1. Learning Objectives for Section 14
Learning Objectives for Section Applications in Business/Economics
1. The student will be able to construct and interpret probability density functions.
2, The student will be able to evaluate a continuous income stream.
3. The student will be able to evaluate consumers’ and producers’ surplus.
Barnett/Ziegler/Byleen Business Calculus 11e

2. Probability Density Functions
A probability density function must satisfy:
f (x)  0 for all x
The area under the graph of f (x) is 1
If [c, d] is a subinterval then Probability (c  x  d) =
Barnett/Ziegler/Byleen Business Calculus 11e

3. Probability Density Functions (continued)
Sample probability density function
Barnett/Ziegler/Byleen Business Calculus 11e

4. Example
In a certain city, the daily use of water in hundreds of gallons per household is a continuous random variable with probability density function
Find the probability that a household chosen at random will use between 300 and 600 gallons.
Barnett/Ziegler/Byleen Business Calculus 11e

5. Insight
The probability that a household in the previous example uses exactly 300 gallons is given by:
In fact, for any continuous random variable x with probability density function f (x), the probability that x is exactly equal to a constant c is equal to 0.
Barnett/Ziegler/Byleen Business Calculus 11e

6. Continuous Income Stream
Total Income for a Continuous Income Stream:
If f (t) is the rate of flow of a continuous income stream, the total income produced during the time period from t = a to t = b is
a Total Income b
Barnett/Ziegler/Byleen Business Calculus 11e

7. Example
Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is
f (t) = 600 e 0.06t
Barnett/Ziegler/Byleen Business Calculus 11e

8. Example
Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is
f (t) = 600 e 0.06t
Barnett/Ziegler/Byleen Business Calculus 11e

9. Future Value of a Continuous Income Stream
From previous work we are familiar with the continuous compound interest formula
A = Pert.
If f (t) is the rate of flow of a continuous income stream, 0  t  T, and if the income is continuously invested at a rate r compounded continuously, the the future value FV at the end of T years is given by
Barnett/Ziegler/Byleen Business Calculus 11e

10. Example Let’s continue the previous example where f (t) = 600 e0.06 t
Find the future value in 2 years at a rate of 10%.
Barnett/Ziegler/Byleen Business Calculus 11e

11. Example Let’s continue the previous example where f (t) = 600 e0.06 t
Find the future value in 2 years at a rate of 10%.
r = 0.10, T = 2, f (t) = 600 e 0.06t
Barnett/Ziegler/Byleen Business Calculus 11e

12. Consumers’ Surplus
If is a point on the graph of the price-demand equation
P = D(x), the consumers’ surplus CS at a price level of is
which is the area between p =
and p = D(x) from x = 0 to x = CS x p
The consumers’ surplus represents the total savings to consumers who are willing to pay more than for the product but are still able to buy the product for .
Barnett/Ziegler/Byleen Business Calculus 11e

13. Example
Find the consumers’ surplus at a price level of for the price-demand equation
p = D (x) = 200 – 0.02x
Barnett/Ziegler/Byleen Business Calculus 11e

14. Example
Find the consumers’ surplus at a price level of for the price-demand equation
p = D (x) = 200 – 0.02x
Step 1. Find the demand when the price is
Barnett/Ziegler/Byleen Business Calculus 11e

15. Example (continued) Step 2. Find the consumers’ surplus:
Barnett/Ziegler/Byleen Business Calculus 11e

16. Producers’ Surplus
If is a point on the graph of the price-supply equation p = S(x), then the producers’ surplus PS at a price level of is x p CS
which is the area between and p = S(x) from x = 0 to
The producers’ surplus represents the total gain to producers who are willing to supply units at a lower price than but are able to sell them at price .
Barnett/Ziegler/Byleen Business Calculus 11e

17. Example
Find the producers’ surplus at a price level of for the price-supply equation
p = S(x) = x
Barnett/Ziegler/Byleen Business Calculus 11e

18. Example
Find the producers’ surplus at a price level of for the price-supply equation
p = S(x) = x x2
Step 1. Find , the supply when the price is
Solving for using a graphing utility:
Barnett/Ziegler/Byleen Business Calculus 11e

19. Example (continued) Step 2. Find the producers’ surplus:
Barnett/Ziegler/Byleen Business Calculus 11e

20. Summary We learned how to use a probability density function.
We defined and used a continuous income stream.
We found the future value of a continuous income stream.
We defined and calculated a consumer’s surplus.
We defined and calculated a producer’s surplus.
Barnett/Ziegler/Byleen Business Calculus 11e