{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# DiscreteDP Example: Job Search"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Daisuke Oyama**\n",
    "\n",
    "*Faculty of Economics, University of Tokyo*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We study an optimal stopping problem, in the context of job search as discussed in\n",
    "[http://quant-econ.net/py/lake_model.html](http://quant-econ.net/py/lake_model.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from __future__ import division, print_function\n",
    "import numpy as np\n",
    "import scipy.stats\n",
    "import scipy.optimize\n",
    "import scipy.sparse\n",
    "from numba import jit\n",
    "import matplotlib.pyplot as plt\n",
    "from quantecon.markov import DiscreteDP"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimal solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We skip the description of the model, just writing down the Bellman equation:\n",
    "$$\n",
    "\\begin{aligned}\n",
    "U   &= u(c) + \\beta \\left[(1 - \\gamma) U + \\gamma E[V_s]\\right], \\\\\n",
    "V_s &= \\max\\left\\{U,\n",
    "                  u(w_s) + \\beta \\left[(1 - \\alpha) V_s + \\alpha U\\right]\n",
    "           \\right\\}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "For this class of problem, we can characterize the solution analytically."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The optimal policy $\\sigma^*$ is monotone;\n",
    "it is characterized by a threshold $s^*$, for which\n",
    "$\\sigma^*(s) = 1$ if and only if $s \\geq s^*$,\n",
    "where actions $0$ and $1$ represent \"reject\" and \"accept\", respectively.\n",
    "The threshold is defined as follows:\n",
    "Let\n",
    "$$\n",
    "\\begin{aligned}\n",
    "g(s) &= u(w_s) - u(c), \\\\\n",
    "h(s) &= \\frac{\\beta \\gamma}{1 - \\beta (1 - \\alpha)}\n",
    "        \\sum_{s' \\geq s} p_s u(w_s).\n",
    "\\end{aligned}\n",
    "$$\n",
    "It is easy to see that $g$ is increasing and $h$ is decreasing.\n",
    "Then the threshold $s^*$ is such that $s \\geq s^*$ if and only if $g(s) > h(s)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given $s^*$, the optimal values can be computed as follows:\n",
    "$$\n",
    "\\begin{aligned}\n",
    "U &=\n",
    "\\frac{\\{1 - (1 - \\alpha) \\beta\\} u(c) + \\beta \\gamma \\sum_{s \\geq s^*} p_s u(w_s)}\n",
    "     {(1 - \\beta) \\left[\\{1 - (1 - \\alpha) \\beta\\} +\n",
    "                        \\beta \\gamma \\sum_{s \\geq s^*} p_s\\right]}, \\\\\n",
    "V_s &=\n",
    "\\begin{cases}\n",
    "U & \\text{if $s < s^*$} \\\\\n",
    "\\dfrac{u(w_s) + \\alpha \\beta U}{1 - (1 - \\alpha) \\beta} & \\text{if $s \\geq s^*$}.\n",
    "\\end{cases}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The optimal policy defines a Markov chain over $\\{\\text{unemployed}, \\text{employed}\\}$.\n",
    "Its stationary distribution is\n",
    "$\\pi = \\left(\\frac{\\alpha}{\\alpha + \\lambda}, \\frac{\\lambda}{\\alpha + \\lambda}\\right)$,\n",
    "where\n",
    "$\\lambda = \\gamma \\sum_{s \\geq s^*} p(w_s)$;\n",
    "note that the flow from unemployed to employed is $\\lambda$,\n",
    "while the flow from employed to unemployed is $\\alpha$.\n",
    "\n",
    "The expected value at the stationary distribution is\n",
    "$$\n",
    "\\pi_0 U + \\pi_1 \\frac{\\sum_{s \\geq s^*} p_s V_s}{\\sum_{s \\geq s^*} p_s}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following implements the job search problem with the analytical solution above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class JobSearchModel(object):\n",
    "    \"\"\"\n",
    "    Job search model.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    w : array_like(float, ndim=1)\n",
    "        Array containing wage levels. Must be ordered in ascending order.\n",
    "    \n",
    "    pdf : array_like(float, ndim=1)\n",
    "        Wage distribution.\n",
    "        \n",
    "    beta : scalar(float)\n",
    "        Discount factor\n",
    "        \n",
    "    alaph :scalar(float)\n",
    "        Firing probability.\n",
    "        \n",
    "    gamma : scalar(float)\n",
    "        Wage offer arrival probability.\n",
    "        \n",
    "    rho: scalar(float)\n",
    "        Degree of (constant) relative risk aversion.\n",
    "        \n",
    "    \"\"\"\n",
    "    def __init__(self, w, pdf, beta, alpha=0, gamma=1, rho=0):\n",
    "        w = np.asarray(w)\n",
    "        self.pdf = np.asarray(pdf)\n",
    "        self.beta, self.alpha, self.gamma, self.rho = beta, alpha, gamma, rho\n",
    "        self.u_w = self.u(w)\n",
    "        \n",
    "    def u(self, y):\n",
    "        \"\"\"\n",
    "        y must be array_like.\n",
    "        \n",
    "        \"\"\"\n",
    "        rho = self.rho\n",
    "        small_number = -9999999\n",
    "        y = np.asarray(y, dtype=float)\n",
    "        nonpositive = (y <= 0)\n",
    "        if rho == 1:\n",
    "            util = np.log(y)\n",
    "        else:\n",
    "            util = (y**(1 - rho) - 1)/(1 - rho)\n",
    "        util[nonpositive] = small_number\n",
    "        return util\n",
    "    \n",
    "    def solve(self, c, *args, **kwargs):\n",
    "        \"\"\"\n",
    "        Solve directly s_star and U and V_s.\n",
    "        \n",
    "        \"\"\"\n",
    "        S = len(self.u_w)\n",
    "        \n",
    "        a0 = 1 - (1 - self.alpha) * self.beta\n",
    "        a1 = self.beta * self.gamma\n",
    "        coeff = a1 / a0\n",
    "        u_c = self.u(np.array([c]))[0]\n",
    "        s_star = _bisect(self.u_w, self.pdf, u_c, coeff)\n",
    "        \n",
    "        C = np.zeros(S, dtype=int)\n",
    "        C[s_star:] = 1\n",
    "        \n",
    "        U = a0 * u_c + a1 * self.u_w[s_star:].dot(self.pdf[s_star:])\n",
    "        U /= a0 + a1 * self.pdf[s_star:].sum()\n",
    "        U /= 1 - self.beta\n",
    "        \n",
    "        V = np.empty(S)\n",
    "        V[:s_star] = U\n",
    "        V[s_star:] = (self.u_w[s_star:] + self.alpha * self.beta * U) / a0\n",
    "        \n",
    "        return V, U, C\n",
    "    \n",
    "    def stationary_distribution(self, C):\n",
    "        lamb = self.pdf.dot(C) * self.gamma \n",
    "        pi = np.array([self.alpha, lamb])\n",
    "        pi /= pi.sum()\n",
    "        return pi\n",
    "        \n",
    "            \n",
    "@jit(nopython=True)\n",
    "def _bisect(u_w, pdf, u_c, coeff):\n",
    "    lo = -1\n",
    "    hi = len(u_w)\n",
    "    while(lo < hi-1):\n",
    "        m = (lo + hi) // 2\n",
    "        lhs = u_w[m] - u_c\n",
    "        rhs = 0\n",
    "        for i in range(m+1, len(u_w)):\n",
    "            rhs += (u_w[i] - u_w[m]) * pdf[i]\n",
    "        rhs *= coeff\n",
    "        if lhs > rhs:\n",
    "            hi = m\n",
    "        else:\n",
    "            lo = m\n",
    "    return hi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For comparison, let us also consider the implementation with the `DiscreteDP` class:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class JobSearchModelDiscreteDP(JobSearchModel):\n",
    "    \"\"\"\n",
    "    Job search model with DiscreteDP.\n",
    "        \n",
    "    \"\"\"\n",
    "    def __init__(self, w, pdf, beta, alpha=0, gamma=1, rho=0):\n",
    "        super(JobSearchModelDiscreteDP, self).__init__(w, pdf, beta, alpha, gamma, rho)\n",
    "        \n",
    "        # Number of states\n",
    "        # s = 0, ..., len(w)-1: wage w[s] offered, s = len(w): no offer\n",
    "        num_states = len(w) + 1\n",
    "        \n",
    "        # Number of actions: 0: reject, 1: accept\n",
    "        num_actions = 2\n",
    "        \n",
    "        L = num_states*num_actions - 1\n",
    "        s_indices, a_indices = np.empty(L), np.empty(L)\n",
    "        s_indices[-1], a_indices[-1] = len(w), 0\n",
    "        s_indices[:-1] = np.repeat(np.arange(len(w)), num_actions)\n",
    "        a_indices[:-1] = np.tile(np.arange(num_actions), len(w))\n",
    "        \n",
    "        R0 = np.zeros(L)\n",
    "        R0[[num_actions*i+1 for i in range(len(w))]] = self.u_w\n",
    "        \n",
    "        Q = scipy.sparse.lil_matrix((L, num_states))\n",
    "        it = np.nditer((s_indices, a_indices))\n",
    "        for (s, a) in it:\n",
    "            i = it.iterindex\n",
    "            if a == 0:\n",
    "                Q[i, -1] = 1 - self.gamma\n",
    "                Q[i, :len(w)] = self.pdf*self.gamma\n",
    "            else:  # if a == 1\n",
    "                Q[i, s], Q[i, -1] = 1 - self.alpha, self.alpha\n",
    "                \n",
    "        self.ddp = DiscreteDP(R0, Q, beta, s_indices, a_indices)\n",
    "        \n",
    "        self.num_iter = None\n",
    "        \n",
    "    def solve(self, c, *args, **kwargs):\n",
    "        n, m = self.ddp.num_states, self.ddp.num_actions\n",
    "        self.ddp.R[[m*i for i in range(n)]] = self.u(np.array([c]))[0]\n",
    "        res = self.ddp.solve(*args, **kwargs)\n",
    "        V = res.v[:-1]  # Values of jobs\n",
    "        U = res.v[-1]  # Value of unemployed\n",
    "        C = res.sigma[:-1]\n",
    "        self.num_iter = res.num_iter\n",
    "        \n",
    "        return V, U, C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following paramter values are from [lakemodel_example.py](https://github.com/QuantEcon/QuantEcon.py/blob/master/examples/lakemodel_example.py)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "w = np.linspace(0, 175, 201)  # wage grid\n",
    "\n",
    "# compute probability of each wage level \n",
    "logw_dist = scipy.stats.norm(np.log(20.),1)\n",
    "cdf = logw_dist.cdf(np.log(w))\n",
    "pdf = cdf[1:]-cdf[:-1]\n",
    "pdf /= pdf.sum()\n",
    "w = (w[1:] + w[:-1])/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "gamma = 1\n",
    "alpha = 0.013  # Monthly\n",
    "alpha_q = (1-(1-alpha)**3)  # Quarterly\n",
    "beta = 0.99\n",
    "rho = 2  # risk-aversion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "js = JobSearchModel(w, pdf, beta, alpha_q, gamma, rho)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "js_ddp = JobSearchModelDiscreteDP(w, pdf, beta, alpha_q, gamma, rho)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us check that the results coincide:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    }
   ],
   "source": [
    "cs = np.linspace(1, 75, 25)\n",
    "bools = []\n",
    "for c in cs:\n",
    "    V, U, C = js.solve(c=c)\n",
    "    V1, U1, C1 = js_ddp.solve(c=c)\n",
    "    bools.append(np.allclose(V, V1))\n",
    "    bools.append(np.allclose(U, U1))\n",
    "    bools.append(np.array_equal(C, C1))\n",
    "print(all(bools))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Take a look at the optimal solution for $c = 40$ for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "c = 40\n",
    "V, U, C = js.solve(c=c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimal policy: Accept if and only if w >= 65.1875\n"
     ]
    }
   ],
   "source": [
    "s_star = len(w) - C.sum()\n",
    "print(r\"Optimal policy: Accept if and only if w >= {0}\".format(w[s_star]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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OTJMK3XorfOc7eUdjZvUpaDIgaaCklyW9Imlgtm8bSU9LmihpgqTeDZw7VdJL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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dad2128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "ax.plot(w, V, label=r'$V$')\n",
    "ax.plot((w[0], w[-1]), (U, U), 'r--', label=r'$U$')\n",
    "ax.set_xlabel('Wage')\n",
    "ax.set_ylabel('Value')\n",
    "ax.set_title('Optimal value function')\n",
    "plt.legend(loc=2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Performance comparison:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10000 loops, best of 3: 58.7 µs per loop\n"
     ]
    }
   ],
   "source": [
    "c = 40\n",
    "%timeit js.solve(c=c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 loops, best of 3: 8.04 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit js_ddp.solve(c=c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimal unemployment insurance policy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We compute the optimal level of unemployment insurance\n",
    "as in the [lecture](http://quant-econ.net/py/lake_model.html#fiscal-policy),\n",
    "mimicking [lakemodel_example.py](https://github.com/QuantEcon/QuantEcon.py/blob/master/examples/lakemodel_example.py)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class UnemploymentInsurancePolicy(object):\n",
    "    def __init__(self, w, pdf, beta, alpha=0, gamma=1, rho=0):\n",
    "        self.w, self.pdf, self.beta, self.alpha, self.gamma, self.rho = \\\n",
    "            w, pdf, beta, alpha, gamma, rho\n",
    "    \n",
    "    def solve_job_search_model(self, c, T):\n",
    "        js = JobSearchModel(self.w-T, self.pdf, self.beta,\n",
    "                            self.alpha, self.gamma, self.rho)\n",
    "        V, U, C = js.solve(c=c-T)\n",
    "        pi = js.stationary_distribution(C)\n",
    "        \n",
    "        return V, U, C, pi\n",
    "        \n",
    "    def implement(self, c):\n",
    "        \n",
    "        def budget_balance(T):\n",
    "            _, _, _, pi = self.solve_job_search_model(c, T)\n",
    "            return T - pi[0]*c\n",
    "        \n",
    "        # Budget balancing tax given c\n",
    "        T = scipy.optimize.brentq(budget_balance, 0, c)\n",
    "        \n",
    "        V, U, C, pi = self.solve_job_search_model(c, T)\n",
    "        \n",
    "        EV = (C*V).dot(self.pdf)/(C.dot(self.pdf))\n",
    "        W = pi[0] * U + pi[1] * EV\n",
    "        \n",
    "        return T, W, pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "uip = UnemploymentInsurancePolicy(w, pdf, beta, alpha_q, gamma, rho)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimal unemployment benefit: 67.4\n"
     ]
    }
   ],
   "source": [
    "grid_size = 501  #25\n",
    "cvec = np.linspace(5, 135, grid_size)\n",
    "Ts, Ws = np.empty(grid_size), np.empty(grid_size)\n",
    "pis = np.empty((grid_size, 2))\n",
    "for i, c in enumerate(cvec):\n",
    "    T, W, pi = uip.implement(c=c)\n",
    "    Ts[i], Ws[i], pis[i] = T, W, pi\n",
    "i_max = Ws.argmax()\n",
    "print('Optimal unemployment benefit:', cvec[i_max])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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RybJddoHPPtO0byKSX+bNg3PPhX32iX7jgQOhU6e0o8q+Bt1iUVISV28+/DD8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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106c570f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot(ax, y_vec, title):\n",
    "    ax.plot(cvec, y_vec)\n",
    "    ax.set_xlabel(r\"$c$\")\n",
    "    ax.vlines(cvec[i_max], ax.get_ylim()[0], y_vec[i_max], \"k\", \"-.\")\n",
    "    ax.set_title(title)\n",
    "\n",
    "fig, axes = plt.subplots(2, 2, figsize=(10, 6))\n",
    "plot(axes[0, 0], Ws, \"Welfare\")\n",
    "plot(axes[0, 1], Ts, \"Taxes\")\n",
    "plot(axes[1, 0], pis[:, 1], \"Employment Rate\")\n",
    "plot(axes[1, 1], pis[:, 0], \"Unemployment Rate\")\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ISL0oqVfjpJNiLPrw4ak8vYhIvWk99dWMHQvPPw+vvZZ2JCIijavoOkqXLIkN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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dace470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "plot(ax, cvec-Ts, \"Net Compensation\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "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": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}