For the production function f(x, y)=60 x^3 / 4 y^1 / 4...

For the production function $f(x, y)=60 x^{3 / 4} y^{1 / 4}$ considered in Example $8,$ think of $f(x, y)$ as the revenue when $x$ units of labor and $y$ units of capital are used. Under actual operating conditions, say, $x=a$ and $y=b, \frac{\partial f}{\partial x}(a, b)$ is referred to as the wage per unit of labor and $\frac{\partial f}{\partial y}(a, b)$ is referred to as the wage per unit of capital. Show that $$ f(a, b)=a \cdot\left[\frac{\partial f}{\partial x}(a, b)\right]+b \cdot\left[\frac{\partial f}{\partial y}(a, b)\right] $$ (This equation shows how the revenue is distributed between labor and capital.)

For the production function $f(x, y)=60 x^{3 / 4} y^{1 / 4}$ considered in Example $8,$ think of $f(x, y)$ as the revenue when $x$ units of labor and $y$ units of capital are used. Under actual operating conditions, say, $x=a$ and $y=b, \frac{\partial f}{\partial x}(a, b)$ is referred to as the wage per unit of labor and $\frac{\partial f}{\partial y}(a, b)$ is referred to as the wage per unit of capital. Show that
$$
f(a, b)=a \cdot\left[\frac{\partial f}{\partial x}(a, b)\right]+b \cdot\left[\frac{\partial f}{\partial y}(a, b)\right]
$$
(This equation shows how the revenue is distributed between labor and capital.)

Key Concepts

Homogeneity and Returns to Scale

A function is homogeneous if scaling all inputs by a common factor results in the output being scaled by some power of that factor. This concept is closely linked to returns to scale, where a function exhibiting constant returns to scale means that the proportional change in output is equal to the proportional change in all inputs combined.

Euler's Theorem for Homogeneous Functions

Euler's Theorem states that for a homogeneous function of degree k, the sum of each input multiplied by its corresponding partial derivative is equal to k times the value of the function. In the case of a function with constant returns to scale (homogeneous of degree one), this theorem shows that the total revenue can be exactly distributed among the inputs in proportion to their marginal contributions.

Production Function

A production function describes the technical relationship between the inputs used in production (like labor and capital) and the resulting output or revenue. It models the way in which resources are transformed into goods or services in an economic setting.

Partial Derivatives in Production Economics

Partial derivatives measure the change in output resulting from a change in one input while keeping other inputs constant. In the context of production, the partial derivative with respect to labor or capital indicates the marginal product, which can be interpreted as the additional revenue generated by an incremental unit of that respective input.

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