Answer to Question #223358 in Economics of Enterprise for Hafizh

Question

A monopolistic producer of two goods, G1 and G2, has a total cost function
TC = 5Q1 + 10Q2
where Q1 and Q2 denote the quantities of G1 and G2, respectively. If P1 and P2 denote
the corresponding prices, then the demand equations are
P1 = 50 − Q1 − Q2
P2 = 100 − Q1 − 4Q2
Use Lagrange Multiplier to find the maximum profit if the firm’s total costs are fixed at $100. Estimate the new optimal profit if total costs rise to $101

Answer 1

Solution:

Formulate the profit function:

Profit = TR – TC

Total revenue = P $\times$ Q

Since cost is fixed at 100, maximizing profit is the same as maximizing profits.

So, the total revenue will be maximized subject to the total cost of 100.

 

L (Q1, Q2, λ) = P1Q1 + P2Q2 – λ (5Q1 + 10Q2 – 100)

Profit = ((50 – Q1 – Q2) Q1 + (100 – Q1 – 4Q2) Q2) – (5Q1 + 10Q2 – 100)

Profit = (50Q1 – Q1² – Q1Q2 + 100Q2 – Q1Q2 – 4Q²) – (5Q1 + 10Q2 – 100)

LQ1 = $\frac{\partial L} {\partial Q1}$ = 50 – 2Q1 – Q2 – Q2 - 5λ = 0

                        = 50 – 2Q1 – 2Q2 = 5λ ………..(1)

LQ2 = $\frac{\partial L} {\partial Q2}$ = -Q1 + 100 – Q1 – 8Q2 – 10λ = 0

                        = 100 – 2Q1 – 8Q2 - 10λ = 0 (divide by two)

                        = 50 – Q1 – 4Q2 - 5λ

                        = 50 – Q1 – 4Q2 = 5λ ………….(2)

Lλ = $\frac{\partial L} {\partial \lambda }$ = 5Q1 + 10Q2 – 100 = 0

Equate equations 1 and 2:

50 – 2Q1 – 2Q2 = 50 – Q1 – 4Q2

-2Q1 + Q1 = -4Q2 + 2Q2

Q1 = 2Q2

Substitute in the total cost constraint to derive Q2:

TC = 5Q1 + 10Q2 – 100 = 0

5(2Q2) + 10Q2 – 100 = 0

20Q2 = 100

Q2 = 5

Q1 = 2Q2 = 2 $\times$ 5 = 10

Q1 = 10

P1 = 50 – Q1 – Q2 = 50 – 10 – 5 = 35

P2 = 100 – Q1 – 4Q2 = 100 – 10 – (4 $\times$5) = 100 – 10 – 20 = 70

Profit = (P1Q1 + P2Q2) – TC

Profit = (35$\times$10) + (70$\times$5) – 100

Profit = 350 + 350 – 100 = 600

Profit = 600

If total costs rise to 101:

Profit = (P1Q1 + P2Q2) – TC

Profit = (35$\times$10) + (70$\times$5) – 101

Profit = 350 + 350 – 101 = 599

Profit = 599