Chapter 7 Additional Integration Topics Section 1 Area Between Curves

2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 7.1 Area Between Two Curves The student will be able to: Day 1: Determine the unsigned area between a curve and the x axis. Determine the unsigned area between two curves. Day 2: Solve applications involving area between curves

3 Application: Oil Production Barnett/Ziegler/Byleen Business Calculus 12e

4 Oil Production (continued) Barnett/Ziegler/Byleen Business Calculus 12e From the end of the 5 th year to the end of the 15 th year, about 120 thousand barrels of oil were produced.

Barnett/Ziegler/Byleen Business Calculus 12e

6 Application: Income Distribution The U.S. Bureau of the Census compiles data on distribution of income among households. Here’s data for 2011.

7 Barnett/Ziegler/Byleen Business Calculus 12e Application: Income Distribution Using regression analysis, this data can be fit to a curve called a Lorenz curve. It was developed by economist Max O. Lorenz in To make a scatter plot of the curve: X-axis: Percentage of households (20% increments) Y-axis: Percentage of total income (10% increments)

8 Barnett/Ziegler/Byleen Business Calculus 12e A Comparison of 1968 and Income equality

9 Barnett/Ziegler/Byleen Business Calculus 12e Income Distribution In a perfect society, everyone would have equal shares of the income. The bottom 20% of households would receive 20% of the total income. The bottom 40% of households would receive 40% of the total income. And so on… If we have absolute equality of income, the Lorenz curve is the line y = x.

10 Barnett/Ziegler/Byleen Business Calculus 12e Income Distribution If the area between the two curves decreases, then there is more income equality. If the area between the curves increases, there is more income inequality. In the worst case, L(x) will be the line y = 0 which would mean that all the income would be in the hands of one family and the rest of the families would have nothing

11 Barnett/Ziegler/Byleen Business Calculus 12e Income Distribution Economists use the area between the line y = x and the Lorenz curve to determine the fairness of distribution of wealth. What is the maximum area between the two curves? What is the minimum area? So the area between the Lorenz curve and the line y = x is between. Economists multiply the area by 2 so that their answer falls between 0 and 1. This area is known as the Gini Index. 1/2 0 0 – ½

12 Barnett/Ziegler/Byleen Business Calculus 12e Gini Index The Gini Index is named after the Italian sociologist Corrado Gini ( ). The Gini Index is two times the area between the Lorenz curve and the line y = x. It will result in values between 0 and 1.

13 Barnett/Ziegler/Byleen Business Calculus 12e Gini Index Gini Index of Income Concentration If y = f (x) is the Lorenz curve, then Index of income concentration = A Gini index of 0 indicates absolute equality (all people share equally in the income). A Gini index of 1 indicates absolute inequality (one person has all the income and the rest have none).

Barnett/Ziegler/Byleen Business Calculus 12e

15 Changes in U.S. Gini Index  According to the Census Bureau, the distribution of income has become somewhat more unequal in the past 40 years.  Possible reasons? Changes in family structure (the % of households headed by single moms has doubled since 1968) Technological change (there is a higher demand for skilled workers who have college degrees) Barnett/Ziegler/Byleen Business Calculus 12e

16 Barnett/Ziegler/Byleen Business Calculus 12e Example A country is planning changes in its tax structure in order to provide a more equitable distribution of income. The current Lorenz curve is: f (x) = x 2.3 With the new tax structure, the Lorenz curve would be: g(x) = 0.4x + 0.6x 2 Will the proposed changes work?

17 Barnett/Ziegler/Byleen Business Calculus 12e Example (continued) Current: Gini Index of income concentration Projected: Gini Index of income concentration The Gini index will decrease, so the projected distribution of income will be more equitable, therefore the changes to the tax structure will be beneficial. (Use your graphing calc to evaluate)

18 U.S. Gini Indices  The top 10 states with the highest Gini index numbers are: New York – highest income inequality in the U.S. Connecticut Louisiana Massachusetts Florida Alabama California Texas Georgia Mississippi Barnett/Ziegler/Byleen Business Calculus 12e U.S. Cities with the lowest Gini index numbers (highest income equality): Salt Lake City Virginia Beach Minneapolis

19 Barnett/Ziegler/Byleen Business Calculus 12e Summary ■ Definite integrals can be used to solve problems involving rates of change. ■ They can also be used to calculate the Gini index of income concentration.

20 Homework Barnett/Ziegler/Byleen Business Calculus 12e