Monotonic and convex technology

Summary

Monotonic technology

• $y=f\left( x_1,x_2 \right)$ is a production function
• We say that the technology is (strictly) monotonic if $f$ is (strictly) increasing in both arguments.
• If technology is strictly monotonic then each isoquant can be described mathematically by a strictly decreasing function, $x_2=g\left( x_1 \right)$
• Example:
• $y=100\sqrt{x_1x_2}$
• $MP_1=50\sqrt{x_2/x_1}$
• $MP_2=50\sqrt{x_1/x_2}$
• Technology is strictly monotonic

Convex technology

• $y=f\left( x_1,x_2 \right)$ is a production function
• $\left( x_1,x_2 \right)$ and $\left( y_1,y_2 \right)$ are two different arbitrary factor bundles on the same isoquant .
• If $f\left( z_1,z_2 \right)≥\left( x_1,x_2 \right)$ for all bundles $\left( z_1,z_2 \right)$ located on a straight line connecting $\left( x_1,x_2 \right)$ and $\left( y_1,y_2 \right)$ then technology is said to be convex .
• If $f\left( z_1,z_2 \right)>\left( x_1,x_2 \right)$ for all a bundles $\left( z_1,z_2 \right)$ located on a straight line connecting $\left( x_1,x_2 \right)$ and $\left( y_1,y_2 \right)$ , with the exception of $\left( x_1,x_2 \right)$ and $\left( y_1,y_2 \right)$ then preferences are said to be strictly convex .

• Given
• A two factors production model
• Production technology which is strictly monotonic and (strictly) convex
• $x_2=g\left( x_1 \right)$ is the equation for a given isoquant
• Then
• $g$ is strictly decreasing and (strictly) convex .
• If $g$ is differentiable then $g'\left( x_1 \right)<0$ and $g^{''}\left( x_1 \right)≥0$ for all $x_1$
• We say that technology is well-behaved if it is strictly monotonic and strictly convex