Question

A monopolist’s inverse demand function is P = 150 – 3Q. The company produces output at two facilities; the marginal cost of producing at facility 1 is MC1(Q1) = 6Q1, and the marginal cost of producing at facility 2 is MC2(Q2) = 2Q2.a. Provide the equation for the monopolist’s marginal revenue function. (Hint: Recall that Q1 + Q2 = Q.)MR(Q) = 150 - 6 Q1 - 3 Q2b. Determine the profit-maximizing level of output for each facility.Output for facility 1: Output for facility 2: c. Determine the profit-maximizing price.$

Answer 1

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 = $150Q - 3Q^{2}$

Where $Q = Q_{1} + Q_{2}$

$MR = \frac{\delta T.R}{\delta (Q_{1}+ Q_{2})}$

(a) MR = 150 - 6Q

$MR = 150 - 6(Q_{1} + Q_{2})$

(b) Since we know that profit maximizing condition is given as :

MR = MC

Therefore , profit maximizing condition for facility 1 is

$150 - 6(Q_{1} + Q_{2})$ = $6Q_{1}$

$150 - 12Q_{1} - 6Q_{2}$

Similary profit maximizing condition for facility 2 is

$150 - 6(Q_{1} + Q_{2})$ = $2Q_{2}$

$150 - 6Q_{1} - 8Q_{2}$

Now, evaluating these two equations. We get ;

$150 - 12Q_{1} - 6Q_{2}$ - $150 - 6Q_{1} - 8Q_{2}$

$Q_{2} = 3Q_{1}$

Therefore, the profit maximizing level of output for facility 1 is

$Q_{1} = 5$

$Q_{2} = 15$

(c)The profit maximizing price is

P = 150 - 3Q

$P = 150 - 3(Q_{1}+Q_{2})$

$P = 150 - 3(5 + 15)$

$P = 150 - 60$

P = 90