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# Using Calculus To Calculate Income Elasticity of Demand

## 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.

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