Economics Dynamics Problems 138

# Economics Dynamics Problems 138 - ak . Letting y t + 1 = Ay...

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122 Economic Dynamics (3) Information demand has a constant elasticity, so: (iii) y t + 1 y t y t =− ε ± p t + 1 p t p t ² Substituting (i) into (ii) and the result into (iii), we obtain y t + 1 y t y t =− εν ( a + by t ) Assume εν = k > 0 then y t + 1 y t y t =− k ( a + by t ) i.e. y t + 1 = (1 ak ) y t kby 2 t which is no more than a logistic equation. In equilibrium y t = y for all t , hence y = (1 ak ) y kby 2 y ( ak + kby ) = 0 and y = 0or y =− a b It is possible to consider the stability in the locality of the equilibrium. Since y t + 1 = (1 ak ) y t kby 2 t let y t + 1 = y and y t = x , then y = (1 ak ) x kbx 2 = f ( x ) f ± ( x ) = (1 ak ) 2 kbx and dy dx ³ ³ ³ ³ y =− a / b = dy t + 1 dy t ³ ³ ³ ³ y =− a / b = (1 ak ) 2 kb ( a / b ) = 1 + ak Hence the stability is very dependent on the sign/value of
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Unformatted text preview: ak . Letting y t + 1 = Ay t − By 2 t , A = (1 − ak ) , B = kb then using our earlier approximation (equation (3.22)) we have y t = Ay By + (1 + A ) − t ( A − By ) i.e. y t = (1 − ak ) y kby + (2 − ak ) − t (1 − ak − kby ) Various paths for this solution are possible depending on the values of v and ε . For instance, if v = 1 and ε ≤ 2, then k = εν ≤ 2, and if a < 1 then ak < 2, which...
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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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