105/replication_package/UNIT_IES/define_model.m

%-------------------------------------------------------------------------
% The model: Safe Assets - the case of unit IES (THETA = 1)
%
% This file defines the baseline model (see Appendix for the full derivation).
% Bonds are short term perfectly safe.
%
% 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 TAU real
syms f1 f2 f1p f2p x1 x2 x1p x2p real
syms logq logqp tilp tilpp real
syms state1 state1p state2 state2p hatyp deltap k1 tilb1 real
syms tila1 tila1p tila2 invtila1 invtila2 invtilp rbp rep c1 c2 c1p c2p q qp real
syms invc1 invc1p invc2 invc2p invf1 invf2 r1p r2p logu1p logu2p u1p u2p logf1 logf1p logf2 logf2p real
syms term1p term2p invr1p invr2p real

%% Parameters

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

%% State variables

state = [tila1]; % current period
statep = [tila1p]; % future period

%% Control variables

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

%% shocks

shocks = [hatyp];

%% auxiliary variables

tilp = 1/RHO; % price-dividend ratio for unit IES
tilpp = tilp; % next period

c1 = RHO/(1 + RHO); % consumption/wealth ratio of agent 1 for unit IES
c1p = c1; % next period

c2 = c1; % consumption/wealth ratio of agent 2 for unit IES
c2p = c2; % next period

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

u1p_ = exp(logu1p);
u2p_ = exp(logu2p);

%% MODEL CONDITIONS

tila2_ = tilp + 1 - tila1;

k1_ = x1*(1 - c1)*tila1/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_ = r1p^(1 - GAMMA1)*((1 - NU*(1 - MU))*u1p^(1 - GAMMA1)...
    + NU*(1 - MU)*u2p^(1 - GAMMA1))*invf1^(1 - GAMMA1);

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

eq2 = -1 + term1p; % define f1 = (E(r1p*u1p)^(1-GAMMA1))^(1/(1-GAMMA1))

eq3 = -1 + term2p; % define f2 similarly

logu1p_ = RHO/(1 + RHO)*logc1p + 1/(1 + RHO)*log(1 - c1p) + 1/(1 + RHO)*logf1p;

logu2p_ = RHO/(1 + RHO)*logc2p + 1/(1 + RHO)*log(1 - c2p) + 1/(1 + RHO)*logf2p;

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

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

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

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

%% law of motion of state variables

Phi_fun = (1 + tilp)*(k1 - NU*(k1 - MU)) + (1 - NU)*tilb1/(hatyp*q); % law of motion of tila1

%% 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')