Minimum Consumption and Borrowing

Natural Borrowing Constraint

In the standard borrowing problem--natural borrowing constraint--with uncertainty in the future. Due to negative infinite utility when consumption is zero, households' borrowing is constrained by endowment in the worst state of future shocks. This leads to the following result:

Given a two period economy where endowment are:

, and,

where

is the probability of the low state

The optimal choice is:

, households would borrow exactly

, but because

, as

converges to 0, the limit of the optimal borrowing choice not

, but

This results happens when households are required to always repay their debts. Non-full-repayment will be avoided even the probability of that occuring is very small.

Minimum Consumption

We can allow for higher borrowing choices by allowing for minimum consumption. Minimum consumption allows for default. We can write the problem as:

where

, and

and Minimum Consumption:

Without additional constraints, the introduction of minimum consumption gives households the possibility of increasing utility by borrowing ever more (increasing current utility), or saving towards infinity (increasing future utility).

Borrowing Constraint: To prevent households from borrowing infinitely, we impose a limit a borrowing that is non-binding:

(non-binding) borrowing constraint:

Debt Continuation Condition: To prevent households from saving infinitely, we modify the second period budget constraints as below:

updated budget constraints:

, and

where:

The Debt Continuation Condition is important also for delaying payment when households are unable to fully repay debts due for models that have no-roll-over conditions. In finite horizon model here, this condition will allow for possible delay in repayments until the final period.

Graphical Illustration of the Natural Borrowing Constraint as Probability of Bad State Diminishes

The natural borrowing constraint. The graph below demonstrates the limitation on borrowing when there is no minimum consumption.

% Define Some Parameters

e1 = 0;

e2h = 3;

e2l = 1;

beta = 1;

r = 0;

% Solve for Optimal Choices as the probability of the low state happening decreases

p_n = 1000;

p_vector = linspace(0, 1, p_n);

b_o_vector = zeros(size(p_vector));

% Define function

fu_bp = @(b, p, e2h) (log(e1 - b) + beta*(p*log(e2l + (1+r)*b) + (1-p)*log(e2h + (1+r)*b)));

for p_i=1:length(p_vector)

% Grab p

p = p_vector(p_i);

% Function of b only

fu_b = @(b) (-1)*fu_bp(b, p, e2h);

% Find Optimal Choice

options = optimset('Display', 'off');

b_o_vector(p_i) = fminsearch(fu_b, -0.1, options);

end

% Plot Optimal Choices

figure();

hold on;

plot(p_vector(2:p_n), b_o_vector(2:p_n));

scatter(p_vector(1), b_o_vector(1), 100, 'k', 'filled');

% Graph Bounds

b_o_max = max(b_o_vector);

b_o_min = min(b_o_vector);

b_o_margin = (b_o_max - b_o_min)/10;

b_o_graph_max = b_o_max + b_o_margin;

b_o_graph_min = b_o_min - b_o_margin;

p_vector_max = max(p_vector);

p_vector_min = min(p_vector);

p_vector_margin = (p_vector_max - p_vector_min)/10;

p_vector_graph_max = p_vector_max + p_vector_margin;

p_vector_graph_min = p_vector_min - p_vector_margin;

xlim([p_vector_graph_min p_vector_graph_max]);

ylim([b_o_graph_min b_o_graph_max]);

% Grids

grid on;

grid minor;

% Titling

title({['Natual Borrowing Constraint'], ['Optimal Borrowing Choice']});

xlabel('Probability of the Bad State');

ylabel('Optimal Borrowing Level');

legend({['Optimal Borrowing as P_{low} gets Small']...

,['Optimal Borrowing when P_{low} = 0']}...

,'Location', 'southeast');

Effects of Utility of a Minimum Consumption

What happens to the utility objective when we allow for minimum consumption? The example below illustrates this graphically. Minimum consumption opens up new set of b choices that were previously not available to households. But now there are two local maximums. Searching for optimal choices is trickier in this context.

Visually, we can also see if borrowing and savings bounds are not considered, then households would find it optimal to borrow or save infinitely.

% r and beta

beta = 1;

r= 0;

c_min_v = 0.09;

% Define the conditional consumption function

fc = @(c, cmin) (c).*(c >= cmin) + (cmin).*(c < cmin);

% Utility Function

fu_bp_cmin = @(b, p, e2h, cmin) log(fc(e1-b, cmin))...

+ beta*(p*log(fc(e2l+b*(1+r), cmin))...

+ (1-p)*log(fc(e2h+b*(1+r), cmin)));

% Bad State has 50 percent chance of happening

p = 0.50;

% A vector of b choice grid, include very large borrowing choices

b_min = -4;

b_max = 1;

b_margin = (b_max - b_min)/10;

b_n = 1000;

b_vector = linspace(b_min, b_max, b_n);

% Evaluate Utility

util_vector = zeros(size(b_vector));

% Evaluate Function

util_cmin = fu_bp_cmin(b_vector, p, e2h, c_min_v);

[max_val_cmin, max_idx_cmin] = max(util_cmin);

opti_b_cmin = b_vector(max_idx_cmin);

util_orig = fu_bp(b_vector, p, e2h);

util_orig_real_idx = arrayfun(@(x) ~any(imag(x)), util_orig);

% Graph Utility

figure();

hold on;

scatter(opti_b_cmin, max_val_cmin, 100, 'black', 'filled');

plot(b_vector, util_cmin, 'LineWidth', 2);

plot(b_vector(util_orig_real_idx), util_orig(util_orig_real_idx), 'b--', 'LineWidth', 2);

% Y and X limits

b_grh_max = b_max + b_margin;

b_grh_min = b_min - b_margin;

util_max = max(max([util_orig(util_orig_real_idx) util_cmin]));

util_min = min(min([util_orig(util_orig_real_idx) util_cmin]));

util_margin = (util_max - util_min)/5;

util_grh_max = util_max + util_margin;

util_grh_min = util_min - util_margin;

xlim([b_grh_min b_grh_max]);

ylim([util_grh_min util_grh_max]);

% Titling

title(['opti b with cmin, opti b=' num2str(opti_b_cmin)])

% Grids

grid on;

grid minor;

ylabel('Utility');

xlabel('B choices');

legend({['Utility with Cmin = 0']...

,['Utility with Cmin > 0']}...

,'Location', 'southwest');