Question

A monopolist's inverse demand function is estimated as P=100−2Q. The company produces output at two facilities; the marginal cost of producing at facility 1 is MC1​(Q1​)=4Q1​, 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)=___−____Q1 − ____​Q2​ b. Determine the profit-maximizing level of output for each facility. Instructions: Round your response to two decimal places. Output for facility 1: Output for facility 2: c. Determine the profit-maximizing price. Instructions: Round your response to the nearest penny (two decimal places).

Answer 1

The monopolist determines the profit-maximizing outputs and price by setting marginal revenue equal to marginal cost for each facility and using the inverse demand function to find the price.

Step-by-step explanation:

The student has asked about the profit-maximizing level of output and price for a monopolist with a given inverse demand function and marginal costs for producing at two facilities. Calculating the marginal revenue function and determining the equilibrium quantity and price involve both mathematical analysis and economic principles.

Calculating the Marginal Revenue Function

To find the monopolist's marginal revenue (MR) function, we first need to derive the total revenue (TR) function from the inverse demand function P = 100 - 2Q. The TR function is P times Q, which gives TR = (100 - 2Q)Q = 100Q - 2Q2. Then, the MR function is the derivative of the TR function with respect to Q, which yields MR = 100 - 4Q.

Determining Profit-Maximizing Output for Each Facility

The profit-maximizing output occurs where MR equals the marginal cost (MC) for each facility. Setting MR = MC for each facility and solving for Q1 and Q2 will give us the profit-maximizing outputs. For facility 1: 100 - 4Q = 4Q1, and for facility 2: 100 - 4Q = 2Q2. These equations will give us the values of Q1 and Q2 once we factor in that Q = Q1 + Q2.

Profit-Maximizing Price

The monopolist will charge the price corresponding to the aggregate quantity (Q) on their perceived demand curve, which is found by substituting the value of Q into the inverse demand function.