Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Call a function \(f(x, y)\) homogeneous of degree 1 if \(f(t x, t y)=t f(x, y)\) for all \(t>0\). For example, \(f(x, y)=\) \(x+y e^{y / x}\) satisfies this criterion. Prove Euler's Theorem that such a function satisfies $$ f(x, y)=x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y} $$ Note: Let \(f(x, y)\) denote the value of production from \(x\) units of capital and \(y\) units of labor. Then \(f\) is a homogeneous function (e.g., doubling capital and labor doubles production). Euler's Theorem then asserts an important law of economics that may be phrased as follows: The value of production \(f(x, y)\) equals the cost of capital plus the cost of labor provided that they are paid for at their respective marginal rates \(\partial f / \partial x\) and \(\partial f / \partial y\).

Step-by-Step Solution

Answer

Euler's Theorem is proven: \(f(x, y) = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}\).

Step 1: Understand the definition of homogeneous functions

A function \(f(x, y)\) is homogeneous of degree 1 if for every scalar \(t > 0\), \(f(t x, t y) = t f(x, y)\). This means scaling both variables by a factor \(t\) causes the function to scale by the same factor \(t\).

Step 2: Define the function in terms of a parameter

Consider the function \(g(t) = f(t x, t y)\) for some function \(f\). By the definition of homogeneity, we know \(g(t) = t f(x, y)\). This creates a new function where \(x\) and \(y\) are treated as fixed and \(t\) is the variable.

Step 3: Differentiate the new function with respect to the parameter

Differentiate \(g(t) = t f(x, y)\) with respect to \(t\): \[ \frac{d}{dt}[g(t)] = \frac{d}{dt}[t f(x, y)] = f(x, y) \] Similarly by chain rule, \[ \frac{d}{dt}[g(t)] = \frac{\partial f}{\partial x} \cdot \frac{d}{dt}[t x] + \frac{\partial f}{\partial y} \cdot \frac{d}{dt}[t y] = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} \] Set these two results equal to each other: \[ f(x, y) = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} \]

Step 4: Verify both methods yield the same result

The differentiation approach shows that the equality holds for \(f(x, y)\). So the function satisfies Euler's Theorem, confirming \(f(x, y) = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}\).

Key Concepts

Homogeneous Functions

Homogeneous functions are incredibly useful in the field of mathematics, especially in economics and production analysis. Understanding a homogeneous function is key to solving many problems involving scale and proportion.
In simple terms, a function is homogeneous of degree 1 if it scales uniformly. This means that if you multiply every input in the function by some positive scalar \(t\), the output will be equally scaled by \(t\). For example, if the function is \(f(x, y)\) and you replace \(x\) and \(y\) with \(tx\) and \(ty\), then the function becomes \(t \cdot f(x, y)\). This characteristic is essential when dealing with proportional change.
One application of homogeneous functions is in production functions. Here, they illustrate how changes in input quantities affect output production. If production is homogeneous of degree 1, doubling inputs like labor and capital results in a doubling of output, which is a fundamental assumption in some economic models.

Partial Derivatives

Partial derivatives are a fundamental concept in calculus that deal with functions of multiple variables. They represent the rate of change of a function with respect to one variable while keeping other variables constant. This concept is often applied in various fields such as physics, engineering, and economics.
When dealing with a function \(f(x, y)\), the partial derivative with respect to \(x\), denoted as \(\frac{\partial f}{\partial x}\), measures how \(f\) changes as \(x\) changes while \(y\) remains constant. Similarly, \(\frac{\partial f}{\partial y}\) shows how \(f\) changes with respect to changes in \(y\) with \(x\) held constant. These partial derivatives are essential for finding critical points and understanding the behavior of multivariable functions.
In the context of Euler's Theorem, using partial derivatives helps us express a homogeneous function as a weighted sum of its partial derivatives multiplied by their respective variables. This leads directly to the theorem:\[f(x, y) = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}\] where each term reflects the contribution of the change in each variable to the total change in the function.

Marginal Analysis

Marginal analysis is a crucial concept, especially in economics, where it is used to analyze the changes in costs and benefits due to a small change in decision-making variables. It involves exploring the impact of marginal or incremental changes to optimize decisions.
In production and economics, marginal analysis often involves looking at the partial derivatives of production functions. Here, the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) are identified as marginal products of capital and labor, respectively. This means they measure the additional output produced when one more unit of capital or labor is deployed.
Euler's Theorem connects marginal analysis directly to economic efficiency. According to it, the value of total production \(f(x, y)\) should align with the cost of inputs weighted by their marginal rates. This highlights the importance of understanding how each incremental change in input affects overall productivity and helps businesses and economists make informed decisions about resource allocation.