A Firm Sells Its Product In A Perfectly Competitive Market
A Firm Sells Its Product In A Perfectly Competitive Market Where Other
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 given by C(Q) = 50 + 10Q + 2Q². The assignment asks to determine (a) the optimal output quantity the firm should produce in the short run, (b) the price the firm should charge in the short run, and (c) the firm’s short-run profits.
Paper For Above instruction
In perfectly competitive markets, individual firms are price takers, meaning they accept the prevailing market price. The primary goal for a firm is to maximize profit, which involves choosing the level of output where marginal cost (MC) equals the market price, provided the firm covers its variable costs. This paper analyzes these aspects using the given cost function and market conditions to determine the optimal output, the short-run price, and the profits earned by the firm.
Determining the Optimal Output in the Short Run
The first step involves deriving the marginal cost (MC) from the total cost function C(Q) = 50 + 10Q + 2Q². Marginal cost is the additional cost incurred from producing one more unit of output, calculated as the derivative of total cost with respect to quantity Q:
MC = dC/dQ = 10 + 4Q.
In a perfectly competitive market, the profit-maximizing output occurs where marginal cost equals the market price, i.e., MC = P. Given the market price of $90, we set:
10 + 4Q = 90
Solving for Q:
4Q = 80
Q = 20
Therefore, the firm should produce 20 units of output in the short run to maximize profit.
Determining the Short-Run Price
In perfect competition, the firm is a price taker, which means it has no control over the market price. The question asks for the short-run price the firm should charge, which is given by the market condition and remains constant at $90 per unit. Thus, the price the firm charges per unit is $90.
Calculating the Firm’s Short-Run Profits
The profit (π) is calculated as total revenue (TR) minus total cost (C). Total revenue is the product of the market price and the quantity sold:
TR = P × Q = 90 × 20 = $1,800
Next, calculate the total cost for Q = 20:
C(20) = 50 + 10(20) + 2(20)² = 50 + 200 + 2(400) = 50 + 200 + 800 = $1,050
Hence, the profit is:
π = TR - C = 1,800 - 1,050 = $750
This indicates the firm earns a short-run profit of $750 when producing 20 units at the market price of $90.
Conclusion
In conclusion, the firm should produce 20 units to maximize its profit in the short run. The equilibrium price is set by the market at $90 per unit, and the firm’s short-run profit at this level of output is $750. These results align with the typical characteristics of perfect competition, where firms produce where marginal cost equals the market price and earn normal or abnormal profits depending on cost structures.
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