# Determining the optimal price and output combination

1. A firm under monopolistic competition faces the demand curve: P = 500 - 12.5Q. The firm's marginal cost is MC = 200 + 5Q.

a. Find the firm's profit-maximizing output and price.

b. Assuming that the firm is at its long-run equilibrium position, estimate total revenue, total cost, and total profit.

2. A monopolist faces the demand curve: P = 8,400 - 3Q, and has long-run average cost

LAC = 900 - 3Q + Q2.

a. Derive equations for revenue, marginal revenue, total cost, and marginal cost.

b. Determine the monopolist's profit-maximizing output, price, and profit.

3. Industry demand for a good is given by: P = 60 - .5Q. The industry's long-run cost is $10 per unit: LAC = LMC = $10.

a. A monopolist controls the industry. Find its output and price.

b. Instead, suppose that the same industry is perfectly competitive. Find the long-run equilibrium price and output. Comment on the differences between the monopoly and competitive results.

4. A 5-member commodity cartel faces the demand curve: P = 60 - .4Q. Each member can produce output at (constant) LAC = LMC = $20 per unit.

a. Describe how the cartel should operate to maximize its total group profit.

b. Under the group profit-maximizing cartel agreement in part a, one member produces 10 units of output. It is tempted to secretly increase its output to 20 units. Assess this strategy and comment on cartel stability.

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#### Solution Preview

1. A firm under monopolistic competition faces the demand curve: P = 500 - 12.5Q. The firm's marginal cost is MC = 200 + 5Q.

a. Find the firm's profit-maximizing output and price.

P=500-12.5Q

Total Revenue=TR=P*Q=(500-12.5Q)*Q=500Q-12.5Q^2

Marginal Revenue=dTR/dQ=500-25Q

For profit-maximizing output, put MR=MC

500-25Q=200+5Q

30Q=300

Q=10

P=500-12.5Q=500-12.5*10=375

Optimal output=10 units

Optimal price=$375 per unit

b. Assuming that the firm is at its long-run equilibrium position, estimate total revenue, total cost, and total profit.

Total Revenue=TR=P*Q=375*10=$3750

In long run equilibrium P=Average cost=$375

Total Cost=Q*AC=10*375=$3750

Profit=Total Revenue-Total Cost=3750-3750=0

2. A monopolist faces the demand curve: P = 8,400 - 3Q, and has long-run average cost

LAC = 900 - 3Q + Q2.

a. Derive equations for revenue, marginal revenue, total cost, and marginal cost.

P=8400-3Q

Total ...

#### Solution Summary

Solutions depict the methodology to find optimal output and price levels in different market structures.