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

The company is making money. Increasing production will increase average cost but also increase profit. Increase production.

Step 1: Calculate Total Revenue

To determine if the company is making or losing money, we first calculate total revenue. Total revenue is calculated by multiplying the number of units sold by the selling price per unit. Here, revenue is \( 1200 \times 20 = 24,000 \).

Step 2: Calculate Total Cost

Next, we calculate the total cost, which is the number of units produced times the average cost per unit. So, total cost is \( 1200 \times 2 = 2,400 \).

Step 3: Determine Profit or Loss

Profit or loss is determined by subtracting total cost from total revenue. Therefore, profit is \( 24,000 - 2,400 = 21,600 \). Since profit is positive, the company is making money.

Step 4: Assess the Impact of Increased Production on Average Cost

If production is increased, the average cost may increase or decrease. The marginal cost (\\(3 per pair) is greater than the current average cost (\\)2 per pair), indicating that producing more will increase the average cost.

Step 5: Assess the Impact of Increased Production on Profit

To determine the impact on profit, compare marginal revenue to marginal cost. Since slippers sell for \\(20 (marginal revenue) and marginal cost is \\)3, the profit from each additional slipper is positive, thus increasing production would increase total profit.

Step 6: Recommendation on Production

In light of the above analysis, while the average cost would increase, the positive marginal profit suggests that increasing production would still result in an increase in overall profit. Therefore, I recommend increasing production.

Key Concepts

Total Revenue

Total revenue is the total income generated from selling a product before any expenses are subtracted. In the case of slipper production, it is calculated by multiplying the number of slipper pairs sold by their selling price per unit. Since each pair sells for \(20 and the company sells 1200 pairs, the total revenue is computed as:

• Number of slippers: 1200 pairs
• Selling price per pair: \\)20

Thus, the total revenue is given by the formula:\[\text{Total Revenue} = 1200 \times 20 = 24,000 \]This figure represents the total inflow of cash from slipper sales each month, which is crucial for assessing the company’s financial performance.

Average Cost

Average cost is the total cost incurred by a company divided by the number of units produced. This metric helps to understand how much each unit being produced costs. For this slipper company:

• Total pairs produced: 1200
• Average cost per pair: \\(2

The average cost can be detailed by:\[\text{Average Cost} = \frac{ \text{Total Cost} }{ \text{Total Units Produced} } = \frac{2400}{1200} = 2\]If the company decides to produce more, since the marginal cost of an additional pair (\)3) is higher than the current average ($2), overall average cost would increase if production increases.

Profit Calculation

Profit is the financial gain obtained by the difference between total revenue and total costs. To find out if the company is making or losing money, we calculate as follows:\[\text{Profit} = \text{Total Revenue} - \text{Total Cost}\]Where the total revenue is \\(24,000 and the total cost is \\)2,400, thus:\[\text{Profit} = 24,000 - 2,400 = 21,600\]Since the resulting profit is a positive \$21,600, the company is making money, which suggests that the current operation is profitable.

Marginal Revenue

Marginal revenue is the additional income gained from selling one more unit of a product. It is vital for decisions regarding production increases. For every extra pair of slippers sold, the marginal revenue is constant at the price per unit:

• Price per slipper pair: \\(20

The marginal cost is the cost of producing one additional unit, which is \\)3 for this company. Thus, each additional slipper sold brings:\[\text{Marginal Profit} = \text{Marginal Revenue} - \text{Marginal Cost} = 20 - 3 = 17\]This positive marginal profit per additional pair implies that increasing production, even as average cost rises, can enhance overall profits due to the greater marginal benefit.