Elasticities will be important for much of what follows in this course. More importantly, an understanding of elasticities will help you think through decision problems you will encounter in your career. You were introduced to elasticities in your introductory microeconomics course, but let us take a moment to review them now.

The term “elastic” implies flexibility. An elastic material responds readily to force. An inelastic material is less responsive. For example, a rubber band is quite elastic. You can exert force by pulling on the rubber band and it responds easily and stretches according the force you apply. For this reason, the term “elastic band” is sometimes used in lieu of the term “rubber band.” In demand, the interest is in the responsiveness of a product’s quantity to forces that affect demand. The forces of interest were introduced in Chapter 1 and include the product’s own-price, prices of related products, income, and other demand shift variables.

Simply stated, an elasticity is the percentage change in one variable resulting from the percentage change in another. In this course, the symbol ϵϵ (the Greek letter epsilon) will be used to refer to demand elasticities. Subscripts will indicate the product and demand variable in question. The first subscript will always refer to the quantity of the product in question. The second subscript will refer to a price or other variable that affects demand. For example, \(\epsilon_{i j}\) refers to the elasticity of demand for good i with respect to a change in the price of good j. Similarly, \(\epsilon_{i X}\) refers to the elasticity of good i with respect to some non-price demand shift variable \(X\).

There are several types of demand elasticities. *Own-price elasticities* measure the relationship between the quantity of a particular good, say good 1, and its own-price. The own-price elasticity of demand for good 1 is defined as

\(\epsilon_{1 1} = \dfrac{\% \Delta Q_{1}}{\% \Delta P_{1}}\)

where \(\Delta\) is the change operator. The formula for the own-price elasticity of another good, say good 2, would be

\(\epsilon_{2 2} = \dfrac{\% \Delta Q_{2}}{\% \Delta P_{2}}\)

**Cross-price elasticities** measure the relationship between the quantity of one good and the price of a related good. The cross-price elasticity of demand for good ii with respect to the price of good \(j\) would be

\(\epsilon_{i j} = \dfrac{\% \Delta Q_{i}}{\% \Delta P_{j}}, \: for \: i \neq j.\)

As you learned in Chapter 1, variables other than prices also affect demand. One of these variables is income. An **income elasticity** for good 1 would be calculated as

\(\epsilon_{1M} = \dfrac{\% \Delta Q_{1}}{\% \Delta M}\),

where \(M\) represents income.

## Interpretation of Elasticity Numbers

Because elasticities are the ratio of two percentage changes, they are easy to interpret. If \(\epsilon_{i X} = 0.5\), then you can say that quantity demanded increases (decreases) by half a percent as \(X\) increases (decreases) by one percent. You can algebraically rearrange the terms in the elasticity definitions above to use an elasticity to predict the change in quantity that would result from a change in a demand variable of interest. Specifically, the definition of an elasticity suggests the following:

\(\epsilon_{i X} = \dfrac{\% \Delta Q_{i}}{\% \Delta X} \Rightarrow \% \Delta Q = \epsilon_{iX} \times \% \Delta X\).

One nice thing about the fact that elasticities are expressed as a ratio of percentages is that they are independent of units of measurement. It does not matter whether prices or income are reflected in US dollars, Euros, Pesos, or Yen. Similarly, it does not matter whether quantities are measured in bushels, pounds, kilograms, or tons. The interpretation of the elasticity number will be unaffected. That said, one could use an elasticity to predict what would happen to physical units demanded when a demand variable changes. Suppose you knew that \(\epsilon_{11} = -2.5\). If you were interested in the effect of a three percent price increase, this elasticity can be used to tell you that

\(\% \Delta Q_{1} = =2.5 \times 3 = -7.5 \%\).

Now, suppose that you know that before the price increase demand is 1,000 units. You could easily predict what demand would be after the 3% price increase. The new quantity demanded would be

\{1000 \: units \times (1-0.075) = 925 \: units\).

This elasticity could be used in a similar fashion to predict the quantity response given any magnitude of a price change, but there are some practical reasons why predictions will tend to be less accurate when using elasticities to predict large percentage price changes. These limitations will become clear below as you learn more about elasticities.

## Ranges for Demand Elasticities

Because elasticities measure responsiveness, their magnitudes are of importance. Table \(\PageIndex{1}\) presents some magnitude-based classifications of demand elasticities that will be important for you in this course. Notice from the table that own-price elasticities are non-positive. This is because of the law of demand. It is common in introductory microeconomics textbooks and even in some more advanced MBA-level textbooks to take the absolute value of an own-price elasticity and report it as a positive number. This will not be done here. Elasticities will be used to predict quantity changes and to model market responses to demand or supply shocks. It will be important to preserve the direction of the negative own-price effect on quantity demanded. Another reason for not taking the absolute value of own-price demand elasticities is that such practice is uncommon in the empirical literature.

Table \(\PageIndex{1}\): Classifications based on magnitude of demand elasticities.
Type |
Range |
Implication |

Own-price |
\(\epsilon_{ii} \< -1\) |
Demand for good \(i\) is elastic |

Own-price |
\(\epsilon_{ii} = -1\) |
Demand for good \(i\) is unitary elastic |

Own-price |
\(-1 \< \epsilon_{ii} \leq 0\) |
Demand for good \(i\) is inelastic |

Cross-price |
\(\epsilon_{ij} \> 0, i \neq j\) |
Good \(j\) is a substitute for \(i\) |

Cross-price |
\(\epsilon_{ij} \< 0, i \neq j\) |
Good \(j\) is a complement to \(i\) |

Income |
\(\epsilon_{iM} \< 0\) |
Good \(i\) is an inferior good |

Income |
\(0 \leq \epsilon_{iM} \leq 1\) |
Good \(i\) is a normal necessity |

Income |
\(\epsilon_{iM} \> 1\) |
Good \(i\) is a normal luxury |

As shown in Table \(\PageIndex{1}\), own-price elasticities are divided into three ranges. First, when demand is elastic, the numerator of the elasticity is large in absolute value relative to the denominator. This means that demand is responsive in the sense that a change in price induces a more-than-proportional change in quantity. With elastic demand, the quantity change will be large in comparison to the price change. When demand is unitary elastic, the numerator of the elasticity is the negative of the denominator. Unitary elastic demand means that the change in quantity is exactly proportional to the change in price. Finally, when demand is inelastic, there is a less than proportional change in quantity in response to a change in price. With inelastic demand, a large change in price may not change quantity very much.

To give you some intuition about own-price elasticity, consider Demonstration \(\PageIndex{1}\). In this demonstration, the price is fixed at $50 and the quantity at 100 units. The only thing you can change is the slope of the inverse demand curve. When the demonstration first loads, this slope will be -0.5 and the elasticity of demand will be -1 or unitary elastic. If you see something different, set it to -1 or reload the page in your browser. Make the slope steeper by sliding the control of the demonstration to the left. As you do, you will see that demand becomes inelastic. As the slope becomes steeper, price changes quite a bit, but quantity does not change very much. This is what is meant by inelastic demand. Now slowly slide the control of the demonstration to the right to flatten the slope of inverse demand. You will see that demand switches from inelastic, to unitary elastic, to elastic as you do. When the slope of inverse demand is 0, the elasticity of demand is negative infinity. This means that any amount can be sold at the given price of $50, but that no amount can be sold at a slightly higher price (e.g., $50.01). The term “perfectly elastic” is sometimes used in situations like this and means the same thing as infinitely elastic.

It is not likely that a market demand schedule will ever be infinitely elastic, but there are situations where the demand schedule facing an individual firm is infinitely elastic. In fact, the price-taking firms you learned about in Chapter 2 face infinitely elastic demand. If you remember, price taking firms can sell all they want at the going price. This is equivalent to saying that price taking firms face an infinite elasticity. As the course progresses, it will be useful to pay attention to whether an elasticity refers to the market demand or the demand facing the firm. The market demands for agricultural commodities are commonly inelastic but the demands facing the individual farms supplying these commodities may be infinitely elastic.

Demonstration \(\PageIndex{1}\): **Own-price elasticities show the responsiveness of quantity to changes to price.**

As indicated above in Table \(\PageIndex{1}\), cross-price elasticities of demand are used to determine whether related products are substitutes or complements. Consistent with the relationships presented in Chapter 1, positive (negative) cross-price elasticities imply substitutes (complements). The magnitude of the cross-price elasticities can be used to determine which products are strong substitutes or complements to the product in question. For instance, Brand A yogurt might be a very close substitute for Brand B yogurt. There might also be some substitutability between Brand A yogurt and cottage cheese. Nevertheless, cottage cheese is a much weaker substitute for Brand A yogurt than is Brand B yogurt. In this case, one would see a large positive cross-price elasticity of demand for the Brand A yogurt with respect to the price of Brand B yogurt. There would be a positive, albeit considerably smaller, cross-price elasticity of demand for Brand A yogurt with respect to the price of cottage cheese.

Finally, income elasticities can be used to classify goods as normal or inferior based on a positive or negative sign, respectively. As indicated in Table 1, normal goods can be further classified as a normal necessity or normal luxury goods. A normal necessity has an income elasticity that is between zero and one. For normal necessity goods, the change in quantity is less than proportionate to the change in income. Many, if not most, food items have income elasticities in the range of normal necessities. With a normal luxury, the change in quantity is more than proportionate to the change in income.

## Revenue Implications of Own-Price Elasticities

If you know the range of the own-price elasticity – whether demand is elastic, unitary elastic, or inelastic – you can predict what will happen to revenue if there is a change the product’s own-price. Higher prices are not always good for revenue. When there is a price increase, two things happen: First, fewer units of the good or service are sold. This is a quantity effect. Second, a higher price is received for each of the remaining units that continue to be sold. This is a price effect. If the price effect is larger than the quantity effect, revenue will increase as price increases. If, on the other hand, the quantity effect is larger than the price effect, revenue will decrease as price increases. Fortunately, you can easily tell from the own price elasticity of demand which of these effects is largest.

**In the elastic range, an increase (decrease) in price causes a decrease (increase) in revenue.** Because demand is elastic, the quantity effect dominates the price effect. The revenue lost from selling fewer units more than offsets the revenue gained from selling the remaining units at a higher price. Thus, raising price causes revenue to fall. On the other hand, when demand is elastic and price is lowered, the revenue gain from selling additional units more than offsets the revenue lost from selling each unit at a lower price. Lowering price when demand is elastic will cause revenue to rise.

**In the inelastic range, an increase (decrease) in price causes an increase (decrease) in revenue.** When demand is inelastic, the price effect dominates the quantity effect. The decline in quantity is proportionately smaller than the increase in price. In this case, the revenue gained from the higher price more than offsets the revenue lost from selling fewer units. Conversely, if price is lowered when demand is inelastic, the revenue gained from selling more units is not sufficient to offset the revenue lost from selling each unit at a lower price.

**When demand is unitary elastic, revenue is maximized.** When demand is unitary elastic, total revenue is maximized. This is because the price effect and quantity effect are of the same magnitude. The revenue gained (lost) from a price increase (decrease) is exactly offset by the revenue lost (gained) from the sale of fewer (more) units.

Use Demonstration \(\PageIndex{1}\) below to get some intuition about the revenue implications of price changes. As you work through the demonstration, pay attention to the size of the red and blue rectangles that appear when you change the price. The blue rectangle represents revenue gained from the price change. The red rectangle represents revenue lost. Note that whenever you change the price, there is always a blue and red rectangle. This is because the quantity effect and price effect of the change are always in opposite directions (because of the law of demand). Calculate the areas of the red and blue rectangles and notice that when you subtract the area of the red rectangle from the area of the blue rectangle you obtain the net revenue reported in the demonstration. Use the demonstration to verify that the revenue implications of price changes under different elasticity conditions match those outlined above.

Demonstration \(\PageIndex{2}\): **The relationship between price changes, own-price elasticity, and revenue.**

One word of caution is in order. The normal assumption is that firms maximize profit (not revenue). That said, the revenue implications of demand elasticities are important because revenue is a key part of the profit calculation you learned about in Chapter 2. Moreover, firms probably have a good idea about their production cost and so information about what happens to revenue in response to a price change may be all that is needed to make the right pricing decisions to increase profit.

Based on what you have learned up to this point in the course, you should be able to prove that a firm that controls its price (this is not a price-taking firm) will never set its price so that demand is in the inelastic or unitary elastic ranges. How could you prove this? The most straightforward way would be to show that it cannot possibly be true that a firm has maximized its profits if its demand is not in the elastic range. Start by asking the question: Could a firm have maximized its profit if its demand is inelastic or unitary elastic? The answer is no. To see why, suppose this firm raised its price by a small amount. Two things would happen:

- First, the firm’s revenue would go up if demand is inelastic or its revenue would stay the same if demand is unitary elastic. Spend some more time in Demonstration 2 and review the content above if this is not clear to you.
- Because the firm raised its price, it would sell less (because of law of demand) and its cost would go down as a result. Remember from Chapter 2 that total cost is an increasing function of quantity. Thus, a lower quantity sold means lower cost to the firm.

In sum, a small price increase in this situation either increases or does not affect revenue but unambiguously decreases cost. This means that profit must unambiguously increase with a price increase. Thus, a firm facing demand in the inelastic or unitary elastic range cannot have possibly maximized its profit because it can raise its profit simply by raising its price.