Answer to Question #227642 in Calculus for Raz

Question

A firms short run production function is given by Q=c * L + B * L^2 - a * L^3

at a=17.6, b=84.9 and c=73

determine the value of L which maximises the average product of labour rounding your answer to two (2) decimal places

Answer 1

Let us define points where MP(L)=$Q'(L)=0$ .

MP(L)=c+2$\cdot B\cdot L-3\cdot a \cdot L^2=73+169.8\cdot L-52.8\cdot L^2=0$ ;

This is a quadratic equation.

$D=169.8^2+4\cdot 73\cdot 52.8=44269.64$

$L_1={-169.8+\sqrt{44269.64}\over {-2\cdot 52.8}}=-0.384- extraneous\space root$

$L_2={-169.8-\sqrt{44269.64}\over {-2\cdot 52.8}}=3.60$

L₂ is the maximum point of Q(L) because from graph of parabola Q'(L)

we see that Q'(L)>0 on (0, 3.6) and Q(L) increases;

and Q'(L)<0 on (3.6, $\infty)$ Q(L) is decreasing.

So L_max=3.60 is a value of factor L where average product is maximum.