RBC_Matlab.m

%% Basic RBC model with full depreciation
%
% Jesus Fernandez-Villaverde
% Haverford, July 31, 2013

%% 0. Housekeeping

clear all
close all
clc

tic

%%  1. Calibration

aalpha = 1/3;     % Elasticity of output w.r.t. capital
bbeta  = 0.95;    % Discount factor

% Productivity values
vProductivity = [0.9792; 0.9896; 1.0000; 1.0106; 1.0212]';

% Transition matrix
mTransition   = [0.9727, 0.0273, 0.0000, 0.0000, 0.0000;
                 0.0041, 0.9806, 0.0153, 0.0000, 0.0000;
                 0.0000, 0.0082, 0.9837, 0.0082, 0.0000;
                 0.0000, 0.0000, 0.0153, 0.9806, 0.0041;
                 0.0000, 0.0000, 0.0000, 0.0273, 0.9727];

%% 2. Steady State

capitalSteadyState = (aalpha*bbeta)^(1/(1-aalpha));
outputSteadyState = capitalSteadyState^aalpha;
consumptionSteadyState = outputSteadyState-capitalSteadyState;

fprintf(' Output = %2.6f, Capital = %2.6f, Consumption = %2.6f\n', outputSteadyState, capitalSteadyState, consumptionSteadyState); 
fprintf('\n')

% We generate the grid of capital
vGridCapital = 0.5*capitalSteadyState:0.00001:1.5*capitalSteadyState;

nGridCapital = length(vGridCapital);
nGridProductivity = length(vProductivity);

%% 3. Required matrices and vectors

mOutput           = zeros(nGridCapital,nGridProductivity);
mValueFunction    = zeros(nGridCapital,nGridProductivity);
mValueFunctionNew = zeros(nGridCapital,nGridProductivity);
mPolicyFunction   = zeros(nGridCapital,nGridProductivity);
expectedValueFunction = zeros(nGridCapital,nGridProductivity);

%% 4. We pre-build output for each point in the grid

mOutput = (vGridCapital'.^aalpha)*vProductivity;

%% 5. Main iteration

maxDifference = 10.0;
tolerance = 0.0000001;
iteration = 0;

while (maxDifference>tolerance)  
    
    expectedValueFunction = mValueFunction*mTransition';
    
    for nProductivity = 1:nGridProductivity
        
        % We start from previous choice (monotonicity of policy function)
        gridCapitalNextPeriod = 1;
        
        for nCapital = 1:nGridCapital
                        
            valueHighSoFar = -1000.0;
            capitalChoice  = vGridCapital(1);
            
            for nCapitalNextPeriod = gridCapitalNextPeriod:nGridCapital
                
                consumption = mOutput(nCapital,nProductivity)-vGridCapital(nCapitalNextPeriod);
                valueProvisional = (1-bbeta)*log(consumption)+bbeta*expectedValueFunction(nCapitalNextPeriod,nProductivity);
            
                if (valueProvisional>valueHighSoFar)
                    valueHighSoFar = valueProvisional;
                    capitalChoice = vGridCapital(nCapitalNextPeriod);
                    gridCapitalNextPeriod = nCapitalNextPeriod;
                else
                    break; % We break when we have achieved the max
                end    
                  
            end
            
            mValueFunctionNew(nCapital,nProductivity) = valueHighSoFar;
            mPolicyFunction(nCapital,nProductivity) = capitalChoice;
            
        end
        
    end
    
    maxDifference = max(max(abs(mValueFunctionNew-mValueFunction)));
    mValueFunction = mValueFunctionNew;
    
    iteration = iteration+1;
    if (mod(iteration,10)==0 || iteration ==1)
        fprintf(' Iteration = %d, Sup Diff = %2.8f\n', iteration, maxDifference); 
    end
           
end

fprintf(' Iteration = %d, Sup Diff = %2.8f\n', iteration, maxDifference); 
fprintf('\n')

fprintf(' My chek = %2.6f\n', mPolicyFunction(1000,3)); 
fprintf('\n')

toc

%% 6. Plotting results

figure(1)

subplot(3,1,1)
plot(vGridCapital,mValueFunction)
xlim([vGridCapital(1) vGridCapital(nGridCapital)])
title('Value Function')

subplot(3,1,2)
plot(vGridCapital,mPolicyFunction)
xlim([vGridCapital(1) vGridCapital(nGridCapital)])
title('Policy Function')

vExactPolicyFunction = aalpha*bbeta.*(vGridCapital.^aalpha);

subplot(3,1,3)
plot((100.*(vExactPolicyFunction'-mPolicyFunction(:,3))./mPolicyFunction(:,3)))
title('Comparison of Exact and Approximated Policy Function')