Supplemental Material Constant Elasticity Demand and Supply Curves

The price elasticity varies along most demand and supply curves. Along a special type of demand or supply curve, however, the price elasticity is the same at every point along the curve.

Constant Elasticity Demand Curves

The elasticity of demand is the same at every point along a constant elasticity demand curve. Panel a of the figure shows five constant elasticity demand curves. A perfectly inelastic demand curve, where = 0 everywhere, is a vertical straight line. The next most vertical demand curve in the figure is also inelastic, with a demand elasticity of -1/3 everywhere. The unitary elasticity, = -1, demand curve is flatter than the inelastic demand curves. As the elasticity becomes more negative (such as the = -3 demand curve), the demand curves become flatter. The demand curve that is completely flat is perfectly elastic.

The constant elasticity demand function is

Q = Ap,
where A is a positive constant, is the constant elasticity of demand, the price is raised to the power, and we are holding income and other factors constant. Differentiating this expression with respect to price, we find that dQ/dp = Ap-1. Thus, the elasticity of demand, (dQ/dp)(p/Q), is

(Ap- 1) p/Q = ((Ap)/Q) = .

This derivation holds for any price; hence the elasticity of demand is constant at every point along this type of demand curve.

If the demand elasticity at every point is = 0, perfectly inelastic, Q = Ap0 = A, so the demand curve is vertical at Q = A. Similarly, a horizontal demand curve is perfectly elastic at every point. Using algebra, we can rewrite the constant elasticity demand curve as p = (Q/A)1/. As approaches - (perfectly elastic), p approaches (Q/A)0 = 1. This perfectly elastic demand curve may be horizontal at any given price for appropriately specified units of price.

Constant Elasticity Supply Curve

The elasticity of supply is the same at every point on a constant elasticity supply curve. Panel b of the figure shows five constant elasticity supply curves. The perfectly inelastic supply curve, = 0, is a vertical line at Q = Bp0 = B. The inelastic supply curve with = 1/2 is a smooth, moderately steep curve through the origin (p = 0 and Q = 0). The unitary elastic supply curve is a straight line through the origin. An elastic supply curve with = 2 is relatively flat compared to the inelastic curves. The perfectly elastic supply curve is a horizontal line.

The general constant elasticity of supply function is

Q = Bp,

where B is a constant and is the constant elasticity of supply. The elasticity of supply is (dQ/dp)(p/Q) = Bp-1(p/[Bp]) = everywhere along this curve, regardless of price. All the constant elasticity supply curves go through origin: When p = 0, Q = B0 = 0.

The unitary elastic supply curve, Q = Bp1 = Bp, is a straight line with a slope of 1/B. The perfectly inelastic supply curve starts at the origin, runs along the horizontal axis until it reaches Q = B, and then is vertical. Similarly, the perfectly elastic supply curve starts at the origin, rises along the price axis, and then is horizontal at p = 1. Because we have not specified the units of p, the perfectly elastic supply curve can be horizontal at any price.

Figure: Constant Elasticity Curves

a) Constant elasticity demand curves are more vertical, the less elastic is demand, and are more horizontal, the greater is the elasticity of demand. b) Constant elasticity supply curves are more vertical, the closer to zero is the elasticity of supply. They are more horizontal, the greater is the elasticity of supply.

(a) Constant Elasticity Demand Curves

(b) Constant Elasticity Supply Curves