# Elasticity

Elasticity is the measure of responsiveness of the change of one parameter to the change of another, measured in percentage terms.

For example, price elasticity of demand for a good is the responsiveness of the demand of the good when its price changes.

# Mathematical formulas for elasticity of demand

With calculus, elasticity of demand is ${\displaystyle e={\frac {dQ}{dP}}\times {\frac {P}{Q}}}$

where P is the price, Q is the quantity demanded. (see below for a non-calculus formula, as well as formulas for other measures of elasticity).

## Interpretation

In terms of calculus, it represents the responsiveness to an infinitesimal change in the parameter.

For example, price elasticity of demand for a good is the change of the demand of the good when price is increased by an infinitesimal amount.

If we rewrite the formula as ${\displaystyle e={\frac {dQ}{Q}}/{\frac {dP}{P}}}$

this can be roughly interpreted as the fractional change in Q divided by the fraction change in P.

Moreover, note that ${\displaystyle {\frac {dQ}{dP}}={\frac {1}{\frac {dP}{dQ}}}}$ , which is the slope of the inverse demand curve. In other words, elasticity is inversely proportional to the slope of the inverse demand graph.

# Non-calculus Calculation version

For Intro Econ, the "formula" you need to know is that

${\displaystyle \mathrm {(a)\ elasticity\ of\ (b)} ={\frac {\mathrm {\%change\ in\ (a)} }{\mathrm {\%change\ in\ (b)} }}}$.

For example, price of elasticity of good ${\displaystyle 1=-2}$ means that ${\displaystyle {\frac {\mathrm {\%change\ in\ x_{1}} }{\mathrm {\%change\ in\ p_{1}} }}=-2}$. When the price of good 1, p1, increases by 1%, the demand x1 decreases by 2%. If price increases by 2%, then demand decreases by 4%.

To estimate percentage change, we use the formula ${\displaystyle {\dfrac {x_{2}-x_{1}}{\frac {x_{1}+x_{2}}{2}}}\times 100\%}$, where ${\displaystyle x_{1}}$ is the initial value and ${\displaystyle x_{2}}$ is the final value. The numerator ${\displaystyle x_{2}-x_{1}}$ represents the change, while the denominator ${\displaystyle {\frac {x_{1}+x_{2}}{2}}}$ is the average value between the initial and the final value. The average value is used in order to increase the accuracy of the estimation. To be precise, elasticity is meant to be a measure of an instantaneous response at one point and to be used with calculus. Elasticity changes as we move from the initial value to the final value. To use the average between the initial value and the final value in the denominator is to reflect this change and to improve accuracy.

## Interpretation

In lower-level economics courses, elasticity is often interpreted based on a 1% change in the parameter.

For example, price elasticity of demand for a good is the percentage change of the demand for good when the price of good 1 is increased by 1%.

Price elasticity of demand should always be negative (as price goes up, demand comes down). If it is smaller than -1, we call it elastic; if it is between 0 and -1, we call it inelastic; and if it's equal to -1, we call it unitary elastic.

## Definition of Key Terms

Elastic

• When price changes, the quantity demanded changes by a larger percentage. For example, if the good is said to be elastic, a 5% change in price can lead to a 20% change in quantity demanded. This is likely for goods with many substitutes and if it is a luxury (people can easily switch away from the good).

Inelastic

• When price changes, the quantity demanded changes by a minimal percentage. For example, if the good is inelastic, a 5% change in price only leads to a 1% change in the quantity demanded. This is likely for goods with few substitutes and if it is a necessity (it is not easy for people to switch away from the good).

Unit Elasticity

• When price changes, the quantity demanded changes by the exact same amount. In other words, the percentage change in the quantity demanded is equal to the percentage change to the price

# Determinants of Elasticity

Demand tends to be more elastic

• the larger the number of close substitutes
• if the good is a luxury
• the more narrowly defined the market
• the longer the time horizon

Demand tends to be more inelastic

• the fewer the number of close substitutes
• if the good is a necessity
• the more broadly defined the market
• the shorter the time horizon

# Commonly Used Elasticity Measures

## Income elasticity

Income elasticity of a good is a measure of the responsiveness of the demand of a certain good to the changes in income.

Most goods are normal goods. Higher income raises quantity demanded. Because quantity demanded and income moves in the same direction, normal goods have positive income elasticity.

Inferior goods are the higher income, lower the quantity demanded such as bus ride. Because quantity demanded and income move in opposite directions, inferior goods have negative income elasticity. Even among normal goods, income elasticity varies substantially in size. Necessities, such as food and clothing, tend to have small income elasticity because consumers, regardless of how low their incomes, choose to buy some of these goods. Income elasticity is between 0 and 1 for necessities. Luxuries, such as caviar and diamonds, tend to have large income elasticity because consumers feel that they can do without these goods altogether if their income is too low. Income elasticity is larger than 1 for luxuries.

## Cross Price Elasticity

Cross price elasticity is the responsiveness of demand of one good to the changes in price of another good. For example, if the cross price elasticity of demand for good A on price B is -2, then when the price of good B increases by a small amount (say 1%), then the demand for good A drops by two times the amount (2%).

When the cross price elasticity is negative, the two goods are (gross) complement. To understand this, when price of B increases, by the Law of Demand the quantity of B must decrease. If good A has a negative cross price elasticity on good B, then quantity demanded for good A must also decrease. As quantities of A and B tend to move in the same direction, they tend to be consumed together and therefore are (gross) complements.

Similarly, if cross price elasticity is positive, then the two goods are (gross) substitutes.

## Product of percentage changes

We can also find the elasticity of a product. For example, for a firm, we can define the price elasticity of revenue, which would be the percentage change in revenue as price changes (by 1%). When we look for percentage change of a product ( ${\displaystyle {\text{revenue}}={\text{price}}\times {\text{quantity}}}$ ), we can convert it into the sum of two percentage changes. In our case,

${\displaystyle \mathrm {\%change\ in\ revenue} =\mathrm {\%change\ in\ price} +\mathrm {\%change\ in\ quantity} }$.

For example, what is the elasticity of revenue when price changes by 3% and price elasticity of demand is -2? With price elasticity = -2, we know that when price goes up by 3%, demand falls by 6%. So

${\displaystyle \mathrm {\%change\ in\ revenue} =\mathrm {\%change\ in\ price} +\mathrm {\%change\ in\ quantity} =3\%-6\%=-3\%}$.

The math behind it is that elasticity of revenue is defined as

${\displaystyle \epsilon _{\text{revenue}}={\frac {dR}{dP}}\times {\frac {P}{R}}}$.

When we do the differentiation, we have to use the product rule for ${\displaystyle R=PQ}$, so

${\displaystyle {\frac {dR}{dP}}=dR/dP=P\times {\frac {dQ}{dP}}+Q}$.

Plug that into the definition to get

${\displaystyle \epsilon _{\text{revenue}}=(P\times {\frac {dQ}{dP}}+Q)\times {\frac {P}{Q}}=({\frac {P}{Q}}\times {\frac {dP}{dQ}}+1)=\epsilon +1}$ where ${\displaystyle \epsilon }$ = price elasticity of demand. Multiply the whole equation by "% change in P% to get the desired result.

## Challenging questions

• Q True or False: if the supply curve is a straight line that passes through the origin, the elasticity of supply does not depend on the price.
• A A line passing through the origin has the form p = aq, or q = p/a. With the formula of elasticity,

${\displaystyle \epsilon ={\frac {dp}{dq}}\times {\frac {p}{q}}={\frac {1}{a}}\times {\frac {aq}{q}}=1}$ .

The elasticity of such a supply curve is always 1.

(revenue change)

• Q: Let ${\displaystyle P=10-0.2Q}$. What is the elasticity if P = 6?
• A: First change the equation into a function in terms of Q. So it will be ${\displaystyle Q=50-5P}$. If P = 6 then ${\displaystyle Q=50-(5\times 6)=20}$. Use the formula for elasticity ${\displaystyle nd={\frac {dQ}{dP}}\times {\frac {P}{Q}}}$.

We know from the rewritten equation (${\displaystyle Q=50-5P}$) that ${\displaystyle {\frac {dQ}{dP}}=-5}$ because it is the slope of the equation.

${\displaystyle {\frac {P}{Q}}={\frac {6}{20}}=0.3}$

${\displaystyle -5\times 0.3=-1.5\ }$.

# Elasticity and Real Life Examples

## Rent Control

Rent control is a kind of price ceiling that the government imposes in order to control the amount of rent a landlord can charge the tenants When a rent control is imposed, the price will be kept at a certain rate. More people are going to demand housing because the rent is not high. However, landlords will not want to increase the supply of housing because they are less likely to earn a decent rent from renting out their houses. This therefore creates a housing shortage.

The amount of the shortage however varies with elasticity.

In the short run, the prices are sticky and therefore are inelastic.Landlords cannot immediately decrease the amount of housing they put out for rent. When the prices are inelastic, the supply and demand curves are steeper sloped. This therefore creates less of a shortage.

In the long run, the prices are not sticky anymore and therefore the prices are elastic. When the prices are elastic, the slopes of the supply and demand curves are more flat. This therefore creates more of a shortage.

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