economists have considered the output of a manulacturing process as function of the size of the

First, we are given the function $y = kn^p$, where $0 < p < 1$. We want to find the marginal pro

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Economists have considered the output of a manufacturing process as a function of the size of the labor force using the function y = kn^p, where 0 < p < 1. The marginal product of labor, defined as the measure of the rate at which output increases with the size of the labor force, is a measure of labor productivity. Complete parts (a) and (b) below: (a) Show that by the power rule, the derivative of y with respect to n is k * p * n^(p-1). Then use the power rule to write the derivative as dy/dn = k * p * n^(p-1). (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 p, as n increases, the denominator of the derivative increases, which means the value of the derivative decreases.

          Economists have considered the output of a manufacturing process as a function of the size of the labor force using the function y = kn^p, where 0 < p < 1. The marginal product of labor, defined as the measure of the rate at which output increases with the size of the labor force, is a measure of labor productivity. Complete parts (a) and (b) below:

(a) Show that by the power rule, the derivative of y with respect to n is k * p * n^(p-1). Then use the power rule to write the derivative as dy/dn = k * p * n^(p-1).

(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 p, as n increases, the denominator of the derivative increases, which means the value of the derivative decreases.

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