1. Economic Faculty Differential Equations and Economic Applications LESSON 1 prof. Beatrice Venturi

2. DIFFERENTIAL EQUATIONS ECONOMIC APPLICATIONS Beatrice Venturi

3. FIRST ORDER DIFFERENTIAL EQUATIONS DEFINITION: Let • y(x) =“ unknown function” • x = free variable • y' = firstderivative First order Ordinary Differential Equation . Beatrice Venturi

4. FIRST ORDER DIFFERENTIAL EQUATIONS DEFINITION: An ordinary differential equation (or ODE) is an equation involving derivates of: y(x) (the unknown function) a real value function (of only one independent variable x) defined in y:(a,b) R an open interval (a,b). Beatrice Venturi

5. FIRST ORDERDIFFERENTIAL EQUATIONS • More generally we may consider the following equation: • Where f is the known function. (*) Beatrice Venturi

6. SolutionofE.D.O. • Definition: A solution or integral curve of an EDO is a function g(x) suchthatwhenitissubstitutedinto (*) itreduces (*) toanidentity in a certain open interval (a,b) in R. • We find a solution of an EDO by integration. matematica per economisti Beatrice Venturi

7. 1.EXAMPLE Beatrice Venturi

8. The Domar’s Growth Model Beatrice Venturi

9. InvestmentI and Capital Stock K • Capital accumulation = process for which new shares of capital stock K are added to a previous stock . Beatrice Venturi

10. Connection betweenCapital Stock and Investment Capitalstock= Investment = matematica per economisti Beatrice Venturi

11. Connection betweenCapital and Investment matematica per economisti Beatrice Venturi

12. Connection betweenCapital and Investment matematica per economisti B eatriceVenturi

13. Connection betweenCapital and Investment matematica per economisti Beatrice Venturi

14. Connection betweenCapital and Investment matematica per economisti Beatrice Venturi

15. Price adjustment in the market • Weconsider the demandfunction: and the supplyfunction: for a commodity matematica per economisti Beatrice Venturi

16. Price adjustment in the market • At the equilibriumwhensupplybalancesdemand , the equilibriumpricessatisfies: matematica per economisti Beatrice Venturi

17. Price adjustment in the market Suppose the market not in equilibrium initially. We study the way in which price varies over time in response to the inequalities between supply and demand. matematica per economisti Beatrice Venturi

18. Price adjustment in the market matematica per economisti Beatrice Venturi

19. Price adjustment in the market • Weuse the methodofintegrantingfactors. • Wemultiplyby the factor matematica per economisti Beatrice Venturi

20. Price adjustment in the market To find c put t=0 Solution = matematica per economisti Beatrice Venturi

21. The equilibrium price P isasymptoticallystableequilibrium matematica per economisti Beatrice Venturi

22. SEPARATION OF VARIABLES. This differential equation can be solved by separation of variables. The method “ separates” the two variables y and x placing them in diffent sides of the equation: matematica per economisti Beatrice Venturi

23. Each sides is then integrated: matematica per economisti Beatrice Venturi

24. The Domar Model s(t)= marginal propensity to save is a function of t Beatrice Venturi

25. PARTICULAR SOLUTION • DEFINITION • Theparticularintegralor • solutionofE.D.O. is a function : obtainedbyassigningparticularvaluesto the arbitraryconstant Beatrice Venturi

26. Example • Given the initialcondition • the solutionisunique Beatrice Venturi

27. matematica per economisti Beatrice Venturi

28. The graphof the particularsolution Beatrice Venturi

29. Case: C₁= 0 y=(1/3)x³ Beatrice Venturi

30. INTEGRALE SINGOLARE Wehavesolutionthatcannotbeobtainedby assigning a valueto a the constant c. Beatrice Venturi

31. Example: Beatrice Venturi

32. y=0 is a solution butthissolutioncannotbeabtainedbyassing a valueto c from the generale solution. Beatrice Venturi