Minimizing Average Variable Cost (AVC) and Marginal Cost (MC) in Cost Functions

In economics and business management, understanding cost functions is crucial for optimizing operations. This article will guide you through the process of finding the minimum values of Average Variable Cost (AVC) and Marginal Cost (MC) using a specific cost function. The given cost function is:

The Given Cost Function

The cost function provided is:

Q 1/3Q3 - 2Q2 - 60Q 100

To find the minimum values of AVC and MC, we need to perform several steps involving differentiation and analysis. Let's dive into the process.

Step 1: Identify Total Cost (TC)

The given cost function represents the total cost (TC).

Step 2: Calculate Marginal Cost (MC)

Marginal Cost (MC) is the derivative of the Total Cost function with respect to Q:

MC d(TC)/dQ

Let's calculate the derivative:

MC  Q^2 - 4Q - 60

Step 3: Calculate Average Variable Cost (AVC)

Average Variable Cost (AVC) is calculated by taking the variable costs and dividing by Q. Variable costs (VC) are the total costs minus fixed costs. Here, the fixed cost is the constant term 100 so:

VC  TC - 100  (1/3)Q^3 - 2Q^2 - 60Q

Therefore, the Average Variable Cost (AVC) is:

AVC  (1/3)Q^2 - 2Q - 60

Step 4: Find the Minimum AVC and MC

Now, we proceed to find the minimum values of AVC and MC by taking the derivatives and setting them to zero.

Finding Minimum AVC

To find the minimum AVC, we take the derivative of AVC and set it to zero:

Average Variable Cost (AVC) derivative  (2/3)Q - 2

Setting the derivative equal to zero:

(2/3)Q - 2  0

Solving for Q:

Q  3

Now, we substitute Q 3 back into the AVC function:

AVC  (1/3)(3)^2 - 2(3) - 60  (1/3)(9) - 6 - 60  3 - 6 - 60  57

Finding Minimum MC

To find the minimum MC, we take the derivative of MC and set it to zero:

Marginal Cost (MC) derivative  2Q - 4

Setting the derivative equal to zero:

2Q - 4  0

Solving for Q:

Q  2

Now, we substitute Q 2 back into the MC function:

MC  (2)^2 - 4(2) - 60  4 - 8 - 60  -56

Summary of Results:

Minimum AVC: Q 3, AVC 57 Minimum MC: Q 2, MC 56

Conclusion

In summary, the minimum value of AVC is 57 when Q 3, and the minimum value of MC is 56 when Q 2. These calculations provide crucial insights into the optimal production levels for minimizing costs.

Key Takeaways:

Average Variable Cost (AVC) is the variable cost per unit. Marginal Cost (MC) is the change in Total Cost when one additional unit is produced. The minimum values of AVC and MC can be found using differentiation techniques to find critical points.

Understanding these concepts can significantly impact business efficiency and profitability.