Question

The average cost function, AC for a commodity is given by AC = `x + 5 + 36/x` in terms of output x. Find

1) The total cost, C and marginal cost, MC as a function of x.

2) The outputs for which AC increases

Answer 1

Let C be the total cost function. Then

Average cost (AC) = `C/x => AC.x => C = (x + 5 + 36/x) x`

`=> C = x^2 + 5x + 36`

Let MC be the marginal cost function. Then,

`MC = (dC)/dx = d/dx (x^2 + 5x + 36) = 2x + 5`

Now, `d/dx (AC) = d/dx (x + 5 + 36/x) = 1 - 36/x^2`

For AC to be increasing `d/dx(AC) > 0 => 1 - 36/x^2 > 0`

`=> x^2 - 36 > 0 => (x - 6) (x + 6) > 0`

`=> x < -6 or x > 6 => x > 6`    [∵ x > 0]