Using Calculus To Calculate Income Elasticity of Demand

Suppose you're given the following question:

Demand is Q = -110P +0.32I, where P is the price of the good and I is the consumers income. What is the income elasticity of demand when income is 20,000 and price is $5?

We saw that we can calculate any elasticity by the formula:

Elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)

In the case of income elasticity of demand, we are interested in the elasticity of quantity demand with respect to income. Thus we can use the following equation:

Price elasticity of income: = (dQ / dI)*(I/Q)

In order to use this equation, we must have quantity alone on the left-hand side, and the right-hand side be some function of income. That is the case in our demand equation of Q = -110P +0.32I. Thus we differentiate with respect to I and get:

dQ/dI = 0.32

So we substitute dQ/dP = -4 and Q = -110P +0.32I into our price elasticity of income equation:

Income elasticity of demand: = (dQ / dI)*(I/Q)
Income elasticity of demand: = (0.32)*(I/(-110P +0.32I))
Income elasticity of demand: = 0.32I/(-110P +0.32I)

We're interested in finding what the income elasticity is at P = 5 and I = 20,000, so we substitute this into our income elasticity of demand equation:

Income elasticity of demand: = 0.32I/(-110P +0.32I)
Income elasticity of demand: = 6400/(-550 + 6400)
Income elasticity of demand: = 6400/5850
Income elasticity of demand: = 1.094

Thus our income elasticity of demand is 1.094. Since it is greater than 1 in absolute terms, we say that Demand is Income Elastic, which also means that our good is a luxury good.

Next: Using Calculus To Calculate Cross-Price Elasticity of Demand

Other Price Elasticity Equations

1. Using Calculus To Calculate Price Elasticity of Demand
2. Using Calculus To Calculate Income Elasticity of Demand
3. Using Calculus To Calculate Cross-Price Elasticity of Demand
4. Using Calculus To Calculate Price Elasticity of Supply