The Lorenz curve y=F(r) is used by economists to study...

The Lorenz curve $y=F(r)$ is used by economists to study income distribution in a given country(see Figure 15). By definition, $F(r)$ is the fraction of the total income that goes to the bottom $\mathrm{rth}$ part of the population, where $0 \leq r \leq 1 .$ For example, if $F(0.4)=0.245,$ then the bottom $40 \%$ of households receive $24.5 \%$ of the total income. Note that $F(0)=0$ and $F(1)=1$ The following table provides values of $F(r)$ for the United States in $2010 .$ Assume that the national average income was $A=\$ 66,000$. (a) What was the average income in the lowest $40 \%$ of households? (b) Show that the average income of the households belonging to the interval [0.4,0.6] was $\$ 48,180$ (c) Estimate $F^{\prime}(0.5) .$ Estimate the income of households in the 50 th percentile. Was it greater or less than the national average?

The Lorenz curve $y=F(r)$ is used by economists to study income distribution in a given country(see Figure 15). By definition, $F(r)$ is the fraction of the total income that goes to the bottom $\mathrm{rth}$ part of the population, where $0 \leq r \leq 1 .$ For example, if $F(0.4)=0.245,$ then the bottom $40 \%$ of households receive $24.5 \%$ of the total income. Note that $F(0)=0$ and $F(1)=1$
The following table provides values of $F(r)$ for the United States in $2010 .$ Assume that the national average income was $A=\$ 66,000$.
(a) What was the average income in the lowest $40 \%$ of households?
(b) Show that the average income of the households belonging to the interval [0.4,0.6] was $\$ 48,180$
(c) Estimate $F^{\prime}(0.5) .$ Estimate the income of households in the 50 th percentile. Was it greater or less than the national average?

Key Concepts

Lorenz Curve

The Lorenz curve is a graphical tool used in economics to depict the cumulative distribution of income or wealth within a population. It plots the cumulative percentage of total income received versus the cumulative percentage of the population, starting from the poorest. This curve is crucial for visualizing inequality and understanding how evenly (or unevenly) income is distributed among households.

Income Distribution

Income distribution refers to how a nation’s total income is divided among its population. It is a fundamental concept in economics, often analyzed to assess economic inequality and social welfare. Measuring income distribution helps in comparing economic health, designing fiscal policies, and targeting social programs.

Average Income Calculation

Determining the average income within a subset of households involves using the proportion of total income received by that group along with the national average income. This calculation typically requires integrating or averaging the income shares identified by the Lorenz curve over a particular segment of the population, allowing for meaningful comparisons between subgroups and the overall population.

Differentiation of the Lorenz Curve

The derivative of the Lorenz curve at a specific point indicates how quickly income accumulates across the population. Economically, this derivative can be interpreted as the local rate of change of income share with respect to the population share, providing an estimate of the average income for households at that percentile. It helps in understanding marginal income differences and the distribution dynamics around particular sections of the population.

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