For a straight-line demand curve intercepting the quantity axis at 200 units, the elasticity equals 1 at a quantity of

For a straight-line demand curve intercepting the quantity axis at 200 units, the elasticity equals 1 at a quantity of

• A. 100 units
• B. 200 units
• C. 50 units
• D. 0 units

Answer: (A)

Unit elasticity is where the percentage change in quantity demanded equals the percentage change in price, so the elasticity equals 1 in magnitude. For a linear demand curve that hits zero price at a quantity of 200, we can write Q = 200 − bP. The slope is dQ/dP = −b, so elasticity ε = (dQ/dP)·(P/Q) = (−b)·(P/(200 − bP)). Setting ε = −1 (unit elasticity) gives (−b)·P/(200 − bP) = −1, which simplifies to bP = 200 − bP, so 2bP = 200 and P = 100/b. The corresponding quantity is Q = 200 − bP = 200 − 100 = 100. So unit elasticity occurs at a quantity of 100 units (half of the Q-intercept, 200).

Unit elasticity is where the percentage change in quantity demanded equals the percentage change in price, so the elasticity equals 1 in magnitude. For a linear demand curve that hits zero price at a quantity of 200, we can write Q = 200 − bP. The slope is dQ/dP = −b, so elasticity ε = (dQ/dP)·(P/Q) = (−b)·(P/(200 − bP)).

Setting ε = −1 (unit elasticity) gives (−b)·P/(200 − bP) = −1, which simplifies to bP = 200 − bP, so 2bP = 200 and P = 100/b. The corresponding quantity is Q = 200 − bP = 200 − 100 = 100.

So unit elasticity occurs at a quantity of 100 units (half of the Q-intercept, 200).