Problem 1 - Applied Calculus

Question

The function \(C(q)\) gives the cost in dollars to produce \(q\) barrels of olive oil. (a) What are the units of marginal cost? (b) What is the practical meaning of the statement \(M C=3\) for \(q=100 ?\)

Step-by-Step Solution

Answer

(a) Dollars per barrel; (b) The cost to produce the 101st barrel at 100 barrels is $3.

Step 1: Understanding Marginal Cost

In economics, the marginal cost (MC) is the derivative of the cost function \(C(q)\) with respect to the quantity \(q\). It represents the cost of producing one additional unit of output. Thus, the units of marginal cost are the units of cost (dollars) per unit of output (barrels), which can be written as dollars per barrel.

Step 2: Interpreting Given Marginal Cost

The statement \(MC = 3\) for \(q = 100\) means that when 100 barrels of olive oil are produced, the cost of producing one more barrel (the 101st barrel) is 3 dollars.

Key Concepts

Cost Function

In economics, a **cost function** is a mathematical formula that helps you determine how much it costs to produce a certain number of goods. The function is typically expressed as \( C(q) \), where \( q \) stands for the quantity of items produced, and \( C(q) \) represents the total cost to produce that quantity. This tool is crucial because it provides businesses a way to understand the relationship between production volume and corresponding costs.
By analyzing a cost function, companies can make informed decisions about production levels and budgeting.

• For example, with olive oil, if producing 50 barrels costs $5000, \( C(50) = 5000 \).
• To find out the cost for any other number of barrels, you simply plug in that number into the cost function.

Derivative

A **derivative** is a key concept in calculus that helps to measure how a function changes as its input changes. Specifically in our context, the derivative of the cost function \( C(q) \) gives us the marginal cost.
To put it plainly, the derivative tells you the rate at which costs change with respect to a change in quantity.

• If you imagine the cost as a curve on a graph, the derivative is the slope of that curve at any given point.
• This concept allows businesses to predict how much it will cost to increase production by just one unit.

For instance, if the derivative of the cost function at \( q = 100 \) is 3, then this tells us that producing one more barrel of olive oil will cost an extra 3 dollars.

Economics

**Economics** is the study of how people make choices when faced with scarcity. Understanding cost functions and their derivatives plays a vital role in economic decision-making, especially for businesses.
Through concepts like marginal cost, businesses can decide how much to produce for maximizing profits or minimizing costs.

• Cost considerations, such as materials and labor, are economic factors influenced by marginal cost.
• Forecasting production levels with economic principles helps to ensure efficient resource allocation and cost control.

When the marginal cost is understood, businesses can adjust their operations to optimize output and profitability.

Units of Measurement

Units of measurement are crucial for clarity in calculations and communication.
In this exercise, the marginal cost, which is the change in cost for producing one more unit, is measured in units of cost per unit of output.
For example, if we are producing olive oil:

• Cost is measured in dollars.
• Output is measured in barrels.

Therefore, the units of marginal cost here are **dollars per barrel**. This unit tells us how much more it costs to produce each additional barrel, providing clear, understandable metrics for decision-making.