Problem 161 - Calculus Volume 3

Question

A Cobb-Douglas production function is \(f(x, y)=200 x^{0.7} y^{0.3}, \quad\) where \(x\) and \(y\) represent the amount of labor and capital available. Let \(x=500\) and \(y=1000\). Find \(\frac{\delta f}{\delta x}\) and \(\frac{\delta f}{\delta y}\) at these values, which represent the marginal productivity of labor and capital, respectively.

Step-by-Step Solution

Answer

The marginal products are \(170.8\) for labor and \(156.8\) for capital.

Step 1: Understand the Cobb-Douglas Function

The given function is a Cobb-Douglas production function, \(f(x, y) = 200 x^{0.7} y^{0.3}\), where \(x\) is labor and \(y\) is capital. We want to find the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) to determine the marginal products.

Step 2: Find \(\frac{\partial f}{\partial x}\)

To find the marginal productivity of labor, calculate the partial derivative of the function with respect to \(x\). Using the power rule, we get:\[\frac{\partial f}{\partial x} = 200 \cdot 0.7 \cdot x^{0.7-1} \cdot y^{0.3}\]\[= 140 \cdot x^{-0.3} \cdot y^{0.3}\]Substitute \(x=500\) and \(y=1000\) into the equation:\[\frac{\partial f}{\partial x} = 140 \cdot (500)^{-0.3} \cdot (1000)^{0.3}\]

Step 3: Calculate \(\frac{\partial f}{\partial x}\)

Evaluate \(\frac{\partial f}{\partial x}\) using the substituted values:\[\frac{\partial f}{\partial x} = 140 \cdot (500)^{-0.3} \cdot (1000)^{0.3}\]First, compute each power:\((500)^{-0.3} \approx 0.122\) and \((1000)^{0.3} \approx 10\)So, \[\frac{\partial f}{\partial x} = 140 \cdot 0.122 \cdot 10 \approx 170.8\]

Step 4: Find \(\frac{\partial f}{\partial y}\)

Now, calculate the partial derivative of the function with respect to \(y\) to find the marginal productivity of capital:\[\frac{\partial f}{\partial y} = 200 \cdot 0.3 \cdot x^{0.7} \cdot y^{0.3-1}\]\[= 60 \cdot x^{0.7} \cdot y^{-0.7}\]Substitute \(x=500\) and \(y=1000\) into the equation:\[\frac{\partial f}{\partial y} = 60 \cdot (500)^{0.7} \cdot (1000)^{-0.7}\]

Step 5: Calculate \(\frac{\partial f}{\partial y}\)

Evaluate \(\frac{\partial f}{\partial y}\) using the substituted values:\[(500)^{0.7} \approx 82.54\) and \((1000)^{-0.7} \approx 0.0316\]So, \[\frac{\partial f}{\partial y} = 60 \cdot 82.54 \cdot 0.0316 \approx 156.8\]

Step 6: Conclusion

The marginal productivity of labor \(\frac{\partial f}{\partial x}\) is approximately 170.8, and the marginal productivity of capital \(\frac{\partial f}{\partial y}\) is approximately 156.8 for the given values of labor and capital.

Key Concepts

Marginal Productivity

The concept of marginal productivity is crucial in understanding production functions like the Cobb-Douglas function. Marginal productivity measures the additional output gained from using an extra unit of an input, while keeping other inputs constant. In the context of our exercise, it refers to how much more output is generated by either increasing labor or capital individually.

For example, when we analyze the marginal productivity of labor, we look at how much more output can be produced with one more unit of labor, holding capital constant. Similarly, the marginal productivity of capital assesses the additional output with more capital, while labor remains the same. Understanding these marginal changes allows firms to make informed decisions about resource allocation to optimize production.

Partial Derivatives

Partial derivatives play a vital role in finding the marginal productivity of each input in a production function. By taking the partial derivative of a function with respect to one input, we isolate and study the effect of changing that input alone. In our problem, the function is given as:\[f(x, y) = 200 x^{0.7} y^{0.3}\]We want to find \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).

• To find \( \frac{\partial f}{\partial x} \), we differentiate with respect to \( x \), treating \( y \) as a constant:

Using the power rule: \[\frac{\partial f}{\partial x} = 200 \cdot 0.7 \cdot x^{0.7-1} \cdot y^{0.3}\]

• For \( \frac{\partial f}{\partial y} \), the process is similar, but we differentiate with respect to \( y \):

\[\frac{\partial f}{\partial y} = 200 \cdot 0.3 \cdot x^{0.7} \cdot y^{0.3-1}\]

By calculating these derivatives, we can understand the marginal impact of labor and capital intuitively.

Labor and Capital

Labor and capital are two fundamental components in production functions such as the Cobb-Douglas model. These inputs are investments for producing goods and services. Though the amount of each can vary, their combination drives productivity.

• **Labor** refers to human effort used in the production of goods and services. It encompasses both physical and mental work that workers provide.
• **Capital** includes tools, machinery, and infrastructure that a firm uses to produce goods and services. It's everything that aids labor in production.

In the Cobb-Douglas function, adjusting labor and capital affects output, illustrating their importance in economics. The exponents in the function, such as \(0.7\) for labor and \(0.3\) for capital, show the relative importance or contribution of each input to the production process. These values suggest the elasticity of output with respect to each input—basically how responsive the output is to changes in the input levels.

Understanding how to balance labor and capital effectively allows for more efficient and productive operations.