Question

8. [Own Price Elasticity of Demand] Given a demand function Q = f(P), the own price elasticity of demand is defined as

? = (dQ/dP) · (P/Q)

What is the own price elasticity of demand ?

(a) for the linear demand function Q = 100?5P when P = 10.

(b) for the linear inverse demand function P = 100?4Q when (i) Q = 10; (ii) Q = 20; (iii) Q = 12.5.

(c) for the demand function Q = P ^?0.5 , when (i) Q = 10; (ii) Q = 20.

(d) for the inverse demand function P = Q ^?1 , when (i) Q = 10; (ii) Q = 20.

Answer 1

Use the relation ? = (dQ/dP) · (P/Q)

(a) PED for the linear demand function Q = 100?5P when P = 10 has Q = 100 - 5*10 = 50. (dQ/dP) = -5 so we have

? = -5 x (10/50) = -1

(b) PED for the linear inverse demand function P = 100?4Q when Q = 10 has P = 100 - 40 = 60. Demand function is Q = 100/4 - P/4 so that  (dQ/dP) = -1/4 or -0.25

? = -0.25 x (60/10) = -1.5

(ii) Q = 20; PED for the linear inverse demand function P = 100?4Q when Q = 20 has P = 100 - 80 = 20.

? = -0.25 x (20/20) = -0.25

(iii) Q = 12.5. PED for the linear inverse demand function P = 100?4Q when Q = 12.5 has P = 100 - 50 = 50

? = -0.25 x (50/12.5) = -1

(c) PED for the demand function Q = P ^?0.5 when has (dQ/dP) = -0.5*(P^-1.5). When Q = 10, we have 10 = P^-0.5 or P = 1/100 = 0.01

? = -0.5*(0.01^(-1.5)) x (0.01/10) = -0.5

For Q = 20, we have 20 = P^-0.5 or P = 0.0025

? = -0.5*(0.0025^(-1.5)) x (0.0025/20) = -0.5

(d) for the inverse demand function P = Q ^?1 , we have Q = P^-1. When Q = 10, P = 0.1 dQ/dP = -P^-2. When Q = 20, P = 1/20 = 0.05

For Q = 10, ? = -(0.05^(-2))*(0.05/20) = -1

For Q = 20, ? = -(0.1^(-2))*(0.1/10) = -1