Problem 12 - Applied Calculus

Question

The marginal cost and marginal revenue of a company are \(M C(q)=0.03 q^{2}-1.4 q+34\) and \(M R(q)=30\), where \(q\) is the number of items manufactured. To increase profits, should the company increase or decrease production from each of the following levels? (a) 25 items (b) 50 items (c) 80 items

Step-by-Step Solution

Answer

(a) Increase production at 25 items; (b) Decrease production at 50 items; (c) Decrease production at 80 items.

Step 1: Understand Marginal Concepts

The marginal cost (MC) is the cost of producing one more unit, given by the function \( MC(q) = 0.03q^2 - 1.4q + 34 \). The marginal revenue (MR) is the revenue of selling one more unit and is given by \( MR(q) = 30 \). To determine whether to increase or decrease production, compare the marginal cost to the marginal revenue at different production levels \( q \).

Step 2: Evaluate at 25 Items

First, substitute \( q = 25 \) into the marginal cost equation to find \( MC(25) \).\[ MC(25) = 0.03(25)^2 - 1.4(25) + 34 \]Calculate:\[ MC(25) = 0.03(625) - 35 + 34 = 18.75 - 35 + 34 = 17.75 \]Here, \( MC(25) = 17.75 \) and \( MR = 30 \).Since \( MC < MR \), the company should increase production at this level.

Step 3: Evaluate at 50 Items

Substitute \( q = 50 \) into the marginal cost equation to find \( MC(50) \).\[ MC(50) = 0.03(50)^2 - 1.4(50) + 34 \]Calculate:\[ MC(50) = 0.03(2500) - 70 + 34 = 75 - 70 + 34 = 39 \]Here, \( MC(50) = 39 \) and \( MR = 30 \).Since \( MC > MR \), the company should decrease production at this level.

Step 4: Evaluate at 80 Items

Substitute \( q = 80 \) into the marginal cost equation to find \( MC(80) \).\[ MC(80) = 0.03(80)^2 - 1.4(80) + 34 \]Calculate:\[ MC(80) = 0.03(6400) - 112 + 34 = 192 - 112 + 34 = 114 \]Here, \( MC(80) = 114 \) and \( MR = 30 \).Since \( MC > MR \), the company should decrease production at this level.

Key Concepts

Understanding Marginal Cost

The concept of marginal cost (MC) is essential in economic decision-making and is defined as the additional cost incurred in producing one more unit of a good or service. In the given exercise, the marginal cost function provided is:\[ MC(q) = 0.03q^2 - 1.4q + 34 \]This quadratic equation helps to determine how cost behaves as you adjust your production levels \( q \):

• The term \(0.03q^2\) indicates that costs increase with the square of the quantity produced. This means costs escalate as production scales.
• The term \(-1.4q\) implies a reduction in cost related to linear production levels, often associated with initial efficiencies gained as production begins.
• The constant \(34\) adds a fixed cost component that impacts the total cost regardless of production quantity.

When evaluating production changes, assess the marginal cost for different levels of \( q \) based on this equation to understand its fluctuations and potential impacts.

Decoding Marginal Revenue

Marginal revenue (MR) is the additional revenue that a firm gains by selling one more unit. In this exercise, it is simply expressed as:\[ MR(q) = 30 \]This constant implies that each additional unit produced and sold brings in exactly $30. Here, it does not depend on the quantity \( q \), reflecting a regular pricing strategy or a perfectly competitive market scenario where products sell consistently at a fixed price.In real-world terms:

• Marginal revenue that remains constant suggests predictability in how revenue changes when varying production levels.
• Stability in MR makes it easier for a company to determine optimal production levels since variability can complicate revenue forecasting.
• Comparing this constant value against the calculated marginal cost helps in strategic decision-making about scaling production up or down.

Understanding this balance is critical in maintaining efficient operations and capturing profit potential without overshooting operational capacity.

Principles of Profit Maximization

Profit maximization is a fundamental goal for firms, typically achieved by aligning production with levels that maximize the difference between total revenue and total cost. This requires understanding both marginal cost and marginal revenue.When addressing the exercise's scenarios:

• At lower production levels, if \( MC < MR \), increasing production is beneficial because each additional unit adds more to revenue than it costs — boosting profit.
• Conversely, when \( MC > MR \), each additional unit costs more than it adds to revenue, thus reducing profit. Cutting back production mitigates surplus expenses.
• The ideal production point is where \( MC = MR \), as this is where profit is maximized. Beyond this point, increasing or decreasing production will reduce profit margins.

By examining the equations and determining the relationship between \( MC \) and \( MR \) at different \( q \) values, firms can strategically adjust production to ensure optimal resource use and maximal profitability. Whether in theory or practice, owning the insights gleaned from these calculations empowers smarter economic choices.