Income and price elasticity when a fixed share of income goes on clothing

Set-up. Let Anita's income be M and the price of clothing P. She always spends one-third of her income on clothing, so her spending on clothing is P x Q = M/3, and her demand for clothing is

Q = M / (3P).

We assume, as the question implies, that the one-third share holds whatever her income and whatever the price.

Income elasticity. Income elasticity of demand measures how quantity demanded responds to income: eI = (% change in Q) ÷ (% change in M). From Q = M/(3P), with the price unchanged, quantity is proportional to income: if M rises by 10 per cent, Q rises by 10 per cent. Check (our figures): with P = ₹500, M = ₹30,000 gives Q = 20; M = ₹33,000 gives Q = 22. The rise in Q is 10 per cent for a 10 per cent rise in M.

Income elasticity = 10% / 10% = 1. Clothing is a normal good for her, with spending rising exactly in step with income.

Price elasticity. Price elasticity of demand measures how quantity demanded responds to the good's own price: eD = (% change in Q) ÷ (% change in P). Since P x Q = M/3 is fixed whatever the price, total spending on clothing does not change when the price changes. A demand curve along which spending stays constant is a rectangular hyperbola, and its elasticity is 1 in absolute value at every point; when spending does not change with price, eD = -1. Exactly: Q = (M/3) x P^(-1), so a 1 per cent rise in price lowers quantity by 1 per cent.

Price elasticity = -1 (unitary elastic; |eD| = 1).

Answer. Anita's income elasticity of demand for clothing is 1 and her price elasticity is -1, i.e. unitary elastic: both follow from her spending a constant one-third of her income on clothing.