Intermediate_Macroeonomics_Review.ipynb

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    "# <font color=\"880000\">Intermediate Macroeconomics Review</font>\n",
    "\n",
    "<https://github.com/braddelong/LSF18E101B/blob/master/Intermediate_Macroeonomics_Review.ipynb>\n",
    "\n",
    "<http://nbviewer.jupyter.org/github/braddelong/LSF18E101B/blob/master/Intermediate_Macroeonomics_Review.ipynb>\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "## <font color=\"880000\">The Structure of Economic Models</font>\n",
    "\n",
    "* **Variables**: Quantities of interest that we can measure, calculate, or at least hypothesize about...\n",
    "* **Identities**: Relationships between variables that are true \"by the metaphysical necessity of the case\"; they do not tell us what happens or how it happens, but they constrain the possibilities that might happen.\n",
    "* **Behavioral Relationships**: Descriptions of (1) what humans, given their opportunities and opportunity costs presented to them by the economic environment and economic policies, decide what to do; and (2) thus determine the values of economic variables.\n",
    "* **Equilibrium Conditions**: Conditions that, if satisfied, place the economy or at least some subset of economic variables at stable values, or in some sort of balance.\n",
    "    * **Convergence Mechanisms**: A dynamic process in the economy that drives the economy toward an equilibrium position of some sort. \n",
    "    \n",
    "&nbsp;"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## <font color=\"880000\">Solow Growth Model</font>\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Framework</font>\n",
    "\n",
    "The Solow Growth Model (SGM) system of equations:\n",
    "\n",
    ">$ g_L = \\frac{d\\left(L_t\\right)}{dt} = nL_t $ :: labor-force growth equation\n",
    "\n",
    ">$ g_E = \\frac{d\\left(E_t\\right)}{dt} = gE_t $ :: efficiency-of-labor growth equation\n",
    "\n",
    ">$ g_K = \\frac{d\\left(K_t\\right)}{dt} = sY_t - \\delta{K_t} $ :: capital stock growth equation\n",
    "\n",
    ">$ Y_t = \\left(K_t\\right)^{\\alpha}\\left(L_tE_t\\right)^{1-\\alpha} $ :: production function\n",
    "\n",
    "&nbsp;"
   ]
  },
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   "source": [
    "#### <font color=\"008800\">Some Facts: Looking Across Countries</font>\n",
    "\n",
    "<img style=\"display:block; margin-left:auto; margin-right:auto;\" src=\"https://delong.typepad.com/.a/6a00e551f080038834022ad37cda53200c-pi\" alt=\"Banners and Alerts and Macroeconomics Review MRE Macro key\" title=\"Banners_and_Alerts_and_Macroeconomics—Review__MRE___Macro_key.png\" border=\"0\" width=\"600\" height=\"636\" />\n",
    "\n",
    "* U.S. today:\n",
    "    * 3x richer than Mexico\n",
    "    * 9x richer than Nigeria\n",
    "    * 20x richer than Afghanistan\n",
    "\n",
    "* Afghans back at where U.S. was 200 years ago\n",
    "    * Differences between Afghanistan and U.S. then much smaller than now\n",
    "    * In spite of improved communications, transport...\n",
    "\n",
    "&nbsp;"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### <font color=\"008800\">Some Facts: Looking Over Time</font>\n",
    "\n",
    "<img style=\"display:block; margin-left:auto; margin-right:auto;\" src=\"https://delong.typepad.com/.a/6a00e551f080038834022ad3c2a3c6200b-pi\" alt=\"Macroeconomics Review MRE Macro key\" title=\"Macroeconomics—Review__MRE___Macro_key.png\" border=\"0\" width=\"600\" height=\"505\" />\n",
    "\n",
    "* 15x in productivity since 1870\n",
    "    * And this might be an underestimate\n",
    "    * Doubling of life expectancy since 1870\n",
    "* Where do you think the 1919 “Spanish Influenza” epidemic is?\n",
    "* Divide $4000 by 13 and you are at a dollar a day\n",
    "    * Anything less, and you are starving to death\n",
    "* ln(60000/4000)/150 = what? &lt;<a href=\"http://www.wolframalpha.com/input/?i=ln(60000%2F4000)%2F150\">https://www.wolframalpha.com/input/?i=ln(60000%2F4000)%2F150</a>>\n",
    "\n",
    "&nbsp;"
   ]
  },
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   "metadata": {},
   "source": [
    "### <font color=\"000088\">Balanced-Growth Path</font>\n",
    "\n",
    "* $ \\left(\\frac{K}{Y}\\right)^* = \\lim\\limits_{t\\to\\infty}\\left(\\frac{K_t}{Y_t}\\right) = \\frac{s}{n+g+\\delta} $ :: steady-state balanced-growth path capital-output ratio\n",
    "\n",
    "* $ \\left(\\frac{Y_t}{L_tE_t}\\right)^* = \\lim\\limits_{t\\to\\infty}\\left(\\frac{Y_t}{L_tE_t}\\right) = \\left(\\frac{s}{n+g+\\delta}\\right)^{\\frac{\\alpha}{1-\\alpha}} $ :: steady-state balanced-growth path output-per-worker ratio\n",
    "\n",
    "* $ \\left(\\frac{K_t}{L_tE_t}\\right)^* = \\lim\\limits_{t\\to\\infty}\\left(\\frac{K_t}{L_tE_t}\\right) = \\left(\\frac{s}{n+g+\\delta}\\right)^{\\frac{1}{1-\\alpha}} $ :: steady-state balanced-growth path capital-worker ratio\n",
    "\n",
    "    * $ E_t = \\left(E_o{e^{gt}}\\right) $ \n",
    "    * $ E_t = (1+g)^tE_o $\n",
    "\n",
    "&nbsp;"
   ]
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   "source": [
    "### <font color=\"000088\">Convergence</font>\n",
    "\n",
    "convergence rate $ = -(1-\\alpha)(n+g+\\delta) $\n",
    "\n",
    "convergence to the steady-state balanced-growth capital-output ratio:\n",
    "\n",
    ">$ \\frac{K_t}{Y_t} = \\left(1- e^{-(1-\\alpha)(n+g+\\delta)t}\\right)\\left(\\frac{K}{Y}\\right)^* + \\left(e^{-(1-\\alpha)(n+g+\\delta)t}\\right)\\left(\\frac{K_o}{Y_o}\\right) $\n",
    "\n",
    "> $ \\frac{K_t}{Y_t} = \\left(1 - \\frac{1}{[(1+(1-\\alpha)(n+g+\\delta)]^t}\\right)\\left(\\frac{K}{Y}\\right)^* + \\left(\\frac{1}{[(1+(1-\\alpha)(n+g+\\delta)]^t}\\right)\\left(\\frac{K_o}{Y_o}\\right) $\n",
    "\n",
    "&nbsp;"
   ]
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   "metadata": {},
   "source": [
    "#### <font color=\"008800\">Exercises</font>\n",
    "\n",
    "Start with an economy on its balanced-growth path, with:\n",
    "\n",
    "* $ \\alpha $\n",
    "* $ \\delta $\n",
    "* $ n_{ini}, s_{ini}, g_{ini} $\n",
    "* $ L_o, E_o, K_o $\n",
    "\n",
    "And let's change something: $ s_{alt} = s_{ini} + {\\Delta}s $...\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "What is $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* $ going to be?\n",
    "\n",
    "A. $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* = \\left(\\frac{{\\Delta}s}{n+g+\\delta}\\right)^{\\alpha} $\n",
    "\n",
    "B. $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* = \\left(\\frac{{\\Delta}s}{n+g+\\delta}\\right)^\\left({\\frac{\\alpha}{1-\\alpha}}\\right) $\n",
    "\n",
    "C. $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* = \\frac{{\\Delta}s}{n+g+\\delta} $\n",
    "\n",
    "D. $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* = \\frac{(1-\\alpha){\\Delta}s}{n+g+\\delta} $\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "If $ \\alpha = 2/3, \\delta = 0.025, n_{ini} = 0.005, $   \n",
    "$ s_{ini} = 0.16, g_{ini} = 0.01,$ and $ {\\Delta}s = 0.04, $      \n",
    "then $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* $ is going to be?\n",
    "\n",
    "A. 9\n",
    "\n",
    "B. 0.404\n",
    "\n",
    "C. 1\n",
    "\n",
    "D. 1/3\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "3 years after a jump $ {\\Delta}s $ in the savings rate, an economy that started out on its initial balanced-growth path would have seen its capital-output ratio reduce approximately what fraction of the gap between the initial and the alternative balanced-growth path values?\n",
    "\n",
    "A. $ (1-\\alpha)(n+g+\\delta) $\n",
    "\n",
    "B. $ e^{(1-\\alpha)(n+g+\\delta)} $\n",
    "\n",
    "C. $ e^{3(1-\\alpha)(n+g+\\delta)} $\n",
    "\n",
    "D. $ 3(1-\\alpha)(n+g+\\delta) $\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;"
   ]
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   "source": [
    "###  <font color=\"000088\">The Shape of World Economic History</font>\n",
    "\n",
    "<img style=\"display:block; margin-left:auto; margin-right:auto;\" src=\"https://delong.typepad.com/.a/6a00e551f080038834022ad37cdb4a200c-pi\" alt=\"Macroeconomics Review MRE Macro key\" title=\"Macroeconomics—Review__MRE___Macro_key.png\" border=\"0\" width=\"600\" height=\"642\" />\n",
    "\n",
    "* The Solow growth model was built to account for the experience of the modern economic growth era\n",
    "    * 1900-today\n",
    "* But there was a lot of history before then\n",
    "* And things were different then: the past is another country\n",
    "* Slower population growth, and much much slower growth in real standards of living\n",
    "\n",
    "&nbsp;"
   ]
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    "### <font color=\"880000\">Malthus and Efficiency-of-Labor Growth</font>\n",
    "\n",
    "H: ideas: non-rival factors: growth rate h\n",
    "\n",
    "E: efficiency of labor: growth rate g\n",
    "\n",
    "L: labor force: growth rate n\n",
    "\n",
    "N: natural resources: rival factors: growth rate 0 (in most applications)\n",
    "\n",
    ">$ g = \\left(\\frac{\\gamma}{1+\\gamma}\\right)h - \\left(\\frac{1}{1+\\gamma}\\right)n $\n",
    "\n",
    ">$ n = {\\phi}\\ln\\left(\\frac{Y/L}{y^s}\\right) $ :: Malthusian population growth\n",
    "\n",
    ">$ g = \\left(\\frac{\\gamma}{1+\\gamma}\\right)h - {\\phi}\\left(\\frac{1}{1+\\gamma}\\right)\\ln\\left(\\frac{Y/L}{y^{s}}\\right) $\n",
    "\n",
    "&nbsp;"
   ]
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   "source": [
    "#### <font color=\"008800\">Exercises</font>\n",
    "    \n",
    "In a Malthusian equilibrium, what is   \n",
    "the relationship between the rate of   \n",
    "growth $ h $ of the useful ideas stock   \n",
    "and the rate of population growth $ n $?\n",
    "\n",
    "A. $ n = h $\n",
    "\n",
    "B. $ n = {\\gamma}h $\n",
    "\n",
    "C. $ n = \\frac{h}{\\gamma} $\n",
    "\n",
    "D. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Between the year 1 and the year 1800,   \n",
    "$ \\gamma = 3, n = 0.075% $ per year, and   \n",
    "the economy was in Malthusian equilibrium.   \n",
    "What, roughly, was the rate of growth of h?\n",
    "\n",
    "A. 0.025% per year\n",
    "\n",
    "B. 0.075% per year\n",
    "\n",
    "C. 0.225% per year\n",
    "\n",
    "D. None of the above \n",
    "\n",
    "&nbsp;\n",
    "\n",
    "In Malthusian equilibrium, what is the    \n",
    "Malthusian balanced-growth path level    \n",
    "of output per worker (Y/L)*?\n",
    "\n",
    "A. $ \\left(Y/L\\right)^* = e^{\\left(\\frac{{\\gamma}h}{\\phi}\\right)}y^s $\n",
    "\n",
    "B. $ \\left(Y/L\\right)^* = {\\left(\\frac{{\\gamma}h}{\\phi}\\right)}y^s $\n",
    "\n",
    "C. $ \\left(Y/L\\right)^* = {\\phi}{\\gamma}hy^s $\n",
    "\n",
    "D. None of the above \n",
    "\n",
    "&nbsp;"
   ]
  },
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   "metadata": {},
   "source": [
    "### <font color=\"000088\">How Did We Escape?</font>\n",
    "\n",
    "Two sets of theories for escape:\n",
    "\n",
    "* Eye of the needle\n",
    "    * Cultural-scientific\n",
    "    * Resource-technology\n",
    "    * Plunder-exploitation\n",
    "    * Variants: \"We almost got there many times\" and \"we never got close before\" variants\n",
    "    * Variants: Commercial Revolution, Industrial Revolution, or Modern Economic Growth?\n",
    "\n",
    "Or:\n",
    "\n",
    "* Two heads are better than one...\n",
    "    * $ h = \\left(h_1\\right)L^{\\lambda} $ :: idea generation\n",
    "    \n",
    "Plus:\n",
    "\n",
    "* Demographic transition...\n",
    "    * $ n = \\min\\left({\\phi}\\ln\\left(\\frac{Y/L}{y^s}\\right), \\frac{n_1}{Y/L}\\right) $\n",
    "\n",
    "&nbsp;"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Escape: Industrial Revolution and Modern Economic Growth</font>\n",
    "\n",
    "* Elasticity of Demand as a Key (not on final)\n",
    "* Productivity Trends in the North Atlantic\n",
    "    * Britain the First Industrial Nation\n",
    "    * Britain richer—but with low real wages\n",
    "    * British growth acceleration\n",
    "    * But America growing faster from 1800\n",
    "    * And American growth acceleration—modern economic growth and the industrial research lab\n",
    "    * Until the productivity growth slowdon of the 1970s\n",
    "        * And then the speed up of the new-economy 1990s\n",
    "        * And then the growth collapse of the Great Recession\n",
    "        \n",
    "&nbsp;"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Income and Wealth Inequality</font>\n",
    "\n",
    "(not on exam)\n",
    "\n",
    "Kaldor facts:\n",
    "\n",
    "* Constant r (=αK/Y)\n",
    "*C onstant wL/Y (= 1-α)\n",
    "* Constant K/Y\n",
    "* Constant g\n",
    "* d(ln(w))/dt = g\n",
    "\n",
    "Piketty facts:\n",
    "\n",
    "* Increase in W/K\n",
    "* Increase in market-to-book ratio for K\n",
    "* Divergence between marginal product of capital and average return\n",
    "* Substantial decrease in real interest rates in financial markets\n",
    "\n",
    "Plutocracy and its fear of creative destruction\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Measuring Economic Growth Truly</font>\n",
    "\n",
    "(not on exam)\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Global Patterns</font>\n",
    "\n",
    "Divergence, 1800-1975\n",
    "\n",
    "* Britain and U.S. growing together\n",
    "* OECD convergence 1945-present\n",
    "* Behind Iron Curtain divergence\n",
    "* General divergence 1800-1975\n",
    "* From a fivefold to a fifty-fold divergence\n",
    "\n",
    "Convergence 1975-present?\n",
    "\n",
    "* East Asia\n",
    "* Japan\n",
    "* China\n",
    "\n",
    "\n",
    "How to understand?\n",
    "\n",
    "* $ \\alpha = 3/5 $\n",
    "* Schooling very important for the efficiency of labor\n",
    "\n",
    "&nbsp;"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Modeling Global Patterns</font>\n",
    "\n",
    "We need a high capital share α:\n",
    "\n",
    "* To make “convergence” take a long time\n",
    "* To amplify the effects of differences in (K/Y)* on prosperity\n",
    "\n",
    "We need n to be inversely and s strongly correlated with E\n",
    "\n",
    "* Demographic transition\n",
    "* Favorable relative price structure\n",
    "\n",
    "And we need education to be a key link:\n",
    "\n",
    "* We need technology transfer to a poorly educated population to be nearly impossible…\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font color=\"880000\">Business Cycles</font>\n",
    "\n",
    "<img style=\"display:block; margin-left:auto; margin-right:auto;\" src=\"http://delong.typepad.com/.a/6a00e551f0800388340223c8472ebd200c-pi\" alt=\"2018 03 08 Macroeconomics Flexprice MRE key\" title=\"2018-03-08_Macroeconomics_Flexprice__MRE_key.png\" border=\"0\" width=\"600\" />\n",
    "\n",
    "* Okun's Law\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font color=\"880000\">Flexible-Price Models</font>\n",
    "\n",
    "Full employment (because of flexible wages and prices and debt)\n",
    "\n",
    "* Unemployment rate equal to NAIRU\n",
    "* Production equal to potential output\n",
    "\n",
    "Shifts of production and spending across categories\n",
    "\n",
    "* In response to changes in the economic environment\n",
    "* And in response to changes in economic policy\n",
    "* As a result of shifts in the long-term real risky interest rate r\n",
    "\n",
    "&nbsp;"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">The Business-Cycle NIPA Framework</font>\n",
    "\n",
    "* $ Y = C + I + G + (GX - IM) $ :: national income and product\n",
    "* $ C = c_o + c_y(1-t)Y $ :: consumption function—consumer confidence; marginal propensity to consume; net taxes-less-transfers rate\n",
    "* $ I = I_o - I_r{r} $ :: investment spending; \"animal spirits\"\n",
    "* $ G $\n",
    "* $ IM = im_y{Y} $ :: imports\n",
    "* $ \\epsilon = \\epsilon_o + \\epsilon_r(r^f - r) $ :: exchange rate; foreign exchange speculators; \"gnomes of Zurich\"\n",
    "* $ GX = x_f{Y^f} + x_\\epsilon{\\epsilon} $ :: gross exports\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">The Flexible-Price Model IS Curve Equation</font>\n",
    "\n",
    "$ Y^* = Y = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $\n",
    "\n",
    "Alternatively:\n",
    "\n",
    "$ Y^* = Y = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)(i - \\pi + \\rho) $\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "#### <font color=\"008800\">Exercises</font>\n",
    "\n",
    "$ Y^* = Y = AD = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $\n",
    "\n",
    "* flexprice causation runs from left to right...\n",
    "* real wage W/P adjusts to get full employment and Y = Y*\n",
    "* price level P adjusts to get AD = Y\n",
    "* interest rate r adjusts to get the components of AD to match up...\n",
    "\n",
    "Start from one equilibrium \"ini\" and consider a shift to another \"alt\"...\n",
    "\n",
    "What things change?\n",
    "\n",
    "* The initial shock to economic policy or the economic environment...\n",
    "* The interest rate: $ {\\Delta}r ≠ 0 $\n",
    "* Anything determined, directly or indirectly, by r:\n",
    "    * $ I $ :: investment spending\n",
    "    * $ \\epsilon $ :: the exchange rate\n",
    "    * $ GX $ :: exports\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "#### A Rise in Interest Rates Abroad\n",
    "\n",
    "$ Y^* = Y = AD = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $\n",
    "\n",
    "$ 0 = \\mu\\left(x_{\\epsilon}{\\epsilon}_r{{\\Delta}r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right){\\Delta}r $\n",
    "\n",
    "&nbsp;"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is $ {\\Delta}r $?\n",
    "\n",
    "A. $ {\\Delta}r = {\\Delta}r^f $\n",
    "\n",
    "B. $ {\\Delta}r =  \\frac{x_{\\epsilon}{\\epsilon}_r}{I_r + x_{\\epsilon}{\\epsilon}_r}{\\Delta}r^f $\n",
    "\n",
    "C. $ {\\Delta}r =  \\frac{I_r}{I_r + x_{\\epsilon}{\\epsilon}_r}{\\Delta}r^f $\n",
    "\n",
    "D. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "If $ I_r = 20 , \\mu = 2, x_{\\epsilon} = 1, $ and   \n",
    "$ {\\epsilon_r = 10 } $, then $ {\\Delta}r^f = + 0.03 $ will    \n",
    "have what effect on domestic investment I?\n",
    "\n",
    "A. $ {\\Delta}I = + 0.6 $\n",
    "\n",
    "B. $ {\\Delta}I = - 0.6 $\n",
    "\n",
    "C. $ {\\Delta}I = - 0.2 $\n",
    "\n",
    "D. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "If $ I_r = 20 , \\mu = 2, x_{\\epsilon} = 1, $ and       \n",
    "$ {\\epsilon_r = 10 } $, then $ {\\Delta}G = + 0.3 $ will have    \n",
    "what effect on domestic investment I?\n",
    "\n",
    "A. $ {\\Delta}I = \\left(x_{\\epsilon}{\\epsilon_r}{I_r + x_{\\epsilon}{\\epsilon}_r}\\right){\\Delta}G $\n",
    "\n",
    "B. $ {\\Delta}I = \\left(\\frac{I_r}{I_r + x_{\\epsilon}{\\epsilon}_r}\\right){\\Delta}G $\n",
    "\n",
    "C. $ {\\Delta}I = \\left(\\frac{1}{I_r + x_{\\epsilon}{\\epsilon}_r}\\right){\\Delta}G $\n",
    "\n",
    "D. None of the above"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font color=\"880000\">Sticky-Price Models</font>\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">The Sticky-Price Model IS Curve Equation</font>\n",
    "\n",
    "$ Y = AD = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $\n",
    "\n",
    "Alternatively:\n",
    "\n",
    "$ Y = Y = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)(i - \\pi + \\rho) $\n",
    "\n",
    "Causation from right to left:\n",
    "\n",
    "* Spending determines aggregate demand\n",
    "* Aggregage demand via the inventory adjustment channel determines national income and product\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Influences on spending from:\n",
    "\n",
    "* Policy variables:\n",
    "    * Fiscal: G, t\n",
    "    * Monetary: $ r = i - \\pi +\\rho $\n",
    "        * risk premium $ \\rho $ sum of term $ \\rho^T $ and counterparty risk $ \\rho^r $\n",
    "* Expectations: confidence/animal spirits/greed vs. fear: $c_o, I_o, \\epsilon_o $\n",
    "* Foreign economic conditions: $ Y^f, r^f $\n",
    "* The multiplier: $ \\mu = \\frac{1}{1 - c_y(1-t) + im_y} $\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">The Keynesian Multiplier</font>\n",
    "\n",
    "$ Y = C + I + G + (GX - IM) $\n",
    "\n",
    "$ Y = (c_o + c_y(1-t)Y) + I + G + (GX - im_y{Y}) $\n",
    "\n",
    "$ (1 - c_y(1-t) + im_y)Y = c_o + I + G + GX $\n",
    "\n",
    "$ Y = \\frac{c_o + I + G + GX}{(1 - c_y(1-t) + im_y)} $\n",
    "\n",
    "$ Y = {\\mu}(c_o + I + G + GX) $\n",
    "\n",
    "$ \\mu = \\frac{1}{(1 - c_y(1-t) + im_y)} $\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Playing with the Multiplier</font>\n",
    "\n",
    "**The Simple Multiplier**: $ \\frac{{\\Delta}Y}{{\\Delta}G} = \\mu $ :: holding r, t fixed\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "**The Balanced Budget Multiplier**:\n",
    "\n",
    "> Allowing t to vary with G, so that: $ {\\Delta}t = \\frac{{\\Delta}G}{Y} $ so that $ {\\Delta}(G - T) = 0 $... \n",
    "\n",
    ">Then: $ {\\Delta}Y = \\frac{{\\Delta}G - c_y(1-t){\\Delta}G}{1-c_y(1-t)+im_y} = \\frac{1-c_y(1-t)}{1-c_y(1-t)+im_y}{\\Delta}G $\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "**The Monetary-Offset Multiplier**:\n",
    "\n",
    "> Allowing r to vary with G, so that: $ {\\Delta}r = \\frac{{\\Delta}G}{I_r + x_{\\epsilon}{\\epsilon}_r} $... \n",
    "\n",
    ">Then: $ {\\Delta}Y = 0{\\Delta}G $\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### <font color=\"008800\">Exercises</font>\n",
    "\n",
    "$ \\mu = \\frac{1}{(1 - c_y(1-t) + im_y)} $\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Suppose $ c_y = 5/6, t = 1/4, im_y = 1/8 $,    \n",
    "then if $ {\\Delta}G = 0.4 $ and r and t are fixed:\n",
    "\n",
    "A. $ {\\Delta}Y = 0.4 $\n",
    "\n",
    "B. $ {\\Delta}Y = 0.5 $\n",
    "\n",
    "C. $ {\\Delta}Y = 0.8 $\n",
    "\n",
    "D. $ {\\Delta}Y = 0.6 $\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Suppose $ c_y = 2/3, t = 1/4, im_y = 1/4 $,   \n",
    "then if $ {\\Delta}G = 0.3 $ and r and t are fixed:\n",
    "\n",
    "A. $ {\\Delta}Y = 0.4 $\n",
    "\n",
    "B. $ {\\Delta}Y = 0.5 $\n",
    "\n",
    "C. $ {\\Delta}Y = 0.8 $\n",
    "\n",
    "D. $ {\\Delta}Y = 0.6 $\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Suppose $ c_y = 1/3, t = 1/4, im_y = 1/4 $,   \n",
    "then if $ {\\Delta}G = 0.6 $ and r and t are fixed:\n",
    "\n",
    "A. $ {\\Delta}Y = 0.4 $\n",
    "\n",
    "B. $ {\\Delta}Y = 0.5 $\n",
    "\n",
    "C. $ {\\Delta}Y = 0.8 $\n",
    "\n",
    "D. $ {\\Delta}Y = 0.6 $\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Solving IS-Curve Problems</font>\n",
    "\n",
    "$ Y = AD = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $\n",
    "\n",
    "* Causation from right to left...\n",
    "\n",
    "* Government policy determines G, t via fiscal policy and (influences) r via monetary policy that determines i in $ r = i + \\rho - \\pi $\n",
    "\n",
    "* The economic environment:\n",
    "    * Expectations: $ c_o, I_o, {\\epsilon}_o $\n",
    "    * International influences: $ Y^f, r^f $\n",
    "    * Parameters in behavioral relationships: $ I_r, c_y $ (influences $ \\mu $ ), $ x_f, x_{\\epsilon}, {\\epsilon}_r $\n",
    "    \n",
    "* Add these up (and apply the multiplier $ \\mu $) to calculate aggregate demand AD\n",
    "* Aggregate demand AD determines production Y through the inventory-adjustment process\n",
    "    * And consequent hiring and firing alters income so it equals Y—and at that point $ I = S^p + S^g + S^f $ no matter what the interest rate r is.\n",
    "    \n",
    "* And potential output $ Y^* $ is nowheresville here...\n",
    "    \n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### <font color=\"008800\">Exercises</font>\n",
    "\n",
    "$ Y = AD = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $\n",
    "\n",
    "Suppose $ \\mu = 2 $ and $ {\\Delta}G = 0.3, $ then:\n",
    "\n",
    "A. $ {\\Delta}Y = 0.2 $\n",
    "\n",
    "B. $ {\\Delta}Y = 0.3 $\n",
    "\n",
    "C. $ {\\Delta}Y = 0.5 $\n",
    "\n",
    "D. $ {\\Delta}Y = 0.6 $\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Suppose $ \\mu = 2, I_r = 20, x_{\\epsilon} = 4, $ and $ {\\epsilon}_r = 5 $.    \n",
    "Then if $ {\\Delta}r = -0.015 $ :\n",
    "\n",
    "A. $ {\\Delta}Y = 0.2 $\n",
    "\n",
    "B. $ {\\Delta}Y = 0.3 $\n",
    "\n",
    "C. $ {\\Delta}Y = 0.5 $\n",
    "\n",
    "D. $ {\\Delta}Y = 0.6 $\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Suppose $ \\mu = 2, {\\Delta}G = 0.3, I_r = 20, x_{\\epsilon} = 4, $ and $ {\\epsilon}_r = 5 $.    \n",
    "Then if $ {\\Delta}r = -0.015 $ :\n",
    "\n",
    "A. $ {\\Delta}Y = 0.2 $\n",
    "\n",
    "B. $ {\\Delta}Y = 0.3 $\n",
    "\n",
    "C. $ {\\Delta}Y = 0.5 $\n",
    "\n",
    "D. $ {\\Delta}Y = 0.6 $\n",
    "\n",
    "E. None of the above\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Phillips Curve</font>\n",
    "\n",
    "$ {\\pi_t} = {\\pi_t}^e - \\beta\\left(u_t - u^*\\right) + SS_t$\n",
    "\n",
    "Expectations:\n",
    "\n",
    "* Static: $ {\\pi_t}^e = \\pi^{*} $\n",
    "* Adaptive: $ {\\pi_t}^e = \\pi_{t-1} $\n",
    "* Rational: $ {\\pi_t}^e = \\pi_{t} $\n",
    "* Hybrids: $ {\\pi_t}^e = \\lambda(\\pi_{t}) + (1-\\lambda)(\\pi_{t-1}) $ or $ {\\pi_t}^e = (1-\\lambda)(\\pi^*) + \\lambda(\\pi_{t-1}) $\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Inflation Dynamics</font>\n",
    "\n",
    "* Static: $ {\\pi_t} = \\pi^* - \\beta\\left(u_t - u^*\\right) + SS_t$\n",
    "* Adaptive: $ {\\pi_t} = {\\pi_{t-1}} - \\beta\\left(u_t - u^*\\right) + SS_t $\n",
    "* Rational: $ {\\pi_t} = {\\pi_t}^e $ and $ u_t = u^* - \\frac{SS_t}{\\beta} $\n",
    "* Hybrid: Adaptive and Rational:\n",
    "    * $ {\\pi_t} = \\lambda{\\pi_t} + (1-\\lambda)\\pi_{t-1} - \\beta\\left(u_t - u^*\\right) + SS_t$\n",
    "    * $ {\\pi_t} = {\\pi_{t-1}} - \\frac{\\beta\\left(u_t - u^*\\right) + SS_t}{1-\\lambda} $\n",
    "    \n",
    "* Hybrid: Adaptive and Static:\n",
    "    * $ {\\pi_t} - \\pi^* = \\lambda({\\pi_{t-1}}-\\pi^*) - \\beta\\left(u_t - u^*\\right) + SS_t $\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Monetary Policy Reaction Function</font>\n",
    "\n",
    "$ r_t = r^{n} + r_{\\pi}(\\pi_t - \\pi^T) $\n",
    "\n",
    "$ u_t - u^* = {u_r}({r_t}^n - r^*) + \\delta_t $\n",
    "\n",
    "$ \\pi_t = {\\pi_t}^e - \\beta(u_t - u^*) + SS_t $\n",
    "\n",
    "Combine the MPRF with the \"inflation dynamics\" version of the Phillips Curve...\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "$ u_t - u^* = {u_r}r_{\\pi}(\\pi_t - \\pi^T) $\n",
    "\n",
    "$ \\pi_t = {\\pi_t}^e - \\beta{u_r}r_{\\pi}(\\pi_t - \\pi^T) + SS_t  $\n",
    "\n",
    "$ \\pi_t = \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}{\\pi_t}^e + \\frac{\\beta{u_r}r_{\\pi}}{1 + \\beta{u_r}r_{\\pi}}\\pi^T\\right) + \n",
    "\\frac{1}{1 + \\beta{u_r}r_{\\pi}}SS_t $\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "**Expectations: Static**:\n",
    "\n",
    "* $ \\pi_t = \\pi^T + \\frac{1}{1 + \\beta{u_r}r_{\\pi}}\\left(\\pi^* - \\pi^T\\right) + \n",
    "\\frac{1}{1 + \\beta{u_r}r_{\\pi}}SS_t $\n",
    "* $ u_t = u^* + {u_r}r_{\\pi}(\\pi_t - \\pi^T) $\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "**Expectations: Adaptive**: \n",
    "\n",
    "* $ \\pi_t = \\pi^T + \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}({\\pi_{t-1}}- \\pi^T) + \n",
    "\\frac{1}{1 + \\beta{u_r}r_{\\pi}}SS_t\\right) $\n",
    "* $ u_t = u^* + {u_r}r_{\\pi}(\\pi_t - \\pi^T) $\n",
    "* $ \\pi_t = \\pi^T + \\frac{1}{1 + \\beta{u_r}r_{\\pi}}SS_t + \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}\\right)^2SS_{t-1} + \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}\\right)^3SS_{t-2} + \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}\\right)^4SS_{t-3}... $\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">The \"Neutral\" Interest Rate</font>\n",
    "\n",
    "$ Y^* = Y = AD = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $ :: Flexprice IS\n",
    "\n",
    "$ r^* = \\frac{\\left[\\frac{Y^*}{\\mu} - \\left(c_o + I_o + G\\right) + \\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right)\\right]}{I_r + x_\\epsilon\\epsilon_r} $  :: r-star\n",
    "\n",
    "* Drive the long-term risky real interest rate to this $ r^* $ value...\n",
    "    * Do so as $ r^* $ is pushed around by shocks to the economic environment and to fiscal policy...\n",
    "\n",
    "<img style=\"display:block; margin-left:auto; margin-right:auto;\" src=\"https://delong.typepad.com/.a/6a00e551f080038834022ad3c398c1200b-pi\" alt=\"Consumer Price Index for All Urban Consumers All Items Less Food and Energy FRED St Louis Fed\" title=\"Consumer_Price_Index_for_All_Urban_Consumers__All_Items_Less_Food_and_Energy___FRED___St__Louis_Fed.png\" border=\"0\" width=\"600\" height=\"450\" />\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"000088\">Monetary Policy and the Zero Lower Bound</font>\n",
    "\n",
    "The interest rate in the IS Curve is the long-term risky real interest rate: r\n",
    "\n",
    "The interest rate the central bank controls is the short-term safe nominal interest rate: i\n",
    "\n",
    "* $ r = i - \\pi + \\rho $ subject to $ i ≥ 0 $\n",
    "* $ \\rho = \\rho^R + \\rho^T $\n",
    "    * $ \\rho^R $ :: the risk premium for lending to privates rather than to the government\n",
    "        * Moral hazard\n",
    "        * Adverse selection\n",
    "        * \"Skin in the game\" from borrowers\n",
    "        * Financial crises\n",
    "    * $ \\rho^T $ :: lack of confidence that the central bank will keep i where it currently is\n",
    "\n",
    "&nbsp;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hysteresis and Budget Arithmetic in a Depression\n",
    "\n",
    "Boost government purchases by ΔG—if no Federal Reserve offset because at ZLB \n",
    "\n",
    "* Get boost to real GDP by μΔG\n",
    "* Get boost to taxes by tμΔG\n",
    "* Increase in debt of (1 - tμ)ΔG = ΔD\n",
    "* Financing cost of this debt: (r-g)ΔD = (r-g)(1 - tμ)ΔG\n",
    "\n",
    "“Hysteresis” parameter η\n",
    "\n",
    "* Gain tημΔG in tax revenue from heading off “hysteresis”\n",
    "* (r-g)(1 - tμ)ΔG greater or less than ηtμΔG?\n",
    "    * t = 0.33\n",
    "    * μ = 2\n",
    "    * 0.33(r - g) greater or less than 0.66η?\n",
    "\n",
    "r - g greater or less than 2η?\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font color=\"880000\">Catch Our Breath</font>\n",
    "\n",
    "<img src=\"https://tinyurl.com/20181029a-delong\" width=\"300\" style=\"float:right\" />\n",
    "\n",
    "* Ask me two questions…\n",
    "* Make two comments…\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "* Further reading…\n",
    "\n",
    "<br clear=\"all\" />\n",
    "\n",
    "----\n",
    "\n",
    "Keynote: <https://www.icloud.com/keynote/0HA_6uYorQfMYi_hRNLo23p5Q>\n",
    "\n",
    "Lecture Support:<http://nbviewer.jupyter.org/github/braddelong/LSF18E101B/blob/master/Intermediate_Macroeonomics_Review.ipynb> <https://github.com/braddelong/LSF18E101B/blob/master/Intermediate_Macroeonomics_Review.ipynb>\n",
    "\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "----"
   ]
  }
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