Income Elasticity, Price Elasticity, and Cross Elasticity

Elasticity measures the sensitivity or responsiveness of one variable to another. There are three main forms of elasticity – price elasticity, income elasticity, and cross-price elasticity.

Price Elasticity

Price elasticity of demand is a measure of how a product’s demand changes in response to changes in its price. It is measured in percentage changes in each of the variables. Thus, we calculate the price elasticity of demand  using the following:

$$ E_{px}^d=\frac{\%\Delta Q_x^d}{\%\Delta P_x} $$

Where:

\(\%ΔQ_x^d\) = the percentage change in quantity demanded

\(\%ΔP_x\) = the percentage change in price

\(\%ΔQ_x^d\) can also be written as \(\frac{ΔQ_x^d}{Q_x^d}\) while \(\%ΔP_x\) can be broken down to \(\frac{ΔP_x }{P_x}\). Hence elasticity of price can be rewritten as:

$$ E_{px}^d=\frac{\%\Delta Q_x^d}{\%\Delta P_x}=\frac{\frac{\Delta Q_x^d}{Q_x^d}}{\frac{\Delta P_x}{P_x}}=\left(\frac{\Delta Q_x^d}{\Delta P_x}\right)\left(\frac{P_x}{Q_x^d}\right) $$

Example: Price Elasticity

The annual premium of a certain life insurance company increased from $20 to $25. This year, the number of policies sold decreased from 1000 to 900. Calculate the price elasticity of demand.

From the information given in the question:

$$E_{px}^d=\left(\frac{\Delta Q_x^d}{Q_x^d}\right)\left(\frac{P_x}{\Delta P_x}\right) =\left(\frac{900-1000}{1000}\right)\left( \frac{20}{25-20}\right)=-0.4$$

The main determinant of price elasticity is the change in price itself.

Classifications of Price Elasticities of Demand

• Demand is elastic if the price elasticity of demand (PED) is greater than one. That is PED > 1. This implies that the quantity demanded changes by a larger proportion than the price.
• Demand is inelastic if PED < 1. This implies that the quantity demanded changes by a smaller proportion than the price.
• Demand has a unit elasticity if PED= -1. This implies that the percentage change in the quantity demanded and the price changes are equal and opposite.

Factors That Affect the Price Elasticity of Demand

• Availability of substitute goods: More substitutes translate to a more elastic demand.
• Time horizon: Generally, it is time-consuming to find alternative goods, and hence, the more time utilized, the more elastic the demand. The price elasticity of demand for a shorter period is always low, or it can even be inelastic.
• Income: this is based on the proportion of income on a good. Intuitively from the formulas, if the product makes a larger proportion of an individual’s income, this translates to more elastic demand.

Income Elasticity

The income elasticity is defined as the percentage change in quantity demanded divided by the percentage change in the income of the customers ceteris paribus (holding all other things constant).

Hence the income elasticity is given by:

$$E_I^d=\frac{\%\Delta Q_x^d}{\%\Delta I}$$

The calculation of income elasticity is similar to price elasticity. However, “own” price elasticity is always negative when the law of demand holds, whereas income elasticity could either be negative, positive or zero.

When an increase in income leads to increased consumption/quantity demanded, there is positive income elasticity. In contrast, negative income elasticity means that an increase in income leads to a decrease in the quantity demanded. Examples of goods possessing positive income elasticity are normal goods, while negative income elasticity goods are inferior goods.

Holding every other factor constant, the main determinant of income elasticity is the income of the consumers.

Example: Income Elasticity

An individual consumer’s monthly demand for apples and their by-products in an African country is given by the equation:

$$Q_a^d=5-0.5P_a+0.005I+0.25P_{ju}$$

Where:

\(Q_a^d\) = number of apples demanded each month

\(P_a\) = the price of apples

\(I\) = estimated household monthly income

\(P_{ju}\) = price of the apple juice

Given that the apple price is $10, that of juice is $20, and the estimated household income is $2,000, calculate the income elasticity of demand for the apples.

Now, using the same analogy as that price elasticity of demand:

$$E_{px}^d=\left(\frac{\Delta Q_x^d}{\Delta P_x}\right)\left(\frac{P_x}{Q_x^d}\right)$$

The income elasticity of demand is given by:

$$E_I^d=\left(\frac{\Delta Q_a^d}{\Delta I}\right)\left(\frac{I}{Q_a^d}\right)$$

(Note that we just swapped the denominators.)

Now,

$$\frac{\Delta Q_a^d}{\Delta I}=\frac{\partial}{\partial I}\left(Q_a^d=5-0.5P_a+0.005I+0.25P_{ju}\right)=0.005$$

and

$$Q_a^d=5-0.5\times 10+0.005\times 2000+0.25\times 20=15$$

$$\Rightarrow E_I^d=0.005\times \frac{2000}{15}=0.6667$$

Cross-price Elasticity

Other than the price of a product and the income of the consumers, the prices of other products can also affect the demand for the product. The cross-price elasticity is defined on this basis. Here, we evaluate the effect of the percentage change in the prices of other products on the quantity of demand for a particular good. This notion is represented mathematically as:

$$E_{P_y}^d=\frac{\%\Delta Q_x^d}{\%\Delta P_y}$$

Where P_y represents the price of other products.

Cross-price elasticity is mostly found in goods with substitutes and complements.

When the price of a good with a close substitute, say cauliflower, increases, the demand for that particular product will likely shift to another vegetable, say broccoli. This relationship describes positive cross-price elasticity.

Conversely,  complementary goods, like cell phones and chargers, have negative cross-price elasticity. In other words, an increase in the price of phones may reduce the quantity demanded of phones; consequently, the quantity demanded of phone chargers will also decline.

Example: Cross-price Elasticity

The cross-price elasticity of demand for Good B  with respect to good A is 0.65. 1000kg of Good B is demanded when the cost of good A is $60 per kg. The cost of Good A rises to $100. Calculate the corresponding quantity of Good B demanded.

Recall that:

$$E_{P_y}^d=\frac{\%\Delta Q_x^d}{\%\Delta P_y}$$

But

$$\%\Delta P_y =\frac{100-60}{60}\times 100=66.6667\%$$

Substituting in the formula:

$$0.65=\frac{\%\Delta Q_x^d}{66.6667\%} \Rightarrow \%\Delta Q_x^d=43.333\%$$

This implies that we expect an increase in demand for Good B, so the corresponding quantity is:

$$1.43333\times 1000=1,433.33\approx 1,433$$

Question

Given the demand function \(Q_x^d\) = 40 – 5P_x, calculate the price elasticity of demand at a price of $1.50.

1. 0.23
2. 0.45
3. 3.45

The correct answer is A.

A unit change (an increase) in price will lead to a 5 unit decrease in demand. If, for instance, the price changes to $1.5, the elasticity or percentage change can be calculated as shown below.

Differentiating the demand function to get the elasticity of demand will give us -5.

Multiplying the demand elasticity with the ratio of price to quantity will give us:

$$Q_x^d = 40 – (5)(1.5) = 32.5$$

Hence, the elasticity of demand at a price of $1.5 is:

$$\left(\frac{\Delta Q_x^d}{\Delta P_x}\right)\left(\frac{P_x}{Q_x^d}\right) =-\left(5\right) \frac{1.5}{32.5}=0.23$$