Problem 100 - Applied Calculus

Question

BUSINESS: Isoquant Curves An isoquant curve (iso means "same" and quant is short for "quantity") shows the various combinations of labor and capital (the invested value of factory buildings, machinery, and raw materials) a company could use to achieve the same total production level. For a given production level, an isoquant curve can be written in the form \(K=a L^{b}\) where \(K\) is the amount of capital, \(L\) is the amount of labor, and \(a\) and \(b\) are constants. For each isoquant curve, find the value of \(K\) corresponding to the given value of \(L\). $$ K=4000 L^{-2 / 3} \text { and } L=125 $$

Step-by-Step Solution

Answer

The value of \( K \) is 160.

Step 1: Understanding the Equation

The given isoquant curve equation is \( K = 4000 L^{-2/3} \). To find \( K \), you need to substitute the given value of \( L \) into this equation. Here, you have \( L = 125 \).

Step 2: Substitute the Value of L

Substitute \( L = 125 \) into the equation: \( K = 4000 (125)^{-2/3} \).

Step 3: Calculate L^{-2/3}

Calculate \( 125^{-2/3} \). This means taking the cube root of 125, which is 5, then raising it to the power of -2: \( 5^{-2} \).

Step 4: Simplifying the Exponent

\( 5^{-2} \) is the same as \( \frac{1}{5^2} = \frac{1}{25} \).

Step 5: Calculate K Using the Base Value

Now, substitute back into the equation: \( K = 4000 \times \frac{1}{25} \).

Step 6: Final Calculation

Divide \( 4000 \) by \( 25 \) to find \( K \): \( K = 160 \).

Key Concepts

Labor and Capital

In economics, labor and capital are two crucial inputs used in the production of goods and services. Labor refers to the human effort used in production, such as working hours, skills, and abilities brought in by workers. Capital, on the other hand, encompasses the tools, equipment, machinery, and buildings that aid in production.
These two inputs can often be substituted for one another to some degree. For example, a company might use more machinery (capital) to reduce the number of workers needed. However, this is not always a perfect substitution and the relationship between them is captured using mathematical models such as isoquant curves. Understanding these relationships can help businesses optimize production to maximize efficiency and output.

Production Level

The concept of production level is central to the understanding of isoquant curves. It refers to the total output a company produces using a combination of labor and capital. Each isoquant curve represents different combinations of inputs that result in the same production level, or quantity of output.
This concept is important because it helps businesses determine how to effectively allocate resources. By analyzing isoquant curves, a business can identify the most cost-effective way to produce a certain level of output by modifying the mix of labor and capital inputs. This can further aid in decision-making and strategic planning.

Mathematical Substitution

Mathematical substitution is a technique used to solve equations by replacing variables with their given values. In the context of isoquant curves, this involves taking the given equation, which defines the relationship between labor and capital, and substituting the known value of one variable to solve for the other.
For example, in the problem provided, the isoquant equation is given as \( K = 4000 L^{-2/3} \). If the value of \( L \) is known, such as \( L = 125 \), this value can be substituted directly into the equation to solve for \( K \).

• This step simplifies the problem and allows us to find the desired output, which is the amount of capital needed for a specific quantity of labor to achieve a particular production level.

Exponent Calculation

In mathematical terms, an exponent indicates how many times a number (also known as the base) is multiplied by itself. Calculating exponents involves a few key steps, especially with fractional exponents like \(-2/3\).
Let’s break down the calculation of \( L^{-2/3} \) where \( L = 125 \):

• First, compute the cube root of 125. The cube root is the number which when multiplied by itself three times gives 125. Here, it is 5, since \( 5 \times 5 \times 5 = 125 \).
• Next, apply the negative exponent, \(-2\), to this base. This implies finding the reciprocal after squaring it: \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \).

Understanding exponent calculations is crucial in analyzing mathematical models like isoquant curves, helping to determine the relationship between inputs and output.