Demand and Elasticity

From Preferences to Demand

The demand curve traces how the optimal quantity of a good changes as its price varies (holding income and other prices constant). It is derived from the utility-maximisation problem:

1. At each price $p_x$, solve for $x^*(p_x, p_y, M)$.
2. Plot $(p_x, x^*)$ — this is the demand curve.

For most goods, demand slopes downward: higher price, lower quantity demanded. The law of demand holds unless the good is a Giffen good (an extreme inferior good where the income effect overwhelms the substitution effect).

Price elasticity of demand measures responsiveness:

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

• $|\varepsilon| > 1$: elastic — quantity responds more than proportionally to price.
• $|\varepsilon| < 1$: inelastic — quantity responds less.
• $|\varepsilon| = 1$: unit elastic — proportional response.

Revenue implications: total revenue $R = P \times Q$.

• Elastic demand: price increase lowers revenue (quantity falls proportionally more).
• Inelastic demand: price increase raises revenue.
• Unit elastic: revenue is maximised.

Determinants of elasticity: availability of substitutes (more substitutes = more elastic), time horizon (long run = more elastic), necessity vs luxury (necessities = more inelastic), share of budget (larger share = more elastic).

Elasticity vocabulary

Common pitfall: Confusing slope with elasticity. Slope is $dQ/dP$ in units; elasticity re-scales it by $P/Q$ into percentages. Two curves with identical slopes can have wildly different elasticities at the same price.

Elasticity Along the Demand Curve

A crucial subtlety: elasticity changes along a linear demand curve, even though the slope is constant.

For $Q = a - bP$: the slope $dQ/dP = -b$ is constant, but elasticity $\varepsilon = -b \cdot P/Q$ varies because $P/Q$ changes.

$$\varepsilon = -b \cdot \frac{P}{a - bP}$$

• At the top of the demand curve (high $P$, low $Q$): $P/Q$ is large, so $|\varepsilon| > 1$ — elastic.
• At the bottom (low $P$, high $Q$): $P/Q$ is small, so $|\varepsilon| < 1$ — inelastic.
• At the midpoint: $|\varepsilon| = 1$ — unit elastic, revenue maximised.

$$\text{Revenue} = P \times Q = P(a - bP) = aP - bP^2$$

This is a parabola in $P$, maximised at $P^* = a/(2b)$ — exactly the midpoint of the demand curve.

Cross-price elasticity: $\varepsilon_{xy} = (\%\Delta Q_x)/(\%\Delta P_y)$. Positive for substitutes (Coke price rises, Pepsi demand rises), negative for complements (petrol price rises, car demand falls).

Income elasticity: $\varepsilon_M = (\%\Delta Q)/(\%\Delta M)$. Positive for normal goods, negative for inferior goods. Luxuries have $\varepsilon_M > 1$ (demand rises faster than income).

Tip: On any linear demand curve, find the midpoint — everything above it is elastic territory (cut price to raise revenue), everything below is inelastic (raise price to raise revenue), and the midpoint itself is where revenue peaks.

Common pitfall: "Constant slope means constant elasticity." The ratio $P/Q$ silently changes as you slide down the line — elasticity runs from infinite at the choke price to zero where the curve hits the quantity axis.