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sim_ves_int.m
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sim_ves_int.m
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%% Elasticity of Substitution Sim with error bars on sigma estimate
close all; clear; clc;
% Simulation params
n = 5000;
cost_multiplier = linspace(1,2,n);
sigma_range = [0.8847];
m = length(sigma_range);
% Exogenous params
c_1 = 104.3;
c_2 = 60;
alpha = [0.6, 0.4];
xi_1 = [1, 1];
xi_2 = [1, 0.01];
budget = 1;
figure('Renderer', 'painters', 'Position', [100 100 900 400])
hold on;
for j = 1:m
sigma = sigma_range(j);
results = zeros(n,2);
for i = 1:n
phi = (sigma - 1)/sigma;
x_1_cost_param = c_1*cost_multiplier(i);
x_2_cost_param = c_2;
% Prices
xi_mat = [xi_1; xi_2];
cost_mat = [x_1_cost_param; x_2_cost_param];
prices = xi_mat\cost_mat;
if any(prices<0)
continue
end
% Price Index
P = ((1/2) * (prices'.^(1-sigma))*(alpha'.^sigma)).^(1/(1-sigma));
if sigma == 1
P = 1;
end
% Quantities
Y = ((alpha'./prices).^(sigma)) * (budget/P);
X = (xi_mat')\Y;
results(i,:) = X';
end
output = [];
output(1,:) = c_1*cost_multiplier'./c_2;
output(2,:) = results(:,1)./results(:,2);
% subset to positive quantities
ind = ~any(output <= 0);
output = output(:,ind);
% relationship between e and the ratio of quantities
hold on;
plot(output(2,2:end), ...
diff(log(output(2,:)))./diff(-log(output(1,:))), ...
'LineWidth', 2, 'Color', 'k');
% store data for average sigma assumption
mean_output = output;
end
%% Get linear approximation of CES (VES)
% Output from optimal equilibrium
output = mean_output;
Y = diff(log(output(2,:)))./diff(-log(output(1,:)));
X = output(2,2:end);
% Drop NaNs
output = [X;Y];
ind = ~any(isnan(output));
output = output(:,ind);
X = output(1,:);
Y = output(2,:);
% OLS
beta = X'\(Y'-1);
X_range = linspace(0, 50, 100);
Y_hat = 1 + X_range.*beta;
plot(X_range, Y_hat, 'LineStyle', '-.', 'LineWidth', 2.5, ...
'Color', [1, 1, 1]*0.5);
%% Plot formatting
annotation('textarrow', [0.45 .34], [.375 .375], ...
'String', {['$\hat{e}_{1,2} \approx 1 + ', num2str(round(beta, 2)), ...
'(X_1/X_2)$']}, ...
'Interpreter', 'latex', 'fontsize', 14)
% Format plot
legend({'Elasticity of Substitution with \sigma = 0.8847'}, ...
'VES Approximation')
xlabel({'Ratio of Quantities', 'X_1/X_2'})
ylabel({'Elasticity of Substitution', ...
'between Technologies', 'e_{1, 2}',})
xlim([0, 5])
grid('on')
% Format legend
[hleg,att] = legend('show');
legend('Location', 'northwest')
% Save figure
print(gcf,'../../figures/fig_ves_approx_int.png','-dpng','-r300')