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Mathe III Lecture 2

This lecture delves into the Cobweb Model, focusing on profit-maximizing farms that raise pigs. It examines the cost of raising pigs, demand dynamics, and how farmers make decisions on production levels over time. Key concepts include the stationary state price, present value of consumption and income streams, and mortgage repayments. The session also explores the implications of linear equations with variable coefficients and the principles of existence and uniqueness in economic models. Gain insight into economic modeling and decision-making processes.

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Mathe III Lecture 2

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  1. Mathe III Lecture 2 Mathe III Lecture 2

  2. Mathematik III Chong-Dae Kim Donnerstag 9.15 – 10.45 Uhr – HS A Donnerstag 10.45 – 12.15 Uhr – HS A Donnerstag 12.15 – 13.45 Uhr – HS A Freitag 13.00 – 14.30 Uhr – HS G

  3. The Cobweb Model

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

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

  6. The Stationary State p* :

  7. Graphic illustration:

  8. etc. etc

  9. Savings Equation: Its solution: a reminder

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

  11. After 1 period Etc.

  12. After t periods is the Present Value of is the Present Value of

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

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

  15. Mortgage Repayments

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

  17. We found that: and:

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

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

  20. Linear Equations with a Variable Coefficient

  21. Linear Equations with a Variable Coefficient

  22. The solution to: is: or: example

  23. The general solution becomes:

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

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

  26. Second Order Equations etc. etc.

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

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