Question

Last week you sold 6 units at a price of $5.43. This week you sold 17 units at a price of $3.79. If your marginal cost is $2, what price should you charge next week to maximize profit? (exact answer please)

Answer 1

Linear demand function: P = a - bQ

When P = 5.43, Q = 6 and when P = 3.79, Q = 17.

5.43 = a - 6b............(1)

3.79 = a - 17b..........(2)

(1) - (2) gives

11b = 1.64

b = 0.15

a = 5.43 + 6b [from (1)] = 5.43 + 6 x 0.15 = 5.43 + 0.90 = 6.33

So,

Demand: P = 6.33 - 0.15Q

Total revenue (TR) = P x Q = 6.33Q - 0.15Q2

Marginal revenue (MR) = dTR/dQ = 6.33 - 0.3Q

Profit is maximized when MR is equal to MC.

6.33 - 0.3Q = 2

0.3Q = 4.33

Q = 14.43

P = 6.33 - (0.15 x 14.43) = 6.33 - 2.16 = $4.17