Formal Informal Cost Minimizer
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Contents
- FFS_FIBS_MIN_C_COST Minimizes to consumption given aprime.
- Default and Parse Parameters
- Initialize Output Arrays
- Find if ar_aprime contains Saving
- Get consumption and savings choices if savings
- Proceed to Process Borrowing if aprime Array has borrowing
- Compute Consumption Informal Borrowing only
- Compute Formal Block Sizes
- Compute Consumption Values Formal + Informal Borrowing Jointly
- Compute Consumption Values Formal Borrowing + Formal Savings
- Maximize Consumption For non-Bridge Loan Component
- Borrowing count up borrowing from different sources
- Display Final
function [mt_max_c_nobridge, mt_inf_borr_nobridge, mt_for_borr, mt_for_save] = ffs_fibs_min_c_cost(varargin)
FFS_FIBS_MIN_C_COST Minimizes to consumption given aprime.
Suppose I want to borrow 5 dollar today. That will increase current consumption by 5 dollars, but its cost to next period cash-on-hand is a function of how I get the 5 dollar of borrowing, whether I borrow from formal sources, informal or joint sources. Suppose the formal borrowing grid has three points, you can borrow 0, 3 or 6, these are the options:
- borrow all 5 informally
- borrow 3 formally and 2 informally
- borrow 6 formally and save 1
- additionally, if 5 is actually on the grid, you would borrow 5 only formally if that offers the lower interest rate.
This function here finds which one of of these options is optimal conditional on aprime choice. Households choose among the different aprime choices given these within aprime choice optimal formal/informal allocations.
ar_aprime is the borrowing choices that do not include bridge loan. The additional borrowing on top of bridge loans.
The function outputs matrixes in N by M dimensions. N is the dimension of the ar_aprime array, and M is the dimension fo the ar_r_inf interest rate array. Solves the optimal formal and informal joint optimization prblem for N by M combinations of aggregate borrowing levels and informal borrowing interest rate. The idea is that there are different aggregate borrowing levels households are choosing from, and they face different informal borrowing interest rate shocks.
@param bl_b_is_principle boolean solving with aggregate savings as savings + debt principles + interests, or just principles no interests. if true, principels only, no interests.
@param ar_r_inf array 1 by M array of informal interest rate
@param fl_r_fsv float (formal) savings interest rate
@param fl_r_fbr float borrowing interest rate
@param ar_aprime array N by 1 level of aggregate borrowing excluding bridge loan. Note that bridge loan is needed if coh is negative and households can not pay back principle and interests. This must be negative.
@return mt_max_c_nobridge array N by M next period consumption cost (bl_b_is_principle == true), or this period consumption gain (bl_b_is_principle == false) based on choosing optimally between formal and informal, borrowing and savings joint categories. This considers both interests as well as principles.
@return mt_inf_borr_nobridge array N by M informal borrowing choices (Excluding Informal Bridge loans, calculated elsewhere) which could come from informal borrowing only if that minimizes consumption cost, or joint formal and informal borrowing if that is the cost minimizing choice. ZIf bl_b_is_principle == true, then this includes just the principles, no intrest rates. if bl_b_is_principle == false, that means this includes interest rates costs as well as principles costs.
@return mt_for_borr array N by M formal borrowing choice that minimizes consumption costs given fixed ar_aprime. Could come from formal borrowing alone (which shows up as joint formal and something else where the other choice is 0), or formal + informal joint borrow, or formal borrowing and formal savings. If bl_b_is_principle == true, then this includes just the principles, no intrest rates. if bl_b_is_principle == false, that means this includes interest rates costs as well as principles costs.
@return mt_for_save array N by M this is the formal savings choice when households are borrowing. Households coulds save just for savings, no borrowing as well, that is not captured here. If bl_b_is_principle == true, then this includes just the principles, no intrest rates. if bl_b_is_principle == false, that means this includes interest rates costs as well as principles costs.
@example
bl_input_override = true; [ar_max_c_nobridge, ar_inf_borr_nobridge, ar_for_borr, ar_for_save] = ... ffs_fibs_min_c_cost(bl_b_is_principle, fl_r_inf, fl_r_fsv, ... ar_forbrblk_r, ar_forbrblk, ... fl_ap, bl_display_minccost, bl_input_override);
@include
@seealso
- Formal Borrowing Grid: ffs_for_br_block_gen
- Informal Interest Rates: ffs_r_inf
- Match Borrowing to Formal Grid: ffs_for_br_block_match
- Optimize Formal and Informal, Borrowing and Savings Joint Choices: ffs_fibs_min_c_cost
- Bridge Loan: ffs_fibs_inf_bridge
- Overall Optimization: ffs_fibs_min_c_cost_bridge
- Discrete Choices: ffs_fibs_identify_discrete
Default and Parse Parameters
if (~isempty(varargin)) % override when called from outside [bl_b_is_principle, ar_r_inf, fl_r_fsv, ... ar_forbrblk_r, ar_forbrblk, ar_aprime_nobridge, bl_display_minccost] = varargin{:}; else close all % Default it_param_set = 4; bl_input_override = true; [param_map, support_map] = ffs_ipwkbz_fibs_set_default_param(it_param_set); % Gather Inputs from armt_map params_group = values(param_map, ... {'fl_r_fbr', 'st_forbrblk_type', 'fl_forbrblk_brmost', 'fl_forbrblk_brleast', 'fl_forbrblk_gap'}); [fl_r_fbr, st_forbrblk_type, fl_forbrblk_brmost, fl_forbrblk_brleast, fl_forbrblk_gap] = params_group{:}; % st_forbrblk_type = 'unif'; % fl_forbrblk_gap = -1.5; [ar_forbrblk, ar_forbrblk_r] = ... ffs_for_br_block_gen(fl_r_fbr, st_forbrblk_type, fl_forbrblk_brmost, fl_forbrblk_brleast, fl_forbrblk_gap); % Gather Inputs from param_map params_group = values(param_map, {'bl_b_is_principle', 'fl_r_fsv'}); [bl_b_is_principle, fl_r_fsv] = params_group{:}; % Informal Interest Rate Array, works with 1 by 1 or 1 by N ar_r_inf = [0.00, 0.085, 0.90]; % ar_r_inf = [0.085]; % Setting interest rate, if r_inf is very high, that means informal % option would generally never be chosen, or options that involve % informal options never choice, either here, or outside. % fl_r_inf = 10000; % Testing COH and Aprime Vectors ar_aprime_nobridge = [-20, -14, -11, -6.8, ... -5.5, -4.5, -4.1, -1.1, ... -0.1, ... 0.1, 1, 2, 10]'; % Set Display Control bl_display_minccost = true; end
Initialize Output Arrays
it_N = length(ar_aprime_nobridge); it_M = length(ar_r_inf); mt_max_c_nobridge = zeros(it_N, it_M); mt_inf_borr_nobridge = zeros(it_N, it_M); mt_for_borr = zeros(it_N, it_M); mt_for_save = zeros(it_N, it_M);
Find if ar_aprime contains Saving
only one savings option, function here meant for borrowing, but can deal with savings as well if they appear.
ar_aprime_nobridge_pos_idx = (ar_aprime_nobridge >= 0); ar_aprime_nobridge_br = ar_aprime_nobridge(~ar_aprime_nobridge_pos_idx);
Get consumption and savings choices if savings
When households overall save, they could still have had to first pay informal lender for bridge loan. For households with positive cash-on-hand, if they save, they are only savings, and not borrowing at the same time. In practice, some households coule borrow from one sector and lend/save in another. This is not a mechanism in this model, but is discussed in Wang (2019)
if (sum(ar_aprime_nobridge_pos_idx)) % savings = savings (including or not interest rates) mt_for_save(ar_aprime_nobridge_pos_idx) = ar_aprime_nobridge(ar_aprime_nobridge_pos_idx); if (bl_b_is_principle) % consumption gain next period due to savings mt_max_c_nobridge(ar_aprime_nobridge_pos_idx, :) = ... zeros(1,it_M) + ar_aprime_nobridge(ar_aprime_nobridge_pos_idx)*(1+fl_r_fsv); else % consumption loss today due to savings mt_max_c_nobridge(ar_aprime_nobridge_pos_idx, :) = ... zeros(1,it_M) + (-1)*ar_aprime_nobridge(ar_aprime_nobridge_pos_idx)./(1+fl_r_fsv); end end
Proceed to Process Borrowing if aprime Array has borrowing
if (sum(~ar_aprime_nobridge_pos_idx))
Compute Consumption Informal Borrowing only
% Generate c Vectors if (bl_b_is_principle) % c_infonly is cost of borrowing in next period consumption mt_b_infonly_inf = (ar_aprime_nobridge_br); mt_c_infonly = mt_b_infonly_inf.*(1+ar_r_inf); else % c_infonly is the gain from borrowing in current period consumption mt_b_infonly_inf = (ar_aprime_nobridge_br); mt_c_infonly = (-1).*mt_b_infonly_inf./(1+ar_r_inf); end % Display if (bl_display_minccost) tab_c_infonly = table(ar_aprime_nobridge_br, mt_b_infonly_inf, mt_c_infonly); disp(['informal borrow interest (fl_r_inf):', num2str(ar_r_inf)]); disp(tab_c_infonly); end
informal borrow interest (fl_r_inf):0 0.085 0.9
ar_aprime_nobridge_br mt_b_infonly_inf mt_c_infonly
_____________________ ________________ __________________________
-20 -20 -20 -21.7 -38
-14 -14 -14 -15.19 -26.6
-11 -11 -11 -11.935 -20.9
-6.8 -6.8 -6.8 -7.378 -12.92
-5.5 -5.5 -5.5 -5.9675 -10.45
-4.5 -4.5 -4.5 -4.8825 -8.55
-4.1 -4.1 -4.1 -4.4485 -7.79
-1.1 -1.1 -1.1 -1.1935 -2.09
-0.1 -0.1 -0.1 -0.1085 -0.19
Compute Formal Block Sizes
Divide each asset choice by each element of the formal choice grid. The numbers closest to 1 above and below indicates the floor and ceil formal borrowing quantity for joint credit market participation choices.
[ar_a_grid_ceil_principle, ar_a_grid_ceil_wthr, ar_a_grid_floor_principle, ar_a_grid_floor_wthr] = ...
ffs_for_br_block_match(ar_aprime_nobridge_br, ar_forbrblk, ar_forbrblk_r, bl_b_is_principle);
Compute Consumption Values Formal + Informal Borrowing Jointly
if right on grid, informal could be zero.
% Generate c Vectors if (bl_b_is_principle) % c_infforb is cost of borrowing in next period consumption ar_b_infforb_inf = (ar_aprime_nobridge_br - ar_a_grid_ceil_principle); ar_b_infforb_for = (ar_a_grid_ceil_principle); mt_c_infforb = (ar_b_infforb_inf.*(1+ar_r_inf) + ar_a_grid_ceil_wthr); else % c_infforb is the gain from borrowing in current period consumption ar_b_infforb_inf = ar_aprime_nobridge_br - ar_a_grid_ceil_wthr; ar_b_infforb_for = ar_a_grid_ceil_wthr; mt_c_infforb = -1*((ar_b_infforb_inf./(1+ar_r_inf) + ar_a_grid_ceil_principle)); end % Display if (bl_display_minccost) tab_c_infforb = table(ar_aprime_nobridge_br, ar_b_infforb_for, ar_b_infforb_inf, mt_c_infforb); disp(['formal block size (ar_forbrblk):', num2str(ar_forbrblk)]); disp(['formal borrow interest (ar_forbrblk_r):', num2str(ar_forbrblk_r)]); disp(['informal borrow interest (fl_r_inf):', num2str(ar_r_inf)]); disp(tab_c_infforb); end
formal block size (ar_forbrblk):-19 -14.5 -10 -7 -5.5 -4 -2.5 -1 0
formal borrow interest (ar_forbrblk_r):0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.065
informal borrow interest (fl_r_inf):0 0.085 0.9
ar_aprime_nobridge_br ar_b_infforb_for ar_b_infforb_inf mt_c_infforb
_____________________ ________________ ________________ _____________________________
-20 -19 -1 -21.235 -21.32 -22.135
-14 -10 -4 -14.65 -14.99 -18.25
-11 -10 -1 -11.65 -11.735 -12.55
-6.8 -5.5 -1.3 -7.1575 -7.268 -8.3275
-5.5 -5.5 0 -5.8575 -5.8575 -5.8575
-4.5 -4 -0.5 -4.76 -4.8025 -5.21
-4.1 -4 -0.1 -4.36 -4.3685 -4.45
-1.1 -1 -0.1 -1.165 -1.1735 -1.255
-0.1 0 -0.1 -0.1 -0.1085 -0.19
Compute Consumption Values Formal Borrowing + Formal Savings
% Generate c Vectors if (bl_b_is_principle) % c_forbrsv is cost of borrowing in next period consumption ar_b_forbrsv_sav = (ar_aprime_nobridge_br - ar_a_grid_floor_principle); ar_b_forbrsv_brr = (ar_a_grid_floor_principle); ar_c_forbrsv = (ar_b_forbrsv_sav*(1+fl_r_fsv) + ar_a_grid_floor_wthr); else % c_forbrsv is the gain from borrowing in current period consumption ar_b_forbrsv_sav = ar_aprime_nobridge_br - ar_a_grid_floor_wthr; ar_b_forbrsv_brr = ar_a_grid_floor_wthr; ar_c_forbrsv = -1*((ar_b_forbrsv_sav./(1+fl_r_fsv) + ar_a_grid_floor_principle)); end % if b_forbrsv_sav < 0, largest formal borrow grid not large enough. set to % 0 so addition of different formal and informal choice categories work. ar_b_forbrsv_brr(ar_b_forbrsv_sav < 0) = 0; ar_c_forbrsv(ar_b_forbrsv_sav < 0) = nan; ar_b_forbrsv_sav(ar_b_forbrsv_sav < 0) = 0; % Display if (bl_display_minccost) tab_c_forbrsv = table(ar_aprime_nobridge_br, ar_b_forbrsv_brr, ar_b_forbrsv_sav, ar_c_forbrsv); disp(['formal block size (ar_forbrblk):', num2str(ar_forbrblk)]); disp(['formal borrow interest (ar_forbrblk_r):', num2str(ar_forbrblk_r)]); disp(['savings interest (fl_r_fsv):', num2str(fl_r_fsv)]); disp(tab_c_forbrsv); end
formal block size (ar_forbrblk):-19 -14.5 -10 -7 -5.5 -4 -2.5 -1 0
formal borrow interest (ar_forbrblk_r):0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.065
savings interest (fl_r_fsv):0.025
ar_aprime_nobridge_br ar_b_forbrsv_brr ar_b_forbrsv_sav ar_c_forbrsv
_____________________ ________________ ________________ ____________
-20 0 0 NaN
-14 -14.5 0.5 -14.93
-11 -14.5 3.5 -11.855
-6.8 -7 0.2 -7.25
-5.5 -7 1.5 -5.9175
-4.5 -5.5 1 -4.8325
-4.1 -5.5 1.4 -4.4225
-1.1 -2.5 1.4 -1.2275
-0.1 -1 0.9 -0.1425
Maximize Consumption For non-Bridge Loan Component
mt_c_forbrsv = repmat(ar_c_forbrsv, [1, length(ar_r_inf)]);
[ar_max_c_nobridge_br, ar_max_idx] = max([mt_c_infonly(:) mt_c_infforb(:) mt_c_forbrsv(:)],[], 2);
mt_max_c_nobridge_br = reshape(ar_max_c_nobridge_br, size(mt_c_infonly));
mt_max_idx = reshape(ar_max_idx, size(mt_c_infonly));
if (bl_display_minccost)
disp('ar_r_inf')
disp(ar_r_inf)
disp('ar_aprime_nobridge''')
disp(ar_aprime_nobridge')
tab_c_max = table(ar_aprime_nobridge_br, mt_max_c_nobridge_br, mt_max_idx, mt_c_infonly, mt_c_infforb, ar_c_forbrsv);
disp(tab_c_max);
end
ar_r_inf
0 0.0850 0.9000
ar_aprime_nobridge'
Columns 1 through 7
-20.0000 -14.0000 -11.0000 -6.8000 -5.5000 -4.5000 -4.1000
Columns 8 through 13
-1.1000 -0.1000 0.1000 1.0000 2.0000 10.0000
ar_aprime_nobridge_br mt_max_c_nobridge_br mt_max_idx mt_c_infonly mt_c_infforb ar_c_forbrsv
_____________________ __________________________ ___________ __________________________ _____________________________ ____________
-20 -20 -21.32 -22.135 1 2 2 -20 -21.7 -38 -21.235 -21.32 -22.135 NaN
-14 -14 -14.93 -14.93 1 3 3 -14 -15.19 -26.6 -14.65 -14.99 -18.25 -14.93
-11 -11 -11.735 -11.855 1 2 3 -11 -11.935 -20.9 -11.65 -11.735 -12.55 -11.855
-6.8 -6.8 -7.25 -7.25 1 3 3 -6.8 -7.378 -12.92 -7.1575 -7.268 -8.3275 -7.25
-5.5 -5.5 -5.8575 -5.8575 1 2 2 -5.5 -5.9675 -10.45 -5.8575 -5.8575 -5.8575 -5.9175
-4.5 -4.5 -4.8025 -4.8325 1 2 3 -4.5 -4.8825 -8.55 -4.76 -4.8025 -5.21 -4.8325
-4.1 -4.1 -4.3685 -4.4225 1 2 3 -4.1 -4.4485 -7.79 -4.36 -4.3685 -4.45 -4.4225
-1.1 -1.1 -1.1735 -1.2275 1 2 3 -1.1 -1.1935 -2.09 -1.165 -1.1735 -1.255 -1.2275
-0.1 -0.1 -0.1085 -0.1425 1 1 3 -0.1 -0.1085 -0.19 -0.1 -0.1085 -0.19 -0.1425
Borrowing count up borrowing from different sources
Informal borrowing comes from informal only or inffor both
% Consumption when borrowing mt_max_c_nobridge(~ar_aprime_nobridge_pos_idx, :) = mt_max_c_nobridge_br; % Informal Borrowing mt_inf_borr_nobridge(~ar_aprime_nobridge_pos_idx, :) = ... mt_b_infonly_inf.*(mt_max_idx == 1) + ar_b_infforb_inf.*(mt_max_idx == 2); % Formal Borrowing mt_for_borr(~ar_aprime_nobridge_pos_idx, :) = ... ar_b_infforb_for.*(mt_max_idx == 2) + ar_b_forbrsv_brr.*(mt_max_idx == 3); % Formal Savings mt_for_save(~ar_aprime_nobridge_pos_idx, :) = ... ar_b_forbrsv_sav.*(mt_max_idx == 3);
end
Display Final
if (bl_display_minccost) if (bl_b_is_principle) ar_average_r = (-1)*(mt_max_c_nobridge./ar_aprime_nobridge); else ar_average_r = (-1)*(ar_aprime_nobridge./mt_max_c_nobridge); end tab_opti_borrow = table(ar_aprime_nobridge, mt_max_c_nobridge, ... ar_average_r, mt_inf_borr_nobridge, mt_for_borr, mt_for_save); disp(tab_opti_borrow); end
ar_aprime_nobridge mt_max_c_nobridge ar_average_r mt_inf_borr_nobridge mt_for_borr mt_for_save
__________________ ____________________________ ____________________________ ____________________ ___________________ _________________
-20 -20 -21.32 -22.135 -1 -1.066 -1.1067 -20 -1 -1 0 -19 -19 0 0 0
-14 -14 -14.93 -14.93 -1 -1.0664 -1.0664 -14 0 0 0 -14.5 -14.5 0 0.5 0.5
-11 -11 -11.735 -11.855 -1 -1.0668 -1.0777 -11 -1 0 0 -10 -14.5 0 0 3.5
-6.8 -6.8 -7.25 -7.25 -1 -1.0662 -1.0662 -6.8 0 0 0 -7 -7 0 0.2 0.2
-5.5 -5.5 -5.8575 -5.8575 -1 -1.065 -1.065 -5.5 0 0 0 -5.5 -5.5 0 0 0
-4.5 -4.5 -4.8025 -4.8325 -1 -1.0672 -1.0739 -4.5 -0.5 0 0 -4 -5.5 0 0 1
-4.1 -4.1 -4.3685 -4.4225 -1 -1.0655 -1.0787 -4.1 -0.1 0 0 -4 -5.5 0 0 1.4
-1.1 -1.1 -1.1735 -1.2275 -1 -1.0668 -1.1159 -1.1 -0.1 0 0 -1 -2.5 0 0 1.4
-0.1 -0.1 -0.1085 -0.1425 -1 -1.085 -1.425 -0.1 -0.1 0 0 0 -1 0 0 0.9
0.1 0.1025 0.1025 0.1025 -1.025 -1.025 -1.025 0 0 0 0 0 0 0.1 0 0
1 1.025 1.025 1.025 -1.025 -1.025 -1.025 0 0 0 0 0 0 1 0 0
2 2.05 2.05 2.05 -1.025 -1.025 -1.025 0 0 0 0 0 0 2 0 0
10 10.25 10.25 10.25 -1.025 -1.025 -1.025 0 0 0 0 0 0 10 0 0
end
ans =
-20.0000 -21.3200 -22.1350
-14.0000 -14.9300 -14.9300
-11.0000 -11.7350 -11.8550
-6.8000 -7.2500 -7.2500
-5.5000 -5.8575 -5.8575
-4.5000 -4.8025 -4.8325
-4.1000 -4.3685 -4.4225
-1.1000 -1.1735 -1.2275
-0.1000 -0.1085 -0.1425
0.1025 0.1025 0.1025
1.0250 1.0250 1.0250
2.0500 2.0500 2.0500
10.2500 10.2500 10.2500