## Other Elasticity Measures

On the demand-side of the market, elasticities can be calculated for any of the relevant exogenous variables. In all cases, this calculation measures the percentage change in quantity demanded relative to a percentage change in one of the exogenous variables.

#### Income Elasticity of Demand

Income elasticities measure the response in quantity demanded to a change in consumer income. In this case, the corresponding elasticity may be positive for normal goods, negative for inferior goods, or equal to zero for income neutral goods. This elasticity is computed as follows:

ηM = (%ΔQD)/(%ΔM)

= (ΔQ/Q)
(ΔM/M)

If both numerator and denominator are of the same sign (both increase or both decrease) then the corresponding good or service is a normal good. If numerator and denominator are opposite in sign (one increases as the other decreases), then the good is an inferior good. Finally if the value of the numerator is zero (quantity demanded does not change with income), then the good is income neutral. The table below summarizes these results:

 Income Elasticity Type of Good hM < 0 An Inferior Good hM = 0 An Income-Neutral Good 0 < hM < 1.0 A Normal (necessity) Good hM > 1.0 A Normal (luxury) Good

#### Cross-Price Elasticity of Demand

Cross-Price elasticity of demand measures the response of quantity demanded of one good to changes in the price of a second (related) good. This elasticity is computed as follows:

ηxy = (%ΔQxD)/ (%ΔPy)

If the two goods are substitutes we would expect the following:

Py ,QyD ; QxD In this case the price of good-y and the quantity demanded of good-x move in the same direction and thus the cross-price elasticity would be positive.

If the two goods are complements, then the relationship between the price of one good and quantity demanded of the other would be:

Py↑,QyD ; QxD

Where in this case the price of good-y and quantity demanded of good-x move in opposite directions. The corresponding cross-price elasticity would be negative.

Finally if changes in the price of one good has no effect on the quantity demanded of the other, then the cross-price elasticity would be zero and the two goods are unrelated. The following table summarizes these results:

 Cross-Price Elasticity Goods 'x' & 'y' are: hxy < 0 Complement Goods hxy = 0 Unrelated Goods hxy > 0 Substitute Goods

#### Elasticities and Non-Linear Demand

The following demand equation represents a non-linear relationship between quantity demanded and market price.

QxD = APxα

This expression is a valid demand relationship if the parameter α is strictly less than zero. In addition, the coefficient 'A', in this equation, is a measure all of the other exogenous influences on demand (income, tastes and preferences, price of related goods, number of consumers in the market). Changes in any of these exogenous variables will affect the value of this coefficient.

Noting that the price-elasticity of demand may be written as:

ηp = (%ΔQ) / (%ΔP)

= (ΔQ/Q)
(ΔP/P)

= (DQ) (P)
(ΔP) (Q)

In this last expression the (ΔQ / ΔP) term represents the slope of the demand equation. Thus, if we differentiate the non-linear demand equation given above and substitute, we have the following:

ηp = αAPα - 1 (P)
(Q)

= (αAPα )/Q = α

The parameter (exponent) 'α' is the price elasticity of demand.

A more specific non-linear demand equation (one where many of the exogenous variables are explicitly defined) may be defined as follows:

Qxd = APxα Mβ Pyf Pzγ .

It can be shown (through partial differentiation) that the parameters a,b,f,g all represents various types of elasticity measures. That is:

α = ηp -- price elasticity of demand,
β = ηM -- income elasticity of demand,
f = ηxy, and
γ = ηxz -- cross-price elasticities.

Thus if we have the following (estimated) demand equation:

ln(QxD) = β0 + β1(Px) +β2ln(M) + β3ln(Py) + β4 ln(Pz) .
ln(QxD) = 5.01 - 0.75 ln(Px) + 0.50 ln(M) + 1.0 ln(Py) - 1.25 ln(Pz) + ε .
QxD = 150Px-0.75M0.50PyPz-1.25 .

we can state that demand for this particular good-x is price inelastic (|ηp| < 1.0), a normal good (0.0 < ηM < 1.0), a substitute with good-yxy > 0), and a complement with good-zxz < 0).

#### Example

Ski Lift Tickets Suppose that the equation above represents the estimated demand for lift tickets at a Colorado resort.

QDtickets = 150(Pticket)-0.75 (M)0.50 (Ptropical resort lodging) (Pequipment rentals)-1.25 .

Ptickets is the own price of a lift ticket, Py the vacation price in a warmer climate -- a substitute and Pequipment rentals the rental price of skiis, boots and poles -- a complement. Now suppose that personal income is forcasted to increase by 3% next year, the price of tropical vacations is reported to fall by 2% and equipment prices increase by 5%. How much will demand change for lift tickets?

The answer will be: 0.50 x 3% + 1.0 x (-2%) - 1.25 x 5% = - 6.75%

The ski resort may want to reduce the price of lift tickets if it's goal is to maintain market share among all resorts.

Concepts for Review
• Complement (goods)
• Cross-price Elascitity
• Income Elasticity
• Income-neutral (goods)
• Inferior Goods
• Normal Goods
• Substitute (goods)