Marginal Productivity Consider the Cobb-Douglas production...

Marginal Productivity Consider the Cobb-Douglas production function $f(x, y)=200 x^{0.7} y^{0.3} .$ When $x=1000$ and $y=500,$ find
(a) the marginal productivity of labor, $\partial f / \partial x$
(b) the marginal productivity of capital, $\partial f / \partial y .$

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Key Concepts

Cobb-Douglas Production Function

This is a specific functional form used in production theory to represent the relationship between inputs (such as labor and capital) and output. It is characterized by inputs raised to constant exponents and a multiplicative constant, which reflects the technology level. The function exhibits constant returns to scale if the sum of the exponents is one and provides insights into the contribution of each input to the production process.

Marginal Productivity

Marginal productivity refers to the additional output that is produced by an incremental increase in one of the inputs while holding all other inputs constant. It is a central concept in production theory and is used to decide the optimal allocation of resources, as it indicates the change in output resulting from a small change in the input level.

Partial Differentiation

Partial differentiation involves computing the derivative of a multivariable function with respect to one variable while treating the other variables as constants. This mathematical tool is essential for finding the marginal products of inputs in production functions, such as determining how output changes when only one input is varied.

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