equilibrium_constrainedborrow.m

%% Equilibrium Interest Rate and Tax
% *back to* <https://fanwangecon.github.io *Fan*>*'s* <https://fanwangecon.github.io/Math4Econ/ 
% *Intro Math for Econ*>*,*  <https://fanwangecon.github.io/M4Econ/ *Matlab Examples*>*, 
% or* <https://fanwangecon.github.io/MEconTools/ *MEconTools*> *Repositories*
%% 
% We have previous solved the household's asset supply problem with a <https://fanwangecon.github.io/Math4Econ/optimization_application/household_borrow_constrained.html 
% borrowing constraint>. And also the firm's <https://fanwangecon.github.io/Math4Econ/derivative_application/K_borrow_firm.html 
% asset demand problem>. We used first order taylor approximation to solve for 
% the <https://fanwangecon.github.io/Math4Econ/matrix_application/demand_supply_taylor_approximate_capital.html 
% approximate equilibrium interest rate> before for the firm's asset demand problem 
% and for the households' <https://fanwangecon.github.io/Math4Econ/derivative_application/K_save_households.html 
% savings problem without borrowing constraint>. Here we find equilibrium interest 
% rate with the constrained borrowing problem. I will analyze the effect of an 
% interest rate (borrowing) rate subsidy for firms and borrowing households that 
% is paid for by savings tax.
%% How do households with different $\beta$ respond to changes in $r$ given Borrowing Constraint?
% Following our previous <https://fanwangecon.github.io/Math4Econ/optimization_application/household_borrow_constrained.html 
% discussions>, the household's borrowing constrained problem is: 
%% 
% * specifically: $\max_b \log(Z_1-b) + \beta_i \cdot \log(Z_2 + b\cdot (1+r))$
% * with: $b\ge \bar{b}$
%% 
% I introduce now heterogeneity in $\beta$. There are $N=3$ households, each 
% with a different $\beta_i$. Note that lower $\beta$ household at the same interest 
% rate $r$ will be more interested in borrowing rather than saving. The households 
% have the same $Z$ and face the same $\bar{b}$. Look at the graph below, at some 
% interest rate, all three households want to borrow, at other rates, some want 
% to borrow and others want to save. 

clear all
% Parameters
z1 = 12;
z2 = 10;
b_bar_num = -1; % borrow up to 1 dollar

% Vector of 3 betas
beta_vec = [0.75 0.85 0.95];
% Vector of interest rates
r_vec = linspace(0.6, 1.40, 100);

% What we had from before to use fmincon
A = [-1];
q = -b_bar_num;
b0 = [0]; % starting value to search for optimal choice

% A vector to store optimal choices
rows = length(r_vec);
cols = length(beta_vec);
b_opti_mat = zeros(rows, cols);
% Solving for optimal choices as we change Z2
for j=1:1:length(beta_vec)
    for i=1:1:length(r_vec)
        U_neg = @(x) -1*(log(z1 - x(1)) + beta_vec(j)*log(z2 + x(1)*r_vec(i)));
        options = optimoptions('FMINCON','Display','off');
        [x_opti,U_at_x_opti] = fmincon(U_neg, b0, A, q, [], [], [], [], [], options);
        b_opti_mat(i, j) = x_opti(1);
    end
end
% Plot Results
legendCell = cellstr(num2str(beta_vec', '\beta=%3.2f'));
figure()
% Individual Demands at different Interest Rate Points
plot(r_vec, b_opti_mat)
ylim([-1.1 1]);
xlim([min(r_vec) max(r_vec)]);
hold on
plot(r_vec,ones(size(r_vec)) * 0, 'k-.');
grid on;
title('Individual Borrowing/Savings')
ylabel('Optimal Savings Choice')
xlabel('interest rate')
legend(legendCell, 'Location','northwest');
%% Aggregate Household Excess Supply along Interest Rate
% When we solved for the <https://fanwangecon.github.io/Math4Econ/matrix_application/demand_supply_taylor_approximate_capital.html 
% equilibrium interest rate before>, we had a firm that demanded credit and a 
% household that supplied credit. Now we are more flexible, as shown in the chart 
% above, households could be supplying or demanding credit. The equilibrium now 
% is about clearing the aggregate demand and supply for the credit market considering 
% both firms and households where households now could either be on demand or 
% supply side. At a particular $r$, if households all want to borrow, there will 
% be no lending, so that particular interest rate will not clear market. We will 
% increase interest rate until some households are willing to save. Eventually, 
% we find the market clearning interest rate.
% 
% If the economy has on the household side exactly these three households, we 
% can sum the aggregate demand and supply for credit at each $r$ from the households 
% by summing across the $b^*( r, \beta_i)$. If households with these three different 
% discount factors are in different proportions in the data for a particular country, 
% we can sum up the weighted average. 
%% 
% * *Aggregate Household Excess Supply*: $B^*_{hh} = \sum_{i=1}^3 b^*(r, \beta_i)$

figure()
hold on;
% Aggregate demand (borrow meaning negative) and supply (saving positive) for households,
% just add sum (the 2 means sum over columns), it will sum across columns, each column is a different individual
plot(r_vec, sum(b_opti_mat, 2))
plot(r_vec,ones(size(r_vec)) * 0, 'k-.');
ylim([-5 5]);
xlim([min(r_vec) max(r_vec)]);
grid on;
title('Aggregates Households')
ylabel({['Aggregate Household Net Saving Borrowing'], ['(over 3 household \beta types)']})
xlabel('interest rate')
legend({'HH Net B Supply'}, 'Location','northwest');
%% Firm Demand for Credit
% We also have the aggregate Demand for the firm side based on the <https://fanwangecon.github.io/Math4Econ/matrix_application/KL_borrowhire_firm.html 
% firm's capital only problem>, with $\alpha_l$ for elasticity of labor, and $\alpha_k$ 
% for elasticity of capital, and $L$ is fixed at 1:
%% 
% * *Firm Demand For Capital*: $K^*_{firm} = \left( \frac{r}{p\cdot A \cdot 
% \alpha \cdot L^{\alpha_l}}\right)^{\frac{1}{\alpha_k -1}}$

figure()
% Aggregate demand from firms (borrowing from firms)
p = 1;
A = 2.5;
alpha_K = 0.36;
alpha_L = 0.5;
L = 1;
FIRM_K = (r_vec./(p*A*alpha_K*(L^alpha_L))).^(1/(alpha_K-1));
% Individual Demands at different Interest Rate Points
plot(r_vec, (-1)*FIRM_K)
ylim([-5 5]);
xlim([min(r_vec) max(r_vec)]);
hold on
plot(r_vec,ones(size(r_vec)) * 0, 'k-.');
grid on;
title('Firm demand for Capital (borrowing)')
ylabel('Optimal Firm Choice')
xlabel('interest rate')
%% Economy Wide Excess Supply for Credit (Firm + Households)
% The firm is demanding credit (it is borrowing), so we put a negative sign 
% in front of $K$ demanded:
%% 
% * *Economy-wide excess supply of Credit*: $\text{ExcesCreditSupply}(r) = B^*_{hh}(r)  
% - K^*_{firm}(r)$
% * If the term above is positive that means total saving from households is 
% greater than total borrowing from households and firms. 
%% 
% Equilibrium interest rate is the interest rate where excess credit supply 
% is equal to zero:
%% 
% * to find *equilibrium interest rate*: find $r^{equi}$, where at this interest 
% rate: $B^*_{hh}(r^{equi})  - K^*_{firm}(r^{equi}) = 0$

% Summing up to get excess credit supply
excess_credit_supply = (sum(b_opti_mat, 2) + (-1)*FIRM_K');
% find at which interest rate we are closest to zero
[excess_credit_supply_at_equi, equi_idx_in_rvec] = min(abs(excess_credit_supply));
equilibrium_r = r_vec(equi_idx_in_rvec);
lowbeta_hh_b_equi = b_opti_mat(equi_idx_in_rvec, 1);
midbeta_hh_b_equi = b_opti_mat(equi_idx_in_rvec, 2);
highbeta_hh_b_equi = b_opti_mat(equi_idx_in_rvec, 3);
FIRM_K_equi = -FIRM_K(equi_idx_in_rvec);
results_withno_tax = table(equilibrium_r, excess_credit_supply_at_equi, equi_idx_in_rvec, FIRM_K_equi, lowbeta_hh_b_equi, midbeta_hh_b_equi, highbeta_hh_b_equi);
disp(results_withno_tax)
%% 
% Note that our equilibrium is an approximation, because we only had a grid 
% of $r$, the excess total supply is $0.023$, which is close to $0$, but not actualy 
% $0$. The numbers above show that our equilibrium interest rate is approximately 
% $1.036$, and at this equilibrium out of our three households and firm:
%% 
% # the firm borrows: $0.80217$ (this is $K$, based on which we can find total 
% output $Y$)
% # household 1 borrow: $0.37$
% # household 2 saves: $0.30$
% # household 3 saves: $0.89$
%% 
% These sum up to approximately $0$. Note that none of our three households 
% is borrowing constrained at the equilibrium. We can redraw the chart earlier 
% and show the aggregate demand and supply for credit $B_{hh}$ from the household 
% side:

figure()
subplot(1,2,1)
hold on;
% Aggregate demand (borrow meaning negative) and supply (saving positive) for households, 
% just add sum (the 2 means sum over columns), it will sum across columns, each column is a different individual
plot(r_vec, sum(b_opti_mat, 2))
plot(r_vec, (-1)*FIRM_K)
plot(r_vec,ones(size(r_vec)) * 0, 'k-.');
ylim([-5 5]);
xlim([min(r_vec) max(r_vec)]);
grid on;
title('Aggregates HH and Firms')
ylabel({['Aggregate Household Net Saving Borrowing (blue)'], ['Firm demand for Capital (borrowing, orange)']})
xlabel('interest rate')
legend({'HH Net B Supply', 'Firm Demand for K'}, 'Location','northwest');
subplot(1,2,2)
hold on;
% Total Aggregate Net Demand for Credit at each R
plot(r_vec, excess_credit_supply, 'k', 'LineWidth', 2);
% Plot equilibrium interest rate line
plot(equilibrium_r*ones(1,10), linspace(min(excess_credit_supply), max(excess_credit_supply),10), '--', 'LineWidth', 2);
% Zero line
plot(r_vec,ones(size(r_vec)) * 0, 'k-.');
ylim([-5 5]);
xlim([min(r_vec) max(r_vec)]);
grid on;
title('Net Excess Supply of Credit')
ylabel({['Economy Wide Excess Supply for Credit'], ['(over 3 household \beta types + Firm)']})
xlabel('interest rate')
legend({'Excess Supply', ['Equilibrium r=' num2str(equilibrium_r)]}, 'Location','northwest');
%% Demand and Supply for Credit with a Tax on Interest Rate
% Suppose some government officials thinks we need to subsidize borrowing. They 
% want to make it easier for households to borrow and also for firms to borrow. 
% This sounds fantastic. Because if borrowing rate is lower for firm, the firm 
% can borrow more in physical capital and increase output. How to pay for it? 
% The officials decide to pay for it by taxing savings. Perhaps people with too 
% much savings can take a cut on their interest rate earnings. 
% 
% For the firm, this is just a discount on the borrowing rate. For households, 
% this means if you save, you get your principle back next period, but you only 
% get $1-\tau$ fraction of the interest rate earning. But if you borrow, you only 
% pay $1-\tau$ fraction of your interest, rather than the full amount. Our problem 
% remains the same as before, except that we need to resolve given the tax rate 
% now. Let's apply the discount in borrowing to both households that borrow and 
% firms that borrow:
% 
% *Household problem* with borrowing discount and saving tax:
%% 
% * $\max_b \log(Z_1-b) + \beta_i \cdot \log(Z_2 + b + b\cdot (r) (1-\tau))$
%% 
% *Firm* optimal Policy function with borrowing discount:
%% 
% * $K^*_{firm} = \left( \frac{r\cdot(1-\tau)}{p\cdot A \cdot \alpha \cdot L^{\alpha_l}}\right)^{\frac{1}{\alpha_k 
% -1}}$
%% 
% _Note:_ this policy *pays for itself* because the credit market clears, so 
% government income from the savings tax will pay for exactly its subsidy on borrowing.
% 
% Suppose $\tau = 0.10$, let's solve for the new optimal choices and equilibrium 
% given this tax policy. We re-use our previous codes but include now the tax:

tau = 0.10;
% Households' problem with Interest Tax and Subsidy
% A vector to store optimal choices
b_opti_mat = zeros(rows, cols);
% Solving for optimal choices as we change Z2
for j=1:1:length(beta_vec)
    for i=1:1:length(r_vec)
        U_neg = @(x) -1*(log(z1 - x(1)) + beta_vec(j)*log(z2 + x(1)*r_vec(i)*(1-tau)));
        options = optimoptions('FMINCON','Display','off');
        [x_opti,U_at_x_opti] = fmincon(U_neg, b0, A, q, [], [], [], [], [], options);
        b_opti_mat(i, j) = x_opti(1);
    end
end

% Firm's problem with interest tax and subsidy
FIRM_K = ((r_vec*(1-tau))./(p*A*alpha_K*(L^alpha_L))).^(1/(alpha_K-1));

% Approximate equilibrium
excess_credit_supply = (sum(b_opti_mat, 2) + (-1)*FIRM_K');
[excess_credit_supply_at_equi, equi_idx_in_rvec] = min(abs(excess_credit_supply));
equilibrium_r = r_vec(equi_idx_in_rvec);

% Grab Results
lowbeta_hh_b_equi = b_opti_mat(equi_idx_in_rvec, 1);
midbeta_hh_b_equi = b_opti_mat(equi_idx_in_rvec, 2);
highbeta_hh_b_equi = b_opti_mat(equi_idx_in_rvec, 3);
FIRM_K_equi = -FIRM_K(equi_idx_in_rvec);
results_with_tax = table(equilibrium_r, excess_credit_supply_at_equi, equi_idx_in_rvec, FIRM_K_equi, lowbeta_hh_b_equi, midbeta_hh_b_equi, highbeta_hh_b_equi);
results_table = [results_withno_tax;results_with_tax];
results_table.Properties.RowNames = {'no r tax/subsidy', ['r tax/subsidy tau=' num2str(tau)]};
disp(results_table)
%% 
% The table above compares the results with the tax and without. With the tax, 
% the approximate equilibrium interest rate has to be higher because with the 
% tax rate at the previous interest rate more people want to borrow (given the 
% borrowing discount) and less people want to save (given the tax). To find the 
% point where demand equals supply, interest rate increases to incentivize households 
% to save despite the tax. In equilibrium, we now have a much higher interest 
% rate that clears that market. Note that compared to before, firms are borrowing 
% less, so output is now lower due to the higher equilibrium interest rate. The 
% policy is potentially having the opposite of its intended effect. We solve for 
% the general equilibrium effects of policies to help us think about these unintended 
% consequences of policies.

figure()
subplot(1,2,1)
hold on;
% Aggregate demand (borrow meaning negative) and supply (saving positive) for households, 
% % just add sum (the 2 means sum over columns), it will sum across columns, each column is a different individual
plot(r_vec, sum(b_opti_mat, 2))
plot(r_vec, (-1)*FIRM_K)
plot(r_vec,ones(size(r_vec)) * 0, 'k-.');
ylim([-7 2]);
xlim([min(r_vec) max(r_vec)]);
grid on;
title(['Aggregates HH and Firms, tau=' num2str(tau)])
ylabel({['Aggregate Household Net Saving Borrowing (blue)'], ['Firm demand for Capital (borrowing, orange)']})
xlabel('interest rate')
legend({'HH Net B Supply', 'Firm Demand for K'}, 'Location','northwest');
subplot(1,2,2)
hold on;
% Total Aggregate Net Demand for Credit at each R
plot(r_vec, excess_credit_supply, 'k', 'LineWidth', 2);
% Plot equilibrium interest rate line
plot(equilibrium_r*ones(1,10), linspace(min(excess_credit_supply), max(excess_credit_supply),10), '--', 'LineWidth', 2);
% Zero line
plot(r_vec,ones(size(r_vec)) * 0, 'k-.');
ylim([-7 2]);
xlim([min(r_vec) max(r_vec)]);
grid on;
title(['Net Excess Supply of Credit, tau=' num2str(tau)])
ylabel({['Economy Wide Excess Supply for Credit'], ['(over 3 household \beta types + Firm)']})
xlabel('interest rate')
legend({'Excess Supply', ['Equilibrium r=' num2str(equilibrium_r)]}, 'Location','northwest');