Problem 92 - 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 \(L\). \(K=4000 L^{-2 / 3}\) and \(L=125\)

Step-by-Step Solution

Answer

The value of \( K \) is 160.

Step 1: Substitute the Given Value of L

We are given the equation for the isoquant curve as \( K = 4000 L^{-2/3} \) and a specific value \( L = 125 \). The first step is to substitute \( L \) with 125 in this equation: \( K = 4000 \times 125^{-2/3} \).

Step 2: Calculate Exponentiation of L

Calculate \( 125^{-2/3} \). First, find the cube root of 125, which is 5, since \( 5^3 = 125 \). Then calculate \( 5^{-2} \): \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \). So, \( 125^{-2/3} = \frac{1}{25} \).

Step 3: Calculate K

Now replace \( 125^{-2/3} \) with \( \frac{1}{25} \) in the equation from Step 1. This gives us: \( K = 4000 \times \frac{1}{25} \). Calculate the multiplication to find \( K \): \( K = \frac{4000}{25} = 160 \).

Key Concepts

Labor and Capital

In economics, labor and capital are two intrinsic components of the production process. Labor refers to the human effort used in the process of production, encompassing physical and mental activities. Capital, on the other hand, involves the assets used to improve productivity like factory buildings, machinery, and raw materials. These two elements are often invested together to generate goods and services efficiently.

An isoquant curve is a helpful tool in this realm, representing different combinations of labor and capital that can produce the same level of output. The curve illustrates the trade-off between labor and capital, showing that if you have more of one, you might need less of the other to achieve the same production level. This understanding helps businesses optimize their resources, finding the most cost-effective way to produce goods.

Understanding how labor and capital work together along an isoquant curve is vital for businesses aiming to maximize efficiency while minimizing costs.

Production Level

The production level refers to the amount of goods or services that a company outputs within a given period. It is a critical aspect of business operations because it directly relates to the profitability and efficiency of an organization.

The isoquant curve plays a vital role in determining how resources like labor and capital are allocated to maintain a steady production level. For a specific production level, the isoquant curve shows combinations of labor and capital that result in the same output. This allows businesses to understand the flexibility they have in resource allocation, helping to adjust inputs without altering the production quantity.

By analyzing different points along an isoquant curve, businesses can determine the most efficient way to operate, considering cost constraints and resource availability. This understanding ensures that production levels are optimized for profitability and sustainability.

Exponentiation

Exponentiation, often utilized in mathematics, involves raising a number (the base) to a power (the exponent) to get another number. In economic modeling and equations for isoquant curves, exponentiation helps express the relationship between labor and capital succinctly.

In the given solution, the equation for the isoquant curve includes exponentiation: \( K = 4000 L^{-2/3} \). Here, labor \( L \) is raised to the power of \(-2/3\). This exponent shows how the quantity of labor affects capital, or vice versa, on the isoquant curve.

Breaking down exponentiation helps in not only calculating exact values but also understanding the rate of substitution, which is vital in observing how much one input needs to change to compensate for changes in another to keep production constant. This mathematical concept thus aids businesses in formulating strategies for optimal resource allocation.