Question

Inverse demand function: P=40-5Q

Using the demand function above, Assuming that marginal cost is $10 and price elasticity of demand is -1.667, what is the optimal price a seller should charge to maximize profit?

Answer 1

Answer: The inverse demand function is the same as the average revenue function, since P = AR. We have P=40-5Q, marginal cost is $10 and price elasticity of demand is -1.667 given.

First we will find the quantity of the firm for which we will multiply each side of the inverse demand function by Q.

i.e. P*Q = 40Q - 5Q²

Next, take the derivative with respect to Q to get the MR function:

dTR/dQ = MR

i.e. dTR/dQ = 40 - 10Q = MR

Equating MR to MC and solving for Q gives 40-10Q = 10 => Q = 3

So 3 is the profit maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P.

i.e. P= 40- 5Q => 40 - 5 * 3 = 25

Also we are given elasticity of demand = -1.667

The formula for elasticity is (?Q/?P) × (P/Q).

profit = P*Q = 25* 3 = 75$

To know whether this price will maximize the profit or not the second derivative should be less than 0

i.e. dTR²/ dQ²  = -10 < 0

therfore the optimal price a seller should charge to maximize profit is $25.