A firm produces x tonnes of output at a total cost of C(x) = `1/10x^3 - 4x^2 - 20x + 7` find the

- average cost
- average variable cost
- average fixed cost
- marginal cost and
- marginal average cost.

#### Solution

c(x) = f(x) + x

c(x) = `1/10x^3 - 4x^2 - 20x + 7`

Then f(x) = `1/10x^3 - 4x^2 - 20x` and k = 7

**(i)** Average Cost (AC) = `"Total cost"/"Output" = ("C"(x))/x = ("f"(x) + k)/x`

`= (1/10x^3 - 4x^2 - 20x + 7)/x`

`= 1/10 x^3/x - (4x^2)/x - (20x)/x + 7/x`

`= 1/10 x^2 - 4x - 20 + 7/x`

**(ii) **Average Variable Cost (AVC) = `"Variable cost"/"Output" = ("f"(x))/x`

`= (1/10 x^3 - 4x^2 - 20x)/x`

`= 1/10 x^3/x - (4x^2)/x - (20x)/x`

`= 1/10 x^2 - 4x - 20`

**(iii)** Average Fixed Cost (AFC) = `"Fixed cost"/"Output" = k/x = 7/x`

**(iv)** Marginal Cost (MC) = `"dC"/"dx"`

`= "d"/"dx"(1/10 x^3 - 4x^2 - 20x + 7)`

`= "d"/"dx" (1/10 x^3) - "d"/"dx"(4x^2) - "d"/"dx" (20x) + "d"/"dx" (7)`

`= 1/10 "d"/"dx" (x^3) - 4"d"/"dx"(x^2) - 20"d"/"dx" (x) + 0`

= `1/10 (3)x^(3-1) - 4(2)^(2-1) - 20(1)`

`= 3/10 x^2 - 8x - 20`

**(v)** Marginal Average Cost (MAC) = `"d"/"dx"`(AC)

`= "d"/"dx"(1/10 x^2 - 4x - 20) + 7/x`

`= 1/10"d"/"dx"(x^2) - 4 "d"/"dx" (x) - "d"/"dx"(20) + "d"/"dx"(7/x)`

`= 1/10 (2x^(2-1)) - 4(1) - 0 - 7/x^2`

`= 1/5x - 4 - 7/x^2`