Economics Dynamics Problems 117

Economics Dynamics Problems 117 - Consider the following...

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Discrete dynamic systems 101 where p 0 is the initial population size. If population is growing at all, k > 0, then this population will grow over time becoming ever larger. We shall discuss population more fully in chapter 14. Example 3.4 As a second example, consider the Harrod–Domar growth model in discrete time S t = sY t I t = v ( Y t Y t 1 ) S t = I t This gives a Frst-order homogeneous difference equation of the form Y t = ± v v s ² Y t 1 with solution Y t = ± v v s ² t Y 0 If v > 0 and v > s then v / ( v s ) > 1 and the solution is explosive and nonoscil- latory. On the other hand, even if v > 0if s > v then the solution oscillates, being damped if s < 2 v , explosive if s > 2 v or constant if s = 2 v . The analytical solution to the Frst-order linear homogeneous equation is use- ful because it also helps to solve Frst-order linear nonhomogeneous equations.
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Unformatted text preview: Consider the following general Frst-order linear nonhomogeneous equation y t + 1 = ay t + c (3.10) Asimplewaytosolvesuchequations,andoneparticularlyusefulfortheeconomist, is to transform the system into deviations from its Fxed point, deviations from equilibrium. Let y denote the Fxed point of the system, then y = ay + c y = c 1 a Subtracting the equilibrium equation from the recursive equation gives y t + 1 y = a ( y t y ) Letting x t + 1 = y t + 1 y and x t = y t y then this is no more than a simple ho-mogeneous difference equation in x x t + 1 = ax t with solution x t = a t x Hence, y t y = a t ( y y )...
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