1. MAT 125 – Applied Calculus 6.7 - Applications of the Definite Integral to Business and Economics

2. Today’s Class  We will be learning the following concepts today:  Consumers’ and Producers’ Surplus  The Future and Present Value of an Income Stream  Lorenz Curves and Income Distributions Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 2

3. Introduction  In this section, we will be considering several applications of the definite integral in the fields of business and economics. Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 3

4. Consumers’ Surplus  The consumers’ surplus is the difference between what the consumers would be willing to pay for a commodity and what they actually pay for them. Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 4

5. Producers’ Surplus  The producers’ surplus is the difference between what the suppliers actually receive and what they would be willing to receive. Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 5

6. Example 1 Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 6

7. Example 2 Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 7

8. Example 3 Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 8

9. Annuities  The amount of an annuity is the sum of the payments plus the interest earned. Where P = size of each payment in the annuity r = interest rate compounded continuously T = term of the annuity in years m = number of payments per year Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 9

10. Annuities Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 10

11. Example 4 Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 11

12. Example 5 Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 12

13. Lorenz Curve  Let f (x) denote the proportion of the total income received by the poorest 100x% of the population for 0 ≤ x ≤ 1.  Using this terminology, f (0.3) = 0.1 simply states that the lowest 30% of the income recipients receive 10% of the total income. Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 13

14. Lorenz Curve  The function f has the following properties: 1.The domain of f is [0, 1]. 2.The range of f is [0, 1]. 3. f (0) = 0 and f (1) = 1. 4. f (x) ≤ x for every x in [0, 1]. 5. f is increasing on [0, 1]. Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 14

15. Lorenz Curve Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 15

16. Example 6 Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 16

17. Example 7 Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 17

18. Next Class  Review/Make-up/Final Exam  THE END! Dr. Erickson 6.7 - Applications of the Definite Integral to Business and Economics 18