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:

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.

Moffatt, Mike. "Using Calculus To Calculate Income Elasticity of Demand." ThoughtCo, Apr. 3, 2017, thoughtco.com/calculate-income-elasticity-of-demand-1146249.Moffatt, Mike. (2017, April 3). Using Calculus To Calculate Income Elasticity of Demand. Retrieved from https://www.thoughtco.com/calculate-income-elasticity-of-demand-1146249Moffatt, Mike. "Using Calculus To Calculate Income Elasticity of Demand." ThoughtCo. https://www.thoughtco.com/calculate-income-elasticity-of-demand-1146249 (accessed March 18, 2018).