Question

Lend products limited manufactures two products A and B. Experimental sales programs have shown that when the price per unit of A is p, the number of x units of A that will be sold is given by X = 12 - P/3, called the demand function for A. The demand function for B is Y = 8 - Q/5 where q is the price for each unit of B. In addition, the cost C of manufacturing and selling x units of A and y units of B is given by C = X² + 2XY + 3Y².

Determine the prices to set per unit of A and B in order to maximize profit and the maximum profit.

Answer 1

Total cost, C = X² + 2XY + 3Y²

Unit A

Demand function, X = 12 – P/3

Inverse demand function, P = 36 – 3X

Unit B

Demand function, Y = 8 – Q/5

Inverse demand function, Q = 40 – 5Y

Total revenue = P₁Q₁ + P₂Q₂ = PX + QY

Total revenue = (36-3X)X + (40-5Y)Y

Profit = Total revenue – Total cost

Profit = (PX + PQ) – Tocal cost

π = [(36-3X)X + (40 – 5Y)Y] – [X² + 2XY + 3Y²]

π = 36X – 3x² + 40Y – 5Y² – X² – 2XY – 3Y²

π = 36X – 4X² + 40 – 8Y² – 2XY

∂π/∂x = 36 - 8X - 2Y = 0

8X = 36 – 2Y

X = 4.5 – 0.25Y

∂π/∂x = 40 - 16Y - 2X = 0

2X = 40 – 16Y

X = 20 – 8Y

Thus, X = X

4.5 – 0.25Y = 20 – 8Y

7.75Y = 15.5

Y = 2

X = 20 – 8Y

X = 20 – 8(2)

X = 4

Price of Unit A

P = 36 – 3X

P = 36 – 3(4)

P = 24

Price of Unit B

Q = 40 – 5Y

Q = 40 – 5(2)

Q = 30

Maximum Profit

Profit = Total revenue – Total cost

Π = 36X – 4X² + 40Y – 8Y² – 2XY

Π= 36(4) – 4(4²) + 40(2) – 8(2²) – 2(4)(2)

Π = 112

Profit maximization is the short run or long run process by which a firm may determine the price, input and output levels that lead to the highest profit.