Marginal product

Summary

Marginal product

• Given: a production function \(y=f\left( x_1,x_2 \right)\)
• The marginal product of factor 1, \(MP_1\) , is defined as

\[{MP}_1= \frac{∂f}{∂x_1}\]

• \(MP_2\) is defined similarly.
• Example:
• \(y=f\left( x_1,x_2 \right)=100\sqrt{x_1x_2}\)
• \(MP_1=50\sqrt{x_2/x_1}\)
• \(MP_2=50\sqrt{x_1/x_2}\)
• Marginal product for small changes in inputs

\[MP_1≈ \frac{Δy}{Δx_1}\]

• If \(x_1\) increases by \(Δx_1\) (small) and \(x_2\) is fixed then \(y\) increases by approximately \(Δy≈MP_1⋅Δx_1\) .

Diminishing marginal product

• Law of diminishing marginal product: \(MP_1\) is eventually decreasing in \(x_1\) for fixed \(x_2\) (Similarly for \(x_2\) )
• Example:
• \(y=100\sqrt{x_1x_2}\)
• \(MP_1=50\sqrt{x_2/x_1}\) decreasing in \(x_1\)