1.2.3a Concepts and calculation of elasticities of demand

Elasticities of Demand

$$\%\Delta =\frac{change}{original\; value}\times 100$$

1. Each elasticity is the percentage change in quantity demanded divided by the percentage change in its cause.
   1. Always keep the sign of the coefficient, because for YED and XED the sign carries the economic meaning interpreted in 1.2.3b.

Price Elasticity (PED)

$$PED=\frac{\%\Delta Q_d}{\%\Delta P}$$

Income Elasticity (YED)

$$YED=\frac{\%\Delta Q_d}{\%\Delta Y}$$

Cross Elasticity (XED)

$$XED=\frac{\%\Delta Q_{dA}}{\%\Delta P_B}$$

1.2.3b Interpreting values and influencing factors

Interpreting PED

$$PED=\frac{\%\Delta Q_d}{\%\Delta P}$$

1. The size of the coefficient, ignoring the sign, shows how responsive quantity demanded is to price.

Interpreting YED

$$YED=\frac{\%\Delta Q_d}{\%\Delta Y}$$

1. Read the sign first for the type of good, then read the size for how strongly demand responds to income.

Interpreting XED

$$XED=\frac{\%\Delta Q_{dA}}{\%\Delta P_B}$$

1. The size of XED shows the strength of the link, so a +0.5 between Coca-Cola and Pepsi is a weak substitute relationship (XED > 0), while a −0.8 between a printer and its toner is a strong complement relationship (XED < 0).

Factors Influencing Elasticity

1. PED: the main factor is the availability of close substitutes, since many close substitutes let buyers switch easily and so make demand more elastic.
2. Demand is more inelastic when a good is a necessity, takes a small share of income, or is habit-forming, addictive or strongly branded.
3. Demand becomes more elastic the longer the time period, as buyers find alternatives, so petrol is inelastic in the short run but more elastic over time.
4. YED and XED: YED depends on whether the good is a necessity, luxury or inferior, while XED depends on how closely the two goods are related.

1.2.3c Significance of elasticities and total revenue

Elasticities and Revenue

$$TR=P\times Q$$

1. Whether a price change raises or lowers total revenue depends on the price elasticity of demand.
2. When demand is inelastic, quantity falls proportionately less than price rises, so a price rise raises total revenue.
3. When demand is elastic, quantity falls proportionately more than price rises, so a price rise lowers total revenue.
4. When demand is unitary elastic, total revenue does not change and is at its maximum, where PED = 1.

Pricing and Taxation

1. A firm should raise price where demand is inelastic and cut price where demand is elastic if the aim is to raise revenue.
2. For an indirect tax, the more inelastic demand is, the more of the tax consumers pay and the more revenue the tax raises.
   1. This is why government taxes inelastic goods such as tobacco and fuel, where consumption falls only a little.
3. For a subsidy, the more inelastic demand is, the more of the benefit passes to consumers as a lower price rather than to producers.

Income and Cross Elasticity

1. For changes in real income, income elasticity of demand shows which industries grow or decline over the economic cycle.
   1. Luxuries with high YED boom in an upturn and suffer in a downturn, while inferior goods do the reverse, so firms plan product ranges accordingly.
2. For changes in the prices of substitutes and complements, cross elasticity of demand shows how strongly a firm's sales respond to a rival's or a complement's price change.
   1. A firm facing a close substitute, such as Coca-Cola against Pepsi (XED > 0), is vulnerable to a rival's price cut, while complements such as printers and ink (XED < 0) can be priced jointly.

Is elasticity a reliable guide?

1. It holds because a firm or government that knows the elasticity can predict how revenue responds, so taxing inelastic goods raises stable revenue.
2. But elasticity values are estimates from past data, and they shift as new substitutes appear or habits change, so the guide can mislead.
3. But elasticity also varies along the demand curve and over time, so a value that fits a small price change may fail for a large one.
4. On balance, elasticity is a useful guide, but its reliability depends on the quality of the data, the size of the price change and the time horizon.