Question

7. When the iPinch is priced at $68.95, sales will be 20,000 units/week. Market research has found that for each decrease in price of $1/unit, sales should increase by 400 units/week. This product costs $14.00/unit to make and that plant has weekly overhead costs of $12,500. Let x represent the number of units produced and then sold each week. a) Develop the linear demand function. b) Develop the revenue function.[ Note that Revenue = Price per item * Number of items sold] c) Develop the linear cost function. d) Develop the profit function. e) Develop the marginal revenue function. f) Develop the marginal profit function. g) What price should they charge for an iPinch in order to maximize their weekly profits?

Answer 1

(a)

Linear demand: P = a - bQ

When P = 68.95, Q = 20,000

68.95 = a - 20,000b..........(1)

When P = 67.95, Q = 20,400

67.95 = a - 20,400b..........(2)

(2) - (1) yields:

400b = 1

b = 0.0025

a = 68.95 + 20,000 x 0.0025 = 68.95 + 50 = 118.95

Demand: P = 118.95 - 0.0025Q

(b)

Revenue (R) = PQ

R = 118.95Q - 0.0025Q²

(c)

Cost: C = FC + Q x AVC

C = 12,500 + 14Q

(d)

Profit = R - C

Profit (Z) = 118.95Q - 0.0025Q² - 12,500 + 14Q = 132.95Q - 0.0025Q² - 12,500

(e)

MR = dR/dQ = 118.95 - 0.005Q

(f)

Marginal profit = dZ/dQ = 132.95 - 0.005Q

(g)

Profit is maximized when Marginal profit = 0

132.95 - 0.005Q = 0

0.005Q = 132.95

Q = 26,590

P = 118.95 - 0.0025 x 26,590 = 118.95 - 66.475 = 52.475