With Interest Rate Shocks, What is the Distribution of the Borrowing Interest Rate

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@seealso

Contents

Solve the Model

Wide interest rate difference between min and max, uniform exogenous distribution.

clear all;
close all;

it_param_set = 9;
[param_map, support_map] = ffs_abz_set_default_param(it_param_set);
% param_map('it_a_n') = 50;
% param_map('it_z_wage_n') = 15;
param_map('fl_z_r_borr_max') = 0.50;
param_map('fl_z_r_borr_min') = 0.00;
% param_map('st_z_r_borr_drv_prb_type') = 'unif';
param_map('st_z_r_borr_drv_prb_type') = 'unif';
% param_map('fl_z_r_borr_poiss_mean') = 10;
param_map('fl_z_r_borr_n') = 15;
param_map('it_z_n') = param_map('it_z_wage_n') * param_map('fl_z_r_borr_n');

[armt_map, func_map] = ffs_abz_get_funcgrid(param_map, support_map);
result_map = ff_abz_vf_vecsv(param_map, support_map, armt_map, func_map);
result_map = ff_az_ds_vecsv(param_map, support_map, armt_map, func_map, result_map);

Elapsed time is 18.751154 seconds.
Elapsed time is 48.188193 seconds.
                    mean       sd       coefofvar      min       max       pYis0       pYls0       pYgr0      pYisMINY      pYisMAXY        p0_1         p1           p5         p10        p15        p20        p25         p35        p50        p65       p75       p80       p85       p90       p95       p99      p99_9     fl_cov_cl_mt_val    fl_cor_cl_mt_val    fl_cov_cl_mt_pol_a    fl_cor_cl_mt_pol_a    fl_cov_cl_mt_coh    fl_cor_cl_mt_coh    fl_cov_cl_mt_pol_c    fl_cor_cl_mt_pol_c    fracByP0_1      fracByP1       fracByP5     fracByP10     fracByP15     fracByP20     fracByP25     fracByP35    fracByP50    fracByP65    fracByP75    fracByP80    fracByP85    fracByP90    fracByP95    fracByP99    fracByP99_9
                   ______    _______    _________    _______    ______    _______    _________    _______    __________    ___________    ________    _________    ________    _______    _______    _______    ________    _______    _______    _______    ______    ______    ______    ______    ______    ______    ______    ________________    ________________    __________________    __________________    ________________    ________________    __________________    __________________    ___________    ___________    __________    __________    __________    __________    __________    _________    _________    _________    _________    _________    _________    _________    _________    _________    ___________

    cl_mt_val      3.1995      1.643     0.51351     -18.519    14.897          0     0.019248    0.98075    4.3953e-12    -5.2878e-35     -1.8002     -0.62557     0.28635     1.1389     1.4318     1.9591      1.9803     2.7385     3.1357     3.7797    4.3132    4.5532    4.9143    5.3215    5.9502    7.0885    8.1805         2.6993                   1               1.5788               0.78191               1.4798             0.70622              0.57217               0.99367          -0.00067087     -0.0040562    -0.0026623        0.0181      0.032661      0.079252      0.093494     0.18667      0.29779      0.46322      0.58939      0.65585      0.73385      0.80884      0.89763      0.97633       0.99734  
    cl_mt_pol_a    0.8659      1.229      1.4193      -6.557    48.412    0.16861      0.04284    0.78855    5.2025e-12    -5.2878e-35    -0.20568    -0.086712           0          0          0          0    0.020638    0.16817    0.38947    0.75831    1.2009     1.496    1.8648    2.4549    3.4139    5.6269    8.3563         1.5788             0.78191               1.5104                     1               1.5259             0.97352              0.34276               0.79576           -0.0003154     -0.0016358    -0.0034988    -0.0034988    -0.0034988    -0.0034988    -0.0014641    0.015835     0.054796      0.14903      0.26322      0.34955      0.43602      0.56896      0.73179      0.92345       0.98916  
    cl_mt_coh      1.2072     1.2754      1.0564      -6.557     50.64    0.01124    0.0069353    0.98182      4.42e-12     1.8295e-35    -0.13007            0    0.091429    0.18286     0.2638    0.32534     0.40269    0.52401    0.73677     1.1447    1.5808    1.8709    2.2771    2.8504    3.8368    6.0876    8.9292         1.4798             0.70622               1.5259               0.97352               1.6265                   1              0.32449               0.72598          -0.00015061    -0.00034518     0.0016025     0.0077435      0.016098      0.028278      0.043822    0.082239       0.1603      0.27453      0.38654      0.45755      0.54277      0.64849      0.78471       0.9394       0.99174  
    cl_mt_pol_c    1.3013    0.35047     0.26932        0.01    4.8668          0            0          1    7.0231e-11    -5.2878e-35     0.46518      0.54193     0.71274    0.80847    0.96021    0.98747       1.045     1.1855     1.2951     1.4128    1.5103    1.5951    1.6839    1.7777    1.8899     2.117    2.3938        0.57217             0.99367              0.34276               0.79576              0.32449             0.72598              0.12283                     1           0.00033915      0.0040513      0.023875      0.059899       0.08984       0.14074       0.16471     0.25987      0.39194       0.5529      0.66434      0.73114      0.78749      0.85261      0.92212      0.98282       0.99811  

Obtain The Joint Distribution of Shocks and Savings/Borrowing

Even though the interest rate is exogenous, borrowing and savings decisions are endogenous and are a function of the borrowing rate. The realized borrowing interest rate distribution is only for households that are borrowing.

ar_a = armt_map('ar_a');
ar_z_r_borr_mesh_wage = armt_map('ar_z_r_borr_mesh_wage');
ar_z_wage_mesh_r_borr = armt_map('ar_z_wage_mesh_r_borr');
ar_z_r_borr_prob = armt_map('ar_z_r_borr_prob');
ar_z_r_borr = armt_map('ar_z_r_borr');

tb_outcomes = result_map('tb_outcomes');
cl_mt_pol_a = result_map('cl_mt_pol_a');
ds_stats_pol_a_map = cl_mt_pol_a{2};
ds_stats_pol_a_map_keys = ds_stats_pol_a_map.keys;

mt_choice_prob_byYZ = ds_stats_pol_a_map('mt_choice_prob_byYZ');
ar_choice_unique_sorted_byY = ds_stats_pol_a_map('ar_choice_unique_sorted_byY');

% disp(mt_choice_prob_byYZ);
if (norm(sum(mt_choice_prob_byYZ,'all') - 1) <= -1e-12)
    error('error:sum(mt_choice_prob_byYZ,''all'') ~= 1');
end

Marginal Distributional of Realized Borrowing Interest Rate

Adding up across borrowing levels for each borrowing rate shock

it_z_wage_n = param_map('it_z_wage_n');
fl_z_r_borr_n = param_map('fl_z_r_borr_n');
[it_mt_row_n, it_mt_col_n] = size(mt_choice_prob_byYZ);

mt_choice_prob_byYrZ = zeros(it_mt_row_n, fl_z_r_borr_n);
ar_z_r_borr_dup = zeros(1, fl_z_r_borr_n);
for it_z_r_borr_ctr=1:1:fl_z_r_borr_n

    it_start_col = it_z_wage_n*(it_z_r_borr_ctr-1) + 1;
    it_end_col = it_start_col + it_z_wage_n - 1;

    ar_z_r_borr_dup(it_z_r_borr_ctr) = ar_z_r_borr_mesh_wage(it_start_col);
    mt_choice_prob_byYrZ(:, it_z_r_borr_ctr) = sum(mt_choice_prob_byYZ(:, it_start_col:it_end_col),2);
end


% R must equal to what was set
if (ar_z_r_borr_dup ~= ar_z_r_borr)
    error('error:ar_z_r_borr_dup ~= ar_z_r_borr');
end

% Sum Up Marginal Probability f(z), must equal to
if (norm(sum(mt_choice_prob_byYrZ,1)- ar_z_r_borr_prob) <= -1e-12)
    error('error:sum(mt_choice_prob_byYrZ,1) ~= ar_z_r_borr_prob');
end

if (norm(sum(mt_choice_prob_byYrZ,'all') - 1) <= -1e-12)
    error('error:sum(mt_choice_prob_byYrZ,''all'') ~= 1');
end

Frac of Borrower for each R: P(a <= 0 | z)

ar_it_choice_unique_sorted_byY_neg = (ar_choice_unique_sorted_byY < 0);
mt_choice_prob_byYrZ_borr = mt_choice_prob_byYrZ(ar_it_choice_unique_sorted_byY_neg,:);

% ar_p_a_le_zr_r_borr = sum(P(a, z) 1(a <=0))
ar_p_a_le_zr_r_borr = sum(mt_choice_prob_byYrZ_borr, 1);

% ar_p_a_le_zr_r_borr = sum(P( z | a <=0))
ar_p_r_borr = ar_p_a_le_zr_r_borr/sum(ar_p_a_le_zr_r_borr);

% ar_e_a_le_zr_r_borr = sum(a*P(a, z) 1(a <=0))
ar_Ebr_le_zr_r_borr = sum((ar_choice_unique_sorted_byY(ar_choice_unique_sorted_byY < 0) ...
                          + zeros(1,fl_z_r_borr_n)) .* mt_choice_prob_byYrZ_borr, 1);

% share of total borrowing by r
ar_Ebr_share_r_borr = ar_Ebr_le_zr_r_borr/sum(ar_Ebr_le_zr_r_borr);


disp(ar_z_r_borr_dup);
disp(ar_z_r_borr_prob);
disp(ar_p_r_borr);
disp(ar_Ebr_share_r_borr);

  Columns 1 through 7

         0    0.0357    0.0714    0.1071    0.1429    0.1786    0.2143

  Columns 8 through 14

    0.2500    0.2857    0.3214    0.3571    0.3929    0.4286    0.4643

  Column 15

    0.5000

  Columns 1 through 7

    0.0667    0.0667    0.0667    0.0667    0.0667    0.0667    0.0667

  Columns 8 through 14

    0.0667    0.0667    0.0667    0.0667    0.0667    0.0667    0.0667

  Column 15

    0.0667

  Columns 1 through 7

    0.2010    0.1838    0.1128    0.0966    0.0966    0.0635    0.0466

  Columns 8 through 14

    0.0453    0.0362    0.0233    0.0231    0.0203    0.0201    0.0185

  Column 15

    0.0126

  Columns 1 through 7

    0.2063    0.1849    0.1133    0.0984    0.0890    0.0593    0.0463

  Columns 8 through 14

    0.0442    0.0367    0.0253    0.0245    0.0205    0.0191    0.0178

  Column 15

    0.0144

Graph

figure();
hold on;
bar(ar_z_r_borr', [ar_z_r_borr_prob;ar_p_r_borr;ar_Ebr_share_r_borr]');
legend({'exogenous borrow rate distribution', 'borrow rate distribution conditional on borrowing', 'borrow volumn conditional on rate'}, ...
        'Location', 'northeast', 'color', 'none');
title({'Model With Exogenous Uniform Interest Rate Shocks', 'Borrowing is Endogenous, Realized R dist. differs from exo rate dist'})
ylabel('Fractions')
xlabel('Borrowing Interest Rates')
grid on;
grid minor;

Published with MATLAB® R2019a