Question

. Given the Production Function Q = 72X + 15X2 - X3, where Q =Output and X=Input (5

marks)

a) What is the Marginal Product (MP) when X = 8?

b) What is the Average Product (AP) when X = 6?

c) At what value of X will Q be at its maximum?

d) At what value of X will Diminishing Returns set in?

Answer 1

Production function:

$Q=72x+15x^2-x^3$

(a) Marginal Product $(MP)=\dfrac{dQ}{dx}=72+30x-3x^2$

   at x=8,

      $MP=72+30(8)-3(8)^2=72+240-192=120$

(b) Average Product $(AP)=\dfrac{Q}{X}=72+15X-X^2$

    

    At x=6,

      $AP=72+15(6)-(6)^2=72+90-36=126$

(c) Maximizing Q:

    $\dfrac{dQ}{dx}=0\Rightarrow 72+30x-3x^2=0\\[9pt]x^2-10x-24=0\\(x-12)(x+2)=0$

 x=12,-2

At x=12, Output maximized.

(d)For Diminishing returns:

    $\dfrac{d MP}{dx}=\dfrac{d^2Q}{dx^2}=30-6X<0$

       $\dfrac{30}{6}<X\\[9pt] 5<X$

  After 5 level of inputs diminishing returns set in.