You Are The Manager Of A Monopoly And Your Demand And Cost

You Are The Manager Of A Monopoly And Your Demand And Cost Functions

You are the manager of a monopoly, and your demand and cost functions are given by P = 300 – 3 Q and C ( Q ) = 1,500 + 2 Q 2, respectively.

a. What price–quantity combination maximizes your firm’s profits? Price: $ Quantity: units

b. Calculate the maximum profits. $

c. What price–quantity combination maximizes revenue? Price: $ Quantity: units

d. Calculate the maximum revenues. $

A firm sells its product in a perfectly competitive market where other firms charge a price of $90 per unit. The firm’s total costs are C ( Q ) = 40 + 10 Q + 2 Q 2.

a. How much output should the firm produce in the short run? units

b. What price should the firm charge in the short run? $

c. What are the firm’s short-run profits? $

Paper For Above instruction

Analyzing monopoly and perfectly competitive market scenarios involves understanding the behavior of firms in response to demand and cost functions. This paper explores the profit-maximizing strategies of a monopolist given specific demand and cost functions and examines the short-run decisions of a firm operating in perfect competition.

Part A: Monopoly Profit Maximization

The demand function provided is P = 300 – 3Q, where P represents the price and Q the quantity sold. The cost function is C(Q) = 1500 + 2Q². To find the profit-maximizing output, the first step is to determine the firm's total revenue (TR) and total cost (TC). The total revenue function is TR = P × Q = (300 – 3Q)Q = 300Q – 3Q². The profit function π(Q) is then TR – TC = (300Q – 3Q²) – (1500 + 2Q²) = 300Q – 3Q² – 1500 – 2Q² = 300Q – 5Q² – 1500.

Maximizing Profits

To maximize profit, take the first derivative of the profit function with respect to Q and set it equal to zero:

dπ/dQ = 300 – 10Q = 0

Solving for Q gives Q = 30 units. Substituting Q = 30 back into the demand function provides the profit-maximizing price:

P = 300 – 3(30) = 300 – 90 = $210.

Thus, the profit-maximizing price-quantity combination is $210 at 30 units.

Maximum Profits

The maximum profit is obtained by substituting Q = 30 into the profit function:

π(30) = 300(30) – 5(30)² – 1500 = 9000 – 5(900) – 1500 = 9000 – 4500 – 1500 = $3000.

Part B: Revenue Maximization

Revenue is maximized when the marginal revenue (MR) equals zero. The MR function is derived from TR:

TR = 300Q – 3Q², so MR = d(TR)/dQ = 300 – 6Q.

Set MR = 0 to find the revenue-maximizing quantity:

300 – 6Q = 0 → Q = 50 units.

Corresponding price is:

P = 300 – 3(50) = 300 – 150 = $150.

Maximum revenue is TR at Q = 50:

TR = (150)(50) = $7500.

Part C: Perfect Competition Decision-Making

Given that the market price is $90, the firm’s cost function is C(Q) = 40 + 10Q + 2Q². The firm's short-run decision involves producing where marginal cost (MC) equals market price. The marginal cost is the derivative of the total cost:

MC = dC/dQ = 10 + 4Q.

Set MC = 90 to find the quantity:

10 + 4Q = 90 → Q = 20 units.

Price and Profit Calculation

Since the market price is given as $90, the firm will charge this in the short run. The profit is the difference between total revenue and total cost at Q = 20:

TR = 90 × 20 = $1800.

TC = 40 + 10(20) + 2(20)² = 40 + 200 + 2(400) = 40 + 200 + 800 = $1040.

Profit = TR – TC = $1800 – $1040 = $760.

Conclusion

This analysis illustrates the contrasting strategies in monopoly and perfect competition scenarios. The monopolist sets the price based on maximizing profits where marginal revenue equals marginal cost, resulting in higher prices and lower quantities than in perfect competition, where firms produce where price equals marginal cost. Understanding these behaviors is crucial for strategic decision-making in different market structures.

References

• Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance. McGraw-Hill Education.