Question

1. Suppose the inverse demand function for a monopolistically competitive firm’s product is given by

? = 100 − 2?
and the cost function is given by ?? = 52 + 4?

1. Determine the profit-maximizing price and quantity

2. Determine the maximum profits.

3. Can we say that this firm is operating in the long-run or short-run equilibrium at the equilibrium price and quantity?

2. Suppose the inverse demand for a monopolist’s product is given by ? = 110 −

The monopolist can produce output in two plants. The marginal cost of producing in plant 1 is ??! = 3?!, and the marginal cost of producing in plant 2 is ??" = 2?".

1. How much output should be produced in each plant to maximize profits?

2. What price should be charged for the product?

Answer 1

II. 1.Given, ? = 100 − 2?

We know Total Revenue (TR) = Price X quantity. Thus TR= (100-2Q)Q

TR= 100Q - 2Q2

Marginal Revenue is defined as the change in total revenue when one more unit is sold. Mathematically, MR= d/dq (TR). That is, MR= d/dq (100Q - 2Q2)

MR= 100 - 4Q

Given, ?? = 52 + 4?. Marginal Cost is defined as the change in total cost when one more unit of output is produced. Mathematically, MC= d/dq (TC)

MC= d/dq (52 + 4Q)

MC= 4

Equating MR= MC, 100 - 4Q= 4

On solving, Q= 24. Putting Q=24 in ? = 100 − 2?, we get Price= 52.

The profit-maximizing price is $52 and quantity is 24 units.

2. Maximum profits= TR - TC = (100Q - 2Q2) - (52+ 4Q)

Putting Q=24, we get : 1248 - 118= 1130

3. In monopoly, long- run equilibrium occurs at the point of intersection between the monopolist's marginal revenue (MR) and long-run marginal cost (LMC) curves. That is, LAC = LMC (at the minimum point of the LAC curve).

Since, MC=4

Average Cost (AC) is the average cost per unit of output. AC is calculated by TC/q. That is, (52 + 4Q)/Q= 52/Q + 4

Putting Q=24, AC= 2.16 + 4 = 6.16

Since AC > MC, the firm is in short run equilibrium.

2. The inverse demand function is incomplete. The monopolist will produce where MC1= MC2. MC1 is marginal cost of first plant and MC2 is marginal cost of second plant. The total output would be then q1 + q2.