Problem 109 - Applied Calculus

Question

ECONOMICS: Pareto's Law The economist Vilfredo Pareto \((1848-1923)\) estimated that the number of people who have an income between \(A\) and \(B\) dollars \((A

Step-by-Step Solution

Answer

The evaluated integral is \( \frac{a}{1-b} \left( B^{1-b} - A^{1-b} \right) \).

Step 1: Understand the Problem

We need to evaluate the definite integral \( N = \int_{A}^{B} a x^{-b} \, dx \) where \( a \) and \( b \) are constants, and \( b eq 1 \).

Step 2: Set Up the Integral

The integral to evaluate is \( \int_{A}^{B} a x^{-b} \, dx \). Notice that \( a \) is a constant and can be factored out of the integral.

Step 3: Apply Integration Formula

Use the power rule of integration for \( x^n \), which is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \). Our integrand is \( x^{-b} \), so \( n = -b \). The integral of \( x^{-b} \) is \( \frac{x^{-b+1}}{-b+1} \).

Step 4: Evaluate the Definite Integral

The definite integral becomes \( a \left[ \frac{x^{-b+1}}{-b+1} \right]_{A}^{B} \). Replace \( x \) with the bounds \( B \) and \( A \).

Step 5: Compute the Result

Substitute the limits into the antiderivative: \( a \left[ \frac{B^{-b+1}}{-b+1} - \frac{A^{-b+1}}{-b+1} \right] = \frac{a}{1-b} \left( B^{1-b} - A^{1-b} \right) \). Ensure \( b eq 1 \) to avoid division by zero.

Key Concepts

Definite Integral

In calculus, a definite integral represents the accumulation of quantities, such as areas under a curve. It is expressed by the integral sign with upper and lower boundaries, in this case, \( \int_{A}^{B} \). The definite integral takes the form of \( \int_{a}^{b} f(x) \, dx \), which calculates the net area between the curve of the function \( f(x) \) and the x-axis from \( x = a \) to \( x = b \). This is extremely useful in various fields to assess total change or accumulated value over a certain interval.
To solve a definite integral, one must evaluate the antiderivative of the function at the upper limit and subtract its evaluation at the lower limit. The result provides the net accumulation of the function over that specific domain. It is crucial to remember that the process of integration reverses differentiation.
In the context of Pareto's Law, the definite integral \( \int_{A}^{B} a x^{-b} \, dx \) allows economists to estimate the distribution of wealth or income within a given range, offering insight into the socioeconomic structure within that range.

Power Rule of Integration

The power rule of integration is a handy tool for integrating functions of the form \( x^n \). This rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \) and \( C \) is the constant of integration. Essentially, the power rule involves increasing the exponent by one and then dividing by this new exponent.
When applied to the problem of Pareto's Law, the function \( x^{-b} \) necessitates using the power rule. By identifying \( n = -b \), the integration results in \( \frac{x^{-b+1}}{-b+1} \).
It's important to note that with definite integrals, the constant of integration \( C \) becomes irrelevant because it cancels itself out when evaluating the antiderivative at specific bounds. This rule streamlines solving definite integrals, making it essential for quick evaluations of polynomial and power functions.

Economics

Pareto's Law is central to economics, particularly when analyzing income distribution. Vilfredo Pareto, an influential economic theorist, observed that wealth is unevenly distributed, where typically a small percentage of the population controls a large portion of total income. This distribution is often represented by a power law, which highlights significant economic inequalities.
The formula \( N = \int_{A}^{B} a x^{-b} \, dx \) specifically measures the number of individuals earning between \( A \) and \( B \) dollars. The constants \( a \) and \( b \) reflect parameters derived from observed data, representing income proportions and socio-economic tendencies.
Through the computation of this integral, economists can gain insights into the quantitative structure of a given economy. This helps policymakers in devising strategies to address inequality and redistribute wealth more equitably. Understanding these integral applications improves economic decision-making, illustrating how mathematical models can influence and illuminate sociopolitical practices.