Technical rate of substitution

Summary

Technical rate of substitution

  • \(y=f\left( x_1,x_2 \right)\) is a production function representing well behaved technology
  • \(x_2=g\left( x_1 \right)\) is the equation of an isoquant
  • The slope of the isoquant at \(\left( x_1,x_2 \right)\) is called the technical rate of substitution or the m arginal rate of technical substitution , \(TRS\) , of factor 2 for factor 1.

\[TRS= \frac{dx_2}{dx_1}=g'\left( x_1 \right)\]

  • Technical rate of substitution for small changes in inputs:

\[TRS≈ \frac{Δx_2}{Δx_1}\]

  • \(TRS\) measures the approximate increase in \(x_2\) , \(Δx_2\) , required to stay on the isoquant for a small increase in \(x_1\) by \(Δx_1\) .
  • \(TRS\) is typically negative since \(Δx_2<0\) if \(Δx_1>0\)
  • Example
    • \(y=100\sqrt{x_1x_2}\)
    • \(x_1=1\) and \(x_2=4\) produces \(y=200\)
    • The isoquant for \(y=200\) has the equation \(x_2=g\left( x_1 \right)=4/x_1\)

\[TRS= \frac{dx_2}{dx_1}=- \frac{4}{x_1^2}=-4\]

    • At \(\left( 1,4 \right)\) , if \(x_1\) is increased by \(Δx_1\) and \(x_2\) is reduced by four times as much , production will remain approximately constant .
  • With (strictly) convex technology:
    • \(TRS\) (strictly) increasing in \(x_2\)
    • \(\left| TRS \right|\) (strictly) decreasing in \(x_2\)
    • We have (strictly) diminishing technical rate of substitution

Marginal product and technical rate of substitution

  • Important result:

\[TRS= \frac{dx_2}{dx_1}=- \frac{MP_1}{MP_2}\]

  • Example
    • \(y=100\sqrt{x_1x_2}\)
    • \(x_1=1\) and \(x_2=4\) produces \(y=200\)
    • The isoquant for \(y=200\) has the equation \(x_2=g\left( x_1 \right)=4/x_1\)

\[TRS= \frac{dx_2}{dx_1}=- \frac{4}{x_1^2}=-4\]

\[MP_1= \frac{50\sqrt{x_2}}{\sqrt{x_1}}=100\]

\[MP_2= \frac{50\sqrt{x_1}}{\sqrt{x_2}}=25\]

\[- \frac{MP_1}{MP_2}=- \frac{100}{25}=-4=TRS\]