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Added incomplete AuerbachKotlikoff1987 which replicates Chapter 5 (incomplete as does transition paths, but not yet doing them with the welfare-compensating transfers)
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‎AuerbachKotlikoff1987/AuerbachKotlikoff1987.m

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% Replication of: Auerbach & Kotlikoff (1987) - Dynamic Fiscal Policy,
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% Chapter 5.
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% INCOMPLETE:Calculate all the equilibria, and have transition paths but
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% they do not yet incorporate the welfare-compenstating transfers.
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% A few lines needed for running on the Server
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addpath(genpath('./MatlabToolkits/'))
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try % Server has 16 cores, but is shared with other users, so use max of 8.
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parpool(8)
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gpuDevice(1)
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catch % Desktop has less than 8, so will give error, on desktop it is fine to use all available cores.
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parpool
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end
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PoolDetails=gcp;
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NCores=PoolDetails.NumWorkers;
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%%
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n_l=51;
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n_a=251;
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N_j=55;
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n_z=1; z_grid=0; pi_z=1; % Deterministic model, so no exogenous shocks z.
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%% Parameters
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Params.J=N_j;
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Params.jj=1:1:Params.J;
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% Discount rate
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Params.delta=0.015;
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Params.beta=1/(1+Params.delta);
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% Constant population growth rate
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Params.n=0.01;
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% Utility function parameters
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Params.rho=0.8;
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Params.gamma=0.25;
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Params.alpha=1.5;
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% Params.ej=4.47+0.033*Params.jj-0.00067*Params.jj.^2; % Footnote on pg 52.
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% Given Figure 5.2 I suspected may be a typo (that or noone is working more than 0.1 of time)
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% Lines 47-50, 62 & 1393 of aktax3.for in codes https://ideas.repec.org/c/dge/qmrbcd/90.html confirm that it should actually be the following
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Params.ej=exp(0.47+0.033*Params.jj-0.00067*Params.jj.^2)/(exp(0.47+0.033-0.00067));
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% That is, they normalize ej at age 1 to take value of 1.
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% Tax parameters for baseline
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Params.tau_phi=0.15; % Proportional income tax. This is the only tax used as part of baseline.
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Params.tau_pi=0; % Progressivity of income tax.
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Params.tau_c=0; % Consumption tax.
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Params.tau_w=0; % Earnings (wage income) tax.
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Params.tau_k=0; % Capital income tax
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Params.z=0; % Fraction of investment which can be deducted from corporate income tax.
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% Firm production function
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Params.epsilon=0.25; % Capital share in the Cobb-Douglas production fn
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Params.sigma=1; % Not actually used (have hard-coded the Cobb-Douglas production function rather than more general CES prodn fn)
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Params.A=0.892657593; % Technology level
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% Investment adjustment-costs
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b=0; % Set to b=10 for Chapter 9
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% Note: there is no capital depreciation.
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%% Grids
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a_grid=gpuArray(linspace(0,7,n_a(1))'); % Is impossition of NO BORROWING CORRECT HERE OR DO THEY ALLOW IT TO BE NEGATIVE???
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l_grid=gpuArray(linspace(0,1,n_l)');
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%% General eqm variables: give some initial values
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GEPriceParamNames={'wt','rt','Gt'};
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Params.wt=1; % wage rate
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Params.rt=0.067; % interest rate (after corporate taxes, but before income taxes)
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Params.Gt=0.7; % Government consumption
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%% Initial distribution of agents at age 20 (jj=1)
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jequaloneDist=zeros(n_a,1);
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jequaloneDist(1,1)=1; % All agents born with zero assets
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%% Return Function
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DiscountFactorParamNames={'beta'};
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ReturnFn=@(l,aprime,a,z,jj,wt,rt,rho,gamma,alpha,ej,tau_phi,tau_pi,tau_c,tau_w) AuerbachKotlikoff1987_ReturnFn(l,aprime,a,z,jj,wt,rt,rho,gamma,alpha,ej,tau_phi,tau_pi,tau_c,tau_w)
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ReturnFnParamNames={'jj','wt','rt','rho','gamma','alpha','ej','tau_phi','tau_pi','tau_c','tau_w'}; %It is important that these are in same order as they appear in 'AuerbachKotlikoff1987_ReturnFn'
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%% Misc
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n_d=n_l;
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d_grid=l_grid;
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sprintf('Grid sizes:')
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n_a
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n_z
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vfoptions.verbose=1;
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vfoptions.policy_forceintegertype=1; % Policy was not being treated as integers (one of the elements was 10^(-15) different from an integer)
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%% Now solve the value function iteration problem (this is just for trying out codes before jumping to general eqm)
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tic;
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[V, Policy]=ValueFnIter_Case1_FHorz(n_d,n_a,n_z,N_j,d_grid, a_grid, z_grid, pi_z, ReturnFn, Params, DiscountFactorParamNames, ReturnFnParamNames, vfoptions);
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toc
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%% Agents stationary distribution
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% Based on the constant population growth rate n, we get that the
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% population weights will be
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Params.mewj=ones(1,Params.J);
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for jj=2:length(Params.mewj)
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Params.mewj(jj)=Params.mewj(jj-1)/(1+Params.n);
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end
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Params.mewj=Params.mewj./sum(Params.mewj);
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simoptions.nsims=4*10^5;
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simoptions.ncores=NCores;
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simoptions.iterate=1;
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AgeWeights={'mewj'}; % Many finite horizon models apply different weights to different 'ages'; eg., due to survival rates or population growth rates.
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StationaryDist=StationaryDist_FHorz_Case1(jequaloneDist,AgeWeights,Policy,n_d,n_a,n_z,N_j,pi_z,Params,simoptions);
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%% General Equilibrium Conditions (aka. marketclearance)
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% Steady State Aggregates (important that ordering of Names and Functions is the same)
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SSvalueParamNames=struct();
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SSvalueParamNames(1).Names={};
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SSvalueParamNames(2).Names={'ej'};
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SSvalueParamNames(3).Names={'wt','rt','ej','tau_phi','tau_pi','tau_c','tau_w','tau_k'};
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SSvaluesFn_1 = @(d_val,aprime_val,a_val,z_val) a_val; % Aggregate assets (which is this periods state)
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SSvaluesFn_2 = @(d_val,aprime_val,a_val,z_val,ej) (1-d_val)*ej; % Aggregate labour supply (in efficiency units)
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SSvaluesFn_3 = @(d_val,aprime_val,a_val,z_val, wt,rt,ej,tau_phi,tau_pi,tau_c,tau_w,tau_k) AuerbachKotlikoff1987_TotalTaxRevenue(d_val,aprime_val,a_val,z_val,wt,rt,ej,tau_phi,tau_pi,tau_c,tau_w,tau_k); % Total tax revenues
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SSvaluesFn={SSvaluesFn_1,SSvaluesFn_2,SSvaluesFn_3};
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% General Equilibrium Equations
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GeneralEqmEqnParamNames=struct();
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GeneralEqmEqnParamNames(1).Names={'epsilon'};
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GeneralEqmEqn_1 = @(AggVars,p,epsilon) p(1)-(1-epsilon)*(AggVars(1)^epsilon)*(AggVars(2)^(-epsilon)); % The requirement that the wage rate equals the marginal product of (efficiency units of) labor.
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GeneralEqmEqnParamNames(2).Names={'epsilon'};
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GeneralEqmEqn_2 = @(AggVars,p,epsilon) p(2)-(epsilon)*(AggVars(1)^(epsilon-1))*(AggVars(2)^(1-epsilon)); % Rate of return on assets is related to Marginal Product of Capital and Tobin's q
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%^PRESENTLY IGNORES TOBINS q
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GeneralEqmEqnParamNames(3).Names={};
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GeneralEqmEqn_3 = @(AggVars,p) p(3)-AggVars(3); % Government balanced budget constraint
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GeneralEqmEqns={GeneralEqmEqn_1,GeneralEqmEqn_2,GeneralEqmEqn_3};
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%% Test
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SSvalues_AggVars=SSvalues_AggVars_FHorz_Case1(StationaryDist, Policy, SSvaluesFn, Params, SSvalueParamNames, n_d, n_a, n_z,N_j, d_grid, a_grid, z_grid, 2); % The 2 is for Parallel (use GPU)
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%% Solve for the General Equilibrium
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% Use the toolkit to find the equilibrium price index
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% GEPriceParamNames={'wt','rt','Gt'}; % Already delared above.
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wt_grid=linspace(0.5,2,21)'*Params.wt; %bad
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rt_grid=linspace(0.5,2,21)'*Params.rt; %okay
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Gt_grid=linspace(0.5,2,21)'*Params.Gt; %good
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p_grid=[wt_grid,rt_grid,Gt_grid];
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disp('Calculating price vector corresponding to the stationary eqm')
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% tic;
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n_p=[length(wt_grid),length(rt_grid),length(Gt_grid)];
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heteroagentoptions.pgrid=p_grid;
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heteroagentoptions.verbose=1;
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[p_eqm_init,p_eqm_index, GeneralEqmEqnsValues]=HeteroAgentStationaryEqm_Case1_FHorz(jequaloneDist,AgeWeights,n_d, n_a, n_z, N_j, n_p, pi_z, d_grid, a_grid, z_grid, ReturnFn, SSvaluesFn, GeneralEqmEqns, Params, DiscountFactorParamNames, ReturnFnParamNames, SSvalueParamNames, GeneralEqmEqnParamNames, GEPriceParamNames, heteroagentoptions, simoptions, vfoptions);
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% findeqmtime=toc
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Params.wt=p_eqm_init(1);
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Params.rt=p_eqm_init(2);
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Params.Gt=p_eqm_init(3);
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save ./SavedOutput/AuerbachKotlikoff1987_GE.mat Params
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%% Get Value fn, policy fn, agents dist in GE.
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%Params.wt=1; % wage rate
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%Params.rt=0.067; % interest rate (after corporate taxes, but before income taxes)
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[V_init, Policy_init]=ValueFnIter_Case1_FHorz(n_d,n_a,n_z,N_j,d_grid, a_grid, z_grid, pi_z, ReturnFn, Params, DiscountFactorParamNames, ReturnFnParamNames, vfoptions);
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StationaryDist_init=StationaryDist_FHorz_Case1(jequaloneDist,AgeWeights,Policy_init,n_d,n_a,n_z,N_j,pi_z,Params,simoptions);
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% Rather than have the population normalized to 1 (I prefer this as then
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% the model can be thought of in terms of probability distributions)
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% Auerbach & Kotlikoff (1987) normalize their population so that the mass
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% of agents of 'age' j=1 is mass one. The following is a correcting factor:
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Params.PopnCorrection=gather(1/sum(sum(sum(StationaryDist_init(:,:,1)))));
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%% For baseline (initial) steady-state, replicate Table 5.1 and Figure 5.2
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SSvalueParamNames(4).Names={'wt','rt','ej','tau_phi','tau_pi','tau_c','tau_w'};
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SSvaluesFn_4 = @(d_val,aprime_val,a_val,z_val,wt,rt,ej,tau_phi,tau_pi,tau_c,tau_w) AuerbachKotlikoff1987_ConsumptionFn(d_val,aprime_val,a_val,z_val,wt,rt,ej,tau_phi,tau_pi,tau_c,tau_w); % Private Consumption
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SSvalueParamNames(5).Names={'delta'};
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SSvaluesFn_5 = @(d_val,aprime_val,a_val,z_val,delta) aprime_val-(1-delta)*a_val; % National Savings rate (gross rate)
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SSvalueParamNames(6).Names={'wt','ej'};
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SSvaluesFn_6 = @(d_val,aprime_val,a_val,z_val,wt,ej) wt*(1-d_val)*ej; % Earnings (pre-tax)
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SSvaluesFn={SSvaluesFn_1,SSvaluesFn_2,SSvaluesFn_3,SSvaluesFn_4,SSvaluesFn_5,SSvaluesFn_6};
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SSvalues_AggVars=SSvalues_AggVars_FHorz_Case1(StationaryDist_init, Policy_init, SSvaluesFn, Params, SSvalueParamNames, n_d, n_a, n_z,N_j, d_grid, a_grid, z_grid, 2); % The 2 is for Parallel (use GPU)
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SSvalues_AggVars=Params.PopnCorrection*SSvalues_AggVars; % Renormalize to AK1987 population size
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if Params.sigma==1 % Cobb-Douglas
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Y=Params.A*(SSvalues_AggVars(1)^Params.epsilon)*(SSvalues_AggVars(2)^(1-Params.epsilon));
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end
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% Table 5_1
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FID = fopen('./SavedOutput/LatexInputs/AuerbachKotlikoff_Table5_1.tex', 'w');
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fprintf(FID, 'The Base Case Steady-State \\\\ \n');
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fprintf(FID, '\\begin{tabular*}{1.00\\textwidth}{@{\\extracolsep{\\fill}}lcclc} \n \\hline \\hline \n');
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fprintf(FID, 'Capital Stock & %8.1f & \\quad & Private Consumption & %8.2f \\\\ \n \\hline \n', SSvalues_AggVars(1),SSvalues_AggVars(4));
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fprintf(FID, 'Labor Supply & %8.1f & & Capital-Output ratio & %8.2f \\\\ \n \\hline \n', SSvalues_AggVars(2),SSvalues_AggVars(1)/Y);
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fprintf(FID, 'Wage & %8.3f & & National Savings rate & %8.2f \\%% \\\\ \n \\hline \n', Params.wt,(Params.delta*SSvalues_AggVars(1))/Y);
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fprintf(FID, 'Pre-tax interest rate & %8.2f \\%% & & Income Tax rate & %8.2f \\%% \\\\ \n \\hline \n', 100*Params.rt,100*Params.tau_phi);
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fprintf(FID, 'National Income & %8.2f & & Social Security Tax rate & %8.2f \\%% \\\\ \n \\hline \n', Y,0);
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fprintf(FID, 'Government Consumption & %8.2f & & Social Security replacement rate & %8.2f \\%% \\\\ \n \\hline \n', Params.PopnCorrection*Params.Gt,0);
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fprintf(FID, '\\hline \n \\end{tabular*} \n');
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fprintf(FID, '\\begin{minipage}[t]{1.00\\textwidth}{\\baselineskip=.5\\baselineskip \\vspace{.3cm} \\footnotesize{ \n');
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fprintf(FID, 'Note: Based on grid sizes of $n_l=%d$ for leisure, and $n_a=%d$ for assets. \\\\ \n', n_d, n_a);
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fprintf(FID, '}} \\end{minipage}');
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fclose(FID);
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SimLifeCycleProfiles=SimLifeCycleProfiles_FHorz_Case1(jequaloneDist,Policy_init, SSvaluesFn,SSvalueParamNames,Params,n_d,n_a,n_z,N_j,d_grid,a_grid,z_grid,pi_z, simoptions);
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% Figure 5.2
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figure(1)
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plot(1:1:N_j, SimLifeCycleProfiles(4,:,1),1:1:N_j, SimLifeCycleProfiles(6,:,1))
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legend('Consumption','Earnings')
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saveas(gcf,'./SavedOutput/Graphs/Figure_5_2.pdf')
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% plot(Params.ej)
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%% Calculate some transition paths
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% Auerbach & Kotlikoff (1987) define the tax reforms in Chapter 5
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% based on keeping government spending the same and setting tax rates to
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% raise this revenue.
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% To implement this we have to switch our definition of the GE prices, so
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% that instead of finding G we find the tax rate (for the relevant reform).
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%% Consumption Tax
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%% Calculate the final equilibrium
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Params.tau_phi=0;
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GEPriceParamNames={'wt','rt','tau_c'};
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% First, calculate the final general equilibrium.
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GeneralEqmEqnParamNames(3).Names={'Gt'};
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GeneralEqmEqn_3 = @(AggVars,p, Gt) Gt-AggVars(3); % Government balanced budget constraint (role of GEPriceParamNames already in AggVars(3) which is total tax revenues)
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GeneralEqmEqns={GeneralEqmEqn_1,GeneralEqmEqn_2,GeneralEqmEqn_3};
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tauc_grid=linspace(0,0.2,21)';
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p_grid=[wt_grid,rt_grid,tauc_grid];
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disp('Calculating price vector corresponding to the final stationary eqm')
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n_p=[length(wt_grid),length(rt_grid),length(tauc_grid)];
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heteroagentoptions.pgrid=p_grid;
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heteroagentoptions.verbose=1;
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[p_eqm_final,p_eqm_index, GeneralEqmEqnsValues]=HeteroAgentStationaryEqm_Case1_FHorz(jequaloneDist,AgeWeights,n_d, n_a, n_z, N_j, n_p, pi_z, d_grid, a_grid, z_grid, ReturnFn, SSvaluesFn, GeneralEqmEqns, Params, DiscountFactorParamNames, ReturnFnParamNames, SSvalueParamNames, GeneralEqmEqnParamNames, GEPriceParamNames, heteroagentoptions, simoptions, vfoptions);
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Params.wt=p_eqm_final(1);
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Params.rt=p_eqm_final(2);
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Params.tau_c=p_eqm_final(3);
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[V_final, Policy_final]=ValueFnIter_Case1_FHorz(n_d,n_a,n_z,N_j,d_grid, a_grid, z_grid, pi_z, ReturnFn, Params, DiscountFactorParamNames, ReturnFnParamNames, vfoptions);
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%%
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save ./SavedOutput/AuerbachKotlikoff1987_TransitionPathSetup.mat V_final StationaryDist_init n_d n_a n_z N_j pi_z d_grid a_grid z_grid ReturnFn SSvaluesFn GeneralEqmEqns Params DiscountFactorParamNames ReturnFnParamNames SSvalueParamNames GeneralEqmEqnParamNames p_eqm_init p_eqm_final
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% load ./SavedOutput/AuerbachKotlikoff1987_TransitionPathSetup.mat V_final StationaryDist_init n_d n_a n_z N_j pi_z d_grid a_grid z_grid ReturnFn SSvaluesFn GeneralEqmEqns Params DiscountFactorParamNames ReturnFnParamNames SSvalueParamNames GeneralEqmEqnParamNames p_eqm_init p_eqm_final
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%% Now, the transition path
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% For this we need the following extra objects: PricePathOld, PriceParamNames, ParamPath, ParamPathNames, T, V_final, StationaryDist_init
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% (already calculated V_final & StationaryDist_init above)
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Params.tau_phi=0.15;
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Params.wt=p_eqm_init(1);
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Params.rt=p_eqm_init(2);
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Params.tau_c=0;
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p_init=[p_eqm_init(1);p_eqm_init(2);0];
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p_final=p_eqm_final;
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% Number of time periods to allow for the transition (if you set T too low
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% it will cause problems, too high just means run-time will be longer).
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T=150
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% We want to look at a one off unanticipated change of tax rate tau_phi. ParamPath & PathParamNames are thus given by
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ParamPath=zeros(T,1); % ParamPath is matrix of size T-by-'number of parameters that change over path'
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% (the way ParamPath is set is designed to allow for a series of changes in the parameters)
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ParamPathNames={'tau_phi'}; % This is the parameter that gets changed 'away' from it's initial value.
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% Consumption tax is considered a GEPriceParam, hence why it is not here.
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% We need to give an initial guess for the price path on interest rates
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PricePath0_1=[linspace(p_init(1), p_final(1), floor(T/2))'; p_final(1)*ones(T-floor(T/2),1)];
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PricePath0_2=[linspace(p_init(2), p_final(2), floor(T/2))'; p_final(2)*ones(T-floor(T/2),1)];
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PricePath0_3=[linspace(p_init(3), p_final(3), floor(T/2))'; p_final(3)*ones(T-floor(T/2),1)];
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PricePath0=[PricePath0_1,PricePath0_2,PricePath0_3];% PricePath0 is matrix of size T-by-'number of prices'
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PricePathNames={'rt','wt','tau_c'};
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% Now just run the TransitionPath_Case1 command (all of the other inputs
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% are things we had already had to define to be able to solve for the
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% initial and final equilibria)
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transpathoptions.weightscheme=1;
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transpathoptions.verbose=1;
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%%
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vfoptions.policy_forceintegertype=1
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PricePathNew=TransitionPath_Case1_Fhorz(PricePath0, PricePathNames, ParamPath, ParamPathNames, T, V_final, StationaryDist_init, n_d, n_a, n_z, N_j, pi_z, d_grid,a_grid,z_grid, ReturnFn, SSvaluesFn, GeneralEqmEqns, Params, DiscountFactorParamNames, ReturnFnParamNames, AgeWeights, SSvalueParamNames, GeneralEqmEqnParamNames,transpathoptions);
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‎AuerbachKotlikoff1987/AuerbachKotlikoff1987_ConsumptionFn.m

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function c=AuerbachKotlikoff1987_ConsumptionFn(l,aprime,a,z,wt,rt,ej,tau_phi,tau_pi,tau_c,tau_w)
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% z is unused as model is deterministic
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% F=-Inf;
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%c=(1+rt)*a+wt*ej*(1-l)-aprime; % in absence of taxes
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TaxableIncome=rt*a+(1-tau_w)*wt*ej*(1-l);
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c=(1/(1+tau_c))*((1-(tau_phi+tau_pi*TaxableIncome/2))*TaxableIncome+a-aprime);
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% ((tau_phi+tau_pi*TaxableIncome/2) is average tax rate on taxable income
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%
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% if c>0
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% Finner=(c^(1-1/rho)+alpha*l^(1-1/rho))^(1/(1-1/rho));
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% F=(Finner^(1-1/gamma))/(1-1/gamma);
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% end
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%
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% if jj==55 && aprime<0
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% F=-Inf; % Impose a_J>=0
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% end
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end

‎AuerbachKotlikoff1987/AuerbachKotlikoff1987_ReturnFn.m

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function F=AuerbachKotlikoff1987_ReturnFn(l,aprime,a,z,jj,wt,rt,rho,gamma,alpha,ej,tau_phi,tau_pi,tau_c,tau_w)
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% z is unused as model is deterministic
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F=-Inf;
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%c=(1+rt)*a+wt*ej*(1-l)-aprime; % in absence of taxes
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TaxableIncome=rt*a+(1-tau_w)*wt*ej*(1-l);
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c=(1/(1+tau_c))*((1-(tau_phi+tau_pi*TaxableIncome/2))*TaxableIncome+a-aprime);
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% ((tau_phi+tau_pi*TaxableIncome/2) is average tax rate on taxable income
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if c>0
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Finner=(c^(1-1/rho)+alpha*l^(1-1/rho))^(1/(1-1/rho));
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F=(Finner^(1-1/gamma))/(1-1/gamma);
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end
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if jj==55 && aprime<0
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F=-Inf; % Impose a_J>=0
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end
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end

‎AuerbachKotlikoff1987/AuerbachKotlikoff1987_TotalTaxRevenue.m

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function TotalRevenue=AuerbachKotlikoff1987_TotalTaxRevenue(l,aprime,a,z,wt,rt,ej,tau_phi,tau_pi,tau_c,tau_w,tau_k)
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% z is unused as model is deterministic
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%c=(1+rt)*a+wt*ej*(1-l)-aprime; % in absence of taxes
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% Revenue from earnings tax:
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Revenue_w=tau_w*wt*ej*(1-l);
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% Revenue from capital income tax
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Revenue_capital=tau_k*(1/(1-tau_k)*rt)*a; %rt is after-capital-tax (but before income tax). So 1/(1-tau_k)*rt give the before-capital-tax rate of return on capital
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TaxableIncome=rt*a+(1-tau_w)*wt*ej*(1-l);
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% Revenue from income tax:
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Revenue_income=(tau_phi+tau_pi*TaxableIncome/2)*TaxableIncome; % ((tau_phi+tau_pi*TaxableIncome/2) is average tax rate on taxable income
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c=(1/(1+tau_c))*((1-(tau_phi+tau_pi*TaxableIncome/2))*TaxableIncome+a-aprime);
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% Revenue from consumption tax:
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Revenue_c=tau_c*c;
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TotalRevenue=Revenue_w+Revenue_capital+Revenue_income+Revenue_c;
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end

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