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Economics Dynamics Problems 231

# Economics Dynamics Problems 231 - x t pro-vided by...

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Discrete systems of equations 215 then the instructions in each case are: Mathematica equ={x[t+1]==-8-x[t]+y[t], y[t+1]==4-0.3x[t]+0.9y[t], x[0]==2, y[0]==8} var={x[t],y[t]} RSolve[equ,var,t] Maple equ:=x(t+1)=-8-x(t)+y(t), y(t+1)=4-0.3*x(t)+0.9*y(t); init:=x(0)=2, y(0)=8; var:={x(t),y(t)}; rsolve({equ,init},var); The output from each programme looks, on the face of it, quite different–even after using the evalf command in Maple to convert the answer to ﬂoating point arithmetic. Maple gives a single solution to both x ( t ) and y ( t ). Mathematica ,how- ever, gives a whole series of possible solutions depending on the value of t being greater than or equal to 1, 2 and 3, respectively, and further additional conditional statements. In economics, with t representing time, the value of t must be the same for all variables. This means we can ignore the additional conditional statements. What it does mean, however, is that only for t 3 will the solution for
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Unformatted text preview: x ( t ) pro-vided by Mathematica and Maple coincide; while y ( t ) will coincide for t ≥ 2. This should act as a warning to be careful in interpreting the output provided by these packages. Turning to the three-equation system (example 5.5) with initial condition ( x , y , z ) = (3 , − 4 , 3) x t = x t − 1 + 2 y t − 1 + z t − 1 y t = − x t − 1 + y t − 1 z t = 3 x t − 1 − 6 y t − 1 − z t − 1 x = 3 , y = − 4 , z = 3 then we would enter the following commands in each programme: Mathematica equ={x[t]==x[t-1]+2y[t-1]+z[t-1], y[t]==-x[t-1]+y[t-1], z[t]==3x[t-1]-6y[t-1]-z[t-1] x[0]==3, y[0]==-4,z[0]==3} var={x[t],y[t],z[t]} RSolve[equ,var,t] Maple equ:=x(t)=x(t-1)+2*y(t-1)+z(t-1), y(t)=-x(t-1)+y(t-1), z(t)=3*x(t-1)-6*y(t-1)-z(t-1); init:=x(0)=3, y(0)=-4, z(0)=3; var:={x(t),y(t),z(t)}; rsolve({equ,init},var);...
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