Problem 67 - Calculus with Concepts in Calculus

Question

If \(f(x, y)\) is the amount of a commodity produced from \(x\) units of capital and \(y\) units of labor, then \(f\) is called a production function. If $$ f(x, y)=x^{\alpha} y^{\beta} \quad \text { for } x>0 \text { and } y>0 $$ where \(\alpha\) and \(\beta\) are positive constants less than 1, then \(f\) is called a Cobb-Douglas production function. a. Show that \(f(t x, t y)=t^{\alpha+\beta} f(x, y)\). b. If \(z=f(x, y)\), show that $$ \frac{1}{z} \frac{\partial z}{\partial x}=\frac{\alpha}{x} \quad \text { and } \quad \frac{1}{z} \frac{\partial z}{\partial y}=\frac{\beta}{y} $$ and that $$ x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=(\alpha+\beta) z $$

Step-by-Step Solution

Answer

a. \(f(tx, ty) = t^{\alpha + \beta} f(x, y)\). b. Partial derivatives as \(\frac{\alpha}{x}\) and \(\frac{\beta}{y}\); identity \(x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = (\alpha+\beta)z\).

Step 1: Express f(tx, ty)

Start with the Cobb-Douglas production function definition, \(f(x, y) = x^{\alpha} y^{\beta}\). We need to determine \(f(tx, ty)\):\[ f(tx, ty) = (tx)^{\alpha} (ty)^{\beta} \]

Step 2: Simplify f(tx, ty)

Simplify the expression from the previous step using properties of exponents:\[ (tx)^{\alpha} (ty)^{\beta} = t^{\alpha} x^{\alpha} t^{\beta} y^{\beta} = t^{\alpha + \beta} x^{\alpha} y^{\beta} \]

Step 3: Compare with original function

Notice that the expression \( t^{\alpha + \beta} x^{\alpha} y^{\beta} \) can be rewritten as:\[ t^{\alpha + \beta} f(x, y) \]This shows that \( f(tx, ty) = t^{\alpha + \beta} f(x, y) \) as required.

Step 4: Compute \(\frac{\partial z}{\partial x}\)

To find \(\frac{\partial z}{\partial x}\), differentiate \(z = x^{\alpha} y^{\beta}\) with respect to \(x\):\[ \frac{\partial z}{\partial x} = \alpha x^{\alpha - 1} y^{\beta} \]

Step 5: Simplify \(\frac{1}{z} \frac{\partial z}{\partial x}\)

Substitute \(z = x^{\alpha} y^{\beta}\) into \(\frac{\partial z}{\partial x}\) and simplify:\[ \frac{1}{z} \frac{\partial z}{\partial x} = \frac{1}{x^{\alpha} y^{\beta}} \alpha x^{\alpha - 1} y^{\beta} = \frac{\alpha}{x} \]

Step 6: Compute \(\frac{\partial z}{\partial y}\)

Differentiate \(z = x^{\alpha} y^{\beta}\) with respect to \(y\):\[ \frac{\partial z}{\partial y} = \beta x^{\alpha} y^{\beta - 1} \]

Step 7: Simplify \(\frac{1}{z} \frac{\partial z}{\partial y}\)

Substitute \(z = x^{\alpha} y^{\beta}\) into \(\frac{\partial z}{\partial y}\) and simplify:\[ \frac{1}{z} \frac{\partial z}{\partial y} = \frac{1}{x^{\alpha} y^{\beta}} \beta x^{\alpha} y^{\beta - 1} = \frac{\beta}{y} \]

Step 8: Express \(x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y}\)

Utilize the partial derivatives found, \(\frac{\partial z}{\partial x} = \alpha x^{\alpha - 1} y^{\beta}\) and \(\frac{\partial z}{\partial y} = \beta x^{\alpha} y^{\beta - 1}\), and write:\[ x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = x(\alpha x^{\alpha - 1} y^{\beta}) + y(\beta x^{\alpha} y^{\beta - 1}) \]

Step 9: Simplify the expression

Simplify the expression from Step 8:\[ x(\alpha x^{\alpha - 1} y^{\beta}) + y(\beta x^{\alpha} y^{\beta - 1}) = \alpha x^{\alpha} y^{\beta} + \beta x^{\alpha} y^{\beta} \]Combine terms:\[ (\alpha + \beta) x^{\alpha} y^{\beta} = (\alpha + \beta) z \]Therefore, \(x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = (\alpha + \beta) z\).

Key Concepts

Partial Derivatives

Partial derivatives are a way to determine how a function changes as one of its variables changes, while keeping the other variables constant. They are crucial in analyzing functions with more than one variable, like the Cobb-Douglas production function. For a function \( f(x, y) = x^{\alpha} y^{\beta} \), where \( x \) and \( y \) represent different resources such as capital and labor, partial derivatives can show the rate of change of the production output with respect to each variable separately.
To compute the partial derivative with respect to \( x \), we differentiate the function treating \( y \) as a constant. This results in: \( \frac{\partial f}{\partial x} = \alpha x^{\alpha - 1} y^{\beta} \).
Similarly, the partial derivative with respect to \( y \) is found by treating \( x \) as constant, leading to: \( \frac{\partial f}{\partial y} = \beta x^{\alpha} y^{\beta - 1} \).
Understanding partial derivatives helps analyze how sensitive the production output is to changes in each input. In business and economics, this is important for optimizing resource allocation.

Production Function

A production function describes the relationship between inputs and outputs in a production process. In simple terms, it explains how different quantities of resources like capital and labor contribute to the production of goods and services.
The Cobb-Douglas production function, expressed as \( f(x, y) = x^{\alpha} y^{\beta} \), is one of the most well-known functions used for this purpose. It captures the essence of how inputs are combined to produce an output, considering the diminishing returns of inputs. Here, \( \alpha \) and \( \beta \) are constants that represent the output elasticity of capital and labor, respectively. These exponents are normally positive and less than one, reflecting the realistic economic circumstance that doubling the input won't necessarily double the output due to inefficiencies and limitations in production processes.
In calculating economic measures, the Cobb-Douglas production function is essential as it can depict scale properties such as constant, increasing, or diminishing returns to scale based on the value of \( \alpha + \beta \).

• If \( \alpha + \beta = 1 \), the function has constant returns to scale.
• If \( \alpha + \beta > 1 \), there are increasing returns to scale.
• If \( \alpha + \beta < 1 \), the function indicates diminishing returns to scale.

Understanding production functions is crucial for businesses to maximize efficiency and for economists to model economic growth.

Exponents in Calculus

Exponents in calculus play a major role in functions like the Cobb-Douglas production function. They represent the power to which a variable is raised, significantly affecting the behavior and properties of the function. In the context of the production function \( f(x, y) = x^{\alpha} y^{\beta} \), \( \alpha \) and \( \beta \) are exponents that provide insight into how each input contributes to the overall output.
When manipulating such functions, properties of exponents are pivotal. For instance, when scalar factors are introduced to variables, exponent rules allow simplification of expressions like \( (tx)^{\alpha} (ty)^{\beta} \) to \( t^{\alpha+\beta} x^{\alpha} y^{\beta} \).
The exponents \( \alpha \) and \( \beta \), being less than 1, suggest diminishing marginal returns, which means each additional unit of input will contribute to smaller increases in output. This is a realistic aspect of production in the real world, highlighting the limitations of endlessly increasing inputs.
In calculus, understanding how to work with exponents not only aids in solving derivatives and integrals but also in comprehending how different components of a function interact, enhancing both mathematical and practical applications.