A firm faces the following Average Cost function

AC=1500Q^-1 + 300-27Q+1.5Q^2

Calculate the output level that minimizes:

a) Marginal Cost
b)Average Variable cost

I need some help on this question. Thanks.

If the average cost is

AC=1500Q^-1 + 300-27Q+1.5Q^2 ,
then that is the TOTAL cost of selling Q items, divided by Q.
The total cost is then
TC = 1500 + 300 Q -27 Q^2 + 1.5 Q^3
The marginal cost is the derivative of that, d(TC)/dQ
= 300 - 54 Q +4.5 Q^2

Set the derivative of that equal to zero to get the Q value (Quantity) for minimum marginal cost.

To get the minimum average cost, differentiate
1500Q^-1 + 300-27Q+1.5Q^2
directly and set that equal to zero, and solve for Q.

b) askes for average variable cost, does that implie that the average cost function also include a fixed portion? If so, would you do part b differently from what you did?

Thanks.

I am not sure what they meant by "average variable cost". I assumed is is the same as average cost.

what is the answer for above question

In that case, we can calculate the average variable cost using the same formula as the average cost function. The average variable cost represents the variable cost per unit of output.

To find the output level that minimizes the average variable cost, we need to differentiate the average variable cost function with respect to the quantity and set it equal to zero.

So, let's differentiate the average variable cost function:

AVC = 1500Q^-1 + 300 - 27Q + 1.5Q^2

AVC' = d(AVC)/dQ = -1500Q^-2 - 27 + 3Q

Setting AVC' equal to zero:

-1500Q^-2 - 27 + 3Q = 0

To solve this equation for Q, we would need to rearrange it:

1500Q^-2 + 3Q - 27 = 0

Since this is a quadratic equation, we can solve it by factoring or using the quadratic formula. Once we find the values of Q that satisfy this equation, we can evaluate the average variable cost at those values to find the minimum.