When a wholesaler sold a product at $40 per unit, sales were 240 units per week. After a price increase of $5, however, the average number of units sold dropped to 220 per week.

Question

Assuming that the demand function is linear, what price per unit will yield a maximum total revenue?

Answer 1

(240,40) and (220,45) are two points on the demand curve or line where points are (Q,P) with Q= quantity sold and P = price in dollars

slope = (45-40)/(220-240) = 5/-20 = -1/4

P=(-1/4)Q + b, solve for b by plugging in either point

40=(-1/4)(240) + b

b = 40+60 = 100

Demand curve is P=(-1/4)Q + 100

Revenue = PQ = -Q^2/4 + 100Q

take the derivative and set = 0

R' = -Q/2 + 100 = 0

Q =200

plug 200 into the price equation

P=(1/4)(200) + 100 = 100-50 = $50 = revenue maximizing price

at $50 price, the revenue = -(50)^2/4 +5000 = 5000-2500/4 = $4,375