Economics Dynamics Problems 127

Economics Dynamics Problems 127 - is y n = c 1 (2) n + c 2...

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Discrete dynamic systems 111 c 1 and c 2 . Since y 0 = c 1 r 0 + c 2 s 0 = c 1 + c 2 y 1 = c 1 r + c 2 s then c 1 = y 1 sy 0 r s and c 2 = y 1 ry 0 s r The solution values r and s to the characteristic equation of the dynamic system are the solutions to a quadratic. As in all quadratics, three possibilities can occur: (i) distinct real roots (ii) identical real roots (iii) complex conjugate roots Since the solution values to the quadratic equation are r , s = a ± a 2 + 4 b 2 then we have distinct real roots if a 2 > 4 b , identical roots if a 2 =− 4 b , and complex conjugate roots if a 2 < 4 b . Example 3.10 (real distinct roots) Suppose y n + 2 = y n + 1 + 2 y n The characteristic equation is given by x 2 x 2 = 0 i.e. ( x 2)( x + 1) = 0 Hence, we have two real distinct roots, x = 2 and x =− 1, and the general solution
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Unformatted text preview: is y n = c 1 (2) n + c 2 ( − 1) n If we know y = 5 and y 1 = 4, then c 1 = y 1 − sy r − s = 4 − ( − 1)(5) 2 − ( − 1) = 3 c 2 = y 1 − ry s − r = 4 − (2)(5) ( − 1) − 2 = 2 Hence, the particular solution satisfying these initial conditions is given by y n = 3(2) n + 2( − 1) n As Fgure 3.13 makes clear, this is an explosive system that tends to inFnity over time. The limiting behaviour of the general solution y n = c 1 r n + c 2 s n is determined by the behaviour of the dominant solution. If, for example, r is the dominant...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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