Home > CFA Level 1 > Price, income, and cross elasticities of demand

# Price, income, and cross elasticities of demand

As the price of a good rises, its demand falls. This fundamental observation is called the law of demand. A demand function for a good is an equation that links demand with its determinants, i.e. the product’s price (called own price), price of substitutes/complements, consumer income, etc. The positive (negative) signs of different factors represent their direct (indirect) relationship with demand.

If we hold all factors other than own price constant, we arrive at a demand function (of the form Q = Constant – m × P) which shows the relationship between quantity demanded and the product’s own price. If the demand function is expressed such that price is the dependent variable and quantity is independent, it is called an inverse demand function (of the form P = C/m – (1/m) × D) and it helps in graphing a demand curve, which plots price on y-axis and quantity on the x-axis.

## Own-price elasticity of demand

Own-price elasticity of demand measures the sensitivity of quantity demanded of a product to a change in its own-price. It equals the percentage change in quantity divided by the percentage change in price:

$Own Price Elasticity = \frac{∆Q}{Q}÷\frac{∆P}{P} = \frac{∆Q}{∆P}×\frac{P}{Q}$

Where ∆Q/∆P is the slope of the demand function. It shows that elasticity depends on the slope of the demand function and the ratio of price P to quantity.

An elasticity of -n means that a 1% increase in price results in an n percent decrease in demand. Demand is elastic if, in absolute terms, price elasticity is greater than 1, unit elastic when it is equal to 1, and inelastic if it is less than 1.

### Elasticity of demand and demand curve

The elasticity of a product does not remain constant but changes as we move along the demand curve. It is because it depends not only on the slope of the demand curve but on the ratio of P/Q. Higher P/Q means elasticity is high, hence elasticity should be high on the curve close to the y-axis, and vice versa.

### Perfectly elastic and perfectly inelastic demand

Own-price elasticity of demand is constant only if it is either perfectly inelastic or perfectly elastic. Demand is perfectly inelastic if the demand curve is vertical, the own-price elasticity of demand is zero. Demand is perfectly elastic if the demand curve is horizontal and own-price elasticity of demand is infinite, for example in perfect competition.

### Determinants of price elasticity of demand

A product’s own-price elasticity of demand depends on:

• Existence of substitutes: if a product has more substitutes, its elasticity is high, and vice versa.
• Portion of the budget spent on the product: if the highest budget is spent, elasticity is high.
• Time allowed to respond to price change: low time means lower elasticity.
• Extent to which it is treated as necessary: necessities have low elasticity.

The elasticity of a product is higher over the long run, but it is opposite in case of durable goods.

When demand is elastic, change in price and change in total expenditure are opposite but if it is inelastic, they move in the same direction. Expenditure is maximum at the midpoint of the demand curve. This applies to all types of demand (individual vs market, etc.) curves and producers too.

## Income elasticity of demand

Income elasticity of demand refers to the percentage change in quantity demanded in response to a percentage change in income.

$Income Elasticity=\frac{∆Q}{Q}÷\frac{∆I}{I}=\frac{∆Q}{∆I}×\frac{I}{Q}$

While own-price elasticity of demand is typically negative and hence sign is generally ignored, the sign of income elasticity of demand matters. When the sign is positive (negative), it means that that the quantity demanded of the good increases (decreases) with an increase in income, and hence the good is a normal (inferior) good. In case of normal (inferior) good, an increase in income shifts demand outwards (inwards).

## Cross elasticity of demand

Cross elasticity of demand measures the sensitivity of quantity demanded of a good (say X) to change in the price of another good (say Y). A positive (negative) cross elasticity means that the products are substitutes (complements).

Substitutes are goods that can be replaced with each other, hence when the price of once increases, the quantity demanded of the other increases (because people move to the other good) and hence cross elasticity is positive.

Complements are goods which are consumed together, say car and gasoline, hence when the price of one good increases, the quantity demanded of the other good falls and hence it has negative cross elasticity of demand.

$Cross Elasticity=\frac{∆Q_x}{Q_x} ÷ \frac{∆P_y}{P_y} = \frac{∆Q_x}{∆P_y}×\frac{P_y}{Q_x}$

Own-price, income, and cross-elasticities can be calculated from demand function. If demand function is the form Q = M – a × Px + b × I − c × Py, the coefficient a, b and c represent the relevant differentials in each elasticity equation.

## Test

In case of a downward sloping demand curve, own-price elasticity of demand of a product:

1. is constant unless there is a change in income level, consumer preferences, etc.
2. is the highest close to the y-axis and the lowest close to the x-axis.
3. is a positive number.