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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. Solution of E.D.O. • Definition: A solution or integral curve of an EDO is a function g(x) such that when it is substituted into (*) it reduces (*) to an identity in a certain open interval (a,b) in R. • We find a solution of an EDO by integration. 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 Capital stock= Investment = Beatrice Venturi

  11. Connection betweenCapital and Investment Beatrice Venturi

  12. Connection betweenCapital and Investment B eatrice Venturi

  13. Connection betweenCapital and Investment Beatrice Venturi

  14. Connection betweenCapital and Investment Beatrice Venturi

  15. Price adjustment in the market • We consider the demand function: and the supply function : for a commodity Beatrice Venturi

  16. Price adjustment in the market • At the equilibrium when supply balances demand , the equilibrium prices satisfies: 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. Beatrice Venturi

  18. Price adjustment in the market Beatrice Venturi

  19. Price adjustment in the market • We use the method of integranting factors. • We multiply by the factor Beatrice Venturi

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

  21. The equilibrium price P is asymptotically stable equilibrium 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: Beatrice Venturi

  23. Each sides is then integrated: Beatrice Venturi

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

  25. PARTICULAR SOLUTION • DEFINITION • The particularintegral or • solution of E.D.O. is a function : obtained by assigning particular values to the arbitrary constant Beatrice Venturi

  26. Example • Given the initial condition • the solution is unique Beatrice Venturi

  27. Beatrice Venturi

  28. The graph of the particular solution Beatrice Venturi

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

  30. INTEGRALE SINGOLARE We have solution that cannot be obtained by assigning a value to a the constant c. Beatrice Venturi

  31. Example: Beatrice Venturi

  32. y=0 is a solution but this solution cannot be abtained by assing a value to c from the generale solution. Beatrice Venturi

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