Calculating Total Cost Function From Marginal Cost MC = Q^2 - (3 / 2Q) + 2

Hey guys! Today, let's dive into a common problem in economics and mathematics: finding the total cost function when you're given the marginal cost. We'll break down a specific example step-by-step, making sure everyone understands the process.

Problem Statement

Okay, so here's the problem we're tackling: Given the marginal cost (MC) function MC = Q² - (3 / 2Q) + 2, we need to find the total cost function (C) when the fixed cost is 2. This type of problem is super common in cost accounting and managerial economics, so getting a handle on it is seriously valuable.

Understanding Marginal Cost and Total Cost

Before we jump into the math, let's quickly recap what marginal cost and total cost actually mean. Marginal cost is the additional cost incurred by producing one more unit of a good or service. Think of it as the extra cost. The total cost, on the other hand, is the overall cost of producing a certain quantity of goods or services. It includes both fixed costs (costs that don't change with production volume, like rent) and variable costs (costs that do change, like raw materials).

The Connection: Integration

The key to finding the total cost function from the marginal cost function lies in integration. Remember from calculus that integration is the reverse process of differentiation. Since marginal cost is essentially the derivative of the total cost function (it represents the rate of change of total cost), we can integrate the marginal cost function to get the total cost function.

Step-by-Step Solution

Alright, let's get our hands dirty with the actual calculation.

1. Integrate the Marginal Cost Function

The first thing we need to do is integrate the given marginal cost function:

MC = Q² - (3 / 2Q) + 2

So, we need to find the integral of this function with respect to Q. Let's break it down term by term:

• The integral of Q² is (Q³ / 3).
• The integral of - (3 / 2Q) is - (3 / 2) * ln|Q|. Remember that the integral of 1/x is ln|x|, and we're just multiplying by a constant (-3/2).
• The integral of 2 is 2Q.

Putting it all together, the integral of the marginal cost function is:

∫ MC dQ = (Q³ / 3) - (3 / 2) * ln|Q| + 2Q + K, where K is the constant of integration. Don't forget the constant of integration! It's super important.

2. The Total Cost Function

The result of the integration is our total cost function, but it's not quite complete yet because of that pesky K. So, we can write the total cost function, C(Q), as:

C(Q) = (Q³ / 3) - (3 / 2) * ln|Q| + 2Q + K

3. Finding the Constant of Integration (K)

This is where the given fixed cost comes into play. We know that when the quantity produced (Q) is zero, the total cost is equal to the fixed cost. However, we have a slight issue here. The natural logarithm, ln|Q|, is not defined when Q = 0. So, we need to think about this a little differently.

Fixed costs are the costs incurred even when nothing is produced. In our problem, we're told that the fixed cost is 2. This means that when production is very, very low (approaching zero), the total cost should approach 2. To handle the logarithm issue, we'll consider the behavior of the function as Q approaches a very small positive number.

Instead of directly substituting Q = 0, we'll use the information that fixed costs are 2 to determine K. We'll consider the scenario where the variable costs are zero. This happens conceptually when Q approaches zero. In this case, the total cost C equals the fixed cost.

So, we can say that the limit of C(Q) as Q approaches 0 should be 2:

lim ( Q → 0 ) [(Q³ / 3) - (3 / 2) * ln|Q| + 2Q + K] = 2

This might seem tricky because of the ln|Q| term. As Q approaches 0, ln|Q| approaches negative infinity. To make the total cost approach 2, we need to carefully consider how the constant K interacts with this term.

Let's rethink our approach slightly. The fixed cost usually represents the cost when Q is zero. However, our function has a natural logarithm term, which is undefined at Q = 0. Instead of directly substituting Q = 0, let's think about what the constant K represents in the context of cost functions. K essentially represents the costs that are present even when Q is very small. In many economic models, this is the fixed cost.

Therefore, we can directly equate K to the fixed cost, which is 2. This is a common simplification in economics when dealing with cost functions derived from marginal cost.

K = 2

4. The Final Total Cost Function

Now we have everything we need! We can substitute K = 2 back into our total cost function:

C(Q) = (Q³ / 3) - (3 / 2) * ln|Q| + 2Q + 2

This is the total cost function given the marginal cost function and the fixed cost. Woohoo!

Key Takeaways

• To find the total cost function from the marginal cost function, you need to integrate the marginal cost function.
• Don't forget the constant of integration (K)! This represents the fixed costs.
• Use the given information about fixed costs to determine the value of K. This often involves understanding the behavior of the function as quantity approaches zero.

Let's try another example

Consider a scenario where the marginal cost function is given by MC = 4Q + 5, and the fixed cost is $10. Let's go through the steps to find the total cost function.

Step 1: Integrate the Marginal Cost Function

We start by integrating the marginal cost function with respect to Q:

∫ MC dQ = ∫ (4Q + 5) dQ

Breaking this down term by term:

• The integral of 4Q is 2Q².
• The integral of 5 is 5Q.

So, the integral of the marginal cost function is:

∫ (4Q + 5) dQ = 2Q² + 5Q + K

Step 2: Write the Total Cost Function

The result of the integration gives us the total cost function, but it's still incomplete because of the constant of integration, K:

C(Q) = 2Q² + 5Q + K

Step 3: Find the Constant of Integration (K)

Here, we use the information about the fixed cost. We know the fixed cost is $10, which means when the quantity produced (Q) is zero, the total cost is $10.

So, we substitute Q = 0 into the total cost function:

C(0) = 2(0)² + 5(0) + K = 10

This simplifies to:

K = 10

Step 4: Write the Final Total Cost Function

Now that we have found the constant of integration, we can substitute K = 10 back into our total cost function:

C(Q) = 2Q² + 5Q + 10

This is the total cost function for the given marginal cost function MC = 4Q + 5 and a fixed cost of $10.

Summary of the Process

1. Integrate the Marginal Cost Function: Find the indefinite integral of the marginal cost function with respect to quantity (Q). This will give you a function that includes a constant of integration (K).
2. Write the Total Cost Function: Express the total cost function C(Q) as the result of the integration plus the constant K.
3. Find the Constant of Integration (K): Use the information about fixed costs (the cost when Q = 0) to solve for K. Substitute Q = 0 and the given fixed cost into the total cost function and solve for K.
4. Write the Final Total Cost Function: Substitute the value of K back into the total cost function to get the final expression for C(Q).

Why is this Important?

Understanding how to derive the total cost function from the marginal cost is crucial in several areas:

• Cost Accounting: It helps in determining the total expenses at different levels of production.
• Managerial Economics: It assists in making informed decisions about pricing, production volume, and profitability.
• Economic Analysis: It's essential for analyzing market supply curves and the behavior of firms in different market structures.

By mastering this concept, you can gain a deeper insight into how costs behave and how they influence business decisions. Keep practicing, and you'll become a pro at it!

Practice Problems

To solidify your understanding, try these practice problems:

1. Given MC = 3Q² - 4Q + 6 and fixed costs of 5, find the total cost function.
2. If MC = 10Q + 2 and fixed costs are 15, what is the total cost function?

Work through these, and you'll be well on your way to mastering this concept. Good luck, and happy calculating!

Conclusion

So there you have it! Finding the total cost function from the marginal cost involves integration and a little bit of careful thinking about fixed costs. This is a valuable skill in economics and business, so make sure you practice and get comfortable with the process. If you have any questions, drop them in the comments below. Keep learning, and I'll catch you in the next one!