Question

A firm has two plants and wishes to maximize profits. The marginal cost curves for the two plants are: MC₁ = 2Q₁ and MC₂ = 3Q₂. The demand is P = 100 - .4Q. To maximize profits, how much output should be produced in plant#1 and plant#2, respectively?

A.       Q = 40;10.

B.       Q = 10; 40.

C.       Q = 20; 30.

D.       Q = 30; 20.

E.        None of the above.

Answer 1

MC1 = 2Q1 and MC2 = 3Q2

The demand is P = 100 - 0.4Q

For MR:

Calculate TR= P*Q= 100-0.4Q2

Differentiate TR with respect to Q to calculate MR

MR=100-0.8Q= 100-0.8Q1-0.8Q2

Maximizing profit condition:

For plant 1: MR=MC1

100-0.8Q1-0.8Q2=2Q1

2.8Q1+0.8Q2=100 Equation 1

For plant 2:

100-0.8Q1-0.8Q2=3Q2

0.8Q1+3.8Q2=100 Equation 2

Solve equation 1 and 2:

Multiply 0.8 in equation 1 and 2.8 in equation 2 and then subtract them:

2.24Q1+0.64Q2-2.24Q1-10.64Q2= 80-280

-10Q2= -200

Q2= 20

Put Q2=20 in equation 1:

2.8Q1+0.8(20)=100

2.8Q1= 84

Q1= 30

Correct answer is D. Q = 30; 20.