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NGDP_2024_nonlinear_SIM_transition_LAMBDA.m
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NGDP_2024_nonlinear_SIM_transition_LAMBDA.m
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%NGDP_2024_nonlinear_sim_transition_LAMBDA
%Insert for the file NGDP_2024_nonlinear_transition_LOOP_LAMBDA.m which
%allows fast social welfare analysis for multiple social discount factors
%Nonlinear simulation of transition to NIT in the simple model (Algorithm in Supp Appendix).
%Written by Michael Hatcher (m.c.hatcher@soton.ac.uk). Any errors are my own.
pi = NaN(T_fin,1); tau = pi; c1 = pi; c2 = pi; Index = pi; Max_Resid = pi; Lambda_NIT = pi;
pi_prime = NaN(n_states^2,1); cprime = pi_prime; c_power = pi_prime; sdf_adj = pi_prime;
R = NaN(T_fin,1); EV = R; EV_reverse = R; Utility = R; U_IT = Utility; U_sum = R;
U_sum_1 = R; U_sum_2 = R; U_sum_3 = R; U_sum_4 = R; Resid_check = R;
Dummy_init = 1;
for t=1:T_sim
dummy_IT = 1;
if t==1
R_lag = R_init;
y_lag = y_init;
else
R_lag = R(t-1);
y_lag = y(t-1);
end
pi(t) = pistar*exp(e_vec(t))/(y(t)/y_lag)^(1-dummy_IT);
tau(t) = ( gbar + (R_lag/((1+n)*pi(t)) - 1)*bstar ) / y(t);
c1(t) = (1-alfa)*y(t) - tau(t)*y(t) - phi*tau(t)^2 - bstar;
c2(t) = alfa*(1+n)*y(t) + R_lag/pi(t)*bstar;
pi_prime = pistar*exp(states(:,1))*y(t)^(1-dummy_IT).*y_prime.^(-(1-dummy_IT));
Resid0 = NaN(N_guess0,1); Resid = NaN(N_guess,1);
for k0=1:N_guess0
Dum = 0;
R_guess = R_guess_stack0(k0);
cprime0 = alfa*(1+n)*y_prime + R_guess*bstar.*pi_prime.^(-1);
c_power0 = cprime0.^(1-gama);
E_c_power0 = prob*c_power0;
E_reverse0 = E_c_power0^(1/(1-gama));
sdf_adj0 = betta*c1(t)^(1-eps).*cprime0.^(eps-1)*( 1 / E_reverse0 )^(1-gama-eps).*cprime0.^(1-gama-eps).*pi_prime.^(-1);
Resid0(k0) = abs(1 - R_guess*prob*sdf_adj0);
if k0 > 1 && Resid0(k0) > Resid0(k0-1)
Dum = 1;
break
end
end
[Resid_mini0,Index_min0] = min(Resid0);
R_guess_stack = R_guess_stack0(Index_min0) + linspace(R_l,R_u,N_guess);
for k=1:N_guess
R_guess = R_guess_stack(k);
cprime = alfa*(1+n)*y_prime + R_guess*bstar.*pi_prime.^(-1);
c_power = cprime.^(1-gama);
E_c_power = prob*c_power;
E_reverse = E_c_power^(1/(1-gama));
sdf_adj = betta*c1(t)^(1-eps).*cprime.^(eps-1)*( 1 / E_reverse )^(1-gama-eps).*cprime.^(1-gama-eps).*pi_prime.^(-1);
Resid(k) = abs(1 - R_guess*prob*sdf_adj);
if k>1 && Resid(k) > Resid(k-1)
break
end
end
[Resid_mini,Index_min] = min(Resid);
R(t) = R_guess_stack(Index_min);
Max_Resid(t) = Resid_mini;
Index(t) = Index_min;
cprime = alfa*(1+n).*y_prime + R(t)*bstar.*pi_prime.^(-1);
c_power = cprime.^(1-gama);
EV(t) = prob*c_power;
Utility(t) = 1/(1-gama)*( c1(t)^eps + betta*( EV(t) )^(eps/(1-gama)) )^((1-gama)/eps);
end
if dummy_NIT==1
dummy_IT = 0;
end
if Announced == 1
Resid_init = NaN(N_guess_init,1);
pi_prime = pistar*exp(states(:,1))*y(T_sim)^(1-dummy_IT).*y_prime.^(-(1-dummy_IT));
for k_init=1:N_guess_init
R_guess = R_guess_stack_init(k_init);
cprime_init = alfa*(1+n)*y_prime + R_guess*bstar.*pi_prime.^(-1);
c_power_init = cprime_init.^(1-gama);
E_c_power_init = prob*c_power_init;
E_reverse_init = E_c_power_init^(1/(1-gama));
sdf_adj_init = betta*c1(T_sim)^(1-eps).*cprime_init.^(eps-1)*( 1 / E_reverse_init )^(1-gama-eps).*cprime_init.^(1-gama-eps).*pi_prime.^(-1);
Resid_init(k_init) = abs(1 - R_guess*prob*sdf_adj_init);
if k_init > 1 && Resid_init(k_init) > Resid_init(k_init-1)
break
end
end
[Resid_init,Index_init] = min(Resid_init);
R_ann = R_guess_stack_init(Index_init);
R(T_sim) = R_ann;
%-------------
%Initial old
%-------------
cprime_init = alfa*(1+n)*y_prime + R_ann*bstar.*pi_prime.^(-1);
c_power_init = cprime_init.^(1-gama);
E_c_power_init = prob*c_power_init;
Utility_init = 1/(1-gama)*( c1(T_sim)^eps + betta*( E_c_power_init )^(eps/(1-gama)) )^((1-gama)/eps);
else
%-------------
%Initial old
%-------------
pi_prime_init = pistar*exp(e_vec(T_sim+1))/(y(T_sim+1)/y(T_sim))^(1-dummy_IT);
cprime_init = alfa*(1+n)*y(T_sim+1) + R(T_sim)*bstar/pi_prime_init;
Utility_init = 1/(1-gama)*( c1(T_sim)^eps + betta*( cprime_init^(1-gama) )^(eps/(1-gama)) )^((1-gama)/eps);
%Utility_init = Utility(T_sim); %if use expected U of initial old
end
if dummy_IT == 1
U_IT_init = Utility_init;
else
load("test_lambda.mat","U_IT","U_IT_init")
%Utility_init = U_IT_init;
end
%------------
%Reform
%------------
for t=T_sim+1:T_fin
R_lag = R(t-1);
y_lag = y(t-1);
pi(t) = pistar*exp(e_vec(t))/(y(t)/y_lag)^(1-dummy_IT);
tau(t) = ( gbar + (R_lag/((1+n)*pi(t)) - 1)*bstar ) / y(t);
c1(t) = (1-alfa)*y(t) - tau(t)*y(t) - phi*tau(t)^2 - bstar;
c2(t) = alfa*(1+n)*y(t) + R_lag/pi(t)*bstar;
pi_prime = pistar*exp(states(:,1))*y(t)^(1-dummy_IT).*y_prime.^(-(1-dummy_IT));
Resid0 = NaN(N_guess0,1); Resid = NaN(N_guess,1);
for k0=1:N_guess0
Dum = 0;
R_guess = R_guess_stack0(k0);
cprime0 = alfa*(1+n)*y_prime + R_guess*bstar.*pi_prime.^(-1);
c_power0 = cprime0.^(1-gama);
E_c_power0 = prob*c_power0;
E_reverse0 = E_c_power0^(1/(1-gama));
sdf_adj0 = betta*c1(t)^(1-eps).*cprime0.^(eps-1)*( 1 / E_reverse0 )^(1-gama-eps).*cprime0.^(1-gama-eps).*pi_prime.^(-1);
Resid0(k0) = abs(1 - R_guess*prob*sdf_adj0);
if k0 > 1 && Resid0(k0) > Resid0(k0-1)
Dum = 1;
break
end
end
[Resid_mini0,Index_min0] = min(Resid0);
R_guess_stack = R_guess_stack0(Index_min0) + linspace(R_l,R_u,N_guess);
for k=1:N_guess
R_guess = R_guess_stack(k);
cprime = alfa*(1+n)*y_prime + R_guess*bstar.*pi_prime.^(-1);
c_power = cprime.^(1-gama);
E_c_power = prob*c_power;
E_reverse = E_c_power^(1/(1-gama));
sdf_adj = betta*c1(t)^(1-eps).*cprime.^(eps-1)*( 1 / E_reverse )^(1-gama-eps).*cprime.^(1-gama-eps).*pi_prime.^(-1);
Resid(k) = abs(1 - R_guess*prob*sdf_adj);
if k>1 && Resid(k) > Resid(k-1)
break
end
end
[Resid_mini,Index_min] = min(Resid);
R(t) = R_guess_stack(Index_min);
Max_Resid(t) = Resid_mini;
Index(t) = Index_min;
cprime = alfa*(1+n).*y_prime + R(t)*bstar.*pi_prime.^(-1);
c_power = cprime.^(1-gama);
E_c_power = prob*c_power;
E_reverse = E_c_power^(1/(1-gama));
SDF_check = betta*c1(t)^(1-eps).*cprime.^(eps-1)*( 1 / E_reverse )^(1-gama-eps).*cprime.^(1-gama-eps).*pi_prime.^(-1);
Resid_check(t) = abs(1 - R(t)*prob*SDF_check);
EV(t) = prob*c_power;
Utility(t) = 1/(1-gama)*( c1(t)^eps + betta*( EV(t) )^(eps/(1-gama)) )^((1-gama)/eps);
if t==T_sim+1
U_sum(t) = omega^(t-1-T_sim)*Utility(t) + omega^(-1)*Utility_init;
U_sum_1(t) = omega_1^(t-1-T_sim)*Utility(t) + omega_1^(-1)*Utility_init;
U_sum_2(t) = omega_2^(t-1-T_sim)*Utility(t) + omega_2^(-1)*Utility_init;
U_sum_3(t) = omega_3^(t-1-T_sim)*Utility(t) + omega_3^(-1)*Utility_init;
U_sum_4(t) = omega_4^(t-1-T_sim)*Utility(t) + omega_4^(-1)*Utility_init;
else
U_sum(t) = omega^(t-1-T_sim)*Utility(t);
U_sum_1(t) = omega_1^(t-1-T_sim)*Utility(t);
U_sum_2(t) = omega_2^(t-1-T_sim)*Utility(t);
U_sum_3(t) = omega_3^(t-1-T_sim)*Utility(t);
U_sum_4(t) = omega_4^(t-1-T_sim)*Utility(t);
end
if dummy_IT == 0
Lambda_NIT(t) = 100*( (Utility(t)/U_IT(t))^(1/(1-gama)) - 1); %Consumption equiv. welfare gain
Lambda_init = 100*( (Utility_init/U_IT_init)^(1/(1-gama)) - 1);
end
end
U_sum_tot = sum(U_sum(T_sim+1:T_fin));
U_sum_tot_1 = sum(U_sum_1(T_sim+1:T_fin));
U_sum_tot_2 = sum(U_sum_2(T_sim+1:T_fin));
U_sum_tot_3 = sum(U_sum_3(T_sim+1:T_fin));
U_sum_tot_4 = sum(U_sum_4(T_sim+1:T_fin));
if dummy_IT == 1
U_IT = Utility;
U_sum_IT = U_sum_tot;
U_sum_IT_1 = U_sum_tot_1;
U_sum_IT_2 = U_sum_tot_2;
U_sum_IT_3 = U_sum_tot_3;
U_sum_IT_4 = U_sum_tot_4;
end