Question

For a positive integer \(n,\) let \(a_{n}\) be the fraction of the US population with income less than or equal to \(\$ n\) thousand dollars. (a) Which is larger, \(a_{25}\) or \(a_{50}\) ? Why? (b) What does the quantity \(a_{50}-a_{25}\) represent in terms of US population? (c) Is there any value of \(n\) with \(a_{n}=0 ?\) Explain. (d) What happens to the value of \(a_{n}\) as \(n\) increases?

Short answer

Answer: \(a_{50}\) is larger than \(a_{25}\), and the difference between these two values (\(a_{50} - a_{25}\)) represents the proportion of the US population with an income between \(\$25,000\) and \(\$50,000\).

Step-by-step solution

a) Comparing \(a_{25}\) and \(a_{50}\)

Since \(a_{25}\) represents the fraction of the US population with an income less than or equal to \(\$25\) thousand dollars and \(a_{50}\) represents the fraction of the US population with an income less than or equal to \(\$50\) thousand dollars, we can determine which one is larger. As \(50\) is greater than \(25\), clearly, anyone who has an income less than or equal to \(\$25\) thousand dollars will also have an income less than or equal to \(\$50\) thousand dollars. Therefore, \(a_{50}\) contains all those people in \(a_{25}\) plus some more. Hence, \(a_{50}\) is larger than \(a_{25}\).

b) The meaning of \(a_{50} - a_{25}\)

The quantity \(a_{50} - a_{25}\) represents the difference in fractions of the US population with incomes less than or equal to \(\$50\) thousand dollars versus less than or equal to \(\$25\) thousand dollars. In practical terms, this quantity shows the proportion of people who have an income between \(\$25,000\) and \(\$50,000\). Thus, \(a_{50} - a_{25}\) tells us the percentage of the US population with income in that range.

c) The existence of \(n\) with \(a_{n} = 0\)

We are asked to determine if there exists a value of \(n\) for which \(a_{n} = 0\). This would mean that, for some positive integer \(n\), no one in the US has an income less than or equal to \(\$n\) thousand dollars. Since there exist people within the US with very low or no income, it is impossible to have \(a_n = 0\), as there will always be a fraction of the population with income less than or equal to any given positive integer \(n\). Thus, there is no value of \(n\) for which \(a_n = 0\).

d) The trend of \(a_{n}\) as \(n\) increases

As \(n\) increases, the value of \(a_n\) will also increase, since a larger income threshold would encompass a greater fraction of the US population. For example, \(a_{25}\) includes everyone earning less than or equal to \(\$25,000\), while \(a_{50}\) includes everyone earning less than or equal to \(\$50,000\). As \(n\) gets larger, \(a_n\) will include a greater percentage of the US population. Eventually, as n gets sufficiently large enough, the value of \(a_n\) would approach 1, meaning that almost the entire US population has an income less than or equal to \(\$n\) thousand dollars.

Key concepts

Economic Analysis

Economic analysis involves studying how resources are distributed within a society, focusing on factors like income distribution. It's crucial to understand how inequities in income can impact broader economic trends.

When analysts examine income distribution, they evaluate how wealth is spread across different segments of the population. This allows them to identify patterns of inequality and understand who holds more economic power.

For example, by analyzing income thresholds like \\(25,000 and \\)50,000, analysts can discern shifts in economic well-being across the population.

• They can identify the proportion of people living below certain income levels.
• This can highlight economic inequalities and help policymakers focus on areas that need improvement.
• Tracking changes over time helps to analyze economic growth or downturns.

Understanding these aspects is vital for comprehensively assessing economic health and developing strategies for more equitable resource distribution.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a statistical tool used to describe the distribution of a variable by showing the probability that this variable will take a value less than or equal to a specific number.

In the context of income, the CDF can demonstrate how many individuals earn less than a certain amount. For instance, the function we discussed in the problem, denoted as \(a_n\), gives the fraction of the population with income less than or equal to \$\ housand.

Essentially, the CDF for income helps illustrate the spread of income throughout a population.

• It provides a graphical representation of how income is distributed among people.
• Higher CDF values indicate a higher fraction of the population earning less than a specific income threshold.
• The CDF is a tool for visualizing inequalities in income and can guide socioeconomic reforms.

This serves as a powerful method for analyzing and visualizing income inequality effectively.

Income Thresholds

Income thresholds act as benchmarks for comparing different income levels within a population. They are essential for identifying economic disparities and understanding how wealth is distributed.

Consider the given thresholds of \\(25,000 and \\)50,000. These thresholds allow us to compare different segments of the population and determine the proportion of people earning below these amounts.

Unfortunately, many people visualize income thresholds as just static numbers, but they have dynamic implications:

• They dictate who is below the poverty line and who qualifies for certain benefits.
• By examining the differences in population segments below various thresholds, you can gain insights into economic well-being and poverty levels.
• Income thresholds help in policy-making, allowing for targeted interventions to support those in need.

The careful analysis of such thresholds can lead to more informed economic policies aimed at reducing inequality and promoting fair economic opportunities for everyone.