additive_fixed_costs.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 247,
   "metadata": {},
   "outputs": [],
   "source": [
    "grid_size = 350\n",
    "\n",
    "δ = 1.5\n",
    "θ = 15.0\n",
    "κ = 50.0\n",
    "\n",
    "def c(x):\n",
    "    return np.exp(θ * x) - 1 #x**(θ + 1) / (θ + 1)\n",
    "\n",
    "s_grid = np.linspace(0, 1, grid_size)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "metadata": {},
   "outputs": [],
   "source": [
    "def T(p):\n",
    "    \n",
    "    pf = lambda x: np.interp(x, s_grid, p)\n",
    "    Tp = np.empty_like(p)\n",
    "    \n",
    "    for i, s in enumerate(s_grid):\n",
    "        ell_grid = np.linspace(0, s, grid_size)\n",
    "        Tp[i] = np.min(c(ell_grid) + δ * pf(s - ell_grid) + κ * (ell_grid > 0.0))\n",
    "        \n",
    "    return Tp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p_init = c(s_grid)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(s_grid, p_init)\n",
    "Tp = T(p_init)\n",
    "ax.plot(s_grid, Tp)\n",
    "TTp = T(Tp)\n",
    "ax.plot(s_grid, TTp)\n",
    "\n",
    "plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 250,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 3264539.8986586174\n",
      "final error = 0.0\n"
     ]
    }
   ],
   "source": [
    "p = c(s_grid)\n",
    "\n",
    "tol = 1e-4\n",
    "error = tol + 1\n",
    "max_iter = 2000\n",
    "i = 0\n",
    "\n",
    "while error > tol and i < max_iter:\n",
    "    Tp = T(p)\n",
    "    error = np.max(np.abs(Tp - p))\n",
    "    \n",
    "    if i % 25 == 0:\n",
    "        print(i, error)\n",
    "    i += 1\n",
    "    p = Tp\n",
    "    \n",
    "print(f\"final error = {error}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fix, ax = plt.subplots(figsize=(8, 6))\n",
    "ax.plot(s_grid, p)\n",
    "ax.plot(s_grid, c(s_grid) + κ)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "metadata": {},
   "outputs": [],
   "source": [
    "pf = lambda x: np.interp(x, s_grid, p)\n",
    "\n",
    "def t_star(s):\n",
    "    ell_grid = np.linspace(0, s, grid_size)\n",
    "    i = np.argmin(c(ell_grid) + δ * pf(s - ell_grid) + κ * (ell_grid > 0.0))\n",
    "    return s - ell_grid[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 253,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_stages():\n",
    "    s = 1.0\n",
    "    transaction_stages = [s]\n",
    "    while s > 0:\n",
    "        s = t_star(s)\n",
    "        transaction_stages.append(s)\n",
    "    return transaction_stages\n",
    "\n",
    "def plot_prices(plottype='-', label=None):\n",
    "    plt.plot(s_grid, p, plottype, label=label)\n",
    "    transaction_stages = compute_stages()\n",
    "    for s in transaction_stages:\n",
    "        plt.axvline(x=s, c=\"0.5\")\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 254,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAD8CAYAAAB5Pm/hAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvOIA7rQAAIABJREFUeJzt3Xl8VOXZ//HPlYRAgLATQEIISBAsO0FR6ortIy5Fq1i1KlAUa926/Fqt9qmPbW2tv6e12sVKtQJaRUVUVFqrKLUuKKvsS0AggUDCFoIh61zPHzm0qQYykEkmM/m+X695zTn33JO5TgjfnNxzz7nN3RERkfiVEO0CRESkYSnoRUTinIJeRCTOKehFROKcgl5EJM4p6EVE4pyCXkQkzinoRUTinIJeRCTOJUW7AIAuXbp4ZmbmMT9vz549AHTu3DnCFcWG5n78ddH3p/70PWxY9f3+LlmyZLe7d62rX5MI+szMTBYvXnzMz5s+fToAkyZNimxBMaK5H39d9P2pP30PG1Z9v79mtjWcfhq6ERGJcwp6EZE4p6AXEYlzCnoRkTinoBcRiXMKehGROKegFxGJcwp6EZEoqKwKsXVPCWWVoQZ/rTqD3sxOMrPlNW4HzOzbZtbJzN4ws43Bfcegv5nZw2aWY2YrzGxEgx+FiEgMCYWcH8xeQX7RIfaXlDf469UZ9O6+3t2HufswYCRQArwI3AnMd/csYH6wDzAOyApuU4FHGqJwEZFY5O7c/dIq5izbTq+OrenWrlWDv+axDt2MBTa5+1ZgPDAjaJ8BXBJsjwdmerWFQAcz6xGRakVEYpi7c+8ra3jmo2186+wT6dkxpVFe91iD/krgmWC7m7vnAwT3aUF7TyC3xnPygrb/YGZTzWyxmS0uLCw8xjJERGKLu/PA6+uZ/v4WJo/J5Pv/dVKjvXbYQW9mycBXgOfr6lpLm3+uwX2au2e7e3bXrnVefE1EJKY9PD+HRxZs4upTM/jxRSdjVltUNoxjOaMfByx1913B/q7DQzLBfUHQngf0qvG8dGBHfQsVEYlVv3trIw++uYGvjujJz8YPatSQh2ML+qv497ANwFxgYrA9EXi5Rvt1weyb0UDR4SEeEZHmxN359d/X879/38Alw07ggcuGkJDQuCEPYV6P3sxaA18CbqzRfD/wnJlNAbYBE4L2ecAFQA7VM3QmR6xaEZEY4e7c/7d1PPqPzUwYmc79lw0hMQohD2EGvbuXAJ0/07aH6lk4n+3rwM0RqU5EJAa5Oz95dQ1PvLeFr5+awU/HD4rKmfxhTWKFKRGReBEKOT+eu4qnFm5j0umZ3HNx477xWhsFvYhIhFRWhbhzzkpmL8njxjP7cue4AVEPeVDQi4hExKHyKm55einz1xVw+9gsvn1eVpMIeVDQi4jUW1FJBVNmLGLJtn389JJBXDu6d7RL+g8KehGRethZVMp1f/6QLbtL+N1VI7hwSNO74ouCXkTkOG0qPMh1j3/E/pJypk8exen9ukS7pFop6EVEjsOHm/dw41NLSDRj1tTTGJzePtolHZGCXkTkGM1ZmscdL6ygV6fWPDFpFL07t4l2SUeloBcRCZO78+CbG3l4/kZO69uZP14zkvatW0S7rDop6EVEwlBaUcUPZq9g7sc7mDAynfsuHUxyUmysxqqgFxGpQ2FxGTc9tYTFW/fx/f86iW+dfWKTmSMfDgW9iMhRfJy7nxufXML+Q+X87urhXDTkhGiXdMwU9CIiR/D84lzufmkVaakteeGm0/nCCU13Zs3RKOhFRD6joirEfa+tZfr7Wzj9xM787uoRdGqTHO2yjpuCXkSkhsLiMm55eikffrKX67/YhzvHDSApMTbedD0SBb2ISOD9Tbu5fdZyiksr+M3XhnHJ8J7RLikiFPQi0uxVhZzfv53Db97cQJ8ubXhqyqmc1D012mVFjIJeRJq1wuIyvvPsct7N2c2lw3vys0sG0aZlfEVjuGvGdgAeAwYBDnwDWA88C2QCW4Ar3H2fVU8ufYjqdWNLgEnuvjTilYuI1NP7Obu5/dnlHDhUwS8vG8wV2b1ian58uMJ9h+Eh4G/uPgAYCqwF7gTmu3sWMD/YBxgHZAW3qcAjEa1YRKSeyiqruO+1NVz92Ie0a5XEy7eM4WujMuIy5CGMM3ozawecCUwCcPdyoNzMxgNnB91mAAuAO4DxwMxgkfCFZtbBzHq4e37EqxcROUbrdxZz+6xlrNtZzLWje3PXBQNJSU6MdlkNKpyhm75AIfCEmQ0FlgC3A90Oh7e755tZWtC/J5Bb4/l5QZuCXkSiJhRynnh/C7/82zratUriz5OyOXdAt2iX1SjCCfokYARwq7t/aGYP8e9hmtrU9rePf66T2VSqh3bIyMgIowwRkeNTFXJue2YZr63M57yBadx/2RC6tG0Z7bIaTThj9HlAnrt/GOzPpjr4d5lZD4DgvqBG/141np8O7PjsF3X3ae6e7e7ZXbt2Pd76RUTq9ND8jby2Mp87zh/An67LblYhD2EEvbvvBHLN7KSgaSywBpgLTAzaJgIvB9tzgeus2migSOPzIhItG3YV84e3c/jq8J7cFGNXnYyUcCeL3gr8xcySgc3AZKp/STxnZlOAbcCEoO88qqdW5lA9vXJyRCsWEQlTKOTc/eJK2rZK4kcXnRztcqImrKB39+VAdi0Pja2lrwM317MuEZF6m70kj0Vb9vHAZUNi+qJk9RXbV+oRETmCPQfL+Plf1zIqsyOXj0yPdjlRpaAXkbh07ytr+LSskvsuHUxCQvMbl69JQS8icefNNbuY+/EObjkni/7d4ufiZMdLQS8icaXoUAV3v7SSAd1TuensE6NdTpMQX5doE5Fm7xfz1lJYXMafrssmOUnnsqAzehGJI+/l7GbWolxuOKMvQ9I7RLucJkNBLyJx4UBpBT+YvYI+XdrwnS/1j3Y5TYqGbkQkLtw7dw35RYeYfdPptGoR31ejPFY6oxeRmPfXlfm8sDSPW87px4iMjtEup8lR0ItITCs4UMpdL65kcM/23Do2K9rlNEkKehGJWe7OD15YQUl5FQ9+bRgtEhVptdF3RURi1l8+3MaC9YXcdcFA+qW1jXY5TZaCXkRiUk5BMfe9tpYzsrpw7eje0S6nSVPQi0jMOVRexc1/WUbr5ET+d8LQZn8tm7poeqWIxJx7X1nN+l3FzPjGKXRr1yra5TR5OqMXkZjy8vLtzFqUy7fOPpGz+msZ0nAo6EUkZmwuPMhdc1YyKrMj39WnX8OmoBeRmFBaUcXNTy8jOSmBh68aTpKmUoZNY/Qi0uS5Oz9+eRVr8w/w50nZ9GifEu2SYkpYvxLNbIuZrTSz5Wa2OGjrZGZvmNnG4L5j0G5m9rCZ5ZjZCjMb0ZAHICLx76mFW3lucR63ntuPcwd0i3Y5MedY/vY5x92HufvhRcLvBOa7exYwP9gHGAdkBbepwCORKlZEmp8PN+/h3lfWMHZAGt85T+Pyx6M+g1zjgRnB9gzgkhrtM73aQqCDmfWox+uISDO1Y/8hbn56KRmdWvPglcM0X/44hRv0DvzdzJaY2dSgrZu75wME92lBe08gt8Zz84I2EZGwlVZU8c2nllBaEWLadSNp16pFtEuKWeG+GTvG3XeYWRrwhpmtO0rf2n7l+uc6Vf/CmAqQkZERZhki0hy4O3fNWcmKvCKmXTuSfmla4Ls+wjqjd/cdwX0B8CJwCrDr8JBMcF8QdM8DetV4ejqwo5avOc3ds909u2tXfehBRP7tt2/lMGfZdr5zXn++/IXu0S4n5tUZ9GbWxsxSD28DXwZWAXOBiUG3icDLwfZc4Lpg9s1ooOjwEI+ISF1eWradX7+xga+O6MltY/tFu5y4EM7QTTfgRTM73P9pd/+bmS0CnjOzKcA2YELQfx5wAZADlACTI161iMSljz7Zyw9mr2B0307c/9UhBLkj9VRn0Lv7ZmBoLe17gLG1tDtwc0SqE5FmY3PhQaY+uZj0Tik8ek02yUn65Guk6DspIlG3+2AZk6cvItGM6ZNOoX1rzbCJJF0CQUSi6kBpBRP//BG7DpTy9A2jyejcOtolxR2d0YtI1JRWVHH9jMWs31nMH68ZyYiMjtEuKS7pjF5EoqKiKsQtTy9l0Za9PHTlcM4+Ka3uJ8lx0Rm9iDS6UMi5Y/YK3lxbwE/GD+IrQ0+IdklxTUEvIo3K3bln7mrmLNvO977UXwt7NwIFvYg0msMh/+TCrdx4Zl9uOVcfiGoMCnoRaRTuzv/MXc3MD7Zywxl9uHPcAH0gqpEo6EWkwbk7976yhhkfbOX6L/bhrgsGKuQbkYJeRBrU4ZCf/v4WpnyxD3dfqJBvbJpeKSINpirk/HDOCp5bnMc3xvThRwr5qFDQi0iDKKus4jvPLmfeyp3cNjaL75yXpZCPEgW9iERcSXkl33xqKe9sKORHFw7k+jP6RrukZk1BLyIRVVRSwZQZi1i6bR8PXDaEK0b1qvtJ0qAU9CISMbl7S5j0xEfk7j3Eb68awYVDekS7JEFBLyIR8nHufqbMWER5ZYiZU05hdN/O0S5JAgp6Eam3N9bs4rZnltG5bTKzpo7WYt5NjIJeRI6bu/PEe1v42WtrGNSzPY9NzCYttVW0y5LPUNCLyHEpq6ziRy+u4vkleXzp5G48dOUwWicrUpqisD8Za2aJZrbMzF4N9vuY2YdmttHMnjWz5KC9ZbCfEzye2TCli0i0FBwo5cppC3l+SR63jc3i0WtGKuSbsGO5BMLtwNoa+78EHnT3LGAfMCVonwLsc/d+wINBPxGJE8u27ePi373L+p3FPPL1EXz3S/1JSNAHoZqysILezNKBC4HHgn0DzgVmB11mAJcE2+ODfYLHx5o+DicS89ydGe9v4WuPLiQ5KYE53zqdcYM1fTIWhPu31m+AHwCH30rvDOx398pgPw/oGWz3BHIB3L3SzIqC/rsjUrGINLoDpRXcMXsFf121k3MHpPGrCUPp2CY52mVJmOoMejO7CChw9yVmdvbh5lq6ehiP1fy6U4GpABkZGWEVKyKN79OySi56+F227z/EXRcM4Pov9tVQTYwJ54x+DPAVM7sAaAW0o/oMv4OZJQVn9enAjqB/HtALyDOzJKA9sPezX9TdpwHTALKzsz/3i0BEosvd2VlUyra9JVS2CvHcjaMZ2btTtMuS41DnGL27/9Dd0909E7gSeMvdvw68DVwedJsIvBxszw32CR5/y90V5CIxpOBAKZOnL2LLnk9pn9KC1247QyEfw+ozH+oOYJaZ/QxYBjwetD8OPGlmOVSfyV9ZvxJFpDG9umIHP3ppFaUVVUzq2obu7VppPD7GHVPQu/sCYEGwvRk4pZY+pcCECNQmIo1of0k5P355NXM/3sGwXh349RVDeee1XdEuSyJAn3AQEd5at4sfzlnJnoPl/L8v9+ebZ51IUmIC70S7MIkIBb1IM1ZwoJR7X1nDayvz6d+tLY9PHMWgnu2jXZZEmIJepBkKhZxZi3L5xV/XUlYZ4vv/dRI3nNGX5KRj+bC8xAoFvUgzs2FXMXe/uJJFW/Zx+omdue/SwfTp0ibaZUkDUtCLNBP7S8r5zZsbeXLhVlJbJfG/E4Zy2YieWrC7GVDQi8S5yqoQzyzK5dd/X0/RoQquPjWD737pJDppymSzoaAXiWPv5ezmp6+uYd3OYkb37cQ9F3+BgT3aRbssaWQKepE4tDKviAdeX8c/N+6mZ4cUHvn6CM4f1F3DNM2Ugl4kjmwuPMiv3tjAayvy6dC6BT+6cCDXjO5NqxaJ0S5NokhBLxIH8vaV8Pu3N/Hc4lxaJiVw27n9uP7MvrRr1SLapUkToKAXiWFbdn/KHxbkMGfpdszg2tG9ufmcfnRNbRnt0qQJUdCLxKCcgmJ+//YmXl6+nRaJCVwzujc3ntWXHu1Tol2aNEEKepEY4e589MleHnv3E95cu4tWSYlM+WIfbjizL2mpraJdnjRhCnqRJq6iKsS8lfk89s9PWLm9iI6tW3Dz2f2YPCaTzm01RCN1U9CLNFH7S8qZtSiX6e9tYeeBUvp2bcN9lw7iq8PTSUnWLBoJn4JepAlxdxZv3cfTH27jtZX5lFeGGNOvM7/46mDO6t9Va7XKcVHQizQBRSUVvLA0j2c+2sbGgoOktkziylG9uPrUDAZ01ydZpX4U9CJRVFxawa/+voFnPtpGWWWIob068MBlQ7hoaA9aJ+u/p0SGfpJEoiQUcm6ftZwF6wu4fGQ6E0/P5AsnaNEPibw6g97MWgHvAC2D/rPd/R4z6wPMAjoBS4Fr3b3czFoCM4GRwB7ga+6+pYHqF4lZv30rh7fWFfCT8V/gutMyo12OxLFwlpMpA85196HAMOB8MxsN/BJ40N2zgH3AlKD/FGCfu/cDHgz6iUgNr6/eyYNvbuCyEelcO7p3tMuROFdn0Hu1g8Fui+DmwLnA7KB9BnBJsD0+2Cd4fKzpknki/7JhVzHffXY5Q9Pbc9+lg3RFSWlwYS0QaWaJZrYcKADeADYB+929MuiSB/QMtnsCuQDB40VA51q+5lQzW2xmiwsLC+t3FCIxoqikgqkzF5OSnMQfrx2pq0pKowgr6N29yt2HAenAKcDA2roF97WdnvjnGtynuXu2u2d37do13HpFYlZFVYibn17K9v2HePTaEboujTSaY1ry3d33AwuA0UAHMzv8Zm46sCPYzgN6AQSPtwf2RqJYkVjl7tz5wkrezdnNfZcOZmTvTtEuSZqROoPezLqaWYdgOwU4D1gLvA1cHnSbCLwcbM8N9gkef8vdP3dGL9KcPPjmRl5Ymse3z8viiuxe0S5Hmplw5tH3AGaYWSLVvxiec/dXzWwNMMvMfgYsAx4P+j8OPGlmOVSfyV/ZAHWLxIxZH23j4fkbuSI7ndvHZkW7HGmG6gx6d18BDK+lfTPV4/WfbS8FJkSkOpEY9/fVO7n7pVWc1b8r9106WDNsJCqOaYxeRML3jw2F3PL0Mgb3bM/vvz6CFon67ybRoZ88kQbwwaY9TJ25mH5pbZkx+RTattTVRiR6FPQiEbZk6z6mzFhERqfWPDnlFNq31gLdEl06zRCJoA837+Eb0xeRltqSv1x/qlaAkiZBZ/QiEfKPDYVMfOIjurdvxaypp5HWTuu4StOgM3qRCHh99U5ufXoZ/dLaMnPKKXTRmbw0IQp6kXp6blEuP3xxJYN7tmfGZI3JS9OjoBc5Tu7Og29u5OH5GzkjqwuPXDNSs2ukSdJPpchxqKgK8cM5K5m9JI8JI9P5+VcHa568NFkKepFjtPfTcm55einvb9rDt8/L4vaxWfrEqzRpCnqRY7BmxwGmPrmYguIyfjVhKJeNTI92SSJ1UtCLhGnuxzv4weyP6ZCSzHM3nsawXh2iXZJIWBT0InUoraji5/PWMvODrYzK7Mjvvz6CtFTNkZfYoaAXOYqNu4q59ZllrNtZzJQv9uGO8weQnKQ3XSW2KOhFauHuzFqUy72vrKZNchJPTBrFOQPSol2WyHFR0It8RkFxKf/90ipeX72LMf068+AVw3Q5A4lpCnqRgLszZ+l2fvLqGg5VVPHDcQO44Yy+JCRo6qTENgW9CLBj/yHuenElC9YXMrJ3Rx64fAgndm0b7bJEIkJBL81aWWUVT7y3hd/O30jI4Z6LT+a60zJJ1Fm8xJE6g97MegEzge5ACJjm7g+ZWSfgWSAT2AJc4e77rPojgg8BFwAlwCR3X9ow5Yscv7fXF/CTV9bwye5POW9gGvdc/AV6dWod7bJEIi6cM/pK4HvuvtTMUoElZvYGMAmY7+73m9mdwJ3AHcA4ICu4nQo8EtyLNAk5BQf5xby1zF9XQN8ubXhi8ijOOUkzaiR+1Rn07p4P5AfbxWa2FugJjAfODrrNABZQHfTjgZnu7sBCM+tgZj2CryMSNdv3H+KhNzcwe0keKS0SuXPcAL4xpo/mxUvcO6YxejPLBIYDHwLdDoe3u+eb2eFTop5Abo2n5QVt/xH0ZjYVmAqQkZFxHKWLhGfPwTIeWbCJmQu3gsOk0/tw8zknapk/aTbCDnozawu8AHzb3Q8c5Wp9tT3gn2twnwZMA8jOzv7c4yL1VV4ZIr/oEGN++RbllSEuG5HO7edlkd5R4/DSvIQV9GbWguqQ/4u7zwmadx0ekjGzHkBB0J4H9Krx9HRgR6QKFqnLpsKDPPqPTRzM3Y8DFw4+gZvOPpF+aZouKc1TOLNuDHgcWOvuv67x0FxgInB/cP9yjfZbzGwW1W/CFml8XhpaKOT8Y2MhM97fwoL1hbRMSuCqDi3p0SGFG68YGu3yRKIqnDP6McC1wEozWx603UV1wD9nZlOAbcCE4LF5VE+tzKF6euXkiFYsUkNRSQVzluUx84OtfLL7U7qmtuT2sVlcM7o3r72wK9rliTQJ4cy6eZfax90BxtbS34Gb61mXyBFVhZz3cnbz/JI8Xl+9k/LKEMMzOvDQlcMYN6iHZtGIfIY+GSsxY1PhQV5cup0XluaRX1RK+5QWXDmqFxNG9mJwevtolyfSZCnopUnLKTjIvJX5zFuZz7qdxSQYnJHVlbsvHMh5A7vRqkVitEsUafIU9NKkhELOqh1FvLWugL+t2sm6ncUAZPfuyD0Xn8wFg3vQTZcMFjkmCnqJuoNllby7sZC31hXw9vpCCovLMIORGdXhPm5QD7q3V7iLHC8FvTS6Q+VVLNm6jw827+aDTXtYkVdEZchp1yqJM/t35ZyT0jj7pK765KpIhCjopcEVlVSwPG8/S7buY+HmPSzftp/yqhCJCcaQ9PZMPbMvZ/XvysjeHUlK1IwZkUhT0EtEVVSFWL+zmGW5+1m2bR/Lc/ezufBTAMxg0AntmTQmk9NO7MyozE60bakfQZGGpv9lctyKDlWwNv8A6/IPsDa/mLU7D7B+ZzFllSEAurRNZlivjlw2Ip3hvTowOL09qa1aRLlqkeZHQS/HZH9JOf/98mqWbt3H9v2H/tXeqU0yA3ukcs3o3gzt1YHhvTqQ3jGFo1z8TkQaiYJewlZaUcUNMxfzcW4R5w/qzjWjezOwRyoDe7QjLbWlQl2kiVLQS1hCIed7z33Moi37+N3Vw7loyAnRLklEwqQpDhKWn89by2sr87n7goEKeZEYo6CXOj32z8089u4nTDo9k+vP6BPtckTkGCno5ahmL8njZ6+t5fwvdOe/LzpZ4/AiMUhBL0f0+uqd3PHCCsb068xDVw0jMUEhLxKLFPRSq7fXF3DrM8sY3LM9067NpmWSrhIpEqsU9PI581bmM3XmYrLS2vLEpFG00adXRWKagl7+w/OLc7nl6aUMTe/AM1NH07FNcrRLEpF6qjPozezPZlZgZqtqtHUyszfMbGNw3zFoNzN72MxyzGyFmY1oyOIlckIh54G/reP7s1cwpl8XZk45hXa6XIFIXAjnjH46cP5n2u4E5rt7FjA/2AcYB2QFt6nAI5EpUxrSp2WVfPOpJfxhwSauOiWDxyeOonWyhmtE4kU4i4O/Y2aZn2keD5wdbM8AFgB3BO0zgwXCF5pZBzPr4e75kSpYImvV9iJue2YZW/Z8yj0Xn8yk0zM1hVIkzhzvaVu3w+Ht7vlmlha09wRya/TLC9oU9E1MKORMf38L9/91HR3btOAv14/mtBM7R7ssEWkAkf77vLZTQa+1o9lUqod3yMjIiHAZcjSbCg/ywxdW8tGWvZw3MI0HLh9KJ73pKhK3jjfodx0ekjGzHkBB0J4H9KrRLx3YUdsXcPdpwDSA7OzsWn8ZSGSVlFfy6D8288iCTaQkJ/LA5UOYMDJdQzUice54g34uMBG4P7h/uUb7LWY2CzgVKNL4fPRVhZzZS3L51d83UFBcxkVDevDji08mLVULbos0B3UGvZk9Q/Ubr13MLA+4h+qAf87MpgDbgAlB93nABUAOUAJMboCaJUyVVSH+umonv3srh/W7ihme0YFHrhnByN6dol2aiDSicGbdXHWEh8bW0teBm+tblNRPaUUVzy/J40/vbGbb3hL6dm3DH74+gnGDumuYRqQZ0mTpOLI2/wCzPtrGi8u2c6C0kmG9OnD3hQP50sBuJOiCZCLNloI+xpVVhnjsn5t55eMdfJxXRHJiAucP6s7Vp2Zwap9OOoMXEQV9rKkKOat3FPHPjbv5ZHsRB8sq+VvOWk7u0Y4fX3Qylw7vqevTiMh/UNA3caGQk1N4kCVb9/Fezm7ey9nNvpIKAC5PhYxOrVlwzdlkdmkT5UpFpKlS0DchVSFn655PWb+zmDX5B1i2bT8f5+6nuKwSgLTUlpw7oBtn9u/CmH5deHV2IYBCXkSOSkHfyNydPZ+Wk7u3hG17S8jbd4hNhQdZv7OYnIKDlFWGAEgwGNC9HeOHn8DwXh0ZltGBvl3aaMxdRI6Zgj5C3J2DZZXs+7SCguJSCovLKCguozC4FRSXkl9Uyra9JZSUV/3Hc9NSW3JS91SuO603/bulclL3VLLSUklJ1qpOIlJ/zS7o3Z2KKqeiKkRllVNeFaKssopD5VUcqqi+L6moorS8ipKgrbSierukvIqiQxUcOFTBgdIKig5V1NivpCr0+Ss5JBh0aduSrqktSe+YwmkndiajU2t6dWxNRufWpHdM0SWBRaRBxXzC7C+pYMIf36e8yqmoDFEZCv0ryGuG+eHtylrCOFzJiQm0S2lB+5Qk2qW0oFObZDI7t6F9Sgvap7SgXUoSHVonk5bakrTUVnRNbUmnNslaVFtEoirmg37vp2WsKCpidN/OtEg0WiQmkJSYUL2dkECLJCMpIYHkpASSEqofr7ndItFo2SKRlODWOjmRVsnV9yktEklJ/vdjSYlaeVFEYk/MB31lyOnTpQ0zvnFKtEsREWmSYv4UtTLkWttUROQoYj7oq6qcdikx/4eJiEiDifmg1xm9iMjRxXzQV4VCtEtR0IuIHEnsB7077Vpp6EZE5EhiOuirQo47OqMXETmKmA76wx9+0hi9iMiRxXTQH77kgGbdiIgcWYMEvZmdb2brzSzHzO5siNcAqAxVX+lRZ/QiIkcW8aA3s0Tg98A44GTgKjM7OdKvA9Vz6EFj9CIiR9MQZ/SnADnuvtndy4FZwPgGeB2N0YuIhKEhgr4nkFtjPy9oi7jfK1wAAAAEgUlEQVRKjdGLiNSpIYK+tmvyfu7awGY21cwWm9niwsLC43qhVkkJdGqTTNuWCnoRkSNpiKDPA3rV2E8Hdny2k7tPc/dsd8/u2rXrcb1QxzbJ9O+WqssHi4gcRUMk5CIgy8z6mFkycCUwtwFeR0REwhDxMQ93rzSzW4DXgUTgz+6+OtKvIyIi4WmQwW13nwfMa4ivLSIix0aD2yIicU5BLyIS5xT0IiJxTkEvIhLnFPQiInHO3D/3odXGL8KsENh6nE/vAuyOYDmxQMfcPOiYm4f6HHNvd6/zE6dNIujrw8wWu3t2tOtoTDrm5kHH3Dw0xjFr6EZEJM4p6EVE4lw8BP20aBcQBTrm5kHH3Dw0+DHH/Bi9iIgcXTyc0YuIyFHETNDXteC4mbU0s2eDxz80s8zGrzKywjjm75rZGjNbYWbzzax3NOqMpHAXljezy83MzSzmZ2iEc8xmdkXwb73azJ5u7BojLYyf7Qwze9vMlgU/3xdEo85IMbM/m1mBma06wuNmZg8H348VZjYiogW4e5O/UX25401AXyAZ+Bg4+TN9vgX8Mdi+Eng22nU3wjGfA7QOtm9qDscc9EsF3gEWAtnRrrsR/p2zgGVAx2A/Ldp1N8IxTwNuCrZPBrZEu+56HvOZwAhg1REevwD4K9Ur9I0GPozk68fKGX04C46PB2YE27OBsWZW27KGsaLOY3b3t929JNhdSPVqXrEs3IXlfwo8AJQ2ZnENJJxjvgH4vbvvA3D3gkauMdLCOWYH2gXb7alllbpY4u7vAHuP0mU8MNOrLQQ6mFmPSL1+rAR9OAuO/6uPu1cCRUDnRqmuYRzrIutTqD4jiGV1HrOZDQd6ufurjVlYAwrn37k/0N/M3jOzhWZ2fqNV1zDCOeb/Aa4xszyq17a4tXFKi5pj/f9+TGJlVe1wFhwPa1HyGBL28ZjZNUA2cFaDVtTwjnrMZpYAPAhMaqyCGkE4/85JVA/fnE31X23/NLNB7r6/gWtrKOEc81XAdHf/lZmdBjwZHHOo4cuLigbNr1g5ow9nwfF/9TGzJKr/3Dvan0pNXViLrJvZecDdwFfcvayRamsodR1zKjAIWGBmW6gey5wb42/Ihvuz/bK7V7j7J8B6qoM/VoVzzFOA5wDc/QOgFdXXhIlXYf1/P16xEvThLDg+F5gYbF8OvOXBuxwxqs5jDoYxHqU65GN93BbqOGZ3L3L3Lu6e6e6ZVL8v8RV3XxydciMinJ/tl6h+4x0z60L1UM7mRq0yssI55m3AWAAzG0h10Bc2apWNay5wXTD7ZjRQ5O75kfriMTF040dYcNzMfgIsdve5wONU/3mXQ/WZ/JXRq7j+wjzm/w+0BZ4P3nfe5u5fiVrR9RTmMceVMI/5deDLZrYGqAK+7+57old1/YR5zN8D/mRm36F6CGNSLJ+4mdkzVA+9dQned7gHaAHg7n+k+n2IC4AcoASYHNHXj+HvnYiIhCFWhm5EROQ4KehFROKcgl5EJM4p6EVE4pyCXkQkzinoRUTinIJeRCTOKehFROLc/wGqOBpf0uXpTwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_prices()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.33524355, -0.3333306 , -0.33142585])"
      ]
     },
     "execution_count": 221,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diff(compute_stages())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1.0, 0.664756446991404, 0.3314258503624765, 0.0]"
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "compute_stages()"
   ]
  },
  {
   "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}