Question

Suppose that the output of a factory is given by $P=80 K^{1 / 4} L^{3 / 4},$ where $K$ is the capital investment in thousands of dollars and $L$ is the labor force in thousands of workers. If $K=256$ and $L=10,000,$ use a partial derivative to estimate the effect of increasing capital by one thousand dollars.

Name: Problem 60, Chapter 12: Calculus: Early Transcendental Functions
Uploaded: 2020-06-25T03:20:31-08:00
Duration: 2 min 4 s
Channel: Foster Wisusik
Description: Suppose that the output of a factory is given by P=80 K^1 / 4 L^3 / 4, where K is the capital investment in thousands of dollars and L is the labor force in thousands of workers. If K=256 and L=10,000, use a partial derivative to estimate the effect of increasing capital by one thousand dollars.

   Suppose that the output of a factory is given by $P=80 K^{1 / 4} L^{3 / 4},$ where $K$ is the capital investment in thousands of dollars and $L$ is the labor force in thousands of workers. If $K=256$ and $L=10,000,$ use a partial derivative to estimate the effect of increasing capital by one thousand dollars.

Calculus: Early Transcendental Functions

Robert T. Smith,… 3rd Edition

Chapter 12, Problem 60

Instant Answer

Solved by Expert Foster Wisusik

06/25/2020

Step 1

The production function is given by $P=80 K^{1 / 4} L^{3 / 4}$. The partial derivative of $P$ with respect to $K$ is given by: \[\frac{\partial P}{\partial K} = 80 \times \frac{1}{4} K^{-3/4} L^{3/4} = 20 L^{3/4} K^{-3/4}\] Show more…

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Suppose that the output of a factory is given by $P=80 K^{1 / 4} L^{3 / 4},$ where $K$ is the capital investment in thousands of dollars and $L$ is the labor force in thousands of workers. If $K=256$ and $L=10,000,$ use a partial derivative to estimate the effect of increasing capital by one thousand dollars.

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Key Concepts

Production Function

A production function represents the relationship between inputs and outputs in an economic process. It quantifies how the inputs, like capital and labor, are transformed into output, thereby providing a framework for analyzing how changes in input levels affect total production.

Partial Derivatives

Partial derivatives measure how a change in one independent variable affects the outcome of a multivariable function while holding all other variables constant. In the context of production functions, they are used to evaluate the marginal impact of a specific input, such as capital, on the total output.

Marginal Product of Capital

The marginal product of capital is the additional output generated from an incremental increase in the capital input, keeping the labor input constant. This concept is a specific application of partial derivatives in economics, where it provides insight into how capital investment influences overall production.

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