{ "metadata": { "name": "OLG Solutions" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# OLG Homework solutions (Python)\n", "### Written by Chase Coleman" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from __future__ import division\n", "import numpy as np\n", "import scipy.optimize as opt\n", "import math\n", "import pandas as pd\n", "from numpy.matlib import repmat\n", "import matplotlib.pyplot as plt" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "code", "collapsed": true, "input": [ "# Here we define the functions necessary to solve the problem. See Documentation\n", "def ksssolve(kvec, params):\n", " '''\n", " Used in solving for steady state. Uses model equations to find the\n", " Euler Errors and then returns the difference in Euler Errors.\n", "\n", " Parameters\n", " ----------\n", " kvec : np.ndarray(1dim)\n", " 1 x 2 vector of capital distribution for current period\n", "\n", " params : np.ndarray(1dim)\n", " Vector that contains the parameters necessary for calculation\n", "\n", " Returns\n", " -------\n", " dif : np.ndarray(1dim)\n", " Vector that contains the two Euler Errors produced by kvec\n", " '''\n", "\n", " k2 = kvec[0]\n", " k3 = kvec[1]\n", "\n", " beta = params[0]\n", " delta = params[1]\n", " gamma = params[2]\n", " A = params[3]\n", " alpha = params[4]\n", "\n", " L = 2\n", " K = k2 + k3\n", " w = (1-alpha) * A * (K/L)**alpha\n", " r = alpha * A * (L/K)**(1-alpha)\n", "\n", " c1 = w - k2\n", " c2 = w + (1 + r - delta) * k2 - k3\n", " c3 = (1 + r - delta) * k3\n", "\n", " MU1 = c1**(-gamma)\n", " MU2 = c2**(-gamma)\n", " MU3 = c3**(-gamma)\n", "\n", " Eul1 = MU1 - beta * (1 + r - delta) * MU2\n", " Eul2 = MU2 - beta * (1 + r - delta) * MU3\n", "\n", " diff = np.array([Eul1, Eul2])\n", " return diff\n", "\n", "\n", "def k32solve(k32, params, pathvals):\n", " '''\n", " Used in solving for K32. Uses model equations to find the\n", " Euler Errors and then returns the difference in Euler Errors.\n", "\n", " Parameters\n", " ----------\n", " k32 : float\n", " intial guess for k32\n", "\n", " params : np.ndarray(1dim)\n", " Vector that contains the parameters necessary for calculation\n", "\n", " pathvals : np.ndarray(1dim)\n", " Vector that contains the paths that are established in OLG3per.py\n", "\n", " Returns\n", " -------\n", " dif : float\n", " The Euler Error produced by Euler Equation 2\n", " '''\n", "\n", " beta = params[0]\n", " delta = params[1]\n", " gamma = params[2]\n", " A = params[3]\n", " alpha = params[4]\n", "\n", " w1 = pathvals[0]\n", " r1 = pathvals[1]\n", " r2 = pathvals[2]\n", " k21 = pathvals[3]\n", "\n", " c2 = w1 + (1 + r1 - delta) * k21 - k32\n", " c3 = (1 + r2 - delta) * k32\n", "\n", " MU2 = c2**(-gamma)\n", " MU3 = c3**(-gamma)\n", "\n", " Eul2 = MU2 - beta * (1 + r2 - delta) * MU3\n", "\n", " diff = Eul2\n", "\n", " return diff\n", "\n", "\n", "def ktsolve(kvec, params, pathvals):\n", " '''\n", " Used in solving for kt. Uses model equations to find the\n", " Euler Errors and then returns the difference in Euler Errors.\n", "\n", " Parameters\n", " ----------\n", " kvec : np.ndarray(1dim)\n", " intial guess for k32\n", "\n", " params : np.ndarray(1dim)\n", " Vector that contains the parameters necessary for calculation\n", "\n", " pathvals : np.ndarray(1dim)\n", " Vector that contains the paths that are established in OLG3per.py\n", "\n", " Returns\n", " -------\n", " dif : np.ndarray(1dim)\n", " Vector that contains the two Euler Errors produced by kvec\n", " '''\n", "\n", " k2tp1 = kvec[0]\n", " k3tp2 = kvec[1]\n", "\n", " beta = params[0]\n", " delta = params[1]\n", " gamma = params[2]\n", " A = params[3]\n", " alpha = params[4]\n", "\n", " wt = pathvals[0]\n", " wtp1 = pathvals[1]\n", " rtp1 = pathvals[2]\n", " rtp2 = pathvals[3]\n", "\n", " c1 = wt - k2tp1\n", " c2 = wtp1 + (1 + rtp1 - delta) * k2tp1 - k3tp2\n", " c3 = (1 + rtp2 - delta) * k3tp2\n", "\n", " MU1 = c1**(-gamma)\n", " MU2 = c2**(-gamma)\n", " MU3 = c3**(-gamma)\n", "\n", " Eul1 = MU1 - beta * (1 + rtp1 - delta) * MU2\n", " Eul2 = MU2 - beta * (1 + rtp2 - delta) * MU3\n", " diff = np.array([Eul1, Eul2])\n", "\n", " return diff" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here we solve for the steady state with the intial $\\beta$, where $\\beta = .96^{20}$" ] }, { "cell_type": "code", "collapsed": true, "input": [ "#-------------------------------------------------------------------------#\n", "# Set parameters before we begin\n", "#-------------------------------------------------------------------------#\n", "# beta = 20-year discount factor from lifetime utility function\n", "# delta = 20-year depreciation rate of capital investment\n", "# gamma = coefficient of relative risk aversion from CRRA utility function\n", "# A = productivity parameter from production function\n", "# alpha = capital share of income from production function\n", "# xi = parameter for convex combination updating of Kpaths in TPI\n", "#-------------------------------------------------------------------------#\n", "beta = 0.96**20\n", "delta = 1 - (1 - 0.05)**20\n", "gamma = 3\n", "A = 1\n", "alpha = 0.35\n", "xi = 0.25" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "#-------------------------------------------------------------------------#\n", "# Solve for steady-state\n", "#-------------------------------------------------------------------------#\n", "# kinit = 1 x 2 vector, initial guess for steady state values of steady-\n", "# state distribution of capital savings (k_2,k_3)\n", "# params = 1 x 5 vector of parameters to be passed into the fsolve file\n", "# options = options for fsolve file\n", "# kssvec = 1 x 2 vector of steady-state equilibrium distribution of\n", "# capital\n", "# k2ss = scalar, steady-state savings of young for middle-age\n", "# k3ss = scalar, steady-state savings of middle-age for old\n", "# Kss = scalar, steady-state aggregate capital stock\n", "# Lss = scalar, steady-state aggregate labor demand\n", "# Yss = scalar, steady-state aggregate output\n", "# wss = scalar, steady-state real wage\n", "# rss = scalar, steady-state net interest rate\n", "# c1ss = scalar, steady-state consumption when young (i=1)\n", "# c2ss = scalar, steady-state consumption when middle age (i=2)\n", "# c3ss = scalar, steady-state consumption when old (i=3)\n", "# Css = scalar, steady-state aggregate consumption\n", "#-------------------------------------------------------------------------#\n", "\n", "kinit = [0.1, 0.1]\n", "params = np.array([beta, delta, gamma, A, alpha])\n", "\n", "# options = optimset('Display','off','MaxFunEvals',100000,..\n", "# 'MaxIter',1000,'TolFun',1e-15)\n", "\n", "kssvec = opt.fsolve(ksssolve, kinit, args=(params), xtol=1e-10)\n", "\n", "k2ss = kssvec[0]\n", "k3ss = kssvec[1]\n", "Kss = k2ss + k3ss\n", "Lss = 2\n", "Yss = A*(Kss**alpha) * (Lss**(1-alpha))\n", "wss = (1-alpha) * A * (Kss/Lss)**alpha\n", "rss = alpha * A * (Lss/Kss)**(1-alpha)\n", "c1ss = wss - k2ss\n", "c2ss = wss + (1 + rss - delta) * k2ss - k3ss\n", "c3ss = (1 + rss - delta) * k3ss\n", "Css = c1ss + c2ss + c3ss" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here we solve for steady state when $\\beta = .55$" ] }, { "cell_type": "code", "collapsed": true, "input": [ "beta2 = .55\n", "#-------------------------------------------------------------------------#\n", "# Solve for steady-state\n", "#-------------------------------------------------------------------------#\n", "# kinit = 1 x 2 vector, initial guess for steady state values of steady-\n", "# state distribution of capital savings (k_2,k_3)\n", "# params = 1 x 5 vector of parameters to be passed into the fsolve file\n", "# options = options for fsolve file\n", "# kssvec = 1 x 2 vector of steady-state equilibrium distribution of\n", "# capital\n", "# k2ss = scalar, steady-state savings of young for middle-age\n", "# k3ss = scalar, steady-state savings of middle-age for old\n", "# Kss = scalar, steady-state aggregate capital stock\n", "# Lss = scalar, steady-state aggregate labor demand\n", "# Yss = scalar, steady-state aggregate output\n", "# wss = scalar, steady-state real wage\n", "# rss = scalar, steady-state net interest rate\n", "# c1ss = scalar, steady-state consumption when young (i=1)\n", "# c2ss = scalar, steady-state consumption when middle age (i=2)\n", "# c3ss = scalar, steady-state consumption when old (i=3)\n", "# Css = scalar, steady-state aggregate consumption\n", "#-------------------------------------------------------------------------#\n", "\n", "kinit2 = [0.1, 0.1]\n", "params2 = np.array([beta2, delta, gamma, A, alpha])\n", "\n", "# options = optimset('Display','off','MaxFunEvals',100000,..\n", "# 'MaxIter',1000,'TolFun',1e-15)\n", "\n", "kssvec2 = opt.fsolve(ksssolve, kinit2, args=(params2), xtol=1e-10)\n", "\n", "k2ss2 = kssvec2[0]\n", "k3ss2 = kssvec2[1]\n", "Kss2 = k2ss2 + k3ss2\n", "Lss2 = 2\n", "Yss2 = A*(Kss2**alpha) * (Lss2**(1-alpha))\n", "wss2 = (1-alpha) * A * (Kss2/Lss2)**alpha\n", "rss2 = alpha * A * (Lss2/Kss2)**(1-alpha)\n", "c1ss2 = wss2 - k2ss\n", "c2ss2 = wss2 + (1 + rss2 - delta) * k2ss2 - k3ss2\n", "c3ss2 = (1 + rss2 - delta) * k3ss2\n", "Css2 = c1ss2 + c2ss2 + c3ss2" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "# This will solve for problem 3\n", "k21 = .8 * k2ss\n", "k31 = 1.1 * k3ss\n", "K1 = k21 + k31\n", "T = 30\n", "\n", "Kpath = np.hstack([np.linspace(K1, Kss, T), np.ones(T)*Kss])\n", "wpath = ((1-alpha) * A) * (Kpath/Lss)**alpha\n", "rpath = (alpha*A) * (Lss/Kpath)**(1-alpha)\n", "\n", "i = 0\n", "maxit = 50\n", "dist = 10\n", "toler = 10**(-9)\n", "\n", "while dist > toler and i <= maxit:\n", " i = i + 1\n", "\n", " Knewmat = np.zeros((2*T, 3))\n", " Knewmat[0, :] = np.array([k21, k31, K1])\n", "\n", " # solve for initial middle age savings decision k_{3,2}\n", " k32init = 0.01\n", " pathvals = np.array([wpath[0], rpath[0], rpath[1], k21])\n", " k32 = opt.fsolve(k32solve, k32init, args=(params, pathvals), xtol=1e-10)\n", " Knewmat[1, 1] = k32\n", "\n", " # Solve 2-period problems for rest of households\n", " for time in xrange(2*T-2):\n", " kinits = [.01, .01]\n", " pathvals = np.array([wpath[time], wpath[time+1],\n", " rpath[time+1], rpath[time+2]])\n", " kvec = opt.fsolve(ktsolve, kinits, args=(params, pathvals), xtol=1e-10)\n", " Knewmat[time+1, 0] = kvec[0]\n", " Knewmat[time+2, 1] = kvec[1]\n", " Knewmat[time+1, 2] = Knewmat[time+1, 0] + Knewmat[time+1, 1]\n", "\n", " dist = sum((Knewmat[0: T+5, 2] - Kpath[0: T+5])**2)\n", " Kpath = xi * Knewmat[0: T+5, 2] + (1-xi) * Kpath[0: T+5]\n", " Kpath = np.hstack([Kpath, Kss * np.ones(T-5)])\n", " wpath = ((1-alpha) * A) * (Kpath/Lss)**alpha\n", " rpath = (alpha * A) * (Lss/Kpath)**(1-alpha)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "(Kpath).size" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 7, "text": [ "60" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Problem 1\n", "\n", "The steady-state equilibrium with the initial parameter values are listed here.\n", " \n", "Steady State Consumption levels\n", "\n", "$\\bar{C_1} = .2140$\n", "\n", "$\\bar{C_2} = .2227$\n", "\n", "$\\bar{C_3} = .2318$\n", "\n", "Steady State Capital levels\n", "\n", "$\\bar{K_2} = .0286$\n", "\n", "$\\bar{K_3} = .0909$\n", "\n", "Steady State Wage and Interest Rate\n", "\n", "$\\bar{W} = .2421$\n", "\n", "$\\bar{r} = 2.1916$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Problem 2\n", "\n", "The steady-state equilibrium with the initial parameter values are listed here.\n", " \n", "Steady State Consumption levels\n", "\n", "$\\bar{C_1} = .2396$\n", "\n", "$\\bar{C_2} = .2404$\n", "\n", "$\\bar{C_3} = .2552$\n", "\n", "Steady State Capital levels\n", "\n", "$\\bar{K_2} = .0413$\n", "\n", "$\\bar{K_3} = .1173$\n", "\n", "Steady State Wage and Interest Rate\n", "\n", "$\\bar{W} = .2677$\n", "\n", "$\\bar{r} = 1.8179$\n", "\n", "We see that the consumption goes down in early periods of life and that investment in capital goes up.\n", "This makes sense because as people become more patient then they are more willing to wait to consume\n", "their goods." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Problem 3\n", "\n", "The TPI was done using the code above." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Problem 4\n", "\n", "The equilibrium time path is plotted below coming from $.8k_2$ and $1.1k_3$ and moves to \n", "the steady state." ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.figure(figsize=(8, 6))\n", "plt.plot(Kpath[0:T+5])\n", "plt.plot(np.ones_like(Kpath[0:T+5]) * Kss, 'r--')\n", "plt.title('Aggregate Captial Time Path')\n", "plt.xlabel('Time (t)')\n", "plt.ylabel('Capital (K)')\n", "plt.legend(['K(t)', 'Steady-state'], loc=0)\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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E95umHPf4NEAiIjPk4OCAoqIiU5dBLcze3h7Xrl1rkXWxBYCIyAzx76B50rdfm7K/eZ9R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\ndqwQkZFCPPGENGz21z3/lsA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} ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "It takes the capital path 6 periods to be within .0001 of the steady state." ] }, { "cell_type": "code", "collapsed": false, "input": [ "df = pd.DataFrame({'Kpath':Kpath, 'Kpath- ss':Kpath- Kss}, index=range(1, Kpath.size + 1))\n", "df.index.name = 'time'\n", "df" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "
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KpathKpath- ss
time
1 0.122427 0.003478
2 0.116197-0.002752
3 0.119004 0.000055
4 0.118401-0.000548
5 0.118834-0.000115
6 0.118815-0.000134
7 0.118901-0.000048
8 0.118918-0.000031
9 0.118940-0.000009
10 0.118949 0.000000
11 0.118956 0.000007
12 0.118959 0.000010
13 0.118961 0.000012
14 0.118962 0.000013
15 0.118962 0.000013
16 0.118962 0.000012
17 0.118961 0.000012
18 0.118961 0.000012
19 0.118960 0.000011
20 0.118959 0.000010
21 0.118959 0.000010
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