Question

The total cost function of a firm is given as : C = 3 + 2q + 5q2. Find AC and MC and hence show that slope of AC = 1 q (MC – AC).

Answer 1

Basic Concept Used :-

$\rm :\longmapsto\:\boxed{ \tt{ \: AC = \frac{C}{q} \: }}$

$\rm :\longmapsto\:\boxed{ \tt{ \: MC \: = \: \dfrac{d}{dq} \: C \: }}$

$\rm :\longmapsto\:\boxed{ \tt{ \: Slope \: of \: AC \: = \: \dfrac{d}{dq} \: AC \: }}$

$\large\underline{\sf{Solution-}}$

Cost function of a firm is given by

$\rm :\longmapsto\:C = 3 + 2q + {5q}^{2}$

So, we know,

$\rm :\longmapsto\:\boxed{ \tt{ \: MC = \dfrac{d}{dq}C \: }}$

So,

$\rm :\longmapsto\:MC$

$\rm \: = \:\dfrac{d}{dq}(3 + 2q + {5q}^{2})$

We know,

$\boxed{ \tt{ \: \dfrac{d}{dx}k \: = \: 0 \: }}$

and

$\boxed{ \tt{ \: \dfrac{d}{dx} {x}^{n} \: = \: {nx}^{n - 1} \: }}$

So, using these, we get

$\rm \: = \:0 + 2 + 10q$

$\bf\implies \:MC = 10q + 2 - - - (1)$

Now, We know that,

$\rm :\longmapsto\:\boxed{ \tt{ \: AC = \frac{C}{q} \: }}$

So,

$\rm :\longmapsto\:AC$

$\rm \: = \:\dfrac{3 + 2q + {5q}^{2} }{q}$

$\rm \: = \:\dfrac{3}{q} + 2 + 5q$

$\bf\implies \:AC = \:\dfrac{3}{q} + 2 + 5q - - - (2)$

Now,

Slope of AC is

$\rm \: = \:\dfrac{d}{dq}AC$

$\rm \: = \:\dfrac{d}{dq}\bigg[\dfrac{3}{q} + 2 + 5q \bigg]$

$\rm \: = \: - \: \dfrac{3}{ {q}^{2} } + 0 + 5$

$\rm \: = \: - \: \dfrac{3}{ {q}^{2} } + 5$

Hence,

$\bf\implies \:\dfrac{d}{dq}AC = \: - \: \dfrac{3}{ {q}^{2} } + 5 - - - (3)$

Now, Consider

$\rm :\longmapsto\:\dfrac{1}{q} \bigg[MC - AC\bigg]$

$\rm \: = \:\dfrac{1}{q} \bigg[10q + 2 - \dfrac{3}{q} - 2 - 5q\bigg]$

$\rm \: = \:\dfrac{1}{q} \bigg[5q - \dfrac{3}{q} \bigg]$

$\rm \: = \:5 - \dfrac{3}{ {q}^{2} }$

$\bf\implies \:\dfrac{1}{q} \bigg[MC - AC\bigg] = 3 - \dfrac{5}{ {q}^{2} } - - - (4)$

So, Equation (3) and (4), we concluded that

$\bf\implies \:\dfrac{d}{dq}AC = \dfrac{1}{q} \bigg[MC - AC\bigg]$

More to know :-

1. Revenue

$\rm :\longmapsto\:\boxed{ \tt{ \: R \: = \: pq \: }}$

2. Marginal Revenue

$\rm :\longmapsto\:\boxed{ \tt{ \: MR = \dfrac{d}{dq}R \: }}$

3. Average Revenue

$\rm :\longmapsto\:\boxed{ \tt{ \: AR = \frac{R}{q} \: }}$

4. Break Even Point

$\rm :\longmapsto\:\boxed{ \tt{ \: R \: = \: C \: }}$

5. Profit

$\rm :\longmapsto\:\boxed{ \tt{ \: P \: = \: R \: - \: C \: }}$