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

Given $72X + 15X^2 - X^3$

a) $MP=\frac {dQ} {dX} =72+30X-3X^2$

When X=8;

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

$MP=120$

b) $AP=\frac {Q} {X} =72+15X-X^2$

When X=6;

$AP=72+15(6)-(6×6)=126$

$AP=126$

c) Q is maximum when $\frac {dQ} {dX} =0$

$72+30X-3X^2=0$

$3X^2-30X-72=0$

$X^2-10X-24=0$

$X^2-12X+2X-24=0$

$X(X-12)+2(X-12)=0$

$X=12$ or (X=$-2$ which is inadmissible)

d) Diminishing returns means, MP is decreasing i.e

$$\frac {dMP} {dX} \lt 0$$

$\frac {dMP} {dX} =; 0$

$30-9X=0$

$9X=30$

$X=3.33$

Thus diminishing returns sets is when $X\gt 3.33$