Question

Consider the following production function.

TPL = 12L2 – 0.8L3

Determine the marginal product function(MPL)

Determine the average product function (APL)

Find the value of L that maximizes TPL

Find the value of L that maximizes APL

Find the value of L that maximizes MPL

Answer 1

$TP_L=12L^2 -0.8L^3 \\ i) \; MP_L= \frac{∂(TP_L)}{∂(L)} \\ MP_L= 2 \times 12L^{(2-1)} – 3 \times 0.8L^{3-1} \\ MP_L = 24 L -2.4L^2 \\ ii) \; AP_L= \frac{TP_L}{L} \\ AP_L= \frac{12L^2 -0.8L^3}{L} \\ AP_L= 12L -0.8L^2$

iii) Maximozation of $TP_L$

Put $\frac{∂(TP_L)}{∂(L)} = 0$ for maximum value

$24L-2.4L^2=0 \\ L(24-2.4L)=0 \\ 24=2.4L \\ L= \frac{24}{2.4}=10 \;units$

iv) Put $\frac{∂(AP_L)}{∂(L)} =0$ , for maximum value

$AP_L= 12L -0.8L^2 \\ \frac{∂(AP_L)}{∂(L)} = 12 -1.6L \\ 12-1.6L=0 \\ L=\frac{12}{1.6}= 7.5 \;units$

v) Put $\frac{∂(MP_L)}{∂(L)}=0$ for maximum value

$MP_L=24L -2.4L^2 \\ \frac{∂(MP_L)}{∂(L)} = 24-4.8L \\ 24-4.8L=0 \\ L=\frac{24}{4.8}= 5 \;units$