Economists have found that the output of a finished product,...

Economists have found that the output of a finished product, $f(x, y),$ is sometimes described by the function $$ f(x, y)=a x^{b} y^{1-b} $$ where $x$ stands for the amount of money expended on labor, $y$ stands for the amount expended on capital, and $a$ and $b$ are positive constants with $0 < b < 1$ b. Use the result of part (a) to show that if the amount of money expended for labor and capital are both increased by $r \%,$ then the output is also increased by $r \%$

Economists have found that the output of a finished product, $f(x, y),$ is sometimes described by the function
$$
f(x, y)=a x^{b} y^{1-b}
$$
where $x$ stands for the amount of money expended on labor, $y$ stands for the amount expended on capital, and $a$ and $b$ are positive constants with $0 < b < 1$
b. Use the result of part (a) to show that if the amount of money expended for labor and capital are both increased by $r \%,$ then the output is also increased by $r \%$

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Key Concepts

Production Function

A production function shows the relationship between inputs and the resulting output in an economic system. It represents how a combination of inputs, such as labor and capital, is transformed into useful goods or services, capturing the efficiency of the production process.

Cobb-Douglas Production Function

The Cobb-Douglas production function is a specific functional form that represents the output as a product of inputs raised to constant powers. In this form, the parameters indicate the output elasticities of the inputs and reflect the contribution each input makes to the overall production process.

Homogeneity

Homogeneity is a property of functions wherein scaling all inputs by a constant factor results in the output being scaled by a predictable factor. For a production function, if it is homogeneous of degree k, then multiplying all inputs by a constant will multiply the output by that constant raised to the power k.

Constant Returns to Scale

Constant returns to scale occur when a proportional increase in all inputs leads to an equal proportional increase in output. In the context of the Cobb-Douglas function with exponents summing to one, scaling both inputs by the same percentage results in the output increasing by that same percentage, thereby demonstrating constant returns to scale.

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