{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Productividad Específica y Decisión de Aceptación Endógena"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El modelo esta caracterizado completamente por tres ecuaciones. La primera es la que caracteriza la productividad de reserva (PR):\n",
    "$$y_{R} =b+\\frac{m(\\theta )\\beta }{r+\\lambda }\\int_{y_{R}}^{\\overline{y}}\\left( y-y_{R}\\right) f(y)dy$$\n",
    "\n",
    "La segunda es la condición de creación de empleo (CCE):\n",
    "$$c =\\frac{m(\\theta )}{\\theta }\\frac{1-\\beta }{r+\\lambda }\\int_{y_{R}}^{\\overline{y}}(y-y_{R})f(y)dy$$\n",
    "\n",
    "La tercera es la curva de Beveridge:\n",
    "$$m\\left( \\theta \\right) \\left( 1-F\\left( y_{R}\\right) \\right) u=\\lambda\\left( 1-u\\right)$$\n",
    "\n",
    "La parametrización que usamos es la siguiente: (1) $m(\\theta)= A \\theta ^\\alpha$ y (2) $\\log x \\sim N(\\mu, \\sigma)$.\n",
    "\n",
    "El algoritmo de solución es el siguiente:\n",
    "\n",
    "1. Conjeturar $\\theta_0$.\n",
    "2. Resolver por $y_R$, el punto fijo de PR, dado $\\theta_0$.\n",
    "3. Resolver $\\theta_1$ usando CCE y $y_R$.\n",
    "4. Si $\\theta_1$ = $\\theta_0$ tenemos la solución, caso contrario retornar a 1 con $\\theta_1$ como nueva conjetura."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Importamos los módulos necesarios\n",
    "import numpy as np\n",
    "import scipy.stats as stats\n",
    "from scipy import integrate\n",
    "from scipy.optimize import root"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definimos la ecuación de Bellman para $y_R$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def TyR(yr, θ, prms):\n",
    "    b, λ, A, α, β, r, c, F = prms\n",
    "    \n",
    "    intf = lambda y: (y - yr)*F.pdf(y)\n",
    "    \n",
    "    yr1 = b + ((A*(θ**α)*β)/(λ+r))*integrate.quad(intf,yr,np.inf)[0]\n",
    "    \n",
    "    return yr1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definimos la ecuación de creación de empleo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def JCC(θ, yr, prms):\n",
    "    b, λ, A, α, β, r, c, F = prms\n",
    "    \n",
    "    intf = lambda y: (y - yr)*F.pdf(y)\n",
    "    \n",
    "    out = c - (A*(θ**α)*(1 - β))/(θ*(r+λ))*integrate.quad(intf,yr,np.inf)[0]\n",
    "    \n",
    "    return out"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definimos el procedimiento que resuelve el modelo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def SolveModel(prms, tol = 1e-6, step=0.5):\n",
    "    b, λ, A, α, β, r, c, F = prms\n",
    "    \n",
    "    yr0, θ0 = 1, 1\n",
    "    \n",
    "    diftheta = 1\n",
    "    \n",
    "    while diftheta>tol:\n",
    "           \n",
    "        difyr = 1\n",
    "        \n",
    "        print('**** Solving for yr given theta')\n",
    "    \n",
    "        while difyr>tol:\n",
    "            yr1 = TyR(yr0, θ0, prms)\n",
    "            difyr = np.abs(yr1-yr0)\n",
    "            print('**** convergence: ', difyr)\n",
    "            yr0 = step*yr1 + (1-step)*yr0\n",
    "        \n",
    "        yrout = yr1\n",
    "        \n",
    "        print('Solving for theta')\n",
    "        \n",
    "        θ1 = root(lambda x: JCC(x, yrout, prms), θ0).x[0]\n",
    "        \n",
    "        diftheta = np.abs(θ1-θ0)\n",
    "        print('convergence: ',diftheta)\n",
    "        θ0 = step*θ1 + (1-step)*θ0\n",
    "    \n",
    "    θout = θ1\n",
    "    \n",
    "    uout = λ / ((A*(θout**α))*(1-F.cdf(yrout)) + λ)\n",
    "    vout = θout*uout\n",
    "    \n",
    "    return yrout, θout, uout, vout"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Resolvemos el modelo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "**** Solving for yr given theta\n",
      "**** convergence:  5.10538612284\n",
      "**** convergence:  1.83481533174\n",
      "**** convergence:  0.121075272636\n",
      "**** convergence:  0.0154280079264\n",
      "**** convergence:  0.00216479530974\n",
      "**** convergence:  0.000300212613543\n",
      "**** convergence:  4.1702290142e-05\n",
      "**** convergence:  5.79150266988e-06\n",
      "**** convergence:  8.04334050919e-07\n",
      "Solving for theta\n",
      "convergence:  1.50117938333\n",
      "**** Solving for yr given theta\n",
      "**** convergence:  0.510984047018\n",
      "**** convergence:  0.136094085393\n",
      "**** convergence:  0.0286789138869\n",
      "**** convergence:  0.00648005247797\n",
      "**** convergence:  0.00144352764124\n",
      "**** convergence:  0.000322608141384\n",
      "**** convergence:  7.20465772801e-05\n",
      "**** convergence:  1.60924133095e-05\n",
      "**** convergence:  3.5942925054e-06\n",
      "**** convergence:  8.02803278521e-07\n",
      "Solving for theta\n",
      "convergence:  0.0606289017393\n",
      "**** Solving for yr given theta\n",
      "**** convergence:  0.0153514657114\n",
      "**** convergence:  0.00349056513261\n",
      "**** convergence:  0.00078770951815\n",
      "**** convergence:  0.000178067218056\n",
      "**** convergence:  4.02377142041e-05\n",
      "**** convergence:  9.09328366783e-06\n",
      "**** convergence:  2.05494198635e-06\n",
      "**** convergence:  4.64387304167e-07\n",
      "Solving for theta\n",
      "convergence:  0.0120067630238\n",
      "**** Solving for yr given theta\n",
      "**** convergence:  0.0030093628574\n",
      "**** convergence:  0.000682446755244\n",
      "**** convergence:  0.000154532803086\n",
      "**** convergence:  3.50040548653e-05\n",
      "**** convergence:  7.92835390673e-06\n",
      "**** convergence:  1.79578845705e-06\n",
      "**** convergence:  4.06748191573e-07\n",
      "Solving for theta\n",
      "convergence:  0.00243198234351\n",
      "**** Solving for yr given theta\n",
      "**** convergence:  0.000608213687441\n",
      "**** convergence:  0.000137858237381\n",
      "**** convergence:  3.12377275331e-05\n",
      "**** convergence:  7.07873367745e-06\n",
      "**** convergence:  1.60407638639e-06\n",
      "**** convergence:  3.63492980604e-07\n",
      "Solving for theta\n",
      "convergence:  0.0004964307938\n",
      "**** Solving for yr given theta\n",
      "**** convergence:  0.000124202969024\n",
      "**** convergence:  2.81491366598e-05\n",
      "**** convergence:  6.37928008596e-06\n",
      "**** convergence:  1.44572027061e-06\n",
      "**** convergence:  3.2763893687e-07\n",
      "Solving for theta\n",
      "convergence:  9.99643596398e-05\n",
      "**** Solving for yr given theta\n",
      "**** convergence:  2.491736526e-05\n",
      "**** convergence:  5.64710880457e-06\n",
      "**** convergence:  1.27980815723e-06\n",
      "**** convergence:  2.90044608331e-07\n",
      "Solving for theta\n",
      "convergence:  2.1436310905e-05\n",
      "**** Solving for yr given theta\n",
      "**** convergence:  5.42483433152e-06\n",
      "**** convergence:  1.2294443783e-06\n",
      "**** convergence:  2.78631449024e-07\n",
      "Solving for theta\n",
      "convergence:  3.38976274139e-06\n",
      "**** Solving for yr given theta\n",
      "**** convergence:  7.84294539979e-07\n",
      "Solving for theta\n",
      "convergence:  1.79768486985e-07\n"
     ]
    }
   ],
   "source": [
    "# Parametros\n",
    "b = 1\n",
    "λ = 0.05\n",
    "A = 1\n",
    "α = 0.5\n",
    "β = 0.5\n",
    "r = 0.1\n",
    "μ = 0.8\n",
    "σ = 0.5\n",
    "c = 1.0\n",
    "F = stats.lognorm(s = σ, loc = 0, scale = np.exp(μ))    \n",
    "\n",
    "parameters = [b, λ, A, α, β, r, c, F]        \n",
    "\n",
    "# Solución\n",
    "yR, θ, u, v = SolveModel(parameters)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Productividad de reserva =  2.78843517574\n",
      "Tightness                =  1.78843394607\n",
      "Tasa de desempleo        =  0.102885066646\n",
      "Tasa de vacancias        =  0.184003145733\n"
     ]
    }
   ],
   "source": [
    "print('Productividad de reserva = ', yR)\n",
    "print('Tightness                = ', θ)\n",
    "print('Tasa de desempleo        = ', u)\n",
    "print('Tasa de vacancias        = ', v)"
   ]
  }
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