Problem 44 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The productivity of a country in Western Europe is given by the function $$f(x, y)=40 x^{4 / 5} y^{1 / 5}$$ when \(x\) units of labor and \(y\) units of capital are used. a. What is the marginal productivity of labor and the marginal productivity of capital when the amounts expended on labor and capital are 32 units and 243 units, respectively? b. Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country's productivity?

Step-by-Step Solution

Answer

The marginal productivity of labor is approximately 35.61, and the marginal productivity of capital is approximately 1.99. Since the marginal productivity of labor is significantly higher compared to the marginal productivity of capital, the government should focus on increasing expenditure on labor at this time rather than capital investment in order to boost the country's productivity.

Step 1: Calculate the partial derivative with respect to labor (x)

The given function is: $$f(x,y)=40x^{\frac{4}{5}}y^{\frac{1}{5}}$$ Now, let's find the partial derivative of the function with respect to labor (x): $$\frac{\partial f(x, y)}{\partial x} = 40\frac{4}{5}x^{\frac{4}{5}-1}y^{\frac{1}{5}}$$

Step 2: Calculate the partial derivative with respect to capital (y)

Now, let's find the partial derivative of the function with respect to capital (y): $$\frac{\partial f(x, y)}{\partial y} = 40x^{\frac{4}{5}}\frac{1}{5}y^{\frac{1}{5}-1}$$

Step 3: Find the marginal productivity of labor and capital

Given that the labor and capital units are 32 and 243, respectively, we now substitute these values into the partial derivative expressions to find the marginal productivity of labor and capital: $$\frac{\partial f(32, 243)}{\partial x} = 40\frac{4}{5}(32)^{\frac{4}{5}-1}(243)^{\frac{1}{5}}$$ $$\frac{\partial f(32, 243)}{\partial y} = 40(32)^{\frac{4}{5}}\frac{1}{5}(243)^{\frac{1}{5}-1}$$ After calculating, we get: $$\frac{\partial f(32, 243)}{\partial x} \approx 35.6066$$ $$\frac{\partial f(32, 243)}{\partial y} \approx 1.9922$$ Thus, the marginal productivity of labor is approximately 35.61, and the marginal productivity of capital is approximately 1.99.

Step 4: Determine the optimal strategy for the government

As we see, the marginal productivity of labor (35.61) is significantly higher compared to the marginal productivity of capital (1.99). This implies that at this point, increasing expenditure on labor would yield a more significant increase in productivity than capital investment. Therefore, the government should focus on increasing expenditure on labor at this time rather than capital investment in order to boost the country's productivity.

Key Concepts

Understanding Partial Derivatives

Partial derivatives are an essential tool in mathematics, particularly when dealing with functions with more than one variable. In the context of economic productivity, we often use them to understand how changes in one input, like labor or capital, affect the overall output of a production process.
To find a partial derivative, we hold one variable constant while differentiating with respect to the other variable. This helps in identifying the sensitivity or rate of change of the function concerning each variable separately.
For example, in a function that models productivity, the partial derivative with respect to labor (\(\frac{\partial f(x,y)}{\partial x}\)) shows the change in productivity when labor changes slightly, keeping capital constant.

• This provides insight into how efficient additional labor units are in increasing output.
• Similarly, the partial derivative with respect to capital (\(\frac{\partial f(x,y)}{\partial y}\)) indicates the behavior of output with changes in capital, holding labor constant.

By using partial derivatives, businesses and policymakers can make informed decisions on how best to allocate resources and improve productivity.

Exploring Economic Productivity

Economic productivity refers to how efficiently a country, company, or business converts resources into goods and services. It's measured as the output per unit of input, whether those inputs are labor, capital, or other resources.
Higher productivity means more output from the same amount of resources, which is desirable for increasing economic growth and raising living standards.

• Improving productivity involves optimizing the mix of inputs like labor and capital.
• Understanding the role of each input through concepts like marginal productivity can reveal where efficiencies can be gained.

Marginal productivity is particularly important. It tells us how much additional output is generated by one more unit of an input, like one more worker or one more machine. If one input shows a higher marginal productivity compared to another, it might be wise to focus enhancements in that area to maximize output.
In our exercise, the partial derivatives reveal the marginal productivity, helping decide whether to increase labor or capital investment for boosting output. This strategic approach aids in enhancing economic productivity effectively.

The Relationship Between Capital and Labor

Capital and labor are two fundamental inputs in the production process of any economy or organization. Understanding their interplay is crucial for optimizing production and improving productivity.
Labor refers to the human effort involved in production, while capital covers the physical assets injected into the process, like machinery and tools. A profit-maximizing firm must decide how much of each to use for optimal production:

• The relationship is often not linear; adding more of one input without adjusting the other may reduce efficiency or returns.
• The balance between capital and labor dictates the production technology and influences costs and productivity.

There's a concept in economics called the substitution effect. This refers to how we might replace labor with capital or vice versa if one becomes more expensive or less effective. By using marginal productivity, businesses decide whether to substitute one input for another to maintain or improve efficiency.
In the given exercise, finding the marginal productivities of labor and capital helps determine where to focus investments. Here, labor exhibited a higher marginal productivity compared to capital. Therefore, increasing labor might improve productivity more than investing in capital, showcasing the delicate balance industries must maintain between these two pivotal resources.