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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### DEMDP01b\n",
    "# Timber Harvesting -- using cubic splines"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from compecon import NLP, demo, BasisSpline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "price = 1.0  # price of biomass\n",
    "kappa = 0.2  # clearcut-replant cost\n",
    "smax  = 0.5  # stand carrying capacity\n",
    "gamma = 0.1  # biomass growth parameter\n",
    "delta = 0.9  # discount factor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Code the growth function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## SOLUTION\n",
    "\n",
    "### Code the approximant and the residual"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Value function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Conditional value functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Residual"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve collocation equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compute critical biomass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ANALYSIS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compute refined state grid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ss = np.linspace(0,smax,1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot Conditional Value Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig1 =demo.figure('Conditional Value Functions','Biomass','Value of Stand')\n",
    "plt.plot(ss,vhat0(ss),label='Grow')\n",
    "plt.plot(ss,vhat1(ss),label='Clear-Cut')\n",
    "plt.legend()\n",
    "\n",
    "vcrit = vhat(scrit)\n",
    "ymin = plt.ylim()[0]\n",
    "plt.vlines(scrit, ymin,vcrit,'grey',linestyles='--')\n",
    "demo.annotate(scrit,ymin,'$s^*$',ms=10)\n",
    "demo.bullet(scrit,vcrit)\n",
    "print(f'Optimal Biomass Harvest Level = {scrit:.4f}') "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot Value Function Residual"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig2 = demo.figure('Bellman Equation Residual', 'Biomass', 'Percent Residual')\n",
    "plt.plot(ss, 100*resid(cc,ss) / vhat(ss))\n",
    "plt.hlines(0,0,smax,linestyles='--')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compute ergodic mean annual harvest "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = h(0)\n",
    "for n in range(100):\n",
    "    if s > scrit: break\n",
    "    s = h(s)\n",
    "    \n",
    "print(f'Ergodic Mean Annual Harvest = {s/n:.4f} after {n+1} iterations') "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#demo.savefig([fig1,fig2])"
   ]
  }
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