Question

1 - A firm sells mid-range snowboards and estimates its demand curve to be Q = 15,700 – 17(P). Variable costs are $265.

At what price does marginal revenue equal 0? Round your answer to the nearest dollar.

2 - A firm sells mid-range snowboards and estimates its demand curve to be Q = 12,700 – 32(P). Variable costs are $240. At what price does the firm's total contribution margin equal zero?

Answer 1

Solution:

1. We are given the demand curve as: Q = 15700 - 17*P, so inverse demand curve becomes: P = (15700 - Q)/17

Total revenue = price*quantity

TR = ((15700 - Q)/17)*Q = (15700*Q - Q^2)/17

Then, marginal revenue = dTR/dQ

MR = (15700/17) - 2*Q/17

For marginal revenue to be 0: MR = 0

(15700/17) - 2Q/17 = 0

Q = 15700/2 = 7850 units

So, price at which MR is 0 is (15700 - 7850)/17 = $461.76

2. Firm's total contribution margin = total sales - total variable cost

Inverse demand curve is P = (12700 - Q)/32, so total sales = (12700Q - Q^2)/32

Total variable cost = 240*Q

Then, total contribution margin = 0 implies:

(12700*Q - Q^2)/32 - 240*Q = 0

(Q/32)*(12700 - Q - 7680) = 0

Q*(5020 - Q) = 0

So, either Q = 0, or Q = 5020 units

Then corresponding price (for positive quantity level) is (12700 - 5020)/32 = $240

(Shortcut: total sales = price*quantity; total variable cost = variable cost per unit*quantity, so for firm's total contribution margin to be 0, price = variable cost per unit.)