Problem 8 - Applied Calculus

Question

Marginal revenue and marginal cost are given in the following table. Estimate the production levels that could maximize profit. Explain. $$ \begin{array}{c|c|c|c|c|c|c} \hline q & 1000 & 2000 & 3000 & 4000 & 5000 & 6000 \\ \hline M R & 78 & 76 & 74 & 72 & 70 & 68 \\ \hline M C & 100 & 80 & 70 & 65 & 75 & 90 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Maximize profit at a production level of 4000 units.

Step 1: Understanding the Problem

We need to find the production level where the marginal revenue (MR) is closest to marginal cost (MC), as this is where profit is maximized. An examination of the table for each production level will help identify this point.

Step 2: Review the Table Data

The table provides marginal revenue and marginal cost for different production levels. We'll compare MR and MC at each production quantity to find when MR equals MC, or when MR is just above MC before it starts to drop below MC.

Step 3: Analyze Marginal Revenue and Marginal Cost

Check each production level: - At 1000 units, MR (78) < MC (100) - At 2000 units, MR (76) < MC (80) - At 3000 units, MR (74) > MC (70) - At 4000 units, MR (72) > MC (65) - At 5000 units, MR (70) < MC (75) - At 6000 units, MR (68) < MC (90) The closest production level at which MR still exceeds MC is 4000 units.

Step 4: Identify Profit-Maximizing Output

At 4000 units, MR exceeds MC and profit is not yet diminishing compared to MC at 5000 units. Production at 3000 units is also a possibility if increments are expected to turn negative shortly after.

Step 5: Conclude Optimal Production

The production level that maximizes profit is where MR approaches MC from above without crossing below it. Between 3000 and 4000 units, the best estimate would be around 4000 units as it is the largest production where MR > MC.

Key Concepts

Understanding Marginal Revenue

Marginal Revenue (MR) is the additional revenue a seller gains from selling one more unit of a product. This concept is vital in determining how production decisions affect overall revenue. It tells us how much money we make from selling an additional unit. Think of MR as a guiding star that tells you how each extra item sold contributes to revenue growth. It helps businesses understand their earnings path and assess whether they are on track to maximization. As production levels increase, MR generally decreases due to market saturation or competitive pricing pressures. In the given table, MR decreases steadily as production increases from 1000 to 6000 units, illustrating a typical scenario where each additional unit generates progressively less revenue. This decline in MR often triggers businesses to optimize their output to maintain profitability.

Marginal Cost Explained

Marginal Cost (MC) represents the expense of producing one more unit of a product. It's crucial for determining at what point increases in production might become unprofitable. MC helps businesses decide if producing one more unit is beneficial. MC accounts for both fixed and variable costs associated with production. While fixed costs remain constant, variable costs change with the level of output. Therefore, understanding MC allows companies to make informed production decisions and manage costs effectively. In our table, MC starts higher than MR at low production levels (1000 units), then declines, reaching its lowest at 4000 units before increasing again. This indicates that after a certain point, increasing production could lead to higher costs than revenue, signaling a need to optimize to avoid losses.

Finding Optimal Production Levels

Production Levels are pivotal in the journey towards maximizing profit. They indicate the number of units a company decides to produce to achieve desired financial outcomes. The sweet spot for maximizing profit occurs when Marginal Revenue equals Marginal Cost. Up to a certain point, as long as MR is greater than MC, producing additional units contributes to profits. Once MR falls below MC, it signals a decrease in profitability. Based on the exercise data, production at 3000 and 4000 units presents opportunities where MR exceeds MC, suggesting they are close to the profit-maximizing level. However, closer analysis shows that at 4000 units, MR is $72 and MC is $65. This is the point where production is most profitable before costs surpass revenue, making it optimal for the company.