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:

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**

- Using Calculus To Calculate Price Elasticity of Demand
- Using Calculus To Calculate Income Elasticity of Demand
- Using Calculus To Calculate Cross-Price Elasticity of Demand
- Using Calculus To Calculate Price Elasticity of Supply