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 $$P=1.014651K^{0.254124}L^{0.745876}\approx 1.01K^{0.25}L^{0.75}$$

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Solution  (a) The marginal productivity of manufacturing output with respect to capital input $K$ is $$\dfrac{\partial P}{\partial K}\approx 1.01( 0.25K^{-0.75}) L^{0.75}=0.2525\left( \dfrac{L}{K}\right) ^{0.75}$$

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.