{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Malthusian Model: Convergence \n", "\n", "Recall our Solow-Malthus model. The rate of growth of the labor force and population depends on luxury-adjusted income per worker $ y/\\phi $ divided by subsistence $ y^{sub} $:\n", "\n", ">(1) $ \\frac{dL/dt}{L} = \\frac{d\\ln(L)}{dt} = n = \\beta \\left( \\frac{y}{\\phi y^{sub}}-1 \\right) $\n", "\n", "Income per worker depends on the capital-output ratio $ \\kappa $, the level of the useful ideas stock $ H $, and the amount of resource scarcity induced by the labor force $ L $:\n", "\n", ">(2) $ \\ln(y) = \\theta\\ln(\\kappa) + \\ln(H) - \\ln(L)/\\gamma $\n", "\n", ">(3) $ \\frac{d\\kappa}{dt} = (1-\\alpha)s - (1-\\alpha)(h + (1 - 1/\\gamma)n + ֿֿ\\delta)\\kappa $\n", "\n", ">(3') $ \\frac{dH/dt}{H} = h $\n", "\n", "With the parameters $ \\alpha $ and $\\theta $—the capital share of income and the capital-intensity elasticity of income—related by:\n", "\n", ">(4) $ \\theta = \\frac{\\alpha}{1-\\alpha}$ and $ \\alpha = \\frac{\\theta}{1+\\theta} $\n", "\n", "Substitute:\n", "\n", ">(5) $ \\frac{1}{L}\\frac{dL}{dt} = \\frac{d\\ln(L)}{dt} = n = \\beta \\left( \\frac{\\kappa^\\theta H L^{-1/\\gamma}}{\\phi y^{sub}}-1 \\right) $\n", "\n", ">(6) $ \\frac{d\\kappa}{dt} = -(1-\\alpha)(h + (1-1/\\gamma)n +\\delta)\\kappa + (1-\\alpha)s $\n", "\n", "Define ideas-adjusted-for-population $ I $:\n", "\n", ">(7) $ I = H L^{-1/\\gamma} $\n", "\n", ">(8) $ i = h - n/\\gamma $\n", "\n", ">(9) $ \\frac{d\\kappa}{dt} = -(1-\\alpha)(\\gamma h - (\\gamma-1)i +\\delta)\\kappa + (1-\\alpha)s $\n", "\n", ">(10) $ \\frac{d\\kappa}{dt} = (1-\\alpha)s -(1-\\alpha)(\\gamma h +\\delta)\\kappa + (1-\\alpha) (\\gamma-1)i\\kappa $\n", "\n", ">(11) $ \\frac{1}{I}\\frac{dI}{dt} = i = h - n/\\gamma = h - \\frac{\\beta}{\\gamma} \\left( \\frac{\\kappa^\\theta I}{\\phi y^{sub}}-1 \\right) $\n", "\n", "Then we have two state variables—capital-intensity $ \\kappa $, the capital-output ratio, and ideas-adjusted-for-population $ I $. We have two dynamic equations: The rate of change of ideas-adjusted-for-population $ I $ is a function of the capital-output ratio and itself. And the rate of change of capital-intensity $ \\kappa $ is a function of itself and of the rate of change of ideas-adjusted-for-population $ I $." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The steady state is then:\n", "\n", ">(12) $ I^{*mal} = \\frac{H}{L^{1/\\gamma}} = \\phi y^{sub}\\left(\\frac{\\delta}{s}\\right)^{\\theta}\\left(1+ \\frac{\\gamma h}{\\delta}\\right)^{\\theta}\\left( 1 + \\gamma h/\\beta \\right) $\n", "\n", ">(13) $ \\kappa^{*mal} = \\frac{s}{\\gamma h + \\delta} $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "Define:\n", "\n", ">(14) $ I = (1 + \\xi) I^{*mal} $\n", "\n", ">(15) $ \\kappa = (1 + k) \\kappa^{*mal} = (1 + k) (s/(\\delta + \\gamma h)) $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Then:\n", "\n", "From (11):\n", "\n", ">(16) $ \\frac{1}{1 + \\xi}\\frac{d\\xi}{dt} = h - \\frac{\\beta}{\\gamma} \\left( \\frac{(1+k)^\\theta\\left(\\kappa^{*mal}\\right)^\\theta (1+\\xi) I^{*mal}}{\\phi y^{sub}}-1 \\right) $\n", "\n", ">(17) $ \\frac{1}{1+\\xi}\\frac{d\\xi}{dt} = i = h - \\frac{\\beta}{\\gamma} \\left(( 1 + \\gamma h/\\beta )(1+k)^\\theta (1+\\xi) - 1 \\right) $\n", "\n", ">(18) $ \\frac{1}{1+\\xi}\\frac{d\\xi}{dt} = i = h - \\left(( h + \\frac{\\beta}{\\gamma})(1+k)^\\theta (1+\\xi) - \\frac{\\beta}{\\gamma} \\right) $\n", "\n", "Using the approximation:\n", "\n", "> $ 1 + \\theta k = (1+k)^{\\theta} $\n", "\n", ">(19) $ \\frac{1}{1+\\xi}\\frac{d\\xi}{dt} = h - h - \\frac{\\beta}{\\gamma} - h \\theta k - \\frac{\\theta \\beta}{\\gamma}k - h \\xi - \\frac{ \\beta}{\\gamma}\\xi + \\frac{\\beta}{\\gamma} $\n", "\n", ">(20) $ \\frac{d\\xi}{dt} = \\left[ h - h - \\frac{\\beta}{\\gamma} - h \\theta k - \\frac{\\theta \\beta}{\\gamma}k - h \\xi - \\frac{ \\beta}{\\gamma}\\xi + \\frac{\\beta}{\\gamma} \\right](1+\\xi) $\n", "\n", ">(21) $ \\frac{d\\xi}{dt} = -(h \\theta + \\theta \\beta / \\gamma)k - (h + \\beta/\\gamma)\\xi $\n", "\n", "This is our linearized exponential-convergence equation for the deviation of ideas-adjusted-for-the-population $ \\xi $.\n", "\n", "Now on to the capital-instensity. Recall:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">(10) $ \\frac{d\\kappa}{dt} = (1-\\alpha)s -(1-\\alpha)(\\gamma h +\\delta)\\kappa + (1-\\alpha) (\\gamma-1)i\\kappa $\n", "\n", "And from our definition of $ k $ we get:\n", "\n", ">(22) $ \\frac{d\\kappa}{dt} = \\frac{dk}{dt}\\kappa^{*mal} $\n", "\n", ">(23) $ \\kappa^{*mal}\\frac{dk}{dt} = (1-\\alpha)s -(1-\\alpha)(\\gamma h +\\delta)(1+k)\\kappa^{*mal} + (1-\\alpha) (\\gamma-1)i(1+k)\\kappa^{*mal} $\n", "\n", ">(24) $ \\kappa^{*mal}\\frac{dk}{dt} = (1-\\alpha)s -(1-\\alpha)(\\gamma h +\\delta)\\kappa^{*mal} -(1-\\alpha)(\\gamma h +\\delta)k\\kappa^{*mal} + (1-\\alpha) (\\gamma-1)i(1+k)\\kappa^{*mal} $\n", "\n", ">(25) $ \\kappa^{*mal}\\frac{dk}{dt} = -(1-\\alpha)(\\gamma h +\\delta)k\\kappa^{*mal} + (1-\\alpha) (\\gamma-1)i(1+k)\\kappa^{*mal} $\n", "\n", ">(26) $ \\kappa^{*mal}\\frac{dk}{dt} = -(1-\\alpha)sk - (1-\\alpha) (\\gamma -1)(h\\theta + \\theta \\beta / \\gamma)k + (h + \\beta/\\gamma)\\xi)\\kappa^{*mal} $\n", "\n", ">(27) $ \\frac{dk}{dt} = -(1-\\alpha)(\\delta + \\gamma h)k - (1-\\alpha) (\\gamma-1)(h\\theta + \\theta \\beta / \\gamma)k - (1-\\alpha) (\\gamma-1)(h + \\beta/\\gamma)\\xi$\n", "\n", ">(28) $ \\frac{dk}{dt} = -(1-\\alpha)\\left[\\delta + \\gamma h + (\\gamma-1)(h\\theta + \\theta \\beta / \\gamma) \\right]k - (1-\\alpha) (\\gamma-1)( h + \\beta/\\gamma)\\xi$\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">(21) $ \\frac{d\\xi}{dt} = -(h \\theta + \\theta \\beta / \\gamma)k - (h + \\beta/\\gamma)\\xi $\n", "\n", ">(28) $ \\frac{dk}{dt} = -(1-\\alpha)\\left[\\delta + \\gamma h + (\\gamma-1)(h\\theta + \\theta \\beta / \\gamma) \\right]k - (1-\\alpha) (\\gamma-1)( h + \\beta/\\gamma)\\xi$\n", "\n", "This is our linearized exponential-convergence system for the deviation of ideas-adjusted-for-the-population $ \\xi $ and the deviation of capital-intensity $ k $ from steady-state Malthusian equilibrium.\n", "\n", " " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import pandas as pd\n", "\n", "P_vector = []\n", "kappa_vector = []\n", "\n", "P_vector = P_vector + [1.941371*.666]\n", "kappa_vector = kappa_vector + [1.980198*1.166]\n", "\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " P kappa\n", "0 1.292953 2.308911\n", "1 1.317706 2.300611\n", "2 1.341619 2.292520\n", "3 1.364721 2.284634\n", "4 1.387037 2.276947\n", "5 1.408591 2.269454\n", "6 1.429408 2.262151\n", "7 1.449512 2.255031\n", "8 1.468925 2.248092\n", "9 1.487670 2.241327\n", "10 1.505767 2.234734\n", "11 1.523239 2.228307\n", "12 1.540104 2.222042\n", "13 1.556382 2.215936\n", "14 1.572093 2.209983\n", "15 1.587254 2.204181\n", "16 1.601884 2.198526\n", "17 1.615999 2.193013\n", "18 1.629616 2.187639\n", "19 1.642752 2.182401\n", "20 1.655422 2.177296\n", "21 1.667641 2.172319\n", "22 1.679424 2.167468\n", "23 1.690785 2.162739\n", "24 1.701739 2.158130\n", "25 1.712298 2.153637\n", "26 1.722475 2.149258\n", "27 1.732283 2.144989\n", "28 1.741735 2.140828\n", "29 1.750841 2.136772\n", ".. ... ...\n", "71 1.934296 2.033684\n", "72 1.935814 2.032333\n", "73 1.937259 2.031017\n", "74 1.938634 2.029734\n", "75 1.939942 2.028483\n", "76 1.941186 2.027264\n", "77 1.942367 2.026075\n", "78 1.943489 2.024917\n", "79 1.944553 2.023788\n", "80 1.945561 2.022687\n", "81 1.946517 2.021614\n", "82 1.947421 2.020568\n", "83 1.948276 2.019549\n", "84 1.949084 2.018555\n", "85 1.949846 2.017587\n", "86 1.950564 2.016643\n", "87 1.951241 2.015723\n", "88 1.951877 2.014826\n", "89 1.952475 2.013951\n", "90 1.953036 2.013099\n", "91 1.953561 2.012268\n", "92 1.954052 2.011458\n", "93 1.954510 2.010669\n", "94 1.954937 2.009900\n", "95 1.955333 2.009150\n", "96 1.955701 2.008419\n", "97 1.956041 2.007706\n", "98 1.956355 2.007012\n", "99 1.956643 2.006335\n", "100 1.956907 2.005675\n", "\n", "[101 rows x 2 columns]\n" ] } ], "source": [ "for i in range(0,100):\n", " P = (1-0.025-0.0005)*P_vector[i] + 0.025*kappa_vector[i]\n", " kappa = (1-0.02525)*kappa_vector[i]+0.05\n", " P_vector = P_vector + [P]\n", " kappa_vector = kappa_vector + [kappa]\n", " \n", "malthus_converge_df = pd.DataFrame()\n", "malthus_converge_df['P'] = P_vector\n", "malthus_converge_df['kappa'] = kappa_vector\n", "\n", "print(malthus_converge_df)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Malthusian: Convergence \n", "\n", "\n", "\n", "### Catch Our Breath—Further Notes:\n", "\n", "* Now in Xinmiao Wang's hands...\n", "\n", "----\n", "\n", "\n", "\n", "\n", "\n", "
\n", "\n", "----\n", "\n", "* Weblog Support \n", "* nbViewer \n", "\n", " " ] } ], "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": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 4 }