Respuesta :
Answer:
Given : Inverse demand function : P = 150 - 3Q
Marginal cost of producing at facility 1: MC1(Q1) = 6Q1
Marginal cost of producing at facility 2: MC2(Q2) = 2Q2
Here we will first find Total Revenue.
i.e. Total Revenue(T.R) = P*Q
T.R(Q) = (150 - 3Q)*Q = [tex]150Q - 3Q^{2}[/tex]
Where [tex]Q = Q_{1} + Q_{2}[/tex]
[tex]MR = \frac{\delta T.R}{\delta (Q_{1}+ Q_{2})}[/tex]
(a) MR = 150 - 6Q
[tex]MR = 150 - 6(Q_{1} + Q_{2})[/tex]
(b) Since we know that profit maximizing condition is given as :
MR = MC
Therefore , profit maximizing condition for facility 1 is
[tex]150 - 6(Q_{1} + Q_{2})[/tex] = [tex]6Q_{1}[/tex]
[tex]150 - 12Q_{1} - 6Q_{2}[/tex]
Similary profit maximizing condition for facility 2 is
[tex]150 - 6(Q_{1} + Q_{2})[/tex] = [tex]2Q_{2}[/tex]
[tex]150 - 6Q_{1} - 8Q_{2}[/tex]
Now, evaluating these two equations. We get ;
[tex]150 - 12Q_{1} - 6Q_{2}[/tex] - [tex]150 - 6Q_{1} - 8Q_{2}[/tex]
[tex]Q_{2} = 3Q_{1}[/tex]
Therefore, the profit maximizing level of output for facility 1 is
[tex]Q_{1} = 5[/tex]
[tex]Q_{2} = 15[/tex]
(c)The profit maximizing price is
P = 150 - 3Q
[tex]P = 150 - 3(Q_{1}+Q_{2})[/tex]
[tex]P = 150 - 3(5 + 15)[/tex]
[tex]P = 150 - 60[/tex]
P = 90
