Demand and Elasticity

Demand Curve & Elasticity

The demand curve traces optimal quantity as price varies ($p_x, x^*$), holding income constant. It slopes downward by the law of demand, unless the good is a Giffen good (an extreme inferior good where income outweighs substitution).

Price elasticity of demand measures responsiveness:

$$\varepsilon_p = \frac{\%\Delta Q}{\%\Delta P} = \frac{dQ}{dP} \cdot \frac{P}{Q}$$

Determinants: More substitutes, longer time horizons, luxuries, and larger budget shares all increase elasticity.

Pitfall: Confusing slope ($dQ/dP$) with elasticity. Slope uses units; elasticity uses percentages. Constant slope does NOT mean constant elasticity.

Elasticity Along a Curve

For $Q = a - bP$, slope is constant at $-b$, but elasticity $\varepsilon = -b \cdot (P/Q)$ changes as $P/Q$ shifts.

• Top: High $P$, low $Q$ $\rightarrow$ $|\varepsilon| > 1$ (elastic).
• Bottom: Low $P$, high $Q$ $\rightarrow$ $|\varepsilon| < 1$ (inelastic).
• Midpoint: $|\varepsilon| = 1$ (unit elastic, revenue peaks).

Total revenue $R = P(a - bP)$ is a parabola, maximised at $P^* = a/(2b)$.

Cross-price elasticity measures how good $x$ demand reacts to good $y$ price: $\varepsilon_{xy} = (\%\Delta Q_x)/(\%\Delta P_y)$. Positive for substitutes, negative for complements.

Income elasticity is $\varepsilon_M = (\%\Delta Q)/(\%\Delta M)$. Positive for normal goods, negative for inferior goods. Luxuries have $\varepsilon_M > 1$.