Problem 91 - Brief Applied Calculus

Question

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 \(L\). \(K=3000 L^{-1 / 2}\) and \(L=225\)

Step-by-Step Solution

Answer

The value of \(K\) is 200 when \(L = 225\).

Step 1: Understand the Problem

We are given the isoquant equation \(K = 3000 L^{-1/2}\) and need to find the capital \(K\) when the labor \(L\) is 225.

Step 2: Substitute the Given Value of L

Substitute \(L = 225\) into the isoquant equation: \[ K = 3000 \times 225^{-1/2} \]

Step 3: Calculate the Power of L

Calculate \(225^{-1/2}\). This means taking the square root of 225 and then taking the reciprocal.\(225^{1/2} = 15\), so \(225^{-1/2} = \frac{1}{15}\).

Step 4: Calculate the Resulting Value of K

Substitute the calculated value of \(225^{-1/2}\) back into the equation: \[ K = 3000 \times \frac{1}{15} = 3000 \div 15 = 200 \]

Step 5: Verify the Calculation

Double-check the division step to ensure the value of \(K\) was computed correctly: Divide 3000 by 15 to get 200, confirming \(K = 200\).

Key Concepts

Labor and Capital

In economics, labor and capital are two essential inputs in production processes. **Labor** refers to the human effort put into production, while **capital** encompasses tools and machinery like buildings, equipment, and raw materials. These inputs are used in various combinations to achieve a desired output level. An isoquant curve serves as a graphical representation of how labor and capital can be combined to maintain the same production level.

The idea is that, by increasing one input—either labor or capital—you might reduce reliance on the other while keeping production constant. For instance, if a factory introduces more advanced machinery, it might require fewer employees. Understanding this allows businesses to optimize their operations, balancing the inputs in ways that might be more cost-effective or productive.

Production Level

Production level refers to the amount of goods or services a company aims to produce. It is a central element in determining how labor and capital resources will be allocated. On an isoquant curve, each point represents a combination of labor and capital that results in the same production output.

For example, if we consider a production level that results from 50 units of labor and 30 units of capital, moving along the isoquant curve could mean using 40 units of labor with a slightly higher level of capital input but achieving the same total production. Companies use these insights to adapt and plan how resources should be utilized to meet demand without necessarily increasing costs.

Mathematical Substitution

Mathematical substitution is a key step in solving isoquant curve equations. To find out the value of one variable, such as capital (\(K\)), when given a specific value for another variable, such as labor (\(L\)), substitution is necessary.

In the given problem, the equation \(K = 3000 L^{-1/2}\) is used. To solve for \(K\) when \(L = 225\), the value 225 is substituted into the equation to find the specific amount of capital needed under this condition. This means replacing \(L\) in the equation with the number 225 and simplifying the expression to get the result for \(K\).

Substitution helps transform equations into solvable problems, making it easier to apply static conditions, like a fixed level of labor, to dynamic groups, such as varying capital requirements.

Reciprocal and Square Root

Understanding the concepts of reciprocal and square root is crucial for interpreting equations like \(K = 3000 L^{-1/2}\). The term \(L^{-1/2}\) may seem complex, but it can be broken down:

- **Square Root**: The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, \(\sqrt{225} = 15\).
- **Reciprocal**: The reciprocal of a number is 1 divided by that number. If you have the number 15, its reciprocal is \(\frac{1}{15}\).

Combining these, \(225^{-1/2}\) means finding the reciprocal of the square root of 225. First, calculate \(225^{1/2} = 15\), then the reciprocal gives \(\frac{1}{15}\). Understanding these mathematical operations can demystify how to handle exponents in expressions, leading to easier manipulation and solution of equations.