If the demand curve is Qppa where a

If the demand curve is Q(p)=p^a (where a<0), what is the elasticity of demand? If the marginal cost is $1, and a=-3, what is the profit-maximizing price?

Solution

If “a” is smaller than zero, then “a” would be -1, -2, -3 …and so on.

Suppose taking (a = -2); if (P = 5),

Q = P ^a

    = 5 ^(-2)

    = 1 / 5 ^2

    = 1 / 25

    = 0.04

Again taking (a = -2); if price increases to 10,

Q = P ^a

    = 10 ^(-2)

    = 1 / 10 ^2

    = 1 / 100

    = 0.01

By increasing price, quantity consumption decreases.

Elasticity of demand = (Delta Q / Delta P) × (P / Q)

                                    = {(0.01 – 0.04) / (10 – 5) × (5 / 0.04)

                                    = (-0.03 / 5) × 125

                                    = -0.75

Since elasticity of demand is less than 1, it is inelastic.

Answer: The elasticity of demand is inelastic.

Profit maximizing price could be found where (MR = MC)

Given, Q = P ^a

Rearranging, P = Q ^(1/a)

TR = PQ = Q ^ {(1 + a)/a}

MR = Derivative of TR with respect to Q

       = {(1 + a)/a} Q ^(1/a)

Now given (a = -3),

MR = (2/3) Q ^(-1/3)

Condition: MR = MC

(2/3) Q ^(-1/3) = 1

Q ^(-1/3) = 1.5

Q = 1/3.375

Q = 0.296296

Now by putting this value in the price function,

P = Q ^(1/a)

   = 0.296296 ^(-1/3)

   = 1 / (0.296296 ^0.33333)

   = 1/ 0.66666

   = 1.50

Answer: The profit maximizing price is 1.50.