Economics Let w=2 x^1 / 2 y^1 / 3 z^1 / 6 be a production...

Economics Let $w=2 x^{1 / 2} y^{1 / 3} z^{1 / 6}$ be a production function that depends on three inputs: $x, y,$ and $z .$ Find the marginal productivity with respect to $x$, the marginal productivity with respect to $y,$ and the marginal productivity with respect to $z$.

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Key Concepts

Marginal Productivity

Marginal productivity refers to the additional output that is generated by increasing one input while holding the other inputs constant. It is an important concept in economics because it helps determine the optimal allocation of resources and informs decisions related to input utilization and cost efficiency.

Partial Derivatives

Partial derivatives are used in the context of multivariable functions to measure the rate of change of the function with respect to one variable while keeping the other variables constant. In economic analysis, they are essential for computing marginal productivities in production functions that depend on several inputs, as they isolate the impact of a change in a single input on the overall output.

Production Function

A production function represents the relationship between the inputs used in production and the output produced. It mathematically captures how different amounts of inputs, such as labor, capital, and materials, translate into the quantity of goods or services produced, providing insights into efficiency and technological capabilities.

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