In 1928 the mathematician Charles Cobb and the economist Paul Douglas empirically (from data) derived a production model for the manufacturing sector of the U.S. economy for the period 1899–1922. Using the model \(P=aK^{b}L^{1-b},\) where \(P\) is manufacturing productivity, \(K\) is capital input, and \(L\) is labor input, and multiple regression techniques, Cobb and Douglas determined that manufacturing productivity was represented by the function \[ P=1.014651K^{0.254124}L^{0.745876}\approx 1.01K^{0.25}L^{0.75} \]
834
For every unit increase in capital input, there is an increase of \( 0.2525\left( \dfrac{L}{K}\right) ^{0.75}\) units in manufacturing productivity.
(b) The marginal productivity of manufacturing output with respect to labor input \(L\) is \[ \dfrac{\partial P}{\partial L}\approx 1.01K^{0.25}\left( 0.75L^{-0.25}\right) =0.7575\left( \dfrac{K}{L}\right) ^{0.25} \]
For every unit increase in labor input, there is an increase of \( 0.7575\left(\dfrac{K}{L}\right)^{0.25}\) units in manufacturing productivity.