auto publishing updates to notebooks

Author: QuantEcon

BCG_complete_mkts.ipynb

@@ -1406,7 +1406,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402672.4190016,
+  "date": 1619661940.1660957,
   "filename": "BCG_complete_mkts.rst",
   "kernelspec": {
    "display_name": "Python",

BCG_incomplete_mkts.ipynb

@@ -2215,7 +2215,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402672.684707,
+  "date": 1619661940.5953467,
   "filename": "BCG_incomplete_mkts.rst",
   "kernelspec": {
    "display_name": "Python",

about_lectures.ipynb

@@ -28,7 +28,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402672.8388484,
+  "date": 1619661940.744541,
   "filename": "about_lectures.rst",
   "kernelspec": {
    "display_name": "Python",

additive_functionals.ipynb

@@ -1498,7 +1498,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402672.9367144,
+  "date": 1619661940.8426106,
   "filename": "additive_functionals.rst",
   "kernelspec": {
    "display_name": "Python",

amss.ipynb

@@ -159,7 +159,7 @@
     "\n",
     "AMSS allow the government to issue only one-period risk-free debt each period.\n",
     "\n",
-    "Ruling out complete markets in this way is a step in the direction of making total tax collections behave more like that prescribed in [[Bar79]](https://python-programming.quantecon.org/zreferences.html#barro1979) than they do in [[LS83]](https://python-programming.quantecon.org/zreferences.html#lucasstokey1983)."
+    "Ruling out complete markets in this way is a step in the direction of making total tax collections behave more like that prescribed in Robert Barro (1979) [[Bar79]](https://python-programming.quantecon.org/zreferences.html#barro1979) than they do in Lucas and Stokey (1983) [[LS83]](https://python-programming.quantecon.org/zreferences.html#lucasstokey1983)."
    ]
   },
   {
@@ -186,14 +186,14 @@
     "$$\n",
     "\\begin{aligned}\n",
     "b_t(s^{t-1})\n",
-    "    & =    \\tau^n_t(s^t) n_t(s^t) - g_t(s_t) - T_t(s^t) +\n",
+    "    & =    \\tau^n_t(s^t) n_t(s^t) - g(s_t) - T_t(s^t) +\n",
     "                   {b_{t+1}(s^t) \\over R_t(s^t )}\n",
     "    \\\\\n",
-    "    & \\equiv z(s^t) + {b_{t+1}(s^t) \\over R_t(s^t )},\n",
+    "    & \\equiv z_t(s^t) + {b_{t+1}(s^t) \\over R_t(s^t )},\n",
     "\\end{aligned} \\tag{4}\n",
     "$$\n",
     "\n",
-    "where $ z(s^t) $ is the net-of-interest government surplus.\n",
+    "where $ z_t(s^t) $ is the net-of-interest government surplus.\n",
     "\n",
     "To rule out Ponzi schemes, we assume that the government is subject to a **natural debt limit** (to be discussed in a forthcoming lecture).\n",
     "\n",
@@ -212,16 +212,16 @@
     "\n",
     "<a id='equation-ts-gov-wo2'></a>\n",
     "$$\n",
-    "b_t(s^{t-1}) =  z(s^t) + \\beta  \\sum_{s^{t+1}\\vert s^t}  \\pi_{t+1}(s^{t+1} | s^t)\n",
+    "b_t(s^{t-1}) =  z_t(s^t) + \\beta  \\sum_{s^{t+1}\\vert s^t}  \\pi_{t+1}(s^{t+1} | s^t)\n",
     "                       { u_c(s^{t+1}) \\over u_c(s^{t}) } \\; b_{t+1}(s^t) \\tag{5}\n",
     "$$\n",
     "\n",
-    "Components of $ z(s^t) $ on the right side depend on $ s^t $, but the left side is required to depend only\n",
+    "Components of $ z_t(s^t) $ on the right side depend on $ s^t $, but the left side is required to depend only\n",
     "on $ s^{t-1} $ .\n",
     "\n",
     "**This is what it means for one-period government debt to be risk-free**.\n",
     "\n",
-    "Therefore, the sum on the right side of equation [(5)](#equation-ts-gov-wo2) also has to depend only on $ s^{t-1} $.\n",
+    "Therefore, the right side of equation [(5)](#equation-ts-gov-wo2) also has to depend only on $ s^{t-1} $.\n",
     "\n",
     "This requirement will give rise to **measurability constraints** on the Ramsey allocation to be discussed soon.\n",
     "\n",
@@ -230,13 +230,13 @@
     "particular realization $ s_{t} $) we get\n",
     "\n",
     "$$\n",
-    "b_t(s^{t-1}) =  z(s^t) + \\sum_{s^{t+1}\\vert s^t} \\beta  \\pi_{t+1}(s^{t+1} | s^t)\n",
+    "b_t(s^{t-1}) =  z_t(s^t) + \\sum_{s^{t+1}\\vert s^t} \\beta  \\pi_{t+1}(s^{t+1} | s^t)\n",
     "                       { u_c(s^{t+1}) \\over u_c(s^{t}) }\n",
-    "\\, \\left[z(s^{t+1}) + {b_{t+2}(s^{t+1}) \\over R_{t+1}(s^{t+1})}\\right]\n",
+    "\\, \\left[z_{t+1}(s^{t+1}) + {b_{t+2}(s^{t+1}) \\over R_{t+1}(s^{t+1})}\\right]\n",
     "$$\n",
     "\n",
     "After making similar repeated substitutions for all future occurrences of\n",
-    "government indebtedness, and by invoking the natural debt limit, we\n",
+    "government indebtedness, and by invoking a natural debt limit, we\n",
     "arrive at:\n",
     "\n",
     "\n",
@@ -245,7 +245,7 @@
     "\\begin{aligned}\n",
     "b_t(s^{t-1})\n",
     "    &=  \\sum_{j=0}^\\infty \\sum_{s^{t+j} | s^t} \\beta^j  \\pi_{t+j}(s^{t+j} | s^t)\n",
-    "              { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;z(s^{t+j})\n",
+    "              { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;z_{t+j}(s^{t+j})\n",
     "        \\end{aligned} \\tag{6}\n",
     "$$\n",
     "\n",
@@ -258,17 +258,17 @@
     "- use the household’s first-order condition $ 1-\\tau^n_t(s^t)= u_{\\ell}(s^t) /u_c(s^t) $ to eliminate the labor tax rate  \n",
     "\n",
     "\n",
-    "so that we can express the net-of-interest government surplus $ z(s^t) $ as\n",
+    "so that we can express the net-of-interest government surplus $ z_t(s^t) $ as\n",
     "\n",
     "\n",
     "<a id='equation-amss-44-2'></a>\n",
     "$$\n",
-    "z(s^t)\n",
-    "    = \\left[1 - {u_{\\ell}(s^t) \\over u_c(s^t)}\\right] \\left[c_t(s^t)+g_t(s_t)\\right]\n",
-    "        -g_t(s_t) - T_t(s^t)\\,. \\tag{7}\n",
+    "z_t(s^t)\n",
+    "    = \\left[1 - {u_{\\ell}(s^t) \\over u_c(s^t)}\\right] \\left[c_t(s^t)+g(s_t)\\right]\n",
+    "        -g(s_t) - T_t(s^t)\\,. \\tag{7}\n",
     "$$\n",
     "\n",
-    "If we substitute  appropriate versions of the right side of [(7)](#equation-amss-44-2) for $ z(s^{t+j}) $ into equation [(6)](#equation-ts-gov-wo3),\n",
+    "If we substitute  appropriate versions of the right side of [(7)](#equation-amss-44-2) for $ z_{t+j}(s^{t+j}) $ into equation [(6)](#equation-ts-gov-wo3),\n",
     "we obtain a sequence of *implementability constraints* on a Ramsey allocation in an AMSS economy.\n",
     "\n",
     "Expression [(6)](#equation-ts-gov-wo3) at time $ t=0 $ and initial state $ s^0 $\n",
@@ -278,7 +278,7 @@
     "<a id='equation-ts-gov-wo4'></a>\n",
     "$$\n",
     "b_0(s^{-1}) = \\mathbb E_0 \\sum_{j=0}^\\infty \\beta^j\n",
-    "               { u_c(s^{j}) \\over u_c(s^{0}) } \\;z(s^{j}) \\tag{8}\n",
+    "               { u_c(s^{j}) \\over u_c(s^{0}) } \\;z_j(s^{j}) \\tag{8}\n",
     "$$\n",
     "\n",
     "Indeed, it was the *only* implementability constraint there.\n",
@@ -289,7 +289,7 @@
     "<a id='equation-ts-gov-wo4a'></a>\n",
     "$$\n",
     "b_t(s^{t-1}) =  \\mathbb E_t \\sum_{j=0}^\\infty \\beta^j\n",
-    "              { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;z(s^{t+j}) \\tag{9}\n",
+    "              { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;z_{t+j}(s^{t+j}) \\tag{9}\n",
     "$$\n",
     "\n",
     "Equation [(9)](#equation-ts-gov-wo4a) must hold for each $ s^t $ for each $ t \\geq 1 $."
@@ -321,7 +321,7 @@
     "$$\n",
     "\\max_{\\{c_t(s^t),b_{t+1}(s^t)\\}}\n",
     "\\mathbb E_0 \\sum_{t=0}^\\infty \\beta^t\n",
-    "                        u\\left(c_t(s^t),1-c_t(s^t)-g_t(s_t)\\right)\n",
+    "                        u\\left(c_t(s^t),1-c_t(s^t)-g(s_t)\\right)\n",
     "$$\n",
     "\n",
     "where the maximization is subject to\n",
@@ -330,7 +330,7 @@
     "<a id='equation-amss-44'></a>\n",
     "$$\n",
     "\\mathbb E_{0} \\sum_{j=0}^\\infty \\beta^j\n",
-    "      { u_c(s^{j}) \\over u_c(s^{0}) } \\;z(s^{j}) \\geq b_0(s^{-1}) \\tag{10}\n",
+    "      { u_c(s^{j}) \\over u_c(s^{0}) } \\;z_j(s^{j}) \\geq b_0(s^{-1}) \\tag{10}\n",
     "$$\n",
     "\n",
     "and\n",
@@ -340,7 +340,7 @@
     "$$\n",
     "\\mathbb E_{t} \\sum_{j=0}^\\infty \\beta^j\n",
     "    { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;\n",
-    "    z(s^{t+j}) = b_t(s^{t-1})\n",
+    "    z_{t+j}(s^{t+j}) = b_t(s^{t-1})\n",
     "      \\quad \\forall \\, t,  s^t \\tag{11}\n",
     "$$\n",
     "\n",
@@ -369,7 +369,7 @@
     "   &\\;\\geq\\; (\\leq)\\;\\, 0 \\quad \\text{if the constraint binds in the following direction }\n",
     "   \\\\\n",
     "   & \\mathbb E_{t} \\sum_{j=0}^\\infty \\beta^j\n",
-    "    { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;z(s^{t+j}) \\;\\geq \\;(\\leq)\\;\\, b_t(s^{t-1})\n",
+    "    { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;z_{t+j}(s^{t+j}) \\;\\geq \\;(\\leq)\\;\\, b_t(s^{t-1})\n",
     "\\end{aligned}\n",
     "$$\n",
     "\n",
@@ -399,14 +399,14 @@
     "$$\n",
     "\\begin{aligned}\n",
     "   J &= \\mathbb E_{0} \\sum_{t=0}^\\infty \\beta^t\n",
-    "                        \\biggl\\{ u\\left(c_t(s^t), 1-c_t(s^t)-g_t(s_t)\\right)\\\\\n",
+    "                        \\biggl\\{ u\\left(c_t(s^t), 1-c_t(s^t)-g(s_t)\\right)\\\\\n",
     "   &  \\qquad + \\gamma_t(s^t) \\Bigl[ \\mathbb E_{t} \\sum_{j=0}^\\infty \\beta^j\n",
-    "         u_c(s^{t+j}) \\,z(s^{t+j}) - u_c(s^{t}) \\,b_t(s^{t-1}) \\biggr\\}\n",
+    "         u_c(s^{t+j}) \\,z_{t+j}(s^{t+j}) - u_c(s^{t}) \\,b_t(s^{t-1}) \\biggr\\}\n",
     "         \\\\\n",
     "   &= \\mathbb E_{0} \\sum_{t=0}^\\infty \\beta^t\n",
-    "                         \\biggl\\{ u\\left(c_t(s^t), 1-c_t(s^t)-g_t(s_t)\\right)\n",
+    "                         \\biggl\\{ u\\left(c_t(s^t), 1-c_t(s^t)-g(s_t)\\right)\n",
     "        \\\\\n",
-    "   &  \\qquad + \\Psi_t(s^t)\\, u_c(s^{t}) \\,z(s^{t}) -\n",
+    "   &  \\qquad + \\Psi_t(s^t)\\, u_c(s^{t}) \\,z_t(s^{t}) -\n",
     "                   \\gamma_t(s^t)\\, u_c(s^{t}) \\, b_t(s^{t-1})  \\biggr\\}\n",
     "\\end{aligned} \\tag{12}\n",
     "$$\n",
@@ -433,7 +433,7 @@
     "$$\n",
     "\\begin{aligned}\n",
     "  u_c(s^t)-u_{\\ell}(s^t) &+ \\Psi_t(s^t)\\left\\{ \\left[\n",
-    "    u_{cc}(s^t) - u_{c\\ell}(s^{t})\\right]z(s^{t}) +\n",
+    "    u_{cc}(s^t) - u_{c\\ell}(s^{t})\\right]z_t(s^{t}) +\n",
     "    u_{c}(s^{t})\\,z_c(s^{t}) \\right\\}\n",
     "    \\\\\n",
     "    & \\hspace{35mm} - \\gamma_t(s^t)\\left[\n",
@@ -449,7 +449,7 @@
     "\\mathbb E_{t} \\left[\\gamma_{t+1}(s^{t+1})\\,u_c(s^{t+1})\\right] = 0 \\tag{15}\n",
     "$$\n",
     "\n",
-    "If we substitute $ z(s^t) $ from [(7)](#equation-amss-44-2) and its derivative\n",
+    "If we substitute $ z_t(s^t) $ from [(7)](#equation-amss-44-2) and its derivative\n",
     "$ z_c(s^t) $ into the first-order condition [(14)](#equation-amss-foc-a), we  find  two\n",
     "differences from the corresponding condition for the optimal allocation\n",
     "in a Lucas-Stokey economy with state-contingent government debt.\n",
@@ -686,8 +686,7 @@
     "where $ R_t(s^t) $ is the gross risk-free rate of interest between $ t $\n",
     "and $ t+1 $ at history $ s^t $ and $ T_t(s^t) $ are non-negative transfers.\n",
     "\n",
-    "Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this makes a possibly\n",
-    "important  difference from AMSS [[AMSSeppala02]](https://python-programming.quantecon.org/zreferences.html#aiyagari2002optimal), who restricted transfers\n",
+    "Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this is  possibly an important  difference from AMSS [[AMSSeppala02]](https://python-programming.quantecon.org/zreferences.html#aiyagari2002optimal), who restricted transfers\n",
     "to be non-negative).\n",
     "\n",
     "In this case, the household faces a sequence of budget constraints\n",
@@ -849,7 +848,7 @@
     "\\beta V_x(x(s),s) = \\mu(s|s_-) \\tag{26}\n",
     "$$\n",
     "\n",
-    "Applying the envelope theorem to Bellman equation [(22)](#equation-eqn-amssapp5) gives\n",
+    "Applying an envelope theorem to Bellman equation [(22)](#equation-eqn-amssapp5) gives\n",
     "\n",
     "\n",
     "<a id='equation-eqn-amssapp8'></a>\n",
@@ -864,7 +863,7 @@
     "<a id='equation-eqn-amssapp9'></a>\n",
     "$$\n",
     "V_x(x_-, s_-) = \\sum_{s} \\left( \\Pi(s|s_-) {\\frac{u_c(s)}{\\sum_{\\tilde s}\n",
-    "\\Pi(\\tilde s| s_-) u_c(\\tilde s)}} \\right) V_x(x(s), s) \\tag{28}\n",
+    "\\Pi(\\tilde s| s_-) u_c(\\tilde s)}} \\right) V_x(x, s) \\tag{28}\n",
     "$$\n",
     "\n",
     "Equation [(28)](#equation-eqn-amssapp9) states that $ V_x(x, s) $ is a *risk-adjusted martingale*.\n",
@@ -899,7 +898,7 @@
     "- a counterpart to $ V_x(x,s) $ is time-invariant and equal to\n",
     "  the Lagrange multiplier on the Lucas-Stokey implementability constraint  \n",
     "- time invariance of $ V_x(x,s) $  is the source of a key\n",
-    "  feature of the Lucas-Stokey model, namely, **state variable degeneracy** in which $ x_t $ is an exact time-invariant function of $ s_t $)  \n",
+    "  feature of the Lucas-Stokey model, namely, **state variable degeneracy** in which $ x_t $ is an exact time-invariant function of $ s_t $.  \n",
     "\n",
     "\n",
     "That $ V_x(x,s) $ varies over time according to a twisted martingale\n",
@@ -1374,7 +1373,7 @@
     ">For convenience in  matching our computer code, we have expressed\n",
     "utility as a function of $ n $ rather than leisure $ l $.\n",
     "\n",
-    "We consider the same government expenditure process studied in the lecture on\n",
+    "We first consider a government expenditure process that we  studied earlier in a lecture on\n",
     "[optimal taxation with state-contingent debt](https://python-programming.quantecon.org/opt_tax_recur.html).\n",
     "\n",
     "Government expenditures are known for sure in all periods except one.\n",
@@ -1470,11 +1469,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The following figure plots the Ramsey plan under both complete and incomplete\n",
+    "The following figure plots  Ramsey plans under complete and incomplete\n",
     "markets for both possible realizations of the state at time $ t=3 $.\n",
     "\n",
-    "Optimal policies when  the government has  access to state-contingent debt are\n",
-    "represented by black lines, while the optimal policies when there is only a risk-free bond are in red.\n",
+    "Ramsey outcomes and  policies when  the government has  access to state-contingent debt are\n",
+    "represented by black lines and  by red lines when there is only a risk-free bond.\n",
     "\n",
     "Paths with circles are histories in which there is peace, while those with\n",
     "triangle denote war."
@@ -1553,9 +1552,9 @@
     "\n",
     "If it is able to trade state-contingent debt, then at time $ t=2 $\n",
     "\n",
-    "- the government purchases an Arrow security that pays off when $ g_3 = g_h $  \n",
-    "- the government sells an Arrow security that  pays off when $ g_3 = g_l $  \n",
-    "- these purchases are designed in such a way that regardless of whether or not there is a war at $ t=3 $, the government will begin  period $ t=4 $ with the *same* government debt  \n",
+    "- the government **purchases** an Arrow security that pays off when $ g_3 = g_h $  \n",
+    "- the government **sells** an Arrow security that  pays off when $ g_3 = g_l $  \n",
+    "- the Ramsey planner designs these purchases and sales  designed so  that, regardless of whether or not there is a war at $ t=3 $, the government  begins  period $ t=4 $ with the *same* government debt  \n",
     "\n",
     "\n",
     "This pattern facilities smoothing tax rates across  states.\n",
@@ -1564,9 +1563,11 @@
     "\n",
     "Instead, it must enter   time $ t=3 $ with the same level of debt falling due whether there is peace or war at $ t=3 $.\n",
     "\n",
-    "It responds to this constraint by smoothing tax rates across time.\n",
+    "The risk-free rate between time $ 2 $ and time $ 3 $ is unusually **low** because at time $ 2 $ consumption at time $ 3 $ is expected to be unusually **low**.\n",
     "\n",
-    "To finance a war it raises taxes and issues more debt.\n",
+    "A **low** risk-free rate of return on government debt between time $ 2 $ and time $ 3 $ allows the government to enter period $ 3 $ with **lower** government debt than it entered period $ 2 $.\n",
+    "\n",
+    "To finance a war at time $ 3 $ it raises taxes and issues more debt to carry into perpetual peace that begins in period $ 4 $.\n",
     "\n",
     "To service the additional debt burden, it raises taxes in all future periods.\n",
     "\n",
@@ -1786,7 +1787,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402673.1077754,
+  "date": 1619661941.0074658,
   "filename": "amss.rst",
   "kernelspec": {
    "display_name": "Python",

amss2.ipynb

@@ -1610,7 +1610,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402673.2683423,
+  "date": 1619661941.1767492,
   "filename": "amss2.rst",
   "kernelspec": {
    "display_name": "Python",

amss3.ipynb

@@ -1651,7 +1651,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402673.464876,
+  "date": 1619661941.3859596,
   "filename": "amss3.rst",
   "kernelspec": {
    "display_name": "Python",

arellano.ipynb

@@ -1037,7 +1037,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402673.7797248,
+  "date": 1619661941.6302557,
   "filename": "arellano.rst",
   "kernelspec": {
    "display_name": "Python",

arma.ipynb

@@ -1183,7 +1183,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402673.9790573,
+  "date": 1619661942.0405178,
   "filename": "arma.rst",
   "kernelspec": {
    "display_name": "Python",

black_litterman.ipynb

@@ -1582,7 +1582,7 @@
   }
  ],
  "metadata": {
-  "date": 1619402674.204994,
+  "date": 1619661942.2040126,
   "filename": "black_litterman.rst",
   "kernelspec": {
    "display_name": "Python",