Chapter 1 Savings Dynamic Programming
1.1 FF_VFI_AZ_LOOP Savings Loop Grid Examples
Go back to fan’s MEconTools Toolbox (bookdown), Matlab Code Examples Repository (bookdown), or Math for Econ with Matlab Repository (bookdown).
Examples](https://fanwangecon.github.io/M4Econ/), or** Dynamic Asset This is the example vignette for function: ff_vfi_az_loop from the MEconTools Package. This function solves the dynamic programming problem for a (a,z) model. Households can save a, and face AR(1) shock z. The problem is solved over the infinite horizon.
This is the looped code, it is slow for larger state-space problems. The code uses common grid, with the same state space and choice space grids.
Links to Other Code:
Core Savings/Borrowing Dynamic Programming Solution Functions that are functions in the MEconTools Package. :
Common Choice and States Grid Loop: ff_vfi_az_loop
Common Choice and States Grid Vectorized: ff_vfi_az_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC, Loop:ff_vfi_az_bisec_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC Vectorized: ff_vfi_az_bisec_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Loop:ff_vfi_az_mzoom_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Vectorized: ff_vfi_az_mzoom_vec
The sample codes are written for the standard dynamic savings problem. The code can be adapted for multiple assets, savings and borrowing, discrete and continuous choice, etc. A large proportion of dynamic economic models are based on the underlying structure of solving a model with endogenous states and exogenous shocks, and that is what the (a,z) model does. In general, one can write looped code first to make sure the economics is correct, then vectorized code can be adopted to increase speed.
1.1.1 Test FF_VFI_AZ_LOOP Defaults
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('fl_crra') = 1.5;
mp_params('fl_beta') = 0.94;
% call function
ff_vfi_az_loop(mp_params);
Elapsed time is 2.378952 seconds.
----------------------------------------
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ ___
ap 1 1 2 700 100 7 9855.1 14.079 14.408 1.0234 0 50
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
______ ______ ______ ________ _______ _______ ______
r1 0 0 0 0.045213 0.25576 0.61095 1.0362
r2 0 0 0 0.045213 0.25576 0.61095 1.0362
r3 0 0 0 0.045213 0.25576 0.61095 1.0362
r4 0 0 0 0.06647 0.25576 0.61095 1.0362
r5 0 0 0 0.06647 0.25576 0.61095 1.164
r96 43.924 43.924 43.924 43.924 43.924 45.102 45.102
r97 45.102 45.102 45.102 45.102 45.102 46.298 46.298
r98 46.298 46.298 46.298 46.298 46.298 47.513 47.513
r99 47.513 47.513 47.513 47.513 47.513 48.747 48.747
r100 48.747 48.747 48.747 48.747 48.747 50 501.1.2 Test FF_VFI_AZ_BISEC_VEC Speed Tests
Call the function with different a and z grid size, print out speed:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('ls_ffcmd') = {};
% A grid 50, shock grid 5:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 0.715890 seconds.
% A grid 750, shock grid 15:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 15;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 300.576571 seconds.
% A grid 600, shock grid 45:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 600;
mp_params('it_z_n') = 45;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 910.111661 seconds.1.1.3 Test FF_VFI_AZ_LOOP Control Outputs
Run the function first without any outputs, but only the timer.
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {};
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 0.400105 seconds.Run the function and show policy function for savings choice. For ls_ffcmd, ls_ffsna, ls_ffgrh, can include these: ‘v,’ ‘ap,’ ‘c,’ ‘y,’ ‘coh,’ ‘savefraccoh.’ These are value, aprime savings choice, consumption, income, cash on hand, and savings fraction as cash-on-hand.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 0.410866 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_4858 mean_z_0_67798 mean_z_0_9462 mean_z_1_3205 mean_z_1_8429
_____ ________ _____________ ______________ _____________ _____________ _____________
1 0 0 0 0.071865 0.20862 0.36462
2 0.002975 0 0 0.071698 0.20827 0.36418
3 0.016829 0 0 0.070928 0.20666 0.36216
4 0.046375 0 0.0029827 0.069341 0.20331 0.35793
5 0.095198 0.0038183 0.044243 0.11681 0.27649 0.35114
6 0.1663 0.054362 0.084837 0.17517 0.26637 0.34171
7 0.26234 0.099899 0.13609 0.16422 0.25383 0.41847
8 0.38568 0.15381 0.19428 0.22348 0.32132 0.40047
9 0.53852 0.21153 0.25554 0.28573 0.39055 0.47258
10 0.72291 0.26934 0.31659 0.34814 0.36175 0.44538
11 0.94076 0.3247 0.37504 0.40848 0.42229 0.50941
12 1.1939 0.37617 0.42941 0.46521 0.4802 0.57087
13 1.484 0.53695 0.47898 0.51743 0.5344 0.5291
14 1.8128 0.57847 0.52356 0.56473 0.58429 0.58056
15 2.1817 0.61468 0.56329 0.6071 0.62958 0.62823
16 2.5924 0.6462 0.5985 0.64475 0.67028 0.67186
17 3.0463 0.67365 0.62963 0.67804 0.60721 0.71141
18 3.5449 0.69762 0.65713 0.70737 0.6404 0.65255
19 4.0894 0.71859 0.68142 0.73318 0.67021 0.68509
20 4.6813 0.73701 0.70293 0.75587 0.6969 0.71446
21 5.3218 0.75325 0.722 0.77584 0.72078 0.74089
22 6.0121 0.76763 0.73895 0.79344 0.74211 0.76461
23 6.7536 0.7804 0.75407 0.80897 0.76119 0.78587
24 7.5474 0.7918 0.76759 0.8227 0.77824 0.80491
25 8.3948 0.80201 0.77972 0.83486 0.79351 0.82194
26 9.2967 0.81119 0.79063 0.84567 0.80719 0.83719
27 10.254 0.81947 0.80049 0.8553 0.81948 0.85083
28 11.269 0.82697 0.80941 0.86389 0.83053 0.86306
29 12.342 0.83379 0.81752 0.87159 0.84048 0.87401
30 13.473 0.84001 0.8249 0.87849 0.84946 0.88384
31 14.665 0.84569 0.83165 0.8847 0.85759 0.82241
32 15.918 0.8509 0.83782 0.8903 0.86495 0.83188
33 17.233 0.8557 0.8435 0.89536 0.87163 0.84053
34 18.611 0.86012 0.84872 0.89995 0.8777 0.84844
35 20.053 0.86421 0.85354 0.90411 0.88324 0.85568
36 21.56 0.86799 0.858 0.9079 0.8883 0.86231
37 23.133 0.87151 0.86214 0.91136 0.89292 0.86841
38 24.773 0.87479 0.86598 0.91452 0.89716 0.87401
39 26.481 0.87784 0.86955 0.91741 0.90105 0.87917
40 28.258 0.8807 0.87289 0.92007 0.90463 0.88393
41 30.104 0.88337 0.87601 0.92251 0.90793 0.88833
42 32.021 0.88588 0.87893 0.92475 0.91097 0.8924
43 34.01 0.88824 0.88166 0.92683 0.91378 0.89617
44 36.07 0.89046 0.88423 0.92874 0.91638 0.89966
45 38.204 0.89256 0.88665 0.93052 0.91879 0.90291
46 40.412 0.89453 0.9403 0.93216 0.92102 0.90592
47 42.695 0.8964 0.94141 0.93368 0.9231 0.90873
48 45.053 0.89817 0.94245 0.9351 0.92504 0.91135
49 47.488 0.89985 0.94341 0.93642 0.92684 0.9138
50 50 0.90144 0.9443 0.93765 0.92853 0.91608 
Run the function and show summaries for savings and fraction of coh saved:
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 9;
mp_support('ls_ffcmd') = {'ap', 'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_support('bl_vfi_store_all') = true; % store c(a,z), y(a,z)
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 3.281815 seconds.
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
ap 1 1 2 900 100 9 12904 14.338 14.524 1.013 0 50
savefraccoh 2 2 2 900 100 9 619.51 0.68834 0.26953 0.39157 0 0.93151
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
______ ______ ______ __________ ________ _______ _______ ______ ______
r1 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r2 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r3 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r4 0 0 0 0.00051272 0.092813 0.25576 0.61095 1.0362 1.6023
r5 0 0 0 0.0029004 0.092813 0.25576 0.61095 1.0362 1.6023
r96 43.924 43.924 43.924 43.924 43.924 45.102 45.102 45.102 46.298
r97 45.102 45.102 45.102 45.102 45.102 46.298 46.298 46.298 47.513
r98 46.298 46.298 46.298 46.298 46.298 47.513 47.513 47.513 48.747
r99 47.513 47.513 47.513 47.513 47.513 48.747 48.747 48.747 50
r100 48.747 48.747 48.747 48.747 48.747 50 50 50 50
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ __________ ________ _______ _______ _______ _______
r1 0 0 0 0 0.070073 0.15255 0.28789 0.38573 0.47121
r2 0 0 0 0 0.070045 0.1525 0.28781 0.38565 0.47114
r3 0 0 0 0 0.069914 0.15228 0.28748 0.3853 0.4708
r4 0 0 0 0.00048613 0.069636 0.1518 0.28676 0.38454 0.47007
r5 0 0 0 0.0027273 0.069182 0.15101 0.28559 0.38329 0.46886
r96 0.92625 0.92358 0.92022 0.916 0.91072 0.92836 0.91992 0.90945 0.92033
r97 0.92676 0.92416 0.92088 0.91677 0.91162 0.92918 0.92095 0.91073 0.92169
r98 0.92727 0.92473 0.92153 0.91752 0.91249 0.92998 0.92194 0.91196 0.923
r99 0.92776 0.92528 0.92216 0.91824 0.91333 0.93076 0.92291 0.91315 0.92426
r100 0.92823 0.92581 0.92277 0.91895 0.91416 0.93151 0.92384 0.91431 0.902521.1.4 Test FF_VFI_AZ_LOOP Change Interest Rate and Discount
Show only save fraction of cash on hand:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with several different interest rates and discount factor:
% Lower Savings Incentives
mp_params('fl_beta') = 0.80;
mp_params('fl_r') = 0.01;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 0.825240 seconds.
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 357.49 0.5107 0.2755 0.53945 0 0.80531
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _______ _______ __________ ________
r1 0 0 0 0 0 0.0002246 0.041573
r2 0 0 0 0 0 0.00022455 0.041566
r3 0 0 0 0 0 0.0012689 0.041533
r4 0 0 0 0 0 0.001266 0.041462
r5 0 0 0 0 0 0.0034759 0.041345
r96 0.78455 0.78145 0.79995 0.79456 0.7876 0.77865 0.76719
r97 0.78669 0.78366 0.77972 0.79679 0.78998 0.78122 0.77001
r98 0.78878 0.78582 0.78197 0.79897 0.79231 0.78374 0.77276
r99 0.79084 0.78794 0.78417 0.77927 0.79459 0.7862 0.77545
r100 0.79285 0.79001 0.78633 0.78154 0.79682 0.7886 0.77808
% Higher Savings Incentives
mp_params('fl_beta') = 0.95;
mp_params('fl_r') = 0.04;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 2.386791 seconds.
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 479.94 0.68563 0.27152 0.39602 0 0.93121
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ __________ ________ _______ _______ _______
r1 0 0 0 0.07007 0.17967 0.30874 0.43404
r2 0 0 0 0.070042 0.17961 0.30866 0.43396
r3 0 0 0 0.069911 0.17935 0.30833 0.4336
r4 0 0 0 0.069633 0.17881 0.30762 0.43284
r5 0 0 0.00049972 0.069179 0.17792 0.30645 0.43158
r96 0.92489 0.92134 0.91672 0.91072 0.92717 0.91691 0.92776
r97 0.92544 0.92198 0.91747 0.91162 0.92802 0.91801 0.92895
r98 0.92598 0.9226 0.9182 0.91249 0.92885 0.91908 0.9301
r99 0.9265 0.9232 0.91891 0.91333 0.92965 0.92011 0.93121
r100 0.927 0.92379 0.9196 0.91416 0.93042 0.9211 0.909141.1.5 Test FF_VFI_AZ_LOOP Changing Risk Aversion
Here, again, show fraction of coh saved in summary tabular form, but also show it graphically.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {'savefraccoh'};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with different risk aversion levels, higher preferences for risk:
% Lower Risk Aversion
mp_params('fl_crra') = 0.5;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 1.327261 seconds.
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ ______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 450.35 0.64336 0.2803 0.43568 0 0.90711
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _________ ________ _______ _______
r1 0 0 0 0.0060341 0.093241 0.19572 0.30604
r2 0 0 0 0.0060316 0.093213 0.19567 0.30599
r3 0 0 0 0.0060204 0.09308 0.19546 0.30574
r4 0 0 0 0.0059964 0.092798 0.19501 0.3052
r5 0 0 0 0.012229 0.092335 0.19427 0.30431
r96 0.90049 0.89703 0.89253 0.88669 0.90296 0.89297 0.90379
r97 0.90128 0.89791 0.89351 0.88781 0.90404 0.89429 0.88181
r98 0.90205 0.89876 0.89447 0.88891 0.9051 0.89557 0.88337
r99 0.9028 0.89959 0.89541 0.88998 0.90612 0.89681 0.88489
r100 0.90354 0.9004 0.89632 0.89101 0.90711 0.89802 0.88636
When risk aversion increases, at every state-space point, the household wants to save more.
% Higher Risk Aversion
mp_params('fl_crra') = 5;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 2.680109 seconds.
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 500.59 0.71513 0.25488 0.35641 0 0.94324
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ ________ _______ _______ _______ _______
r1 0 0 0.044811 0.15534 0.25694 0.40177 0.48276
r2 0 0 0.044787 0.15528 0.25686 0.40168 0.48268
r3 0 0 0.044678 0.15499 0.2565 0.40124 0.48228
r4 0 0 0.044445 0.15437 0.25572 0.40032 0.48143
r5 0 0 0.064784 0.15337 0.25445 0.39879 0.48003
r96 0.92489 0.92134 0.94129 0.93513 0.92717 0.91691 0.92776
r97 0.92544 0.92198 0.9418 0.9358 0.92802 0.91801 0.92895
r98 0.92598 0.9226 0.9423 0.93644 0.92885 0.91908 0.9301
r99 0.9265 0.9232 0.94278 0.93706 0.92965 0.92011 0.93121
r100 0.927 0.92379 0.94324 0.93765 0.93042 0.9211 0.90914
1.1.6 Test FF_VFI_AZ_LOOP with Higher Uncertainty
Increase the standard deviation of the Shock.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 150;
mp_params('it_z_n') = 15;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';
% graph color spectrum
mp_params('cl_colors') = 'copper';Lower standard deviation of shock:
% Lower Risk Aversion
mp_params('fl_shk_std') = 0.10;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 13.492999 seconds.
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1506.3 0.66947 0.28673 0.4283 0 0.93222
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.14061 0.1891 0.24154 0.2699 0.32439
r2 0 0 0 0 0 0.1406 0.18908 0.24152 0.26988 0.32437
r3 0 0 0 0 0 0.14053 0.189 0.24142 0.26977 0.32426
r4 0 0 0 0 0 0.14038 0.18881 0.2412 0.26956 0.32402
r5 0 0 0 0 0 0.14013 0.18851 0.24085 0.2692 0.32362
r146 0.93087 0.92957 0.92815 0.92661 0.92492 0.92712 0.92403 0.92069 0.91706 0.91312
r147 0.93121 0.92994 0.92854 0.92702 0.92537 0.92768 0.92465 0.92135 0.91778 0.91391
r148 0.93156 0.9303 0.92893 0.92743 0.92581 0.92823 0.92525 0.92201 0.91849 0.91467
r149 0.93189 0.93065 0.9293 0.92783 0.92623 0.92878 0.92584 0.92264 0.91918 0.91542
r150 0.93222 0.931 0.92967 0.92823 0.92665 0.9293 0.92641 0.92327 0.91986 0.91616Higher shock standard deviation: low shock high asset save more, high shock more asset save less, high shock low asset save more:
% Higher Risk Aversion
mp_params('fl_shk_std') = 0.40;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 18.680264 seconds.
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1678.8 0.74614 0.22779 0.30529 0 0.93141
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.53612 0.59853 0.67884 0.73891 0.77675
r2 0 0 0 0 0 0.53609 0.5985 0.67882 0.73889 0.77674
r3 0 0 0 0 0 0.53594 0.59839 0.67873 0.73883 0.77669
r4 0 0 0 0 0 0.53563 0.59814 0.67853 0.73868 0.77658
r5 0 0 0 0 0 0.53511 0.59774 0.67821 0.73843 0.7764
r146 0.92696 0.9262 0.92513 0.92359 0.92142 0.91653 0.9078 0.88992 0.86057 0.80415
r147 0.92721 0.92647 0.92541 0.9239 0.92176 0.91741 0.90895 0.89144 0.84828 0.79341
r148 0.92746 0.92673 0.92569 0.92421 0.9221 0.91827 0.91007 0.87813 0.83621 0.78284
r149 0.9277 0.92698 0.92596 0.9245 0.92243 0.9191 0.89605 0.86507 0.82436 0.77245
r150 0.92794 0.92724 0.92623 0.9248 0.92276 0.90467 0.88233 0.85227 0.81273 0.762231.2 FF_VFI_AZ_VEC Savings Vectorized Grid Examples
Go back to fan’s MEconTools Toolbox (bookdown), Matlab Code Examples Repository (bookdown), or Math for Econ with Matlab Repository (bookdown).
Examples](https://fanwangecon.github.io/M4Econ/), or** Dynamic Asset This is the example vignette for function: ff_vfi_az_vec from the MEconTools Package. This function solves the dynamic programming problem for a (a,z) model. Households can save a, and face AR(1) shock z. The problem is solved over the infinite horizon.
This is the vectorized code, its speed is much faster than the looped code. The function is designed to have small memory footprint and requires low computing resources, yet is fast.
The code uses common grid, with the same state space and choice space grids. ff_vfi_az_bisec_vec from the MEconTools Package solves the same problem but using continuous exact percentage asset choices, which is more precise than the solution here, and perhaps a little bit slower and relies on First Order Conditions. The ff_vfi_az_mzoom_vec also solves the same class of problems with continuous exact percentage asset choices, and does not rely on First Order Conditions, but is slower than ff_vfi_az_bisec_vec.
Links to Other Code:
Core Savings/Borrowing Dynamic Programming Solution Functions that are functions in the MEconTools Package. :
Common Choice and States Grid Loop: ff_vfi_az_loop
Common Choice and States Grid Vectorized: ff_vfi_az_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC, Loop:ff_vfi_az_bisec_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC Vectorized: ff_vfi_az_bisec_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Loop:ff_vfi_az_mzoom_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Vectorized: ff_vfi_az_mzoom_vec
1.2.1 Test FF_VFI_AZ_VEC Defaults
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('fl_crra') = 1.5;
mp_params('fl_beta') = 0.94;
ff_vfi_az_vec(mp_params);
Elapsed time is 0.136223 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ ___
ap 1 1 2 700 100 7 9855.1 14.079 14.408 1.0234 0 50
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
______ ______ ______ ________ _______ _______ ______
r1 0 0 0 0.045213 0.25576 0.61095 1.0362
r2 0 0 0 0.045213 0.25576 0.61095 1.0362
r3 0 0 0 0.045213 0.25576 0.61095 1.0362
r4 0 0 0 0.06647 0.25576 0.61095 1.0362
r5 0 0 0 0.06647 0.25576 0.61095 1.164
r96 43.924 43.924 43.924 43.924 43.924 45.102 45.102
r97 45.102 45.102 45.102 45.102 45.102 46.298 46.298
r98 46.298 46.298 46.298 46.298 46.298 47.513 47.513
r99 47.513 47.513 47.513 47.513 47.513 48.747 48.747
r100 48.747 48.747 48.747 48.747 48.747 50 501.2.2 Test FF_VFI_AZ_BISEC_VEC Speed Tests
Call the function with different a and z grid size, print out speed:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('ls_ffcmd') = {};
% A grid 50, shock grid 5:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.025309 seconds.
% A grid 750, shock grid 15:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 15;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 4.855482 seconds.
% A grid 600, shock grid 45:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 600;
mp_params('it_z_n') = 45;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 12.201130 seconds.1.2.3 Test FF_VFI_AZ_VEC Control Outputs
Run the function first without any outputs, but only the timer.
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {};
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.022504 seconds.Run the function and show policy function for savings choice. For ls_ffcmd, ls_ffsna, ls_ffgrh, can include these: ‘v,’ ‘ap,’ ‘c,’ ‘y,’ ‘coh,’ ‘savefraccoh.’ These are value, aprime savings choice, consumption, income, cash on hand, and savings fraction as cash-on-hand.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.041571 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_4858 mean_z_0_67798 mean_z_0_9462 mean_z_1_3205 mean_z_1_8429
_____ ________ _____________ ______________ _____________ _____________ _____________
1 0 0 0 0.071865 0.20862 0.36462
2 0.002975 0 0 0.071698 0.20827 0.36418
3 0.016829 0 0 0.070928 0.20666 0.36216
4 0.046375 0 0.0029827 0.069341 0.20331 0.35793
5 0.095198 0.0038183 0.044243 0.11681 0.27649 0.35114
6 0.1663 0.054362 0.084837 0.17517 0.26637 0.34171
7 0.26234 0.099899 0.13609 0.16422 0.25383 0.41847
8 0.38568 0.15381 0.19428 0.22348 0.32132 0.40047
9 0.53852 0.21153 0.25554 0.28573 0.39055 0.47258
10 0.72291 0.26934 0.31659 0.34814 0.36175 0.44538
11 0.94076 0.3247 0.37504 0.40848 0.42229 0.50941
12 1.1939 0.37617 0.42941 0.46521 0.4802 0.57087
13 1.484 0.53695 0.47898 0.51743 0.5344 0.5291
14 1.8128 0.57847 0.52356 0.56473 0.58429 0.58056
15 2.1817 0.61468 0.56329 0.6071 0.62958 0.62823
16 2.5924 0.6462 0.5985 0.64475 0.67028 0.67186
17 3.0463 0.67365 0.62963 0.67804 0.60721 0.71141
18 3.5449 0.69762 0.65713 0.70737 0.6404 0.65255
19 4.0894 0.71859 0.68142 0.73318 0.67021 0.68509
20 4.6813 0.73701 0.70293 0.75587 0.6969 0.71446
21 5.3218 0.75325 0.722 0.77584 0.72078 0.74089
22 6.0121 0.76763 0.73895 0.79344 0.74211 0.76461
23 6.7536 0.7804 0.75407 0.80897 0.76119 0.78587
24 7.5474 0.7918 0.76759 0.8227 0.77824 0.80491
25 8.3948 0.80201 0.77972 0.83486 0.79351 0.82194
26 9.2967 0.81119 0.79063 0.84567 0.80719 0.83719
27 10.254 0.81947 0.80049 0.8553 0.81948 0.85083
28 11.269 0.82697 0.80941 0.86389 0.83053 0.86306
29 12.342 0.83379 0.81752 0.87159 0.84048 0.87401
30 13.473 0.84001 0.8249 0.87849 0.84946 0.88384
31 14.665 0.84569 0.83165 0.8847 0.85759 0.82241
32 15.918 0.8509 0.83782 0.8903 0.86495 0.83188
33 17.233 0.8557 0.8435 0.89536 0.87163 0.84053
34 18.611 0.86012 0.84872 0.89995 0.8777 0.84844
35 20.053 0.86421 0.85354 0.90411 0.88324 0.85568
36 21.56 0.86799 0.858 0.9079 0.8883 0.86231
37 23.133 0.87151 0.86214 0.91136 0.89292 0.86841
38 24.773 0.87479 0.86598 0.91452 0.89716 0.87401
39 26.481 0.87784 0.86955 0.91741 0.90105 0.87917
40 28.258 0.8807 0.87289 0.92007 0.90463 0.88393
41 30.104 0.88337 0.87601 0.92251 0.90793 0.88833
42 32.021 0.88588 0.87893 0.92475 0.91097 0.8924
43 34.01 0.88824 0.88166 0.92683 0.91378 0.89617
44 36.07 0.89046 0.88423 0.92874 0.91638 0.89966
45 38.204 0.89256 0.88665 0.93052 0.91879 0.90291
46 40.412 0.89453 0.9403 0.93216 0.92102 0.90592
47 42.695 0.8964 0.94141 0.93368 0.9231 0.90873
48 45.053 0.89817 0.94245 0.9351 0.92504 0.91135
49 47.488 0.89985 0.94341 0.93642 0.92684 0.9138
50 50 0.90144 0.9443 0.93765 0.92853 0.91608 
Run the function and show summaries for savings and fraction of coh saved:
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 9;
mp_support('ls_ffcmd') = {'ap', 'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_support('bl_vfi_store_all') = true; % store c(a,z), y(a,z)
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.230510 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
ap 1 1 2 900 100 9 12904 14.338 14.524 1.013 0 50
savefraccoh 2 2 2 900 100 9 619.51 0.68834 0.26953 0.39157 0 0.93151
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
______ ______ ______ __________ ________ _______ _______ ______ ______
r1 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r2 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r3 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r4 0 0 0 0.00051272 0.092813 0.25576 0.61095 1.0362 1.6023
r5 0 0 0 0.0029004 0.092813 0.25576 0.61095 1.0362 1.6023
r96 43.924 43.924 43.924 43.924 43.924 45.102 45.102 45.102 46.298
r97 45.102 45.102 45.102 45.102 45.102 46.298 46.298 46.298 47.513
r98 46.298 46.298 46.298 46.298 46.298 47.513 47.513 47.513 48.747
r99 47.513 47.513 47.513 47.513 47.513 48.747 48.747 48.747 50
r100 48.747 48.747 48.747 48.747 48.747 50 50 50 50
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ __________ ________ _______ _______ _______ _______
r1 0 0 0 0 0.070073 0.15255 0.28789 0.38573 0.47121
r2 0 0 0 0 0.070045 0.1525 0.28781 0.38565 0.47114
r3 0 0 0 0 0.069914 0.15228 0.28748 0.3853 0.4708
r4 0 0 0 0.00048613 0.069636 0.1518 0.28676 0.38454 0.47007
r5 0 0 0 0.0027273 0.069182 0.15101 0.28559 0.38329 0.46886
r96 0.92625 0.92358 0.92022 0.916 0.91072 0.92836 0.91992 0.90945 0.92033
r97 0.92676 0.92416 0.92088 0.91677 0.91162 0.92918 0.92095 0.91073 0.92169
r98 0.92727 0.92473 0.92153 0.91752 0.91249 0.92998 0.92194 0.91196 0.923
r99 0.92776 0.92528 0.92216 0.91824 0.91333 0.93076 0.92291 0.91315 0.92426
r100 0.92823 0.92581 0.92277 0.91895 0.91416 0.93151 0.92384 0.91431 0.902521.2.4 Test FF_VFI_AZ_VEC Change Interest Rate and Discount
Show only save fraction of cash on hand:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with several different interest rates and discount factor:
% Lower Savings Incentives
mp_params('fl_beta') = 0.80;
mp_params('fl_r') = 0.01;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.058079 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 357.49 0.5107 0.2755 0.53945 0 0.80531
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _______ _______ __________ ________
r1 0 0 0 0 0 0.0002246 0.041573
r2 0 0 0 0 0 0.00022455 0.041566
r3 0 0 0 0 0 0.0012689 0.041533
r4 0 0 0 0 0 0.001266 0.041462
r5 0 0 0 0 0 0.0034759 0.041345
r96 0.78455 0.78145 0.79995 0.79456 0.7876 0.77865 0.76719
r97 0.78669 0.78366 0.77972 0.79679 0.78998 0.78122 0.77001
r98 0.78878 0.78582 0.78197 0.79897 0.79231 0.78374 0.77276
r99 0.79084 0.78794 0.78417 0.77927 0.79459 0.7862 0.77545
r100 0.79285 0.79001 0.78633 0.78154 0.79682 0.7886 0.77808
% Higher Savings Incentives
mp_params('fl_beta') = 0.95;
mp_params('fl_r') = 0.04;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.177867 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 479.94 0.68563 0.27152 0.39602 0 0.93121
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ __________ ________ _______ _______ _______
r1 0 0 0 0.07007 0.17967 0.30874 0.43404
r2 0 0 0 0.070042 0.17961 0.30866 0.43396
r3 0 0 0 0.069911 0.17935 0.30833 0.4336
r4 0 0 0 0.069633 0.17881 0.30762 0.43284
r5 0 0 0.00049972 0.069179 0.17792 0.30645 0.43158
r96 0.92489 0.92134 0.91672 0.91072 0.92717 0.91691 0.92776
r97 0.92544 0.92198 0.91747 0.91162 0.92802 0.91801 0.92895
r98 0.92598 0.9226 0.9182 0.91249 0.92885 0.91908 0.9301
r99 0.9265 0.9232 0.91891 0.91333 0.92965 0.92011 0.93121
r100 0.927 0.92379 0.9196 0.91416 0.93042 0.9211 0.909141.2.5 Test FF_VFI_AZ_VEC Changing Risk Aversion
Here, again, show fraction of coh saved in summary tabular form, but also show it graphically.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {'savefraccoh'};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with different risk aversion levels, higher preferences for risk:
% Lower Risk Aversion
mp_params('fl_crra') = 0.5;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.181638 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ ______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 450.35 0.64336 0.2803 0.43568 0 0.90711
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _________ ________ _______ _______
r1 0 0 0 0.0060341 0.093241 0.19572 0.30604
r2 0 0 0 0.0060316 0.093213 0.19567 0.30599
r3 0 0 0 0.0060204 0.09308 0.19546 0.30574
r4 0 0 0 0.0059964 0.092798 0.19501 0.3052
r5 0 0 0 0.012229 0.092335 0.19427 0.30431
r96 0.90049 0.89703 0.89253 0.88669 0.90296 0.89297 0.90379
r97 0.90128 0.89791 0.89351 0.88781 0.90404 0.89429 0.88181
r98 0.90205 0.89876 0.89447 0.88891 0.9051 0.89557 0.88337
r99 0.9028 0.89959 0.89541 0.88998 0.90612 0.89681 0.88489
r100 0.90354 0.9004 0.89632 0.89101 0.90711 0.89802 0.88636
When risk aversion increases, at every state-space point, the household wants to save more.
% Higher Risk Aversion
mp_params('fl_crra') = 5;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.152901 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 500.59 0.71513 0.25488 0.35641 0 0.94324
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ ________ _______ _______ _______ _______
r1 0 0 0.044811 0.15534 0.25694 0.40177 0.48276
r2 0 0 0.044787 0.15528 0.25686 0.40168 0.48268
r3 0 0 0.044678 0.15499 0.2565 0.40124 0.48228
r4 0 0 0.044445 0.15437 0.25572 0.40032 0.48143
r5 0 0 0.064784 0.15337 0.25445 0.39879 0.48003
r96 0.92489 0.92134 0.94129 0.93513 0.92717 0.91691 0.92776
r97 0.92544 0.92198 0.9418 0.9358 0.92802 0.91801 0.92895
r98 0.92598 0.9226 0.9423 0.93644 0.92885 0.91908 0.9301
r99 0.9265 0.9232 0.94278 0.93706 0.92965 0.92011 0.93121
r100 0.927 0.92379 0.94324 0.93765 0.93042 0.9211 0.90914
1.2.6 Test FF_VFI_AZ_VEC with Higher Uncertainty
Increase the standard deviation of the Shock.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 150;
mp_params('it_z_n') = 15;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';
% graph color spectrum
mp_params('cl_colors') = 'copper';Lower standard deviation of shock:
% Lower Risk Aversion
mp_params('fl_shk_std') = 0.10;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.544499 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1506.3 0.66947 0.28673 0.4283 0 0.93222
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.14061 0.1891 0.24154 0.2699 0.32439
r2 0 0 0 0 0 0.1406 0.18908 0.24152 0.26988 0.32437
r3 0 0 0 0 0 0.14053 0.189 0.24142 0.26977 0.32426
r4 0 0 0 0 0 0.14038 0.18881 0.2412 0.26956 0.32402
r5 0 0 0 0 0 0.14013 0.18851 0.24085 0.2692 0.32362
r146 0.93087 0.92957 0.92815 0.92661 0.92492 0.92712 0.92403 0.92069 0.91706 0.91312
r147 0.93121 0.92994 0.92854 0.92702 0.92537 0.92768 0.92465 0.92135 0.91778 0.91391
r148 0.93156 0.9303 0.92893 0.92743 0.92581 0.92823 0.92525 0.92201 0.91849 0.91467
r149 0.93189 0.93065 0.9293 0.92783 0.92623 0.92878 0.92584 0.92264 0.91918 0.91542
r150 0.93222 0.931 0.92967 0.92823 0.92665 0.9293 0.92641 0.92327 0.91986 0.91616Higher shock standard deviation: low shock high asset save more, high shock more asset save less, high shock low asset save more:
% Higher Risk Aversion
mp_params('fl_shk_std') = 0.40;
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.515060 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1678.8 0.74614 0.22779 0.30529 0 0.93141
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.53612 0.59853 0.67884 0.73891 0.77675
r2 0 0 0 0 0 0.53609 0.5985 0.67882 0.73889 0.77674
r3 0 0 0 0 0 0.53594 0.59839 0.67873 0.73883 0.77669
r4 0 0 0 0 0 0.53563 0.59814 0.67853 0.73868 0.77658
r5 0 0 0 0 0 0.53511 0.59774 0.67821 0.73843 0.7764
r146 0.92696 0.9262 0.92513 0.92359 0.92142 0.91653 0.9078 0.88992 0.86057 0.80415
r147 0.92721 0.92647 0.92541 0.9239 0.92176 0.91741 0.90895 0.89144 0.84828 0.79341
r148 0.92746 0.92673 0.92569 0.92421 0.9221 0.91827 0.91007 0.87813 0.83621 0.78284
r149 0.9277 0.92698 0.92596 0.9245 0.92243 0.9191 0.89605 0.86507 0.82436 0.77245
r150 0.92794 0.92724 0.92623 0.9248 0.92276 0.90467 0.88233 0.85227 0.81273 0.762231.3 FF_VFI_AZ_BISEC_LOOP Savings Loop Exact (FOC) Examples
Go back to fan’s MEconTools Toolbox (bookdown), Matlab Code Examples Repository (bookdown), or Math for Econ with Matlab Repository (bookdown).
Examples](https://fanwangecon.github.io/M4Econ/), or** Dynamic Asset This is the example vignette for function:ff_vfi_az_bisec_loopfrom the MEconTools Package. This function solves the dynamic programming problem for a (a,z) model. Households can save a, and face AR(1) shock z. The problem is solved over the infinite horizon.
This is the looped code, it is slow for larger state-space problems.
The code uses continuous choices, solved with bisection. The state-space is on a grid, but choice grids are in terms of percentage of resources to save and solved exactly.
Links to Other Code:
Core Savings/Borrowing Dynamic Programming Solution Functions that are functions in the MEconTools Package. :
Common Choice and States Grid Loop: ff_vfi_az_loop
Common Choice and States Grid Vectorized: ff_vfi_az_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC, Loop:ff_vfi_az_bisec_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC Vectorized: ff_vfi_az_bisec_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Loop:ff_vfi_az_mzoom_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Vectorized: ff_vfi_az_mzoom_vec
1.3.1 Test FF_VFI_AZ_BISEC_LOOP Defaults
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('fl_crra') = 1.5;
mp_params('fl_beta') = 0.94;
% call function
ff_vfi_az_bisec_loop(mp_params);
Elapsed time is 33.158577 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ ______
ap 1 1 2 700 100 7 9863.4 14.091 14.388 1.0211 0 50.117
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
______ ______ ______ ________ _______ _______ ______
r1 0 0 0 0.053491 0.25574 0.60604 1.1157
r2 0 0 0 0.053998 0.25571 0.6066 1.1163
r3 0 0 0 0.056449 0.25576 0.60907 1.1187
r4 0 0 0 0.061799 0.26016 0.6109 1.1239
r5 0 0 0 0.066463 0.26897 0.61141 1.1327
r96 43.388 43.52 43.701 43.925 44.222 44.68 45.228
r97 44.566 44.695 44.878 45.103 45.398 45.856 46.403
r98 45.761 45.892 46.072 46.298 46.592 47.05 47.597
r99 46.973 47.107 47.286 47.514 47.806 48.263 48.815
r100 48.206 48.338 48.519 48.746 49.037 49.497 50.1171.3.2 Test FF_VFI_AZ_BISEC_LOOP Speed Tests
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('ls_ffcmd') = {};
% A grid 50, shock grid 5:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 14.819629 seconds.
% A grid 750, shock grid 15:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 15;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 783.169420 seconds.
%A grid 600, shock grid 45:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 600;
mp_params('it_z_n') = 45;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 1955.142516 seconds.1.3.3 Test FF_VFI_AZ_BISEC_LOOP Control Outputs
Run the function first without any outputs;
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {};
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.122166 seconds.Run the function and show policy function for savings choice. For ls_ffcmd, ls_ffsna, ls_ffgrh, can include these: ‘v,’ ‘ap,’ ‘c,’ ‘y,’ ‘coh,’ ‘savefraccoh.’ These are value, aprime savings choice, consumption, income, cash on hand, and savings fraction as cash-on-hand.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 20.812511 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_4858 mean_z_0_67798 mean_z_0_9462 mean_z_1_3205 mean_z_1_8429
_____ ________ _____________ ______________ _____________ _____________ _____________
1 0 0 0 0.067239 0.20859 0.35953
2 0.002975 0 0 0.069375 0.20829 0.36032
3 0.016829 0 0 0.070901 0.2139 0.36215
4 0.046375 0 0.0061439 0.087319 0.2266 0.36264
5 0.095198 0.0087684 0.034403 0.1168 0.2468 0.37473
6 0.1663 0.054361 0.077248 0.1522 0.26639 0.39151
7 0.26234 0.099892 0.13132 0.19388 0.29929 0.41281
8 0.38568 0.15958 0.19309 0.24112 0.33017 0.43088
9 0.53852 0.23417 0.25553 0.29215 0.37436 0.45969
10 0.72291 0.3071 0.31656 0.34812 0.41153 0.48386
11 0.94076 0.37595 0.37503 0.40842 0.44925 0.50992
12 1.1939 0.43881 0.42941 0.45755 0.48697 0.54367
13 1.484 0.49509 0.48129 0.50381 0.53262 0.56979
14 1.8128 0.54489 0.53018 0.54642 0.56778 0.59634
15 2.1817 0.58871 0.57382 0.58548 0.60055 0.6282
16 2.5924 0.62716 0.61258 0.62076 0.63101 0.65249
17 3.0463 0.66079 0.64682 0.65243 0.65884 0.6752
18 3.5449 0.69027 0.67709 0.68069 0.68423 0.69638
19 4.0894 0.71621 0.70376 0.70596 0.70724 0.71591
20 4.6813 0.73703 0.72732 0.72848 0.72799 0.73385
21 5.3218 0.75326 0.74813 0.7485 0.74673 0.75021
22 6.0121 0.76913 0.76657 0.76632 0.76364 0.76535
23 6.7536 0.78536 0.78286 0.78231 0.77889 0.7842
24 7.5474 0.79983 0.79745 0.79653 0.79269 0.79678
25 8.3948 0.81271 0.81039 0.80929 0.80514 0.80831
26 9.2967 0.82418 0.82198 0.82076 0.81637 0.81875
27 10.254 0.8345 0.83242 0.83114 0.82656 0.82833
28 11.269 0.84377 0.84176 0.84042 0.83584 0.83706
29 12.342 0.85214 0.85024 0.84884 0.8442 0.84499
30 13.473 0.85964 0.85781 0.85647 0.85183 0.85232
31 14.665 0.86648 0.86471 0.86337 0.85879 0.85897
32 15.918 0.87264 0.87099 0.86965 0.86507 0.86507
33 17.233 0.87826 0.87667 0.87533 0.87161 0.87063
34 18.611 0.88338 0.88186 0.88052 0.87771 0.87582
35 20.053 0.88802 0.88656 0.88528 0.88326 0.88052
36 21.56 0.8923 0.89089 0.88967 0.88833 0.88485
37 23.133 0.89614 0.89486 0.89364 0.8926 0.88888
38 24.773 0.89974 0.89852 0.8973 0.89626 0.8926
39 26.481 0.90304 0.90182 0.90072 0.89968 0.89608
40 28.258 0.90603 0.90493 0.90383 0.90279 0.89925
41 30.104 0.90884 0.90774 0.9067 0.90572 0.90218
42 32.021 0.9114 0.91036 0.90932 0.90841 0.90493
43 34.01 0.91378 0.9128 0.91183 0.91091 0.90749
44 36.07 0.91598 0.91506 0.91408 0.91317 0.90987
45 38.204 0.91805 0.91714 0.91622 0.91537 0.91207
46 40.412 0.91994 0.91909 0.91817 0.91732 0.91415
47 42.695 0.92171 0.92086 0.92001 0.91921 0.9161
48 45.053 0.92336 0.92257 0.92171 0.92092 0.91799
49 47.488 0.92489 0.92409 0.92336 0.92257 0.92025
50 50 0.92629 0.92562 0.92489 0.92428 0.92403 
Run the function and show summaries for savings and fraction of coh saved:
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 9;
mp_support('ls_ffcmd') = {'ap', 'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_support('bl_vfi_store_all') = true; % store c(a,z), y(a,z)
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 57.010652 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
ap 1 1 2 900 100 9 12926 14.362 14.544 1.0127 0 51.171
savefraccoh 2 2 2 900 100 9 621.24 0.69027 0.26896 0.38965 0 0.92739
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
______ ______ ______ __________ ________ _______ _______ ______ ______
r1 0 0 0 0 0.087442 0.27778 0.58243 1.0038 1.5724
r2 0 0 0 0 0.087962 0.27828 0.58297 1.0044 1.5731
r3 0 0 0 0 0.090477 0.28074 0.58547 1.0069 1.5755
r4 0 0 0 0.00055771 0.09279 0.28605 0.5907 1.0122 1.5808
r5 0 0 0 0.0059496 0.09602 0.29477 0.59952 1.0209 1.5895
r96 43.845 43.923 44.022 44.198 44.428 44.722 45.103 45.546 46.186
r97 45.031 45.101 45.208 45.384 45.613 45.91 46.293 46.735 47.382
r98 46.237 46.297 46.411 46.59 46.818 47.115 47.501 47.948 48.605
r99 47.46 47.512 47.635 47.812 48.041 48.34 48.726 49.191 49.869
r100 48.703 48.746 48.878 49.055 49.283 49.586 49.978 50.495 51.171
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ __________ ________ _______ _______ _______ _______
r1 0 0 0 0 0.066018 0.16569 0.27445 0.37369 0.46243
r2 0 0 0 0 0.066384 0.16593 0.27463 0.37381 0.46256
r3 0 0 0 0 0.068154 0.16715 0.27549 0.37442 0.46292
r4 0 0 0 0.00052879 0.069619 0.16978 0.27726 0.37564 0.46378
r5 0 0 0 0.0055946 0.071572 0.17405 0.28025 0.37766 0.46512
r96 0.92458 0.92354 0.92226 0.92171 0.92116 0.92055 0.91994 0.91842 0.91811
r97 0.92531 0.92416 0.92306 0.92251 0.92196 0.92141 0.92086 0.91933 0.91915
r98 0.92605 0.9247 0.92379 0.9233 0.92275 0.9222 0.92171 0.92031 0.92031
r99 0.92672 0.92525 0.92452 0.92403 0.92348 0.923 0.92251 0.92147 0.92184
r100 0.92739 0.9258 0.92525 0.92477 0.92422 0.92379 0.92342 0.92336 0.923671.3.4 Test FF_VFI_AZ_BISEC_LOOP Change Interest Rate and Discount
Show only save fraction of cash on hand:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with several different interest rates and discount factor:
% Lower Savings Incentives
mp_params('fl_beta') = 0.80;
mp_params('fl_r') = 0.01;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 10.824225 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 357.85 0.51122 0.27528 0.53848 0 0.79965
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _______ _______ __________ ________
r1 0 0 0 0 0 0.00022362 0.041544
r2 0 0 0 0 0 0.00022362 0.041544
r3 0 0 0 0 0 0.0011391 0.041544
r4 0 0 0 0 0 0.0016884 0.041483
r5 0 0 0 0 0 0.0034584 0.04136
r96 0.79586 0.79275 0.78945 0.78591 0.78225 0.77853 0.77059
r97 0.79684 0.79379 0.79055 0.78713 0.78359 0.77993 0.77212
r98 0.79782 0.79482 0.79171 0.78835 0.78488 0.78127 0.77365
r99 0.79873 0.79586 0.79275 0.78951 0.7861 0.78262 0.77548
r100 0.79965 0.79684 0.79385 0.79061 0.78732 0.7839 0.7781
% Higher Savings Incentives
mp_params('fl_beta') = 0.95;
mp_params('fl_r') = 0.04;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 53.369195 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 481.37 0.68768 0.27118 0.39435 0 0.92702
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ ________ _______ _______ _______
r1 0 0 0 0.065774 0.18076 0.30655 0.41654
r2 0 0 0 0.066201 0.18101 0.30674 0.4166
r3 0 0 0 0.06791 0.18223 0.30747 0.41709
r4 0 0 0 0.069619 0.18467 0.30759 0.41812
r5 0 0 0 0.071694 0.18876 0.30838 0.41983
r96 0.92428 0.92245 0.92178 0.92116 0.92049 0.91872 0.91824
r97 0.92501 0.92324 0.92257 0.92196 0.92129 0.91958 0.91921
r98 0.92574 0.92397 0.92336 0.92275 0.92208 0.92049 0.92025
r99 0.92647 0.9247 0.92409 0.92348 0.92287 0.92147 0.92159
r100 0.92702 0.92544 0.92483 0.92422 0.92373 0.92336 0.923481.3.5 Test FF_VFI_AZ_BISEC_LOOP Changing Risk Aversion
Here, again, show fraction of coh saved in summary tabular form, but also show it graphically.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {'savefraccoh'};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with different risk aversion levels, higher preferences for risk:
% Lower Risk Aversion
mp_params('fl_crra') = 0.5;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 47.635241 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 452.13 0.6459 0.28031 0.43398 0 0.90359
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _________ ________ _______ _______
r1 0 0 0 0.0047401 0.089089 0.19822 0.30783
r2 0 0 0 0.0051674 0.089394 0.1984 0.30796
r3 0 0 0 0.0060218 0.090676 0.19926 0.30851
r4 0 0 0 0.0082801 0.092812 0.20115 0.30973
r5 0 0 0 0.012247 0.092995 0.2042 0.31174
r96 0.90047 0.89925 0.89828 0.8973 0.89632 0.89376 0.89297
r97 0.90127 0.90017 0.89919 0.89828 0.8973 0.8948 0.89394
r98 0.90206 0.90102 0.90011 0.89919 0.89828 0.89577 0.89498
r99 0.90279 0.90188 0.90102 0.90011 0.89919 0.89681 0.8959
r100 0.90359 0.90273 0.90188 0.90096 0.90011 0.89803 0.89687
When risk aversion increases, at every state-space point, the household wants to save more.
% Higher Risk Aversion
mp_params('fl_crra') = 5;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 46.937845 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 502.71 0.71816 0.25437 0.3542 0 0.93587
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ ________ _______ _______ _______ _______
r1 0 0 0.047037 0.15537 0.27573 0.3909 0.48782
r2 0 0 0.047525 0.15531 0.27591 0.39102 0.48795
r3 0 0 0.049844 0.1569 0.27695 0.3917 0.48837
r4 0 0 0.054788 0.16025 0.27915 0.3931 0.48929
r5 0 0 0.062905 0.16569 0.28275 0.39542 0.49075
r96 0.93307 0.93258 0.93203 0.93154 0.9302 0.92995 0.92971
r97 0.93374 0.93325 0.93276 0.93227 0.93111 0.93105 0.93117
r98 0.93441 0.93398 0.93349 0.93307 0.93209 0.93227 0.9327
r99 0.93508 0.93465 0.93423 0.93392 0.93331 0.93368 0.93435
r100 0.93575 0.93539 0.93508 0.9349 0.93496 0.93526 0.93587
1.3.6 Test FF_VFI_AZ_BISEC_LOOP with Higher Uncertainty
Increase the standard deviation of the Shock.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 150;
mp_params('it_z_n') = 15;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';
% graph color spectrum
mp_params('cl_colors') = 'copper';Lower standard deviation of shock:
% Lower Risk Aversion
mp_params('fl_shk_std') = 0.10;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 150.979328 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1507.5 0.67001 0.28668 0.42788 0 0.92568
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.13847 0.18485 0.23026 0.27378 0.31729
r2 0 0 0 0 0 0.13853 0.18491 0.23032 0.27384 0.31736
r3 0 0 0 0 0 0.13895 0.18528 0.23063 0.27408 0.3176
r4 0 0 0 0 0 0.13987 0.18607 0.2313 0.27469 0.31809
r5 0 0 0 0 0 0.14011 0.18735 0.2324 0.27567 0.31888
r146 0.92373 0.92354 0.9233 0.92312 0.92287 0.92086 0.92068 0.92049 0.91952 0.91933
r147 0.92422 0.92403 0.92385 0.92361 0.92342 0.92141 0.92123 0.92098 0.92007 0.91988
r148 0.9247 0.92452 0.92434 0.92409 0.92391 0.9219 0.92171 0.92153 0.92062 0.92043
r149 0.92519 0.92501 0.92483 0.92458 0.9244 0.92245 0.92226 0.92208 0.92116 0.9211
r150 0.92568 0.9255 0.92531 0.92507 0.92489 0.92293 0.92275 0.92257 0.92245 0.92232Higher shock standard deviation: low shock high asset save more, high shock more asset save less, high shock low asset save more:
% Higher Risk Aversion
mp_params('fl_shk_std') = 0.40;
ff_vfi_az_bisec_loop(mp_params, mp_support);
Elapsed time is 136.803951 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1685.6 0.74914 0.22909 0.3058 0 0.93679
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.5264 0.61264 0.68271 0.73922 0.78433
r2 0 0 0 0 0 0.52646 0.61264 0.68271 0.73922 0.78433
r3 0 0 0 0 0 0.52658 0.6127 0.68271 0.73922 0.78433
r4 0 0 0 0 0 0.52682 0.61288 0.68283 0.73928 0.78439
r5 0 0 0 0 0 0.52731 0.61313 0.68295 0.73934 0.78439
r146 0.92983 0.92928 0.92873 0.92806 0.92739 0.92269 0.92354 0.9258 0.92904 0.93331
r147 0.9302 0.92971 0.9291 0.92849 0.92788 0.92361 0.92477 0.9269 0.93001 0.93423
r148 0.93056 0.93008 0.92953 0.92892 0.92831 0.92458 0.92593 0.928 0.93105 0.93508
r149 0.93093 0.93044 0.92995 0.92934 0.92873 0.9258 0.92702 0.9291 0.93203 0.936
r150 0.9313 0.93087 0.93032 0.92977 0.92916 0.92696 0.92818 0.93014 0.93294 0.936791.4 FF_VFI_AZ_BISEC_VEC Savings Vectorized Exact (FOC) Examples
Go back to fan’s MEconTools Toolbox (bookdown), Matlab Code Examples Repository (bookdown), or Math for Econ with Matlab Repository (bookdown).
Examples](https://fanwangecon.github.io/M4Econ/), or** Dynamic Asset This is the example vignette for function: ff_vfi_az_bisec_vec from the MEconTools Package. This function solves the dynamic programming problem for a (a,z) model. Households can save a, and face AR(1) shock z. The problem is solved over the infinite horizon.
This is the vectorized code, its speed is much faster than the looped code. The function is designed to have small memory footprint and requires low computing resources, yet is fast.
The code uses continuous choices, solved with bi(multi)section. The state-space is on a grid, but choice grids are in terms of percentage of resources available, which is individual specific, to save and solved exactly up to ((1/(2)^16)*100=0.001525878) percentage of cash on hand. The ff_vfi_az_vec from the MEconTools Package solves the same problem using vectorized common grid code where the choice set and state space share the same grid. The common grid function is faster, but less precise for the same number of asset grid points.
Links to Other Code:
Core Savings/Borrowing Dynamic Programming Solution Functions that are functions in the MEconTools Package. :
Common Choice and States Grid Loop: ff_vfi_az_loop
Common Choice and States Grid Vectorized: ff_vfi_az_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC, Loop:ff_vfi_az_bisec_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC Vectorized: ff_vfi_az_bisec_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Loop:ff_vfi_az_mzoom_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Vectorized: ff_vfi_az_mzoom_vec
1.4.1 Test FF_VFI_AZ_BISEC_VEC Defaults
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('fl_crra') = 1.5;
mp_params('fl_beta') = 0.94;
% call function
ff_vfi_az_bisec_vec(mp_params);
Elapsed time is 1.762201 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ ______
ap 1 1 2 700 100 7 9863.4 14.091 14.388 1.0211 0 50.117
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
______ ______ ______ ________ _______ _______ ______
r1 0 0 0 0.053491 0.25574 0.60604 1.1157
r2 0 0 0 0.053998 0.25571 0.6066 1.1163
r3 0 0 0 0.056449 0.25576 0.60907 1.1187
r4 0 0 0 0.061799 0.26016 0.6109 1.1239
r5 0 0 0 0.066463 0.26897 0.61141 1.1327
r96 43.388 43.52 43.701 43.925 44.222 44.68 45.228
r97 44.566 44.695 44.878 45.103 45.398 45.856 46.403
r98 45.761 45.892 46.072 46.298 46.592 47.05 47.597
r99 46.973 47.107 47.286 47.514 47.806 48.263 48.815
r100 48.206 48.338 48.519 48.746 49.037 49.497 50.1171.4.2 Test FF_VFI_AZ_BISEC_VEC Speed Tests
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('ls_ffcmd') = {};
% A grid 50, shock grid 5:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 0.792541 seconds.
% A grid 750, shock grid 15:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 15;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 43.095190 seconds.
% A grid 600, shock grid 45:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 600;
mp_params('it_z_n') = 45;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 80.139775 seconds.1.4.3 Test FF_VFI_AZ_BISEC_VEC Control Outputs
Run the function first without any outputs;
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {};
ff_vfi_az_vec(mp_params, mp_support);
Elapsed time is 0.029901 seconds.Run the function and show policy function for savings choice. For ls_ffcmd, ls_ffsna, ls_ffgrh, can include these: ‘v,’ ‘ap,’ ‘c,’ ‘y,’ ‘coh,’ ‘savefraccoh.’ These are value, aprime savings choice, consumption, income, cash on hand, and savings fraction as cash-on-hand.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 0.494900 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_4858 mean_z_0_67798 mean_z_0_9462 mean_z_1_3205 mean_z_1_8429
_____ ________ _____________ ______________ _____________ _____________ _____________
1 0 0 0 0.067239 0.20859 0.35953
2 0.002975 0 0 0.069375 0.20829 0.36032
3 0.016829 0 0 0.070901 0.2139 0.36215
4 0.046375 0 0.0061439 0.087319 0.2266 0.36264
5 0.095198 0.0087684 0.034403 0.1168 0.2468 0.37473
6 0.1663 0.054361 0.077248 0.1522 0.26639 0.39151
7 0.26234 0.099892 0.13132 0.19388 0.29929 0.41281
8 0.38568 0.15958 0.19309 0.24112 0.33017 0.43088
9 0.53852 0.23417 0.25553 0.29215 0.37436 0.45969
10 0.72291 0.3071 0.31656 0.34812 0.41153 0.48386
11 0.94076 0.37595 0.37503 0.40842 0.44925 0.50992
12 1.1939 0.43881 0.42941 0.45755 0.48697 0.54367
13 1.484 0.49509 0.48129 0.50381 0.53262 0.56979
14 1.8128 0.54489 0.53018 0.54642 0.56778 0.59634
15 2.1817 0.58871 0.57382 0.58548 0.60055 0.6282
16 2.5924 0.62716 0.61258 0.62076 0.63101 0.65249
17 3.0463 0.66079 0.64682 0.65243 0.65884 0.6752
18 3.5449 0.69027 0.67709 0.68069 0.68423 0.69638
19 4.0894 0.71621 0.70376 0.70596 0.70724 0.71591
20 4.6813 0.73703 0.72732 0.72848 0.72799 0.73385
21 5.3218 0.75326 0.74813 0.7485 0.74673 0.75021
22 6.0121 0.76913 0.76657 0.76632 0.76364 0.76535
23 6.7536 0.78536 0.78286 0.78231 0.77889 0.7842
24 7.5474 0.79983 0.79745 0.79653 0.79269 0.79678
25 8.3948 0.81271 0.81039 0.80929 0.80514 0.80831
26 9.2967 0.82418 0.82198 0.82076 0.81637 0.81875
27 10.254 0.8345 0.83242 0.83114 0.82656 0.82833
28 11.269 0.84377 0.84176 0.84042 0.83584 0.83706
29 12.342 0.85214 0.85024 0.84884 0.8442 0.84499
30 13.473 0.85964 0.85781 0.85647 0.85183 0.85232
31 14.665 0.86648 0.86471 0.86337 0.85879 0.85897
32 15.918 0.87264 0.87099 0.86965 0.86507 0.86507
33 17.233 0.87826 0.87667 0.87533 0.87161 0.87063
34 18.611 0.88338 0.88186 0.88052 0.87771 0.87582
35 20.053 0.88802 0.88656 0.88528 0.88326 0.88052
36 21.56 0.8923 0.89089 0.88967 0.88833 0.88485
37 23.133 0.89614 0.89486 0.89364 0.8926 0.88888
38 24.773 0.89974 0.89852 0.8973 0.89626 0.8926
39 26.481 0.90304 0.90182 0.90072 0.89968 0.89608
40 28.258 0.90603 0.90493 0.90383 0.90279 0.89925
41 30.104 0.90884 0.90774 0.9067 0.90572 0.90218
42 32.021 0.9114 0.91036 0.90932 0.90841 0.90493
43 34.01 0.91378 0.9128 0.91183 0.91091 0.90749
44 36.07 0.91598 0.91506 0.91408 0.91317 0.90987
45 38.204 0.91805 0.91714 0.91622 0.91537 0.91207
46 40.412 0.91994 0.91909 0.91817 0.91732 0.91415
47 42.695 0.92171 0.92086 0.92001 0.91921 0.9161
48 45.053 0.92336 0.92257 0.92171 0.92092 0.91799
49 47.488 0.92489 0.92409 0.92336 0.92257 0.92025
50 50 0.92629 0.92562 0.92489 0.92428 0.92403 
Run the function and show summaries for savings and fraction of coh saved:
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 9;
mp_support('ls_ffcmd') = {'ap', 'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_support('bl_vfi_store_all') = true; % store c(a,z), y(a,z)
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 1.164186 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
ap 1 1 2 900 100 9 12926 14.362 14.544 1.0127 0 51.171
savefraccoh 2 2 2 900 100 9 621.24 0.69027 0.26896 0.38965 0 0.92739
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
______ ______ ______ __________ ________ _______ _______ ______ ______
r1 0 0 0 0 0.087442 0.27778 0.58243 1.0038 1.5724
r2 0 0 0 0 0.087962 0.27828 0.58297 1.0044 1.5731
r3 0 0 0 0 0.090477 0.28074 0.58547 1.0069 1.5755
r4 0 0 0 0.00055771 0.09279 0.28605 0.5907 1.0122 1.5808
r5 0 0 0 0.0059496 0.09602 0.29477 0.59952 1.0209 1.5895
r96 43.845 43.923 44.022 44.198 44.428 44.722 45.103 45.546 46.186
r97 45.031 45.101 45.208 45.384 45.613 45.91 46.293 46.735 47.382
r98 46.237 46.297 46.411 46.59 46.818 47.115 47.501 47.948 48.605
r99 47.46 47.512 47.635 47.812 48.041 48.34 48.726 49.191 49.869
r100 48.703 48.746 48.878 49.055 49.283 49.586 49.978 50.495 51.171
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ __________ ________ _______ _______ _______ _______
r1 0 0 0 0 0.066018 0.16569 0.27445 0.37369 0.46243
r2 0 0 0 0 0.066384 0.16593 0.27463 0.37381 0.46256
r3 0 0 0 0 0.068154 0.16715 0.27549 0.37442 0.46292
r4 0 0 0 0.00052879 0.069619 0.16978 0.27726 0.37564 0.46378
r5 0 0 0 0.0055946 0.071572 0.17405 0.28025 0.37766 0.46512
r96 0.92458 0.92354 0.92226 0.92171 0.92116 0.92055 0.91994 0.91842 0.91811
r97 0.92531 0.92416 0.92306 0.92251 0.92196 0.92141 0.92086 0.91933 0.91915
r98 0.92605 0.9247 0.92379 0.9233 0.92275 0.9222 0.92171 0.92031 0.92031
r99 0.92672 0.92525 0.92452 0.92403 0.92348 0.923 0.92251 0.92147 0.92184
r100 0.92739 0.9258 0.92525 0.92477 0.92422 0.92379 0.92342 0.92336 0.923671.4.4 Test FF_VFI_AZ_BISEC_VEC Change Interest Rate and Discount
Show only save fraction of cash on hand:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with several different interest rates and discount factor:
% Lower Savings Incentives
mp_params('fl_beta') = 0.80;
mp_params('fl_r') = 0.01;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 0.271658 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 357.85 0.51122 0.27528 0.53848 0 0.79965
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _______ _______ __________ ________
r1 0 0 0 0 0 0.00022362 0.041544
r2 0 0 0 0 0 0.00022362 0.041544
r3 0 0 0 0 0 0.0011391 0.041544
r4 0 0 0 0 0 0.0016884 0.041483
r5 0 0 0 0 0 0.0034584 0.04136
r96 0.79586 0.79275 0.78945 0.78591 0.78225 0.77853 0.77059
r97 0.79684 0.79379 0.79055 0.78713 0.78359 0.77993 0.77212
r98 0.79782 0.79482 0.79171 0.78835 0.78488 0.78127 0.77365
r99 0.79873 0.79586 0.79275 0.78951 0.7861 0.78262 0.77548
r100 0.79965 0.79684 0.79385 0.79061 0.78732 0.7839 0.7781
% Higher Savings Incentives
mp_params('fl_beta') = 0.95;
mp_params('fl_r') = 0.04;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 0.971218 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 481.37 0.68768 0.27118 0.39435 0 0.92702
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ ________ _______ _______ _______
r1 0 0 0 0.065774 0.18076 0.30655 0.41654
r2 0 0 0 0.066201 0.18101 0.30674 0.4166
r3 0 0 0 0.06791 0.18223 0.30747 0.41709
r4 0 0 0 0.069619 0.18467 0.30759 0.41812
r5 0 0 0 0.071694 0.18876 0.30838 0.41983
r96 0.92428 0.92245 0.92178 0.92116 0.92049 0.91872 0.91824
r97 0.92501 0.92324 0.92257 0.92196 0.92129 0.91958 0.91921
r98 0.92574 0.92397 0.92336 0.92275 0.92208 0.92049 0.92025
r99 0.92647 0.9247 0.92409 0.92348 0.92287 0.92147 0.92159
r100 0.92702 0.92544 0.92483 0.92422 0.92373 0.92336 0.923481.4.5 Test FF_VFI_AZ_BISEC_VEC Changing Risk Aversion
Here, again, show fraction of coh saved in summary tabular form, but also show it graphically.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {'savefraccoh'};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with different risk aversion levels, higher preferences for risk:
% Lower Risk Aversion
mp_params('fl_crra') = 0.5;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 0.873752 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 452.13 0.6459 0.28031 0.43398 0 0.90359
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _________ ________ _______ _______
r1 0 0 0 0.0047401 0.089089 0.19822 0.30783
r2 0 0 0 0.0051674 0.089394 0.1984 0.30796
r3 0 0 0 0.0060218 0.090676 0.19926 0.30851
r4 0 0 0 0.0082801 0.092812 0.20115 0.30973
r5 0 0 0 0.012247 0.092995 0.2042 0.31174
r96 0.90047 0.89925 0.89828 0.8973 0.89632 0.89376 0.89297
r97 0.90127 0.90017 0.89919 0.89828 0.8973 0.8948 0.89394
r98 0.90206 0.90102 0.90011 0.89919 0.89828 0.89577 0.89498
r99 0.90279 0.90188 0.90102 0.90011 0.89919 0.89681 0.8959
r100 0.90359 0.90273 0.90188 0.90096 0.90011 0.89803 0.89687
When risk aversion increases, at every state-space point, the household wants to save more.
% Higher Risk Aversion
mp_params('fl_crra') = 5;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 0.970314 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 502.71 0.71816 0.25437 0.3542 0 0.93587
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ ________ _______ _______ _______ _______
r1 0 0 0.047037 0.15537 0.27573 0.3909 0.48782
r2 0 0 0.047525 0.15531 0.27591 0.39102 0.48795
r3 0 0 0.049844 0.1569 0.27695 0.3917 0.48837
r4 0 0 0.054788 0.16025 0.27915 0.3931 0.48929
r5 0 0 0.062905 0.16569 0.28275 0.39542 0.49075
r96 0.93307 0.93258 0.93203 0.93154 0.9302 0.92995 0.92971
r97 0.93374 0.93325 0.93276 0.93227 0.93111 0.93105 0.93117
r98 0.93441 0.93398 0.93349 0.93307 0.93209 0.93227 0.9327
r99 0.93508 0.93465 0.93423 0.93392 0.93331 0.93368 0.93435
r100 0.93575 0.93539 0.93508 0.9349 0.93496 0.93526 0.93587
1.4.6 Test FF_VFI_AZ_BISEC_VEC with Higher Uncertainty
Increase the standard deviation of the Shock.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 150;
mp_params('it_z_n') = 15;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';
% graph color spectrum
mp_params('cl_colors') = 'copper';Lower standard deviation of shock:
% Lower Risk Aversion
mp_params('fl_shk_std') = 0.10;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 2.595920 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1507.5 0.67001 0.28668 0.42788 0 0.92568
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.13847 0.18485 0.23026 0.27378 0.31729
r2 0 0 0 0 0 0.13853 0.18491 0.23032 0.27384 0.31736
r3 0 0 0 0 0 0.13895 0.18528 0.23063 0.27408 0.3176
r4 0 0 0 0 0 0.13987 0.18607 0.2313 0.27469 0.31809
r5 0 0 0 0 0 0.14011 0.18735 0.2324 0.27567 0.31888
r146 0.92373 0.92354 0.9233 0.92312 0.92287 0.92086 0.92068 0.92049 0.91952 0.91933
r147 0.92422 0.92403 0.92385 0.92361 0.92342 0.92141 0.92123 0.92098 0.92007 0.91988
r148 0.9247 0.92452 0.92434 0.92409 0.92391 0.9219 0.92171 0.92153 0.92062 0.92043
r149 0.92519 0.92501 0.92483 0.92458 0.9244 0.92245 0.92226 0.92208 0.92116 0.9211
r150 0.92568 0.9255 0.92531 0.92507 0.92489 0.92293 0.92275 0.92257 0.92245 0.92232Higher shock standard deviation: low shock high asset save more, high shock more asset save less, high shock low asset save more:
% Higher Risk Aversion
mp_params('fl_shk_std') = 0.40;
ff_vfi_az_bisec_vec(mp_params, mp_support);
Elapsed time is 2.805227 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1685.6 0.74914 0.22909 0.3058 0 0.93679
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.5264 0.61264 0.68271 0.73922 0.78433
r2 0 0 0 0 0 0.52646 0.61264 0.68271 0.73922 0.78433
r3 0 0 0 0 0 0.52658 0.6127 0.68271 0.73922 0.78433
r4 0 0 0 0 0 0.52682 0.61288 0.68283 0.73928 0.78439
r5 0 0 0 0 0 0.52731 0.61313 0.68295 0.73934 0.78439
r146 0.92983 0.92928 0.92873 0.92806 0.92739 0.92269 0.92354 0.9258 0.92904 0.93331
r147 0.9302 0.92971 0.9291 0.92849 0.92788 0.92361 0.92477 0.9269 0.93001 0.93423
r148 0.93056 0.93008 0.92953 0.92892 0.92831 0.92458 0.92593 0.928 0.93105 0.93508
r149 0.93093 0.93044 0.92995 0.92934 0.92873 0.9258 0.92702 0.9291 0.93203 0.936
r150 0.9313 0.93087 0.93032 0.92977 0.92916 0.92696 0.92818 0.93014 0.93294 0.936791.5 FF_VFI_AZ_MZOOM_LOOP Savings Loop Exact (VALUE) Examples
Go back to fan’s MEconTools Toolbox (bookdown), Matlab Code Examples Repository (bookdown), or Math for Econ with Matlab Repository (bookdown).
Examples](https://fanwangecon.github.io/M4Econ/), or** Dynamic Asset This is the example vignette for function:ff_vfi_az_mzoom_loop from the MEconTools Package. This function solves the dynamic programming problem for a (a,z) model. The state-space is on a grid, but choice grids are in terms of percentage of resources to save and solved exactly.
This is a looped code for continuous choices, solved with the mzoom algorithm. In contrast to the bisection based solution, this is slower, but this does not rely on first order conditions.
Links to Other Code:
Core Savings/Borrowing Dynamic Programming Solution Functions that are functions in the MEconTools Package. :
Common Choice and States Grid Loop: ff_vfi_az_loop
Common Choice and States Grid Vectorized: ff_vfi_az_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC, Loop:ff_vfi_az_bisec_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC Vectorized: ff_vfi_az_bisec_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Loop:ff_vfi_az_mzoom_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Vectorized: ff_vfi_az_mzoom_vec
1.5.1 Test FF_VFI_AZ_MZOOM_LOOP Defaults
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('fl_crra') = 1.5;
mp_params('fl_beta') = 0.94;
% call function
ff_vfi_az_mzoom_loop(mp_params);
Elapsed time is 83.956071 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ ______
ap 1 1 2 700 100 7 9861.5 14.088 14.386 1.0212 0 50.115
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
______ ______ ______ ________ _______ _______ ______
r1 0 0 0 0.05343 0.25568 0.60598 1.1155
r2 0 0 0 0.053451 0.25571 0.60652 1.1161
r3 0 0 0 0.056468 0.25574 0.60897 1.1174
r4 0 0 0 0.061232 0.25995 0.61042 1.1238
r5 0 0 0 0.065929 0.2689 0.61091 1.1323
r96 43.387 43.517 43.7 43.922 44.221 44.657 45.225
r97 44.562 44.694 44.876 45.095 45.392 45.847 46.394
r98 45.758 45.89 46.071 46.287 46.583 47.037 47.596
r99 46.972 47.103 47.285 47.5 47.794 48.247 48.812
r100 48.183 48.337 48.518 48.732 49.025 49.478 50.1151.5.2 Test FF_VFI_AZ_MZOOM_LOOP Speed Tests
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('ls_ffcmd') = {};A grid 50, shock grid 5:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 26.554641 seconds.A grid 750, shock grid 15:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 15;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 2148.508425 seconds.A grid 600, shock grid 45:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 600;
mp_params('it_z_n') = 45;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 8507.097739 seconds.1.5.3 Test FF_VFI_AZ_MZOOM_LOOP Control Outputs
Run the function first without any outputs, but only the timer.
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {};
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 24.011245 seconds.Run the function and show policy function for savings choice. For ls_ffcmd, ls_ffsna, ls_ffgrh, can include these: ‘v,’ ‘ap,’ ‘c,’ ‘y,’ ‘coh,’ ‘savefraccoh.’ These are value, aprime savings choice, consumption, income, cash on hand, and savings fraction as cash-on-hand.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 23.773078 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_4858 mean_z_0_67798 mean_z_0_9462 mean_z_1_3205 mean_z_1_8429
_____ ________ _____________ ______________ _____________ _____________ _____________
1 0 0 0 0.067148 0.2084 0.35952
2 0.002975 0 0 0.069345 0.20826 0.36029
3 0.016829 0 0 0.070749 0.2136 0.36206
4 0.046375 0 0.0059631 0.08732 0.22641 0.36263
5 0.095198 0.008725 0.033935 0.11637 0.24674 0.3747
6 0.1663 0.054327 0.077152 0.15198 0.26635 0.39127
7 0.26234 0.099882 0.13131 0.1936 0.29922 0.41248
8 0.38568 0.15954 0.1928 0.24107 0.33005 0.43049
9 0.53852 0.23411 0.25482 0.29164 0.37407 0.4593
10 0.72291 0.30704 0.31604 0.34806 0.41148 0.48371
11 0.94076 0.37567 0.37487 0.40768 0.44925 0.50972
12 1.1939 0.43849 0.42939 0.4573 0.48691 0.54333
13 1.484 0.49491 0.48129 0.50332 0.53253 0.56934
14 1.8128 0.54486 0.53013 0.54642 0.56773 0.59615
15 2.1817 0.58868 0.57335 0.58545 0.60016 0.62817
16 2.5924 0.6271 0.61254 0.62056 0.63057 0.65247
17 3.0463 0.66058 0.6468 0.65237 0.65884 0.67518
18 3.5449 0.69019 0.67699 0.68069 0.68379 0.69636
19 4.0894 0.71615 0.70375 0.7058 0.70719 0.7159
20 4.6813 0.73661 0.72701 0.72843 0.72781 0.73341
21 5.3218 0.75302 0.7481 0.74821 0.74661 0.74981
22 6.0121 0.76912 0.76622 0.76622 0.76342 0.76534
23 6.7536 0.78503 0.78285 0.78223 0.77885 0.78383
24 7.5474 0.79943 0.79703 0.79623 0.79223 0.79677
25 8.3948 0.81264 0.81024 0.8093 0.80504 0.80784
26 9.2967 0.82384 0.82198 0.82064 0.81634 0.81874
27 10.254 0.83447 0.83225 0.83065 0.82653 0.82824
28 11.269 0.84345 0.84174 0.84025 0.83545 0.83703
29 12.342 0.85185 0.85017 0.84865 0.84417 0.84497
30 13.473 0.85962 0.85746 0.85642 0.85178 0.85185
31 14.665 0.86626 0.86466 0.86306 0.85873 0.85895
32 15.918 0.87226 0.87066 0.86959 0.86504 0.86466
33 17.233 0.87786 0.87626 0.87529 0.87146 0.87061
34 18.611 0.88332 0.88182 0.88026 0.87766 0.87546
35 20.053 0.888 0.88656 0.88507 0.88267 0.88026
36 21.56 0.89187 0.89087 0.88947 0.88825 0.88483
37 23.133 0.89587 0.89484 0.89347 0.89256 0.88867
38 24.773 0.8997 0.89827 0.89727 0.89587 0.89259
39 26.481 0.903 0.90147 0.90066 0.89964 0.89587
40 28.258 0.90601 0.90467 0.90376 0.90278 0.89907
41 30.104 0.90881 0.9077 0.90628 0.90547 0.90216
42 32.021 0.91137 0.91035 0.90908 0.90838 0.90467
43 34.01 0.91377 0.91275 0.91148 0.91068 0.90708
44 36.07 0.91595 0.91468 0.91388 0.91308 0.90983
45 38.204 0.91788 0.91708 0.91617 0.91531 0.91204
46 40.412 0.91948 0.91868 0.91788 0.91708 0.91388
47 42.695 0.92168 0.92085 0.91998 0.91915 0.91604
48 45.053 0.92331 0.92251 0.92171 0.92091 0.91788
49 47.488 0.92485 0.92408 0.92331 0.92254 0.9202
50 50 0.92588 0.92555 0.92485 0.92423 0.92402 
Run the function and show summaries for savings and fraction of coh saved:
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
% mp_params('fl_crra') = 1.5;
% mp_params('fl_beta') = 0.94;
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 9;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
% call function
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 111.419370 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_36853 mean_z_0_46648 mean_z_0_59047 mean_z_0_74742 mean_z_0_94608 mean_z_1_1975 mean_z_1_5159 mean_z_1_9188 mean_z_2_4288
_____ __________ ______________ ______________ ______________ ______________ ______________ _____________ _____________ _____________ _____________
1 0 0 0 0 0 0.065547 0.16519 0.27438 0.37327 0.46242
2 0.00051272 0 0 0 0 0.066347 0.16588 0.27443 0.3738 0.4625
3 0.0029004 0 0 0 0 0.067948 0.16679 0.27523 0.37407 0.4625
4 0.0079925 0 0 0 0.00050216 0.069549 0.16975 0.27721 0.37559 0.4633
5 0.016407 0 0 0 0.005563 0.071534 0.17359 0.2802 0.37727 0.4649
6 0.028662 0 0 0 0.011926 0.080274 0.17999 0.28363 0.38054 0.4665
7 0.045213 0 0 0 0.022095 0.090757 0.18443 0.28523 0.38087 0.4665
8 0.06647 0 0 0.0043625 0.033935 0.10076 0.19478 0.29244 0.38621 0.4697
9 0.092813 0.00076108 0.0091251 0.017748 0.047979 0.11397 0.2068 0.30084 0.39207 0.47371
10 0.12459 0.02539 0.027791 0.036336 0.066347 0.13237 0.21681 0.31124 0.39938 0.47898
11 0.16214 0.049062 0.054743 0.057497 0.087289 0.14878 0.23341 0.31965 0.4076 0.48451
12 0.20576 0.080353 0.076351 0.084213 0.11115 0.16729 0.24668 0.33295 0.41295 0.48895
13 0.25576 0.11036 0.10076 0.11357 0.13677 0.1944 0.26643 0.34326 0.42327 0.49539
14 0.31242 0.14798 0.12866 0.14076 0.16483 0.21731 0.28363 0.35886 0.43195 0.50332
15 0.37601 0.17839 0.16439 0.16895 0.194 0.24107 0.30164 0.37165 0.44362 0.51204
16 0.4468 0.2098 0.20032 0.1988 0.22401 0.26563 0.32575 0.38953 0.4565 0.51772
17 0.52503 0.24246 0.23721 0.23371 0.25482 0.29153 0.34564 0.40412 0.46694 0.52773
18 0.61095 0.28123 0.27422 0.26803 0.28577 0.31725 0.36526 0.42169 0.48129 0.5355
19 0.7048 0.31861 0.30964 0.30224 0.31644 0.34326 0.38636 0.43889 0.49228 0.54649
20 0.8068 0.35352 0.34406 0.33561 0.34646 0.37247 0.41168 0.4555 0.50395 0.55454
21 0.91719 0.38727 0.37774 0.36766 0.37639 0.40048 0.43289 0.47181 0.51932 0.56671
22 1.0362 0.42001 0.40688 0.39888 0.40495 0.42569 0.4541 0.49222 0.53173 0.57495
23 1.164 0.4501 0.43289 0.42881 0.43266 0.4501 0.47451 0.50812 0.54573 0.58735
24 1.3008 0.47851 0.45746 0.45719 0.45922 0.47371 0.4948 0.52453 0.5608 0.59678
25 1.4468 0.50572 0.48514 0.48371 0.48451 0.49652 0.51432 0.54067 0.57335 0.60616
26 1.6023 0.53093 0.51118 0.50952 0.50892 0.51852 0.53322 0.56014 0.58575 0.61895
27 1.7673 0.55214 0.53571 0.53333 0.53173 0.53973 0.55258 0.57573 0.60152 0.62867
28 1.9422 0.57052 0.55854 0.55614 0.55374 0.55981 0.57335 0.59055 0.61336 0.63834
29 2.127 0.58782 0.58031 0.57735 0.57415 0.57893 0.59055 0.60534 0.62569 0.65071
30 2.3221 0.60768 0.60016 0.59758 0.59375 0.59695 0.60696 0.6194 0.6375 0.65978
31 2.5275 0.62577 0.61947 0.61496 0.61226 0.61416 0.62256 0.63297 0.65018 0.66898
32 2.7434 0.64351 0.63697 0.63101 0.62956 0.63057 0.6376 0.64617 0.66298 0.68097
33 2.97 0.65976 0.65338 0.64537 0.64591 0.64617 0.65178 0.66218 0.67378 0.68986
34 3.2075 0.67458 0.66898 0.66058 0.66124 0.66058 0.66498 0.67432 0.68445 0.69819
35 3.456 0.68919 0.68379 0.67538 0.67538 0.67458 0.67829 0.68592 0.69419 0.70716
36 3.7158 0.7022 0.69739 0.68939 0.68928 0.68779 0.69019 0.69659 0.7042 0.7154
37 3.9869 0.7146 0.7098 0.7022 0.70205 0.70039 0.7022 0.7074 0.71372 0.72609
38 4.2696 0.72668 0.7218 0.7146 0.7138 0.71209 0.713 0.71779 0.7254 0.73341
39 4.564 0.73741 0.73341 0.7262 0.7254 0.72317 0.7234 0.72748 0.73419 0.74121
40 4.8702 0.74798 0.74381 0.73711 0.73581 0.73341 0.73341 0.73661 0.74221 0.74821
41 5.1884 0.75768 0.75382 0.74727 0.74581 0.74348 0.74301 0.74551 0.75041 0.75542
42 5.5188 0.76679 0.7618 0.75684 0.75542 0.75281 0.75217 0.75382 0.75782 0.76232
43 5.8615 0.77502 0.76862 0.76542 0.76422 0.76165 0.76022 0.76182 0.76502 0.77102
44 6.2166 0.78303 0.77658 0.77422 0.77262 0.76996 0.76862 0.76941 0.77229 0.77742
45 6.5844 0.79063 0.78452 0.78223 0.78063 0.77742 0.77633 0.77661 0.77895 0.7835
46 6.9649 0.79783 0.79196 0.78983 0.78823 0.78529 0.78356 0.78543 0.78503 0.78903
47 7.3583 0.80499 0.79863 0.79695 0.79543 0.79223 0.79043 0.79223 0.79143 0.79463
48 7.7647 0.81024 0.80566 0.80343 0.80231 0.79863 0.79699 0.79863 0.79943 0.80023
49 8.1844 0.81504 0.81184 0.81003 0.80862 0.80504 0.80329 0.80471 0.80504 0.80547
50 8.6173 0.81984 0.81744 0.81584 0.81424 0.81104 0.80997 0.81024 0.81024 0.81024
51 9.0637 0.82544 0.82351 0.82144 0.82031 0.81664 0.81584 0.81573 0.81579 0.81733
52 9.5237 0.83065 0.82881 0.82664 0.82544 0.82224 0.82224 0.82064 0.82064 0.82191
53 9.9975 0.83545 0.83385 0.83217 0.83065 0.82744 0.82704 0.82544 0.82537 0.82624
54 10.485 0.84025 0.83863 0.83697 0.83545 0.83225 0.8321 0.83044 0.82985 0.83054
55 10.987 0.84494 0.84315 0.84155 0.84023 0.83703 0.83625 0.83465 0.83385 0.83457
56 11.502 0.84919 0.84705 0.84585 0.84425 0.84105 0.841 0.83915 0.83785 0.83844
57 12.032 0.85319 0.85156 0.85002 0.84785 0.84562 0.84505 0.84322 0.84185 0.84185
58 12.577 0.85666 0.85506 0.85396 0.85174 0.84945 0.84865 0.84665 0.84585 0.84574
59 13.136 0.86064 0.85906 0.85746 0.85506 0.85338 0.85265 0.85079 0.84945 0.84919
60 13.709 0.86386 0.86226 0.86122 0.85826 0.85666 0.85639 0.85425 0.85265 0.85245
61 14.298 0.86706 0.86596 0.86461 0.86138 0.86042 0.85978 0.85746 0.85586 0.85562
62 14.901 0.87052 0.86906 0.86746 0.86464 0.86372 0.86304 0.86066 0.85906 0.85826
63 15.519 0.87306 0.87215 0.87066 0.86746 0.86682 0.86615 0.86386 0.86226 0.86146
64 16.152 0.87626 0.87466 0.87378 0.87066 0.86981 0.86906 0.86698 0.86535 0.86464
65 16.801 0.87866 0.87779 0.87626 0.8736 0.87226 0.87196 0.86981 0.86812 0.8687
66 17.465 0.88163 0.88026 0.87923 0.87626 0.87538 0.87466 0.87226 0.87066 0.87126
67 18.144 0.88409 0.88267 0.88179 0.87866 0.87786 0.87706 0.87466 0.87386 0.87375
68 18.839 0.88646 0.88507 0.88422 0.88107 0.88026 0.87946 0.8776 0.87706 0.87612
69 19.55 0.88867 0.88747 0.88653 0.88347 0.88267 0.88187 0.87997 0.87946 0.87843
70 20.277 0.89087 0.88947 0.88867 0.88587 0.88507 0.88427 0.88187 0.88187 0.88026
71 21.02 0.89267 0.89187 0.89087 0.88787 0.88736 0.88659 0.88427 0.88419 0.88267
72 21.778 0.89493 0.89347 0.89267 0.89027 0.88945 0.88867 0.88656 0.88587 0.88477
73 22.553 0.89667 0.89582 0.89487 0.89187 0.89107 0.89027 0.88859 0.88825 0.88667
74 23.345 0.89827 0.89747 0.89667 0.89422 0.89336 0.89262 0.89027 0.89016 0.88862
75 24.152 0.90034 0.89907 0.89827 0.89587 0.89507 0.89427 0.89237 0.89187 0.89027
76 24.977 0.90204 0.90111 0.89987 0.89747 0.89667 0.89587 0.89416 0.89347 0.89187
77 25.818 0.90361 0.90274 0.90147 0.89907 0.89827 0.89797 0.89587 0.89507 0.89347
78 26.675 0.90515 0.90387 0.90307 0.90067 0.89987 0.89961 0.89747 0.89711 0.89551
79 27.55 0.90628 0.90547 0.90467 0.90227 0.90147 0.90117 0.89907 0.89827 0.89667
80 28.441 0.90788 0.90708 0.90547 0.90387 0.90307 0.90227 0.90066 0.89987 0.89827
81 29.35 0.90908 0.9086 0.90708 0.90547 0.90467 0.90387 0.90213 0.90147 0.89987
82 30.276 0.91068 0.90988 0.90825 0.90697 0.90623 0.90547 0.90354 0.90307 0.90146
83 31.219 0.91195 0.91121 0.90908 0.90828 0.90758 0.9069 0.90467 0.90444 0.90281
84 32.179 0.91308 0.91228 0.91035 0.90958 0.90887 0.90788 0.90623 0.90547 0.90387
85 33.157 0.91388 0.91361 0.91148 0.91068 0.90988 0.90908 0.90708 0.90703 0.9054
86 34.153 0.91543 0.91468 0.91228 0.91198 0.9113 0.91063 0.90872 0.90788 0.90628
87 35.166 0.91628 0.91548 0.9138 0.91308 0.91228 0.91148 0.90988 0.90908 0.9078
88 36.198 0.91708 0.91688 0.91468 0.91388 0.91355 0.91291 0.91068 0.91057 0.90893
89 37.247 0.91851 0.91786 0.91548 0.91527 0.91463 0.91388 0.91214 0.91148 0.90988
90 38.314 0.91946 0.91868 0.91691 0.91628 0.91548 0.91468 0.91308 0.91228 0.91108
91 39.399 0.92028 0.91948 0.91788 0.91708 0.91628 0.91604 0.91422 0.91374 0.91228
92 40.503 0.92108 0.92028 0.91868 0.91788 0.91761 0.917 0.9154 0.91468 0.91388
93 41.625 0.92188 0.92108 0.91948 0.91868 0.91851 0.91788 0.91628 0.91548 0.91527
94 42.765 0.92268 0.92188 0.92028 0.92001 0.9194 0.91868 0.91771 0.91628 0.9162
95 43.924 0.92348 0.92268 0.92108 0.92085 0.92026 0.91948 0.91868 0.91708 0.91708
96 45.102 0.92428 0.92348 0.92188 0.92168 0.92108 0.92028 0.91948 0.91838 0.91788
97 46.298 0.92508 0.92414 0.92268 0.92248 0.92188 0.92108 0.92028 0.91931 0.91868
98 47.513 0.92588 0.92469 0.92348 0.92325 0.92268 0.92188 0.92168 0.92026 0.92028
99 48.747 0.92668 0.92508 0.92428 0.92398 0.92347 0.92268 0.92248 0.92108 0.9218
100 50 0.92737 0.9258 0.92508 0.92428 0.9242 0.92348 0.92337 0.92334 0.92348 
1.5.4 Test FF_VFI_AZ_MZOOM_LOOP Change Interest Rate and Discount
Show only save fraction of cash on hand:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 9;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with several different interest rates and discount factor:
% Lower Savings Incentives
mp_params('fl_beta') = 0.80;
mp_params('fl_r') = 0.01;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 294.329574 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ _______ ________ ___ _______
savefraccoh 1 1 2 6750 750 9 3468.2 0.5138 0.27192 0.52924 0 0.80103
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ _______ _______ _______ _______ _______ ________
r1 0 0 0 0 0 0 0 0.02073 0.065955
r2 0 0 0 0 0 0 0 0.02073 0.065955
r3 0 0 0 0 0 0 0 0.02073 0.065955
r4 0 0 0 0 0 0 0 0.02073 0.065955
r5 0 0 0 0 0 0 0 0.02073 0.065987
r746 0.8008 0.79843 0.7959 0.79303 0.78983 0.78663 0.78303 0.77903 0.77502
r747 0.80092 0.79855 0.79603 0.79303 0.79058 0.78713 0.78362 0.77953 0.77553
r748 0.80102 0.79863 0.79615 0.7935 0.79063 0.78729 0.78378 0.77972 0.77568
r749 0.80103 0.79863 0.79623 0.79369 0.79063 0.78743 0.78383 0.77983 0.77582
r750 0.80103 0.79904 0.79623 0.79378 0.79063 0.78743 0.78383 0.77983 0.77582
% Higher Savings Incentives
mp_params('fl_beta') = 0.95;
mp_params('fl_r') = 0.04;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 1309.412430 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ _______ ________ ___ _______
savefraccoh 1 1 2 6750 750 9 4667.7 0.6915 0.26685 0.3859 0 0.92668
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ _______ ________ _______ _______ _______ _______
r1 0 0 0 0 0.0647 0.16668 0.27352 0.37327 0.4617
r2 0 0 0 0 0.0647 0.16668 0.27352 0.37327 0.4617
r3 0 0 0 0 0.064731 0.16668 0.27352 0.37327 0.4617
r4 0 0 0 0 0.064731 0.16668 0.27355 0.37327 0.4617
r5 0 0 0 0 0.064747 0.16671 0.27355 0.37327 0.4617
r746 0.92657 0.92588 0.92508 0.92428 0.92348 0.92268 0.92235 0.92188 0.92188
r747 0.92664 0.92588 0.92508 0.92428 0.92402 0.92318 0.92248 0.92188 0.92235
r748 0.92668 0.92588 0.92508 0.92478 0.92411 0.92328 0.9226 0.92188 0.9226
r749 0.92668 0.92588 0.92555 0.92488 0.9242 0.9234 0.92268 0.92254 0.92268
r750 0.92668 0.92588 0.92565 0.92497 0.92427 0.92348 0.92268 0.92268 0.922681.5.5 Test FF_VFI_AZ_MZOOM_LOOP Changing Risk Aversion
Here, again, show fraction of coh saved in summary tabular form, but also show it graphically.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {'savefraccoh'};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with different risk aversion levels, higher preferences for risk:
% Lower Risk Aversion
mp_params('fl_crra') = 0.5;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 84.461743 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 452.03 0.64575 0.28029 0.43406 0 0.90354
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _________ ________ _______ _______
r1 0 0 0 0.0047077 0.089109 0.198 0.30781
r2 0 0 0 0.0051079 0.089156 0.198 0.30793
r3 0 0 0 0.0059631 0.090679 0.1988 0.30848
r4 0 0 0 0.0079639 0.092358 0.20109 0.30964
r5 0 0 0 0.011926 0.092758 0.20413 0.31171
r96 0.90047 0.89907 0.89826 0.89727 0.89587 0.89347 0.89267
r97 0.90127 0.89987 0.89907 0.89822 0.89727 0.89477 0.89394
r98 0.90204 0.90067 0.89987 0.89907 0.89822 0.89573 0.89493
r99 0.90278 0.90147 0.90067 0.89987 0.89907 0.89667 0.89587
r100 0.90354 0.90227 0.90147 0.90067 0.89987 0.89801 0.89667
When risk aversion increases, at every state-space point, the household wants to save more.
% Higher Risk Aversion
mp_params('fl_crra') = 5;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 88.697274 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ _____ _____ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 502.6 0.718 0.25437 0.35427 0 0.93587
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ ________ _______ _______ _______ _______
r1 0 0 0.04674 0.15532 0.27563 0.39047 0.48771
r2 0 0 0.047493 0.15525 0.27563 0.39101 0.48771
r3 0 0 0.049541 0.15685 0.27693 0.39127 0.48834
r4 0 0 0.054343 0.16018 0.27883 0.39287 0.48923
r5 0 0 0.062848 0.16566 0.28272 0.39528 0.49071
r96 0.93269 0.93251 0.93189 0.93108 0.93014 0.92988 0.92968
r97 0.93349 0.93322 0.93269 0.93189 0.93107 0.93104 0.93108
r98 0.93429 0.93349 0.93347 0.93269 0.93189 0.93189 0.93269
r99 0.93507 0.93429 0.93424 0.93349 0.93331 0.93349 0.93429
r100 0.93575 0.93509 0.93507 0.93488 0.93491 0.93509 0.93587
1.5.6 Test FF_VFI_AZ_MZOOM_LOOP with Higher Uncertainty
Increase the standard deviation of the Shock.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 150;
mp_params('it_z_n') = 15;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Lower standard deviation of shock:
% Lower Risk Aversion
mp_params('fl_shk_std') = 0.10;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 304.022067 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1507.2 0.66985 0.28667 0.42796 0 0.92565
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.13838 0.18479 0.23021 0.27363 0.31725
r2 0 0 0 0 0 0.13838 0.18479 0.23027 0.27363 0.31725
r3 0 0 0 0 0 0.13894 0.18526 0.23041 0.27407 0.31725
r4 0 0 0 0 0 0.13987 0.18606 0.23121 0.27443 0.31805
r5 0 0 0 0 0 0.13998 0.18719 0.23201 0.27563 0.31885
r146 0.92348 0.92348 0.92328 0.92268 0.92268 0.92085 0.92028 0.92028 0.91948 0.91931
r147 0.9242 0.92398 0.92348 0.92348 0.92337 0.92108 0.92108 0.92097 0.92001 0.91948
r148 0.92428 0.92428 0.92428 0.92408 0.92348 0.92188 0.92171 0.92108 0.92028 0.92028
r149 0.92508 0.92497 0.92478 0.92428 0.92428 0.92241 0.92188 0.92188 0.92108 0.92107
r150 0.92565 0.92508 0.92508 0.92507 0.92485 0.92268 0.92268 0.92254 0.92238 0.92188Higher shock standard deviation: low shock high asset save more, high shock more asset save less, high shock low asset save more:
% Higher Risk Aversion
mp_params('fl_shk_std') = 0.40;
ff_vfi_az_mzoom_loop(mp_params, mp_support);
Elapsed time is 304.175092 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1685.2 0.74898 0.22908 0.30585 0 0.93669
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.52613 0.61256 0.68259 0.73901 0.78427
r2 0 0 0 0 0 0.52613 0.61256 0.68259 0.73901 0.7843
r3 0 0 0 0 0 0.52613 0.61256 0.68259 0.73901 0.7843
r4 0 0 0 0 0 0.52682 0.61256 0.68259 0.73901 0.78433
r5 0 0 0 0 0 0.52693 0.61309 0.68259 0.73901 0.78439
r146 0.92948 0.92925 0.92828 0.92805 0.92737 0.92263 0.92348 0.92577 0.92901 0.93331
r147 0.93017 0.92948 0.92868 0.92828 0.92748 0.92348 0.92428 0.92668 0.93002 0.93421
r148 0.93028 0.93005 0.92948 0.92891 0.92827 0.92428 0.92587 0.92799 0.93101 0.93507
r149 0.93091 0.93028 0.92948 0.92931 0.92828 0.92574 0.92668 0.92904 0.93189 0.93589
r150 0.93108 0.93082 0.93027 0.92948 0.92868 0.92668 0.92814 0.93008 0.93269 0.936691.6 FF_VFI_AZ_MZOOM_VEC Savings Vectorized Exact (VALUE) Examples
Go back to fan’s MEconTools Toolbox (bookdown), Matlab Code Examples Repository (bookdown), or Math for Econ with Matlab Repository (bookdown).
Examples](https://fanwangecon.github.io/M4Econ/), or** Dynamic Asset This is the example vignette for function:ff_vfi_az_mzoom_vec from the MEconTools Package. This function solves the dynamic programming problem for a (a,z) model. The state-space is on a grid, but choice grids are in terms of percentage of resources to save and solved exactly.
This is a vectorized code for continuous choices, solved with the mzoom algorithm. In contrast to the bisection based solution, this is slower, but this does not rely on first order conditions.
Links to Other Code:
Core Savings/Borrowing Dynamic Programming Solution Functions that are functions in the MEconTools Package. :
Common Choice and States Grid Loop: ff_vfi_az_loop
Common Choice and States Grid Vectorized: ff_vfi_az_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC, Loop:ff_vfi_az_bisec_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC Vectorized: ff_vfi_az_bisec_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Loop:ff_vfi_az_mzoom_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Vectorized: ff_vfi_az_mzoom_vec
1.6.1 Test FF_VFI_AZ_MZOOM_VEC Defaults
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('fl_crra') = 1.5;
mp_params('fl_beta') = 0.94;
% call function
ff_vfi_az_mzoom_vec(mp_params);
Elapsed time is 6.126702 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ ______
ap 1 1 2 700 100 7 9861.5 14.088 14.386 1.0212 0 50.115
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
______ ______ ______ ________ _______ _______ ______
r1 0 0 0 0.05343 0.25568 0.60598 1.1155
r2 0 0 0 0.053451 0.25571 0.60652 1.1161
r3 0 0 0 0.056468 0.25574 0.60897 1.1174
r4 0 0 0 0.061232 0.25995 0.61042 1.1238
r5 0 0 0 0.065929 0.2689 0.61091 1.1323
r96 43.387 43.517 43.7 43.922 44.221 44.657 45.225
r97 44.562 44.694 44.876 45.095 45.392 45.847 46.394
r98 45.758 45.89 46.071 46.287 46.583 47.037 47.596
r99 46.972 47.103 47.285 47.5 47.794 48.247 48.812
r100 48.183 48.337 48.518 48.732 49.025 49.478 50.1151.6.2 Test FF_VFI_AZ_MZOOM_VEC Speed Tests
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('ls_ffcmd') = {};
% A grid 50, shock grid 5:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 1.996365 seconds.
% A grid 750, shock grid 15:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 15;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 337.171768 seconds.
% A grid 600, shock grid 45:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 600;
mp_params('it_z_n') = 45;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 1758.273287 seconds.1.6.3 Test FF_VFI_AZ_MZOOM_VEC Control Outputs
Run the function first without any outputs, but only the timer.
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {};
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 1.091918 seconds.Run the function and show policy function for savings choice. For ls_ffcmd, ls_ffsna, ls_ffgrh, can include these: ‘v,’ ‘ap,’ ‘c,’ ‘y,’ ‘coh,’ ‘savefraccoh.’ These are value, aprime savings choice, consumption, income, cash on hand, and savings fraction as cash-on-hand.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 1.090424 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_4858 mean_z_0_67798 mean_z_0_9462 mean_z_1_3205 mean_z_1_8429
_____ ________ _____________ ______________ _____________ _____________ _____________
1 0 0 0 0.067148 0.2084 0.35952
2 0.002975 0 0 0.069345 0.20826 0.36029
3 0.016829 0 0 0.070749 0.2136 0.36206
4 0.046375 0 0.0059631 0.08732 0.22641 0.36263
5 0.095198 0.008725 0.033935 0.11637 0.24674 0.3747
6 0.1663 0.054327 0.077152 0.15198 0.26635 0.39127
7 0.26234 0.099882 0.13131 0.1936 0.29922 0.41248
8 0.38568 0.15954 0.1928 0.24107 0.33005 0.43049
9 0.53852 0.23411 0.25482 0.29164 0.37407 0.4593
10 0.72291 0.30704 0.31604 0.34806 0.41148 0.48371
11 0.94076 0.37567 0.37487 0.40768 0.44925 0.50972
12 1.1939 0.43849 0.42939 0.4573 0.48691 0.54333
13 1.484 0.49491 0.48129 0.50332 0.53253 0.56934
14 1.8128 0.54486 0.53013 0.54642 0.56773 0.59615
15 2.1817 0.58868 0.57335 0.58545 0.60016 0.62817
16 2.5924 0.6271 0.61254 0.62056 0.63057 0.65247
17 3.0463 0.66058 0.6468 0.65237 0.65884 0.67518
18 3.5449 0.69019 0.67699 0.68069 0.68379 0.69636
19 4.0894 0.71615 0.70375 0.7058 0.70719 0.7159
20 4.6813 0.73661 0.72701 0.72843 0.72781 0.73341
21 5.3218 0.75302 0.7481 0.74821 0.74661 0.74981
22 6.0121 0.76912 0.76622 0.76622 0.76342 0.76534
23 6.7536 0.78503 0.78285 0.78223 0.77885 0.78383
24 7.5474 0.79943 0.79703 0.79623 0.79223 0.79677
25 8.3948 0.81264 0.81024 0.8093 0.80504 0.80784
26 9.2967 0.82384 0.82198 0.82064 0.81634 0.81874
27 10.254 0.83447 0.83225 0.83065 0.82653 0.82824
28 11.269 0.84345 0.84174 0.84025 0.83545 0.83703
29 12.342 0.85185 0.85017 0.84865 0.84417 0.84497
30 13.473 0.85962 0.85746 0.85642 0.85178 0.85185
31 14.665 0.86626 0.86466 0.86306 0.85873 0.85895
32 15.918 0.87226 0.87066 0.86959 0.86504 0.86466
33 17.233 0.87786 0.87626 0.87529 0.87146 0.87061
34 18.611 0.88332 0.88182 0.88026 0.87766 0.87546
35 20.053 0.888 0.88656 0.88507 0.88267 0.88026
36 21.56 0.89187 0.89087 0.88947 0.88825 0.88483
37 23.133 0.89587 0.89484 0.89347 0.89256 0.88867
38 24.773 0.8997 0.89827 0.89727 0.89587 0.89259
39 26.481 0.903 0.90147 0.90066 0.89964 0.89587
40 28.258 0.90601 0.90467 0.90376 0.90278 0.89907
41 30.104 0.90881 0.9077 0.90628 0.90547 0.90216
42 32.021 0.91137 0.91035 0.90908 0.90838 0.90467
43 34.01 0.91377 0.91275 0.91148 0.91068 0.90708
44 36.07 0.91595 0.91468 0.91388 0.91308 0.90983
45 38.204 0.91788 0.91708 0.91617 0.91531 0.91204
46 40.412 0.91948 0.91868 0.91788 0.91708 0.91388
47 42.695 0.92168 0.92085 0.91998 0.91915 0.91604
48 45.053 0.92331 0.92251 0.92171 0.92091 0.91788
49 47.488 0.92485 0.92408 0.92331 0.92254 0.9202
50 50 0.92588 0.92555 0.92485 0.92423 0.92402 
Run the function and show summaries for savings and fraction of coh saved:
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
% mp_params('fl_crra') = 1.5;
% mp_params('fl_beta') = 0.94;
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 9;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
% call function
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 5.173849 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_36853 mean_z_0_46648 mean_z_0_59047 mean_z_0_74742 mean_z_0_94608 mean_z_1_1975 mean_z_1_5159 mean_z_1_9188 mean_z_2_4288
_____ __________ ______________ ______________ ______________ ______________ ______________ _____________ _____________ _____________ _____________
1 0 0 0 0 0 0.065547 0.16519 0.27438 0.37327 0.46242
2 0.00051272 0 0 0 0 0.066347 0.16588 0.27443 0.3738 0.4625
3 0.0029004 0 0 0 0 0.067948 0.16679 0.27523 0.37407 0.4625
4 0.0079925 0 0 0 0.00050216 0.069549 0.16975 0.27721 0.37559 0.4633
5 0.016407 0 0 0 0.005563 0.071534 0.17359 0.2802 0.37727 0.4649
6 0.028662 0 0 0 0.011926 0.080274 0.17999 0.28363 0.38054 0.4665
7 0.045213 0 0 0 0.022095 0.090757 0.18443 0.28523 0.38087 0.4665
8 0.06647 0 0 0.0043625 0.033935 0.10076 0.19478 0.29244 0.38621 0.4697
9 0.092813 0.00076108 0.0091251 0.017748 0.047979 0.11397 0.2068 0.30084 0.39207 0.47371
10 0.12459 0.02539 0.027791 0.036336 0.066347 0.13237 0.21681 0.31124 0.39938 0.47898
11 0.16214 0.049062 0.054743 0.057497 0.087289 0.14878 0.23341 0.31965 0.4076 0.48451
12 0.20576 0.080353 0.076351 0.084213 0.11115 0.16729 0.24668 0.33295 0.41295 0.48895
13 0.25576 0.11036 0.10076 0.11357 0.13677 0.1944 0.26643 0.34326 0.42327 0.49539
14 0.31242 0.14798 0.12866 0.14076 0.16483 0.21731 0.28363 0.35886 0.43195 0.50332
15 0.37601 0.17839 0.16439 0.16895 0.194 0.24107 0.30164 0.37165 0.44362 0.51204
16 0.4468 0.2098 0.20032 0.1988 0.22401 0.26563 0.32575 0.38953 0.4565 0.51772
17 0.52503 0.24246 0.23721 0.23371 0.25482 0.29153 0.34564 0.40412 0.46694 0.52773
18 0.61095 0.28123 0.27422 0.26803 0.28577 0.31725 0.36526 0.42169 0.48129 0.5355
19 0.7048 0.31861 0.30964 0.30224 0.31644 0.34326 0.38636 0.43889 0.49228 0.54649
20 0.8068 0.35352 0.34406 0.33561 0.34646 0.37247 0.41168 0.4555 0.50395 0.55454
21 0.91719 0.38727 0.37774 0.36766 0.37639 0.40048 0.43289 0.47181 0.51932 0.56671
22 1.0362 0.42001 0.40688 0.39888 0.40495 0.42569 0.4541 0.49222 0.53173 0.57495
23 1.164 0.4501 0.43289 0.42881 0.43266 0.4501 0.47451 0.50812 0.54573 0.58735
24 1.3008 0.47851 0.45746 0.45719 0.45922 0.47371 0.4948 0.52453 0.5608 0.59678
25 1.4468 0.50572 0.48514 0.48371 0.48451 0.49652 0.51432 0.54067 0.57335 0.60616
26 1.6023 0.53093 0.51118 0.50952 0.50892 0.51852 0.53322 0.56014 0.58575 0.61895
27 1.7673 0.55214 0.53571 0.53333 0.53173 0.53973 0.55258 0.57573 0.60152 0.62867
28 1.9422 0.57052 0.55854 0.55614 0.55374 0.55981 0.57335 0.59055 0.61336 0.63834
29 2.127 0.58782 0.58031 0.57735 0.57415 0.57893 0.59055 0.60534 0.62569 0.65071
30 2.3221 0.60768 0.60016 0.59758 0.59375 0.59695 0.60696 0.6194 0.6375 0.65978
31 2.5275 0.62577 0.61947 0.61496 0.61226 0.61416 0.62256 0.63297 0.65018 0.66898
32 2.7434 0.64351 0.63697 0.63101 0.62956 0.63057 0.6376 0.64617 0.66298 0.68097
33 2.97 0.65976 0.65338 0.64537 0.64591 0.64617 0.65178 0.66218 0.67378 0.68986
34 3.2075 0.67458 0.66898 0.66058 0.66124 0.66058 0.66498 0.67432 0.68445 0.69819
35 3.456 0.68919 0.68379 0.67538 0.67538 0.67458 0.67829 0.68592 0.69419 0.70716
36 3.7158 0.7022 0.69739 0.68939 0.68928 0.68779 0.69019 0.69659 0.7042 0.7154
37 3.9869 0.7146 0.7098 0.7022 0.70205 0.70039 0.7022 0.7074 0.71372 0.72609
38 4.2696 0.72668 0.7218 0.7146 0.7138 0.71209 0.713 0.71779 0.7254 0.73341
39 4.564 0.73741 0.73341 0.7262 0.7254 0.72317 0.7234 0.72748 0.73419 0.74121
40 4.8702 0.74798 0.74381 0.73711 0.73581 0.73341 0.73341 0.73661 0.74221 0.74821
41 5.1884 0.75768 0.75382 0.74727 0.74581 0.74348 0.74301 0.74551 0.75041 0.75542
42 5.5188 0.76679 0.7618 0.75684 0.75542 0.75281 0.75217 0.75382 0.75782 0.76232
43 5.8615 0.77502 0.76862 0.76542 0.76422 0.76165 0.76022 0.76182 0.76502 0.77102
44 6.2166 0.78303 0.77658 0.77422 0.77262 0.76996 0.76862 0.76941 0.77229 0.77742
45 6.5844 0.79063 0.78452 0.78223 0.78063 0.77742 0.77633 0.77661 0.77895 0.7835
46 6.9649 0.79783 0.79196 0.78983 0.78823 0.78529 0.78356 0.78543 0.78503 0.78903
47 7.3583 0.80499 0.79863 0.79695 0.79543 0.79223 0.79043 0.79223 0.79143 0.79463
48 7.7647 0.81024 0.80566 0.80343 0.80231 0.79863 0.79699 0.79863 0.79943 0.80023
49 8.1844 0.81504 0.81184 0.81003 0.80862 0.80504 0.80329 0.80471 0.80504 0.80547
50 8.6173 0.81984 0.81744 0.81584 0.81424 0.81104 0.80997 0.81024 0.81024 0.81024
51 9.0637 0.82544 0.82351 0.82144 0.82031 0.81664 0.81584 0.81573 0.81579 0.81733
52 9.5237 0.83065 0.82881 0.82664 0.82544 0.82224 0.82224 0.82064 0.82064 0.82191
53 9.9975 0.83545 0.83385 0.83217 0.83065 0.82744 0.82704 0.82544 0.82537 0.82624
54 10.485 0.84025 0.83863 0.83697 0.83545 0.83225 0.8321 0.83044 0.82985 0.83054
55 10.987 0.84494 0.84315 0.84155 0.84023 0.83703 0.83625 0.83465 0.83385 0.83457
56 11.502 0.84919 0.84705 0.84585 0.84425 0.84105 0.841 0.83915 0.83785 0.83844
57 12.032 0.85319 0.85156 0.85002 0.84785 0.84562 0.84505 0.84322 0.84185 0.84185
58 12.577 0.85666 0.85506 0.85396 0.85174 0.84945 0.84865 0.84665 0.84585 0.84574
59 13.136 0.86064 0.85906 0.85746 0.85506 0.85338 0.85265 0.85079 0.84945 0.84919
60 13.709 0.86386 0.86226 0.86122 0.85826 0.85666 0.85639 0.85425 0.85265 0.85245
61 14.298 0.86706 0.86596 0.86461 0.86138 0.86042 0.85978 0.85746 0.85586 0.85562
62 14.901 0.87052 0.86906 0.86746 0.86464 0.86372 0.86304 0.86066 0.85906 0.85826
63 15.519 0.87306 0.87215 0.87066 0.86746 0.86682 0.86615 0.86386 0.86226 0.86146
64 16.152 0.87626 0.87466 0.87378 0.87066 0.86981 0.86906 0.86698 0.86535 0.86464
65 16.801 0.87866 0.87779 0.87626 0.8736 0.87226 0.87196 0.86981 0.86812 0.8687
66 17.465 0.88163 0.88026 0.87923 0.87626 0.87538 0.87466 0.87226 0.87066 0.87126
67 18.144 0.88409 0.88267 0.88179 0.87866 0.87786 0.87706 0.87466 0.87386 0.87375
68 18.839 0.88646 0.88507 0.88422 0.88107 0.88026 0.87946 0.8776 0.87706 0.87612
69 19.55 0.88867 0.88747 0.88653 0.88347 0.88267 0.88187 0.87997 0.87946 0.87843
70 20.277 0.89087 0.88947 0.88867 0.88587 0.88507 0.88427 0.88187 0.88187 0.88026
71 21.02 0.89267 0.89187 0.89087 0.88787 0.88736 0.88659 0.88427 0.88419 0.88267
72 21.778 0.89493 0.89347 0.89267 0.89027 0.88945 0.88867 0.88656 0.88587 0.88477
73 22.553 0.89667 0.89582 0.89487 0.89187 0.89107 0.89027 0.88859 0.88825 0.88667
74 23.345 0.89827 0.89747 0.89667 0.89422 0.89336 0.89262 0.89027 0.89016 0.88862
75 24.152 0.90034 0.89907 0.89827 0.89587 0.89507 0.89427 0.89237 0.89187 0.89027
76 24.977 0.90204 0.90111 0.89987 0.89747 0.89667 0.89587 0.89416 0.89347 0.89187
77 25.818 0.90361 0.90274 0.90147 0.89907 0.89827 0.89797 0.89587 0.89507 0.89347
78 26.675 0.90515 0.90387 0.90307 0.90067 0.89987 0.89961 0.89747 0.89711 0.89551
79 27.55 0.90628 0.90547 0.90467 0.90227 0.90147 0.90117 0.89907 0.89827 0.89667
80 28.441 0.90788 0.90708 0.90547 0.90387 0.90307 0.90227 0.90066 0.89987 0.89827
81 29.35 0.90908 0.9086 0.90708 0.90547 0.90467 0.90387 0.90213 0.90147 0.89987
82 30.276 0.91068 0.90988 0.90825 0.90697 0.90623 0.90547 0.90354 0.90307 0.90146
83 31.219 0.91195 0.91121 0.90908 0.90828 0.90758 0.9069 0.90467 0.90444 0.90281
84 32.179 0.91308 0.91228 0.91035 0.90958 0.90887 0.90788 0.90623 0.90547 0.90387
85 33.157 0.91388 0.91361 0.91148 0.91068 0.90988 0.90908 0.90708 0.90703 0.9054
86 34.153 0.91543 0.91468 0.91228 0.91198 0.9113 0.91063 0.90872 0.90788 0.90628
87 35.166 0.91628 0.91548 0.9138 0.91308 0.91228 0.91148 0.90988 0.90908 0.9078
88 36.198 0.91708 0.91688 0.91468 0.91388 0.91355 0.91291 0.91068 0.91057 0.90893
89 37.247 0.91851 0.91786 0.91548 0.91527 0.91463 0.91388 0.91214 0.91148 0.90988
90 38.314 0.91946 0.91868 0.91691 0.91628 0.91548 0.91468 0.91308 0.91228 0.91108
91 39.399 0.92028 0.91948 0.91788 0.91708 0.91628 0.91604 0.91422 0.91374 0.91228
92 40.503 0.92108 0.92028 0.91868 0.91788 0.91761 0.917 0.9154 0.91468 0.91388
93 41.625 0.92188 0.92108 0.91948 0.91868 0.91851 0.91788 0.91628 0.91548 0.91527
94 42.765 0.92268 0.92188 0.92028 0.92001 0.9194 0.91868 0.91771 0.91628 0.9162
95 43.924 0.92348 0.92268 0.92108 0.92085 0.92026 0.91948 0.91868 0.91708 0.91708
96 45.102 0.92428 0.92348 0.92188 0.92168 0.92108 0.92028 0.91948 0.91838 0.91788
97 46.298 0.92508 0.92414 0.92268 0.92248 0.92188 0.92108 0.92028 0.91931 0.91868
98 47.513 0.92588 0.92469 0.92348 0.92325 0.92268 0.92188 0.92168 0.92026 0.92028
99 48.747 0.92668 0.92508 0.92428 0.92398 0.92347 0.92268 0.92248 0.92108 0.9218
100 50 0.92737 0.9258 0.92508 0.92428 0.9242 0.92348 0.92337 0.92334 0.92348 
1.6.4 Test FF_VFI_AZ_MZOOM_VEC Change Interest Rate and Discount
Show only save fraction of cash on hand:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 9;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with several different interest rates and discount factor:
% Lower Savings Incentives
mp_params('fl_beta') = 0.80;
mp_params('fl_r') = 0.01;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 37.005214 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ _______ ________ ___ _______
savefraccoh 1 1 2 6750 750 9 3468.2 0.5138 0.27192 0.52924 0 0.80103
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ _______ _______ _______ _______ _______ ________
r1 0 0 0 0 0 0 0 0.02073 0.065955
r2 0 0 0 0 0 0 0 0.02073 0.065955
r3 0 0 0 0 0 0 0 0.02073 0.065955
r4 0 0 0 0 0 0 0 0.02073 0.065955
r5 0 0 0 0 0 0 0 0.02073 0.065987
r746 0.8008 0.79843 0.7959 0.79303 0.78983 0.78663 0.78303 0.77903 0.77502
r747 0.80092 0.79855 0.79603 0.79303 0.79058 0.78713 0.78362 0.77953 0.77553
r748 0.80102 0.79863 0.79615 0.7935 0.79063 0.78729 0.78378 0.77972 0.77568
r749 0.80103 0.79863 0.79623 0.79369 0.79063 0.78743 0.78383 0.77983 0.77582
r750 0.80103 0.79904 0.79623 0.79378 0.79063 0.78743 0.78383 0.77983 0.77582
% Higher Savings Incentives
mp_params('fl_beta') = 0.95;
mp_params('fl_r') = 0.04;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 159.606266 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ _______ ________ ___ _______
savefraccoh 1 1 2 6750 750 9 4667.7 0.6915 0.26685 0.3859 0 0.92668
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ _______ ________ _______ _______ _______ _______
r1 0 0 0 0 0.0647 0.16668 0.27352 0.37327 0.4617
r2 0 0 0 0 0.0647 0.16668 0.27352 0.37327 0.4617
r3 0 0 0 0 0.064731 0.16668 0.27352 0.37327 0.4617
r4 0 0 0 0 0.064731 0.16668 0.27355 0.37327 0.4617
r5 0 0 0 0 0.064747 0.16671 0.27355 0.37327 0.4617
r746 0.92657 0.92588 0.92508 0.92428 0.92348 0.92268 0.92235 0.92188 0.92188
r747 0.92664 0.92588 0.92508 0.92428 0.92402 0.92318 0.92248 0.92188 0.92235
r748 0.92668 0.92588 0.92508 0.92478 0.92411 0.92328 0.9226 0.92188 0.9226
r749 0.92668 0.92588 0.92555 0.92488 0.9242 0.9234 0.92268 0.92254 0.92268
r750 0.92668 0.92588 0.92565 0.92497 0.92427 0.92348 0.92268 0.92268 0.922681.6.5 Test FF_VFI_AZ_MZOOM_VEC Changing Risk Aversion
Here, again, show fraction of coh saved in summary tabular form, but also show it graphically.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {'savefraccoh'};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Solve the model with different risk aversion levels, higher preferences for risk:
% Lower Risk Aversion
mp_params('fl_crra') = 0.5;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 3.409484 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 452.03 0.64575 0.28029 0.43406 0 0.90354
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _________ ________ _______ _______
r1 0 0 0 0.0047077 0.089109 0.198 0.30781
r2 0 0 0 0.0051079 0.089156 0.198 0.30793
r3 0 0 0 0.0059631 0.090679 0.1988 0.30848
r4 0 0 0 0.0079639 0.092358 0.20109 0.30964
r5 0 0 0 0.011926 0.092758 0.20413 0.31171
r96 0.90047 0.89907 0.89826 0.89727 0.89587 0.89347 0.89267
r97 0.90127 0.89987 0.89907 0.89822 0.89727 0.89477 0.89394
r98 0.90204 0.90067 0.89987 0.89907 0.89822 0.89573 0.89493
r99 0.90278 0.90147 0.90067 0.89987 0.89907 0.89667 0.89587
r100 0.90354 0.90227 0.90147 0.90067 0.89987 0.89801 0.89667
When risk aversion increases, at every state-space point, the household wants to save more.
% Higher Risk Aversion
mp_params('fl_crra') = 5;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 4.012888 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ _____ _____ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 502.6 0.718 0.25437 0.35427 0 0.93587
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ ________ _______ _______ _______ _______
r1 0 0 0.04674 0.15532 0.27563 0.39047 0.48771
r2 0 0 0.047493 0.15525 0.27563 0.39101 0.48771
r3 0 0 0.049541 0.15685 0.27693 0.39127 0.48834
r4 0 0 0.054343 0.16018 0.27883 0.39287 0.48923
r5 0 0 0.062848 0.16566 0.28272 0.39528 0.49071
r96 0.93269 0.93251 0.93189 0.93108 0.93014 0.92988 0.92968
r97 0.93349 0.93322 0.93269 0.93189 0.93107 0.93104 0.93108
r98 0.93429 0.93349 0.93347 0.93269 0.93189 0.93189 0.93269
r99 0.93507 0.93429 0.93424 0.93349 0.93331 0.93349 0.93429
r100 0.93575 0.93509 0.93507 0.93488 0.93491 0.93509 0.93587
1.6.6 Test FF_VFI_AZ_MZOOM_VEC with Higher Uncertainty
Increase the standard deviation of the Shock.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 150;
mp_params('it_z_n') = 15;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';Lower standard deviation of shock:
% Lower Risk Aversion
mp_params('fl_shk_std') = 0.10;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 16.599473 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1507.2 0.66985 0.28667 0.42796 0 0.92565
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.13838 0.18479 0.23021 0.27363 0.31725
r2 0 0 0 0 0 0.13838 0.18479 0.23027 0.27363 0.31725
r3 0 0 0 0 0 0.13894 0.18526 0.23041 0.27407 0.31725
r4 0 0 0 0 0 0.13987 0.18606 0.23121 0.27443 0.31805
r5 0 0 0 0 0 0.13998 0.18719 0.23201 0.27563 0.31885
r146 0.92348 0.92348 0.92328 0.92268 0.92268 0.92085 0.92028 0.92028 0.91948 0.91931
r147 0.9242 0.92398 0.92348 0.92348 0.92337 0.92108 0.92108 0.92097 0.92001 0.91948
r148 0.92428 0.92428 0.92428 0.92408 0.92348 0.92188 0.92171 0.92108 0.92028 0.92028
r149 0.92508 0.92497 0.92478 0.92428 0.92428 0.92241 0.92188 0.92188 0.92108 0.92107
r150 0.92565 0.92508 0.92508 0.92507 0.92485 0.92268 0.92268 0.92254 0.92238 0.92188Higher shock standard deviation: low shock high asset save more, high shock more asset save less, high shock low asset save more:
% Higher Risk Aversion
mp_params('fl_shk_std') = 0.40;
ff_vfi_az_mzoom_vec(mp_params, mp_support);
Elapsed time is 16.323916 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1685.2 0.74898 0.22908 0.30585 0 0.93669
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.52613 0.61256 0.68259 0.73901 0.78427
r2 0 0 0 0 0 0.52613 0.61256 0.68259 0.73901 0.7843
r3 0 0 0 0 0 0.52613 0.61256 0.68259 0.73901 0.7843
r4 0 0 0 0 0 0.52682 0.61256 0.68259 0.73901 0.78433
r5 0 0 0 0 0 0.52693 0.61309 0.68259 0.73901 0.78439
r146 0.92948 0.92925 0.92828 0.92805 0.92737 0.92263 0.92348 0.92577 0.92901 0.93331
r147 0.93017 0.92948 0.92868 0.92828 0.92748 0.92348 0.92428 0.92668 0.93002 0.93421
r148 0.93028 0.93005 0.92948 0.92891 0.92827 0.92428 0.92587 0.92799 0.93101 0.93507
r149 0.93091 0.93028 0.92948 0.92931 0.92828 0.92574 0.92668 0.92904 0.93189 0.93589
r150 0.93108 0.93082 0.93027 0.92948 0.92868 0.92668 0.92814 0.93008 0.93269 0.93669