The
Mathematica function
RSolve has limitations on the form of recurrence formulas that it will handle (these limitatiions seem to give rise to many similar questions on this discussion group).
What existing parts of
Mathematica would I need to utilize, to try to write a new function to
numerically solve systems of coupled nonlinear recurrence equations?
(call the new function
NRSolve)
These nonlinear systems of recurrence equations often arise in macroeconomics.
To be specific, here is a small model (small by macroeconomic standards) that I want to numerically solve using
Mathematica, written in a
RSolve-type recurrence-equation form:
alpha = 10;
beta = -1/2;
lambda = 1;
epsilon = 3/4;
sigma = 1/2;
At = 200;
Cn = 400;
gL = 2/100;
gMult[t_] := Piecewise[{{1, t <= 1}}, 1 + gL];
mua[t_] := (alpha*Ar[t]^beta - lambda);
eqs = {Cw[t] == (vC[t]*pc[t] + Aw[t - 1]*pa[t] - L[t - 1] +
Ar[t]*(epa[t] - pa[t]))/(epc[t] + pc[t]),
Cr[t] == (vD[t - 1] + Ar[t - 1]*pa[t])/pc[t],
vC[t] == Cw[t] + Cr[t], L[t] == L[t - 1]*gMult[t], vD[t] == L[t],
Aw[t] == At - Ar[t],
Ar[t] == (Cr[t]*pc[t] + Aw[t - 1]*pa[t] - L[t - 1] - vD[t])/pa[t],
pc[t] == pc[t - 1]*gMult[t]*(vC[t]/Cn)^sigma,
pa[t] == epa[t]/(1 - mua[t]),
epc[t] == epc[t - 1]*gMult[t]*(pc[t]/epc[t - 1])^epsilon,
epa[t] == epa[t - 1]*gMult[t]*(pa[t]/epa[t - 1])^epsilon,
Ar[0] == 100, Aw[0] == 100, vC[0] == 400, Cr[0] == 200,
Cw[0] == 200, vD[0] == 100, L[0] == 100, pa[0] == 1, pc[0] == 1,
epa[0] == 1, epc[0] == 1,
Ar[1] == 100, Aw[1] == 100, vC[1] == 400, Cr[1] == 200,
Cw[1] == 200, vD[1] == 100, L[1] == 100, pa[1] == 1, pc[1] == 1,
epa[1] == 1, epc[1] == 1}
vars = {Ar[t], Aw[t], vC[t], Cr[t], Cw[t], vD[t], L[t], pa[t], pc[t],
epa[t], epc[t]};
(* numerical solution desired for t>=2 *)
