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:

\$Ed = \dfrac{\% \Delta Qd}{\% \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.

\$EI = \dfrac{\% \Delta Qd}{\% \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 Qx}{\% \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 📊