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Economics Dynamics Problems 134

# Economics Dynamics Problems 134 - pp 87–8 THEOREM 3.3 The...

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118 Economic Dynamics Given y 0 = 4 and y 1 = 5, then y 0 = c 1 + c 2 = 4 y 1 = c 1 2 c 2 + 4 = 5 with solutions c 1 = 3 and c 2 = 1 Hence, the general solution satisfying the given conditions is y n = 3 + ( 2) n + 4 n For the nonhomogeneous second-order linear difference equation y n + 2 + ay n + 1 + by n = c y n y , where y is the fixed point, if and only if the complementary solution, y c , tends to zero as n tends to infinity; while y n will oscillate about y if and only if the complementary solution oscillates about zero. Since the complementary solution is the solution to the homogeneous part, we have already indicated the stability of these in section 3.8.1. In the case of the second-order linear difference equations, both homogeneous and nonhomogeneous, it is possible to have explicit criteria on the parameters a and b for stability. These are contained in the following theorem (Elaydi 1996,
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Unformatted text preview: pp. 87–8). THEOREM 3.3 The conditions 1 + a + b > , 1 − a + b > , 1 − b > are necessary and sufFcient for the equilibrium point of both homo-geneous and nonhomogeneous second-order difference equations to be asymptotically stable. 3.9 The logistic equation: discrete version Suppose ± y t + 1 = ay t − by 2 t (3.21) where b is the competition coef±cient. 7 Then y t + 1 = (1 + a ) y t − by 2 t This is a nonlinear recursive equation and cannot be solved analytically as it stands. However, with a slight change we can solve the model. 8 Let y 2 t ± y t y t + 1 7 We shall discuss this coef±cient more fully in chapter 14. 8 This approximate solution is taken from Grif±ths and Oldknow (1993, p. 16)....
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