1 Single Capital Model#
This Julia notebook describes the code used to solve the single agent models with one capital stock taken from Sections 4.4 and 4.5 of the paper. Suppose that the logarithm of continuation utility \(V\) evolves as:
\[
d {\widehat V}_t = {\hat \mu}_{v,t} dt + \sigma_{v,t} \cdot dB_t
\]
where \( {\hat \mu}_{v,t}\) is the local mean, \( |\sigma_{v,t}|^2\) is local variance and \(dB_t\) is a Brownian increment, such that:
\[
0 = \left({\frac \delta {1 - \rho}} \right) \left[ (C_t/V_t)^{1 - \rho}
- 1 \right] + {\hat \mu}_{v,t} + \left(\frac{1 - \gamma}{ 2 } \right) \vert \sigma_{v,t} \vert^2
\]
We presume that the stochastic component of technology growth \(Z_1\) and aggregate stochastic volatility \(Z_2\) are governed by the following processes:
\[\begin{split}
d Z_{t}^1 = - \beta_{1} Z_t^1 dt + \sqrt{Z_t^2} \sigma_1 \cdot d B_{t} \\
d Z_t^2 = - \beta_2 ( Z_t^2 - \mu_2) dt + \sqrt{Z_t^2} \sigma_2 \cdot d B_t
\end{split}\]
Capital growth is governed by:
\[
dK_t = K_t\left[ \Phi \left( \frac{I_t}{K_t} \right) + \beta_k Z_t^1 - \eta_k \right] dt + K_t \sqrt{Z_t^2} {\sigma}_k \cdot dB_t
\]
Where:
\[
\Phi(i) = {\frac 1 \phi} \log \left( 1 + \phi i \right)
\]
Output is constrained by:
\[
C_t + I_t = \alpha K_t
\]
From this we can deduce the Hamilton-Jacobi-Bellman (HJB) equation:
\[\begin{split}
0 = \max_{c+i = \alpha} \left\{ \left( \frac{\delta}{1 - \rho} \right) \left( c^{1-\rho} \exp \left[ (\rho - 1) \upsilon \right] - 1 \right) + \Phi(i) + \beta_k z_1 - \eta_k - \frac{1}{2} z_2 |\sigma_k|^2 \right. \\
\left. + \mu_z \cdot \frac {\partial \upsilon}{\partial z} + \frac {z_2}2 {\rm trace} \left\{ {\sigma_z}' \frac {\partial^2 \upsilon}{\partial z \partial z'} \sigma_z \right\} + \frac{(1 - \gamma) z_2 }{2} \left\vert\sigma_k + {\sigma_z}' \frac {\partial \upsilon}{\partial z} \right\vert^2 \right\}
\end{split}\]
Where \(c\) is the consumption-to-capital ratio and \(i\) is the investment-to-capital ratio. We solve this HJB using the algorithm described in Supplemental Appendix C.1.
1.1 Solution Overview#
The function main_one_capital solves the above HJB equation. It uses functions from utils_onecapital, which first-time users should not change. main_onecapital takes in the following inputs:
Input |
Notation in paper |
Description |
|---|---|---|
|
\(\Delta_t\) |
False-transient step-size: lower values (0.01 or 0.001) reduce likelihood of convergence issues |
|
\(\gamma\) |
Uncertainty aversion |
|
\(\delta\) |
Discount rate |
|
\(\rho\) |
Inverse of IES |
|
\(\alpha\) |
Productivity |
The user can also edit the main_onecapital Julia file to change model parameters which are held fixed in the paper, including:
Input |
Notation in paper |
Description |
Default used in paper |
|---|---|---|---|
|
\(\beta_k\) |
Loading factor on the capital stock in the long run risk process |
0.04 |
|
\(\eta\) |
Depreciation |
0.04 |
|
\(\beta_1\) |
Persistence of \(Z_1\) |
0.056 |
|
\(\beta_2\) |
Persistence of \(Z_2\) |
0.194 |
|
\(\mu_2\) |
Mean of \(Z_2\) |
\(6.3\times 10^{-6}\) |
|
\(\phi\) |
Adjustment cost |
8.0 |
|
\(\sigma_k\) |
Shock exposure for \(dK\) |
\(\sqrt{12}[0.92,0.4,0.0]\) |
|
\(\sigma_1\) |
Shock exposure for \(dZ_1\) |
\(\sqrt{12}[0.0, 5.7, 0.0]\) |
|
\(\sigma_2\) |
Shock exposure for \(dZ_2\) |
\(\sqrt{12}[0, 0, 0.00031]\) |
Finally, the following parameters govern the construction of the state space grid and convergence criteria:
Input |
Description |
|---|---|
|
Minimum and maximum values of \(Z_1\) grid; an insufficiently large grid may cause inaccuracies |
|
Number of \(Z_1\) gridpoints |
|
Minimum and maximum values of \(Z_2\) grid; an insufficiently large grid may cause inaccuracies |
|
Number of \(Z_2\) gridpoints |
|
Maximum number of value function iterations |
|
The algorithm stops when the difference between \(V_t\) and \(V_{t+1}\) is less than |
main_onecapital outputs a dictionary that contains the above parameters, as well as the following outputs:
Output |
Notation in paper |
Description |
|---|---|---|
|
\(V\) |
Value function |
|
\(\frac{C}{K}\) |
Consumption-capital ratio |
|
\(\frac{I}{K}\) |
Investment-capital ratio |
|
\(\frac{\partial{v}}{\partial{z_1}}\) |
Derivative of \(V\) with respect to \(Z^1\) |
|
\(\frac{\partial{v}}{\partial{z_2}}\) |
Derivative of \(V\) with respect to \(Z^2\) |
|
\(\mu_{\hat{K}}\) |
Drift of \(\log{K}\) |
|
Stationary density |
|
[ |
\(\sigma_k + \sigma_z' \frac{\partial{v}}{\partial{z}}\) |
Robust control term |
|
\(\mu_2\) |
Mean of \(Z_2\) |
|
\(\phi\) |
Adjustment cost |
|
\(\sigma_k\) |
Shock exposure for \(dK\) |
|
\(\sigma_1\) |
Shock exposure for \(dZ_1\) |
|
\(\sigma_2\) |
Shock exposure for \(dZ_2\) |
1.2 Quick Start#
We begin by importing required libraries:
using LinearAlgebra
using SparseArrays
using Interpolations
using SuiteSparse
using Plots
include("src/1/utils_onecapital.jl")
include("src/1/main_onecapital.jl")
Show code cell output
Activating project at `/project/lhansen/macrofinance`
Next we set parameters, including an action_name which determines where the file is saved:
alpha = 0.0922
delta = 0.01
gamma = 1.0
Delta = 1.0
rho = 1.0
action_name = "test"
Show code cell output
"test"
Now we can run our main function:
results = main(gamma, rho, Delta, delta, alpha, action_name)
Show code cell output
zmax = 5.0, zmin = -5.0, zlength = 201
ymax = 2.0e-5, ymin = 5.0e-7, ylength = 201
----------------------------------
Iteration = 1
Distance = 0.2253588241838127
v max = -5.774641175816187
v min = -6.149840658685116
----------------------------------
----------------------------------
Iteration = 2
Distance = 0.21336164298098925
v max = -5.561279532835198
v min = -6.28844158588914
----------------------------------
----------------------------------
Iteration = 3
Distance = 0.2020913901009722
v max = -5.359188142734226
v min = -6.416520157178582
----------------------------------
----------------------------------
Iteration = 4
Distance = 0.19150073759421993
v max = -5.167687405140006
v min = -6.534747094790304
----------------------------------
----------------------------------
Iteration = 5
Distance = 0.1815472483518974
v max = -4.9861401567881085
v min = -6.643751090716898
----------------------------------
----------------------------------
Iteration = 6
Distance = 0.17219150143816453
v max = -4.813948655349944
v min = -6.744121723370125
----------------------------------
----------------------------------
Iteration = 7
Distance = 0.1633965997756075
v max = -4.6505520555743365
v min = -6.836411926841702
----------------------------------
----------------------------------
Iteration = 8
Distance = 0.15512795340565422
v max = -4.495424102168682
v min = -6.921140261314288
----------------------------------
----------------------------------
Iteration = 9
Distance = 0.14735312333239836
v max = -4.348070978836284
v min = -6.998793036482345
----------------------------------
----------------------------------
Iteration = 10
Distance = 0.1400416845395922
v max = -4.208029294296692
v min = -7.069826303616259
----------------------------------
----------------------------------
Iteration = 11
Distance = 0.13316509964973378
v max = -4.074864194646958
v min = -7.134667725232818
----------------------------------
----------------------------------
Iteration = 12
Distance = 0.12669660102532898
v max = -3.948167593621629
v min = -7.193718329897394
----------------------------------
----------------------------------
Iteration = 13
Distance = 0.12061108042061086
v max = -3.827556513201018
v min = -7.247354159129413
----------------------------------
----------------------------------
Iteration = 14
Distance = 0.11488498560899219
v max = -3.712671527592026
v min = -7.295927812969371
----------------------------------
----------------------------------
Iteration = 15
Distance = 0.10949622351522592
v max = -3.6031753040768
v min = -7.339769900381859
----------------------------------
----------------------------------
Iteration = 16
Distance = 0.10442406943040172
v max = -3.4987512346463983
v min = -7.379190400302026
----------------------------------
----------------------------------
Iteration = 17
Distance = 0.09964908192078292
v max = -3.3991021527256153
v min = -7.4144799387851545
----------------------------------
----------------------------------
Iteration = 18
Distance = 0.09515302306783768
v max = -3.3039491296577777
v min = -7.44591098739077
----------------------------------
----------------------------------
Iteration = 19
Distance = 0.09091878370068551
v max = -3.213030345957092
v min = -7.473738987623915
----------------------------------
----------------------------------
Iteration = 20
Distance = 0.08693031330387813
v max = -3.126100032653214
v min = -7.498203405965285
----------------------------------
----------------------------------
Iteration = 21
Distance = 0.08317255430379822
v max = -3.042927478349416
v min = -7.519528723748951
----------------------------------
----------------------------------
Iteration = 22
Distance = 0.0796313804564881
v max = -2.9632960978929277
v min = -7.53792536588859
----------------------------------
----------------------------------
Iteration = 23
Distance = 0.07629353907716885
v max = -2.887002558815759
v min = -7.5535905722111645
----------------------------------
----------------------------------
Iteration = 24
Distance = 0.07314659686922065
v max = -2.813855961946538
v min = -7.566709214928967
----------------------------------
----------------------------------
Iteration = 25
Distance = 0.07017888912550507
v max = -2.743677072821033
v min = -7.577454565566124
----------------------------------
----------------------------------
Iteration = 26
Distance = 0.06737947209015971
v max = -2.6762976007308734
v min = -7.585989014454071
----------------------------------
----------------------------------
Iteration = 27
Distance = 0.06473807828212008
v max = -2.6115595224487533
v min = -7.592464745719907
----------------------------------
----------------------------------
Iteration = 28
Distance = 0.06224507459499318
v max = -2.54931444785376
v min = -7.597024370513247
----------------------------------
----------------------------------
Iteration = 29
Distance = 0.059891422999243105
v max = -2.489423024854517
v min = -7.599801521049017
----------------------------------
----------------------------------
Iteration = 30
Distance = 0.05766864368421043
v max = -2.4317543811703066
v min = -7.600921407885455
----------------------------------
----------------------------------
Iteration = 31
Distance = 0.05556878048748404
v max = -2.3761856006828226
v min = -7.600501342708155
----------------------------------
----------------------------------
Iteration = 32
Distance = 0.05358436846908354
v max = -2.322601232213739
v min = -7.598651228751441
----------------------------------
----------------------------------
Iteration = 33
Distance = 0.05170840349672812
v max = -2.270892828717011
v min = -7.595474020857287
----------------------------------
----------------------------------
Iteration = 34
Distance = 0.04993431371706736
v max = -2.2209585149999436
v min = -7.591066157048856
----------------------------------
----------------------------------
Iteration = 35
Distance = 0.048255932795452594
v max = -2.172702582204491
v min = -7.5855179633796945
----------------------------------
----------------------------------
Iteration = 36
Distance = 0.04666747481441469
v max = -2.1260351073900763
v min = -7.578914033711891
----------------------------------
----------------------------------
Iteration = 37
Distance = 0.04516351072766778
v max = -2.0808715966624085
v min = -7.571333585973548
----------------------------------
----------------------------------
Iteration = 38
Distance = 0.043738946273102375
v max = -2.037132650389306
v min = -7.562850796350842
----------------------------------
----------------------------------
Iteration = 39
Distance = 0.042389001254186987
v max = -1.9947436491351191
v min = -7.553535112779956
----------------------------------
----------------------------------
Iteration = 40
Distance = 0.041109190104831494
v max = -1.9536344590302877
v min = -7.543451549019663
----------------------------------
----------------------------------
Iteration = 41
Distance = 0.03989530365810845
v max = -1.9137391553721792
v min = -7.53266096050653
----------------------------------
----------------------------------
Iteration = 42
Distance = 0.038743392044169545
v max = -1.8749957633280097
v min = -7.521220303119955
----------------------------------
----------------------------------
Iteration = 43
Distance = 0.03764974864731441
v max = -1.8373460146806952
v min = -7.5091828759150285
----------------------------------
----------------------------------
Iteration = 44
Distance = 0.036610895056615744
v max = -1.8007351196240795
v min = -7.496598548815609
----------------------------------
----------------------------------
Iteration = 45
Distance = 0.03562356694842772
v max = -1.7651115526756518
v min = -7.483513976198523
----------------------------------
----------------------------------
Iteration = 46
Distance = 0.03468470084309461
v max = -1.7304268518325572
v min = -7.469972797242244
----------------------------------
----------------------------------
Iteration = 47
Distance = 0.033791421681681255
v max = -1.696635430150876
v min = -7.456015823859638
----------------------------------
----------------------------------
Iteration = 48
Distance = 0.03294103117188052
v max = -1.6636943989789954
v min = -7.441681216983331
----------------------------------
----------------------------------
Iteration = 49
Distance = 0.03213099685550613
v max = -1.6315634021234893
v min = -7.4270046519247765
----------------------------------
----------------------------------
Iteration = 50
Distance = 0.03135894185280175
v max = -1.6002044602706875
v min = -7.412019473483759
----------------------------------
----------------------------------
Iteration = 51
Distance = 0.030622635241715823
v max = -1.5695818250289717
v min = -7.3967568414425955
----------------------------------
----------------------------------
Iteration = 52
Distance = 0.029919983032741015
v max = -1.5396618419962307
v min = -7.381245867040775
----------------------------------
----------------------------------
Iteration = 53
Distance = 0.029249019702527246
v max = -1.5104128222937034
v min = -7.365513740988268
----------------------------------
----------------------------------
Iteration = 54
Distance = 0.02860790025154758
v max = -1.4818049220421559
v min = -7.349585853541605
----------------------------------
----------------------------------
Iteration = 55
Distance = 0.0279948927534579
v max = -1.453810029288698
v min = -7.333485907134195
----------------------------------
----------------------------------
Iteration = 56
Distance = 0.027408371365667206
v max = -1.4264016579230308
v min = -7.317236022021857
----------------------------------
----------------------------------
Iteration = 57
Distance = 0.026846809772549163
v max = -1.3995548481504816
v min = -7.300856835376259
----------------------------------
----------------------------------
Iteration = 58
Distance = 0.02630877503451834
v max = -1.3732460731159633
v min = -7.284367594231941
----------------------------------
----------------------------------
Iteration = 59
Distance = 0.025792921817834547
v max = -1.3474531512981287
v min = -7.267786242667573
----------------------------------
----------------------------------
Iteration = 60
Distance = 0.02529798698158081
v max = -1.322155164316548
v min = -7.251129503578547
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Iteration = 61
Distance = 0.024822784499650252
v max = -1.2973323798168976
v min = -7.234412955375855
----------------------------------
----------------------------------
Iteration = 62
Distance = 0.024366200697069074
v max = -1.2729661791198286
v min = -7.217651103925473
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----------------------------------
Iteration = 63
Distance = 0.02392718978111419
v max = -1.2490389893387144
v min = -7.2008574500230385
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----------------------------------
Iteration = 64
Distance = 0.023504769649075996
v max = -1.2255342196896384
v min = -7.184044552680418
----------------------------------
----------------------------------
Iteration = 65
Distance = 0.02309801795540123
v max = -1.2024362017342372
v min = -7.167224088483433
----------------------------------
----------------------------------
Iteration = 66
Distance = 0.02270606842229106
v max = -1.179730133311946
v min = -7.150406907264158
----------------------------------
----------------------------------
Iteration = 67
Distance = 0.022328107378614792
v max = -1.1574020259333313
v min = -7.133603084316123
----------------------------------
----------------------------------
Iteration = 68
Distance = 0.02196337051298136
v max = -1.13543865542035
v min = -7.11682196936655
----------------------------------
----------------------------------
Iteration = 69
Distance = 0.021611139827782733
v max = -1.1138275155925672
v min = -7.100072232506306
----------------------------------
----------------------------------
Iteration = 70
Distance = 0.021270740781687225
v max = -1.09255677481088
v min = -7.083361907266405
----------------------------------
----------------------------------
Iteration = 71
Distance = 0.02094153960900469
v max = -1.0716152352018753
v min = -7.066698431017431
----------------------------------
----------------------------------
Iteration = 72
Distance = 0.020622940804844392
v max = -1.050992294397031
v min = -7.05008868285802
----------------------------------
----------------------------------
Iteration = 73
Distance = 0.020314384765976135
v max = -1.0306779096310548
v min = -7.033539019147796
----------------------------------
----------------------------------
Iteration = 74
Distance = 0.020015345577565347
v max = -1.0106625640534894
v min = -7.017055306830777
----------------------------------
----------------------------------
Iteration = 75
Distance = 0.01972532893693102
v max = -0.9909372351165584
v min = -7.000642954686102
----------------------------------
----------------------------------
Iteration = 76
Distance = 0.01944387020568139
v max = -0.971493364910877
v min = -6.984306942634507
----------------------------------
----------------------------------
Iteration = 77
Distance = 0.019170532582414745
v max = -0.9523228323284623
v min = -6.96805184922102
----------------------------------
----------------------------------
Iteration = 78
Distance = 0.018904905388429705
v max = -0.9334179269400326
v min = -6.9518818773869455
----------------------------------
----------------------------------
Iteration = 79
Distance = 0.018646602459504003
v max = -0.9147713244805286
v min = -6.935800878636964
----------------------------------
----------------------------------
Iteration = 80
Distance = 0.01839526063711794
v max = -0.8963760638434106
v min = -6.919812375701071
----------------------------------
----------------------------------
Iteration = 81
Distance = 0.018150538353060797
v max = -0.8782255254903498
v min = -6.903919583784454
----------------------------------
----------------------------------
Iteration = 82
Distance = 0.017912114301523063
v max = -0.8603134111888268
v min = -6.88812543049272
----------------------------------
----------------------------------
Iteration = 83
Distance = 0.01767968619335092
v max = -0.8426337249954758
v min = -6.8724325745149315
----------------------------------
----------------------------------
Iteration = 84
Distance = 0.017452969587335554
v max = -0.8251807554081403
v min = -6.856843423141071
----------------------------------
----------------------------------
Iteration = 85
Distance = 0.0172316967937729
v max = -0.8079490586143674
v min = -6.841360148686399
----------------------------------
----------------------------------
Iteration = 86
Distance = 0.017015615845813237
v max = -0.7909334427685542
v min = -6.825984703890388
----------------------------------
----------------------------------
Iteration = 87
Distance = 0.016804489534440137
v max = -0.774128953234114
v min = -6.810718836353903
----------------------------------
----------------------------------
Iteration = 88
Distance = 0.01659809450308869
v max = -0.7575308587310253
v min = -6.795564102074019
----------------------------------
----------------------------------
Iteration = 89
Distance = 0.01639622039825306
v max = -0.7411346383327723
v min = -6.780521878132813
----------------------------------
----------------------------------
Iteration = 90
Distance = 0.016198669072584826
v max = -0.7249359692601874
v min = -6.765593374592156
----------------------------------
----------------------------------
Iteration = 91
Distance = 0.016005253837276356
v max = -0.7089307154229111
v min = -6.750779645644121
----------------------------------
----------------------------------
Iteration = 92
Distance = 0.01581579876061634
v max = -0.6931149166622947
v min = -6.736081600062917
----------------------------------
----------------------------------
Iteration = 93
Distance = 0.01563013800992663
v max = -0.6774847786523681
v min = -6.721500011001778
----------------------------------
----------------------------------
Iteration = 94
Distance = 0.01544811523414713
v max = -0.662036663418221
v min = -6.707035525175505
----------------------------------
----------------------------------
Iteration = 95
Distance = 0.015269582984566377
v max = -0.6467670804336546
v min = -6.692688671466613
----------------------------------
----------------------------------
Iteration = 96
Distance = 0.015094402171357713
v max = -0.6316726782622969
v min = -6.678459868990997
----------------------------------
----------------------------------
Iteration = 97
Distance = 0.014922441553671506
v max = -0.6167502367086254
v min = -6.6643494346566134
----------------------------------
----------------------------------
Iteration = 98
Distance = 0.014753577261219508
v max = -0.6019966594474059
v min = -6.65035759024651
----------------------------------
----------------------------------
Iteration = 99
Distance = 0.014587692345422343
v max = -0.5874089671019835
v min = -6.6364844690559694
----------------------------------
----------------------------------
Iteration = 100
Distance = 0.014424676358253175
v max = -0.5729842907437304
v min = -6.62273012211131
----------------------------------
----------------------------------
Iteration = 101
Distance = 0.01426442495710134
v max = -0.558719865786629
v min = -6.60909452399623
----------------------------------
----------------------------------
Iteration = 102
Distance = 0.01410683953401115
v max = -0.5446130262526179
v min = -6.5955775783102695
----------------------------------
----------------------------------
Iteration = 103
Distance = 0.013951826867819817
v max = -0.530661199384798
v min = -6.582179122782088
----------------------------------
----------------------------------
Iteration = 104
Distance = 0.01379929879774977
v max = -0.5168619005870483
v min = -6.56889893405903
----------------------------------
----------------------------------
Iteration = 105
Distance = 0.013649171917140768
v max = -0.5032127286699075
v min = -6.555736732193195
----------------------------------
----------------------------------
Iteration = 106
Distance = 0.013501367286081922
v max = -0.4897113613838256
v min = -6.542692184842672
----------------------------------
----------------------------------
Iteration = 107
Distance = 0.013355810161766402
v max = -0.4763555512220592
v min = -6.529764911205817
----------------------------------
----------------------------------
Iteration = 108
Distance = 0.013212429745457
v max = -0.4631431214766022
v min = -6.516954485705131
----------------------------------
----------------------------------
Iteration = 109
Distance = 0.013071158945074157
v max = -0.45007196253152804
v min = -6.504260441436194
----------------------------------
----------------------------------
Iteration = 110
Distance = 0.012931934152393909
v max = -0.4371400283791341
v min = -6.491682273396496
----------------------------------
----------------------------------
Iteration = 111
Distance = 0.012794695033988834
v max = -0.4243453333451453
v min = -6.479219441507639
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Distance = 3.257301313874805e-5
v max = 0.8377878262295676
v min = -5.222114511404493
----------------------------------
----------------------------------
Iteration = 711
Distance = 3.225050806765495e-5
v max = 0.8378200767375256
v min = -5.222082260896538
----------------------------------
----------------------------------
Iteration = 712
Distance = 3.193119607924899e-5
v max = 0.8378520079335247
v min = -5.22205032970054
----------------------------------
----------------------------------
Iteration = 713
Distance = 3.161504562676498e-5
v max = 0.8378836229790687
v min = -5.222018714655005
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----------------------------------
Iteration = 714
Distance = 3.1302025378820986e-5
v max = 0.8379149250043602
v min = -5.22198741262972
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----------------------------------
Iteration = 715
Distance = 3.0992104322447034e-5
v max = 0.8379459171086093
v min = -5.221956420525478
----------------------------------
----------------------------------
Iteration = 716
Distance = 3.068525182392534e-5
v max = 0.8379766023603397
v min = -5.221925735273743
----------------------------------
----------------------------------
Iteration = 717
Distance = 3.038143744227284e-5
v max = 0.838006983797698
v min = -5.22189535383639
----------------------------------
----------------------------------
Iteration = 718
Distance = 3.0080631117979095e-5
v max = 0.8380370644287456
v min = -5.221865273205349
----------------------------------
----------------------------------
Iteration = 719
Distance = 2.978280310017567e-5
v max = 0.8380668472317621
v min = -5.2218354904023405
----------------------------------
----------------------------------
Iteration = 720
Distance = 2.9487923872473232e-5
v max = 0.8380963351555406
v min = -5.221806002478559
----------------------------------
----------------------------------
Iteration = 721
Distance = 2.919596423289761e-5
v max = 0.8381255311196802
v min = -5.221776806514426
----------------------------------
----------------------------------
Iteration = 722
Distance = 2.8906895268576704e-5
v max = 0.838154438014866
v min = -5.2217478996192375
----------------------------------
----------------------------------
Iteration = 723
Distance = 2.8620688403258043e-5
v max = 0.8381830587031672
v min = -5.221719278930946
----------------------------------
----------------------------------
Iteration = 724
Distance = 2.8337315264082008e-5
v max = 0.8382113960183172
v min = -5.221690941615791
----------------------------------
----------------------------------
Iteration = 725
Distance = 2.8056747763738343e-5
v max = 0.8382394527659908
v min = -5.221662884868121
----------------------------------
----------------------------------
Iteration = 726
Distance = 2.7778958161750467e-5
v max = 0.838267231724083
v min = -5.221635105910031
----------------------------------
----------------------------------
Iteration = 727
Distance = 2.7503918988536213e-5
v max = 0.8382947356429873
v min = -5.2216076019911295
----------------------------------
----------------------------------
Iteration = 728
Distance = 2.7231602963695423e-5
v max = 0.8383219672458615
v min = -5.221580370388255
----------------------------------
----------------------------------
Iteration = 729
Distance = 2.696198310658815e-5
v max = 0.8383489292289072
v min = -5.221553408405211
----------------------------------
----------------------------------
Iteration = 730
Distance = 2.6695032801171692e-5
v max = 0.8383756242616257
v min = -5.2215267133724925
----------------------------------
----------------------------------
Iteration = 731
Distance = 2.6430725560189217e-5
v max = 0.8384020549870879
v min = -5.221500282647021
----------------------------------
----------------------------------
Iteration = 732
Distance = 2.6169035196588197e-5
v max = 0.8384282240221997
v min = -5.221474113611912
----------------------------------
----------------------------------
Iteration = 733
Distance = 2.5909935855494837e-5
v max = 0.8384541339579545
v min = -5.221448203676158
----------------------------------
----------------------------------
Iteration = 734
Distance = 2.5653401812597565e-5
v max = 0.838479787359693
v min = -5.2214225502744185
----------------------------------
----------------------------------
Iteration = 735
Distance = 2.5399407740600566e-5
v max = 0.8385051867673537
v min = -5.221397150866768
----------------------------------
----------------------------------
Iteration = 736
Distance = 2.514792844365843e-5
v max = 0.8385303346957298
v min = -5.221372002938393
----------------------------------
----------------------------------
Iteration = 737
Distance = 2.48989390776444e-5
v max = 0.8385552336347165
v min = -5.2213471039994
----------------------------------
----------------------------------
Iteration = 738
Distance = 2.4652414923664878e-5
v max = 0.8385798860495548
v min = -5.221322451584562
----------------------------------
----------------------------------
Iteration = 739
Distance = 2.4408331611880385e-5
v max = 0.8386042943810783
v min = -5.221298043253036
----------------------------------
----------------------------------
Iteration = 740
Distance = 2.4166664959857087e-5
v max = 0.8386284610459536
v min = -5.2212738765881515
----------------------------------
----------------------------------
Iteration = 741
Distance = 2.3927391041400625e-5
v max = 0.8386523884369181
v min = -5.221249949197183
----------------------------------
----------------------------------
Iteration = 742
Distance = 2.369048618433567e-5
v max = 0.8386760789230212
v min = -5.2212262587110825
----------------------------------
----------------------------------
Iteration = 743
Distance = 2.3455926941196026e-5
v max = 0.8386995348498577
v min = -5.221202802784246
----------------------------------
----------------------------------
Iteration = 744
Distance = 2.3223690005735875e-5
v max = 0.8387227585397939
v min = -5.22117957909431
----------------------------------
----------------------------------
Iteration = 745
Distance = 2.2993752475386486e-5
v max = 0.8387457522922052
v min = -5.221156585341897
----------------------------------
----------------------------------
Iteration = 746
Distance = 2.276609160389853e-5
v max = 0.8387685183837026
v min = -5.221133819250404
----------------------------------
----------------------------------
Iteration = 747
Distance = 2.254068475160409e-5
v max = 0.8387910590683532
v min = -5.2211112785657505
----------------------------------
----------------------------------
Iteration = 748
Distance = 2.2317509637215238e-5
v max = 0.8388133765779074
v min = -5.221088961056197
----------------------------------
----------------------------------
Iteration = 749
Distance = 2.2096544201488655e-5
v max = 0.8388354731220218
v min = -5.22106686451208
----------------------------------
----------------------------------
Iteration = 750
Distance = 2.1877766521960496e-5
v max = 0.8388573508884716
v min = -5.221044986745624
----------------------------------
----------------------------------
Iteration = 751
Distance = 2.1661154981700292e-5
v max = 0.8388790120433727
v min = -5.221023325590724
----------------------------------
----------------------------------
Iteration = 752
Distance = 2.1446688122317425e-5
v max = 0.8389004587313921
v min = -5.221001878902699
----------------------------------
----------------------------------
Iteration = 753
Distance = 2.1234344638187963e-5
v max = 0.8389216930759665
v min = -5.220980644558115
----------------------------------
----------------------------------
Iteration = 754
Distance = 2.1024103717515175e-5
v max = 0.8389427171795059
v min = -5.220959620454573
----------------------------------
----------------------------------
Iteration = 755
Distance = 2.081594418790189e-5
v max = 0.8389635331236043
v min = -5.220938804510471
----------------------------------
----------------------------------
Iteration = 756
Distance = 2.060984573448721e-5
v max = 0.8389841429692457
v min = -5.220918194664828
----------------------------------
----------------------------------
Iteration = 757
Distance = 2.0405787839017364e-5
v max = 0.8390045487570101
v min = -5.2208977888770685
----------------------------------
----------------------------------
Iteration = 758
Distance = 2.0203750354497174e-5
v max = 0.839024752507273
v min = -5.220877585126804
----------------------------------
----------------------------------
Iteration = 759
Distance = 2.000371320365346e-5
v max = 0.8390447562204034
v min = -5.220857581413673
----------------------------------
----------------------------------
Iteration = 760
Distance = 1.980565664494449e-5
v max = 0.8390645618769683
v min = -5.220837775757116
----------------------------------
----------------------------------
Iteration = 761
Distance = 1.9609561084266147e-5
v max = 0.8390841714379232
v min = -5.220818166196165
----------------------------------
----------------------------------
Iteration = 762
Distance = 1.94154069896868e-5
v max = 0.8391035868448091
v min = -5.220798750789283
----------------------------------
----------------------------------
Iteration = 763
Distance = 1.922317524716277e-5
v max = 0.8391228100199436
v min = -5.220779527614145
----------------------------------
----------------------------------
Iteration = 764
Distance = 1.9032846784838853e-5
v max = 0.839141842866611
v min = -5.220760494767475
----------------------------------
----------------------------------
Iteration = 765
Distance = 1.8844402728035448e-5
v max = 0.8391606872692523
v min = -5.220741650364834
----------------------------------
----------------------------------
Iteration = 766
Distance = 1.8657824486734143e-5
v max = 0.83917934509365
v min = -5.220722992540428
----------------------------------
----------------------------------
Iteration = 767
Distance = 1.847309358016247e-5
v max = 0.8391978181871139
v min = -5.220704519446965
----------------------------------
----------------------------------
Iteration = 768
Distance = 1.8290191605707662e-5
v max = 0.839216108378661
v min = -5.220686229255416
----------------------------------
----------------------------------
Iteration = 769
Distance = 1.8109100632379693e-5
v max = 0.8392342174792017
v min = -5.220668120154873
----------------------------------
----------------------------------
Iteration = 770
Distance = 1.7929802616833967e-5
v max = 0.839252147281718
v min = -5.220650190352358
----------------------------------
----------------------------------
Iteration = 771
Distance = 1.7752279788396663e-5
v max = 0.8392698995614386
v min = -5.220632438072633
----------------------------------
----------------------------------
Iteration = 772
Distance = 1.75765146792628e-5
v max = 0.8392874760760134
v min = -5.220614861558057
----------------------------------
----------------------------------
Iteration = 773
Distance = 1.740248976478398e-5
v max = 0.8393048785656891
v min = -5.220597459068377
----------------------------------
----------------------------------
Iteration = 774
Distance = 1.723018788624131e-5
v max = 0.8393221087534869
v min = -5.220580228880578
----------------------------------
----------------------------------
Iteration = 775
Distance = 1.7059591968404675e-5
v max = 0.8393391683453688
v min = -5.220563169288687
----------------------------------
----------------------------------
Iteration = 776
Distance = 1.6890685094139712e-5
v max = 0.8393560590303996
v min = -5.220546278603661
----------------------------------
----------------------------------
Iteration = 777
Distance = 1.6723450609212875e-5
v max = 0.8393727824809251
v min = -5.220529555153137
----------------------------------
----------------------------------
Iteration = 778
Distance = 1.6557871944655744e-5
v max = 0.8393893403527319
v min = -5.220512997281333
----------------------------------
----------------------------------
Iteration = 779
Distance = 1.6393932606106887e-5
v max = 0.8394057342852139
v min = -5.220496603348854
----------------------------------
----------------------------------
Iteration = 780
Distance = 1.6231616418505013e-5
v max = 0.8394219659015334
v min = -5.220480371732542
----------------------------------
----------------------------------
Iteration = 781
Distance = 1.6070907313370242e-5
v max = 0.8394380368087796
v min = -5.2204643008252996
----------------------------------
----------------------------------
Iteration = 782
Distance = 1.591178942605964e-5
v max = 0.8394539485981338
v min = -5.22044838903595
----------------------------------
----------------------------------
Iteration = 783
Distance = 1.5754246965649088e-5
v max = 0.8394697028450182
v min = -5.220432634789064
----------------------------------
----------------------------------
Iteration = 784
Distance = 1.5598264352156832e-5
v max = 0.8394853011092607
v min = -5.220417036524817
----------------------------------
----------------------------------
Iteration = 785
Distance = 1.544382608109629e-5
v max = 0.8395007449352421
v min = -5.220401592698835
----------------------------------
----------------------------------
Iteration = 786
Distance = 1.529091687846318e-5
v max = 0.8395160358520571
v min = -5.220386301782025
----------------------------------
----------------------------------
Iteration = 787
Distance = 1.513952166920518e-5
v max = 0.8395311753736553
v min = -5.220371162260427
----------------------------------
----------------------------------
Iteration = 788
Distance = 1.4989625412020757e-5
v max = 0.8395461649989999
v min = -5.2203561726350785
----------------------------------
----------------------------------
Iteration = 789
Distance = 1.4841213298311118e-5
v max = 0.8395610062122121
v min = -5.220341331421864
----------------------------------
----------------------------------
Iteration = 790
Distance = 1.4694270596748993e-5
v max = 0.8395757004827201
v min = -5.22032663715136
----------------------------------
----------------------------------
Iteration = 791
Distance = 1.454878276874183e-5
v max = 0.8395902492654009
v min = -5.220312088368672
----------------------------------
----------------------------------
Iteration = 792
Distance = 1.440473540359477e-5
v max = 0.8396046540007288
v min = -5.220297683633345
----------------------------------
----------------------------------
Iteration = 793
Distance = 1.4262114287788563e-5
v max = 0.839618916114914
v min = -5.220283421519162
----------------------------------
----------------------------------
Iteration = 794
Distance = 1.412090522912024e-5
v max = 0.8396330370200474
v min = -5.2202693006140235
----------------------------------
----------------------------------
Iteration = 795
Distance = 1.398109431161032e-5
v max = 0.8396470181142395
v min = -5.220255319519833
----------------------------------
----------------------------------
Iteration = 796
Distance = 1.3842667595298508e-5
v max = 0.8396608607817564
v min = -5.220241476852321
----------------------------------
----------------------------------
Iteration = 797
Distance = 1.3705611493275427e-5
v max = 0.8396745663931585
v min = -5.2202277712409195
----------------------------------
----------------------------------
Iteration = 798
Distance = 1.3569912373778692e-5
v max = 0.8396881363054386
v min = -5.220214201328649
----------------------------------
----------------------------------
Iteration = 799
Distance = 1.3435556787122493e-5
v max = 0.8397015718621503
v min = -5.220200765771934
----------------------------------
----------------------------------
Iteration = 800
Distance = 1.3302531473691204e-5
v max = 0.8397148743935487
v min = -5.220187463240527
----------------------------------
----------------------------------
Iteration = 801
Distance = 1.31708232271599e-5
v max = 0.8397280452167153
v min = -5.22017429241735
----------------------------------
----------------------------------
Iteration = 802
Distance = 1.3040419033938377e-5
v max = 0.8397410856356933
v min = -5.220161251998376
----------------------------------
----------------------------------
Iteration = 803
Distance = 1.2911305985241484e-5
v max = 0.8397539969416115
v min = -5.220148340692458
----------------------------------
----------------------------------
Iteration = 804
Distance = 1.2783471310395811e-5
v max = 0.8397667804128178
v min = -5.220135557221251
----------------------------------
----------------------------------
Iteration = 805
Distance = 1.2656902260488323e-5
v max = 0.8397794373150032
v min = -5.2201229003190655
----------------------------------
----------------------------------
Iteration = 806
Distance = 1.2531586429886943e-5
v max = 0.8397919689013256
v min = -5.220110368732742
----------------------------------
----------------------------------
Iteration = 807
Distance = 1.2407511301049112e-5
v max = 0.8398043764125344
v min = -5.220097961221536
----------------------------------
----------------------------------
Iteration = 808
Distance = 1.2284664632211673e-5
v max = 0.8398166610770982
v min = -5.220085676556978
----------------------------------
----------------------------------
Iteration = 809
Distance = 1.2163034281087448e-5
v max = 0.8398288241113206
v min = -5.22007351352275
----------------------------------
----------------------------------
Iteration = 810
Distance = 1.2042608212414763e-5
v max = 0.8398408667194602
v min = -5.220061470914619
----------------------------------
----------------------------------
Iteration = 811
Distance = 1.1923374465538927e-5
v max = 0.8398527900938572
v min = -5.22004954754023
----------------------------------
----------------------------------
Iteration = 812
Distance = 1.18053212672109e-5
v max = 0.8398645954150418
v min = -5.220037742219052
----------------------------------
----------------------------------
Iteration = 813
Distance = 1.1688436881041042e-5
v max = 0.839876283851858
v min = -5.220026053782243
----------------------------------
----------------------------------
Iteration = 814
Distance = 1.1572709856189078e-5
v max = 0.8398878565615764
v min = -5.220014481072527
----------------------------------
----------------------------------
Iteration = 815
Distance = 1.1458128510888343e-5
v max = 0.8398993146900094
v min = -5.220003022944103
----------------------------------
----------------------------------
Iteration = 816
Distance = 1.1344681697167402e-5
v max = 0.8399106593716279
v min = -5.219991678262485
----------------------------------
----------------------------------
Iteration = 817
Distance = 1.1232358126278541e-5
v max = 0.8399218917296665
v min = -5.219980445904442
----------------------------------
----------------------------------
Iteration = 818
Distance = 1.112114669155062e-5
v max = 0.8399330128762382
v min = -5.219969324757872
----------------------------------
----------------------------------
Iteration = 819
Distance = 1.101103629341793e-5
v max = 0.8399440239124479
v min = -5.2199583137216585
----------------------------------
----------------------------------
Iteration = 820
Distance = 1.090201611919639e-5
v max = 0.8399549259284976
v min = -5.2199474117056095
----------------------------------
----------------------------------
Iteration = 821
Distance = 1.0794075417930316e-5
v max = 0.8399657200037959
v min = -5.219936617630315
----------------------------------
----------------------------------
Iteration = 822
Distance = 1.0687203348958008e-5
v max = 0.8399764072070612
v min = -5.219925930427055
----------------------------------
----------------------------------
Iteration = 823
Distance = 1.0581389457087198e-5
v max = 0.8399869885964332
v min = -5.219915349037677
----------------------------------
----------------------------------
Iteration = 824
Distance = 1.0476623223620862e-5
v max = 0.8399974652195727
v min = -5.219904872414527
----------------------------------
----------------------------------
Iteration = 825
Distance = 1.0372894292842716e-5
v max = 0.8400078381137709
v min = -5.219894499520334
----------------------------------
----------------------------------
Iteration = 826
Distance = 1.0270192347228146e-5
v max = 0.8400181083060453
v min = -5.219884229328055
----------------------------------
----------------------------------
Iteration = 827
Distance = 1.0168507330821086e-5
v max = 0.8400282768132484
v min = -5.21987406082086
----------------------------------
----------------------------------
Iteration = 828
Distance = 1.006782903445469e-5
v max = 0.8400383446421616
v min = -5.219863992991953
----------------------------------
----------------------------------
Iteration = 829
Distance = 9.968147520744708e-6
v max = 0.8400483127895997
v min = -5.219854024844514
----------------------------------
Value Function Converged, Iteration = 829 with gamma=1.0 with rho=1.0 and v=-2.2204582637455093
Convergence time (minutes): 1.5750099951166667
Dict{String, Any} with 40 entries:
"ymax" => 2.0e-5
"V" => [-5.21985 -5.21984 … -5.213 -5.213; -5.1899 -5.1899 … -5.18883 -…
"dy" => 9.75e-8
"a22" => 0.194
"zz" => [-5.0 -5.0 … -5.0 -5.0; -4.95 -4.95 … -4.95 -4.95; … ; 4.95 4.95…
"times" => 94.5006
"hz" => [7.30085 7.29732 … 6.15858 6.15821; 13.2831 13.2795 … 11.9034 11…
"delta" => 0.01
"mu_y" => [1.1252e-6 1.10629e-6 … -2.63889e-6 -2.6578e-6; 1.1252e-6 1.1062…
"hy" => [0.097075 0.241849 … 0.0415655 0.014188; -0.00158519 -0.00420511…
"sigma_y" => [0.0, 0.0, 0.00107387]
"phi" => 8.0
"zmin" => -5.0
"maxit" => 50000
"Delta" => 1.0
"crit" => 1.0e-5
"d" => [0.0761111 0.0761111 … 0.0761111 0.0761111; 0.0761111 0.0761111 …
"zmax" => 5.0
"sigma_k" => [3.18697, 1.38564, 0.0]
"dz" => 0.05
"g" => [3.00353e-8 2.13511e-8 … 5.2324e-17 5.08546e-17; 1.44656e-8 1.42…
"hk" => [3.18697 3.18697 … 3.18697 3.18697; 3.18697 3.18697 … 3.18697 3.…
"alpha" => 0.0922
"mu_z" => [0.28 0.28 … 0.28 0.28; 0.2772 0.2772 … 0.2772 0.2772; … ; -0.27…
"cons" => [0.0160889 0.0160889 … 0.0160889 0.0160889; 0.0160889 0.0160889 …
⋮ => ⋮
By default, the solution is saved to the output folder. These results can be used to compute elasticities, which are described in a different section.
1.3 Model Ambiguity#
Now we abstract away from stochastic volatility (set \(\sigma_2\) to 0) and introduce model ambiguity, following Hansen and Sargent (2021). Suppose that the agent’s baseline model follows:
\[\begin{split}
d Z_{t}^1 = - \hat{\beta}_1 Z_t^1 dt + \sqrt{Z_t^2} \sigma_1 \cdot d B_{t}^S, \\
dK_t = K_t\left[ \Phi \left( \frac{I_t}{K_t} \right) + \hat{\beta}_k Z_t^1 - \hat{\eta}_k \right] dt + K_t \sqrt{Z_t^2} {\sigma}_k \cdot dB_t^S
\end{split}\]
Entertaining model ambiguity, the agents form a collection of structured models parameterized by \((\phi_1,\beta_1,\eta_k,\beta_k)\):
\[\begin{split}
d Z_{t}^1 = \phi_1 - \beta_1 Z_t^1 dt + \sqrt{Z_t^2} \sigma_1 \cdot d B_{t}^S, \\
dK_t = K_t\left[ \Phi \left( \frac{I_t}{K_t} \right) + \beta_k Z_t^1 - \eta_k \right] dt + K_t \sqrt{Z_t^2} {\sigma}_k \cdot dB_t^S
\end{split}\]
Where:
\[
dB_t^S = H_t dt + dB_t
\]
And \(H_t\) is the drift distortion implied by the parameters, with the form:
\[
H_t := \psi(Z_t^1) = \psi_0 + \psi_1 (Z_t^1 - \bar{z})
\]
The agent solves:
\[\begin{split}
0 = \max_{c+i = \alpha} \min_{h} \left\{ \left( \frac{\delta}{1 - \rho} \right) \left( c^{1-\rho} \exp \left[ (\rho - 1) \upsilon \right] - 1 \right) + \Phi(i) + \hat{\beta}_k z_1 - \hat{\eta}_k + \sigma_k \cdot h - \frac{1}{2} |\sigma_k|^2 \right. \\
\left. + (-\hat{\beta}_1 + \sigma_1 \cdot h) \cdot \frac {\partial \upsilon}{\partial z_1} + \frac {1}2 {\rm trace} \left\{ {\sigma_1}' \frac {\partial^2 \upsilon}{\partial z_1^2} \sigma_1 \right\} + \frac{(1 - \gamma) }{2} \left\vert\sigma_k + {\sigma_1}' \frac {\partial \upsilon}{\partial z_1} \right\vert^2 \right\}
\end{split}\]
Such that:
\[
(z_1-\bar{z}_1) \omega \left[\hat{\beta}_1(z_1-\bar{z}_1) + \sigma_1 \cdot h\right] + \frac{|\sigma_1|^2}{2} \omega - \frac{q^2}{2} + \frac{h \cdot h}{2} \leq 0
\]
For some relative entropy \(q\) and constant \(\omega\). Thus, in addition to the inputs in the baseline model, the user must also supply:
Input |
Description |
|---|---|
|
Relative entropy constraint |
|
|
We begin again by importing the necessary libraries and setting our inputs:
include("src/2/utils_onecapital.jl")
include("src/2/main_onecapital.jl")
alpha = 0.0922
delta = 0.01
gamma = 1.0
Delta = 1.0
rho = 1.0
q = 0.1
twoparameter = 0
action_name = "test"
phi = 8.0
Show code cell output
Activating project at `/project/lhansen/macrofinance`
"test"
We can then run the main function:
results_ambiguity = model_ambiguity(gamma, rho, Delta, delta, alpha, q, twoparameter, action_name)
Show code cell output
zmax = 5.0, zmin = -5.0, zlength = 401
----------------------------------
Iteration = 1
Distance = 0.18934075639238657
v max = -1.8144044505006767
v min = -2.1893407563923866
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Iteration = 2
Distance = 0.19747279005475749
v max = -1.6652059793580514
v min = -2.386813546447144
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Iteration = 3
Distance = 0.18756926604960356
v max = -1.5283724355162036
v min = -2.5743828124967476
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Iteration = 4
Distance = 0.17827800619270162
v max = -1.4029653154337325
v min = -2.7526608186894492
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Iteration = 5
Distance = 0.16958009302500798
v max = -1.2881276581699697
v min = -2.922240911714457
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Iteration = 6
Distance = 0.1614358518365382
v max = -1.1830544599834418
v min = -3.0836767635509954
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Iteration = 7
Distance = 0.15379332919506128
v max = -1.0869897694566042
v min = -3.2374700927460567
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Iteration = 8
Distance = 0.14660148129779094
v max = -0.9992213603830931
v min = -3.3840715740438476
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Iteration = 9
Distance = 0.13981545026214315
v max = -0.9190817267094171
v min = -3.5238870243059908
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Iteration = 10
Distance = 0.13339750693330021
v max = -0.845947626389654
v min = -3.657284531239291
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Iteration = 11
Distance = 0.12731624266563024
v max = -0.7792396684622421
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Iteration = 12
Distance = 0.12154533176677651
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v min = -3.9061461056716977
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Iteration = 13
Distance = 0.11606239713921429
v max = -0.6629957748385789
v min = -4.022208502810912
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Iteration = 14
Distance = 0.11084811312449983
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v min = -4.133056615935412
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Iteration = 15
Distance = 0.10588553227581521
v max = -0.5665330882536429
v min = -4.238942148211227
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Iteration = 16
Distance = 0.10115958805800052
v max = -0.5246884454863086
v min = -4.340101736269228
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Iteration = 17
Distance = 0.09665672742898668
v max = -0.48661802554026473
v min = -4.436758463698214
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Iteration = 18
Distance = 0.09236463726400146
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Iteration = 19
Distance = 0.08827203825973928
v max = -0.4205262236270087
v min = -4.617395139221955
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Iteration = 20
Distance = 0.08436852751796664
v max = -0.39193415401687687
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Iteration = 21
Distance = 0.08064445650900964
v max = -0.36597085074446517
v min = -4.782408123248931
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Iteration = 22
Distance = 0.07709083500206582
v max = -0.34240813212549637
v min = -4.859498958250997
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Iteration = 23
Distance = 0.07369925426650425
v max = -0.3210374464435844
v min = -4.933198212517501
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Iteration = 24
Distance = 0.07046182474457474
v max = -0.3016682695728244
v min = -5.003660037262076
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Iteration = 25
Distance = 0.0673711247239348
v max = -0.28412662213509515
v min = -5.071031161986011
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Iteration = 26
Distance = 0.06442015747446206
v max = -0.26825369997450565
v min = -5.135451319460473
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Iteration = 27
Distance = 0.06160231497830093
v max = -0.2539046108548898
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Iteration = 28
Distance = 0.0589113468585829
v max = -0.24094720982622347
v min = -5.255964981297357
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Iteration = 29
Distance = 0.05634133345628012
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Iteration = 30
Distance = 0.05388666225606098
v max = -0.21873627057721948
v min = -5.366192977009698
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Iteration = 31
Distance = 0.0515420070469883
v max = -0.20927292726290453
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Iteration = 32
Distance = 0.049302309341291206
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Iteration = 33
Distance = 0.04716276167765532
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Iteration = 34
Distance = 0.045118792513393124
v max = -0.18638048446871552
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Iteration = 35
Distance = 0.04316605246929495
v max = -0.18032982539029282
v min = -5.602484900058321
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Iteration = 36
Distance = 0.04130040173671645
v max = -0.17495969128566
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Iteration = 37
Distance = 0.039517898491985726
v max = -0.1702130800348547
v min = -5.683303200287023
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Iteration = 38
Distance = 0.037814788190901005
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v min = -5.721117988477924
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Iteration = 39
Distance = 0.03618749363790652
v max = -0.16238731728503608
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Iteration = 40
Distance = 0.034632605742026534
v max = -0.15921768278189827
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Iteration = 41
Distance = 0.033146874885452426
v max = -0.1564897795552114
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Iteration = 42
Distance = 0.03172720284199215
v max = -0.1541676451023266
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Iteration = 43
Distance = 0.03037063519174854
v max = -0.15221841178518988
v min = -5.88718280077705
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Iteration = 44
Distance = 0.029074354185895324
v max = -0.15061201431470791
v min = -5.9162571549629455
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Iteration = 45
Distance = 0.02783567202178805
v max = -0.1493209258628858
v min = -5.9440928269847335
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Iteration = 46
Distance = 0.026652024493738935
v max = -0.14831992033836486
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Iteration = 47
Distance = 0.02552096498926648
v max = -0.1475858584431993
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Iteration = 48
Distance = 0.024440158804432954
v max = -0.14709749524270443
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Iteration = 49
Distance = 0.023407377754874226
v max = -0.14683530711878007
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Iteration = 50
Distance = 0.022420495062170254
v max = -0.1467813361320693
v min = -6.066533848089216
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Iteration = 51
Distance = 0.021477480497254398
v max = -0.14691904998169716
v min = -6.088011328586471
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Iteration = 52
Distance = 0.020576395764861033
v max = -0.14723321591623828
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Iteration = 53
Distance = 0.019715390114605214
v max = -0.147709787110491
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Iteration = 54
Distance = 0.018892696165884182
v max = -0.14833580017563494
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Iteration = 55
Distance = 0.01810662593512724
v max = -0.1490992826128264
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Iteration = 56
Distance = 0.01735556705495611
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Iteration = 57
Distance = 0.016637979175886386
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Iteration = 58
Distance = 0.015952390541948525
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Iteration = 59
Distance = 0.0152973947323094
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Iteration = 60
Distance = 0.014671647561618428
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Iteration = 61
Distance = 0.01407386413213274
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Iteration = 62
Distance = 0.01350281603123804
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Iteration = 63
Distance = 0.012957328668169232
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Iteration = 64
Distance = 0.01243627874410791
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Iteration = 65
Distance = 0.011938591849905755
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Iteration = 66
Distance = 0.0114632401860284
v max = -0.16392201177827007
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Iteration = 67
Distance = 0.011009240399296516
v max = -0.16566797163453695
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Iteration = 68
Distance = 0.010575651531191887
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Iteration = 69
Distance = 0.010161573072672958
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Iteration = 70
Distance = 0.009766143120363502
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Iteration = 71
Distance = 0.009388536629261424
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Iteration = 72
Distance = 0.009027963757086255
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Iteration = 73
Distance = 0.008683668295579317
v max = -0.17690354641545789
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Iteration = 74
Distance = 0.008354926184074962
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Iteration = 75
Distance = 0.008041044100872696
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Iteration = 76
Distance = 0.00774135812803145
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Iteration = 77
Distance = 0.007455232485352958
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Iteration = 78
Distance = 0.007182058329433083
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Iteration = 79
Distance = 0.0069212526139308395
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Iteration = 80
Distance = 0.006672257007163118
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Iteration = 81
Distance = 0.006434536863584306
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Iteration = 82
Distance = 0.006207580245634503
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Iteration = 83
Distance = 0.005990896992762451
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Iteration = 84
Distance = 0.0057840178345704985
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Iteration = 85
Distance = 0.005586493545238547
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Iteration = 86
Distance = 0.005397894136527803
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Iteration = 87
Distance = 0.005217808086880105
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Iteration = 88
Distance = 0.005045841604274237
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Iteration = 89
Distance = 0.004881617920718284
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Iteration = 90
Distance = 0.004724776616374271
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Iteration = 91
Distance = 0.0045749729714374965
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Iteration = 92
Distance = 0.004431877344234003
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Iteration = 93
Distance = 0.004295174573820226
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Iteration = 94
Distance = 0.00416456340579785
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Iteration = 95
Distance = 0.0040397559400222605
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Iteration = 96
Distance = 0.003920477099059738
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Iteration = 97
Distance = 0.003806464116273389
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Iteration = 98
Distance = 0.003697466042653197
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Iteration = 99
Distance = 0.003593243271435398
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Iteration = 100
Distance = 0.0034935670797677787
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Iteration = 101
Distance = 0.0033982191866650524
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Iteration = 102
Distance = 0.003306991326600617
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Iteration = 103
Distance = 0.0032196848381174092
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Iteration = 104
Distance = 0.003136110266936498
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Iteration = 105
Distance = 0.0030560869829887594
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Iteration = 106
Distance = 0.0029794428109752857
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Iteration = 107
Distance = 0.0029060136739706977
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Iteration = 108
Distance = 0.0028356432496492445
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Iteration = 109
Distance = 0.0027681826387873087
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Iteration = 110
Distance = 0.002703490045695034
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Iteration = 111
Distance = 0.0026414304701534164
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Iteration = 112
Distance = 0.0025818754106028408
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Iteration = 113
Distance = 0.002524702578273086
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Iteration = 114
Distance = 0.002469795621881765
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Iteration = 115
Distance = 0.0024170438626960333
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Iteration = 116
Distance = 0.0023663420395845236
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Iteration = 117
Distance = 0.0023175900638792157
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Iteration = 118
Distance = 0.0022706927836591007
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Iteration = 119
Distance = 0.0022255597573188624
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Iteration = 120
Distance = 0.0021821050360086858
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Iteration = 121
Distance = 0.002140246954832392
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Iteration = 122
Distance = 0.0020999079324406367
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Iteration = 123
Distance = 0.0020610142788326513
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Iteration = 124
Distance = 0.002023496011060999
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Iteration = 125
Distance = 0.0019872866766261765
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Iteration = 126
Distance = 0.0019523231843319167
v max = -0.27777799366453754
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Iteration = 127
Distance = 0.001918545642296543
v max = -0.2793659299949004
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Iteration = 128
Distance = 0.0018858972029942578
v max = -0.28093972442029047
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Iteration = 129
Distance = 0.0018543239149275692
v max = -0.2824994265478726
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Iteration = 130
Distance = 0.0018237745808677985
v max = -0.2840450906898101
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Iteration = 131
Distance = 0.0017942006223536922
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Iteration = 132
Distance = 0.0017655559501932316
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Iteration = 133
Distance = 0.0017377968408460731
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Iteration = 134
Distance = 0.0017108818183722008
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Iteration = 135
Distance = 0.0016847715418020215
v max = -0.2915650298306881
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Iteration = 136
Distance = 0.0016594286977129613
v max = -0.2930278288616106
v min = -6.622265921085561
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Iteration = 137
Distance = 0.001634817897764762
v max = -0.29447707514551524
v min = -6.623900738983326
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Iteration = 138
Distance = 0.0016109055810575867
v max = -0.2959128497458287
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Iteration = 577
Distance = 1.841987183670568e-5
v max = -0.439114531786653
v min = -6.771637184925924
----------------------------------
----------------------------------
Iteration = 578
Distance = 1.8237496868245273e-5
v max = -0.4391327692835172
v min = -6.771655422422791
----------------------------------
----------------------------------
Iteration = 579
Distance = 1.805692759226929e-5
v max = -0.4391508262111061
v min = -6.771673479350381
----------------------------------
----------------------------------
Iteration = 580
Distance = 1.78781461315225e-5
v max = -0.43916870435723354
v min = -6.771691357496509
----------------------------------
----------------------------------
Iteration = 581
Distance = 1.770113478194446e-5
v max = -0.439186405492013
v min = -6.771709058631288
----------------------------------
----------------------------------
Iteration = 582
Distance = 1.752587602332767e-5
v max = -0.4392039313680325
v min = -6.771726584507306
----------------------------------
----------------------------------
Iteration = 583
Distance = 1.7352352498001267e-5
v max = -0.43922128372052666
v min = -6.771743936859802
----------------------------------
----------------------------------
Iteration = 584
Distance = 1.7180547025930082e-5
v max = -0.43923846426755037
v min = -6.771761117406825
----------------------------------
----------------------------------
Iteration = 585
Distance = 1.701044260205009e-5
v max = -0.4392554747101482
v min = -6.771778127849425
----------------------------------
----------------------------------
Iteration = 586
Distance = 1.684202237850485e-5
v max = -0.43927231673252254
v min = -6.7717949698718
----------------------------------
----------------------------------
Iteration = 587
Distance = 1.6675269684185423e-5
v max = -0.4392889920022002
v min = -6.7718116451414785
----------------------------------
----------------------------------
Iteration = 588
Distance = 1.6510168001637737e-5
v max = -0.4393055021701978
v min = -6.7718281553094775
----------------------------------
----------------------------------
Iteration = 589
Distance = 1.634670099370794e-5
v max = -0.43932184887118564
v min = -6.771844502010467
----------------------------------
----------------------------------
Iteration = 590
Distance = 1.6184852467127087e-5
v max = -0.4393380337236487
v min = -6.77186068686293
----------------------------------
----------------------------------
Iteration = 591
Distance = 1.6024606401821018e-5
v max = -0.43935405833004787
v min = -6.771876711469331
----------------------------------
----------------------------------
Iteration = 592
Distance = 1.586594693492316e-5
v max = -0.43936992427697763
v min = -6.771892577416264
----------------------------------
----------------------------------
Iteration = 593
Distance = 1.5708858348340016e-5
v max = -0.4393856331353245
v min = -6.771908286274609
----------------------------------
----------------------------------
Iteration = 594
Distance = 1.5553325098061066e-5
v max = -0.43940118646042026
v min = -6.771923839599705
----------------------------------
----------------------------------
Iteration = 595
Distance = 1.5399331780407977e-5
v max = -0.43941658579219767
v min = -6.771939238931484
----------------------------------
----------------------------------
Iteration = 596
Distance = 1.5246863148909995e-5
v max = -0.4394318326553433
v min = -6.771954485794628
----------------------------------
----------------------------------
Iteration = 597
Distance = 1.50959041080867e-5
v max = -0.4394469285594481
v min = -6.771969581698735
----------------------------------
----------------------------------
Iteration = 598
Distance = 1.4946439710783466e-5
v max = -0.4394618749991566
v min = -6.771984528138442
----------------------------------
----------------------------------
Iteration = 599
Distance = 1.4798455159059642e-5
v max = -0.43947667345431324
v min = -6.7719993265935985
----------------------------------
----------------------------------
Iteration = 600
Distance = 1.4651935802412197e-5
v max = -0.43949132539011165
v min = -6.772013978529399
----------------------------------
----------------------------------
Iteration = 601
Distance = 1.4506867129782108e-5
v max = -0.43950583225723877
v min = -6.7720284853965245
----------------------------------
----------------------------------
Iteration = 602
Distance = 1.4363234781100687e-5
v max = -0.43952019549201815
v min = -6.772042848631301
----------------------------------
----------------------------------
Iteration = 603
Distance = 1.4221024537519611e-5
v max = -0.4395344165165524
v min = -6.772057069655837
----------------------------------
----------------------------------
Iteration = 604
Distance = 1.4080222314305502e-5
v max = -0.4395484967388634
v min = -6.77207114987815
----------------------------------
----------------------------------
Iteration = 605
Distance = 1.3940814171498062e-5
v max = -0.43956243755303315
v min = -6.772085090692318
----------------------------------
----------------------------------
Iteration = 606
Distance = 1.3802786308581005e-5
v max = -0.4395762403393394
v min = -6.772098893478627
----------------------------------
----------------------------------
Iteration = 607
Distance = 1.366612505826481e-5
v max = -0.43958990646439483
v min = -6.772112559603681
----------------------------------
----------------------------------
Iteration = 608
Distance = 1.3530816890039432e-5
v max = -0.43960343728128154
v min = -6.772126090420566
----------------------------------
----------------------------------
Iteration = 609
Distance = 1.3396848405733408e-5
v max = -0.43961683412968394
v min = -6.77213948726897
----------------------------------
----------------------------------
Iteration = 610
Distance = 1.3264206342178397e-5
v max = -0.43963009833602334
v min = -6.772152751475312
----------------------------------
----------------------------------
Iteration = 611
Distance = 1.3132877567656465e-5
v max = -0.4396432312135868
v min = -6.7721658843528765
----------------------------------
----------------------------------
Iteration = 612
Distance = 1.3002849074794653e-5
v max = -0.43965623406265975
v min = -6.772178887201948
----------------------------------
----------------------------------
Iteration = 613
Distance = 1.2874107996552198e-5
v max = -0.4396691081706528
v min = -6.772191761309943
----------------------------------
----------------------------------
Iteration = 614
Distance = 1.2746641582239704e-5
v max = -0.4396818548122299
v min = -6.77220450795152
----------------------------------
----------------------------------
Iteration = 615
Distance = 1.2620437209065472e-5
v max = -0.43969447524943484
v min = -6.7722171283887285
----------------------------------
----------------------------------
Iteration = 616
Distance = 1.2495482383911849e-5
v max = -0.43970697073181614
v min = -6.772229623871109
----------------------------------
----------------------------------
Iteration = 617
Distance = 1.2371764738894342e-5
v max = -0.43971934249655
v min = -6.7722419956358415
----------------------------------
----------------------------------
Iteration = 618
Distance = 1.2249272018927115e-5
v max = -0.43973159176856413
v min = -6.7722542449078595
----------------------------------
----------------------------------
Iteration = 619
Distance = 1.212799209504567e-5
v max = -0.43974371976065774
v min = -6.772266372899952
----------------------------------
----------------------------------
Iteration = 620
Distance = 1.2007912967071377e-5
v max = -0.4397557276736215
v min = -6.7722783808129146
----------------------------------
----------------------------------
Iteration = 621
Distance = 1.1889022738742483e-5
v max = -0.4397676166963576
v min = -6.772290269835648
----------------------------------
----------------------------------
Iteration = 622
Distance = 1.1771309642583105e-5
v max = -0.4397793880059973
v min = -6.772302041145284
----------------------------------
----------------------------------
Iteration = 623
Distance = 1.1654762023027843e-5
v max = -0.4397910427680167
v min = -6.7723136959073065
----------------------------------
----------------------------------
Iteration = 624
Distance = 1.1539368339086309e-5
v max = -0.4398025821363528
v min = -6.772325235275643
----------------------------------
----------------------------------
Iteration = 625
Distance = 1.1425117166119492e-5
v max = -0.4398140072535172
v min = -6.772336660392806
----------------------------------
----------------------------------
Iteration = 626
Distance = 1.1311997194063395e-5
v max = -0.4398253192507101
v min = -6.7723479723899995
----------------------------------
----------------------------------
Iteration = 627
Distance = 1.1199997222988145e-5
v max = -0.43983651924793105
v min = -6.772359172387219
----------------------------------
----------------------------------
Iteration = 628
Distance = 1.1089106164874352e-5
v max = -0.4398476083540901
v min = -6.7723702614933785
----------------------------------
----------------------------------
Iteration = 629
Distance = 1.0979313032066784e-5
v max = -0.43985858766711866
v min = -6.772381240806405
----------------------------------
----------------------------------
Iteration = 630
Distance = 1.0870606961255191e-5
v max = -0.4398694582740774
v min = -6.7723921114133665
----------------------------------
----------------------------------
Iteration = 631
Distance = 1.0762977188605305e-5
v max = -0.4398802212512639
v min = -6.772402874390554
----------------------------------
----------------------------------
Iteration = 632
Distance = 1.0656413059528802e-5
v max = -0.4398908776643202
v min = -6.772413530803609
----------------------------------
----------------------------------
Iteration = 633
Distance = 1.05509040206897e-5
v max = -0.4399014285683363
v min = -6.772424081707629
----------------------------------
----------------------------------
Iteration = 634
Distance = 1.0446439621780712e-5
v max = -0.43991187500795603
v min = -6.772434528147249
----------------------------------
----------------------------------
Iteration = 635
Distance = 1.0343009527957747e-5
v max = -0.43992221801748066
v min = -6.772444871156775
----------------------------------
----------------------------------
Iteration = 636
Distance = 1.0240603492306377e-5
v max = -0.4399324586209704
v min = -6.772455111760265
----------------------------------
----------------------------------
Iteration = 637
Distance = 1.0139211378934476e-5
v max = -0.4399425978323463
v min = -6.772465250971641
----------------------------------
----------------------------------
Iteration = 638
Distance = 1.003882314698501e-5
v max = -0.43995263665549095
v min = -6.772475289794787
----------------------------------
----------------------------------
Iteration = 639
Distance = 9.939428857297372e-6
v max = -0.4399625760843472
v min = -6.772485229223643
----------------------------------
Value Function Converged, Iteration = 639 with gamma=1.0 with rho=1.0 and v=-2.526997294628275
Convegence time (minutes): 0.01894326488333333