% Yuichiro Waki, 08/02/07
%
% This program solves for the equilibrium of a two-period OLG model, 
% using truncation and Newton's method.
%
% ** Initial old's problem **
%   max c[1,0]   
%       s.t.    c[1,0] = w[1]l_2 + (1+r[1]-\delta)k[1,0]
%               k[1,0] = \overline{k[1,0]} : given
%
% ** Generation t's problem **
%   max \log(c[t,t])+ \beta \log(c[t+1,t]) 
%       s.t.    c[t,t] + k[t+1,t] = w[t]l_1 
%               c[t+1,t] = w[t+1]l_2 + (1+r[t+1]-\delta)k[t+1,t]
%
% ** Firm's FOC (in period t) **
%   r[t] = \alphaAk[t,t-1]^{\alpha-1} (l_1+l_2)^{1-\alpha}
%   w[t] = (1-\alpha)Ak[t,t-1]^\alpha (l_1+l_2)^{-\alpha}
%
% ** Market clearing **
%   c[t,t]+c[t,t-1] +k[t+1,t] 
%       = Ak[t,t-1]^\alpha (l_1+l_2)^{1-\alpha}+(1-\delta)k[t,t-1] 
%
% ** Equilibrium condition **
% (0) 0 = k[1,0] - \overline{k[1,0]}
% For t=1,2,...
% (1) 0 = -c[t,t-1] + (1-\alpha)Ak[t,t-1]^\alpha (l_1+l_2)^{-\alpha}l_2 
%       + (1+(1-\alpha)Ak[t,t-1]^\alpha(l_1+l_2)^{-\alpha}-\delta)k[t,t-1]
% (2) 0 = -c[t,t]-c[t,t-1] -k[t+1,t] 
%        +Ak[t,t-1]^\alpha (l_1+l_2)^{1-\alpha}+(1-\delta)k[t,t-1] 
% (3) 0 = 1-\beta (c[t,t]/c[t+1,t])
%       *(1-\delta+\alpha Ak[t+1,t]^{\alpha-1}(l_1+l_2)^{1-\alpha}) 
%
% [@]** Steady state ** 
% (1) 0 = -c^o + (1-\alpha)Ak^\alpha (l_1+l_2)^{-\alpha}l_2 
%                   + (1+(1-\alpha)Ak^\alpha (l_1+l_2)^{-\alpha}-\delta)k  
% (2) 0 = -c^y -c^o -k 
%       + Ak^\alpha (l_1+l_2)^{1-\alpha}+(1-\delta)k 
% (3) 0 = 1-\beta (c^y/c^o)(1-\delta+\alpha Ak^{\alpha-1}(l_1+l_2)^{1-\alpha}) 
%
% [@@] ** Actual condition we solve **
% Given T (period of truncation), we solve 3(T+1) equations
% (1) and (2) for t=1,2,...,T+1. 
% (3) for t=1,2,...,T.
% (4) k[T+2,T+1] = k (steady state value)
% ((0) is implicit) 
% for 3(T+1) unknowns (c[t,t], c[t-1,t], k[t+1,t], t=1,...T+1).
%
%--------------------------------------------------------------------------