Problem 161 - Calculus 3

Question

\(\mathrm{A}\) 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

\( \frac{\partial f}{\partial x} \approx 64.4 \), \( \frac{\partial f}{\partial y} \approx 250.2 \).

Step 1: Understanding the Problem

The Cobb-Douglas production function given is \( f(x,y) = 200x^{0.7}y^{0.3} \). We need to find the partial derivatives of \( f \) with respect to \( x \) and \( y \), representing the marginal product of labor and capital, respectively. Given \( x = 500 \) and \( y = 1000 \), we will evaluate these derivatives at the specified values.

Step 2: Calculating the Partial Derivative with respect to x

To find \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant. Differentiate the function \( f(x,y) = 200x^{0.7}y^{0.3} \) with respect to \( x \):\[ \frac{\partial f}{\partial x} = 200 \cdot 0.7 \cdot x^{0.7-1} \cdot y^{0.3} = 140x^{-0.3}y^{0.3} \].Substitute \( x = 500 \) and \( y = 1000 \):\[ \frac{\partial f}{\partial x}\bigg|_{x=500,y=1000} = 140 \cdot (500)^{-0.3} \cdot (1000)^{0.3} \].

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

Compute the values by plugging into the derivative:\( 500^{-0.3} \approx 0.046 \) and \( 1000^{0.3} \approx 10 \).Therefore, \[ \frac{\partial f}{\partial x}\bigg|_{x=500,y=1000} = 140 \cdot 0.046 \cdot 10 = 64.4 \].

Step 4: Calculating the Partial Derivative with respect to y

To find \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant. Differentiate the function \( f(x,y) = 200x^{0.7}y^{0.3} \) with respect to \( y \):\[ \frac{\partial f}{\partial y} = 200 \cdot 0.3 \cdot x^{0.7} \cdot y^{0.3-1} = 60x^{0.7}y^{-0.7} \].Substitute \( x = 500 \) and \( y = 1000 \):\[ \frac{\partial f}{\partial y}\bigg|_{x=500,y=1000} = 60 \cdot (500)^{0.7} \cdot (1000)^{-0.7} \].

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

Compute the values by plugging into the derivative:\( 500^{0.7} \approx 131.75 \) and \( 1000^{-0.7} \approx 0.0316 \).Therefore, \[ \frac{\partial f}{\partial y}\bigg|_{x=500,y=1000} = 60 \cdot 131.75 \cdot 0.0316 \approx 250.2 \].

Key Concepts

Cobb-Douglas Production Function

The Cobb-Douglas production function is a mathematical formula used to represent the relationship between inputs, typically labor and capital, and the resulting output in production economics. This specific function is widely appreciated for its simplicity and ability to accurately describe how changes in inputs affect output levels in many economic scenarios. The general form of the Cobb-Douglas production function is: \[ f(x, y) = Ax^{b}y^{c} \] Here, \(x\) and \(y\) denote the quantities of labor and capital, respectively, while \(A\), \(b\), and \(c\) are constants. The constants often reflect the efficiency of input usage and their respective contributions to production.

• \(A\) is a constant that represents the level of technology in the production process.
• \(b\) and \(c\) are the output elasticities of labor and capital, indicating how responding output is to changes in inputs.

In our example, the Cobb-Douglas function is given as \(f(x, y)=200x^{0.7}y^{0.3}\). It indicates the output when 200 units of technology, labor \(x\) raised to the power of 0.7, and capital \(y\) raised to the power of 0.3, are used. This function helps us understand how labor and capital contribute individually to the production process and their combined impact on total output.

Marginal Product of Labor

The marginal product of labor is a measure of how much additional output is generated by adding an extra unit of labor, holding all other inputs constant. In simpler terms, it tells us what happens to production if one more worker is employed.To find this value, we compute the partial derivative of the production function with respect to labor, \(x\). This derivative indicates the rate of change of output relative to changes in labor input, highlighting the incremental effect of labor on total output.

• The formula to find the marginal product of labor involves treating capital \(y\) as a constant.
• The partial derivative is given by \(\frac{\partial f}{\partial x} = 140x^{-0.3}y^{0.3}\).
• Substituting \(x = 500\) and \(y = 1000\), the marginal product of labor is calculated to be approximately 64.4.

This marginal product of labor value helps economists and businesses understand the efficiency of additional labor and aids in decision-making around hiring additional workers.

Marginal Product of Capital

Just like labor, the marginal product of capital measures the additional output resulting from using an extra unit of capital while holding labor constant. This helps in understanding the impact on production when capital investment is incremented.To derive this value, we calculate the partial derivative of the Cobb-Douglas production function with respect to the capital variable, \(y\). This operation highlights how changes in capital influence total production output.

• To find the marginal product of capital, labor \(x\) is kept constant.
• The partial derivative forming the basis for this calculation is \(\frac{\partial f}{\partial y} = 60x^{0.7}y^{-0.7}\).
• By substituting \(x = 500\) and \(y = 1000\), we find the marginal product of capital to be approximately 250.2.

This figure is crucial for organizations seeking to optimize their capital investments, as it helps in assessing the potential yield of investing in further capital when expanding output.