratio of product i's demand to product j's demand. In the discrete case, the diversion ratio is naturally interpreted as the fraction of product j demand which treats product i as a second choice,^[11]^[12] measuring how much of the demand diverting from product j because of a price increase is diverted to product i can be written as the product of the ratio of the cross-elasticity to the own-elasticity and the ratio of the demand for product i to the demand for product j. In some cases, it has a natural interpretation as the proportion of people buying product j who would consider product i their "second choice".

Approximate estimates of the cross price elasticities of preference-independent bundles of goods (e.g. food and education, healthcare and clothing, etc.) can be calculated from the income elasticities of demand and market shares of individual bundles, using established models of demand based on a differential approach.^[13]

Selected cross price elasticities of demand

Below are some examples of the cross-price elasticity of demand (XED) for various goods:^[14]

Good	Good with price change	XED
Butter	Margarine	+0.81
Beef	Pork	+0.28
Entertainment	Food	−0.72

Application and Implication

An enterprise needs to understand the cross-elastic demand for a product or service. Cross-elastic demand can help enterprises set prices and identify the sensitivity of others to their products. For example, a strategic "loss leader" takes advantage of the negative cross elasticity of demand for complementary commodities to price in a counterintuitive way deliberately. A company can sell one of its goods for less than the cost of making it and thus promote sales of its complementary products. Large profits on complementary products can make up for net losses in the business of its main products. Many large companies use this strategy, such as Sony. Sony's PlayStation consoles are sold below the cost of making them encourage the sale of games. Games and consoles are almost perfectly complementary. The reduction in the price of consoles will significantly increase the demand for games. As a result, Sony can make up for its net losses in the console business by making big profits in games ^[15]

Besides, unique and irreplaceable products enable companies to sell their products at higher prices. Because of the uniqueness of the product, companies do not worry too much about consumers switching to other products. However, the specific price setting should also follow the demand curve of the commodity. Suppose the elasticity of demand for the product is greater than 1. In that case, it means that a slight change in the product's price will cause a significant reduction in the consumer demand for the product. Therefore, companies should first make a careful study of the elasticity of demand for their products before setting prices. It ensures a broader profit range for the company. A real-life example is Apple. Apple used iOS, which is different from Android, at the beginning of the launch of their phones. The clean and straightforward interface is an irreplaceable advantage of this system. Apple, meanwhile, has its unique text-message tone and call ringtone. In many small ways, Apple is building uniqueness. Phone users who are used to iOS develop a habit that makes it difficult to adapt to other systems, such as Android.

Finally, the providers of substitutes need to be aware of the competitors of their products through detailed market research. The company can reduce the sensitivity of competitors' products by increasing customer loyalty. For example, the recently hot quality stars are invited to endorse their company's products. It can attract some of the star's loyalists to the product, thus increasing the overall loyalty. Alternatively, the company could spend more money on advertising to make consumers aware of the difference between its product and that of its competitors.

Contribution to policy improvement

The UK and Scottish governments intended to use price-based policy interventions, like setting minimum unit pricing and increasing taxation to reduce alcohol consumption and mediate the related harms among their population.^[16] Estimation of cross-price elasticities of alcohol in respect to other related beverages helps set price-based policy interventions, as it measures the percentage change in demand for one type of alcohol due to a 1% change in the price of another type of beverage. For example, the cross elasticity of demand for wine in respect to the price change of spirit is 0.05, which implies that a 1% price decrease for Spirit will reduce market demand for wine by 5%. Therefore, the cross elasticity of demand enables policymakers to take better control of the policy effects, thus, reducing the risk for mortality, morbidity, and other social harms caused by over-drinking.

Contribution to the sustainable supply chain

A high coefficient of negative cross-price elasticity implies that the sales of product A are decided by the sales of product B. If the demand of A significantly depends on the demand of B, there must be a reduction in the profit of A. In this case, the cross elasticity of demand is a reminder to the firms to cautiously selecting products with high dependence on complements. On the other hand, the high-positive cross elasticity of demand reflects high substitutability of goods, which means customers' demand can be fulfilled by other products easily. Businesses that understand the implications of high-positive cross elasticity of demand can reduce their operating risk by avoiding overstock, thus, maintaining a sustainable supply chain.

Contribution to firm reactions

Knowledge of a firm's cross elasticity of demand and their competitors' allows them to map out the market, enabling them to calculate the number of rivals and the importance of their complementary (and substitute) products relative to their own. Firms can develop strategies to reduce their exposure to the risks they are imposed to by price changes of other firms, such as an increase in the price of a complement or a decrease in the price of a substitute.^[17]

Potential strategies

Horizontal integration

In markets with few competitors, cross elasticity between rivals are likely to be high,^[18] this makes firms in the market vulnerable to price competition. Horizontal integration, usually mergers, could reduce said risks by reducing competition in the market. For example, when Anheuser-Busch InBev (the world's biggest brewer at the time) acquired SABMiller (InBev's closest rival) in 2015, it was one of the biggest takeover of a British firm, creating the world's first global brewer.^[19] The takeover created a brewing empire that produces a third of the world's beer.

Vertical integration

Firms may gain better control of the market by merging with suppliers of complementary products. Developing their own complementary products is another possible solution. For example, Google developing Google Pixel is an attempt by Google to capture the smartphone market share by integrating both its software and hardware features for improved performance while being more resource efficient.^[20]

Alliances and collusion

Competitors may pool resources to create a joint alliance, such as Sony-Ericsson in October of 2001. Sony had a share of less than 1% in the mobile phone market; while Ericsson was the third largest market share holder. Unfortunately, Ericsson relied heavily on a single supplier, and when a fire broke out at a Phillips factory, Ericsson couldn't fulfill their orders. Sony wanted a greater market share and Ericsson wanted to avoid going out of business, hence the Sony-Ericsson joint venture was formed.^[21]

Firms entering into a price fixing agreement in order to avoid price wars means they are involved in a collusion. The chances of collusion to occur is higher in markets with few competitors such as oligopolistic markets. It is illegal according to antitrust laws, even though collusive agreements may be implicit, its implication with cartels are the same.

Notes

^ "OECD Glossary of Statistical Terms – Cross price elasticity of demand Definition". stats.oecd.org. Retrieved 2021-04-17.
^ Hemmati, M.; Fatemi Ghomi, S.M.T.; Sajadieh, Mohsen S. (2017-09-04). "Inventory of complementary products with stock dependent demand under vendor managed inventory with consignment policy". Scientia Iranica: 0. doi:10.24200/sci.2017.4457. ISSN 2345-3605.
^ Das, R. L.; Jana, R. K. (2019-09-01), "Some Studies on EPQ Model of Substitutable Products Under Imprecise Environment", Asset Analytics, Singapore: Springer Singapore, pp. 331–360, doi:10.1007/978-981-13-9698-4_18, ISBN 978-981-13-9697-7, S2CID 202935211, retrieved 2021-04-17
^ Lishan, John M. (1959). "The Cellophane Case and the Cross-Elasticity of Demand". Antitrust Bulletin. 4: 593.
^ Waldman, Don (1980). "The du Pont Cellophane Case Revisited: An Analysis of the Indirect Effects of Antitrust Policy on Market Structure and Performance". Antitrust Bulletin. 25 (4): 805. doi:10.1177/0003603X8002500406. S2CID 240425105. Retrieved 25 April 2021.
^ "Lesson Overview – Cross Price Elasticity and Income Elasticity of Demand (article)". Khan Academy. Retrieved 2021-04-18.
^ Bain, Joe S. (August 1942). "Market Classifications in Modern Price Theory". The Quarterly Journal of Economics. 56 (4): 560–574. doi:10.2307/1883410. ISSN 0033-5533. JSTOR 1883410.
^ "G. J. Stigler The Theory of Price. New York, Macmillan, 1952, VII p. 340 P". Bulletin de l'Institut de recherches économiques et sociales. 19 (1): 97. February 1953. doi:10.1017/S1373971900100782.
^ Bain, Joe S. (August 1942). "Market Classifications in Modern Price Theory". The Quarterly Journal of Economics. 56 (4): 560–574. doi:10.2307/1883410. ISSN 0033-5533. JSTOR 1883410.
^ "G. J. Stigler The Theory of Price. New York, Macmillan, 1952, VII p. 340 P". Bulletin de l'Institut de Recherches Économiques et Sociales. 19 (1): 97. 1953. doi:10.1017/S1373971900100782.
^ Bordley, Robert F. (1985). "Relating Elasticities to Changes in Demand". Journal of Business & Economic Statistics. 3 (2): 156–158. doi:10.2307/1391869. JSTOR 1391869.
^ Capps, O. and Dharmasena, S., "Enhancing the Teaching of Product Substitutes/Complements: A Pedagogical Note on Diversion Ratios",Applied Economics Teaching Resources, Vol. 1, Issue 1, pp. 32–45, (2019), https://www.aaea.org/UserFiles/file/AETR_2019_001ProofFinal_v1.pdf
^ Sabatelli, Lorenzo (2016). "Relationship between the Uncompensated Price Elasticity and the Income Elasticity of Demand under Conditions of Additive Preferences". PLOS ONE. 11 (3): e0151390. arXiv:1602.08644. Bibcode:2016PLoSO..1151390S. doi:10.1371/journal.pone.0151390. PMC 4801373. PMID 26999511.
^ Frank (2008) p. 186.
^ LIU, HONGJU (2010). "Dynamics of Pricing in the Video Game Console Market: Skimming or Penetratio". Journal of Marketing Research. 47 (3): 428–43. doi:10.1509/jmkr.47.3.428. JSTOR 25674441. S2CID 153931398.
^ Meng, Yang; Brennan, Alan; Purshouse, Robin; Hill-Mcmanus, Daniel; Angus, Colin; Holmes, John; Meier, Petra Sylvia (1 March 2014). "Estimation of own and cross price elasticities of alcohol demand in the UK—A pseudo-panel approach using the Living Costs and Food Survey 2001–2009". Journal of Health Economics. 34 (100): 96–103. doi:10.1016/j.jhealeco.2013.12.006. PMC 3991422. PMID 24508846.
^ "Cross elasticity of demand". Economics Online. 13 January 2020. Retrieved 26 April 2021.
^ "Cross elasticity of demand | Economics Online | Economics Online". Economics Online. 13 January 2020. Retrieved 26 April 2021.
^ "SABMiller agrees AB Inbev takeover deal of £68bn". the Guardian. 13 October 2015. Retrieved 26 April 2021.
^ Inc, Spiceworks. "Snapback: Google Pixel and the move towards vertical integration". The Spiceworks Community. Retrieved 26 April 2021. {{cite web}}: |last1= has generic name (help)
^ "History of sony ericsson as a company". AUEssays.com. Retrieved 26 April 2021.

References

Frank, Robert (2008). Microeconomics and Behavior (7th ed.). McGraw-Hill. ISBN 978-0-07-126349-8.
Mankiw, Gregory (2009). Principles of Microeconomics (5 ed.). South-Western College Pub. ISBN 9780324589979.

External links

Database for Commodity Elasticity at Food and Agricultural Policy Research Institute, University of Iowa
"Other Demand Elasticities". Lumen Learning. Retrieved 26 April 2021.
"Cross Elasticity of Demand". Economics Online. 13 January 2020. Retrieved 26 April 2021.

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