Optimizing Cost Functions: Finding Minimum AVC and MC

Businesses often find themselves at a crossroads when deciding how to manage their production costs. An integral part of this process involves understanding and optimizing the Average Variable Cost (AVC) and Marginal Cost (MC). This comprehensive guide will walk you through the steps to find the minimum values of AVC and MC from a total cost function, using the given function as an example.

The Total Cost Function

The total cost function given is:

TC Q^3 - 2Q^2 60Q 100

Step 1: Derive the Variable Cost (VC)

The fixed cost (FC) is the constant term in the total cost function, which is 100. The variable cost (VC) can be found by subtracting the fixed cost from the total cost:

VC TC - FC Q^3 - 2Q^2 60Q

Step 2: Calculate the Average Variable Cost (AVC)

The average variable cost (AVC) is the variable cost divided by the quantity (Q):

AVC frac{VC}{Q} frac{Q^3 - 2Q^2 60Q}{Q} Q^2 - 2Q 60

Step 3: Calculate the Marginal Cost (MC)

The marginal cost (MC) is the derivative of the total cost with respect to quantity (Q):

MC frac{dTC}{dQ} frac{d(Q^3 - 2Q^2 60Q 100)}{dQ} 3Q^2 - 4Q 60

Step 4: Find the Minimum AVC

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

frac{dAVC}{dQ} 2Q - 2

Setting this equal to zero:

2Q - 2 0 implies Q 1

Substituting Q 1 back into the AVC function:

AVC_1 1^2 - 2*1 60 1 - 2 60 59

Step 5: Find the Minimum MC

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

frac{dMC}{dQ} 6Q - 4

Setting this equal to zero:

6Q - 4 0 implies Q frac{4}{6} frac{2}{3}

Substituting Q frac{2}{3} back into the MC function:

MCleft(frac{2}{3}right) 3left(frac{2}{3}right)^2 - 4left(frac{2}{3}right) 60 3left(frac{4}{9}right) - frac{8}{3} 60 frac{12}{9} - frac{24}{9} 60 -frac{12}{9} 60 59.33

Summary

The minimum AVC occurs at Q 1 and is 59. The minimum MC occurs at Q frac{2}{3} and is approximately 59.33.

While the provided total cost function wasn't directly leading to finding the minimum values of AVC and MC, the steps outlined here show a systematic approach to achieving these optimizations. This can be applied to similar scenarios involving cost functions.