1. Mathe III Lecture 2 Mathe III Lecture 2

2. 2 Tutorien für Mathematik III Im WS 05/06 Tutor: ChongDae KIM Mo. 11:00 Uhr - 12.30Uhr HS N. Mo. 12.30 Uhr - 14.00Uhr HS N. Di. 9.30 Uhr - 11.00Uhr HS N. Di.13.30 Uhr - 15.00Uhr HS N.

3. 3 The Cobweb Model

4. 4 Cost of raising q pigs: : Demand for pigs: N (profit maximizing) farms producing pigs Each farm, takes p as given and maximizes: It takes one period to raise a pig

5. 5 Supply: Farmers decide at t-1 on the production at t

6. 6 The Stationary State p* :

7. 7 Graphic illustration:

8. 8 etc. etc

9. 9

10. 10

11. 11 a reminder Savings Equation: Its solution:

12. 12 a 2 nd reminder First order equation with constant coefficient Its solution:

13. 13 Etc. After 1 period

14. 14 After t periods is the Present Value of

15. 15

16. 16 The present values of the streams of consumption and income are equal The present values of the streams of consumption and income are equal

17. 17 Mortgage Repayments Outstanding Balance At time t Repayment per period

18. 18 Mortgage Repayments

19. 19 Ad - Kan

20. 20 The loan equals the present value of T payments of z

21. 21 We found that: and:

22. 22 For t-1 this becomes: Interest on last period’s principal Payment towards the principal

23. 23 Payment towards the principal For t=1: For t=T: Interest: principal repayment: Interest: principal repayment: small large small large

24. 24 Linear Equations with a Variable Coefficient

25. 25 Linear Equations with a Variable Coefficient

26. 26 example The solution to: is: or:

27. 27 The general solution becomes:

28. 28 then: ( the present value, at t = 0, of $1 at t )

29. 29

30. 30 or: the present value at period 0 the present value at period t

31. 31 Second Order Equations etc.

32. 32 Existence and Uniqueness : The equation has a unique solution for any given