Problem 08-07 You Are The Manager Of A Monopolistically Comp

Problem 08-07 You are The Manager Of A Monopolistically Competitive Fir

You are the manager of a monopolistically competitive firm, and your demand and cost functions are given by Q = 36 – 4 P and C ( Q ) = 124 – 16 Q + Q 2. a. Find the inverse demand function for your firm’s product. P = - Q b. Determine the profit-maximizing price and level of production. Instruction: Price should be rounded to the nearest penny (two decimal places). Price: $ Quantity: c. Calculate your firm’s maximum profits. Instruction: Your response should appear to the nearest penny (two decimal places). $ d. What long-run adjustments should you expect? Explain. Neither entry nor exit will occur. Exit will occur until profits rise sufficiently high. Entry will occur until profits are zero.

Paper For Above instruction

Introduction

Monopolistic competition is a market structure characterized by many firms selling differentiated products, easy entry and exit, and some degree of market power. In such markets, individual firms face a downward-sloping demand curve and must determine the optimal price and quantity to maximize profits. This paper aims to analyze a specific scenario where a firm faces given demand and cost functions, determine the profit-maximizing level of output and price, calculate maximum profits, and discuss the long-run implications within the context of monopolistic competition.

Inverse Demand Function

The demand function provided is Q = 36 – 4P. To find the inverse demand function, we solve for P in terms of Q. Starting with the demand function:

Q = 36 – 4P

Rearranging for P:

4P = 36 – Q

P = (36 – Q) / 4

Hence, the inverse demand function is:

P = 9 – (Q / 4)

This function indicates that the price decreases as quantity increases, reflecting typical demand behavior in monopolistic competition.

Profit-Maximizing Price and Quantity

The firm’s total cost function is given by C(Q) = 124 – 16Q + Q². To maximize profit, we need to find the quantity where marginal revenue (MR) equals marginal cost (MC). Since the demand function is linear, the total revenue (TR) is:

TR = P × Q = (9 – Q/4) × Q = 9Q – (Q²)/4

Marginal revenue (MR) is the derivative of TR with respect to Q:

MR = d(TR)/dQ = 9 – (Q/2)

Similarly, the marginal cost (MC) is the derivative of total cost C(Q):

MC = d(C)/dQ = -16 + 2Q

Setting MR equal to MC to find the profit-maximizing quantity:

9 – (Q/2) = -16 + 2Q

Multiplying through by 2 to clear the denominator:

18 – Q = -32 + 4Q

Solving for Q:

18 + 32 = 4Q + Q

50 = 5Q

Q = 10

Substituting Q = 10 into the inverse demand function to find the price:

P = 9 – (10/4) = 9 – 2.5 = 6.50

Thus, the profit-maximizing price is approximately $6.50, and the optimal quantity is 10 units.

Maximum Profits Calculation

To compute maximum profits, we need total revenue and total costs at Q = 10:

Total revenue:

TR = P × Q = 6.50 × 10 = $65.00

Total cost:

C(10) = 124 – 16(10) + (10)² = 124 – 160 + 100 = 64

Profit:

π = TR – C = 65.00 – 64.00 = $1.00

Therefore, the firm’s maximum profit is approximately $1.00.

Long-Run Adjustments

In the context of monopolistic competition, firms tend to earn zero economic profit in the long run due to free entry and exit. If current profits are positive, new firms will enter the market, increasing industry supply and decreasing prices until profits are driven to zero. Conversely, if profits are negative, some firms will exit, reducing supply and increasing prices until remaining firms break even. Since the calculated profit here is low but positive, we can expect that in the long run, entry will occur until profits are eliminated. This equilibrium will be characterized by firms earning just enough to cover their average costs, with no incentive for further entry or exit.

Hence, in the long run, the firm will adjust its output and price such that economic profits are zero, aligning with the typical characteristics of monopolistically competitive markets.

Conclusion

The analysis indicates that the profit-maximizing output is 10 units, with a price of approximately $6.50. The maximum profit achievable under the current conditions is about $1.00. Over time, market dynamics will drive profits toward zero in the long run, due to free entry and exit, leading to an equilibrium where firms cover their costs but earn no excess profit. This outcome underscores the competitive pressures existing within monopolistic markets, shaping firm strategies and market outcomes.

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