A monopolist firm faces a demand curve: Q = 900 – 3P, and the costs of its two plants are:
C1 100Q1 50 and 2
2 C2 0.5Q2 10Q .
To analyze the profit-maximizing output and pricing decision of a monopolist firm, we need to follow these steps:
Step 1: Determine the monopolist's marginal revenue (MR) function.
The monopolist's marginal revenue is the change in total revenue resulting from selling one additional unit of output. In this case, the total revenue (TR) can be calculated as the product of price (P) and quantity (Q): TR = P * Q.
To find the monopolist's marginal revenue, we differentiate the total revenue function with respect to quantity: MR = d(TR)/dQ.
Step 2: Set the marginal revenue equal to marginal cost (MC) to find the profit-maximizing output level.
The monopolist maximizes profit by producing at the level where marginal revenue equals marginal cost. In this case, we have two plants, so we need to consider the marginal cost of each plant.
Step 3: Solve for the equilibrium price.
Step 1: Determine the monopolist's marginal revenue (MR) function.
To find the marginal revenue (MR), we differentiate the total revenue equation (TR) with respect to quantity (Q):
TR = P * Q (Total Revenue)
MR = d(TR)/dQ
Since the demand curve is given as Q = 900 - 3P, we can solve for P in terms of Q and substitute it into the total revenue equation to find MR.
Q = 900 - 3P
Q + 3P = 900
3P = 900 - Q
P = (900 - Q)/3
Now, we substitute this expression for P into the total revenue equation:
TR = P * Q
TR = [(900 - Q)/3] * Q
TR = (900Q - Q^2)/3
Finally, we differentiate the total revenue function with respect to quantity to find the marginal revenue:
MR = d(TR)/dQ
MR = (900 - 2Q)/3
Step 2: Set the marginal revenue equal to marginal cost (MC) to find the profit-maximizing output level.
The monopolist's profit-maximizing output level is where marginal revenue (MR) equals marginal cost (MC). In this case, we have two plants with different cost functions, so we need to consider the marginal cost of each plant.
The marginal cost (MC) is the change in total cost resulting from producing one additional unit of output. The marginal cost functions for the two plants are given as follows:
C1 = 100Q1 + 50
C2 = 0.5Q2 + 10Q
To find the profit-maximizing output levels, we set the marginal cost of each plant equal to the marginal revenue:
MR = MC1 for Plant 1
(900 - 2Q) / 3 = d(C1) / dQ
MR = MC2 for Plant 2
(900 - 2Q) / 3 = d(C2) / dQ
To find the optimal quantities (Q1 and Q2), we solve these two equations simultaneously. Substitute the cost functions into the equations and solve for Q1 and Q2.
Step 3: Solve for the equilibrium price.
Once we have the optimal quantities (Q1 and Q2), we can substitute them into the demand curve equation to determine the equilibrium price (P):
Q = 900 - 3P
P = (900 - Q)/3
Substitute the values of Q1 and Q2 into the price equation to find the equilibrium price.