105/replication_package/GENERAL_IES/define_model.m

%-------------------------------------------------------------------------
% The model: Safe Assets - the case of general IES (THETA not equal 1)
%
% This file defines the model (see Appendix for the full derivation).
% Bonds are perfectly safe short-term assets.
%
% Variables are denoted by small letters and
% parameters by capital letters. Future values are denoted by suffix p.
%-------------------------------------------------------------------------

clear,clc

%% Symbolic variables

syms RHO GAMMA1 GAMMA2 NU MU THETA real
syms f1 f2 f1p f2p x1 x2 x1p x2p real
syms logq logqp tilp tilpp real
syms state1 state1p state2 state2p hatyp k1 tilb1 real
syms tila1 tila2 invtila1 invtila2 invtilp rbp rep c1 c2 c1p c2p q qp real
syms invc1 invc1p invc2 invc2p invf1 invf2 r1p r2p u1p_power u2p_power u1p u2p logf1 logf1p logf2 logf2p real
syms term1p term2p invr1p invr2p real

%% Parameters

symparams = [RHO,GAMMA1,GAMMA2,NU,MU,THETA];

%% State variables

state = [state1,state2]; % current period
statep = [state1p,state2p]; % future period

%% Control variables

control = [f1,f2,x1,x2,logq,tilp]; % current period
controlp = [f1p,f2p,x1p,x2p,logqp,tilpp]; % future period

%% shocks

shocks = hatyp;

%% auxiliary variables

logc1p = log(c1p);
logc2p = log(c2p);

invf1_ = 1/f1;
invf2_ = 1/f2;

logf1p_ = log(f1p);
logf2p_ = log(f2p);

invr1p_ = 1/r1p;
invr2p_ = 1/r2p;

q_ = exp(logq);
qp_ = exp(logqp);

invtila1_ = 1/tila1;
invtila2_ = 1/tila2;

rep_ = (1 + tilpp)/tilp*hatyp; % return on equity
rbp_ = 1/q; % return on bond

%% MODEL CONDITIONS

invc1_ = 1 + 1/RHO*f1^(1 - THETA); 

c1_ = 1/invc1;

invc1p_ = 1 + 1/RHO*f1p^(1 - THETA);

c1p_ = 1/invc1p;

invc2_ = 1 + 1/RHO*f2^(1-THETA); 

c2_ = 1/invc2;

invc2p_ = 1 + 1/RHO*f2p^(1 - THETA);

c2p_ = 1/invc2p;

tila1_ = (1 + tilp)*state1 + state2;

tila2_ = tilp + 1 - tila1;

k1_ = x1*(1 - c1)*tila1/tilp;

eq0 = -(1 - k1) + x2*(1 - c2)*tila2/tilp;

tilb1_ = (1 - x1)*(1 - c1)*tila1;

eq1 = tilb1*invtila2 + (1 - x2)*(1 - c2);

r1p_ = x1*rep + (1 - x1)*rbp;

r2p_ = x2*rep + (1 - x2)*rbp;

term1p_ = ((invc1 - 1)*r1p*invf1)^(1 - GAMMA1)*((1 - NU*(1 - MU))*u1p^(1 - GAMMA1)...
    + NU*(1 - MU)*u2p^(1 - GAMMA1));

term2p_ = ((invc2 - 1)*r2p*invf2)^(1 - GAMMA2)*((1 - NU*MU)*u2p^(1 - GAMMA2)...
    + NU*MU*u1p^(1 - GAMMA2));

eq2 = -1 + term1p; 

eq3 = -1 + term2p; 

u1p_power_ = RHO/(1 + RHO)*c1p^(1 - THETA) + 1/(1 + RHO)*c1p^(1 - THETA)*f1p^(1 - THETA);

u2p_power_ = RHO/(1 + RHO)*c2p^(1 - THETA) + 1/(1 + RHO)*c2p^(1 - THETA)*f2p^(1 - THETA);

u1p_ = u1p_power^(1/(1 - THETA));

u2p_ = u2p_power^(1/(1 - THETA));

eq4 = (rep - rbp)*term1p*invr1p;

eq5 = (rep - rbp)*term2p*invr2p;

%% Function f (Ef = 0 imposes model conditions)

f_fun = [eq0;eq1;eq2;eq3;eq4;eq5];

%% law of motion of state variables

Phi_fun = [k1 - NU*(k1 - MU); % law of motion of state1p
          (1 - NU)*tilb1/(hatyp*q)]; % law of motion of state2p

%% collect auxiliary variables and functions

allvars=who;
auxfuns=[];
auxvars=[];
for i=1:length(allvars)
    if strcmp(allvars{i}(end),'_')
        eval(['tempfun=' allvars{i} ';'])
        eval(['tempvar=' allvars{i}(1:end-1) ';'])
        auxfuns=[auxfuns;tempfun];
        auxvars=[auxvars;tempvar];
    end
end

%% Approximation order (<=4)

order = 4;

%% Preprocess model and save

model = prepare_tp(f_fun,Phi_fun,controlp,control,statep,state,shocks,symparams,order,auxfuns,auxvars);

save('model')