suppose that the manager of a firm operating in a perfectly competitive market average variable cost reaches its minimum value at

Complete Question:

Suppose that the manager of a firm operating in a perfectly competitive market has estimated the average variable cost function to be:

AVC = 4.0 - 0.0024Q + 0.000006Q^2             Fixed costs are $500.

Requirement:

Average variable cost reaches its minimum value at___ units of output, and the minimum value of average variable cost is $___

Average variable cost reaches its minimum value at 200 units of output, and the minimum value of average variable cost is $3.76.

Explanation:

To find the Average Variable Cost we will have to calculate quantity and for that sake we will first of all find the point of intersection of AVC and MC to find the Quantity "Q".

So

AVC  * Quantity = Total Variable Cost  + Total Fixed Cost

Here

AVC = 4.0 - 0.0024Q + 0.000006Q^2

Fixed costs are $500

Total Variable Cost is TVC

Quantity is Q here

By putting values, we have:

(4.0 - 0.0024Q + 0.000006Q^2) * Q = TVC + 500

4Q - .0024Q^2 + .000006Q^3 = TVC + 500

By rearranging the above formula, we have:

TVC = 4Q - .0024Q^2 + .000006Q^3 - 500

By applying derivation rules, we have:

dTC/dQ = 4 - 0.0048Q + 0.000018Q^2

Now this equation is Marginal cost equation.

At the point of intersection of AVC and MC, both equations will equal to each other and thus we can find Q.

Mathematically,

4 - 0.0024Q + 0.000006Q^2 = 4 - .0048Q + .000018Q2

Cancelling 4 on both sides, and netting off the equation, we have:

0.0024Q = .000012Q2

1 = .000012Q2 / 0.0024Q

1 = 0.005Q

Q = 1/ 0.005 = 200 Units

By putting value of Q in AVC equation given above, we have:

AVC = 4 - 0.0024*200 + 0.000006*(200)^2

AVC = 4 - 0.48 + 0.24 = $3.76

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