Question

There are 100 households in the economy of Equalor. Initially, 99 of them have an income of \(\$ 10,000\) each, and one household has an income of \(\$ 1,010,000\). a. What is the median income in this economy? What is the mean income? Through its poverty programs, the government of Equalor now redistributes income: it takes \(\$ 990,000\) away from the richest household and distributes it equally among the remaining 99 households. b. What is the median income in this economy now? What is the mean income? Has the median income changed? Has the mean income changed? Which indicator (mean or median household income) is a better indicator of the typical Equalorian household's income? Explain.

Short answer

#tag_title# Initial Median and Mean Income Calculation #tag_content# Median Income \(= \frac{\$10,000 + \$10,000}{2} = \$10,000\) Mean Income \(= \frac{99(\$ 10,000) + (\$ 1,010,000)}{100} = \frac{\$ 990,000 + (\$ 1,010,000)}{100} = \frac{\$ 2,000,000}{100} = \$20,000\) The initial median household income is \(\$10,000\), and the initial mean household income is \(\$20,000\). #Part b# #tag_title# New Median Income #tag_content# After the government has redistributed income, each household will have an income of \(\$20,000\). Since all incomes are now the same, we can easily find the new median income: New Median Income = \$20,000 #tag_title# New Mean Income #tag_content# Since each household now has an income of \(\$20,000\), the new mean income does not change: New Mean Income = \$20,000 #Part c# #tag_title# Best Indicator for a Typical Equalorian Household's Income #tag_content# After the redistribution, both the median and mean income are the same at \(\$20,000\). This means that the typical Equalorian household's income is best represented by both the median and mean income, which is \(\$20,000\).

Step-by-step solution

Initial Median Income

To find the median income, we must first arrange the incomes in ascending order. Since there are 99 households with an income of \(10,000 each, and one household has an income of \)1,010,000: \$10,000, \$10,000, ..., \$10,000 (99 times), \$1,010,000 As the number of households (100) is an even number, the median income will be the average of the incomes of the 50th and 51st households.

Initial Mean Income

The mean income is calculated by adding up all the incomes and dividing by the number of households. Thus, Mean Income \(= \frac{99(\$ 10,000) + (\$ 1,010,000)}{100}\)

Key concepts

Median Income

The median income is a key concept when discussing income distribution, as it represents the income level at which half the households earn more and half earn less. It's a valuable measure because it isn't skewed by extremely high or low incomes, providing a clearer picture of a typical household's earning within the economy.
Let's look at the example from the Equalor economy. There are 100 households, 99 of them earn $10,000, but one household earns $1,010,000. To find the median income, arrange the incomes in order; because there is an even number of households, the median is the average of the 50th and 51st incomes. Both of these households earn $10,000, so the median income before any redistribution is $10,000.
After redistribution of income through poverty programs, the richest household’s income is significantly reduced, and additional funds are distributed evenly among 99 households. Despite these adjustments, the median remains $10,000 because this measure focuses strictly on the middle value and not extremes or the sum total. Thus, the median often remains stable, reflecting the central trend amidst varied income distribution.

Mean Income

To understand mean income, imagine pooling the entire economy's income and dividing it equally among all households. It provides an average amount based on total economic output, which can easily be swayed by extreme incomes.
In Equalor, initially, the mean income is calculated straightforwardly by summing all household incomes—99 households earning \(10,000 and one earning \)1,010,000—and then dividing by 100. This formula gives the mean: \[\text{Mean Income} = \frac{99(10,000) + 1,010,000}{100} = \frac{10,000,000 + 1,010,000}{100} = \frac{11,010,000}{100} = 110,100\]As evident, this figure is heavily influenced by the single very wealthy household, making it seem like the average household income is significantly higher than most actually earn.
Post-redistribution via poverty programs, the richest household's contribution diminishes, and funds are spread across the others, yielding new averages. Thus reflecting a considerably different mean. While useful, mean income can misrepresent the typical income level due to its sensitivity to high outliers.

Poverty Programs

Poverty programs aim to alleviate income disparities by redistributing wealth. In Equalor's case, these programs demonstrate how governments can intervene to benefit the majority.
Initially, a massive income gap exists due to the disparity created by a single wealthy household amidst generally lower incomes. By reallocating $990,000 from the wealthiest household to the others, such programs elevate living standards for the majority and attempt to level the playing field.
This redistribution doesn't alter the median income due to its nature of focusing on the middle values, but it significantly affects the mean—a reflection of the economic overhaul. Here, the mean becomes more representative of the majority rather than skewed by extremes, showcasing how strategic redistribution can create perceived economic equity.

• These programs highlight the disparity in income distribution.
• They aim to bring support to lower-income households.
• Help reduce economic inequality and improve overall welfare.

Overall, while both mean and median incomes are valuable for different reasons, poverty programs mainly aim to bring major relief to lower-income households than measuring income distributions, shaping a fairer society.