105

105/paper.pdf

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105/replication_package/Financial balance sheets non consolidated SNA 2008/CentralBank_gov_bonds.csv

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105/replication_package/Financial balance sheets non consolidated SNA 2008/CentralBank_liabilities.csv

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105/replication_package/Financial balance sheets non consolidated SNA 2008/Financial_Asset_Details.csv

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105/replication_package/Financial balance sheets non consolidated SNA 2008/Financial_Safe_Liabilities.csv

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105/replication_package/Financial balance sheets non consolidated SNA 2008/Government_Safe_Liabilities.csv

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105/replication_package/Financial balance sheets non consolidated SNA 2008/Total liabilities.csv

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+size 6614345

105/replication_package/GDP/GDP_current_prices.csv

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105/replication_package/GENERAL_IES/Parameters.m

@@ -0,0 +1,22 @@
+period_length = 0.25;
+
+P = 1 - exp(-.04*period_length); % disaster probability
+
+B = -log(1 - .32); % disaster size
+
+meanB = B;
+
+G = 0.025*period_length; % drift of log output
+
+RHO = 0.04*period_length; % time preference rate
+
+NU = 0.02*period_length; % replacement rate
+
+MU = 0.05; % popoulation share of agent 1
+
+ALPHA = 1/3; % capital share in output
+
+TAU = 0; % bond duration - short-term bonds
+
+GAMMA1 = 1.000001; % start with unit risk aversion
+GAMMA2 = GAMMA1;

105/replication_package/GENERAL_IES/Table_6_theta_half.m

@@ -0,0 +1,100 @@
+% THETA = 0.5 
+
+load('model')
+addpath('files')
+
+Parameters;
+THETA = 0.9999; % start with THETA close to 1, for which we have a good initial guess
+
+% make the vector of parameters
+params = eval(symparams);
+
+% distribution of hatyp
+nodes = exp([G,G-B]); % hatyp
+weights = [1-P,P]; % corresponding probabilities
+
+T = 2000/period_length; % simulate 2000 years
+
+% disaster shock 
+rng('default')
+disaster = double(rand(1,T+1)<P) + 1; % 1 for normal, 2 for disaster
+
+GAMMA1 = 2.6;
+GAMMA2 = GAMMA1;
+params(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+params(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+% tolerance for the Newton solver
+tolX=1e-7; tolF=1e-7; maxiter=10; testF=1e-5;
+% tolerance for the least squares solver (if a simple Newton fails)
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',100,'display','iter-detailed'); % use lsqnonlin if a simple Newton algorithm fails
+
+solve_and_simulate;
+
+%% Change THETA to 0.5
+
+THETA = 0.5;
+newparams = params;
+newparams(logical(symparams==sym('THETA'))) = THETA;
+burn=1;
+
+correct_params;
+
+%%
+
+GAMMA1 = 2.6;
+GAMMA2 = 4.15;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+burn=1;
+
+correct_params;
+simulate_with_disasters; % This file simulates the model with disasters. 
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+GAMMA1=2.5;
+GAMMA2=4.29;
+
+burn=1;
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+correct_params;
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%
+GAMMA1=2.4;
+GAMMA2=4.54;
+
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+correct_params;
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+save('Table_6_theta_half','Table*')

105/replication_package/GENERAL_IES/Table_6_theta_two.m

@@ -0,0 +1,100 @@
+% THETA = 0.5 
+
+load('model')
+addpath('files')
+
+Parameters;
+THETA = 0.9999; % start with THETA close to 1, for which we have a good initial guess
+
+% make the vector of parameters
+params = eval(symparams);
+
+% distribution of hatyp
+nodes = exp([G,G-B]); % hatyp
+weights = [1-P,P]; % corresponding probabilities
+
+T = 2000/period_length; % simulate 2000 years
+
+% disaster shock 
+rng('default')
+disaster = double(rand(1,T+1)<P) + 1; % 1 for normal, 2 for disaster
+
+GAMMA1 = 2.6;
+GAMMA2 = GAMMA1;
+params(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+params(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+% tolerance for the Newton solver
+tolX=1e-7; tolF=1e-7; maxiter=10; testF=1e-5;
+% tolerance for the least squares solver (if a simple Newton fails)
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',100,'display','iter-detailed'); % use lsqnonlin if a simple Newton algorithm fails
+
+solve_and_simulate;
+
+%% Change THETA to 0.5
+
+THETA = 2;
+newparams = params;
+newparams(logical(symparams==sym('THETA'))) = THETA;
+burn=1;
+
+correct_params;
+
+%%
+
+GAMMA1 = 2.6;
+GAMMA2 = 4.15;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+burn=1;
+
+correct_params;
+simulate_with_disasters; % This file simulates the model with disasters. 
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+GAMMA1=2.5;
+GAMMA2=4.29;
+
+burn=1;
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+correct_params;
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%
+GAMMA1=2.4;
+GAMMA2=4.54;
+
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+correct_params;
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+save('Table_6_theta_two','Table*')

105/replication_package/GENERAL_IES/correct_params.m

@@ -0,0 +1,45 @@
+% This file changes the parameters gradually from their initival value to
+% the target value
+
+solve = 1;
+stop = 0;
+t = 0;
+
+xt = state0;
+params0 = params;
+while stop==0
+    t = t + 1;
+
+    if t<=burn
+        factor = t/burn;
+        params = (1 - factor)*params0 + factor*newparams;
+    end
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % if residuals are too large solve again
+    if norm(R(:))>testF && solve==1
+        t
+        [coeffs,model] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+        
+        % evaluate the new solution
+        [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+    end
+    
+    newxt = nPhi(:,1); % assume no realized disasters 
+
+    if t>burn+10 % after 10 periods start checking for convergence
+        if max(abs(newxt-xt))<1e-7
+            [coeffs] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+            [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+            
+            newxt = nPhi(:,1);
+            if max(abs(newxt-xt))<1e-7
+                stop = 1;
+                state0 = xt; % solution point
+                coeffs0 = coeffs;
+            end
+        end
+    end
+    xt = newxt;
+end
+

105/replication_package/GENERAL_IES/define_model.m

@@ -0,0 +1,152 @@
+%-------------------------------------------------------------------------
+% 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')
+

105/replication_package/GENERAL_IES/make_Table_6_part_1.m

@@ -0,0 +1,44 @@
+Table_6_theta_half;
+Table_6_theta_two;
+
+%% Display Table 6 (part 1)
+clc
+homefolder = pwd;
+cd ..
+
+diary on
+
+disp('********** Table 6 **********')
+
+load([homefolder '\Table_6_theta_half'])
+
+Table_6 = [round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)];
+
+disp('THETA = 0.5')
+disp(Table_6(3,:))
+
+load([homefolder '\Table_6_theta_two'])
+
+Table_6 = [round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)];
+
+disp('THETA = 2')
+disp(Table_6(3,:))
+
+%% Accuracy Measures
+disp('Appendix Table 2: Accuracy Measures for Table 6')
+
+load([homefolder '\Table_6_theta_half'])
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+
+disp('THETA = 0.5')
+disp(Accuarcy(3,2:end))
+
+load([homefolder '\Table_6_theta_two'])
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+
+disp('THETA = 2')
+disp(Accuarcy(3,2:end))
+
+diary off
+
+cd(homefolder)

105/replication_package/GENERAL_IES/simulate_with_disasters.m

@@ -0,0 +1,19 @@
+% Simulate with disasters
+y_results = zeros(model.n_y,T+1);
+x_results = zeros(model.n_x,T+1);
+R_results = zeros(model.n_f,T+1);
+
+x_results(:,1) = state0;
+
+for t = 1:T
+t
+    xt = x_results(:,t);
+    
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % store results 
+    R_results(:,t) = R;
+    y_results(:,t) = g;
+    
+    x_results(:,t+1) = nPhi(:,disaster(t+1)); 
+end

105/replication_package/GENERAL_IES/solve_and_simulate.m

@@ -0,0 +1,84 @@
+% This file performs the following:
+% 1. Solve the model by Taylor projection at the initial state.
+% 2. Simulate the model without realized disasters.
+
+%% make initial guess for a deterministic version of the model
+
+% in a deterministic economy, the following variables are constant:
+
+x1 = 1; % agents invests only in equity
+x2 = 1;
+tilp = 1/RHO; % price/earning ratio
+hatyp = exp(G-meanB*P); % average growth
+haty = hatyp;
+rep = (1+tilp)/tilp*hatyp; % asset return
+logq = log(1/rep); % price of bond
+c1 = RHO/(1+RHO); % consumption/wealth ratio
+c2 = c1;
+logu1 = (RHO*log(c1)+log(1-c1)+log(rep))/RHO;
+u1 = exp(logu1);
+logu2 = (RHO*log(c2)+log(1-c2)+log(rep))/RHO;
+u2 = exp(logu2);
+f1 = (rep*u1);
+f2 = (rep*u2);
+
+k1 = MU;
+
+tila1 = k1*(1+tilp);
+
+state0 = [k1;0];
+c0 = state0;
+
+derivs0 = [f1;f2;x1;x2;logq;tilp];
+
+derivs1 = zeros(model.n_f,model.n_x);
+derivs2 = zeros(model.n_f,model.n_x^2);
+derivs3 = zeros(model.n_f,model.n_x^3);
+derivs4 = zeros(model.n_f,model.n_x^4);
+
+if order==1
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1 );
+elseif order==2
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2);
+elseif order==3
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2,derivs3 );
+elseif order==4
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2,derivs3,derivs4 );
+end
+
+%% solve the model
+
+[coeffs,model] = tpsolve(initial_guess,state0,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+
+%% simulate the model
+
+solve = 1;
+stop = 0;
+t = 0;
+xt = state0;
+while stop==0
+    t = t+1;
+    % evaluate the previous solution at the new point xt
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % if residuals are too large solve again
+    if norm(R(:))>testF && solve==1
+        t
+        [coeffs] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+        
+        % evaluate the new solution
+        [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+    end
+    
+    newxt = nPhi(:,disaster(t+1));  % new state
+
+    if t>=10 % after 10 periods start checking for convergence
+        if max(abs(newxt-xt))<1e-7
+            stop = 1;
+            state0 = xt; 
+            coeffs0 = coeffs;
+        end
+    end
+    xt = newxt;
+end
+

105/replication_package/GENERAL_IES/summarize_results.m

@@ -0,0 +1,61 @@
+
+normal = logical(disaster==1); % normal periods
+d = logical(disaster>1); % disaster periods
+
+state1 = x_results(1,1:T);
+state2 = x_results(2,1:T);
+
+f1 = y_results(1,1:T);
+x1 = y_results(3,1:T);
+logq = y_results(5,1:T);
+tilp=y_results(6,1:T);
+
+invc1 = 1 + 1/RHO*f1.^(1 - THETA); 
+c1 = 1./invc1;
+
+q = exp(logq);
+
+tila1 = (1 + tilp).*state1(1:T) + state2(1:T);
+
+k1 = x1.*(1 - c1).*tila1./tilp;
+tilb1 = (1 - x1).*(1 - c1).*tila1;
+
+
+W1_share = k1 - NU*(k1 - MU) + (1 - NU)*tilb1./tilp; % wealth share after type changes
+equity = k1 - NU*(k1 - MU);
+
+debt_to_assets = -(1 - NU)*tilb1./tilp; % debt ratio (after type changes)
+debt_to_GDP = -(1 - NU)*tilb1*period_length;
+
+haty = nodes(1,double(disaster(1:T)));
+
+% compute means by iterated expectations
+
+roe = ((1 + tilp(2:T))./tilp(1:T-1).*haty(2:T)); % this is actual return from t to t+1.
+mean_roe = 1/period_length*log((1-P)*mean(roe(normal(2:T)))+P*mean(roe(d(2:T)))); % mean return
+
+period_mean_roe = (1-P)*mean(roe(normal(2:T)))+P*mean(roe(d(2:T)));
+period_var_roe = (1-P)*mean((roe(normal(2:T)) - period_mean_roe).^2)+P*mean((roe(d(2:T)) - period_mean_roe).^2);
+vol_roe = sqrt(period_var_roe/period_length);
+
+rb = log(1./q(1:T-1))/period_length; % this is log return on bonds
+mean_rb = (1-P)*mean(rb(normal(1:T-1)))+P*mean(rb(d(1:T-1)));
+
+Rb = 1./q(2:T-1);
+period_mean_rb = (1-P)*mean(Rb(normal(2:T-1)))+P*mean(Rb(d(2:T-1)));
+period_var_rb = (1-P)*mean((Rb(normal(2:T-1)) - period_mean_rb).^2)+P*mean((Rb(d(2:T-1)) - period_mean_rb).^2);
+vol_rb = sqrt(period_var_rb/period_length);
+
+mean_equity = (1-P)*mean(equity(normal(1:T))) + P*mean(equity(d(1:T)));
+mean_debt_to_assets = (1-P)*mean(debt_to_assets(normal(1:T))) + P*mean(debt_to_assets(d(1:T)));
+mean_debt_to_GDP = (1-P)*mean(debt_to_GDP(normal(1:T))) + P*mean(debt_to_GDP(d(1:T)));
+mean_W1_share = (1-P)*mean(W1_share(normal(1:T))) + P*mean(W1_share(d(1:T)));
+
+% mean_W1_share_excluding_labor = mean_W1_share*(1+L) - MU*L;
+% mean_debt_to_assets_excluding_labor = mean_debt_to_assets*(1+L);
+% mean_debt_to_GDP_including_labor = mean_debt_to_GDP/(1+L);
+% mean_equity_excluding_labor = mean_equity*(1+L) - MU*L;
+
+mean_equity_excluding_labor = mean_equity/ALPHA - MU*(1 - ALPHA)/ALPHA;
+mean_debt_to_assets_excluding_labor = mean_debt_to_assets/ALPHA;
+mean_W1_share_excluding_labor = mean_equity_excluding_labor - mean_debt_to_assets_excluding_labor;

105/replication_package/Make_OECD_data.do

@@ -0,0 +1,684 @@
+clear
+
+*set matsize 3000
+set matsize 800
+clear mata
+set mem 500m 		
+set more off
+set logtype text		
+capture log close
+*set linesize 255
+
+// IMPORTANT!!! change working directory to the folder of this file
+
+* cd "folder name"
+
+// GDP current prices
+clear
+
+local folder="GDP"
+
+insheet using "`folder'\GDP_current_prices.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - current price and ppp */
+tab Transaction
+
+keep if Measure=="Current prices"
+
+keep Country Year Value
+
+rename Value GDP
+
+rename Country country
+rename Year year
+
+
+label var GDP "GDP current prices, Million of national currency"
+
+save GDP, replace
+
+
+// Step 1: make dataset.dta
+
+clear
+
+local folder="Financial balance sheets non consolidated SNA 2008"
+
+* Total liabilities - all sectors
+
+insheet using "`folder'\Total liabilities.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+
+keep if Measure=="National currency, current prices"
+
+keep Country Sector Time Value
+
+rename Time Year
+rename Value totliab
+
+gen Sec=""
+
+replace Sec="Total" if Sector=="Total economy"
+replace Sec="World" if Sector=="Rest of the world"
+replace Sec="Households" if Sector=="Households and NPISH"
+replace Sec="Financials" if Sector=="Financial corporations"
+replace Sec="nonFinancials" if Sector=="Non-financial corporations"
+replace Sec="Gov" if Sector=="General Government"
+
+keep if Sec~=""
+
+keep Country Year Sec totliab
+
+rename Country country
+rename Year year
+rename Sec sector
+rename totliab totliab_
+
+reshape wide totliab_, i(country year) j(sector) string
+
+label var totliab_Total "Tota Liabilities - Total Economy"
+label var totliab_World "Tota Liabilities - Rest of the World"
+label var totliab_Households "Tota Liabilities - Households and NPISH"
+label var totliab_Financials "Tota Liabilities - Financial corporations"
+label var totliab_nonFinancials "Tota Liabilities - Non-financial corporations"
+
+save total_liabilities, replace
+
+
+//* Financial Corporations - safe liabilities
+clear
+
+insheet using "`folder'\Financial_Safe_Liabilities.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+tab Sector /* only financial corporations */
+
+keep if Measure=="National currency, current prices" & Sector=="Financial corporations"
+
+keep Country Time Transaction Value
+
+gen tran=""
+
+replace tran="dep" if Transaction=="Currency and deposits"
+replace tran="secur" if Transaction=="Debt securities"
+replace tran="loan" if Transaction=="Loans"
+replace tran="mmf" if Transaction=="Money market fund shares /units"
+replace tran="trade" if Transaction=="Trade credits and advances"
+
+drop if tran==""
+
+keep Country Time tran Value
+rename Country country
+rename Time year
+rename Value fin_
+
+reshape wide fin_, i(country year) j(tran) string
+
+label var fin_dep "Currency and deposits - Financial corporations"
+label var fin_secur "Debt securities - Financial corporations"
+label var fin_loan "Loans - Financial corporations"
+label var fin_mmf "Money market fund shares /units - Financial corporations"
+label var fin_trade "Trade credits and advances - Financial corporations"
+
+
+save financials_safe_items, replace
+
+
+//* General Government - safe liabilities
+
+clear
+
+insheet using "`folder'\Government_Safe_Liabilities.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+tab Sector /* only general government */
+
+keep if Measure=="National currency, current prices" & Sector=="General Government"
+
+keep Country Time Transaction Value
+
+gen tran=""
+
+replace tran="dep" if Transaction=="Currency and deposits"
+replace tran="secur" if Transaction=="Debt securities"
+replace tran="loan" if Transaction=="Loans"
+replace tran="mmf" if Transaction=="Money market fund shares /units"
+replace tran="trade" if Transaction=="Trade credits and advances"
+
+drop if tran==""
+
+keep Country Time tran Value
+rename Country country
+rename Time year
+rename Value gov_
+
+reshape wide gov_, i(country year) j(tran) string
+
+
+label var gov_dep "Currency and deposits - General Government"
+label var gov_secur "Debt securities - General Government"
+label var gov_loan "Loans - General Government"
+label var gov_mmf "Money market fund shares /units - General Government"
+label var gov_trade "Trade credits and advances - General Government"
+
+save government_safe_items, replace
+
+//* Central bank holdings of bonds (assume most of these holdings are government bonds)
+
+clear
+
+insheet using "`folder'\CentralBank_gov_bonds.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+tab Sector /* only general government */
+
+keep if Measure=="National currency, current prices" & Sector=="Central Bank"
+
+keep Country Time Transaction Value
+
+gen tran=""
+
+replace tran="lbond" if Transaction=="Long-term debt securities"
+replace tran="sbond" if Transaction=="Short-term debt securities"
+
+
+drop if tran==""
+
+keep Country Time tran Value
+rename Country country
+rename Time year
+rename Value cb_
+
+reshape wide cb_, i(country year) j(tran) string
+
+save central_bank_bonds, replace
+
+//* Central bank - safe and total liabilities
+
+clear
+
+insheet using "`folder'\CentralBank_liabilities.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+tab Sector /* only Central Bank */
+
+keep if Measure=="National currency, current prices" & Sector=="Central Bank"
+
+keep Country Time Transaction Value
+
+gen tran=""
+
+replace tran="dep" if Transaction=="Currency and deposits"
+replace tran="secur" if Transaction=="Debt securities"
+replace tran="loan" if Transaction=="Loans"
+replace tran="mmf" if Transaction=="Money market fund shares /units"
+replace tran="trade" if Transaction=="Trade credits and advances"
+replace tran="totliab_Total" if Transaction=="Financial liabilities"
+
+drop if tran==""
+
+keep Country Time tran Value
+rename Country country
+rename Time year
+rename Value cb_
+
+reshape wide cb_, i(country year) j(tran) string
+
+
+label var cb_dep "Currency and deposits - Central Bank"
+label var cb_secur "Debt securities - Central Bank"
+label var cb_loan "Loans - Central Bank"
+label var cb_mmf "Money market fund shares /units - Central Bank"
+label var cb_trade "Trade credits and advances - Central Bank"
+label var cb_totliab_Total "Financial liabilities - Central Bank"
+
+save cb_safe_items, replace
+
+clear
+
+use total_liabilities, replace
+
+joinby country year using financials_safe_items, unmatched(both)
+cap drop _merge
+
+joinby country year using government_safe_items, unmatched(both)
+cap drop _merge
+
+joinby country year using central_bank_bonds, unmatched(both)
+cap drop _merge
+
+joinby country year using GDP, unmatched(master)
+cap drop _merge
+
+joinby country year using cb_safe_items, unmatched(master)
+cap drop _merge
+
+* replace missing values with zeros
+foreach var of varlist fin_* gov_* cb_* {
+replace `var'=0 if `var'==.
+}
+
+label var cb_lbond "Long-term debt securities held by the central bank"
+label var cb_sbond "Short-term debt securities held by the central bank"
+
+
+encode country, gen(countrys)
+
+drop country
+rename countrys country
+
+tsset country year
+
+save dataset, replace
+
+//* Compute variables in USD
+* Total liabilities - all sectors
+
+clear 
+
+insheet using "`folder'\Total liabilities.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+
+
+*keep if Measure=="National currency, current prices"
+keep if Measure=="US $, current prices, current exchange rates, end of period"
+
+keep Country Sector Time Value
+
+rename Time Year
+rename Value totliab
+
+gen Sec=""
+
+replace Sec="Total" if Sector=="Total economy"
+replace Sec="World" if Sector=="Rest of the world"
+replace Sec="Households" if Sector=="Households and NPISH"
+replace Sec="Financials" if Sector=="Financial corporations"
+replace Sec="nonFinancials" if Sector=="Non-financial corporations"
+replace Sec="Gov" if Sector=="General Government"
+
+keep if Sec~=""
+
+keep Country Year Sec totliab
+
+rename Country country
+rename Year year
+rename Sec sector
+rename totliab totliab_
+
+reshape wide totliab_, i(country year) j(sector) string
+
+label var totliab_Total "Tota Liabilities - Total Economy"
+label var totliab_World "Tota Liabilities - Rest of the World"
+label var totliab_Households "Tota Liabilities - Households and NPISH"
+label var totliab_Financials "Tota Liabilities - Financial corporations"
+label var totliab_nonFinancials "Tota Liabilities - Non-financial corporations"
+
+save total_liabilitiesUSD, replace
+
+*/
+
+//* Financial Corporations - safe liabilities
+clear
+
+insheet using "`folder'\Financial_Safe_Liabilities.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+tab Sector /* only financial corporations */
+
+*keep if Measure=="National currency, current prices" & Sector=="Financial corporations"
+keep if Measure=="US $, current prices, current exchange rates, end of period"
+
+keep Country Time Transaction Value
+
+gen tran=""
+
+replace tran="dep" if Transaction=="Currency and deposits"
+replace tran="secur" if Transaction=="Debt securities"
+replace tran="loan" if Transaction=="Loans"
+replace tran="mmf" if Transaction=="Money market fund shares /units"
+replace tran="trade" if Transaction=="Trade credits and advances"
+
+drop if tran==""
+
+keep Country Time tran Value
+rename Country country
+rename Time year
+rename Value fin_
+
+reshape wide fin_, i(country year) j(tran) string
+
+label var fin_dep "Currency and deposits - Financial corporations"
+label var fin_secur "Debt securities - Financial corporations"
+label var fin_loan "Loans - Financial corporations"
+label var fin_mmf "Money market fund shares /units - Financial corporations"
+label var fin_trade "Trade credits and advances - Financial corporations"
+
+
+save financials_safe_itemsUSD, replace
+
+//* General Government - safe liabilities
+
+clear
+
+insheet using "`folder'\Government_Safe_Liabilities.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+tab Sector /* only general government */
+
+*keep if Measure=="National currency, current prices" & Sector=="General Government"
+keep if Measure=="US $, current prices, current exchange rates, end of period"
+
+keep Country Time Transaction Value
+
+gen tran=""
+
+replace tran="dep" if Transaction=="Currency and deposits"
+replace tran="secur" if Transaction=="Debt securities"
+replace tran="loan" if Transaction=="Loans"
+replace tran="mmf" if Transaction=="Money market fund shares /units"
+replace tran="trade" if Transaction=="Trade credits and advances"
+
+drop if tran==""
+
+keep Country Time tran Value
+rename Country country
+rename Time year
+rename Value gov_
+
+reshape wide gov_, i(country year) j(tran) string
+
+
+label var gov_dep "Currency and deposits - General Government"
+label var gov_secur "Debt securities - General Government"
+label var gov_loan "Loans - General Government"
+label var gov_mmf "Money market fund shares /units - General Government"
+label var gov_trade "Trade credits and advances - General Government"
+
+save government_safe_itemsUSD, replace
+
+
+//* Central bank holdings of bonds (assume most of these holdings are government bonds)
+
+clear
+
+insheet using "`folder'\CentralBank_gov_bonds.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+tab Sector /* only general government */
+
+*keep if Measure=="National currency, current prices" & Sector=="Central Bank"
+keep if Measure=="US $, current prices, current exchange rates, end of period"
+
+keep Country Time Transaction Value
+
+gen tran=""
+
+replace tran="lbond" if Transaction=="Long-term debt securities"
+replace tran="sbond" if Transaction=="Short-term debt securities"
+
+
+drop if tran==""
+
+keep Country Time tran Value
+rename Country country
+rename Time year
+rename Value cb_
+
+reshape wide cb_, i(country year) j(tran) string
+
+save central_bank_bondsUSD, replace
+
+
+//* Central bank - safe and total liabilities
+
+clear
+
+insheet using "`folder'\CentralBank_liabilities.csv", n c case
+
+des
+
+tab PowerCode /* all number in millions */
+tab Measure /* two types of measures - national currency and USD */
+tab Transaction
+tab Sector /* only Central Bank */
+
+*keep if Measure=="National currency, current prices" & Sector=="Central Bank"
+keep if Measure=="US $, current prices, current exchange rates, end of period"
+
+keep Country Time Transaction Value
+
+gen tran=""
+
+replace tran="dep" if Transaction=="Currency and deposits"
+replace tran="secur" if Transaction=="Debt securities"
+replace tran="loan" if Transaction=="Loans"
+replace tran="mmf" if Transaction=="Money market fund shares /units"
+replace tran="trade" if Transaction=="Trade credits and advances"
+replace tran="totliab_Total" if Transaction=="Financial liabilities"
+
+drop if tran==""
+
+keep Country Time tran Value
+rename Country country
+rename Time year
+rename Value cb_
+
+reshape wide cb_, i(country year) j(tran) string
+
+
+label var cb_dep "Currency and deposits - Central Bank"
+label var cb_secur "Debt securities - Central Bank"
+label var cb_loan "Loans - Central Bank"
+label var cb_mmf "Money market fund shares /units - Central Bank"
+label var cb_trade "Trade credits and advances - Central Bank"
+label var cb_totliab_Total "Financial liabilities - Central Bank"
+
+save cb_safe_itemsUSD, replace
+
+*** 
+
+clear
+
+use total_liabilitiesUSD, replace
+
+joinby country year using financials_safe_itemsUSD, unmatched(both)
+cap drop _merge
+
+joinby country year using government_safe_itemsUSD, unmatched(both)
+cap drop _merge
+
+joinby country year using central_bank_bondsUSD, unmatched(both)
+cap drop _merge
+
+joinby country year using cb_safe_itemsUSD, unmatched(both)
+cap drop _merge
+
+
+* replace missing values with zeros
+foreach var of varlist fin_* gov_* cb_* {
+replace `var'=0 if `var'==.
+}
+
+label var cb_lbond "Long-term debt securities held by the central bank"
+label var cb_sbond "Short-term debt securities held by the central bank"
+
+
+encode country, gen(countrys)
+
+drop country
+rename countrys country
+
+tsset country year
+
+save datasetUSD, replace
+
+*** do USD aggregates
+
+use datasetUSD, clear
+
+
+gen fin_safeUSD=fin_dep+fin_loan+fin_mmf+fin_secur // remove trade credit and advances - it's a small part of financial debt and seems unrelated
+gen cb_safeUSD=cb_dep+cb_loan+cb_mmf+cb_secur // remove trade credit and advances - it's a small part of financial debt and seems unrelated
+
+
+gen gov_safeUSD=gov_dep+gov_loan+gov_mmf+gov_secur // remove trade credit and advnaces
+
+
+rename totliab_Total totliab_TotalUSD
+
+keep country year fin_safeUSD gov_safeUSD cb_safeUSD totliab_TotalUSD
+
+sort country year
+tsset country year
+
+
+save safeUSD, replace
+
+// Make world table
+
+use safeUSD, clear
+
+gen tempvar=fin_safeUSD+gov_safeUSD+totliab_TotalUSD
+
+bysort country: egen minyear=min(year) if tempvar<.
+bysort country: egen maxyear=max(year) if tempvar<.
+
+keep if minyear<=1995
+keep if year>=1995
+
+keep if maxyear>=2017
+keep if year<=2017
+
+sort country year
+tsset country year // panel is balanced
+
+
+if r(balanced)!="strongly balanced" {
+di "WARNING!!!! panel is not balanced"
+ }
+
+preserve
+
+keep if year==2017
+
+keep country
+
+gen OECD=1
+
+save OECD, replace
+
+restore
+ 
+ 
+decode country, g(country_name)
+
+bysort year: egen fin_safeWorld=total(fin_safeUSD)
+bysort year: egen gov_safeWorld=total(gov_safeUSD)
+bysort year: egen new_totalWorld=total(totliab_TotalUSD)
+
+bysort year: egen cb_safeWorld=total(cb_safeUSD)
+
+gen fin_shareWorld=fin_safeWorld/new_totalWorld
+gen gov_shareWorld=gov_safeWorld/new_totalWorld
+gen shareWorld=fin_shareWorld+gov_shareWorld
+
+gen cb_shareWorld=cb_safeWorld/new_totalWorld
+
+bysort year: egen fin_safeNonUS=total(fin_safeUSD) if country_name!="United States"
+bysort year: egen gov_safeNonUS=total(gov_safeUSD) if country_name!="United States"
+bysort year: egen new_totalNonUS=total(totliab_TotalUSD) if country_name!="United States"
+
+bysort year: egen cb_safeNonUS=total(cb_safeUSD) if country_name!="United States"
+
+gen fin_shareNonUS=fin_safeNonUS/new_totalNonUS
+gen gov_shareNonUS=gov_safeNonUS/new_totalNonUS
+gen shareNonUS=fin_shareNonUS+gov_shareNonUS
+
+gen cb_shareNonUS=cb_safeNonUS/new_totalNonUS
+
+sort country year
+
+keep if country==country[1]
+
+keep fin_shareWorld gov_shareWorld shareWorld cb_shareWorld fin_shareNonUS gov_shareNonUS shareNonUS cb_shareNonUS year
+
+
+keep year fin_shareWorld gov_shareWorld shareWorld  cb_shareWorld  fin_shareNonUS gov_shareNonUS shareNonUS cb_shareNonUS
+
+save World, replace
+
+
+// Make final dataset
+
+use dataset // non-consolidated data
+
+joinby country year using safeUSD, unmatched(both)
+cap drop _merge
+
+joinby year using World, unmatched(both)
+cap drop _merge
+
+joinby country using OECD, unmatched(both)
+cap drop _merge
+
+cap drop fin_safe
+cap drop cb_safe
+
+label var gov_safeUSD "government safe liabilities in USD"
+label var fin_shareWorld "financial liabilities (as share of total assets) for the whole world"
+label var gov_shareWorld "government liabilities (as share of total assets) for the whole world"
+label var cb_shareWorld "central bank liabilities (as share of total assets) for the whole world"
+label var shareWorld "safe liabilities (as share of total assets) for the whole world"
+
+label var fin_shareNonUS "financial liabilities (as share of total assets) for non-US sample"
+label var gov_shareNonUS "government liabilities (as share of total assets) for non-US sample"
+label var cb_shareNonUS "central bank liabilities (as share of total assets) for non-US sample"
+label var shareNonUS "safe liabilities (as share of total assets) for non-US sample"
+
+label var totliab_Gov "Tota Liabilities - Government"
+label var OECD "dummy for OECD countries"
+
+save OECD_data, replace
+

105/replication_package/ReadMe.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9068400c5ca91682b0dd81943ea676402c4cd3e4a0083709b46a65c0d502af1f
+size 42724

105/replication_package/Replicate_Empirical_Results.do

@@ -0,0 +1,16 @@
+*************************************************************************************************************
+* Replicate Tables 1-4 and Figures 1-2 of "Safe Assets" by Barro, Fernandez-Villaverde, Levintal and Mollerus
+*************************************************************************************************************
+
+clear all
+
+clear mata
+set mem 500m 
+set maxvar 32767		
+set more off
+set linesize 255
+cap log close
+
+do make_Tables_1_to_3_Figures_1_and_2
+
+do make_Table_4

105/replication_package/Replicate_Simulation_Results.m

@@ -0,0 +1,40 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Replicate Tables 5-7 and Figure 3 of "Safe Assets"  by Barro, Fernandez-Villaverde, Levintal and Mollerus
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+clear
+
+%% open diary file
+
+FID = fopen('Tables_5_to_7.txt','w');
+fclose(FID);
+
+diary Tables_5_to_7.txt
+diary off
+
+%% add folders to search path
+homefolder = pwd;
+
+addpath(genpath([homefolder '\solution_methods']));
+
+%% define models
+
+run('UNIT_IES\define_model'); % THETA = 1
+run('GENERAL_IES\define_model'); % THETA ~= 1
+run('Variable_Disaster_Size\define_model'); % THETA = 1, variable disaster size, defaultable long-term bonds
+
+
+%% replicate Table 5
+run('UNIT_IES\make_Table_5.m')
+
+%% replicate Table 6
+run('GENERAL_IES\make_Table_6_part_1.m')
+run('UNIT_IES\make_Table_6_part_2.m')
+
+%% replicate Table 7
+run('Variable_Disaster_Size\make_Table_7.m')
+
+%% replicate Figure 3
+run('UNIT_IES\Disaster_IRF.m')
+
+

105/replication_package/UNIT_IES/Disaster_IRF.m

@@ -0,0 +1,46 @@
+% Disaster Impulse response function
+
+clear,clc,close all
+addpath('files')
+
+load('benchmark')
+
+state0 = mean(tila1);
+mean_W1_share0 = mean_W1_share;
+
+T = 10/period_length + 1;
+
+disaster = ones(1,T+1);
+disaster(2) = 2;
+
+simulate_with_disasters;
+summarize_results;
+
+W1_share = [mean_W1_share0;W1_share(:)];
+
+% adjust for human capital
+equity = equity/ALPHA - MU*(1 - ALPHA)/ALPHA;
+debt_to_assets = debt_to_assets/ALPHA;
+W1_share = W1_share/ALPHA - (1 - ALPHA)/ALPHA*MU;
+
+set(0, 'defaultFigurePaperPosition',  [0 0 20 21.5]*30);
+h = figure('color', [1 1 1], 'PaperType', 'A4');
+
+subplot(2,2,1);
+plot(W1_share);
+title('W1/W')
+
+subplot(2,2,2)
+plot(rb);
+title('rf')
+
+subplot(2,2,3)
+plot(equity);
+title('K1')
+
+subplot(2,2,4)
+plot(debt_to_assets);
+title('B1/assets')
+
+cd ..
+saveas(h,'Figure_3','jpg')

105/replication_package/UNIT_IES/Parameters.m

@@ -0,0 +1,22 @@
+period_length = 0.25;
+
+P = 1 - exp(-.04*period_length); % disaster probability
+
+B = -log(1 - .32); % disaster size
+
+meanB = B;
+
+G = 0.025*period_length; % drift of log output
+
+RHO = 0.04*period_length; % time preference rate
+
+NU = 0.02*period_length; % replacement rate
+
+MU = 0.05; % popoulation share of agent 1
+
+ALPHA = 1/3; % capital share in output
+
+TAU = 0; % bond duration - short-term bonds
+
+GAMMA1 = 1.000001; % start with unit risk aversion
+GAMMA2 = GAMMA1;

105/replication_package/UNIT_IES/Table_6_MU.m

@@ -0,0 +1,94 @@
+% MU = 0.1
+
+load('model')
+addpath('files')
+
+Parameters;
+MU = 0.1; % popoulation share of agent 1
+
+% make the vector of parameters
+params = eval(symparams);
+
+% distribution of hatyp
+nodes = exp([G,G-B]); % hatyp
+weights = [1-P,P]; % corresponding probabilities
+
+T = 2000/period_length; % simulate 2000 years
+
+% disaster shock 
+rng('default')
+disaster = double(rand(1,T+1)<P) + 1; % 1 for normal, 2 for disaster
+
+GAMMA1 = 2.6;
+GAMMA2 = GAMMA1;
+params(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+params(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+% tolerance for the Newton solver
+tolX=1e-7; tolF=1e-7; maxiter=10; testF=1e-5;
+% tolerance for the least squares solver (if a simple Newton fails)
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',100,'display','iter-detailed'); % use lsqnonlin if a simple Newton algorithm fails
+
+solve_and_simulate;
+
+%%%%%%%%%%%%%%%%% 
+
+GAMMA1 = 2.6;
+GAMMA2 = 4.15;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+burn=1;
+
+correct_params;
+simulate_with_disasters; % This file simulates the model with disasters. 
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+GAMMA1=2.5;
+GAMMA2=4.29;
+
+burn=1;
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+correct_params;
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%
+GAMMA1=2.4;
+GAMMA2=4.54;
+
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+params=newparams;
+xt=max(x_results);
+[coeffs,model]=tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+save('Table_6_MU','Table*')
+

105/replication_package/UNIT_IES/Table_6_NU.m

@@ -0,0 +1,94 @@
+% NU = .03
+
+load('model')
+addpath('files')
+
+Parameters;
+NU = 0.03*period_length; % replacement rate
+
+% make the vector of parameters
+params = eval(symparams);
+
+% distribution of hatyp
+nodes = exp([G,G-B]); % hatyp
+weights = [1-P,P]; % corresponding probabilities
+
+T = 2000/period_length; % simulate 2000 years
+
+% disaster shock 
+rng('default')
+disaster = double(rand(1,T+1)<P) + 1; % 1 for normal, 2 for disaster
+
+GAMMA1 = 2.6;
+GAMMA2 = GAMMA1;
+params(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+params(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+% tolerance for the Newton solver
+tolX=1e-7; tolF=1e-7; maxiter=10; testF=1e-5;
+% tolerance for the least squares solver (if a simple Newton fails)
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',100,'display','iter-detailed'); % use lsqnonlin if a simple Newton algorithm fails
+
+solve_and_simulate;
+
+%%%%%%%%%%%%%%%%% 
+
+GAMMA1 = 2.6;
+GAMMA2 = 4.15;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+burn=1;
+
+correct_params;
+simulate_with_disasters; % This file simulates the model with disasters. 
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+GAMMA1=2.5;
+GAMMA2=4.29;
+
+burn=1;
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+correct_params;
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%
+GAMMA1=2.4;
+GAMMA2=4.54;
+
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+params=newparams;
+xt=max(x_results);
+[coeffs,model]=tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+
+save('Table_6_NU','Table*')

105/replication_package/UNIT_IES/Table_6_P.m

@@ -0,0 +1,94 @@
+% P = .02
+
+load('model')
+addpath('files')
+
+Parameters;
+P = 1 - exp(-.02*period_length); % disaster probability
+
+% make the vector of parameters
+params = eval(symparams);
+
+% distribution of hatyp
+nodes = exp([G,G-B]); % hatyp
+weights = [1-P,P]; % corresponding probabilities
+
+T = 2000/period_length; % simulate 2000 years
+
+% disaster shock 
+rng('default')
+disaster = double(rand(1,T+1)<P) + 1; % 1 for normal, 2 for disaster
+
+GAMMA1 = 2.6;
+GAMMA2 = GAMMA1;
+params(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+params(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+% tolerance for the Newton solver
+tolX=1e-7; tolF=1e-7; maxiter=10; testF=1e-5;
+% tolerance for the least squares solver (if a simple Newton fails)
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',100,'display','iter-detailed'); % use lsqnonlin if a simple Newton algorithm fails
+
+solve_and_simulate;
+
+%%%%%%%%%%%%%%%%% 
+
+GAMMA1 = 2.6;
+GAMMA2 = 4.15;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+burn=1;
+
+correct_params;
+simulate_with_disasters; % This file simulates the model with disasters. 
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+GAMMA1=2.5;
+GAMMA2=4.29;
+
+burn=1;
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+correct_params;
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%%%%%%%%%%%%%%%%%%%%
+GAMMA1=2.4;
+GAMMA2=4.54;
+
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+params=newparams;
+xt=max(x_results);
+[coeffs,model]=tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+
+simulate_with_disasters;
+
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+save('Table_6_P','Table*')
+

105/replication_package/UNIT_IES/Tranquility.m

@@ -0,0 +1,51 @@
+% 40 years of tranquility
+
+clear,clc,close all
+addpath('files')
+
+load('benchmark')
+
+state0 = mean(tila1);
+mean_W1_share0 = mean_W1_share;
+
+T40 = 40/period_length+1;
+T = 2000;
+
+disaster = ones(1,T+1);
+
+simulate_with_disasters;
+summarize_results;
+
+W1_share = [mean_W1_share0;W1_share(:)];
+
+% adjust for human capital
+equity = equity/ALPHA - MU*(1 - ALPHA)/ALPHA;
+debt_to_assets = debt_to_assets/ALPHA;
+W1_share = W1_share/ALPHA - (1 - ALPHA)/ALPHA*MU;
+
+[W1_share(end),rb(end),equity(end),debt_to_assets(end)]
+
+W1_share = W1_share(1:T40);
+rf = rb(1:T40);
+equity = equity(1:T40);
+debt_to_assets = debt_to_assets(1:T40);
+
+set(0, 'defaultFigurePaperPosition',  [0 0 20 21.5]*30);
+h = figure('color', [1 1 1], 'PaperType', 'A4');
+
+subplot(2,2,1);
+plot(W1_share);
+title('W1/W')
+
+subplot(2,2,2)
+plot(rf);
+title('rf')
+
+subplot(2,2,3)
+plot(equity);
+title('K1')
+
+subplot(2,2,4)
+plot(debt_to_assets);
+title('B1/assets')
+

105/replication_package/UNIT_IES/correct_params.m

@@ -0,0 +1,45 @@
+% This file changes the parameters gradually from their initival value to
+% the target value
+
+solve = 1;
+stop = 0;
+t = 0;
+
+xt = state0;
+params0 = params;
+while stop==0
+    t = t + 1;
+
+    if t<=burn
+        factor = t/burn;
+        params = (1 - factor)*params0 + factor*newparams;
+    end
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % if residuals are too large solve again
+    if norm(R(:))>testF && solve==1
+        t
+        [coeffs,model] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+        
+        % evaluate the new solution
+        [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+    end
+    
+    newxt = nPhi(:,1); % assume no realized disasters 
+
+    if t>burn+10 % after 10 periods start checking for convergence
+        if max(abs(newxt-xt))<1e-7
+            [coeffs] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+            [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+            
+            newxt = nPhi(:,1);
+            if max(abs(newxt-xt))<1e-7
+                stop = 1;
+                state0 = xt; % solution point
+                coeffs0 = coeffs;
+            end
+        end
+    end
+    xt = newxt;
+end
+

105/replication_package/UNIT_IES/define_model.m

@@ -0,0 +1,139 @@
+%-------------------------------------------------------------------------
+% 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')
+

105/replication_package/UNIT_IES/make_Table_5.m

@@ -0,0 +1,210 @@
+% Table 5 
+
+clear,clc
+
+load('model')
+addpath('files')
+
+Parameters;
+
+% make the vector of parameters
+params = eval(symparams);
+
+% distribution of hatyp
+nodes = exp([G,G-B]); % hatyp
+weights = [1-P,P]; % corresponding probabilities
+
+T = 2000/period_length; % simulate 2000 years
+
+% disaster shock 
+rng('default')
+disaster = double(rand(1,T+1)<P) + 1; % 1 for normal, 2 for disaster
+
+GAMMA1 = 3.85;
+GAMMA2 = GAMMA1;
+params(logical(symparams==sym('GAMMA1')))=GAMMA1;
+params(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+step_size=1;
+
+% tolerance for the Newton solver
+tolX = 1e-5; tolF = 1e-5; maxiter = 10; testF = 1e-5;
+% tolerance for the least squares solver (if a simple Newton fails)
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',100,'display','iter-detailed'); % use lsqnonlin if a simple Newton algorithm fails
+
+solve_and_simulate;
+
+simulate_with_disasters; % This file simulates the model with disasters. 
+
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [vol_roe,vol_rb];
+
+%%
+GAMMA1 = 3.3;
+GAMMA2 = 3.89;
+
+burn=1;
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%
+GAMMA1 = 2.9;
+GAMMA2 = 3.98;
+
+burn=5;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%
+GAMMA1 = 2.8;
+GAMMA2 = 4.02;
+
+burn = 5;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%
+GAMMA1 = 2.7;
+GAMMA2 = 4.07;
+
+burn = 5;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%
+GAMMA1 = 2.6;
+GAMMA2 = 4.15;
+
+burn = 5;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%
+GAMMA1 = 2.5;
+GAMMA2 = 4.29;
+
+burn=5;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%%
+
+GAMMA1 = 2.4;
+GAMMA2 = 4.54;
+
+burn=5;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+save('benchmark')
+
+%%
+GAMMA1 = 2.3;
+GAMMA2 = 5.50;
+
+burn=5;
+
+newparams = params;
+newparams(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+newparams(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [Table;GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [Table_labor;GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [Table_vol;vol_roe,vol_rb];
+
+%% display Table 5
+clc
+
+homefolder = pwd;
+cd ..
+diary on
+
+disp('********** Table 5 **********')
+
+Table_5 = [round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)]
+
+
+%% display accuracy measure
+disp('Appendix Table 1: Accuracy Measures for Table 5')
+
+Accuracy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)]
+
+diary off
+
+cd(homefolder)

105/replication_package/UNIT_IES/make_Table_6_part_2.m

@@ -0,0 +1,60 @@
+Table_6_NU;
+Table_6_P;
+Table_6_MU;
+
+%% Display Table 6 (part 2)
+clc
+homefolder = pwd;
+cd ..
+diary on
+
+disp('********** Table 6 (continued) **********')
+
+load([homefolder '\Table_6_NU'],'Table*')
+
+Table_6 = [round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)];
+
+disp('nu = 0.03')
+disp(Table_6(3,:))
+
+load([homefolder '\Table_6_P'],'Table*')
+
+Table_6 = [round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)];
+
+disp('p = 0.02')
+disp(Table_6(3,:))
+
+load([homefolder '\Table_6_MU'],'Table*')
+
+Table_6 = [round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)];
+
+disp('mu = 0.1')
+disp(Table_6(3,:))
+
+%% Accuracy Measures
+disp('Appendix Table 2 (continued): Accuarcy Measures for Table 6')
+
+load([homefolder '\Table_6_NU'],'Table*')
+
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+
+disp('nu = 0.03')
+disp(Accuarcy(3,2:end))
+
+load([homefolder '\Table_6_P'],'Table*')
+
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+
+disp('p = 0.02')
+disp(Accuarcy(3,2:end))
+
+load([homefolder '\Table_6_MU'],'Table*')
+
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+
+disp('mu = 0.1')
+disp(Accuarcy(3,2:end))
+
+diary off
+
+cd(homefolder)

105/replication_package/UNIT_IES/rep_agent.m

@@ -0,0 +1,66 @@
+% Table 5 
+
+clear,clc
+
+load('model')
+
+addpath('files')
+
+Parameters;
+
+% make the vector of parameters
+params = eval(symparams);
+
+% distribution of hatyp and deltap
+nodes = exp([G,G-B]); % hatyp
+weights = [1-P,P]; % corresponding probabilities
+
+rng('default')
+
+T = 10; % for rep agent simulate for 10 periods only
+
+% disaster shock
+rng('default')
+
+disaster = ones(1,T+1); % 1 for normal
+disaster(ceil(T/2)) = 2; % 2 for disaster
+
+% tolerance for the Newton solver
+tolX = 1e-5; tolF = 1e-5; maxiter = 10; testF = 1e-5;
+% tolerance for the least squares solver (if a simple Newton fails)
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',100,'display','iter-detailed'); % use lsqnonlin if a simple Newton algorithm fails
+
+%%%%%%%%%%%%%%%%%
+
+GAMMA1=1.000001;
+GAMMA2=GAMMA1;
+params(logical(symparams==sym('GAMMA1')))=GAMMA1;
+params(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+solve_and_simulate;
+
+simulate_with_disasters; % This file simulates the model with disasters. 
+
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+
+%%
+
+for GAMMA1 = [1.5,2,2.4,2.5:.5:6]
+    GAMMA2 = GAMMA1;
+
+    % update parameter values and solve the model for the new parameters
+    params(logical(symparams==sym('GAMMA1'))) = GAMMA1;
+    params(logical(symparams==sym('GAMMA2'))) = GAMMA2;
+    [coeffs,model] = tpsolve(coeffs,state0,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+
+    simulate_with_disasters; % This file simulates the model with disasters. 
+
+    summarize_results;
+
+    Table = [Table;GAMMA1,GAMMA2,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+end
+
+Table_5 = round(Table(:,[1,3,4]),3)
+

105/replication_package/UNIT_IES/simulate_with_disasters.m

@@ -0,0 +1,19 @@
+% Simulate with disasters
+y_results = zeros(model.n_y,T+1);
+x_results = zeros(model.n_x,T+1);
+R_results = zeros(model.n_f,T+1);
+
+x_results(:,1) = state0;
+
+for t = 1:T
+t
+    xt = x_results(:,t);
+    
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % store results 
+    R_results(:,t) = R;
+    y_results(:,t) = g;
+    
+    x_results(:,t+1) = nPhi(:,disaster(t+1)); 
+end

105/replication_package/UNIT_IES/solve_and_simulate.m

@@ -0,0 +1,84 @@
+% This file performs the following:
+% 1. Solve the model by Taylor projection at the initial state.
+% 2. Simulate the model without realized disasters.
+
+%% make initial guess for a deterministic version of the model
+
+% in a deterministic economy, the following variables are constant:
+
+x1 = 1; % agents invests only in equity
+x2 = 1;
+tilp = 1/RHO; % price/earning ratio
+hatyp = exp(G-meanB*P); % average growth
+haty = hatyp;
+rep = (1+tilp)/tilp*hatyp; % asset return
+logq = log(1/rep); % price of bond
+c1 = RHO/(1+RHO); % consumption/wealth ratio
+c2 = c1;
+logu1 = (RHO*log(c1)+log(1-c1)+log(rep))/RHO;
+u1 = exp(logu1);
+logu2 = (RHO*log(c2)+log(1-c2)+log(rep))/RHO;
+u2 = exp(logu2);
+f1 = (rep*u1);
+f2 = (rep*u2);
+
+k1 = MU;
+
+tila1 = k1*(1+tilp);
+
+state0 = tila1;
+c0 = state0;
+
+derivs0 = [f1;f2;x1;x2;logq];
+
+derivs1 = zeros(model.n_f,model.n_x);
+derivs2 = zeros(model.n_f,model.n_x^2);
+derivs3 = zeros(model.n_f,model.n_x^3);
+derivs4 = zeros(model.n_f,model.n_x^4);
+
+if order==1
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1 );
+elseif order==2
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2);
+elseif order==3
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2,derivs3 );
+elseif order==4
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2,derivs3,derivs4 );
+end
+
+%% solve the model
+
+[coeffs,model] = tpsolve(initial_guess,state0,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+
+%% simulate the model
+
+solve = 1;
+stop = 0;
+t = 0;
+xt = state0;
+while stop==0
+    t = t+1;
+    % evaluate the previous solution at the new point xt
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % if residuals are too large solve again
+    if norm(R(:))>testF && solve==1
+        t
+        [coeffs] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+        
+        % evaluate the new solution
+        [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+    end
+    
+    newxt = nPhi(:,disaster(t+1));  % new state
+
+    if t>=10 % after 10 periods start checking for convergence
+        if max(abs(newxt-xt))<1e-7
+            stop = 1;
+            state0 = xt; 
+            coeffs0 = coeffs;
+        end
+    end
+    xt = newxt;
+end
+

105/replication_package/UNIT_IES/summarize_results.m

@@ -0,0 +1,56 @@
+
+normal = logical(disaster==1); % normal periods
+d = logical(disaster>1); % disaster periods
+
+tila1 = x_results(1,1:T);
+
+x1 = y_results(3,1:T);
+logq = y_results(5,1:T);
+
+c1 = RHO/(1+RHO);
+tilp = 1/RHO;
+tilp = repmat(tilp,1,T);
+q = exp(logq);
+
+k1 = x1.*(1 - c1).*tila1./tilp;
+tilb1 = (1 - x1).*(1 - c1).*tila1;
+
+
+W1_share = k1 - NU*(k1 - MU) + (1 - NU)*tilb1./tilp; % wealth share after type changes
+equity = k1 - NU*(k1 - MU);
+
+debt_to_assets = -(1 - NU)*tilb1./tilp; % debt ratio (after type changes)
+debt_to_GDP = -(1 - NU)*tilb1*period_length;
+
+haty = nodes(1,double(disaster(1:T)));
+
+% compute means by iterated expectations
+
+roe = ((1 + tilp(2:T))./tilp(1:T-1).*haty(2:T)); % this is actual return from t to t+1.
+mean_roe = 1/period_length*log((1-P)*mean(roe(normal(2:T)))+P*mean(roe(d(2:T)))); % mean return
+
+period_mean_roe = (1-P)*mean(roe(normal(2:T)))+P*mean(roe(d(2:T)));
+period_var_roe = (1-P)*mean((roe(normal(2:T)) - period_mean_roe).^2)+P*mean((roe(d(2:T)) - period_mean_roe).^2);
+vol_roe = sqrt(period_var_roe/period_length);
+
+rb = log(1./q(1:T-1))/period_length; % this is log return on bonds
+mean_rb = (1-P)*mean(rb(normal(1:T-1)))+P*mean(rb(d(1:T-1)));
+
+Rb = 1./q(2:T-1);
+period_mean_rb = (1-P)*mean(Rb(normal(2:T-1)))+P*mean(Rb(d(2:T-1)));
+period_var_rb = (1-P)*mean((Rb(normal(2:T-1)) - period_mean_rb).^2)+P*mean((Rb(d(2:T-1)) - period_mean_rb).^2);
+vol_rb = sqrt(period_var_rb/period_length);
+
+mean_equity = (1-P)*mean(equity(normal(1:T))) + P*mean(equity(d(1:T)));
+mean_debt_to_assets = (1-P)*mean(debt_to_assets(normal(1:T))) + P*mean(debt_to_assets(d(1:T)));
+mean_debt_to_GDP = (1-P)*mean(debt_to_GDP(normal(1:T))) + P*mean(debt_to_GDP(d(1:T)));
+mean_W1_share = (1-P)*mean(W1_share(normal(1:T))) + P*mean(W1_share(d(1:T)));
+
+% mean_W1_share_excluding_labor = mean_W1_share*(1+L) - MU*L;
+% mean_debt_to_assets_excluding_labor = mean_debt_to_assets*(1+L);
+% mean_debt_to_GDP_including_labor = mean_debt_to_GDP/(1+L);
+% mean_equity_excluding_labor = mean_equity*(1+L) - MU*L;
+
+mean_equity_excluding_labor = mean_equity/ALPHA - MU*(1 - ALPHA)/ALPHA;
+mean_debt_to_assets_excluding_labor = mean_debt_to_assets/ALPHA;
+mean_W1_share_excluding_labor = mean_equity_excluding_labor - mean_debt_to_assets_excluding_labor;

105/replication_package/User Guide.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4ae3ea65d6ead2e1e273aabe94a8a1f913ec926e7b77f7f75784b25d6db4748f
+size 161394

105/replication_package/Variable_Disaster_Size/Parameters.m

@@ -0,0 +1,46 @@
+period_length = 0.25;
+
+P = 1 - exp(-.04*period_length); % disaster probability
+
+% variable disaster size
+B = -log(1 - [0.1384074;
+    0.2375926;
+    0.335;
+    0.4331111;
+    0.5516667;
+    0.653]);
+
+% distribution of disaster size
+probB = [0.6;
+        0.2;
+        0.088888889;
+        0.066666667;
+        0.022222222;
+        0.022222222];
+
+
+meanB = B(:)'*probB;
+
+Size = 1 - exp(-B(:)');
+meanSize = Size*probB;
+sdSize = sqrt((Size - meanSize).^2*probB(:));
+
+G = 0.021*period_length; % drift of log output
+
+RHO = 0.04*period_length; % time preference rate
+
+NU = 0.02*period_length; % replacement rate
+
+MU = 0.05; % popoulation share of agent 1
+
+ALPHA = 1/3; % capital share in output
+
+TAU = 0; % bond duration - start with short-term bonds
+
+GAMMA1 = 1.000001; % start with unit risk aversion
+GAMMA2 = GAMMA1;
+
+delta_prob = 0.4; % default probability
+delta_size = 0; % default size
+
+

105/replication_package/Variable_Disaster_Size/correct_params.m

@@ -0,0 +1,45 @@
+% This file changes the parameters gradually from their initival value to
+% the target value
+
+solve = 1;
+stop = 0;
+t = 0;
+
+xt = state0;
+params0 = params;
+while stop==0
+    t = t + 1;
+
+    if t<=burn
+        factor = t/burn;
+        params = (1 - factor)*params0 + factor*newparams;
+    end
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % if residuals are too large solve again
+    if norm(R(:))>testF && solve==1
+        t
+        [coeffs,model] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+        
+        % evaluate the new solution
+        [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+    end
+    
+    newxt = nPhi(:,1); % assume no realized disasters 
+
+    if t>burn+10 % after 10 periods start checking for convergence
+        if max(abs(newxt-xt))<1e-7
+            [coeffs] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+            [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+            
+            newxt = nPhi(:,1);
+            if max(abs(newxt-xt))<1e-7
+                stop = 1;
+                state0 = xt; % solution point
+                coeffs0 = coeffs;
+            end
+        end
+    end
+    xt = newxt;
+end
+

105/replication_package/Variable_Disaster_Size/define_model.m

@@ -0,0 +1,142 @@
+%-------------------------------------------------------------------------
+% The model: Safe Assets - the case of unit IES (THETA = 1)
+% Bonds are long term and subject to a default shock deltap
+%
+% This file defines the model (see Appendix for the full derivation).
+%
+% 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 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,TAU];
+
+%% State variables
+
+state = [state1,state2]; % current period
+statep = [state1p,state2p]; % future period
+
+%% Control variables
+
+control = [f1,f2,x1,x2,logq]; % current period
+controlp = [f1p,f2p,x1p,x2p,logqp]; % future period
+
+%% shocks
+
+shocks = [hatyp,deltap];
+
+%% 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 + TAU*qp)/q*(1 - deltap); % return on bond
+
+u1p_ = exp(logu1p);
+u2p_ = exp(logu2p);
+
+%% MODEL CONDITIONS
+
+tila1_ = (1 + tilp)*state1 + state2*(1 + TAU*q);
+
+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 = [k1 - NU*(k1 - MU); % law of motion of state1p
+          (1 - NU)*tilb1*(1 - deltap)/(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')
+

105/replication_package/Variable_Disaster_Size/make_Table_7.m

@@ -0,0 +1,200 @@
+% Variable disaster size, long-term bonds, default probability
+
+clear,clc
+
+load('model')
+addpath('files')
+
+Parameters;
+
+% make the vector of parameters
+params = eval(symparams);
+
+% distribution of hatyp and deltap
+nodes = [exp([G,kron(G-B(:)',ones(1,2))]);
+    0,kron(ones(1,numel(B)),[0,delta_size])];% realizations
+
+weights = [1-P,P*kron(probB(:)',[1-delta_prob,delta_prob])]; % corresponding probabilities
+
+rng('default')
+
+T = 1e5; 
+
+N_disasters = ceil(T*P);
+ptr = 1 + round(cumsum([0;probB(:)])*N_disasters);
+disaster_state = zeros(1,N_disasters);
+for i = 1:numel(probB)
+    disaster_state(ptr(i):ptr(i+1)-1) = i;
+end
+randorder = rand(1,N_disasters);
+[~,I] = sort(randorder);
+disaster_state = disaster_state(I);
+
+% disaster shock (normal = 1, disaster>1)
+
+disaster = zeros(1,T+1);
+disaster(randperm(T,N_disasters)) = 1;
+
+default = double(rand(1,T+1)<delta_prob);
+
+n = 1;
+for t = 1:T+1
+    if disaster(t)==1
+        disaster(t) = 1 + 2*disaster_state(n) - (1 - default(t));
+        n = n + 1;
+    end
+end
+disaster(disaster==0) = 1;
+
+GAMMA1=1.46;
+GAMMA2=GAMMA1;
+params(logical(symparams==sym('GAMMA1')))=GAMMA1;
+params(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+step_size=1;
+
+% tolerance for the Newton solver
+tolX=1e-5; tolF=1e-5; maxiter=10; testF=1e-5;
+% tolerance for the least squares solver (if a simple Newton fails)
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',100,'display','iter-detailed'); % use lsqnonlin if a simple Newton algorithm fails
+
+solve_and_simulate;
+simulate_with_disasters;
+summarize_results;
+
+%%
+GAMMA1=1.46;
+GAMMA2=4.13;
+
+burn=10;
+newparams=params;
+newparams(logical(symparams==sym('GAMMA1')))=GAMMA1;
+newparams(logical(symparams==sym('GAMMA2')))=GAMMA2;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [vol_roe,vol_rb];
+
+save('VariableDisasterSize')
+
+%% add default
+
+for delta_size = .05:.05:.2
+    nodes(2,:) = [0,kron(ones(1,numel(B)),[0,delta_size])];
+    
+    correct_params;
+    simulate_with_disasters;
+    summarize_results;
+
+    Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+    Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+    Table_vol = [vol_roe,vol_rb];
+end
+
+save('ShortDefault')
+
+%% change duration
+
+load('VariableDisasterSize')
+
+for TAU = [0.1:.1:.85,.9,.91:.01:.97,.975]
+
+burn=5;
+
+newparams = params;
+newparams(logical(symparams==sym('TAU'))) = TAU;
+
+correct_params;
+simulate_with_disasters;
+summarize_results;
+
+Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+Table_vol = [vol_roe,vol_rb];
+
+if TAU==.95
+    save('Duration5')
+end
+
+end
+
+%% add default
+
+load('Duration5')
+for delta_size = .05:.05:.2
+    nodes(2,:) = [0,kron(ones(1,numel(B)),[0,delta_size])];
+    
+    correct_params;
+    simulate_with_disasters;
+    summarize_results;
+
+    Table = [GAMMA1,GAMMA2,RHO,mean_roe,mean_rb,mean_equity,mean_W1_share,mean_debt_to_assets,mean_debt_to_GDP,mean(abs(R_results(:))),max(abs(R_results(:)))];
+    Table_labor = [GAMMA1,GAMMA2,mean_equity_excluding_labor,mean_W1_share_excluding_labor,mean_debt_to_assets_excluding_labor,mean_debt_to_GDP];
+    Table_vol = [vol_roe,vol_rb];
+end
+
+save('LongDefault')
+
+%% show results
+clc
+
+homefolder = pwd;
+cd ..
+diary on
+
+disp('********** Table 7 **********')
+
+load([homefolder '\VariableDisasterSize'])
+
+disp('variable disaster size')
+disp([round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)])
+
+load([homefolder '\ShortDefault'])
+
+disp('default probability')
+disp([round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)])
+
+load([homefolder '\Duration5'])
+
+disp('long term bonds, no default')
+disp([round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)])
+
+load([homefolder '\LongDefault'])
+
+disp('long term bonds with default')
+disp([round(Table(:,[1,2,4,5]),3),Table_vol,round(Table_labor(:,[3,4,5]),3),round(Table_labor(:,[6]),2)])
+
+%% accuracy measures
+disp('Appendix Table 3: Accuracy Measures for Table 7')
+
+load([homefolder '\VariableDisasterSize'])
+
+disp('variable disaster size')
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+disp(Accuarcy)
+
+load([homefolder '\ShortDefault'])
+
+disp('default probability')
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+disp(Accuarcy)
+
+load([homefolder '\Duration5'])
+
+disp('long term bonds, no default')
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+disp(Accuarcy)
+
+load([homefolder '\LongDefault'])
+
+disp('long term bonds with default')
+Accuarcy = [round(Table(:,1),3),round(log10(Table(:,end-1:end)),1)];
+disp(Accuarcy)
+
+diary off
+
+cd(homefolder)

105/replication_package/Variable_Disaster_Size/simulate_with_disasters.m

@@ -0,0 +1,19 @@
+% Simulate with disasters
+y_results = zeros(model.n_y,T+1);
+x_results = zeros(model.n_x,T+1);
+R_results = zeros(model.n_f,T+1);
+
+x_results(:,1) = state0;
+
+for t = 1:T
+t
+    xt = x_results(:,t);
+    
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % store results 
+    R_results(:,t) = R;
+    y_results(:,t) = g;
+    
+    x_results(:,t+1) = nPhi(:,disaster(t+1)); 
+end

105/replication_package/Variable_Disaster_Size/solve_and_simulate.m

@@ -0,0 +1,84 @@
+% This file performs the following:
+% 1. Solve the model by Taylor projection at the initial state.
+% 2. Simulate the model without realized disasters.
+
+%% make initial guess for a deterministic version of the model
+
+% in a deterministic economy, the following variables are constant:
+
+x1 = 1; % agents invests only in equity
+x2 = 1;
+tilp = 1/RHO; % price/earning ratio
+hatyp = exp(G-meanB*P); % average growth
+haty = hatyp;
+rep = (1+tilp)/tilp*hatyp; % asset return
+logq = log(1/rep); % price of bond
+c1 = RHO/(1+RHO); % consumption/wealth ratio
+c2 = c1;
+logu1 = (RHO*log(c1)+log(1-c1)+log(rep))/RHO;
+u1 = exp(logu1);
+logu2 = (RHO*log(c2)+log(1-c2)+log(rep))/RHO;
+u2 = exp(logu2);
+f1 = (rep*u1);
+f2 = (rep*u2);
+
+k1 = MU;
+
+tila1 = k1*(1+tilp);
+
+state0 = [k1;0];
+c0 = state0;
+
+derivs0 = [f1;f2;x1;x2;logq];
+
+derivs1 = zeros(model.n_f,model.n_x);
+derivs2 = zeros(model.n_f,model.n_x^2);
+derivs3 = zeros(model.n_f,model.n_x^3);
+derivs4 = zeros(model.n_f,model.n_x^4);
+
+if order==1
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1 );
+elseif order==2
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2);
+elseif order==3
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2,derivs3 );
+elseif order==4
+    [ initial_guess ] = derivs2coeffs( model,derivs0,derivs1,derivs2,derivs3,derivs4 );
+end
+
+%% solve the model
+
+[coeffs,model] = tpsolve(initial_guess,state0,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+
+%% simulate the model
+
+solve = 1;
+stop = 0;
+t = 0;
+xt = state0;
+while stop==0
+    t = t+1;
+    % evaluate the previous solution at the new point xt
+    [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+
+    % if residuals are too large solve again
+    if norm(R(:))>testF && solve==1
+        t
+        [coeffs] = tpsolve(coeffs,xt,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS); % solve
+        
+        % evaluate the new solution
+        [R,g,nPhi] = residual(coeffs,xt,params,c0,nodes,weights);
+    end
+    
+    newxt = nPhi(:,disaster(t+1));  % new state
+
+    if t>=10 % after 10 periods start checking for convergence
+        if max(abs(newxt-xt))<1e-7
+            stop = 1;
+            state0 = xt; 
+            coeffs0 = coeffs;
+        end
+    end
+    xt = newxt;
+end
+

105/replication_package/Variable_Disaster_Size/summarize_results.m

@@ -0,0 +1,55 @@
+
+normal = logical(disaster==1); % normal periods
+d = logical(disaster>1); % disaster periods
+
+state1 = x_results(1,1:T);
+state2 = x_results(2,1:T);
+
+x1 = y_results(3,1:T);
+logq = y_results(5,1:T);
+
+c1 = RHO/(1+RHO);
+tilp = 1/RHO;
+tilp = repmat(tilp,1,T);
+q = exp(logq);
+
+tila1 = (1 + tilp).*state1 + state2.*(1 + TAU*q);
+
+k1 = x1.*(1 - c1).*tila1./tilp;
+tilb1 = (1 - x1).*(1 - c1).*tila1;
+
+
+W1_share = k1 - NU*(k1 - MU) + (1 - NU)*tilb1./tilp; % wealth share after type changes
+equity = k1 - NU*(k1 - MU);
+
+debt_to_assets = -(1 - NU)*tilb1./tilp; % debt ratio (after type changes)
+debt_to_GDP = -(1 - NU)*tilb1*period_length;
+
+haty = nodes(1,double(disaster(1:T)));
+delta = nodes(2,double(disaster(1:T)));
+
+% compute means by iterated expectations
+
+roe = ((1 + tilp(2:T))./tilp(1:T-1).*haty(2:T)); % this is actual return from t to t+1.
+mean_roe = 1/period_length*log((1-P)*mean(roe(normal(2:T)))+P*mean(roe(d(2:T)))); % mean return
+
+period_mean_roe = (1-P)*mean(roe(normal(2:T)))+P*mean(roe(d(2:T)));
+period_var_roe = (1-P)*mean((roe(normal(2:T)) - period_mean_roe).^2)+P*mean((roe(d(2:T)) - period_mean_roe).^2);
+vol_roe = sqrt(period_var_roe/period_length);
+
+rb = log((1 + TAU*q(2:T))./q(1:T-1)).*(1 - delta(2:T))/period_length; % this is log return on bonds
+mean_rb = (1-P)*mean(rb(normal(1:T-1)))+P*mean(rb(d(1:T-1)));
+
+Rb = (1 + TAU*q(3:T))./q(2:T-1).*(1 - delta(3:T));
+period_mean_rb = (1-P)*mean(Rb(normal(2:T-1)))+P*mean(Rb(d(2:T-1)));
+period_var_rb = (1-P)*mean((Rb(normal(2:T-1)) - period_mean_rb).^2)+P*mean((Rb(d(2:T-1)) - period_mean_rb).^2);
+vol_rb = sqrt(period_var_rb/period_length);
+
+mean_equity = (1-P)*mean(equity(normal(1:T))) + P*mean(equity(d(1:T)));
+mean_debt_to_assets = (1-P)*mean(debt_to_assets(normal(1:T))) + P*mean(debt_to_assets(d(1:T)));
+mean_debt_to_GDP = (1-P)*mean(debt_to_GDP(normal(1:T))) + P*mean(debt_to_GDP(d(1:T)));
+mean_W1_share = (1-P)*mean(W1_share(normal(1:T))) + P*mean(W1_share(d(1:T)));
+
+mean_equity_excluding_labor = mean_equity/ALPHA - MU*(1 - ALPHA)/ALPHA;
+mean_debt_to_assets_excluding_labor = mean_debt_to_assets/ALPHA;
+mean_W1_share_excluding_labor = mean_equity_excluding_labor - mean_debt_to_assets_excluding_labor;

105/replication_package/examples/rbc/prepare_model.m

@@ -0,0 +1,69 @@
+%------------------------------------------------------------------
+% This script shows how to solve the RBC model by Taylor projection
+%------------------------------------------------------------------
+
+clear,clc
+
+%-----------------------------------------
+% Define symbolic variables and parameters
+%-----------------------------------------
+
+syms k kp c cp z zp epsp real
+syms BETA GAMMA ALPHA RHO DELTA SIGMA real
+
+%-------------------------------------------------
+% Function f (Euler condition) in a unit-free form
+%-------------------------------------------------
+
+f_fun=BETA*(c/cp)^GAMMA*(ALPHA*exp(zp)*kp^(ALPHA-1)+1-DELTA)-1;
+
+%-------------------------------------------------------
+% Function Phi (law of motion of capital and technology)
+%-------------------------------------------------------
+
+Phi_fun=[exp(z)*k^ALPHA+(1-DELTA)*k-c;
+         RHO*z+SIGMA*epsp];
+
+%--------------------------
+% Vector of state variables
+%--------------------------
+
+x=[k,z]; % current period
+xp=[kp,zp]; % future period
+
+%----------------------------
+% Vector of control variables
+%----------------------------
+
+y=[c]; % current period
+yp=[cp]; % future period
+
+%-----------------
+% Vector of shocks
+%-----------------
+
+shocks=[epsp];
+
+%---------------------
+% Vector of parameters
+%---------------------
+
+symparams=[BETA,GAMMA,ALPHA,RHO,DELTA,SIGMA];
+
+%--------------------
+% Approximation order
+%--------------------
+
+order=4; % fourth order is the maximum possible
+
+%----------------
+% Call prepare_tp
+%----------------
+
+model=prepare_tp(f_fun,Phi_fun,yp,y,xp,x,shocks,symparams,order);
+
+%-----------
+% Save model
+%-----------
+
+save('model') % you will need this later

105/replication_package/examples/rbc/prepare_model_auxiliary_functions.m

@@ -0,0 +1,127 @@
+%----------------------------------------------
+% RBC model with auxiliary functions
+%----------------------------------------------
+
+clear,clc
+
+%-----------------------------------------
+% Define symbolic variables and parameters
+%-----------------------------------------
+
+syms k kp c cp z zp epsp real
+syms BETA GAMMA ALPHA RHO DELTA SIGMA real
+
+%-------------------------------
+% Define the auxiliary functions
+%-------------------------------
+
+% Logs of consumption and capital.
+syms logc logcp logk logkp real
+logc_=log(c);
+logcp_=log(cp);
+logk_=log(k);
+logkp_=log(kp);
+
+% Log and level of future mpk.
+syms mpkp logmpkp real
+logmpkp_=log(ALPHA)+zp+(ALPHA-1)*logkp; 
+mpkp_=exp(logmpkp);
+
+% Log and level of stochastic discount factor.
+syms mp logmp real
+logmp_=log(BETA)+GAMMA*(logc-logcp);
+mp_=exp(logmp);
+
+% Log and level of output.
+syms logoutput output real
+logoutput_=z+ALPHA*logk;
+output_=exp(logoutput);
+
+%-----------------------------
+% Function f (Euler condition)
+%-----------------------------
+f_fun=mp*(mpkp+1-DELTA)-1;
+
+%-------------------------------------------------------
+% Function Phi (law of motion of capital and technology)
+%-------------------------------------------------------
+
+Phi_fun=[output+(1-DELTA)*k-c; 
+         RHO*z+SIGMA*epsp];
+
+%--------------------------
+% Vector of state variables
+%--------------------------
+x=[k,z]; % current period
+xp=[kp,zp]; % future period
+
+%----------------------------
+% Vector of control variables
+%----------------------------
+y=[c]; % current period
+yp=[cp]; % future period
+
+%-----------------
+% Vector of shocks
+%-----------------
+shocks=[epsp];
+
+%---------------------
+% Vector of parameters
+%---------------------
+symparams=[BETA,GAMMA,ALPHA,RHO,DELTA,SIGMA];
+
+%-----------------------------------------------------------
+% Vectors of auxiliary functions and corresponding variables
+%-----------------------------------------------------------
+
+% you can do it manually:
+
+% auxfuns=[logc_;logcp_;logk_;logkp_;logmp_;logmpkp_;logoutput_;mp_;mpkp_;output_];
+% auxvars=[logc;logcp;logk;logkp;logmp;logmpkp;logoutput;mp;mpkp;output];
+
+
+% or automatically by the following code (the names of the
+% auxiliary functions must be the same as the auxiliary variables with an
+% underscore suffix):
+
+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
+
+% Note that f is a function of the model variables and the auxiliary
+% variables. To get f as a function of the model variables only, use the
+% function subsf:
+
+f_noaux = subsf( f_fun,auxvars,auxfuns );
+
+% Compare f with f_noaux
+
+f_fun,f_noaux
+
+% Display the auxiliary equations:
+
+[auxvars,auxfuns]
+
+%--------------------
+% Approximation order
+%--------------------
+order=4; % fourth order is the maximum possible
+
+%----------------
+% Call prepare_tp
+%----------------
+model=prepare_tp(f_fun,Phi_fun,yp,y,xp,x,shocks,symparams,order,auxfuns,auxvars);
+
+%-----------
+% Save model
+%-----------
+save('model') % you will need this later

105/replication_package/examples/rbc/solve_continuation.m

@@ -0,0 +1,102 @@
+%--------------------------------------------
+% Solve the RBC model by continuation method
+%--------------------------------------------
+
+clear,clc
+
+%---------------------------------------------------------
+% Add folder 'files' to the search path and load the model
+%---------------------------------------------------------
+addpath('files'); 
+load('model')
+
+%----------------------------------------------------------------------------
+% Provide nodes and weights for the quadrature that approximates expectations
+%----------------------------------------------------------------------------
+n_e=1; % number of shocks.
+[n_nodes,nodes,weights] = Monomials_2(n_e,eye(n_e)); % this quadrature function was written by Judd, Maliar, Maliar and Valero (2014).
+nodes=nodes'; % transpose to n_e-by-n_nodes
+
+%----------------------------------------------------
+% Choose parameter values with a closed-form solution
+%----------------------------------------------------
+BETA=.96; GAMMA=1; ALPHA=.3; RHO=.8; DELTA=1; SIGMA=.02;
+params=eval(symparams);
+
+%-----------------------------------------------------------------------
+% The closed-form solution for the case GAMMA=1, DELTA=1 for consumption
+%-----------------------------------------------------------------------
+
+g=(1-ALPHA*BETA)*exp(z)*k^ALPHA;
+
+%---------------------------------------------------------
+% Use the closed-form solution to produce an initial guess
+%---------------------------------------------------------
+
+% differentiate the closed-form solution up to fourth order
+gx=jacobian(g,x);
+gxx=jacobian(gx(:),x);
+gxxx=jacobian(gxx(:),x);
+gxxxx=jacobian(gxxx(:),x);
+
+% choose some arbitrary state - I use the steady state of the model of
+% interest (with DELTA=.1)
+
+k0=((1/BETA-1+.1)/ALPHA)^(1/(ALPHA-1));
+z0=0;
+
+x0=[k0;z0];
+
+% compute g(x) and its derivatives at x0
+
+g0=double(subs(g,x(:),x0));
+gx0=double(subs(gx,x(:),x0));
+gxx0=double(subs(gxx,x(:),x0));
+gxxx0=double(subs(gxxx,x(:),x0));
+gxxxx0=double(subs(gxxxx,x(:),x0));
+
+% transform the derivatives into a vector of coefficients
+
+[ initial_guess ] = derivs2coeffs(model,g0,gx0,gxx0,gxxx0,gxxxx0);
+
+% this is for order=4. for lower orders include only the relevant
+% derivatives, e.g. derivs2coeffs(model,g0,gx0,gxx0) is for second order.
+
+% define the center of the initial guess (this is the point at which we computed
+% the derivatives)
+
+c0=x0;
+
+% now we have the initial guess, and we can proceed to solve the model by
+% continuation
+
+%-------------------------------------------------------------------------------
+% solve by Taylor projection and change the parameters gradually to the
+% required level
+%-------------------------------------------------------------------------------
+tolX=1e-6;
+tolF=1e-6;
+maxiter=10;
+
+[coeffs,model]=tpsolve(initial_guess,x0,model,params,c0,nodes,weights,tolX,tolF,maxiter);
+
+% Now change the parameters GAMMA and DELTA gradually to their required levels:
+
+GAMMA_original=GAMMA;
+GAMMA_target=2;
+
+DELTA_original=DELTA;
+DELTA_target=.1;
+
+for h=0:.1:1 % this is the homotopy parameter
+    GAMMA=(1-h)*GAMMA_original+h*GAMMA_target;
+    DELTA=(1-h)*DELTA_original+h*DELTA_target;
+    
+    disp(['GAMMA=' num2str(GAMMA) ' DELTA=' num2str(DELTA)])
+    
+    params(2)=GAMMA;
+    params(5)=DELTA;
+    [coeffs,model]=tpsolve(coeffs,x0,model,params,c0,nodes,weights,tolX,tolF,maxiter);
+
+end
+

105/replication_package/examples/rbc/solve_model.m

@@ -0,0 +1,160 @@
+clear,clc
+
+%---------------------------------------------------------
+% Add folder 'files' to the search path and load the model
+%---------------------------------------------------------
+addpath('files'); 
+load('model')
+
+%----------------------------------------------------------------------------
+% Provide nodes and weights for the quadrature that approximates expectations
+%----------------------------------------------------------------------------
+n_e=1; % number of shocks.
+[n_nodes,nodes,weights] = Monomials_2(n_e,eye(n_e)); % this quadrature function was written by Judd, Maliar, Maliar and Valero (2014).
+nodes=nodes'; % transpose to n_e-by-n_nodes
+
+%----------------------------------
+% Make a vector of parameter values
+%----------------------------------
+BETA=.96; GAMMA=2; ALPHA=.3; RHO=.8; DELTA=.1; SIGMA=.02;
+params=eval(symparams);
+
+%----------------------------------------------------------------------
+% Prepare an initial guess - in this case I use a perturbation solution
+%----------------------------------------------------------------------
+
+% Steady state values
+
+kss=((1/BETA-1+DELTA)/ALPHA)^(1/(ALPHA-1));
+zss=0;
+css=kss^ALPHA-DELTA*kss;
+
+nxss=[kss;zss];
+nyss=css;
+
+% Cross moments of the shocks
+
+M=get_moments(nodes,weights,model.order(2));
+
+% Compute the perturbation solution (keep the 4 outputs):
+
+[derivs,stoch_pert,nonstoch_pert,model]=get_pert(model,params,M,nxss,nyss);
+
+% Explanation of outputs:
+% derivs=structure with the perturbation solution as explained in Levintal
+% (2017): "Fifth-Order Perturbation Solution to DSGE Models".
+% stoch_pert=the perturbation solution in the form of unique polynomial coefficients.
+% nonstoch_pert=same as stoch_pert but without correction for the model volatility (i.e. this is a perturbation solution of a deterministic version of the model)
+
+%-------------------------------------
+% Solve the model by Taylor projection
+%-------------------------------------
+
+x0=nxss; % the approximation point (here we use the steady state, but it could be any arbitrary state)
+c0=nxss; % the center of the initial guess
+
+% tolerance parameters for the Newton solver
+tolX=1e-6;
+tolF=1e-6;
+maxiter=10;
+
+% model.jacobian='exact'; % this is the default
+% model.jacobian='approximate'; % for large models try the approximate jacobian.
+
+initial_guess=stoch_pert;
+[coeffs,model]=tpsolve(initial_guess,x0,model,params,c0,nodes,weights,tolX,tolF,maxiter);
+
+%------------------------------------------------------------------
+% Compute the residual function and the model variables at point x0
+%------------------------------------------------------------------
+
+[R_fun0,g_fun0,Phi_fun0,auxvars0]=residual(coeffs,x0,params,c0,nodes,weights);
+
+% R_fun0 is the residual function at x0.
+% g_fun0 is the control variables at x0, namely, g(x0).
+% Phi_fun0 is the function Phi at x0 and each future node, namely, Phi(x0,g(x0),epsp), for each node of the quadrature.
+% auxvars0 is the auxiliary functions at x0 and each future node.
+
+% compute the function g(x) at x0
+y0=evalg(x0,coeffs,c0);
+
+% compute the function Phi(x,y,epsp) at x0, y0 and epsp0
+epsp0=.02;
+xp0=evalPhi(x0,y0,epsp0,params);
+
+%---------------------------------
+% simulate the model for T periods
+%---------------------------------
+T=100;
+shocks=randn(1,T+1); % draw shocks
+
+% preallocate
+x_simul=zeros(model.n_x,T+1);
+y_simul=zeros(model.n_y,T);
+R_simul=zeros(model.n_y,T);
+
+x_simul(:,1)=x0;
+
+% option=1; % compute only simulated variables
+option=2; % compute model residuals
+
+for t=1:T
+    xt=x_simul(:,t);
+    epsp=shocks(t+1);
+
+    % Option 1 - compute only the simulated variables
+    if option==1
+        yt=evalg(xt,coeffs,c0);
+        
+        y_simul(:,t)=yt;
+        x_simul(:,t+1)=evalPhi(xt,yt,epsp,params);
+    else
+    % Option 2 - compute also model residuals
+        [Rt,yt]=residual(coeffs,xt,params,c0,nodes,weights);
+        
+        y_simul(:,t)=yt;
+        x_simul(:,t+1)=evalPhi(xt,yt,epsp,params);
+        R_simul(:,t)=Rt;
+    end
+end
+
+%-------------------------------------------
+% Solve the model again at a different state
+%-------------------------------------------
+% This is useful when the long run domain of the model is far from the
+% initial state, so we need to approximate the solution at the long run state
+% (e.g. the risky steady state or the mean of the ergodic distribution)
+% rather than the steady state.
+
+x1=x0*1.1; % take some arbitrary state
+[coeffs1,model]=tpsolve(coeffs,x1,model,params,c0,nodes,weights,tolX,tolF,maxiter); % solve at x1
+
+%-----------------------
+% Use a different solver
+%-----------------------
+
+% The function tpsolve uses the Newton method for up to maxiter iterations. If it fails, it
+% switches automatically to fsolve for another maxiter iterations. You can
+% control the parameters of the second solver by optimoptions. The
+% supported solvers are fsolve and lsqnonlin.
+
+% For example, do one Newton iteration and switch to lsqnonlin:
+
+x2=x1*1.1;
+maxiter=1; % one Newton iteration
+OPTIONS = optimoptions('lsqnonlin','TolX',tolX,'TolF',tolF,'MaxIter',10,'Display','iter-detailed'); % 10 more iterations by lsqnonlin
+
+[coeffs2,model]=tpsolve(coeffs,x2,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+
+
+% or switch to fsolve:
+
+maxiter=1; % one Newton iteration
+OPTIONS = optimoptions('fsolve','TolX',tolX,'TolF',tolF,'MaxIter',10,'Display','iter-detailed'); % 10 more iterations by fsolve
+
+[coeffs3,model]=tpsolve(coeffs,x2,model,params,c0,nodes,weights,tolX,tolF,maxiter,OPTIONS);
+
+
+
+
+

105/replication_package/examples/rbc_EZ/prepare_model.m

@@ -0,0 +1,88 @@
+%------------------------------------------------------------------
+% RBC model with Epstein-Zin preferences
+%------------------------------------------------------------------
+
+clear,clc
+
+%-----------------------------------------
+% Define symbolic variables and parameters
+%-----------------------------------------
+
+syms logk logkp logc logcp z zp logxi logxip logq logqp epsp real
+syms BETA GAMMA PSI ALPHA RHO DELTA SIGMA real
+
+
+%----------------------
+% Substituted variables
+%----------------------
+
+c=exp(logc); cp=exp(logcp);
+
+k=exp(logk); kp=exp(logkp);
+
+xi=exp(logxi); xip=exp(logxip);
+
+q=exp(logq);
+
+Vpowerp=(1-BETA)*cp^(1-PSI)+BETA*xip^(1-PSI); % this is Vp^(1-PSI)
+
+logVp=1/(1-PSI)*log( Vpowerp );
+
+Vp=exp(logVp);
+
+logmp=log(BETA)+PSI*(logc-logcp)+(PSI-GAMMA)*(logVp-logxi);
+
+mp=exp(logmp);
+
+%-----------------------
+% Equilibrium conditions
+%-----------------------
+f1=mp*(ALPHA*exp(zp)*kp^(ALPHA-1)+1-DELTA)-1;
+f2=xi^(GAMMA-1)*Vp^(1-GAMMA)-1;
+f3=mp/q-1;
+
+f_fun=[f1;f2;f3];
+
+%-------------------------------------------------------
+% Function Phi (law of motion of log capital and technology)
+%-------------------------------------------------------
+
+Phi_fun=[log(exp(z)*k^ALPHA+(1-DELTA)*k-c);
+         RHO*z+SIGMA*epsp];
+
+%--------------------------
+% Vector of state variables
+%--------------------------
+x=[logk,z]; % current period
+xp=[logkp,zp]; % future period
+
+%----------------------------
+% Vector of control variables
+%----------------------------
+y=[logc,logxi,logq]; % current period
+yp=[logcp,logxip,logqp]; % future period
+
+%-----------------
+% Vector of shocks
+%-----------------
+shocks=[epsp];
+
+%---------------------
+% Vector of parameters
+%---------------------
+symparams=[BETA,GAMMA,PSI,ALPHA,RHO,DELTA,SIGMA];
+
+%--------------------
+% Approximation order
+%--------------------
+order=3; % fourth order is the maximum possible
+
+%----------------
+% Call prepare_tp
+%----------------
+model=prepare_tp(f_fun,Phi_fun,yp,y,xp,x,shocks,symparams,order);
+
+%-----------
+% Save model
+%-----------
+save('model') % you will need this later

105/replication_package/examples/rbc_EZ/prepare_model_auxiliary_functions.m

@@ -0,0 +1,106 @@
+%------------------------------------------------------------------
+% RBC model with Epstein-Zin preferences
+%------------------------------------------------------------------
+
+clear,clc
+
+%-----------------------------------------
+% Define symbolic variables and parameters
+%-----------------------------------------
+
+syms logk logkp logc logcp z zp logxi logxip logq logqp epsp real
+syms BETA GAMMA PSI ALPHA RHO DELTA SIGMA real
+
+
+%-----------------------------------------------------
+% Substituted variables defined by auxiliary functions
+%-----------------------------------------------------
+
+syms c cp k kp xi xip q Vpowerp logVp Vp logmp mp real
+
+c_=exp(logc); cp_=exp(logcp);
+
+k_=exp(logk); kp_=exp(logkp);
+
+xi_=exp(logxi); xip_=exp(logxip);
+
+q_=exp(logq);
+
+Vpowerp_=(1-BETA)*cp^(1-PSI)+BETA*xip^(1-PSI); % this is Vp^(1-PSI)
+
+logVp_=1/(1-PSI)*log( Vpowerp );
+
+Vp_=exp(logVp);
+
+logmp_=log(BETA)+PSI*(logc-logcp)+(PSI-GAMMA)*(logVp-logxi);
+
+mp_=exp(logmp);
+
+%-----------------------
+% Equilibrium conditions
+%-----------------------
+f1=mp*(ALPHA*exp(zp)*kp^(ALPHA-1)+1-DELTA)-1;
+f2=xi^(GAMMA-1)*Vp^(1-GAMMA)-1;
+f3=mp/q-1;
+
+f_fun=[f1;f2;f3];
+
+%-------------------------------------------------------
+% Function Phi (law of motion of log capital and technology)
+%-------------------------------------------------------
+
+Phi_fun=[log(exp(z)*k^ALPHA+(1-DELTA)*k-c);
+         RHO*z+SIGMA*epsp];
+
+%--------------------------
+% Vector of state variables
+%--------------------------
+x=[logk,z]; % current period
+xp=[logkp,zp]; % future period
+
+%----------------------------
+% Vector of control variables
+%----------------------------
+y=[logc,logxi,logq]; % current period
+yp=[logcp,logxip,logqp]; % future period
+
+%-----------------
+% Vector of shocks
+%-----------------
+shocks=[epsp];
+
+%---------------------
+% Vector of parameters
+%---------------------
+symparams=[BETA,GAMMA,PSI,ALPHA,RHO,DELTA,SIGMA];
+
+%------------------------------------------
+% Collect auxiliary functions and variables
+%------------------------------------------
+
+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
+%--------------------
+order=4; % fourth order is the maximum possible
+
+%----------------
+% Call prepare_tp
+%----------------
+model=prepare_tp(f_fun,Phi_fun,yp,y,xp,x,shocks,symparams,order,auxfuns,auxvars);
+
+%-----------
+% Save model
+%-----------
+save('model') % you will need this later

105/replication_package/examples/rbc_EZ/solve_model.m

@@ -0,0 +1,104 @@
+clear,clc
+
+%---------------------------------------------------------
+% Add folder 'files' to the search path and load the model
+%---------------------------------------------------------
+addpath('files'); 
+load('model')
+
+%----------------------------------------------------------------------------
+% Provide nodes and weights for the quadrature that approximates expectations
+%----------------------------------------------------------------------------
+n_e=length(shocks); % number of shocks.
+[n_nodes,nodes,weights] = Monomials_2(n_e,eye(n_e)); % this quadrature function was written by Judd, Maliar, Maliar and Valero (2014).
+nodes=nodes'; % transpose to n_e-by-n_nodes
+
+%----------------------------------
+% Make a vector of parameter values
+%----------------------------------
+BETA=.96; GAMMA=2; PSI=1.5; ALPHA=.3; RHO=.8; DELTA=.1; SIGMA=.08;
+params=eval(symparams);
+
+%----------------------------------------------------------------------
+% Prepare an initial guess - in this case I use a perturbation solution
+%----------------------------------------------------------------------
+
+% Steady state values
+
+kss=((1/BETA-1+DELTA)/ALPHA)^(1/(ALPHA-1));
+zss=0;
+css=kss^ALPHA-DELTA*kss;
+vss=css;
+xiss=vss;
+qss=BETA;
+
+nxss=[log(kss);zss];
+nyss=[log(css);log(xiss);log(qss)];
+
+% Cross moments of the shocks
+
+M=get_moments(nodes,weights,model.order(2));
+
+% Compute the perturbation solution (keep the 4 outputs):
+
+[derivs,stoch_pert,nonstoch_pert,model]=get_pert(model,params,M,nxss,nyss);
+
+% Explanation of outputs:
+% derivs=structure with the perturbation solution as explained in Levintal
+% (2017): "Fifth-Order Perturbation Solution to DSGE Models".
+% stoch_pert=the perturbation solution in the form of unique polynomial coefficients.
+% nonstoch_pert=same as stoch_pert but without correction for the model volatility (i.e. this is a perturbation solution of a deterministic version of the model)
+
+%-------------------------------------
+% Solve the model by Taylor projection
+%-------------------------------------
+
+x0=nxss; % the approximation point (here we use the steady state, but it could be any arbitrary state)
+c0=nxss; % the center of the initial guess
+
+% tolerance parameters for the Newton solver
+tolX=1e-6;
+tolF=1e-6;
+maxiter=10;
+
+% model.jacobian='exact'; % this is the default
+% model.jacobian='approximate'; % for large models try the approximate jacobian.
+
+initial_guess=stoch_pert;
+[coeffs,model]=tpsolve(initial_guess,x0,model,params,c0,nodes,weights,tolX,tolF,maxiter);
+
+%------------------------------------------------------------------
+% Compute the residual function and the model variables at point x0
+%------------------------------------------------------------------
+
+[R_fun0,g_fun0,Phi_fun0,auxvars0]=residual(coeffs,x0,params,c0,nodes,weights);
+
+%------------------------
+% Check the interest rate
+%------------------------
+
+logq=g_fun0(3);
+Rf=exp(-logq)-1
+
+%----------------------------------------------------------------------------
+% Increase risk aversion (gradually) and see how the interest rate falls
+%----------------------------------------------------------------------------
+
+GAMMAvec=2:4:82;
+
+Rfvec=zeros(size(GAMMAvec));
+
+i=0;
+for GAMMA=GAMMAvec
+    i=i+1;
+    params(2)=GAMMA;
+    [coeffs,model]=tpsolve(coeffs,x0,model,params,c0,nodes,weights,tolX,tolF,maxiter);
+    [R_fun0,g_fun0,Phi_fun0,auxvars0]=residual(coeffs,x0,params,c0,nodes,weights);
+    logq=g_fun0(3);
+    Rfvec(i)=exp(-logq)-1;
+end
+
+
+plot(GAMMAvec,Rfvec)
+xlabel('Risk aversion (GAMMA)')
+ylabel('Risk-free interest rate (Rf)')