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    "### <font color=\"000088\"> Malthusian Model: Convergence </font>\n",
    "\n",
    "Recall:\n",
    "\n",
    ">(2.29) $ \\frac{dL/dt}{L} = \\frac{d\\ln(L)}{dt} = n = \\beta  \\left( \\frac{y}{\\phi y^{sub}}-1 \\right) $\n",
    "\n",
    ">(2.39) $ \\ln(y) = \\theta\\ln(\\kappa) + \\ln(H) -\\ln(L)/\\gamma $\n",
    "\n",
    "Substitute:\n",
    "\n",
    ">(2.40) $ \\frac{1}{L}\\frac{dL}{dt} = \\frac{d\\ln(L)}{dt} = n = \\beta  \\left( \\frac{\\kappa^\\theta H L^{-\\gamma}}{\\phi y^{sub}}-1 \\right) $\n",
    "\n",
    ">(2.41) $ \\frac{d\\kappa}{dt} = -(1-\\alpha)(h + (1-1/\\gamma)n +\\delta)\\kappa + (1-\\alpha)s $\n",
    "\n",
    "Define population-adjusted-for-ideas $ P $:\n",
    ">(2.42) $ P = L H^{-\\gamma} $\n",
    "\n",
    "Then we have two state variables—capital-intensity $ \\kappa $, the capital-output ratio, and population-adjusted-for-ideas $ P $. We have two dynamic equations: The rate of change of population-adjusted-for-ideas $ P $ is a function of the capital-output ratio and itself: \n",
    "\n",
    ">(2.43) $ \\frac{1}{P}\\frac{dP}{dt} = p = \\beta  \\left( \\frac{\\kappa^\\theta }{P^{1/\\gamma} \\phi y^{sub}}-1 \\right) -h\\gamma $\n",
    "\n",
    "And the rate of change of capital-intensity $ \\kappa $ is a function of itself andf of the rate of change of population-adjusted-for-ideas $ P $:\n",
    "\n",
    ">(2.44) $ \\frac{d\\kappa}{dt} = (1-\\alpha)s -(1-\\alpha)((1-1/\\gamma)p +\\delta)\\kappa  $\n",
    "\n",
    "With the two parameters $ \\alpha $ and $ \\theta $ related by:\n",
    "\n",
    ">(2.45) $ \\theta = \\frac{\\alpha}{1-\\alpha}  $\n",
    "\n",
    ">(2.46) $ \\alpha = \\frac{\\theta}{1+\\theta}   $\n",
    "\n",
    "And if we set:\n",
    "\n",
    ">(2.47) $ P = \\Pi P^{*mal}     $\n",
    "\n",
    ">(2.48) $ \\frac{1}{\\Pi}\\frac{d\\Pi}{dt} = p = \\beta \\left[ \\pi^{(-1/\\gamma)} \\kappa^\\theta (s/(\\delta + \\gamma h))^{-\\theta} (1 + \\gamma h/\\beta) -1 \\right] - h \\gamma $\n",
    "\n",
    "(2.44) and (2.48) are then our system...\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "#### <font color=\"008800\"> 2.9: Model Simulation Experiments </font>\n",
    "\n",
    "\n",
    "And, with respect to its dynamics, in Python:\n",
    "\n",
    "    delong_classes.malthusian\n",
    "    \n",
    "at:\n",
    "\n",
    "><https://nbviewer.jupyter.org/github/braddelong/LS2019/blob/master/2019-09-06-210a-ancient-intro.ipynb>    \n",
    "<https://github/braddelong/LS2019/blob/master/2019-09-06-210a-ancient-intro.ipynb>\n",
    "    \n",
    "we can examine how the simulated model behaves dynamically.\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "----"
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