Equilibrium Interest Rate and Tax

How do households with different β respond to changes in r given Borrowing Constraint?

• specifically:
• with:

I introduce now heterogeneity in β. There are households, each with a different . Note that lower β 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 . 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

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 . 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:

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 firm's capital only problem, with for elasticity of labor, and for elasticity of capital, and L is fixed at 1:

• Firm Demand For Capital:

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)

• Economy-wide excess supply of Credit:
• If the term above is positive that means total saving from households is greater than total borrowing from households and firms.

• to find equilibrium interest rate: find , where at this interest rate:

Note that our equilibrium is an approximation, because we only had a grid of r, the excess total supply is , which is close to 0, but not actualy 0. The numbers above show that our equilibrium interest rate is approximately , and at this equilibrium out of our three households and firm:

1. the firm borrows: (this is K, based on which we can find total output Y)
2. household 1 borrow:
3. household 2 saves:
4. household 3 saves:

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 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

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 fraction of the interest rate earning. But if you borrow, you only pay 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:

Suppose , 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:

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');