Economics Dynamics Problems 149

Economics Dynamics Problems 149 - to do with t ≥ − 1...

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Discrete dynamic systems 133 When an initial condition is supplied the caveat just alluded to is of no conse- quence. Thus, if we wish to solve x t = ax t 1 x 0 = 2 then the instructions are: Mathematica RSolve[{x[t]==ax[t-1],x[0]==2},x[t],t] with result {{x[t]->2a t }} Maple rsolve({x(t)=a*x(t-1),x(0)=2},x(t)); with result 2a t So no difference arises when initial conditions are supplied. Using either the RSolve command of Mathematica or the rsolve command of Maple , we can readily check the following equations used in this chapter: (i) y t + 1 = ay t (ii) p t + 1 = (1 + k ) p t (iii) Y t = ± v v + s ² Y t 1 (iv) y t + 1 = ay t + c (v) p t = ± a c b ² ± d b ² p t 1 The following observations, however, should be borne in mind. (1) When using both Mathematica and Maple to solve the Harrod–Domar model,problem(iii),therecursiveequationshouldbethoughtofas Y t + 1 = ( v / ( v + s )) Y t and solved accordingly. (2) On some occasions it is necessary to use additional commands, especially the Simplify command( Mathematica )orthe simplify command( Maple ). (3) Mathematica sometimes supplies ‘If’ conditions in the solutions, usually
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Unformatted text preview: to do with t ≥ − 1 for example. This partly arises from the caveat men-tioned above. Many of these can be avoided by writing the equations in terms of future lags, as in the case of the Harrod–Domar model. (4) A number of solutions involve complex output that is not always meaning-ful. This is especially true of general algebraic problems, such as solving y t + 2 = ay t + 1 + by t . (5) Even when results have been simpliFed, it is not always possible to in-terpret the results in an economically meaningful way. ±or instance, in problem (v), it is impossible for a computer software package to ‘know’ that ( a − c ) / ( b + d ) is the equilibrium price and that it is more economi-cally meaningful to take the difference ( p − ( a − c ) / ( b + d )). Economic insight is still a vital element....
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