Human Knowledge, Natural Resources, and Labor Efficiency-Lecture_Support.ipynb

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    "# Extending the Solow Model: Human Knowledge, Natural Resources, and Labor Efficiency: Lecture Support\n",
    "\n",
    "* <https://github.com/braddelong/LSF18E101B/blob/master/Human%20Knowledge%2C%20Natural%20Resources%2C%20and%20Labor%20Efficiency-Lecture_Support.ipynb>\n",
    "* <http://nbviewer.jupyter.org/github/braddelong/LSF18E101B/blob/master/Human%20Knowledge%2C%20Natural%20Resources%2C%20and%20Labor%20Efficiency-Lecture_Support.ipynb>\n",
    "* <http://datahub.berkeley.edu/user-redirect/interact?account=&repo=nbviewer.jupyter.org&branch=braddelong&path=LSF18E101B/blob/master/Human%20Knowledge%2C%20Natural%20Resources%2C%20and%20Labor%20Efficiency-Lecture_Support.ipynb>\n",
    "* <https://www.icloud.com/keynote/0vEcqP380_yG3W4AgUW3lLxNw> "
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    "### From Efficiency to Ideas\n",
    "\n",
    "We have our production function, output as a function of the capital stock, the product of the labor force and the efficiency with which that labor can produce, and the single parameter $ \\alpha $ describing how fast diminishing returns to capital set in as the capital-labor ratio increases:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ Y = K^{\\alpha}(EL)^{1-\\alpha}  $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ ln(Y) = {\\alpha}(ln(K)) + (1-\\alpha)(EL)  $\n",
    "\n",
    "But what determines the efficiency of labor? Ideas, certainly—the level of technological knowledge in its society; its understanding and ability to implement productive organizations; plus the skills, education, and experience of the workforce. But Malthus's insight was that he needed to focus on the role of resources and, in particular, _resource scarcity_ in limiting the efficiency with which labor can work. Ideas, you see, are non-rival: that one person or group has and is using an idea does not mean another person or group cannot productively use the same idea. The same is not true of natural resources. Natural resources are _rival_: crowd twice as many farmers on the same soil, and their productive efficiency is likely to drop.\n",
    "\n",
    "Thus we write the efficiency of labor as a function of society's level of ideas—technology; organizational capability; plus experience, skills, and education—which we will call H, non-rival, and of society's level of resources R divided by the labor force L:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ E = \\left(\\frac{R}{L}\\right)^{\\left(\\frac{1}{1+\\gamma}\\right)}H^{\\left(\\frac{\\gamma}{1+\\gamma}\\right)}  $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ ln(E) = {\\left(\\frac{1}{1+\\gamma}\\right)}ln\\left(\\frac{R}{L}\\right) + \n",
    "{\\left(\\frac{\\gamma}{1+\\gamma}\\right)}H  $\n",
    "\n",
    "With the parameter $ \\gamma $ describing the ratio of the salience of ideas compared to natural resources in generating production efficiency. Why the ratio—and thus all of the $ {\\gamma}/(1-\\gamma) $ terms? Because it will lead to simpler formulas later on, so why not define our production function in terms of it at the start?\n",
    "\n",
    "We thus gain an alternative formulation of the production function when we want to peel back _resource scarcity_ as an influence lowering the efficiency of labor E:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ Y = K^{\\alpha}R^{\\left(\\frac{1-\\alpha}{1+\\gamma}\\right)}(HL)^{\\left(\\frac{\\gamma(1-\\alpha)}{1+\\gamma}\\right)}  $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ ln(Y) = {\\alpha}(ln(K)) + {\\left(\\frac{1-\\alpha}{1+\\gamma}\\right)}ln(R) + \n",
    "{\\left(\\frac{\\gamma(1-\\alpha)}{1+\\gamma}\\right)}ln(HL)  $\n",
    "\n",
    "We also note that that in this setup there is an interaction between the rate of growth of the labor force n and the rate of growth of the efficiency of labor g. Taking the natural log and then the derivative, we get expressions for the growth rate g of the efficiency of labor E:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ \\frac{dln(E)}{dt} = \n",
    "\\left(\\frac{1}{1+\\gamma}\\right)\n",
    "\\left(\\frac{dln(R)}{dt} - \\frac{dln(L)}{dt} \\right) +\n",
    "\\left(\\frac{\\gamma}{1+\\gamma}\\right)\n",
    "\\left(\\frac{dln(H)}{dt}\\right)  $\n",
    "\n",
    "If natural resources are constant, this becomes:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ \\frac{dln(E)}{dt} = \n",
    "\\left(\\frac{\\gamma}{1+\\gamma}\\right)\n",
    "\\left(\\frac{dln(H)}{dt}\\right) -\n",
    "\\left(\\frac{1}{1+\\gamma}\\right)\n",
    "\\left(\\frac{dln(L)}{dt} \\right) $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ g = \\left(\\frac{\\gamma}{1+\\gamma}\\right)h - \\left(\\frac{1}{1+\\gamma}\\right)n  $\n",
    "\n",
    "with \"h\" being the proportional rate of growth of the ideas—engineering technology; business and market organization; plus workers' skills, experience, and education—and g and n being, as before, the proportional rates of growth of labor efficiency and the labor force, respectively.\n",
    "\n",
    "Faster population growth thus exerts a drag on the growth of the efficiency of labor...\n",
    "\n",
    "We can have stagnant living standards if improvements in ideas or just offset by the effects of diminish natural resources per capita. But how likely is it that that just happens?\n",
    "\n",
    "&nbsp;"
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    "### Incorporating Natural Resource Scarcity\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ Y = K^{\\alpha}Y = K^{\\alpha}(EL)^{1-\\alpha} $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ \\frac{Y}{L} =\n",
    "\\left(\\frac{K}{Y}\\right)^\\left(\\frac{\\alpha}{1-\\alpha}\\right)\n",
    "E  $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ E = \\left(\\frac{N}{L}\\right)^{\\left(\\frac{1}{1+\\gamma}\\right)}H^{\\left(\\frac{\\gamma}{1+\\gamma}\\right)}  $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ \\frac{Y}{L} = \n",
    "\\left(\\frac{K}{Y}\\right) \n",
    "^\\left(\\frac{\\alpha}{1-\\alpha}\\right) \n",
    "\\left(\\frac{N}{L}\\right)^{\\left(\\frac{1}{1+\\gamma}\\right)}H^{\\left(\\frac{\\gamma}{1+\\gamma}\\right)} $\n",
    "\n",
    "And if the natural resource stock in the economy is constant,   \n",
    "we might as well choose units that set it equal to 1:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ \\left(\\frac{Y}{L}\\right) =  \\left(\\frac{K}{Y}\\right)^{\\left(\\frac{\\alpha}{1-\\alpha}\\right)}  H^{\\left(\\frac{\\gamma}{1 + \\gamma}\\right)} L^{\\left(-\\frac{1}{1 + \\gamma}\\right)}  $\n",
    "\n",
    "&nbsp;"
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    "### We Need One More Piece...\n",
    "\n",
    "$ \\frac{K}{Y} $ was our equilibrating variable and:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $ \\left(\\frac{K}{Y}\\right)^* = \\frac{s}{n+g+\\delta} $\n",
    "\n",
    "was our equilibrium condition. Now we want y to be a second equilibrating variable and:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $ y = \\bar{y_n} $ to be a second equilibrium condition."
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    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $ ln{\\left(\\frac{Y}{L}\\right)} ={\\left(\\frac{\\alpha}{1-\\alpha}\\right)} ln{\\left(\\frac{K}{Y}\\right)} + ln(E) $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $ ln{\\left(\\frac{Y}{L}\\right)} =\n",
    "{\\left(\\frac{\\alpha}{1-\\alpha}\\right)} ln{\\left(\\frac{s}{n+g+\\delta}\\right)} + ln(E) $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $ ln{\\left(\\frac{Y}{L}\\right)} =\n",
    "{\\left(\\frac{\\alpha}{1-\\alpha}\\right)} ln{\\left(\\frac{s}{n+g+\\delta}\\right)} + \n",
    "{\\left(\\frac{\\gamma}{1+\\gamma}\\right)}ln(H) - {\\left(\\frac{1}{1+\\gamma}\\right)}ln(L) $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $ ln{\\left(\\bar{y_n}\\right)} =\n",
    "{\\left(\\frac{\\alpha}{1-\\alpha}\\right)} ln{\\left(\\frac{s}{n+g+\\delta}\\right)} + \n",
    "{\\left(\\frac{\\gamma}{1+\\gamma}\\right)}ln(H) - {\\left(\\frac{1}{1+\\gamma}\\right)}ln(L) $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $0 = \n",
    "{\\left(\\frac{\\gamma}{1+\\gamma}\\right)}\\frac{d(ln(H))}{dt} - \n",
    "{\\left(\\frac{1}{1+\\gamma}\\right)}\\frac{d(ln(L))}{dt} $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $0 = \n",
    "\\gamma\\frac{d(ln(H))}{dt} - \n",
    "\\frac{d(ln(L))}{dt} $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\n",
    "$ \\frac{dln(L)}{dt} = {\\gamma}\\frac{dln(H)}{dt} $\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$ n = {\\gamma}h $\n",
    "\n",
    "We would not expect to see growth in ideas to show up as increases in output per worker levels if population growth is fast enough to keep increases in the economy's idea stock from showing up in increases in the efficiency of labor.\n",
    "\n",
    "That rate of population growth is simply equal to $ \\gamma $ times the proportional rate of increase in the idea stock.\n",
    "\n",
    "&nbsp;"
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    "### Making Labor-Force Growth Dependent on Living Standards\n",
    "\n",
    "Thomas Robert Malthus's key insight was that prosperity produced faster population growth which generated resource scarcity which in turn curbed prosperity. With output per worker written as:\n",
    "\n",
    "> $ y_t = \\frac{Y_t}{L_t} $\n",
    "\n",
    "we model this by setting the rate of population and labor force growth in each year t equal to:\n",
    "\n",
    "> $ n_t = {\\beta}(y_t - \\bar{y}) $ if $ y_t < y_{peak} $\n",
    "\n",
    "> $ n_{max} = {\\beta}(y_{peak} - \\bar{y}) $\n",
    "\n",
    "> $ n_t = n_{max}\\left(\\frac{y_t}{y_{peak}}\\right)^{-\\eta} $ if $ y_t > y_{peak} $\n",
    "\n",
    "with $ \\bar{y} $ being the \"subsistence\" level of output per worker, $ y_{peak} $ being that level of output per worker at which the society is rich enough with enough women literate and possessing social power to begin taking conscious control over fertility, and $ n_{max} = {\\beta}(y_{peak} - \\bar{y}) $ the maximum rate of population and labor force growth attained.\n",
    "\n",
    "The code cell immediately below presents a \"THIRD MALTHUSIAN FUNCTION\" that implements this dependence of the rate of population growth in year t $ n_t $ on the level of prosperity in this \"demographic transition\" way.\n",
    "\n",
    "Using this third Malthusian function, write code that analyzes what happens with endogenous population growth but no growth in the ideas stock after 1000 BC, h = 0.0, over the period 1000 BC - 1000 AD. Begin with a labor force of 55 million, an efficiency of labor of 500, and a capital-output ratio of 3 in 1000 BC. Set the variables YoverLstagnation1000 and LFstagnation1000 to the value of output-per-worker and of the labor force you calculate for the year 1000, respectively. Use the default parameters for the determinants of population growth:\n",
    "\n",
    ">$ \\bar{y} = 1500 $   \n",
    "$ \\beta = 0.00001 $   \n",
    "$ y_{peak} = 3500 $   \n",
    "$ \\eta = 2 $\n",
    "\n",
    "This means that:\n",
    "\n",
    ">At $ y = 1500 $ ⇒ $ n = 0 $   \n",
    "At $ y = y_{peak} = 3500 $ ⇒ $ n = n_{max} = 0.02 $   \n",
    "At $ y = 2y_{peak} = 7000 $ ⇒ $ n = \\frac{n_{max}}{4} = 0.005 $"
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