Competitive Firm Output And Profit Analysis

Understanding the Competitive Firm's Cost Structure

In this economic analysis, we delve into the decision-making process of a competitive firm operating in a perfectly competitive market. Specifically, we examine a firm with a total cost function of C = 450 + 15q + 2q² and a marginal cost function of MC = 15 + 4q. The total cost function represents the overall expenses incurred by the firm in producing a certain quantity (q) of output. It comprises two main components: fixed costs and variable costs. In this case, the fixed cost is $450, which remains constant regardless of the output level. The variable cost, on the other hand, depends on the quantity produced and is represented by the term 15q + 2q². This indicates that the variable cost increases as the firm produces more output. Marginal cost (MC), a crucial concept in economics, signifies the additional cost incurred by producing one more unit of output. For this firm, the marginal cost function is given by MC = 15 + 4q. This equation reveals that the marginal cost increases with the quantity produced. This phenomenon is known as diminishing returns, where the additional cost of producing an extra unit rises as output expands. The market price, denoted by P, is a crucial factor influencing the firm's production decisions. In a perfectly competitive market, the firm is a price taker, meaning it cannot influence the market price. The firm must accept the prevailing market price, which in this case is $115 per unit. To maximize its profit, the firm must determine the optimal level of output to produce at this given price.

Output Level Determination and Profit Maximization

To determine the level of output produced by the firm, we must consider the fundamental principle of profit maximization in a competitive market. A firm maximizes its profit by producing the quantity of output where its marginal cost (MC) equals the market price (P). This rule stems from the fact that as long as the marginal cost of producing an additional unit is less than the market price, the firm can increase its profit by producing that unit. Conversely, if the marginal cost exceeds the market price, the firm would incur a loss by producing that unit. Therefore, the profit-maximizing output level occurs where the two are equal. In our scenario, the market price is given as $115 per unit, and the marginal cost function is MC = 15 + 4q. To find the optimal output level, we set the marginal cost equal to the market price and solve for q: 115 = 15 + 4q. Subtracting 15 from both sides, we get 100 = 4q. Dividing both sides by 4, we find the profit-maximizing output level: q = 25 units. This result indicates that the firm will maximize its profit by producing 25 units of output. Now that we have determined the output level, we can calculate the firm's profit. Profit is defined as the difference between total revenue (TR) and total cost (TC): Profit = TR - TC. Total revenue is the product of the market price and the quantity sold: TR = P * q. In this case, TR = $115 * 25 = $2875. Total cost is given by the total cost function: TC = 450 + 15q + 2q². Substituting the output level of 25 units, we get TC = 450 + 15(25) + 2(25)² = 450 + 375 + 1250 = $2075. Finally, we calculate the profit: Profit = $2875 - $2075 = $800. Therefore, the firm's profit-maximizing output level is 25 units, and the corresponding profit is $800.

Shutdown Point and Long-Run Equilibrium Price

To determine the shutdown price, it's vital to first understand the concept of the shutdown point. A firm's shutdown point is the price level below which the firm will cease production in the short run. This occurs when the market price falls below the firm's minimum average variable cost (AVC). The average variable cost is calculated by dividing the total variable cost by the quantity of output. In our case, the total variable cost is 15q + 2q², so the average variable cost is AVC = (15q + 2q²)/q = 15 + 2q. To find the minimum AVC, we can take the derivative of AVC with respect to q and set it equal to zero: d(AVC)/dq = 2 = 0. However, in this case, the derivative is a constant (2), indicating that the AVC is always increasing with output. This implies that the minimum AVC occurs at the lowest possible output level, which is essentially zero. However, this isn't a practical shutdown point. A more accurate approach to finding the shutdown price is to recognize that it occurs where marginal cost (MC) intersects the average variable cost (AVC). This is because at the minimum point of AVC, the MC curve will cross it. Setting MC equal to AVC, we have: 15 + 4q = 15 + 2q. Solving for q, we get 2q = 0, which implies q = 0. This result is not particularly helpful in determining the shutdown price. However, we can infer that the shutdown price would be the AVC at the quantity where MC intersects AVC. Since AVC is always increasing, there isn't a clear shutdown point in the typical sense where the firm would cease production immediately. Instead, the firm would incur losses at any price below its minimum average total cost (ATC). To determine the long-run equilibrium price, we need to consider the concept of free entry and exit in a perfectly competitive market. In the long run, firms can enter or exit the market in response to profit opportunities. If firms are earning positive economic profits, new firms will be attracted to enter the market, increasing the market supply and driving down the market price. Conversely, if firms are incurring losses, some firms will exit the market, decreasing the market supply and driving up the market price. This process of entry and exit continues until firms are earning zero economic profit, which occurs when the market price equals the minimum average total cost (ATC). The average total cost (ATC) is calculated by dividing the total cost by the quantity of output: ATC = TC/q = (450 + 15q + 2q²)/q = 450/q + 15 + 2q. To find the minimum ATC, we can take the derivative of ATC with respect to q and set it equal to zero: d(ATC)/dq = -450/q² + 2 = 0. Solving for q, we get 2 = 450/q², which implies q² = 225, and therefore q = 15 (we take the positive root since quantity cannot be negative). Now, we can find the minimum ATC by substituting q = 15 into the ATC equation: ATC = 450/15 + 15 + 2(15) = 30 + 15 + 30 = $75. Therefore, the long-run equilibrium price in this market is $75 per unit. At this price, firms will earn zero economic profit, and there will be no incentive for entry or exit.

In conclusion, this analysis provides a comprehensive understanding of the decision-making process of a competitive firm. By equating marginal cost to market price, the firm determines its profit-maximizing output level. The concept of the shutdown point helps identify the price below which the firm will cease production in the short run. In the long run, the free entry and exit of firms drive the market price towards the minimum average total cost, ensuring zero economic profit for firms in the industry.