Problem 79 - Contemporary Precalculus

Question

Do Exercise 77 when the equation relating output to resources is \(Q=L^{1 / 2} C^{3 / 4}\)

Step-by-Step Solution

Answer

Answer: The production function equation \(Q=L^{\frac{1}{2}}C^{\frac{3}{4}}\) relates output (Q) to inputs, labor (L) and capital (C). The exponents in the equation represent the contribution of labor and capital to the overall output. In this case, capital (C) has a higher exponent, which indicates that it has a more significant impact on the output than labor. Both exponents are less than 1, suggesting diminishing returns. This means that as more labor and capital are added to the production, the additional output starts to decrease. A balanced increase in labor and capital, with more emphasis on capital, would lead to a higher level of output.

Step 1: 1. Understand the production function and the given equation

The production function relates the output (Q) to the inputs or resources like labor (L) and capital (C). In this exercise, the production function is given by the equation \(Q=L^{\frac{1}{2}}C^{\frac{3}{4}}\). The exponents in the equation can be interpreted as the contribution of labor and capital to the overall output.

Step 2: 2. Calculate the output (Q) for given resources (L and C) values

To calculate the output for a given input of labor and capital, substitute the given values of L and C into the equation and evaluate. For example, if L = 4 and C = 8, we need to find the value of Q: \(Q = L^{\frac{1}{2}}C^{\frac{3}{4}}\) Substitute L=4 and C=8 into the equation: \(Q = 4^{\frac{1}{2}} * 8^{\frac{3}{4}}\) Evaluating this expression, we get the output Q.

Step 3: 3. Assessing the effect of changes in labor (L) or capital (C) on output (Q)

To understand how changes in labor and capital affect the output, we can look at the exponents of each variable in the equation. In this case, the exponent of labor (L) is \(\frac{1}{2}\), and that of capital (C) is \(\frac{3}{4}\). A higher exponent means that the corresponding variable has a more significant impact on output. Here, capital (C) has a higher exponent, which indicates that capital has a more significant impact on the output than labor.

Step 4: 4. Understanding diminishing returns

It is important to note that both exponents in the given equation are less than 1. This is an indication of diminishing returns, which means that as more labor and capital are added to the production, the additional output starts to decrease.

Step 5: 5. Find the ratios of labor (L) and capital (C) that maximize output (Q)

This step involves calculus to determine the optimal ratios of labor and capital that maximize output. We would need to use partial derivatives, but this might be more advanced for high school level. However, we can still see the optimal ratios by analyzing how the exponents affect the output. Here, since both exponents are less than 1, it shows that increasing both labor and capital will not result in proportionate increases in the output. This means that a balanced increase in labor and capital, with more emphasis on capital (due to its higher exponent), would lead to a higher level of output.

Key Concepts

output calculation

When discussing production functions, calculating the output is crucial. The equation provided in the exercise is a general form that relates labor (\(L\)) and capital (\(C\)) to output (\(Q\)). To find the output, you simply substitute the values of labor and capital resources into the production function equation.

For example, if the values given are \(L = 4\) and \(C = 8\), you plug them into the equation:

• Calculate \(L^{\frac{1}{2}}\), which means \(4^{\frac{1}{2}} = 2\).
• Calculate \(C^{\frac{3}{4}}\), which means \(8^{\frac{3}{4}}\).

After calculating these, multiply the results to get the output. This step highlights how the resources translate into productivity. Understanding this process is key to making informed decisions in resource allocation.

labor and capital resources

Labor and capital are the two main resources in the production function equation. In this context:

• **Labor (L)** represents the human effort used in production, which may include the number of hours worked or number of employees.
• **Capital (C)** refers to the machinery, tools, buildings, or facilities used in producing goods and services.

These resources are essential for driving output. By manipulating labor and capital, companies can influence how much product is produced. However, the exact contribution of each resource to the output is dictated by their respective exponents in the function. More substantial capital typically indicates more efficient production capabilities, leading to a larger output increase with a smaller resource increase.

diminishing returns

Diminishing returns are a common phenomenon in production. It occurs when adding more of one input, such as labor or capital, leads to smaller increases in output.

In the equation provided, both exponents \(\frac{1}{2}\) for labor and \(\frac{3}{4}\) for capital are less than 1, which signals diminishing returns. This means:

• Initially, adding more labor or capital will increase output significantly.
• As more is added, the output increment becomes less compared to the preceding additions.

Having an understanding of diminishing returns ensures that resources are used efficiently, without over-investing in inputs that yield little additional output.

exponents in production functions

Exponents in production functions serve as indicators of how each resource contributes to the output. In the given equation, we have the following exponents:

• Labor (\(L\)) has the exponent of \(\frac{1}{2}\).
• Capital (\(C\)) has the exponent of \(\frac{3}{4}\).

These exponents tell us that the output's sensitivity to capital is greater than its sensitivity to labor. A higher exponent implies a larger change in output when that input is increased, provided all other factors remain constant.

Therefore, understanding these exponents helps in optimizing resource allocation by focusing on the inputs that maximize returns. In this case, investing more in capital will likely yield a better increase in output compared to investing the same amount in labor.