OLGModel8.m

%% OLG Model 8: Deterministic Economic/Productivity Growth
% Add deterministic productivity growth to the model.
% We have to solve a 'per capita per technology-unit' version of the model
% (same concept as balanced growth path in Solow-Swan neoclassical growth model)
% At the end we 'reinflate' the model to create a time series for aggregate output.
%
% Note that to be able to have a balanced growth path we have to change the
% utility function. See the Introduction to Life-Cycle Models: Life-Cycle
% Model 22 (https://www.vfitoolkit.com/updates-blog/2021/an-introduction-to-life-cycle-models/)
% for an explanation.
%
% As explained in the pdf, most of the variables in this model are in fact
% the 'hat' variables (adjusted to remove the deterministic productivity
% growth) but they have not been renamed to reflect this. At the end the
% original variables are recovered and referred to as 'actual L', 'actual
% w', etc.

%% Begin setting up to use VFI Toolkit to solve
% Lets model agents from age 20 to age 100, so 81 periods

Params.agejshifter=19; % Age 20 minus one. Makes keeping track of actual age easy in terms of model age
Params.J=100-Params.agejshifter; % =81, Number of period in life-cycle

% Grid sizes to use
n_d=51; % Endogenous labour choice (fraction of time worked)
n_a=301; % Endogenous asset holdings
% Exogenous labor productivity units shocks (next two lines)
n_z=15; % AR(1) with age-dependent params
vfoptions.n_e=3; % iid
N_j=Params.J; % Number of periods in finite horizon

%% Parameters

% Deterministic growth
Params.g=0.02;

% Discount rate
Params.beta = 0.96;
% Preferences
Params.sigma1 = 0.3; % Weight on consumption
Params.sigma2 = 2;  % Curvature of utility
% Note: I have not thought seriously about appropriate parameter values for sigma1, sigma2. These are very arbitrary

Params.A=1; % Aggregate TFP. Not actually used anywhere.
% Production function
Params.alpha = 0.3; % Share of capital
Params.delta = 0.1; % Depreciation rate of capital

% Demographics
Params.agej=1:1:Params.J; % Is a vector of all the agej: 1,2,3,...,J
Params.Jr=46;
% Population growth rate
Params.n=0.02; % percentage rate (expressed as fraction) at which population growths

% Age-dependent labor productivity units
Params.kappa_j=[linspace(0.5,2,Params.Jr-15),linspace(2,1,14),zeros(1,Params.J-Params.Jr+1)];
% Life-cycle AR(1) process z, on (log) labor productivity units
% Chosen following Karahan & Ozkan (2013) [as used by Fella, Gallipoli & Pan (2019)]
% Originals just cover ages 24 to 60, so I create these, and then repeat first and last periods to fill it out
Params.rho_z=0.7596+0.2039*((1:1:37)/10)-0.0535*((1:1:37)/10).^2+0.0028*((1:1:37)/10).^3; % Chosen following Karahan & Ozkan (2013) [as used by Fella, Gallipoli & Pan (2019)]
Params.sigma_epsilon_z=0.0518-0.0405*((1:1:37)/10)+0.0105*((1:1:37)/10).^2-0.0002*((1:1:37)/10).^3; % Chosen following Karahan & Ozkan (2013) [as used by Fella, Gallipoli & Pan (2019)]
% Note that 37 covers 24 to 60 inclusive
% Now repeat the first and last values to fill in working age, and put zeros for retirement (where it is anyway irrelevant)
Params.rho_z=[Params.rho_z(1)*ones(1,4),Params.rho_z,Params.rho_z(end)*ones(1,4),zeros(1,100-65+1)];
Params.sigma_epsilon_z=[Params.sigma_epsilon_z(1)*ones(1,4),Params.sigma_epsilon_z,Params.sigma_epsilon_z(end)*ones(1,4),Params.sigma_epsilon_z(end)*ones(1,100-65+1)];
% Transitory iid shock
Params.sigma_e=0.0410+0.0221*((24:1:60)/10)-0.0069*((24:1:60)/10).^2+0.0008*((24:1:60)/10).^3;
% Now repeat the first and last values to fill in working age, and put zeros for retirement (where it is anyway irrelevant)
Params.sigma_e=[Params.sigma_e(1)*ones(1,4),Params.sigma_e,Params.sigma_e(end)*ones(1,4),Params.sigma_e(end)*ones(1,100-65+1)];
% Note: These will interact with the endogenous labor so the final labor
% earnings process will not equal that of Karahan & Ozkan (2013)
% Note: Karahan & Ozkan (2013) also have a fixed effect (which they call alpha) and which I ignore here.

% Conditional survival probabilities: sj is the probability of surviving to be age j+1, given alive at age j
% Most countries have calculations of these (as they are used by the government departments that oversee pensions)
% In fact I will here get data on the conditional death probabilities, and then survival is just 1-death.
% Here I just use them for the US, taken from "National Vital Statistics Report, volume 58, number 10, March 2010."
% I took them from first column (qx) of Table 1 (Total Population)
% Conditional death probabilities
Params.dj=[0.006879, 0.000463, 0.000307, 0.000220, 0.000184, 0.000172, 0.000160, 0.000149, 0.000133, 0.000114, 0.000100, 0.000105, 0.000143, 0.000221, 0.000329, 0.000449, 0.000563, 0.000667, 0.000753, 0.000823,...
    0.000894, 0.000962, 0.001005, 0.001016, 0.001003, 0.000983, 0.000967, 0.000960, 0.000970, 0.000994, 0.001027, 0.001065, 0.001115, 0.001154, 0.001209, 0.001271, 0.001351, 0.001460, 0.001603, 0.001769, 0.001943, 0.002120, 0.002311, 0.002520, 0.002747, 0.002989, 0.003242, 0.003512, 0.003803, 0.004118, 0.004464, 0.004837, 0.005217, 0.005591, 0.005963, 0.006346, 0.006768, 0.007261, 0.007866, 0.008596, 0.009473, 0.010450, 0.011456, 0.012407, 0.013320, 0.014299, 0.015323,...
    0.016558, 0.018029, 0.019723, 0.021607, 0.023723, 0.026143, 0.028892, 0.031988, 0.035476, 0.039238, 0.043382, 0.047941, 0.052953, 0.058457, 0.064494,...
    0.071107, 0.078342, 0.086244, 0.094861, 0.104242, 0.114432, 0.125479, 0.137427, 0.150317, 0.164187, 0.179066, 0.194979, 0.211941, 0.229957, 0.249020, 0.269112, 0.290198, 0.312231, 1.000000]; 
% dj covers Ages 0 to 100
Params.sj=1-Params.dj(21:101); % Conditional survival probabilities
Params.sj(end)=0; % In the present model the last period (j=J) value of sj is actually irrelevant

% Warm glow of bequest
Params.warmglow1=0.3; % (relative) importance of bequests
Params.warmglow2=3; % bliss point of bequests (essentially, the target amount)
Params.warmglow3=Params.sigma2; % By using the same curvature as the utility of consumption it makes it much easier to guess appropraite parameter values for the warm glow

% Taxes
Params.tau = 0.15; % Tax rate on labour income
Params.tau_final=0.1; % We will look at transition path resulting from reduction in tax rate to this level
Params.tau=Params.tau;
% In addition to payroll tax rate tau, which funds the pension system we will add a progressive 
% income tax which funds government spending.
% The progressive income tax takes the functional form:
% IncomeTax=eta1+eta2*log(Income)*Income; % This functional form is empirically a decent fit for the US tax system
% And is determined by the two parameters
Params.eta1=0.09; % eta1 will be determined in equilibrium to balance gov budget constraint
Params.eta2=0.053;

% Government spending
Params.GdivYtarget = 0.15; % Government spending as a fraction of GDP (this is essentially just used as a target to define a general equilibrium condition)

%% Some initial values/guesses for variables that will be determined in general eqm
Params.pension=0.4; % Initial guess (this will be determined in general eqm)
Params.r=0.1;
Params.AccidentBeq=0.03; % Accidental bequests (this is the lump sum transfer)
Params.G=0.2; % Government expenditure
% Params.eta1=0.09; % already set above

%% Grids
a_grid=10*(linspace(0,1,n_a).^3)'; % The ^3 means most points are near zero, which is where the derivative of the value fn changes most.

% First, z, the AR(1) with age-dependent parameters
[z_grid_J, pi_z_J] = discretizeLifeCycleAR1_FellaGallipoliPan(Params.rho_z,Params.sigma_epsilon_z,n_z,Params.J);
% z_grid_J is n_z-by-J, so z_grid_J(:,j) is the grid for age j
% pi_z_J is n_z-by-n_z-by-J, so pi_z_J(:,:,j) is the transition matrix for age j

% Second, e, the iid normal with age-dependent parameters
[e_grid_J, pi_e_J] = discretizeLifeCycleAR1_FellaGallipoliPan(zeros(1,Params.J),Params.sigma_e,vfoptions.n_e,Params.J); % Note: AR(1) with rho=0 is iid normal
% Because e is iid we actually just use
pi_e_J=shiftdim(pi_e_J(1,:,:),1);

% To use exogenous shocks that depend on age you have to add them to vfoptions and simoptions
% Similarly any (iid) e variable always has to go into vfoptions and simoptions
vfoptions.e_grid=e_grid_J;
vfoptions.pi_e=pi_e_J;
simoptions.n_e=vfoptions.n_e;
simoptions.e_grid=e_grid_J;
simoptions.pi_e=pi_e_J;


% Grid for labour choice
h_grid=linspace(0,1,n_d)'; % Notice that it is imposing the 0<=h<=1 condition implicitly
% Switch into toolkit notation
d_grid=h_grid;

%% Now, create the return function 
% The discount factor includes a part that comes from the renormalization
% to remove growth
Params.growthdiscountfactor=(1+Params.g)^(Params.sigma1*(1-Params.sigma2));
DiscountFactorParamNames={'beta','sj','growthdiscountfactor'};

% Notice we use 'OLGModel8_ReturnFn'
ReturnFn=@(h,aprime,a,z,e,sigma1,sigma2,agej,Jr,pension,r,A,delta,alpha,kappa_j,warmglow1,warmglow2,warmglow3,beta,sj,tau,g)...
    OLGModel8_ReturnFn(h,aprime,a,z,e,sigma1,sigma2,agej,Jr,pension,r,A,delta,alpha,kappa_j,warmglow1,warmglow2,warmglow3,beta,sj,tau,g);

%% Now solve the value function iteration problem, just to check that things are working before we go to General Equilbrium
disp('Test ValueFnIter')
tic;
[V, Policy]=ValueFnIter_Case1_FHorz(n_d,n_a,n_z,N_j, d_grid, a_grid, z_grid_J, pi_z_J, ReturnFn, Params, DiscountFactorParamNames, [], vfoptions);
toc

%% Initial distribution of agents at birth (j=1)
% Before we plot the life-cycle profiles we have to define how agents are
% at age j=1. We will give them all zero assets.
jequaloneDist=zeros([n_a,n_z,vfoptions.n_e],'gpuArray'); % Put no households anywhere on grid
jequaloneDist(1,floor((n_z+1)/2),floor((simoptions.n_e+1)/2))=1; % All agents start with zero assets, and the median shock

%% Agents age distribution
% Many OLG models include some kind of population growth, and perhaps
% some other things that create a weighting of different ages that needs to
% be used to calculate the stationary distribution and aggregate variable.
% Many OLG models include some kind of population growth, and perhaps
% some other things that create a weighting of different ages that needs to
% be used to calculate the stationary distribution and aggregate variable.
Params.mewj=ones(1,Params.J); % Marginal distribution of households over age
for jj=2:length(Params.mewj)
    Params.mewj(jj)=Params.sj(jj-1)*Params.mewj(jj-1)/(1+Params.n);
end
Params.mewj=Params.mewj./sum(Params.mewj); % Normalize to one

AgeWeightsParamNames={'mewj'}; % So VFI Toolkit knows which parameter is the mass of agents of each age

%% Test
disp('Test StationaryDist')
StationaryDist=StationaryDist_FHorz_Case1(jequaloneDist,AgeWeightsParamNames,Policy,n_d,n_a,n_z,N_j,pi_z_J,Params,simoptions);

%% General eqm variables
GEPriceParamNames={'r','pension','AccidentBeq','G','eta1'};

%% Set up the General Equilibrium conditions (on assets/interest rate, assuming a representative firm with Cobb-Douglas production function)
% Note: we need to add z & e to FnsToEvaluate inputs.

% Stationary Distribution Aggregates (important that ordering of Names and Functions is the same)
FnsToEvaluate.H = @(h,aprime,a,z,e) h; % Aggregate labour supply
FnsToEvaluate.L = @(h,aprime,a,z,e,kappa_j) kappa_j*exp(z+e)*h;  % Aggregate labour supply in efficiency units 
FnsToEvaluate.K = @(h,aprime,a,z,e) a;% Aggregate  physical capital
FnsToEvaluate.PensionSpending = @(h,aprime,a,z,e,pension,agej,Jr) (agej>=Jr)*pension; % Total spending on pensions
FnsToEvaluate.AccidentalBeqLeft = @(h,aprime,a,z,e,sj) aprime*(1-sj); % Accidental bequests left by people who die
FnsToEvaluate.IncomeTaxRevenue = @(h,aprime,a,z,e,eta1,eta2,kappa_j,r,delta,alpha,A) OLGModel6_ProgressiveIncomeTaxFn(h,aprime,a,z,e,eta1,eta2,kappa_j,r,delta,alpha,A); % Revenue raised by the progressive income tax (needed own function to avoid log(0) causing problems)

% General Equilibrium conditions (these should evaluate to zero in general equilbrium)
GeneralEqmEqns.capitalmarket = @(r,K,L,alpha,delta,A) r-alpha*A*(K^(alpha-1))*(L^(1-alpha)); % interest rate equals marginal product of capital net of depreciation
GeneralEqmEqns.pensions = @(PensionSpending,tau,L,r,A,alpha,delta) PensionSpending-tau*(A*(1-alpha)*((r+delta)/(alpha*A))^(alpha/(alpha-1)))*L; % Retirement benefits equal Payroll tax revenue: pension*fractionretired-tau*w*H
GeneralEqmEqns.bequests = @(AccidentalBeqLeft,AccidentBeq,n,g) AccidentalBeqLeft/((1+n)*(1+g))-AccidentBeq; % Accidental bequests received equal accidental bequests left
GeneralEqmEqns.Gtarget = @(G,GdivYtarget,A,K,L,alpha) G-GdivYtarget*(A*K^(alpha)*(L^(1-alpha))); % G is equal to the target, GdivYtarget*Y
GeneralEqmEqns.govbudget = @(G,IncomeTaxRevenue) G-IncomeTaxRevenue; % Government budget balances (note that pensions are a seperate budget)
% Note: the pensions general eqm condition looks more complicated just because we replaced w with the formula for w in terms of r. It is actually just the same formula as before.

%% Test
% Note: Because we used simoptions we must include this as an input
disp('Test AggVars')
AggVars=EvalFnOnAgentDist_AggVars_FHorz_Case1(StationaryDist, Policy, FnsToEvaluate, Params, [], n_d, n_a, n_z,N_j, d_grid, a_grid, z_grid_J,[],simoptions);

%% Solve for the General Equilibrium
heteroagentoptions.verbose=1;
p_eqm=HeteroAgentStationaryEqm_Case1_FHorz(jequaloneDist,AgeWeightsParamNames,n_d, n_a, n_z, N_j, 0, pi_z_J, d_grid, a_grid, z_grid_J, ReturnFn, FnsToEvaluate, GeneralEqmEqns, Params, DiscountFactorParamNames, [], [], [], GEPriceParamNames, heteroagentoptions, simoptions, vfoptions);
% p_eqm contains the general equilibrium parameter values
% Put this into Params so we can calculate things about the initial equilibrium
Params.r=p_eqm.r;
Params.pension=p_eqm.pension;
Params.AccidentBeq=p_eqm.AccidentBeq;
Params.G=p_eqm.G;
Params.eta1=p_eqm.eta1;

% Calculate a few things related to the general equilibrium.
[V, Policy]=ValueFnIter_Case1_FHorz(n_d,n_a,n_z,N_j, d_grid, a_grid, z_grid_J, pi_z_J, ReturnFn, Params, DiscountFactorParamNames, [], vfoptions);
StationaryDist=StationaryDist_FHorz_Case1(jequaloneDist,AgeWeightsParamNames,Policy,n_d,n_a,n_z,N_j,pi_z_J,Params,simoptions);
% Can just use the same FnsToEvaluate as before.
AgeConditionalStats=LifeCycleProfiles_FHorz_Case1(StationaryDist,Policy,FnsToEvaluate,Params,[],n_d,n_a,n_z,N_j,d_grid,a_grid,z_grid_J,simoptions);

%% Plot the life cycle profiles of capital and labour for the inital and final eqm.

figure(1)
subplot(3,1,1); plot(1:1:Params.J,AgeConditionalStats.H.Mean)
title('Life Cycle Profile: Hours Worked')
subplot(3,1,2); plot(1:1:Params.J,AgeConditionalStats.L.Mean)
title('Life Cycle Profile: Labour Supply')
subplot(3,1,3); plot(1:1:Params.J,AgeConditionalStats.K.Mean)
title('Life Cycle Profile: Assets')
% saveas(figure_c,'./SavedOutput/Graphs/OLGModel6_LifeCycleProfiles','pdf')

%% Calculate some aggregates and print findings about them

% Add consumption to the FnsToEvaluate
FnsToEvaluate.Consumption=@(h,aprime,a,z,e,agej,Jr,r,pension,tau,kappa_j,alpha,delta,A,eta1,eta2,AccidentBeq) OLGModel6_ConsumptionFn(h,aprime,a,z,e,agej,Jr,r,pension,tau,kappa_j,alpha,delta,A,eta1,eta2,AccidentBeq);

AggVars=EvalFnOnAgentDist_AggVars_FHorz_Case1(StationaryDist, Policy, FnsToEvaluate, Params, [], n_d, n_a, n_z,N_j, d_grid, a_grid, z_grid_J,[],simoptions);

% GDP
Y=Params.A*(AggVars.K.Mean^Params.alpha)*(AggVars.L.Mean^(1-Params.alpha));

% wage (note that this is calculation is only valid because we have Cobb-Douglas production function and are looking at a stationary general equilibrium)
KdivL=((Params.r+Params.delta)/(Params.alpha*Params.A))^(1/(Params.alpha-1));
w=Params.A*(1-Params.alpha)*(KdivL^Params.alpha); % wage rate (per effective labour unit)

fprintf('Following are some aggregates of the model economy: \n')
fprintf('Output: Y=%8.2f \n',Y)
fprintf('Capital-Output ratio: K/Y=%8.2f \n',AggVars.K.Mean/Y)
fprintf('Consumption-Output ratio: C/Y=%8.2f \n',AggVars.Consumption.Mean/Y)
fprintf('Average labor productivity: Y/H=%8.2f \n', Y/AggVars.H.Mean)
fprintf('Government-to-Output ratio: G/Y=%8.2f \n', Params.G/Y)
fprintf('Accidental Bequests as fraction of GDP: %8.2f \n',Params.AccidentBeq/Y)
fprintf('Wage: w=%8.2f \n',w)

%% Now, restore the growth into the economy.
% We need an aribitrary starting point.
% We already have Params.A, just add
Params.N0=1;

% Number of time periods
T=60;

% Create a time series for output
% There are essentially just the formulae we used to define the 'hat'
% variables (detrended/removing the deterministic growth), just rearranged.
Y_timeseries=Y*cumprod((1+Params.g)*ones(1,T)).*cumprod((1+Params.n)*ones(1,T))*Params.N0; % Note: this is just how we defined Yhat, but rearranged to instead recover Y

Ypercapita_timeseries=Y*cumprod((1+Params.g)*ones(1,T));

w_timeseries=w*cumprod((1+Params.g)*ones(1,T)); 

r_timeseries=Params.r*ones(1,T);

K_timeseries=AggVars.K.Mean*cumprod((1+Params.g)*ones(1,T)).*cumprod((1+Params.n)*ones(1,T))*Params.N0;

effectiveL_timeseries=AggVars.L.Mean*cumprod((1+Params.g)*ones(1,T)).*cumprod((1+Params.n)*ones(1,T))*Params.N0; % This is the term that goes into the Cobb-Douglas production function [the whole (1+g)^t*L term]

N_timeseries=cumprod((1+Params.n)*ones(1,T))*Params.N0; % Population

figure(2)
subplot(2,2,1); plot(1:1:T, Y_timeseries,1:1:T, K_timeseries,1:1:T, effectiveL_timeseries)
legend('Output','Physical Capital','Effective Labor input')
subplot(2,2,2); plot(1:1:T, Ypercapita_timeseries,1:1:T, w_timeseries)
legend('Output per capita','wage')
subplot(2,2,3); plot(1:1:T, N_timeseries)
legend('Population')
subplot(2,2,4); plot(1:1:T, r_timeseries)
legend('Interest rate')