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Economists have considered the output y of a manufacturing process as a function of the size of the labor forcen using the function y = kn°, where 0

Economists have considered the output y of a manufacturing process as a function of the size of the labor forcen using the function y = kn°, where 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.2: Exponential Functions
Problem 58E
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Economists have considered the output y of a manufacturing process as a function of the size of the labor force n using the function y = kn", where 0<p<1. The marginal product of labor, defined as dy measures the rate that output increases with the size of the labor force, and is a measure of labor productivity. Complete parts (a) and (b) below. dn' dy kp (a) Show that dn %3D 1-p' dy kp By the rule, the derivative of y with respect to n is k• Then use the rule to write the derivative as dn n1 -p" (b) How can you tell from the answer to part (a) that as the size of the labor force increases, the marginal product of labor gets smaller? This is a phenomenon known as the law of diminishing returns. For fixed values of as increases, the denominator of the derivative which means the value of the derivative decreases.
Transcribed Image Text:Economists have considered the output y of a manufacturing process as a function of the size of the labor force n using the function y = kn", where 0<p<1. The marginal product of labor, defined as dy measures the rate that output increases with the size of the labor force, and is a measure of labor productivity. Complete parts (a) and (b) below. dn' dy kp (a) Show that dn %3D 1-p' dy kp By the rule, the derivative of y with respect to n is k• Then use the rule to write the derivative as dn n1 -p" (b) How can you tell from the answer to part (a) that as the size of the labor force increases, the marginal product of labor gets smaller? This is a phenomenon known as the law of diminishing returns. For fixed values of as increases, the denominator of the derivative which means the value of the derivative decreases.
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