renou_dp.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Preliminary Code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import fminbound\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "grid_size = 100\n",
    "grid = np.linspace(0, 1, grid_size)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bellman_operator(v, delta, compute_policy=0):\n",
    "\n",
    "    # === Apply linear interpolation to v === #\n",
    "    v_func = lambda x: np.interp(x, grid, v)\n",
    "    \n",
    "    Tv = np.empty_like(v)\n",
    "\n",
    "    if compute_policy:\n",
    "        σ = np.empty_like(v)\n",
    "\n",
    "    for i, t in enumerate(grid):\n",
    "        \n",
    "        def objective(r):\n",
    "            return - r * t - delta * (1 - r) * v_func(r)\n",
    "\n",
    "        #import pdb; pdb.set_trace()\n",
    "        c_star = fminbound(objective, 0, t)\n",
    "        \n",
    "        if compute_policy:\n",
    "            σ[i] = c_star\n",
    "            \n",
    "        Tv[i] = -objective(c_star)\n",
    "            \n",
    "\n",
    "    if compute_policy:\n",
    "        return Tv, σ\n",
    "    else:\n",
    "        return Tv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "def solve(delta=0.9, tol=1e-5, max_iter=500):\n",
    "\n",
    "    v = np.ones_like(grid)  # Set initial condition\n",
    "    error = tol + 1\n",
    "    i = 0\n",
    "\n",
    "    # Iterate to find solution\n",
    "    while error > tol and i < max_iter:\n",
    "        \n",
    "        v_new = bellman_operator(v, delta)\n",
    "        error = np.max(np.abs(v_new - v))\n",
    "        print(f'i = {i}, error = {error}')\n",
    "        v = v_new\n",
    "        i += 1\n",
    "\n",
    "    return v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Examples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First let's solve the model with low $\\delta$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "i = 0, error = 0.7500013000232614\n",
      "i = 1, error = 0.1875\n",
      "i = 2, error = 0.046875\n",
      "i = 3, error = 0.01171875\n",
      "i = 4, error = 0.0029296875\n",
      "i = 5, error = 0.000732421875\n",
      "i = 6, error = 0.00018310546875\n",
      "i = 7, error = 4.57763671875e-05\n",
      "i = 8, error = 1.1444091796875e-05\n",
      "i = 9, error = 2.86102294921875e-06\n"
     ]
    }
   ],
   "source": [
    "v = solve(delta=0.25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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jTK4xJs8Yc+8JjrnWGJNtjNlljPmzZ8sUERFfq2po5Ud/28E720sJCjR8a/ZI7pqbrtAhp+WUPR7GmEDg18ClQDGw2Rizwlqb3eWYdOA+4AJr7WFjTP/uKlhERLxvR3Etj72bQ01TOwlRIfxo/hgykqN9XZb4IXcutUwD8qy1+wGMMa8Ai4DsLsfcBvzaWnsYwFpb4elCRUTE+6y1vLHtIMs+zcdpYWJKLP99+WhiI4J9XZr4KXeCx2CgqMv9YmD6McdkABhjPgUCgQettf849kTGmCXAEoDU1NQzqVdERLykqc3Brz7cy/q8agCuOTeFG88bSmCApsrKmXMneBzvHWaPc550YBaQAnxijBlvra056pusfQF4ASAzM/PYc4iISA9RWN3ET1fu5mBNM+HBgXz70nRmjEj0dVnSC7gTPIqBIV3upwAlxzlmo7W2HThgjMnFFUQ2e6RKERHxmrW5FTy7Oo9Wh5PUhAh+NH8Mg+PCfV2W9BLuBI/NQLoxZhhwELge+Ooxx7wB3AAsM8Yk4rr0st+ThYqISPc6dqrs7FFJ3DFbS5+LZ50yeFhrHcaYO4F3cY3fWGqt3WWMeRjIstauOPLcZcaYbKAD+IG1tro7CxcREc+pqG/h56ty2VNeT1CgYclFw5k3foCWPhePM9b6ZqhFZmamzcrK8slri4jIv20pOMQT7+6hodVBUnQo914xWlNl5bQZY7ZYazNPdZxWLhUR6aOcTsufNxXyalYR1sK5Q+P57mUZxIRpqqx0HwUPEZE+qKapjSfey+VfRbUEGLjxvKFcc24KAZoqK91MwUNEpI/ZebCWx97N5XBjG7HhwXz/8lFMGqJdZcU7FDxERPoIp9Py960H+cMG1yqkYwfG8IN5o0iM0q6y4j0KHiIifUB9Szu/fH8vm/MPAfDlKYO56fw0rUIqXqfgISLSy+WU1fHYP3KprG8lKjSI71yawbRh/XxdlvRRCh4iIr2UtZY3t5Xw0vp8nE5LenIUP5w3muSYMF+XJn2YgoeISC9U39LOUx/sZdMB16WVRZMGcfOMNIIDA3xcmfR1Ch4iIr1MTlkdj/8jl4r6ViJDA7lnbgbnj0jwdVkigIKHiEiv4XRa3th2kN9vKHBdWukfxQ+v0KUV6VkUPEREeoHa5nZ++f4ethQcBnRpRXouBQ8RET+382AtT7yXS3VDG1GhQXz7knSmD9elFemZFDxERPyU02l5NauIv2wqxGlh9IBofjBvFP2jdWlFei4FDxERP1Td0Mov3t/DjuJajIFrzk3hP6enEqRLK9LDKXiIiPiZLQWHePL9PdQ1O4iLCOa7l2YwOTXe12WJuEXBQ0TET7Q5nPxhQz5vbisBYHJqHN+9NIO4iBDfFiZyGhQ8RET8wMGaZh7/Rw77Khs7t7H/8hRtYy/+R8FDRKSHW51Tzm/X7qOl3UlyTCjfv3wUowfE+LoskTOi4CEi0kM1tTl4bu0+1uRWAnBReiLfmj2SyFB9dIv/0rtXRKQHyimr44l3cymvayU0KIBvXDyCS8b0xxhdWhH/puAhItKDOJ2W17cU8/JnBTgtDE+K5PuXjWJIvwhflybiEQoeIiI9RGV9K0++n8vOg3UAXDV5MDedN5SQIK3NIb2HgoeISA+wbm8Vv16TR0Ora22O71yawRStzSG9kIKHiIgPNbd18MLH+/lgdzkAmWnx3DM3XWtzSK+l4CEi4iO5ZfU88V4uZbUtBAca/uui4VwxfoAGkEqvpuAhIuJlHU7La102dxuW6BpAmpqgAaTS+yl4iIh4UWltM0++t4ecsnoAFk0axNfOT9MAUukzFDxERLzAWssHuyv4v4/309zeQUJUCN++JINJQ+J8XZqIVyl4iIh0s9rmdn6zJo/1+6oBuGBkIt+aPYLosGAfVybifQoeIiLdaEvBIZ76YC81Te2EBwfyzVnDmT1KK5BK36XgISLSDVraO1j66QFW7SgDYNygGL5zaQbJMWE+rkzEtxQ8REQ8LLesniffz6WkpoXAAMNN5w3lPyYP1hb2Iih4iIh4THuHk1c2F/F6VhFOC6kJEXzv0gyGJ0X5ujSRHkPBQ0TEAwqrm/jF+7nsr2zEGPiPyYO5UfusiHyBgoeIyFlwOi1v/usgf9xQQHuHJTkmlG9fksH4wbG+Lk2kR1LwEBE5Q6W1zTz1/l6yS127yV4+LplbLxxOeEigjysT6bkUPERETpO1llU7y1i67gCtDidxEcHcNSedacP6+bo0kR5PwUNE5DRU1rfy9Id72VZUA8DMjES+cfEIYrQYmIhbFDxERNzQueT5J/tpbusgOiyIO2aN5ML0RF+XJuJXFDxERE6hqqGVZ1fnsaXgMADTh/XjzjkjiYsI8XFlIv7HreBhjJkH/AoIBF601v7sBMddA7wGTLXWZnmsShERH7DWsia3guc/2k9TWwdRoUEsuXg4szKStOS5yBk6ZfAwxgQCvwYuBYqBzcaYFdba7GOOiwbuBj7rjkJFRLypqqGVX6/JIyvf1csxNa0f35o9goSoUB9XJuLf3OnxmAbkWWv3AxhjXgEWAdnHHPcT4DHg+x6tUETEiz4fy/HiJ65ejoiQQJbMHM6c0drYTcQT3Akeg4GiLveLgeldDzDGTAaGWGvfNsYoeIiIX6qsb+XZ1Xv5Z6FrxsrUtH7cMXsEierlEPEYd4LH8SK+7XzSmADgl8DiU57ImCXAEoDU1FT3KhQR6WZOp+W97DKWrsunuf3IWI6Zw5k1SmM5RDzNneBRDAzpcj8FKOlyPxoYD6w98h90ALDCGLPw2AGm1toXgBcAMjMzLSIiPlZa28wzq/PYUVwLuGas3DF7JP0iNWNFpDu4Ezw2A+nGmGHAQeB64KufP2mtrQU6J7IbY9YC39esFhHpyZxOy1vbS/jDhgLaHE5iw4NZMnM4F6UnqpdDpBudMnhYax3GmDuBd3FNp11qrd1ljHkYyLLWrujuIkVEPKmwuomnV+8lt6wegIszkrht5nBiw7X6qEh3c2sdD2vtSmDlMY/df4JjZ519WSIintfe4eS1rGJezSqiw2npFxnCHbNGMH14gq9LE+kztHKpiPQJuWX1PP3hXgoPNQEwb/wAFs9IIzJUH4Mi3qT/cSLSqzW3dfCHDfm8s6MUa2FQXBh3zUln/OBYX5cm0icpeIhIr7XpwCF+uzaPqoY2AgxcPWUwN0xPJTQo0NelifRZCh4i0uvUNLXxwsf7+WRvFQDp/aO4c85IhidF+bgyEVHwEJFew7UQWDnL1h+gsbWD0KAAbjp/KFdOHERggKbIivQECh4i0isUVjfx6zV5ZJfWATAlNY47Zo8kOSbMx5WJSFcKHiLi19ocTpZnFfHXLcV0OC1xEcHcdpEWAhPpqRQ8RMRvbS08zG/W7qOstgVwTZH92vlDiQ7TQmAiPZWCh4j4ncONbby4bj8f73ENHk3tF8Eds0cwbpCmyIr0dAoeIuI3nE7LP3aV8fv1+TS1dRASFMD1U4dw1eTBBAcG+Lo8EXGDgoeI+IW8inp+s2YfeysaADh3aDy3zxqhwaMifkbBQ0R6tIZWB3/aWMDKIyuPJkSFcNtFw5kxIkGDR0X8kIKHiPRI1lrW5lay9NMD1DS1E2Bg0eTBfHVaKuEhWnlUxF8peIhIj5Nf1chzH+1jV4lrTY4xA6O5Y9ZI0hIjfVyZiJwtBQ8R6TGa2hz8+bNC3vpXCU4LseHBLJ6RxpzR/QnQyqMivYKCh4j4nLWWtXsqeenTfA43ujZ0+9LEgfzn9FStySHSyyh4iIhP7a9s4PmP9ncudT5qQDS3zxrBCG3oJtIrKXiIiE/Ut7Tz8meFrNpRitNCXEQwN5+vyyoivZ2Ch4h4ldNpeXdXGX/cWEB9i4MAAwvPGcQN01OJCtVHkkhvp//lIuI1Ow/W8sLH+zlQ1QjA+MGxfPPi4QxN0GwVkb5CwUNEul1lfSsvfXqAT/a69lZJig7l1guHaREwkT5IwUNEuk1Lewd/++dB/vrPYtocToIDDdecO4SrpwwmLFiLgIn0RQoeIuJxn0+P/f36fKob2gC4MD2Rr89Io7/2VhHp0xQ8RMSjcsvqefGT/eSU1QMwIimS22YO15b1IgIoeIiIh1TUt/D79fl8vMc1jiMuIpivnZ/GXE2PFZEuFDxE5Kw0t3Xw+pYi/r71IO0dluBAw39MHsw15w7RZm4i8gUKHiJyRjqclvezy3n5swJqmtoBuCg9kcUaxyEiJ6HgISKnxVrLPwsPs3RdPoWHmgDISI7mvy4axpiBMT6uTkR6OgUPEXHb/soGXvo0n21FNQAkx4Ry84w0LhyZqPU4RMQtCh4ickoVdS38aWMBa/dUYi1EhgZy/dRU5k8YSEhQgK/LExE/ouAhIidU39LO61uKeetfJbR3WIICDV+aMJBrpw4hRtvVi8gZUPAQkS9odXTw9r9KeW1LEY2tHQBcnJHETecPJVkDR0XkLCh4iEinDqdldU4FL39W0Lni6ISUWG65II2R/aN9XJ2I9AYKHiKCtZYN+6v508YCig41AzAsMZKbZ6QxJTVOA0dFxGMUPET6uG1FNfxhQz57yxsA10yVm85P46KRiVpxVEQ8TsFDpI/aW17P7zfk86+iWsC1xPl1U4dw+bgBBAdqpoqIdA8FD5E+Jr+qkZc/K2Dj/kMARIQE8uUpKSycNEhb1YtIt1PwEOkjDtY08+fPCvhkbxXWQkhQAAsmDuTL56YQramxIuIlCh4ivVx5XQuvbCpidU45TgtBgYZ54wZwbeYQ4iNDfF2eiPQxCh4ivVRFfQuvZRXzXnY5TqclwMBlY5O5btoQ+kdrLQ4R8Q0FD5Fepqqhldeyinl3VxkdRwLH7NH9uX7qEAbFhfu6PBHp49wKHsaYecCvgEDgRWvtz455/rvAfwEOoBK4xVpb4OFaReQkKutbeX1LMe9ll+HosBjjWm30+mlDSImP8HV5IiKAG8HDGBMI/Bq4FCgGNhtjVlhrs7scthXItNY2GWNuBx4DruuOgkXkaJX1rby2pYj3s8s7A8dF6YlcPzWV1AQFDhHpWdzp8ZgG5Flr9wMYY14BFgGdwcNau6bL8RuBGz1ZpIh8UXldC69lFfHB7go6nAocIuIf3Akeg4GiLveLgeknOf5WYNXxnjDGLAGWAKSmprpZooh0dbCmmVc3F7E2twKnRYFDRPyKO8HjeGsm2+MeaMyNQCZw8fGet9a+ALwAkJmZedxziMjx5Vc18tqWItbtrcJp6Rw0em1misZwiIjfcCd4FANDutxPAUqOPcgYcwnwP8DF1tpWz5QnIjlldbyWVcymA66VRgMCDJeN6c9XMocwIFbTYkXEv7gTPDYD6caYYcBB4Hrgq10PMMZMBp4H5llrKzxepUgfY61lW1ENr28pZnuxay+VkKAALh+XzFWTB2sdDhHxW6cMHtZahzHmTuBdXNNpl1prdxljHgayrLUrgMeBKOC1I9tnF1prF3Zj3SK9ktNpWb+vmte3FLGvshGA8JBArpw4kIXnDCIuQiuNioh/c2sdD2vtSmDlMY/d3+X2JR6uS6RPaXM4WZ1Tzt/+eZDS2hbAtVvsgnMGMX/CQKJCtdafiPQO+jQT8aH6lnZW7ijl7e2l1DS1A5AcE8rVU1KYO6Y/oUHaLVZEehcFDxEfKK9r4c1tB3k/u5yWdicAw5MiuXpKCheOTCQw4HiTyURE/J+Ch4gX5ZTV8cbWEjbsc02JBZicGsfVU1I4JyWWI2OkRER6LQUPkW7mdFo27q/m71sPklNWD7imxM7OSOSqyYMZnhTl4wpFRLxHwUOkmzS2Ong/u5y3t5dQXuda2iYyNJB54wZw5TmDSIwK9XGFIiLep+Ah4mEHa5p5+18lfLi7gub2DgAGxIax4JxBXDommfAQDRgVkb5LwUPEA5xOy9aiw7z1r1K2FBzufHxiSiwLzxnE1LR+BGjAqIiIgofI2WhsdfBhTgXvbC+hpMa1/kZwoOHijP4snDSIYYmRPq5QRKRnUfAQOQMHqhpZuaOUtbkVndNhE6NC+NLEQVw2LpmYsGAfVygi0jMpeIi4qc3hZMP+alZuLyW7tK7z8QkpsVw5YSDThydo/Q0RkVNQ8BA5hbLaFlbtLOWD3eWL7oANAAAVH0lEQVTUNTsACA8OZM6Y/swfP5DUBG1JLyLiLgUPkeNwdDjZdOAQ7+4q45+FNZ2PpyVGcsX4Acwe1V+zU0REzoCCh0gXJTXNvLerjA9zKjr3TgkONFyUnsQVEwYwKjlaq4uKiJwFBQ/p81odHazfV80H2eVsL67tfDy1XwSXjUtmzuj+RGuwqIiIRyh4SJ9krWVfZQPvZZfzUW4lTW2uhb5CggK4KD2Ry8cNYPQA9W6IiHiagof0KYcb21i7p4IPd1dQUN3U+Xh6/yguHZvMRRlJRIXqv4WISHfRJ6z0em0OJ5vzD/Hh7gq2FBzq3BU2OiyIOaP7c8mYZNK00JeIiFcoeEiv5HRaskvrWJtbwbq8KhpbXZdSAgIM09PimTsmmcy0eIIDA3xcqYhI36LgIb1KYXUTH+2p4KM9lZ07wgIMT4pkzuj+zMroT2yEBoqKiPiKgof4vYr6Fj7ZU8VHeyo5UNXY+XhCVAizMpKYMzpZi3yJiPQQCh7ilw43trEur4pP9layu7S+8/HI0EAuGJHIxaOSGD8oVjvCioj0MAoe4jdqmtrYuL+aj/dWsfNgLfbIINHgQMP04QlcnJHElNR4QoI0bkNEpKdS8JAe7VBjG+v3VfFpXjXZJbWdM1KCAg1TUuO5KD2R6cMStHy5iIifUPCQHqestoUN+6vYsK+anLL6zp6NwABD5tA4LhyZyPThCVpvQ0TED+mTW3zOWsuBqkY27j/Exv3VRw0QDT7Ss3HByESmDetHpMKGiIhf06e4+ER7h5OdB2v57MAhNh04RGX9v6e+hgcHkpkWz/kjEsgc2k+XUUREehEFD/GamqY2NucfJiv/EFsLa2hu7+h8Li4imGlp/Th/RAITU+I0QFREpJdS8JBu43Ra9lTU88+CGrLyD7G3ouGo51MTIjhvWD+mD09gZFKUpr6KiPQBCh7iUdUNrWwrqmFLwWG2FtbQ0OrofC440DAxJY6paf2YmhZP/5gwH1YqIiK+oOAhZ6W5rYNdJbVsK6pha2ENhYeajno+OSaMzLR4pqTGMzEllrBgjdcQEenLFDzktLQ5nOwpr2dbUQ3bi2vILW/A+fniGkBYcADjB8cyOTWec4fGMzgu3IfViohIT6PgISfV5nCSW1bPjoO17CypJae0jvaOfweNAAPpyVFMHhLH5NR4Rg2I1o6vIiJyQgoecpSGVgc5pXVkl9aRXVJHbnk9ji5BA2BoQgTnpMQxMSWW8YNjtbaGiIi4TT8x+jBrLSW1LeSU1pFTVs/u0joKDzV1rhQKYAykJUYyflAMEwbHMm5QrLaVF/Gg9vZ2iouLaWlp8XUpIm4JCwsjJSWF4OAz+1mg4NGH1Da3k1dRT25ZA3sr6sktq6e+xXHUMUGBhvT+UYwdGMPYQbGMGRhNdJiChkh3KS4uJjo6mrS0NIzRlHLp2ay1VFdXU1xczLBhw87oHAoevVRdSzv7KhrYV9lIXkUDeRX1lNe1fuG4uIhgxgyMYVRyNKMHRpPeP1qLd4l4UUtLi0KH+A1jDAkJCVRWVp7xORQ8/JzTaSmvb+FAZSMHqhtdf1Y1UlH/xZARGhTAyP5RjOwfRUZyNKMGRNM/OlQfeCI+pv+D4k/O9v2q4OEnrLUcamyj6HAzBdWNFFQ3UVDdRNGhpqOWHv9caFAAwxIjGdE/ihFJrrCR2i+CQK0OKiIiPqTg0cO0tHdQWttCaU0zxTXNFB9upvhQE8WHm48bMMB1uWR4YiTDEiMZlhTFsIRIUuLDtQS5iPiVFStWkJ2dzb333nvW53rqqadYsmQJERERHqjs+BYvXsyVV17JNddcc9bn+sEPfsDKlSuZP38+jz/+uAeqg23btlFSUsL8+fMBz7bv2VDw8DKn01LT3E5FfQtltS2U17VQVttKWV0zpbUtVDe0nfB7o0KDGNIvnKEJkQxNiGBov0hS+0VolomI9DgOh4OgoNP7EbNw4UIWLlzokdd/6qmnuPHGG7s1eHjS888/T2VlJaGhoR4757Zt28jKyuoMHp5s37Ph1rvCGDMP+BUQCLxorf3ZMc+HAn8AzgWqgeustfmeLbXna+9wUtvczuHGNqoa2qhubOXQkdtVDa1U1LVS1dBKh9Oe8ByBAYYBMWEMigtnUFwYKfHhpMRHMCReAUOkt1vwzLpuOe9bd1140uevuuoqioqKaGlp4Z577mHJkiUAREVF8Y1vfIM1a9YQHx/PK6+8QlJSErNmzWLSpEls2rSJuro6li5dyrRp03jwwQcpKSkhPz+fxMREli5dyu23305WVhZBQUE8+eSTzJ49myeffJKdO3eydOlSduzYwQ033MCmTZt49dVXycrK4tlnn2Xx4sWEh4eTk5NDQUEBL730Er///e/ZsGED06dPZ9myZQDcfvvtbN68mebmZq655hoeeughnn76aUpKSpg9ezaJiYmsWbOG9957jwceeIDW1lZGjBjBSy+9RFRUVGcb7N69m5tvvplNmzYBkJ+fz8KFC9m+fTsPP/wwb731Fs3NzcyYMYPnn3/+C+Mc0tLSyMrKIjExkaysLL7//e+zdu1aGhsbueuuu9ixYwcOh4MHH3yQRYsWHfW9CxcupLGxkenTp3PfffexatWqo3pSoqKiaGhoYO3atTz44IMkJiayc+dOzj33XP70pz9hjGHz5s3cc889NDY2Ehoayvvvv8/9999Pc3Mz69at47777qO5ubmzfQsKCrjllluorKwkKSmJl156idTUVBYvXkxMTAxZWVmUlZXx2GOPeaRHp6tTBg9jTCDwa+BSoBjYbIxZYa3N7nLYrcBha+1IY8z1wM+B6zxaqRdZa2lpd9Lc3kFDi4PGNgcNra6v+hYHdc3t1LW0U9fsoLa5ndrmNg43th+1IdrJxIQHkRQVSnJsGANiwhgYG0ZyTBgDY8NJig7VOAwR8aqlS5fSr18/mpubmTp1Kl/+8pdJSEigsbGRKVOm8Itf/IKHH36Yhx56iGeffRaAxsZG1q9fz8cff8wtt9zCzp07AdiyZQvr1q0jPDycX/ziFwDs2LGDnJwcLrvsMvbs2cO3v/1tZs2axd///nceffRRnn/++eP2TBw+fJjVq1ezYsUKFixYwKeffsqLL77I1KlT2bZtG5MmTeLRRx+lX79+dHR0MHfuXLZv387dd9/Nk08+yZo1a0hMTKSqqopHHnmEDz74gMjISH7+85/z5JNPcv/993e+1pgxY2hra2P//v0MHz6c5cuXc+211wJw5513dh5700038fbbb7NgwQK32vbRRx9lzpw5LF26lJqaGqZNm8Yll1xCZGRk5zErVqwgKiqKbdu2AbBq1aoTnm/r1q3s2rWLQYMGccEFF/Dpp58ybdo0rrvuOpYvX87UqVOpq6sjIiKChx9+uDNoAJ1h7fO/09e+9jVuvvlmli5dyt13380bb7wBQGlpKevWrSMnJ4eFCxd6P3gA04A8a+1+AGPMK8AioGvwWAQ8eOT268CzxhhjrT3xr/bdoLHVwV82FWItOK3FcuRPC44OS4fTicNpcTgtbQ4nbR1O158OJ62ODlfYaOugxdHBmVQeYCAmPJj4iBASokJIjAqlX2QICZEhJEaH0j86lMSoUG2UJiLHdaqeie7y9NNP8/e//x2AoqIi9u7dS0JCAgEBAVx3net3yBtvvJGrr76683tuuOEGAGbOnEldXR01NTWA67f38HDXHk3r1q3jrrvuAmD06NEMHTqUPXv2MHHiRJYtW8bEiRP5xje+wQUXXHDcuhYsWIAxhgkTJpCcnMyECRMAGDduHPn5+UyaNIlXX32VF154AYfDQWlpKdnZ2UycOPGo82zcuJHs7OzO12lra+P888//wutde+21vPrqq9x7770sX76c5cuXA7BmzRoee+wxmpqaOHToEOPGjXM7eLz33nusWLGCJ554AnBNny4sLGTMmDFuff+xpk2bRkpKCgCTJk0iPz+f2NhYBg4cyNSpUwGIiYk55Xk2bNjA3/72N8AVpv77v/+787mrrrqKgIAAxo4dS3l5+RnVeTLuBI/BQFGX+8XA9BMdY611GGNqgQSgqutBxpglwBKA1NTUMyz5xFodTt7cVuKRc4UGBRAeEkhkSBBRYUFEhQYRGRpIVGgwseHBxIQHER0WTExYEHERIcRHBBMTFqwBnSLiV9auXcsHH3zAhg0biIiIYNasWSdcRbXr5YVjLzV8fr/rb/In+91z7969REVFUVJy4s/sz8c7BAQEHDX2ISAgAIfDwYEDB3jiiSfYvHkz8fHxLF68+Li1W2u59NJL+ctf/nLC1wK47rrr+MpXvsLVV1+NMYb09HRaWlq44447yMrKYsiQITz44IPHfY2goCCcTifAUc9ba/nrX//KqFGjTvraJzqXtZa2tn+P/evaDoGBgTgcDqy1Zz3Ftev3d32N7ug/cCd4HO9vc2wl7hyDtfYF4AWAzMxMj/9tIkICufXCYRjjasTPM0CAMQQGGIIDDYEBAQQHuO6HBgcSGhRASFAAIYGuoBEe7PpSgBCRvqC2tpb4+HgiIiLIyclh48aNnc85nU5ef/11rr/+ev785z9z4YX/7pFZvnw5s2fPZt26dcTGxhIbG/uFc8+cOZOXX36ZOXPmsGfPHgoLCxk1ahS1tbXcc889fPzxx9x55528/vrrZ9SdX1dXR2RkJLGxsZSXl7Nq1SpmzZoFQHR0NPX19SQmJnLeeefxrW99i7y8PEaOHElTUxPFxcVkZGQcdb4RI0YQGBjIT37yk86ens9DRGJiIg0NDSesNS0tjS1btnDFFVfw17/+tfPxyy+/nGeeeYZnnnkGYwxbt25l8uTJJ/17fX6ua6+9ljfffJP29vaTHj969GhKSkrYvHkzU6dOpb6+nvDw8M42OJ4ZM2bwyiuvcNNNN/Hyyy8f9W/b3dwJHsXAkC73U4BjI+rnxxQbY4KAWOCQRyo8DWHBgVw1ebC3X1ZExG/NmzeP5557jokTJzJq1CjOO++8zuciIyPZtWsX5557LrGxsZ2XHgDi4+OZMWNG5+DS47njjjv45je/yYQJEwgKCmLZsmWEhoZy++23c8cdd5CRkcHvfvc7Zs+ezcyZM0+79nPOOYfJkyczbtw4hg8fftQlmyVLlnDFFVcwcOBA1qxZw7Jly7jhhhtobXUtrvjII498IXiAq9fjBz/4AQcOHAAgLi6O2267jQkTJpCWltZ5OeNYDzzwALfeeis//elPmT793xcFfvzjH/Ptb3+biRMnYq0lLS2Nt99++6R/r9tuu41FixYxbdo05s6de1Qv0vGEhISwfPly7rrrLpqbmwkPD+eDDz5g9uzZ/OxnP2PSpEncd999R33P008/zS233MLjjz/eObjUW8ypulGOBIk9wFzgILAZ+Kq1dleXY74FTLDWfvPI4NKrrbXXnuy8mZmZNisr62zrFxHxa7t37z7j6/3d7fPZFMeaNWsWTzzxBJmZmT6oSnqC471vjTFbrLWnfFOcssfjyJiNO4F3cU2nXWqt3WWMeRjIstauAH4H/NEYk4erp+P6M/h7iIiISC/n1joe1tqVwMpjHru/y+0W4CueLU1ERHzpeL0d4BqQKnKmtA2piIiPeXnlAZGzcrbvVwUPEREfCgsLo7q6WuFD/IK1lurqasLCws74HNqrRUTEh1JSUiguLqaystLXpYi4JSwsrHMRszOh4CEi4kPBwcEMGzbM12WIeI0utYiIiIjXKHiIiIiI1yh4iIiIiNeccuXSbnthYyqBgm46fSLHbFAn3UZt7V1qb+9RW3uP2tp7urOth1prk051kM+CR3cyxmS5s2yrnD21tXepvb1Hbe09amvv6QltrUstIiIi4jUKHiIiIuI1vTV4vODrAvoQtbV3qb29R23tPWpr7/F5W/fKMR4iIiLSM/XWHg8RERHpgRQ8RERExGv8OngYY+YZY3KNMXnGmHuP83yoMWb5kec/M8akeb/K3sGNtv6uMSbbGLPdGPOhMWaoL+rsDU7V1l2Ou8YYY40xmoZ4Ftxpb2PMtUfe37uMMX/2do29hRufI6nGmDXGmK1HPkvm+6LO3sAYs9QYU2GM2XmC540x5ukj/xbbjTFTvFactdYvv4BAYB8wHAgB/gWMPeaYO4Dnjty+Hlju67r98cvNtp4NRBy5fbvauvva+shx0cDHwEYg09d1++uXm+/tdGArEH/kfn9f1+2PX2629QvA7UdujwXyfV23v34BM4EpwM4TPD8fWAUY4DzgM2/V5s89HtOAPGvtfmttG/AKsOiYYxYBvz9y+3VgrjHGeLHG3uKUbW2tXWOtbTpydyNw5nsm923uvK8BfgI8BrR4s7heyJ32vg34tbX2MIC1tsLLNfYW7rS1BWKO3I4FSrxYX69irf0YOHSSQxYBf7AuG4E4Y8xAb9Tmz8FjMFDU5X7xkceOe4y11gHUAgleqa53caetu7oVV5KW03fKtjbGTAaGWGvf9mZhvZQ77+0MIMMY86kxZqMxZp7Xqutd3GnrB4EbjTHFwErgLu+U1ied7ue6xwR540W6yfF6Lo6dG+zOMXJqbrejMeZGIBO4uFsr6r1O2tbGmADgl8BibxXUy7nz3g7CdbllFq6evE+MMeOttTXdXFtv405b3wAss9b+whhzPvDHI23t7P7y+hyf/Xz05x6PYmBIl/spfLFbrvMYY0wQrq67k3U9yfG509YYYy4B/gdYaK1t9VJtvc2p2joaGA+sNcbk47o2u0IDTM+Yu58jb1pr2621B4BcXEFETo87bX0r8CqAtXYDEIZrUzPxPLc+17uDPwePzUC6MWaYMSYE1+DRFcccswK4+cjta4DV9sioGjktp2zrI93/z+MKHboGfuZO2tbW2lprbaK1Ns1am4ZrPM1Ca22Wb8r1e+58jryBa/A0xphEXJde9nu1yt7BnbYuBOYCGGPG4AoelV6tsu9YAXztyOyW84Baa22pN17Yby+1WGsdxpg7gXdxjZZeaq3dZYx5GMiy1q4Afoerqy4PV0/H9b6r2H+52daPA1HAa0fG7xZaaxf6rGg/5WZbi4e42d7vApcZY7KBDuAH1tpq31Xtn9xs6+8B/2eM+Q6ubv/F+mXxzBhj/oLr8mDikTEzDwDBANba53CNoZkP5AFNwNe9Vpv+TUVERMRb/PlSi4iIiPgZBQ8RERHxGgUPERER8RoFDxEREfEaBQ8RERHxGgUPERER8RoFDxEREfGa/w83E6YD3DsfWQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7fe9f863c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(9, 5))\n",
    "\n",
    "ax.plot(grid, v, lw=2, alpha=0.8, label='approximate value function')\n",
    "\n",
    "ax.legend(loc='lower right')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's solve the model with high $\\delta$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "i = 0, error = 0.15000511734478406\n",
      "i = 1, error = 0.12750000000000006\n",
      "i = 2, error = 0.108375\n",
      "i = 3, error = 0.09211875000000003\n",
      "i = 4, error = 0.0783009375\n",
      "i = 5, error = 0.06655579687500002\n",
      "i = 6, error = 0.05657242734375001\n",
      "i = 7, error = 0.04808656324218746\n",
      "i = 8, error = 0.04087357875585937\n",
      "i = 9, error = 0.03474254194248047\n",
      "i = 10, error = 0.02953116065110839\n",
      "i = 11, error = 0.02510148655344213\n",
      "i = 12, error = 0.021336263570425812\n",
      "i = 13, error = 0.01813582403486194\n",
      "i = 14, error = 0.015415450429632646\n",
      "i = 15, error = 0.013103132865187744\n",
      "i = 16, error = 0.01113766293540959\n",
      "i = 17, error = 0.009467013495098146\n",
      "i = 18, error = 0.008046961470833426\n",
      "i = 19, error = 0.0068399172502084105\n",
      "i = 20, error = 0.005813929662677152\n",
      "i = 21, error = 0.004941840213275577\n",
      "i = 22, error = 0.0042005641812842415\n",
      "i = 23, error = 0.003570479554091604\n",
      "i = 24, error = 0.0030349076209778628\n",
      "i = 25, error = 0.002579671477831183\n",
      "i = 26, error = 0.002192720756156506\n",
      "i = 27, error = 0.0018638126427330311\n",
      "i = 28, error = 0.0015842407463230752\n",
      "i = 29, error = 0.001346604634374614\n",
      "i = 30, error = 0.0011446139392184218\n",
      "i = 31, error = 0.0009729218483356585\n",
      "i = 32, error = 0.00082698357108531\n",
      "i = 33, error = 0.0007029360354225134\n",
      "i = 34, error = 0.0005974956301091367\n",
      "i = 35, error = 0.000507871285592766\n",
      "i = 36, error = 0.000431690592753851\n",
      "i = 37, error = 0.0003669370038407735\n",
      "i = 38, error = 0.00031189645326465745\n",
      "i = 39, error = 0.0002651119852749588\n",
      "i = 40, error = 0.00022534518748371503\n",
      "i = 41, error = 0.00019154340936115757\n",
      "i = 42, error = 0.00016281189795698406\n",
      "i = 43, error = 0.00013839011326343645\n",
      "i = 44, error = 0.00011763159627392096\n",
      "i = 45, error = 9.998685683283281e-05\n",
      "i = 46, error = 8.498882830790787e-05\n",
      "i = 47, error = 7.224050406172167e-05\n",
      "i = 48, error = 6.140442845246343e-05\n",
      "i = 49, error = 5.2193764184593934e-05\n",
      "i = 50, error = 4.436469955690483e-05\n",
      "i = 51, error = 3.77099946233691e-05\n",
      "i = 52, error = 3.205349542986373e-05\n",
      "i = 53, error = 2.7245471115384186e-05\n",
      "i = 54, error = 2.3158650448076542e-05\n",
      "i = 55, error = 1.9684852880865063e-05\n",
      "i = 56, error = 1.6732124948735306e-05\n",
      "i = 57, error = 1.4222306206425003e-05\n",
      "i = 58, error = 1.208896027546125e-05\n",
      "i = 59, error = 1.027561623414207e-05\n",
      "i = 60, error = 8.734273799020756e-06\n"
     ]
    }
   ],
   "source": [
    "v = solve(delta=0.85)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7fe9eee898>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(9, 5))\n",
    "\n",
    "ax.plot(grid, v, lw=2, alpha=0.8, label='approximate value function')\n",
    "\n",
    "ax.legend(loc='lower right')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}