Question

For the following demand equation compute the elasticity of demand and determine whether the demand is elastic, unitary, or inelastic at the indicated price. (Round your answer to three decimal places.)
p= 157 − x2 ; p = 20

Answer 1

Answer:

Note: While answering this question, there is a confusion as to whether the correct demand equation is p= 157 − x2 as it appears in the question, or p= 157 − x^2 which is suspected to be the correct equation. Whichever the case may be, answers are provided for the two equations. Just confirm from the original question or your teacher which one is correct and pick the relevant answer out of the two following answers:

1. If p= 157 − x2 is the correct equation:

Elasticity of demand = - 0.15

Since -0.15 in absolute term |-0.15| is less than 1, the demand is inelastic.

2. If p= 157 − x^2 is the correct equation:

Elasticity of demand = - 0.28

Since -0.28 in absolute term |-0.28| is less than 1, the demand is also inelastic.

Explanation:

Elasticity of demand is the degree of responsive of quantity demanded of a good to change in its price.

For this question, elasticity of demand can be computed using the formula for calculating the elasticity of demand as follows:

1. If p= 157 − x2 is the correct equation

E = Elasticity of demand = (p / x) * (dx / dp) ............................... (1)

From the question, we have:

p = 157 − x2.

Therefore, we solve for as follows:

x2 = 157 - p

x = (157 - P) / 2

x = 78.5 - 0.5p ....................................................... (2)

Differentiating equation (2) with respect to p, we have:

dx/dp = -0.5

Substituting values for dx/dp and x into equation (1), we have:

E = [p / (78.5 - 0.5p)] * (-0.5)

Since p = 20, we have:

E = [20 / (78.5 - (0.5 * 20))] * (-0.5)

E = [20 / (78.5 - 10)] * (-0.5)

E = − 0.15

Since -0.15 in absolute term |-0.15| is less than 1, the demand is inelastic.

2. If p = 157 − x^2 is the correct equation

E = (p / x) * (dx / dp) ............................................ (1)

From the question, we have:

p = 157 − x^2.

Therefore, we solve for as follows:

x = 157^(1/2) – p^(1/2)

x = 157^0.5 – p^0.5 ........................................... (2)

Differentiating equation (2) with respect to p, we have:

dx/dp = -0.5p^(-0.5) = -0.5/p^0.5

Substituting values for dx/dp and x into equation (1), we have:

E = (p / x) * (dx / dp)

E = [p /  (157^0.5 – p^0.5)] * [ -0.5/p^0.5]

Since p = 20, we have:

E = [20 /  (157^0.5 – 20^0.5)] * [–0.5/20^0.5]

E = - 0.28

Since -0.28 in absolute term |-0.28| is less than 1, the demand is inelastic.