Monopoly Pricing Strategy with Single Consumer Group

Monopoly Pricing with a Single Consumer Group

This example illustrates how a monopolist maximizes profit when selling to a single group of consumers.

Scenario:

A market consists of two consumer groups (A and B) and a single monopoly producer. The cost of producing Q units for the monopolist is C(Q) = 5Q². While group A has an inverse demand function of p = 100 - q_A, the monopolist is restricted to selling only to group B, whose inverse demand function is p = 10 - q_B.

Objective:

Determine the profit-maximizing unit price and optimal profit for the monopolist using a uniform pricing strategy.

Solution:

1. Market Demand: Since the monopolist sells exclusively to group B, the market demand is represented by q_B = Q. Thus, the inverse demand function becomes p(Q) = 10 - Q.

2. Profit Function: The monopolist's profit (π) is calculated as total revenue (TR) minus total cost (TC): π(Q) = TR(Q) - TC(Q). We know TR(Q) = p(Q)Q and TC(Q) = 5Q².

3. Profit Maximization: To maximize profit, we find the quantity (Q) where the derivative of the profit function with respect to Q is zero:

   dπ(Q)/dQ = d(TR(Q))/dQ - d(TC(Q))/dQ = 10 - 10Q - 10Q = 0

   Solving for Q, we find Q = 1. This means the monopolist should produce and sell 1 unit.

4. Optimal Price: Substitute Q = 1 into group B's inverse demand function to find the profit-maximizing price:

   p(Q) = 10 - Q = 10 - 1 = 9

   Therefore, the optimal unit price is $9.

5. Optimal Profit: Calculate the monopolist's optimal profit by substituting Q = 1 and p = 9 into the profit function:

   π(Q) = TR(Q) - TC(Q) = p(Q)Q - C(Q) = (9)(1) - 5(1)² = $4

Conclusion:

The monopolist achieves maximum profit by selling 1 unit at a price of $9, resulting in a profit of $4. This example demonstrates how a monopolist leverages market power to determine the most profitable output and pricing strategy when dealing with a specific consumer group.