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