Question

Given the functions: ATC= x-6+(30/x) and AR =10+(15/x), where x represents output, find TC, TR, and the output level when net revenue is zero.

Answer 1

$\bold {Answers}$

When net revenue is zero, output = 1 unit or 15 units.

When output = 1 unit, TR = TC = $25, and,

When output = 15 units, TR = TC = $165

$\bold {Solutions}$

We are given:

$ATC = x - 6 + \dfrac {30}{x}$ and $AR = 10 + \dfrac {15}{x}$

Now,

$TC = x × ATC$

$= x (x - 6 + \dfrac {30}{x})$

$= x^2 - 6x + 30$

$TR = x × AR$

$= x (10 + \dfrac {15}{x})$

$= 10x + 15$

When net revenue is zero:

$TR - TC = 0$

$=> x^2 - 6x + 30 -( 10x + 15 )= 0$

$=> x^2 -16x + 15 = 0$

$=> x^2 - x - 15x + 15 = 0$

$=> x(x-1)-15(x-1) = 0$

$=> (x-1)(x-15) = 0$

$\therefore \space x = 1 \space or \space x = 15$

Thus, when net revenue is zero, output is either 1 unit or 15 units.

When $x = 1$

$TR = TC = (1)^2 - 6(1) + 30$

$= 1 - 6 + 30$

$\bold {= \$25}$

When $x = 15$

$TC = TR = 10(15) + 15$

$= 150 + 15$

$\bold {= \$165}$