Suppose the multiplant monopolist having a linear demand function Q=50-p is operating in two plants each has marginal cost function as mc1=20+2Q1 and mc2=10+5Q2 respectively. 1.What is total output and price. 2.What is profit of the monopolist?

To find the total output and price, we need to determine the quantities produced by each plant and the corresponding prices set by the monopolist in each plant.

1. Total Output and Price:
First, we need to find the equilibrium quantity and price for each plant. Equilibrium occurs when marginal cost equals marginal revenue.

For Plant 1 (Q1):
Marginal cost (MC1) = 20 + 2Q1
Marginal revenue (MR1) = Demand function = Q = 50 - P

Setting MC1 equal to MR1:
20 + 2Q1 = 50 - P

Next, we solve for P:
P = 50 - 20 - 2Q1
P = 30 - 2Q1

Now, we can substitute P back into MR1 to find equilibrium quantity (Q1):
Q1 = 50 - P
Q1 = 50 - (30 - 2Q1)
Q1 = 80 - 2Q1
3Q1 = 80
Q1 = 80/3

Substituting Q1 back into the demand function to find price (P):
P = 30 - 2Q1
P = 30 - 2(80/3)
P = 30 - 160/3
P = 90/3 - 160/3
P = -70/3

Since the price cannot be negative, we know that Plant 1 does not produce any output (Q1 = 0) due to the high costs.

For Plant 2 (Q2):
Marginal cost (MC2) = 10 + 5Q2

Setting MC2 equal to MR2:
10 + 5Q2 = 50 - P

Next, we can solve for P:
P = 50 - 10 - 5Q2
P = 40 - 5Q2

Now, we substitute P back into MR2 to find equilibrium quantity (Q2):
Q2 = 50 - P
Q2 = 50 - (40 - 5Q2)
Q2 = 90 - 5Q2
6Q2 = 90
Q2 = 15

Substituting Q2 back into the demand function to find price (P):
P = 40 - 5Q2
P = 40 - 5(15)
P = 40 - 75
P = -35

Again, since the price cannot be negative, we know that Plant 2 also does not produce any output (Q2 = 0) due to the high costs.

As a result, neither Plant 1 nor Plant 2 produces any output in this scenario, and the price is negative (-70/3 and -35) indicating that the monopolist is unable to operate profitably.

2. Profit of the monopolist:
Since no output is produced, the profit of the monopolist is zero.

To find the total output and price, as well as the profit of the monopolist, we need to follow these steps:

Step 1: Find the individual equilibrium output for each plant.
- Apply the profit-maximizing condition by setting marginal cost (MC) equal to marginal revenue (MR).
- The marginal revenue is given by the derivative of the demand function, which is the negative coefficient of p in the demand equation.
For Q = 50 - p, the marginal revenue (MR) is -1.
- Set MC1 = MR (= -1) and solve for Q1.
20 + 2Q1 = -1
2Q1 = -21
Q1 = -21/2
Q1 = -10.5
However, since we can't have a negative quantity, we set Q1 to zero.

- Set MC2 = MR (= -1) and solve for Q2.
10 + 5Q2 = -1
5Q2 = -11
Q2 = -11/5
Q2 ≈ -2.2
Again, since we can't have a negative quantity, we set Q2 to zero.

Therefore, the individual equilibrium outputs for each plant are Q1 = 0 and Q2 = 0.

Step 2: Find the total output and price.
- The total output, denoted as Q, is the sum of the individual outputs for each plant.
Q = Q1 + Q2
Q = 0 + 0
Q = 0

- To find the price, substitute the total output (Q = 0) into the demand function Q = 50 - p and solve for p.
0 = 50 - p
p = 50

Therefore, the total output (Q) is 0, and the price (p) is 50.

Step 3: Find the profit of the monopolist.
- The profit (Π) is calculated using the formula: Π = (p - MC) * Q.
- Substitute the values into the formula:
Π = (50 - (20 + 2Q1)) * Q1 + (50 - (10 + 5Q2)) * Q2
Π = (50 - (20 + 2(0))) * 0 + (50 - (10 + 5(0))) * 0
Π = 0

Therefore, the profit of the monopolist is 0.