Find the elasticity of demand given Q k where k and n are positive constants.(A) Does the elasticity depend on the price in this case? (B) In the special case where n=l what is the shape of the demand curve? what is the point elasticity of demand
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2 stepsLet's find the elasticity of demand and answer your questions.
The demand function is given by Q = k/Pⁿ, which can be written as Q = kP⁻ⁿ.
The formula for the point elasticity of demand (E_d) is:
E_d = (dQ/dP) × (P/Q)
Find the derivative of Q with respect to P (dQ/dP).
Q = kP⁻ⁿ
dQ/dP = d/dP (kP⁻ⁿ)
dQ/dP = k × (-n) × P⁻ⁿ⁻¹
dQ/dP = -nkP⁻ⁿ⁻¹Substitute dQ/dP and Q into the elasticity formula.
E_d = (-nkP⁻ⁿ⁻¹) × (P / (kP⁻ⁿ))
E_d = (-nkP⁻ⁿ⁻¹ × P¹) / (kP⁻ⁿ)
E_d = (-nkP⁻ⁿ⁻¹⁺¹) / (kP⁻ⁿ)
E_d = (-nkP⁻ⁿ) / (kP⁻ⁿ)
E_d = -nThe elasticity of demand is -n. (Often, the absolute value is used, so |E_d| = n).
A) Does the elasticity depend on the price in this case?
No, the elasticity of demand is -n, which is a constant. It does not depend on the price P.B) In the special case where n=1, what is the shape of the demand curve? What is the point elasticity of demand?
If n=1, the demand function becomes Q = k/P.
The shape of this demand curve is a rectangular hyperbola.
The point elasticity of demand when n=1 is E_d = -1. This means the demand is unit elastic.
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