Which statement is true about a demand curve with unit price elasticity at all prices?

Which statement is true about a demand curve with unit price elasticity at all prices?

• A. It has unit price elasticity of demand at all prices.
• B. The elasticity differs along the curve.
• C. It has infinite elasticity at all prices.
• D. It has zero elasticity at all prices.

Answer: (A)

Unit price elasticity means the elasticity value is -1 at every price. For a demand curve to have this property everywhere, the elasticity epsilon = (dQ/dP) * (P/Q) must equal -1 for all P. This leads to the differential equation dQ/dP = -Q/P. Solving by separation: dQ/Q = -dP/P, which integrates to ln Q = -ln P + C, so Q = k/P, a rectangular hyperbola. On this curve, P*Q = k is constant, so total revenue stays the same when price changes. This constant-elasticity shape is why the curve exhibits unit elasticity at all prices. Other shapes don’t maintain elasticity of -1 everywhere (a straight line’s elasticity varies along it, infinite elasticity would be at all prices in a perfectly elastic case, zero elasticity would be perfectly inelastic), so only this hyperbola form has unit elasticity throughout.

Unit price elasticity means the elasticity value is -1 at every price. For a demand curve to have this property everywhere, the elasticity epsilon = (dQ/dP) * (P/Q) must equal -1 for all P. This leads to the differential equation dQ/dP = -Q/P. Solving by separation: dQ/Q = -dP/P, which integrates to ln Q = -ln P + C, so Q = k/P, a rectangular hyperbola. On this curve, P*Q = k is constant, so total revenue stays the same when price changes. This constant-elasticity shape is why the curve exhibits unit elasticity at all prices. Other shapes don’t maintain elasticity of -1 everywhere (a straight line’s elasticity varies along it, infinite elasticity would be at all prices in a perfectly elastic case, zero elasticity would be perfectly inelastic), so only this hyperbola form has unit elasticity throughout.