MAT 125 – Applied Calculus Applications of the Definite Integral to Business and Economics

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 Applications of the Definite Integral to Business and Economics 2

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

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 Applications of the Definite Integral to Business and Economics 4

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

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

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

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

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 Applications of the Definite Integral to Business and Economics 9

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

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

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

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 Applications of the Definite Integral to Business and Economics 13

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) = f (x) ≤ x for every x in [0, 1]. 5. f is increasing on [0, 1]. Dr. Erickson Applications of the Definite Integral to Business and Economics 14

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

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

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

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