significance of relative percentage changes, the size and sign of the coefficient of: cross elasticity of demand

Price Elasticity of Demand 📈

Price elasticity tells us how much the quantity demanded of a good changes when its price changes. Think of it like a rubber band: if the band stretches a lot, the good is elastic; if it barely stretches, the good is inelastic.

Formula

\$E_p = \dfrac{\% \text{ change in quantity demanded}}{\% \text{ change in price}}\$

Interpretation

  • Elastic (|Ep| > 1) – Quantity demanded changes more than price. Example: a small price drop on a trendy gadget leads to a big jump in sales.

  • Unit‑elastic (|Ep| = 1) – Quantity demanded changes proportionally with price.

  • Inelastic (|Ep| < 1) – Quantity demanded changes less than price. Example: salt – people buy almost the same amount even if the price rises.

Income Elasticity of Demand 💰

Income elasticity measures how quantity demanded changes when consumers’ income changes. Imagine a pizza shop: when people earn more, they might order more premium pizzas.

Formula

\$E_I = \dfrac{\% \text{ change in quantity demanded}}{\% \text{ change in income}}\$

Interpretation

  • Normal good (EI > 0) – Demand rises when income rises.

  • Inferior good (EI < 0) – Demand falls when income rises. Example: instant noodles vs. fresh pasta.

  • Luxury good (EI > 1) – Demand increases more than income. Example: designer watches.

Cross Elasticity of Demand 🔄

Cross elasticity looks at how the demand for one product changes when the price of another product changes. Think of it as a friendship: if one friend’s mood changes, the other friend’s mood might change too.

Formula

\$E_{xy} = \dfrac{\% \text{ change in quantity demanded of } x}{\% \text{ change in price of } y}\$

Significance of Relative Percentage Changes

Because we use percentage changes, the coefficient is dimensionless and comparable across products. It tells us the *relative* sensitivity – a 10% price drop in coffee might lead to a 5% rise in tea sales, giving \$E_{tea,coffee}=0.5\$.

Size and Sign of the Coefficient

  • Positive (Exy > 0) – Substitutes. When the price of product y rises, consumers switch to product x. Example: butter and margarine.

  • Negative (Exy < 0) – Complements. When the price of product y rises, demand for product x falls. Example: printers and ink cartridges.

  • Magnitude – The larger the absolute value, the stronger the relationship. A value of 2 means a 1% price increase in y leads to a 2% change in x’s demand.

Quick Reference Table 📊

Elasticity TypeFormulaInterpretationExample
Price Elasticity (Ep)\$E_p = \dfrac{\%ΔQ}{\%ΔP}\$|Ep| > 1: Elastic; < 1: Inelastic; = 1: Unit‑elasticSmartphone price drop → sales surge
Income Elasticity (EI)\$E_I = \dfrac{\%ΔQ}{\%ΔI}\$EI > 0: Normal; < 0: Inferior; > 1: LuxuryHigher salary → more gourmet meals
Cross Elasticity (Exy)\$E{xy} = \dfrac{\%ΔQx}{\%ΔP_y}\$>0: Substitutes; <0: Complements; magnitude shows strengthCoffee price ↑ → tea demand ↑ (substitutes)