1 FF_OPTIM_MLSEC_SAVEZRONE Derivative Multisection
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Examples](https://fanwangecon.github.io/M4Econ/), or** Dynamic Asset This is the example vignette for function: ff_optim_mlsec_savezrone from the MEconTools Package. This functions solves for optimal savings/borrowing level given an anonymous function that provides the derivative of a intertemporal savings problem. This is a vectorized function solved with multi-section (multiple points bisection concurrently).
The vectorized and looped bisection savings problem rely on this function to solve for optimal savings choices:
States Grid + Continuous Exact Savings as Share of Cash-on-Hand Loop:ff_vfi_az_bisec_loop, high precision even with small grid
States Grid + Continuous Exact Savings as Share of Cash-on-Hand Vectorized: ff_vfi_az_bisec_vec, precision and speed
1.1 Test FF_OPTIM_MLSEC_SAVEZRONE One Individual
Bisection for savings choice at one state:
% Generate the state-space and function
[fl_z1, fl_z2, fl_r, fl_beta] = deal(0.4730, 0.6252, 0.0839, 0.7365);
% ffi_intertemporal_max is a function in ff_optim_mlsec_savezrone for testing
fc_deri_wth_uniroot = @(x) ffi_intertemporal_max(x, fl_z1, fl_z2, fl_r, fl_beta);
% Call Function
bl_verbose = false;
bl_timer = true;
% optimally borrowing given the parameters here
mp_mlsec_ctrlinfo = containers.Map('KeyType','char', 'ValueType','any');
mp_mlsec_ctrlinfo('it_mzoom_jnt_pnts') = 10;
mp_mlsec_ctrlinfo('it_mzoom_max_iter') = 4;
[fl_opti_save_frac, fl_opti_save_level] = ...
ff_optim_mlsec_savezrone(fc_deri_wth_uniroot, bl_verbose, bl_timer, mp_mlsec_ctrlinfo)
Elapsed time is 0.011265 seconds.
fl_opti_save_frac = 0.4241
fl_opti_save_level = -0.13161.2 Test FF_OPTIM_MLSEC_SAVEZRONE 5 Individuals 5 Iterations 5 Points Per Iteration
5 grid points per iteration, and 5 iterations.
% Generate the state-space and function
rng(123);
it_draws = 6; % must be even number
ar_z1 = exp(rand([it_draws,1])*3-1.5);
ar_z2 = exp(rand([it_draws,1])*3-1.5);
ar_r = (rand(it_draws,1)*10.0);
ar_beta = [rand(round(it_draws/2),1)*0.9+0.1; rand(round(it_draws/2),1)*0.9+1];
fc_deri_wth_uniroot = @(x) ffi_intertemporal_max(x, ar_z1, ar_z2, ar_r, ar_beta);
% Call Function
bl_verbose = true;
bl_timer = true;
mp_mlsec_ctrlinfo = containers.Map('KeyType','char', 'ValueType','any');
mp_mlsec_ctrlinfo('it_mlsect_jnt_pnts') = 5;
mp_mlsec_ctrlinfo('it_mlsect_max_iter') = 5;
ff_optim_mlsec_savezrone(fc_deri_wth_uniroot, bl_verbose, bl_timer, mp_mlsec_ctrlinfo);
iter cl_row_names_a Var1 Var2 Var3 Var4 Var5 Var6
____ ______________ _______ _______ _______ _______ _______ _______
0 "point=1" 1e-05 1e-05 1e-05 1e-05 1e-05 1e-05
1 "point=1" 1e-05 1e-05 1e-05 1e-05 1e-05 1e-05
1 "point=2" 0.25001 0.25001 0.25001 0.25001 0.25001 0.25001
1 "point=3" 0.5 0.5 0.5 0.5 0.5 0.5
1 "point=4" 0.75 0.75 0.75 0.75 0.75 0.75
1 "point=5" 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999
2 "point=1" 0.29167 0.29167 0.29167 0.54167 0.54167 0.54167
2 "point=2" 0.33334 0.33334 0.33334 0.58333 0.58333 0.58333
2 "point=3" 0.375 0.375 0.375 0.625 0.625 0.625
2 "point=4" 0.41667 0.41667 0.41667 0.66666 0.66666 0.66666
2 "point=5" 0.45833 0.45833 0.45833 0.70833 0.70833 0.70833
3 "point=1" 0.34028 0.34028 0.38195 0.63194 0.59028 0.59028
3 "point=2" 0.34723 0.34723 0.38889 0.63889 0.59722 0.59722
3 "point=3" 0.35417 0.35417 0.39584 0.64583 0.60416 0.60416
3 "point=4" 0.36111 0.36111 0.40278 0.65277 0.61111 0.61111
3 "point=5" 0.36806 0.36806 0.40972 0.65972 0.61805 0.61805
4 "point=1" 0.36227 0.36227 0.39699 0.6331 0.61921 0.60532
4 "point=2" 0.36343 0.36343 0.39815 0.63426 0.62037 0.60648
4 "point=3" 0.36459 0.36459 0.39931 0.63541 0.62153 0.60764
4 "point=4" 0.36574 0.36574 0.40046 0.63657 0.62268 0.60879
4 "point=5" 0.3669 0.3669 0.40162 0.63773 0.62384 0.60995
5 "point=1" 0.36594 0.36594 0.40066 0.63792 0.62288 0.60783
5 "point=2" 0.36613 0.36613 0.40085 0.63811 0.62307 0.60802
5 "point=3" 0.36632 0.36632 0.40104 0.63831 0.62326 0.60822
5 "point=4" 0.36652 0.36652 0.40124 0.6385 0.62345 0.60841
5 "point=5" 0.36671 0.36671 0.40143 0.63869 0.62365 0.6086
Elapsed time is 0.495996 seconds.
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CONTAINER NAME: mp_container_map ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ___________ ___________ __________ ________ ___________ __________
ar_opti_foc_obj 1 1 2 6 6 1 -0.00037648 -6.2746e-05 0.00042601 -6.7894 -0.00067107 0.00055875
ar_opti_save_frac 2 2 2 6 6 1 3.0037 0.50061 0.13506 0.26979 0.36642 0.63821
xxx TABLE:ar_opti_foc_obj xxxxxxxxxxxxxxxxxx
c1
___________
r1 7.0837e-05
r2 -0.0002782
r3 0.00017713
r4 0.00055875
r5 -0.00023392
r6 -0.00067107
xxx TABLE:ar_opti_save_frac xxxxxxxxxxxxxxxxxx
c1
_______
r1 0.36642
r2 0.36661
r3 0.40153
r4 0.63821
r5 0.62297
r6 0.607931.3 Test FF_OPTIM_MLSEC_SAVEZRONE 8 Individuals 3 Iterations 10 Points Per Iteration
10 grid points per iteration, and 3 iterations.
% Generate the state-space and function
rng(123);
it_draws = 8; % must be even number
ar_z1 = exp(rand([it_draws,1])*3-1.5);
ar_z2 = exp(rand([it_draws,1])*3-1.5);
ar_r = (rand(it_draws,1)*10.0);
ar_beta = [rand(round(it_draws/2),1)*0.9+0.1; rand(round(it_draws/2),1)*0.9+1];
fc_deri_wth_uniroot = @(x) ffi_intertemporal_max(x, ar_z1, ar_z2, ar_r, ar_beta);
% Call Function
bl_verbose = true;
bl_timer = true;
mp_mlsec_ctrlinfo = containers.Map('KeyType','char', 'ValueType','any');
mp_mlsec_ctrlinfo('it_mlsect_jnt_pnts') = 10;
mp_mlsec_ctrlinfo('it_mlsect_max_iter') = 3;
ff_optim_mlsec_savezrone(fc_deri_wth_uniroot, bl_verbose, bl_timer, mp_mlsec_ctrlinfo);
iter cl_row_names_a Var1 Var2 Var3 Var4 Var5 Var6 Var7 Var8
____ ______________ _______ _______ _______ _______ _______ _______ _______ _______
0 "point=1" 1e-05 1e-05 1e-05 1e-05 1e-05 1e-05 1e-05 1e-05
1 "point=1" 1e-05 1e-05 1e-05 1e-05 1e-05 1e-05 1e-05 1e-05
1 "point=2" 0.11112 0.11112 0.11112 0.11112 0.11112 0.11112 0.11112 0.11112
1 "point=3" 0.22223 0.22223 0.22223 0.22223 0.22223 0.22223 0.22223 0.22223
1 "point=4" 0.33334 0.33334 0.33334 0.33334 0.33334 0.33334 0.33334 0.33334
1 "point=5" 0.44445 0.44445 0.44445 0.44445 0.44445 0.44445 0.44445 0.44445
1 "point=6" 0.55555 0.55555 0.55555 0.55555 0.55555 0.55555 0.55555 0.55555
1 "point=7" 0.66666 0.66666 0.66666 0.66666 0.66666 0.66666 0.66666 0.66666
1 "point=8" 0.77777 0.77777 0.77777 0.77777 0.77777 0.77777 0.77777 0.77777
1 "point=9" 0.88888 0.88888 0.88888 0.88888 0.88888 0.88888 0.88888 0.88888
1 "point=10" 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999
2 "point=1" 0.34344 0.23233 0.23233 0.23233 0.56566 0.56566 0.45455 0.56566
2 "point=2" 0.35354 0.24243 0.24243 0.24243 0.57576 0.57576 0.46465 0.57576
2 "point=3" 0.36364 0.25253 0.25253 0.25253 0.58586 0.58586 0.47475 0.58586
2 "point=4" 0.37374 0.26263 0.26263 0.26263 0.59596 0.59596 0.48485 0.59596
2 "point=5" 0.38384 0.27273 0.27273 0.27273 0.60606 0.60606 0.49495 0.60606
2 "point=6" 0.39394 0.28283 0.28283 0.28283 0.61616 0.61616 0.50505 0.61616
2 "point=7" 0.40404 0.29293 0.29293 0.29293 0.62626 0.62626 0.51515 0.62626
2 "point=8" 0.41414 0.30303 0.30303 0.30303 0.63636 0.63636 0.52525 0.63636
2 "point=9" 0.42424 0.31314 0.31314 0.31314 0.64646 0.64646 0.53535 0.64646
2 "point=10" 0.43434 0.32324 0.32324 0.32324 0.65656 0.65656 0.54545 0.65656
3 "point=1" 0.42516 0.27365 0.29385 0.23325 0.55647 0.60698 0.51607 0.57667
3 "point=2" 0.42608 0.27457 0.29477 0.23417 0.55739 0.60789 0.51699 0.57759
3 "point=3" 0.427 0.27549 0.29569 0.23508 0.55831 0.60881 0.51791 0.57851
3 "point=4" 0.42792 0.2764 0.29661 0.236 0.55923 0.60973 0.51882 0.57943
3 "point=5" 0.42884 0.27732 0.29752 0.23692 0.56015 0.61065 0.51974 0.58035
3 "point=6" 0.42975 0.27824 0.29844 0.23784 0.56106 0.61157 0.52066 0.58127
3 "point=7" 0.43067 0.27916 0.29936 0.23876 0.56198 0.61249 0.52158 0.58218
3 "point=8" 0.43159 0.28008 0.30028 0.23967 0.5629 0.6134 0.5225 0.5831
3 "point=9" 0.43251 0.281 0.3012 0.24059 0.56382 0.61432 0.52342 0.58402
3 "point=10" 0.43343 0.28191 0.30212 0.24151 0.56474 0.61524 0.52433 0.58494
Elapsed time is 0.486844 seconds.
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CONTAINER NAME: mp_container_map ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ _________ __________ _________ ________ __________ _________
ar_opti_foc_obj 1 1 2 8 8 1 0.0033175 0.00041468 0.0029592 7.1361 -0.0044871 0.0050249
ar_opti_save_frac 2 2 2 8 8 1 3.5124 0.43905 0.15005 0.34177 0.23371 0.61019
xxx TABLE:ar_opti_foc_obj xxxxxxxxxxxxxxxxxx
c1
___________
r1 0.00087102
r2 0.0033354
r3 -0.0044871
r4 0.001317
r5 -0.0017862
r6 0.0050249
r7 -0.00058496
r8 -0.00037273
xxx TABLE:ar_opti_save_frac xxxxxxxxxxxxxxxxxx
c1
_______
r1 0.42838
r2 0.28054
r3 0.2989
r4 0.23371
r5 0.55877
r6 0.61019
r7 0.5202
r8 0.581721.4 Test FF_OPTIM_MLSEC_SAVEZRONE Speed
Test Speed doing 6.25 million multisections for a savings problem:
% Generate the state-space and function
rng(123);
it_draws = 6250000; % must be even number
ar_z1 = exp(rand([it_draws,1])*3-1.5);
ar_z2 = exp(rand([it_draws,1])*3-1.5);
ar_r = (rand(it_draws,1)*10.0);
ar_beta = [rand(round(it_draws/2),1)*0.9+0.1; rand(round(it_draws/2),1)*0.9+1];
% ffi_intertemporal_max is a function in ff_optim_mlsec_savezrone for testing
fc_deri_wth_uniroot = @(x) ffi_intertemporal_max(x, ar_z1, ar_z2, ar_r, ar_beta);
% Call Function
bl_verbose = false;
bl_timer = true;
[ar_opti_save_frac, ar_opti_save_level] = ff_optim_mlsec_savezrone(fc_deri_wth_uniroot, bl_verbose, bl_timer);
Elapsed time is 16.390434 seconds.
mp_container_map = containers.Map('KeyType','char', 'ValueType','any');
mp_container_map('ar_opti_save_frac') = ar_opti_save_frac;
mp_container_map('ar_opti_save_level') = ar_opti_save_level;
mp_container_map('ar_opti_save_frac_notnan') = ar_opti_save_frac(~isnan(ar_opti_save_frac));
ff_container_map_display(mp_container_map);
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CONTAINER NAME: mp_container_map ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ ________ ________ ____ __________ _______ _______ ________ ________ _______
ar_opti_save_frac 1 1 2 6.25e+06 6.25e+06 1 2.884e+06 0.46144 0.15306 0.33171 0.090876 0.65519
ar_opti_save_frac_notnan 2 2 2 6.25e+06 6.25e+06 1 2.884e+06 0.46144 0.15306 0.33171 0.090876 0.65519
ar_opti_save_level 3 3 2 6.25e+06 6.25e+06 1 2.9482e+06 0.47172 0.66667 1.4133 -3.9807 2.922
figure();
histogram(ar_opti_save_frac(~isnan(ar_opti_save_frac)),100);
title('Distribution of Optimal Savings Fractions');
xlabel('Savings Fractions');
grid on;
1.5 Define Two Period Intertemporal FOC Log Utility No Shock
function [ar_deri_zero, ar_saveborr_level] = ffi_intertemporal_max(ar_saveborr_frac, z1, z2, r, beta)
ar_saveborr_level = ar_saveborr_frac.*(z1+z2./(1+r)) - z2./(1+r);
ar_deri_zero = 1./(ar_saveborr_level-z1) + (beta.*(r+1))./(z2 + ar_saveborr_level.*(r+1));
end