What practical lessons can be drawn from the factor factor relationship in the combination of soybean meal and fishmeal for the formulation of layer mash?

We will calculate the values for $x = 0, 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, 440, 480$.

1c.iv) With this production function, how would you advice a poultry farmer to operate within a particular range of the variable input x (maize).

• Stage I (Increasing Returns): TPP increases at an increasing rate, MPP is increasing, and APP is increasing. This stage ends where APP is maximized (and APP = MPP).
• Stage II (Diminishing Returns): TPP increases at a decreasing rate, MPP is positive but decreasing, and APP is decreasing. This stage ends where TPP is maximized (and MPP = 0). This is the rational stage of production.
• Stage III (Negative Returns): TPP decreases, and MPP is negative. It is irrational to operate in this stage.

• End of Stage I (APP is maximized): Set APP = MPP. $1.5 + 0.0084x - 0.000046x^2 = 1.5 + 0.0168x - 0.000138x^2$ $0.0084x - 0.000046x^2 = 0.0168x - 0.000138x^2$ $0 = 0.0084x - 0.000092x^2$ $0 = x(0.0084 - 0.000092x)$ This gives $x=0$ or $x = \frac{0.0084}{0.000092} \approx 91.3$ units.
• End of Stage II (TPP is maximized, MPP = 0): Set MPP = 0. $1.5 + 0.0168x - 0.000138x^2 = 0$ Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $x = \frac{-0.0168 \pm \sqrt{(0.0168)^2 - 4(-0.000138)(1.5)}}{2(-0.000138)}$ $x = \frac{-0.0168 \pm \sqrt{0.00028224 + 0.000828}}{-0.000276}$ $x = \frac{-0.0168 \pm \sqrt{0.00111024}}{-0.000276}$ $x = \frac{-0.0168 \pm 0.0333202}{-0.000276}$ The relevant positive root is $x = \frac{-0.0168 - 0.0333202}{-0.000276} \approx 181.59$ units.

The farmer should operate in Stage II of production. This stage occurs when the input of maize ($x$) is between approximately 91.3 units and 181.59 units. In this range, the total output of eggs (TPP) is increasing, and the average output per unit of maize (APP) is still positive, although it is declining. Operating beyond 181.59 units would lead to a decrease in total egg production, making it economically irrational.