Problem 8 - Applied Calculus

Question

You are the manager of a firm that produces slippers that sell for \(\$ 20\) a pair. You are producing 1200 pairs of slippers each month, at an average cost of \(\$ 2\) each. The marginal cost at a production level of 1200 is \(\$ 3\) per pair. (a) Are you making or losing money? (b) Will increasing production increase or decrease your average cost? Your profit? (c) Would you recommend that production be increased or decreased?

Step-by-Step Solution

Answer

(a) Making money. (b) Increases average cost; increases profit. (c) Increase production.

Step 1: Calculate Total Revenue

The total revenue is calculated by multiplying the number of slippers produced by the selling price per slipper. Thus,\[ \text{Total Revenue} = 1200 \times 20 = \$24000. \]

Step 2: Calculate Total Cost

The total cost is calculated by multiplying the number of slippers produced by the average cost per slipper. Thus,\[ \text{Total Cost} = 1200 \times 2 = \$2400. \]

Step 3: Determine Profit or Loss

Profit is calculated by subtracting the total cost from the total revenue. Thus,\[ \text{Profit} = \text{Total Revenue} - \text{Total Cost} = 24000 - 2400 = \$21600. \]Since the profit is positive, the firm is making money.

Step 4: Analyze Effect of Increasing Production on Average Cost

Since the marginal cost at the current production level is \\(3, which is higher than the average cost of \\)2, increasing production will increase the average cost.

Step 5: Analyze Effect on Profit with Increased Production

The current profit for each additional slipper is the selling price minus the marginal cost: \( 20 - 3 = \$17. \) Since this profit is positive, increasing production will increase total profit despite the rise in average cost.

Step 6: Make Recommendation

Given that increasing production leads to higher total profit, I recommend increasing production despite the rise in average cost.

Key Concepts

Understanding Average Cost

The average cost of producing an item is found by dividing the total cost by the number of items produced. In this scenario, each pair of slippers has an average cost of \\(2. This means for each pair made, overall expenses average out to this amount.

Average cost helps in understanding how efficiently a company uses its resources. If the average cost is lower than the selling price, the company profits from each sale. Here, the slippers sell for \\)20, which is far higher than the \\(2 average cost, indicating a significant profit margin.

When a firm's marginal cost exceeds its average cost, like in this example where the marginal cost is \\)3, it suggests that producing one more unit will raise the average cost. Thus, in this scenario, increasing production will make each slipper slightly more expensive on average, hinting at less efficiency with added production.

Calculating Total Revenue

Total revenue is crucial as it reflects the money earned from selling goods. It's calculated simply by multiplying the quantity of goods sold by their price. Here, the firm sells 1200 pairs of slippers at \\(20 apiece, resulting in a total revenue of \\)24000.

This metric provides a big picture of a company's ability to generate sales income. But, it’s not the end-all measure. It must work with other calculations, like total costs, to truly grasp the financial standing. If your total revenue exceeds your total costs, the business is likely profitable.

Understanding sales trends and setting appropriate pricing can help maximize revenue, essential for financial health.

Profit Analysis Made Simple

Profit analysis involves figuring out if a business is making or losing money. It's the difference between total revenue and total costs. In this case, subtracting the total cost of \\(2400 from the total revenue of \\)24000 results in a healthy profit of \$21600.

This analysis helps businesses determine viability and guides in making better decisions. A positive profit indicates that the company’s activities are favorable, whereas a negative one suggests reevaluation might be needed.

Nonetheless, profit margins can vary significantly across industries, and a key to successful analysis is understanding industry benchmarks and striving for efficiency in costs.

Making a Production Decision

Choosing whether to increase or decrease production hinges upon factors like marginal cost and profit analysis. In this case, the marginal cost of \\(3 is notably less than the selling price of \\)20, suggesting that any additional production still leads to profitability. Each extra pair produced generates a profit of \$17.

When marginal cost exceeds average cost, it raises the average cost with increased production, potentially lowering overall profit margins. However, because the additional profit remains positive, it's wise to boost production.

This scenario shows increasing production is advantageous. It’s crucial, however, to also factor in market demand and capacity limitations before making such decisions to avoid overproduction.