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Economic Faculty

Economic Faculty. Differential Equations and Economic Applications. LESSON 1 prof. Beatrice Venturi. DIFFERENTIAL EQUATIONS ECONOMIC APPLICATIONS. FIRST ORDER DIFFERENTIAL EQUATIONS. DEFINITION : Let y(x) =“ unknown function” x = free variable y ' = first derivative.

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Economic Faculty

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  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

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