{
 "metadata": {
  "name": "",
  "signature": "sha256:255fb4cc32bceceb22af0c76e038632c61c56f21c88996a9bb34c0a881d0830b"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "import numpy as np\n",
      "import quantecon as qe"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 38
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Economic Dynamics, Chapter 4: Exercises"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Exercises using Hamilton's Markov chain:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P = [[0.971, 0.029, 0    ],\n",
      "     [0.145, 0.778, 0.077],\n",
      "     [0    , 0.508, 0.492]]\n",
      "mc_H = qe.MarkovChain(P)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mc_H.P"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 40,
       "text": [
        "array([[ 0.971,  0.029,  0.   ],\n",
        "       [ 0.145,  0.778,  0.077],\n",
        "       [ 0.   ,  0.508,  0.492]])"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "states = {'NG': 0, 'MR': 1, 'SR': 2}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 41
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "4.2 Finite State Markov Chains"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "4.2.2 Marginal Distributions"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.3"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us use the following function from the previous exercise set:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def cross_sectional_dist(mc, init, T, num_reps=100):\n",
      "    \"\"\"\n",
      "    Return a distribution of visits at time T across num_reps sample paths\n",
      "    of mc with an initial state init.\n",
      "    \n",
      "    \"\"\"\n",
      "    x = np.empty(num_reps, dtype=int)\n",
      "    for i in range(num_reps):\n",
      "        x[i] = mc.simulate(init=init, sample_size=T+1)[-1]\n",
      "    bins = np.arange(mc.n+1)\n",
      "    hist, bin_edges = np.histogram(x, bins=bins)\n",
      "    dist = hist/len(x)\n",
      "    return dist"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "psi = [0, 0, 1]\n",
      "psi_10 = cross_sectional_dist(mc_H, init=psi, T=10)\n",
      "print (psi_10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0.57  0.41  0.02]\n"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "psi_10[states['NG']]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 54,
       "text": [
        "0.56999999999999995"
       ]
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "4.2.3 Other Identities"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.12"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "profits = [1000, 0, -1000]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t = 5\n",
      "exp_profits_vec = np.linalg.matrix_power(mc_H.P, t).dot(profits)\n",
      "print (exp_profits_vec)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 885.34763268  388.68267371  217.74304877]\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "exp_profits_vec[states['SR']]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "217.74304876607994"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.13"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t = 1000\n",
      "exp_profits_vec = np.linalg.matrix_power(mc_H.P, t).dot(profits)\n",
      "print (exp_profits_vec)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 788.16  788.16  788.16]\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.14"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t = 5\n",
      "psi=[0.2,0.2,0.6]\n",
      "prob_t_ini=np.dot(psi,np.linalg.matrix_power(mc_H.P, t))\n",
      "\n",
      "exp_profits = prob_t_ini.dot(profits)\n",
      "print (exp_profits)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "385.451890538\n"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "4.2.4 Constructing Joint Distributions"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.15"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def path_prob(mc, init, X):\n",
      "    \"\"\"\n",
      "     calucualtion of the probablility of a path x\n",
      "    \n",
      "    \"\"\"\n",
      "    \n",
      "    prob=init[X[0]]\n",
      "    for t in range(len(X)-1):\n",
      "        prob=prob*mc[X[t],X[t+1]]\n",
      "    return prob\n",
      "\n",
      "Path=[0,1,0]\n",
      "path_010=path_prob(mc_H.P, init=psi, X=Path)\n",
      "print(path_010)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.000841\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.16"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.18"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Y_0=[1,1,1]\n",
      "Y_1=[1,2,1]\n",
      "Y_2=[1,1,2]\n",
      "Y_3=[1,2,2]\n",
      "Y_4=[2,1,1]\n",
      "Y_5=[2,1,2]\n",
      "Y_6=[2,2,1]\n",
      "Y_7=[2,2,2]\n",
      "\n",
      "Y=[Y_0,Y_1,Y_2,Y_3,Y_4,Y_5,Y_6,Y_7]\n",
      "\n",
      "path_recession= np.empty(len(Y))\n",
      "for i in range(len(Y)):\n",
      " path_recession[i]=path_prob(mc_H.P, init=psi, X=Y[i])\n",
      "\n",
      "print(path_recession)\n",
      "\n",
      "I=[1,1,1,1,1,1,1,1]\n",
      "Pr_recession=np.dot(path_recession,I)\n",
      "\n",
      "print(Pr_recession)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0.1210568  0.0078232  0.0119812  0.0075768  0.2371344  0.0234696\n",
        "  0.1499616  0.1452384]\n",
        "0.704242\n"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.19"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.20"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "Feasible_Path_0=[0,0,0]   \n",
      "Feasible_Path_1=[0,0,1] \n",
      "Feasible_Path_2=[0,0,2]\n",
      "Feasible_Path_3=[0,1,0] \n",
      "Feasible_Path_4=[0,1,1]     \n",
      "Feasible_Path_5=[0,1,2]  \n",
      "Feasible_Path_6=[0,2,0]     \n",
      "Feasible_Path_7=[0,2,1]     \n",
      "Feasible_Path_8=[0,2,2]    \n",
      " \n",
      "Feasible_Path_9=[1,0,0]   \n",
      "Feasible_Path_10=[1,0,1] \n",
      "Feasible_Path_11=[1,0,2]\n",
      "Feasible_Path_12=[1,1,0] \n",
      "Feasible_Path_13=[1,1,1]     \n",
      "Feasible_Path_14=[1,1,2]  \n",
      "Feasible_Path_15=[1,2,0]     \n",
      "Feasible_Path_16=[1,2,1]     \n",
      "Feasible_Path_17=[1,2,2]\n",
      "\n",
      "Feasible_Path_18=[2,0,0]   \n",
      "Feasible_Path_19=[2,0,1] \n",
      "Feasible_Path_20=[2,0,2]\n",
      "Feasible_Path_21=[2,1,0] \n",
      "Feasible_Path_22=[2,1,1]     \n",
      "Feasible_Path_23=[2,1,2]  \n",
      "Feasible_Path_24=[2,2,0]     \n",
      "Feasible_Path_25=[2,2,1]     \n",
      "Feasible_Path_26=[2,2,2]\n",
      "\n",
      "Feasible_Path=[Feasible_Path_0,Feasible_Path_1,Feasible_Path_2,Feasible_Path_3\n",
      "               ,Feasible_Path_4,Feasible_Path_5,Feasible_Path_6,Feasible_Path_7\n",
      "               ,Feasible_Path_8,Feasible_Path_9,Feasible_Path_10,Feasible_Path_11\n",
      "               ,Feasible_Path_12,Feasible_Path_13,Feasible_Path_14,Feasible_Path_15\n",
      "               ,Feasible_Path_16,Feasible_Path_17,Feasible_Path_18,Feasible_Path_19\n",
      "               ,Feasible_Path_20,Feasible_Path_21,Feasible_Path_22,Feasible_Path_23\n",
      "               ,Feasible_Path_24,Feasible_Path_25,Feasible_Path_26,]\n",
      "\n",
      "\n",
      "rho=1/(1+0.05)\n",
      "T=2\n",
      "\n",
      "\n",
      "profits_path=np.empty(len(Feasible_Path))\n",
      "for j in range(len(Feasible_Path)):\n",
      "  for i in range(T):\n",
      "    profits_path[j]=profits_path[j]+rho**i*profits[Feasible_Path[j][i]]\n",
      "\n",
      "Feasible_path_prob=np.empty(len(Feasible_Path))\n",
      "\n",
      "for j in range(len(Feasible_Path)):\n",
      "     Feasible_path_prob[j]=path_prob(mc_H.P, init=psi, X=Feasible_Path[j])    \n",
      "\n",
      "\n",
      "print(profits_path)\n",
      "print(Feasible_path_prob)  \n",
      "\n",
      "NPV=np.dot(Feasible_path_prob,profits_path)\n",
      "\n",
      "print(NPV)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 1952.38095238  1952.38095238  1952.38095238  1000.          1000.          1000.\n",
        "    47.61904762    47.61904762    47.61904762   952.38095238   952.38095238\n",
        "   952.38095238     0.             0.             0.          -952.38095238\n",
        "  -952.38095238  -952.38095238   -47.61904762   -47.61904762   -47.61904762\n",
        " -1000.         -1000.         -1000.         -1952.38095238 -1952.38095238\n",
        " -1952.38095238]\n",
        "[ 0.1885682  0.0056318  0.         0.000841   0.0045124  0.0004466  0.         0.\n",
        "  0.         0.028159   0.000841   0.         0.022562   0.1210568\n",
        "  0.0119812  0.         0.0078232  0.0075768  0.         0.         0.\n",
        "  0.044196   0.2371344  0.0234696  0.         0.1499616  0.1452384]\n",
        "-483.238095238\n"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.22"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.2.23"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "T=2\n",
      "NPV=0\n",
      "\n",
      "prob_t_ini=np.dot(psi,np.linalg.matrix_power(mc_H.P, t))\n",
      "exp_profits = prob_t_ini.dot(profits)\n",
      "\n",
      "for t in range(T):\n",
      "    NPV=NPV+rho**t*exp_profits\n",
      "print(NPV)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "-170.638095238\n"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "4.3 Stability of Finite State MCs"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "4.3.1 Stationary Distributions"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.3.9"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "vecs = mc_H.stationary_distributions\n",
      "print(vecs)\n",
      "\n",
      "exp_profits_st=vecs.dot(profits)\n",
      "print(exp_profits_st)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 0.8128   0.16256  0.02464]]\n",
        "[ 788.16]\n"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.3.10"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from numpy import ones, identity,transpose\n",
      "from numpy.linalg import solve\n",
      "\n",
      "I=identity(3)\n",
      "Q,b=ones((3,3)),ones((3,1))\n",
      "\n",
      "A=transpose(I-mc_H.P+Q)\n",
      "print (solve(A,b))\n",
      "\n",
      "\n",
      "psi = solve(A,b)\n",
      "psi_10 = cross_sectional_dist(mc_H, init=psi, T=20)\n",
      "print (psi_10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 0.8128 ]\n",
        " [ 0.16256]\n",
        " [ 0.02464]]\n",
        "[ 0.83  0.15  0.02]\n"
       ]
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "4.3.3 Stability"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.3.27"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "4.3.4 The Law of Large Numbers"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.3.34"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exercise 4.3.36"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}