relationship between price elasticity of demand and total expenditure on a product

Price Elasticity of Demand 📈

Price elasticity measures how much the quantity demanded of a good changes when its price changes. The formula is:

$E_d = \dfrac{\% \Delta Q_d}{\% \Delta P}$

If $|E_d| > 1$ the demand is elastic – quantity changes a lot. If $|E_d| < 1$ the demand is inelastic – quantity changes little. If $|E_d| = 1$ the demand is unit‑elastic – quantity changes proportionally.

Analogy: The Rubber Band

Think of demand like a rubber band. A highly elastic good is a stretchy rubber band – a small tug (price change) pulls it far (big quantity change). An inelastic good is a stiff rubber band – you have to tug hard (big price change) to see little movement (small quantity change).

Income Elasticity of Demand 💰

Income elasticity tells us how demand changes when consumers’ income changes.

$E_I = \dfrac{\% \Delta Q_d}{\% \Delta I}$

• Normal goods – $E_I > 0$. Demand rises when income rises.
• Inferior goods – $E_I < 0$. Demand falls when income rises.
• Luxury goods – $E_I > 1$. Demand rises more than proportionally.

Cross Elasticity of Demand 🔀

Cross elasticity measures how the demand for one good changes when the price of another good changes.

$E_{xy} = \dfrac{\% \Delta Q_x}{\% \Delta P_y}$

• Substitutes – $E_{xy} > 0$. If the price of good Y rises, demand for good X rises.
• Complements – $E_{xy} < 0$. If the price of good Y rises, demand for good X falls.

Total Expenditure and Price Elasticity 💸

Total expenditure on a product is:

$E = P \times Q$

When price changes, the change in expenditure is:

$\Delta E = \Delta P \times Q + P \times \Delta Q$

Using elasticity, we can rewrite the percentage change in expenditure:

$\% \Delta E = \% \Delta P + \% \Delta Q = \% \Delta P (1 + E_d)$

So:

• If $|E_d| > 1$ (elastic), a price increase ($\% \Delta P > 0$) leads to $\% \Delta E < 0$ – total expenditure falls.
• If $|E_d| < 1$ (inelastic), a price increase leads to $\% \Delta E > 0$ – total expenditure rises.
• If $|E_d| = 1$, total expenditure stays the same.

Example: Ice Cream 🍦

Suppose the price of ice cream rises by 10 %. If the demand is elastic (say $E_d = -1.5$), the quantity demanded falls by 15 %. Total expenditure change: $\% \Delta E = 10\% + (-15\%) = -5\%$ – consumers spend 5 % less on ice cream.

If the demand were inelastic ($E_d = -0.5$), the quantity falls by 5 %. $\% \Delta E = 10\% + (-5\%) = 5\%$ – consumers spend 5 % more on ice cream.

Quick Summary Table 📊