As this semester closed, I asked several colleagues who taught introductory economics courses to name the most difficult topics to teach to first-time economics students. There was some variation in their answers, but one concept was mentioned far more often than any other—elasticity. In this Teacher’s Corner, we will define what elasticity means in economics, explain how one particular type of elasticity is calculated, and discuss why the concept is critical to economic agents trying to maximize their revenue. We will also see that although the precise definitions and terminology surrounding elasticity are just a little more than a century old, earlier economists had an understanding of both the idea behind elasticity and its relevance, especially with regard to taxation.

#### Elasticity in Economic Decisions

Economics is all about determining what choices concerning resources, rules, time, and effort will lead to the best outcomes. Knowing how some economic factors will react to a decision made regarding other economic factors is fundamental to figuring out what the best decisions are. For instance, in order for a firm’s manager to know whether he should lower the price on the firm’s product, he needs to have an idea of how many new customers will be attracted by the lower price. If the firm sells several products, he will also need to consider how the change in one product’s price will affect the sales of the other products. A government considering an increase in tax rates needs to know how much the higher rates will shrink the tax base in order to determine whether the amount of revenue generated will rise or fall.

These considerations and countless others you might imagine all involve how one change leads to other changes. In economics, an elasticity is a measurement of the responsiveness of one variable to a change in another variable. There are many different types of elasticities distinguished by the pair of variables that each one considers, but at their core they are all simply comparisons of how one thing changes in response to changes in another.

Because all elasticities perform the same function, just with different variables, focusing on one type allows us to reduce clutter and confusion while exploring how elasticities work and why they are useful. In what follows below, we will emphasize the type of elasticity relevant to the first example given above, in which a manager wants to know how the quantity demanded of a product will change in response to a change in price. This type of elasticity, called the “price elasticity of demand,” is probably the most intuitive and readily accessible type, and so serves as the best introduction into the subject.

#### Elasticity and the Market for Lemonade

To illustrate the importance of a price elasticity of demand, consider a young boy, Henry, who sells lemonade on Saturdays for 50 cents a glass from a stand in his front yard in Austin, Texas. Suppose that on any given Saturday he sells ten glasses, so that his total revenue is \$0.50 × 10 = \$5.00. (For simplicity, we’ll assume that Henry’s mother buys all of the ingredients and gives them to him for free, so his costs are zero and do not change with the number of glasses he sells.)

Henry wonders whether he would have higher or lower total revenue if he lowered his price for a glass of lemonade. A lower price would mean less revenue per glass, but might also mean he would attract more customers and sell more glasses. The size of his total revenue after lowering his price would depend on whether the rise in customers grew proportionately more or less than the fall in price.

Let’s say that Henry lowers his price to 40 cents, a 20% fall from the original price. If Henry still sells ten glasses a day, then his total revenue will be \$0.40 × 10 = \$4.00, also a 20% fall. But when prices fall, sales generally increase, offsetting some of the decrease in revenue. If Henry’s sales increase proportionately more than his price decreases, then the higher sales will more than offset the price reduction, and his revenue will actually increase.

For example, if in response the number of glasses he sells rises, but only by 10% to 11 glasses every Saturday, then his total revenue will be \$0.40 × 11 = \$4.40. If, on the other hand, the number of glasses he sells rises 30% to 13 glasses every Saturday, his total revenue will be \$0.40 × 13 = \$5.20. If the growth in quantity demanded rises proportionately more than the fall in price, then Henry will generate higher total revenue if he lowers his price, as shown in the first case. If the growth in quantity demanded rises proportionately less than the fall in price, as in the second case, then Henry will generate lower total revenue if he lowers his price.1 Knowing how much quantity demanded changes (proportionately, not in absolute terms) in response to a change in price is therefore critical to Henry’s decisions. It can determine whether he will raise, lower, or keep steady his price in his attempt to generate more revenue.

The biography of Alfred Marshall in the Concise Encyclopedia of Economics notes that “To Marshall also goes credit for the concept of price-elasticity of demand, which quantifies buyers’ sensitivity to price.” As we will discuss below, Marshall was the first to quantify that sensitivity, but not the first to be aware of its practical importance.

The definition of a price elasticity of demand was first explicitly laid out by Alfred Marshall in his classic textbook Principles of Economics (1920, first pub. 1890). In the second paragraph of Book III, Chapter 4, he wrote that “The elasticity (or responsiveness) of demand in a market is great or small according as the amount demanded increases much or little for a given fall in price, and diminishes much or little for a given rise in price (italics in the original).” This definition is not very precise, but Marshall provided a more mathematical definition in a footnote to the above passage, footnote 69:

We may say that the elasticity of demand is one, if a small fall in price will cause an equal proportionate increase in the amount demanded: or as we may say roughly, if a fall of one per cent. in price will increase the sales by one per cent.; that it is two or a half, if a fall of one per cent. in price makes an increase of two or one half per cent. respectively in the amount demanded; and so on. (This statement is rough; because 98 does not bear exactly the same proportion to 100 that 100 does to 102.) (The parenthetical statement is in the original.)

This lays out the procedure for calculating the price elasticity of demand for Henry’s lemonade. Let’s take the first case given above. First, we need the percentage change in price, which was given as a 20% fall. Next, we need the percentage change in quantity demanded, which in the first set of numbers was given as a 30% rise. The price elasticity of demand is the ratio of the percentage change in quantity demanded to the percentage change in price, which in this case is 30% / 20% = 1.5. In the second case, in response to a 20% fall in price, quantity demanded rises only 10%, so the price elasticity of demand equals 10% / 20% = 0.5.2 Because elasticities are ratios of percentages, they are “unit-free,” not denominated in dollars or in a specific good such as glasses of lemonade. This allows the elasticities of different markets involving different products to be compared to one another meaningfully.

The above calculations lead us directly to some terminology used in conjunction with elasticities (these terms apply to all types of elasticities, not just price elasticities of demand):

1. When the price elasticity of demand is greater than 1, the change in quantity demanded is proportionately more than the change in price. Demand in this case is said to be elastic (you can think of quantity demanded as having “stretched” more than price).
2. When the price elasticity of demand is less than 1, the change in quantity demanded is proportionately less than the change in price. Demand in this case is said to be inelastic (quantity demanded “stretched” less than price).

With these terms in mind, you can form an intuition linking elasticity to Henry’s original concern, how a lower price would affect his total revenue:

1. If demand is elastic (the first case, where elasticity = 1.5), then a small drop in price results in a proportionately bigger rise in sales, and his revenue will grow (from \$5.00 to \$5.20). Conversely, a small rise in price will cause a proportionately bigger drop in sales, and his revenue will fall.
2. If demand is inelastic (the second case, where elasticity = 0.5), then a small drop in price results in a proportionately smaller rise in sales, and his revenue will fall (from \$5.00 to \$4.40). Conversely, a small rise in price will cause a proportionately smaller drop in sales, and his revenue will rise.

#### Some Caveats

The above section illustrated how the concept of elasticity can be extremely important to economic agents by clarifying the relationships between changes in economically significant variables. However, for elasticities to be useful, it is important to keep in mind exactly what they do and do not measure. The price elasticity of demand for Henry’s lemonade applies to the demand for Henry’s lemonade. It cannot be assumed to apply to the demand for lemonade in general or even to lemonade sold by other young Texan boys. This is because any estimate of price elasticity of demand, or any other type of elasticity, is determined in part by the idiosyncrasies of the particular market from which the information is derived. Additionally, those idiosyncrasies must be stable for a given elasticity to be used over time. Ludwig von Mises, who was critical of the shift toward more quantitative analysis in economics he saw in his lifetime, used this limitation of an elasticity estimate as part of his critique. He wrote in Chapter 2, paragraph 107 of Human Action (1996, first pub. 1949) that

If a statistician determines that a rise of 10 per cent in the supply of potatoes in Atlantis at a definite time was followed by a fall of 8 per cent in the price, he does not establish anything about what happened or may happen with a change in the supply of potatoes in another country or at another time. He has not “measured” the “elasticity of demand” of potatoes. He has established a unique and individual historical fact.

To the extent that the market for lemonade in Austin differs from that for lemonade generally, or that the market for lemonade from one boy’s stand differs from lemonade from another boy’s stand, or any number of other differences, the price elasticity of demand for Henry’s lemonade may not apply to other lemonade. One way of expressing this critique is to note that there is no way to theoretically predict elasticity—it must be calculated for each case at hand, and the elasticity from one case may bear little or no relation to the elasticity in another.

In Book III, Chapter 4, paragraphs 4-5 of Principles of Economics, Marshall described how elasticity varies as you move along a demand curve. His description only holds for certain kinds of demand curves, but the larger point, that the elasticity in a market generally changes as the price changes, remains valid.

Another thing that must be remembered about elasticity is that even in the case of an elasticity estimated for a very specific market like that for Henry’s lemonade, a single elasticity will not hold for that market at all times. In our example, the price elasticity of demand for Henry’s lemonade is likely to vary if the analysis begins at a different price level. Except in a few special cases, quantity demanded will still change in response to a change in price, but the relative sizes of the changes will be different. As a result, it is important to know not just to what markets a given estimate of elasticity applies, but also whether the price levels in those markets have changed significantly from when the estimates were made.

#### Elasticity and It’s Evolution in Tax Analysis

Marshall was the first economist to explicitly define price elasticity of demand and formalize the mathematical derivation of elasticities, but he was not the first to consider the relationship between changes in prices and changes in quantities demanded. Earlier writers displayed an understanding of how the elasticities of different goods affect the revenues related to those goods even without the precise vocabulary and mathematical framework Marshall later provided. One area in which they often employed the idea was tax policy.

That there should have been a conceptual link in the minds of classical economists between tax policy and what we now call price elasticity of demand is perhaps not very surprising. During the eighteenth and nineteenth centuries, European governments raised much of their revenue from taxes on foreign and domestic goods, taxes that raised the final prices of the goods. For a government seeking to raise large amounts of revenue in this manner, an understanding of the idea behind price elasticities of demand was crucial in determining the effects of imposing, removing, raising, or lowering the taxes on specific commodities. Classical economists commenting on the tax policies of governments might naturally consider the price elasticities of different goods when drawing conclusions about what goods should be taxed and at what rates.

In Book V, Chapter 2, paragraphs 178 and 179 An Inquiry into the Nature and Causes of the Wealth of Nations (1904, first pub. 1776), Adam Smith remarked on the possibility that lower taxes can raise tax revenue by lowering the price and encouraging the consumption of the taxed good. He noted that “High taxes, sometimes by diminishing the consumption of the taxed commodities, and sometimes by encouraging smuggling, frequently afford a smaller revenue to government than what might be drawn from more moderate taxes. When the diminution of revenue is the effect of the diminution of consumption there can be but one remedy, and that is the lowering of the tax.” Smith seems to have understood that some goods’ demands are more price elastic than others’, but did not appear to consider it a major point or something worth explaining at length because he did not return to the idea, but spent considerable time discussing the means and impact of smuggling in Britain.

Some related but distinct definitions of necessities and luxuries can be compared by reading Book V, Chapter 2, paragraph 148 of Adam Smith’s An Inquiry into the Nature and Causes of the Wealth of Nations (1904, first pub. 1776), chapter 3, paragraphs 30-33 of Nassau Senior’s Political Economy (1854, first pub. 1850), and Book III, Chapter 5, paragraphs 13-16 of Jean-Baptiste Say’s Treatise on Political Economy (1903, first pub. 1855).

Some of the discussion of other early economists involving elasticity and taxation revolved around the distinction between “necessities” and “luxuries.” Different writers had different opinions of where to draw the line between the two categories, but in general, necessities were those goods that a person needed to stay alive, be physically productive, and in some definitions, remain socially acceptable. Luxuries were all other goods.

For instance, in paragraph 6 of Book III, Chapter 3 of Principles of Political Economy (1909, first pub. 1848), John Stuart Mill essentially described the demand for necessities like bread as inelastic: “There are many articles for which it requires a very considerable rise of price materially to reduce the demand; in particular, articles of necessity, such as the habitual food of the people in England, wheaten bread: of which there is probably almost as much consumed, at the present cost price, as there would be with the present population at a price considerably lower.” In other words, the change in the quantity demanded of bread would be proportionately small relative to a change in price, which is the definition of inelastic demand today.

Today, the terms necessity and luxury are closely associated with another type of elasticity, the income elasticity of demand. This type of elasticity is described on this webpage from Digital Learning Resources.

David Ricardo explored the relative benefits of taxes on luxuries versus taxes on necessities in Chapter 16, paragraph 44 of On the Principles of Political Economy and Taxation (1821):

Taxes on luxuries have some advantage over taxes on necessaries. They are generally paid from income, and therefore do not diminish the productive capital of the country. If wine were much raised in price in consequence of taxation, it is probable that a man would rather forego the enjoyments of wine, than make any important encroachments on his capital, to be enabled to purchase it. They are so identified with price, that the contributor is hardly aware that he is paying a tax. But they have also their disadvantages. First, they never reach capital, and on some extraordinary occasions it may be expedient that even capital should contribute towards the public exigencies; and secondly, there is no certainty as to the amount of the tax, for it may not reach even income. A man intent on saving, will exempt himself from a tax on wine, by giving up the use of it. The income of the country may be undiminished, and yet the State may be unable to raise a shilling by the tax.

Implicit in this discussion is the notion that a man would be less likely to avoid a tax on a necessity by no longer buying it, because he needs the necessity to stay alive and productive. If the price of a necessity were to rise in consequence of taxation, the quantity demanded would not fall as much because fewer people are able to reduce their consumption of necessities.

That earlier economists understood the ideas behind price elasticity of demand does not diminish the importance of Marshall’s work in refining and formalizing the concept. The usefulness of his framework and terminology was as apparent to other economists of Marshall’s day as it is to us today. Just two years after Marshall’s Principles was published, Charles Bastable used Marshall’s treatment of price elasticity of demand in his own analysis of tax policy. Unlike the other economists we have seen, who confined their use of the concept underlying elasticity, Bastable used Marshall’s vocabulary to discuss a further topic. He examined how the price elasticity of demand in part determines how extensively a producer forced to pay a tax on the goods he produces can pass the cost of the tax onto his customers. In Book III, Chapter 5, paragraph 24 of Public Finance (1892, first pub. 1917), he explained that:

Taxation imposed on a necessary article, or one which forms a very small part of the total outlay of the consumer, will, since demand is inelastic, be more likely to pass on at once to the consumer than if the commodity belonged to that large intermediate class, the demand for which is speedily checked or increased by an upward or downward movement of price.

If the price elasticity of demand is high, then a producer that tries to pass along a tax by raising his price will lose a proportionately large amount of sales. If demand is inelastic, then a producer can raise his price without losing as proportionately large an amount of sales, and so pass along the tax without hurting his total revenue.

Though it is still intimately tied to questions of tax policy, Bastable’s analysis represents an application of price elasticity to an economic question to which it had not been applied before. It is difficult from this temporal distance to know what influence Marshall’s exposition on elasticity had on this small but significant innovation, but it is at least plausible that Marshall’s work on the subject played some role in its expanded use into other applications, including lemonade stand revenue.

#### Conclusion

Elasticity in general, and price elasticity of demand in particular, allow economic agents to get a firmer grasp of the actions they should take to improve the economic outcomes that affect them. Classical economists were well aware of basic principle underlying elasticity, but they lacked a vocabulary for discussing it in a unit-free way that was not specific to the market under investigation. Given the limited applicability of any given estimate of elasticity, maybe that should not be so surprising. After Marshall’s refinement and formalization of the concept, an incremental expansion of the range of subjects to which elasticity was applied began, and the number of decisions for which it became a useful tool expanded.

Footnotes

In some extremely narrow cases when the percentage changes in price and quantity demanded are very close, these statements may not hold. In the vast majority of cases, however, these statements are correct.

Strictly speaking, the changes in price are -20% and -10%, and the price elasticities of demand should therefore be -1.5 and -0.5, respectively. By convention, however, price elasticities of demand are expressed in absolute value, so that a larger number (in absolute value) implies a higher degree of elasticity.

*Morgan Rose is a Ph.D. candidate in economics at Washington University in St. Louis, with research interests in industrial organization, corporate governance and economic history.

For more articles by Morgan Rose see the Archive.