Answer to Question #171507 in Microeconomics for Tefera Alemu

Question

Consider a monopoly firm that produce two products Q1&Q2. Suppose that the demands facing the firm as follow:

P1=55-Q1-Q2

P2=70-Q1-2Q2

And Tc function= 

Q1(square)+Q1Q2+Q2(square)

then optimize the profit function, determine the optimal level of output ,price and profit

Answer 1

The level of output is optimal at such level for which MR = MC.

MR1 = TR1'(Q1) = 55 - 2Q1 - Q2,

MR2 = TR2'(Q2) = 70 - Q1 - 4Q2,

MC1 = TC1'(Q1) = 2Q1 + Q2,

MC2 = TC2'(Q2) = 2Q2 + Q1.

We have the system of two equations:

55 - 2Q1 - Q2 = 2Q1 + Q2,

70 - Q1 - 4Q2 = 2Q2 + Q1.

4Q1 + 2Q2 = 55,

2Q1 + 6Q2 = 70 or 4Q1 + 12Q2 = 140.

If we substract the first equation from the second, then:

10Q2 = 85,

Q2 = 8.5 units,

4Q1 + 17 = 55,

Q1 = 9.5 units.

P1 = 55 - 9.5 - 8.5 = 37,

P2 = 70 - 9.5 - 2×8.5 = 43.5.

Total profit is:

$TP = P1Q1 + P2Q2 - (Q1^2+Q1Q2+Q2^2) = 37×9.5 + 43.5×8.5 - (9.5^2+9.5×8.5+8.5^2) = 478.$