auto publishing updates to notebooks

Author: QuantEcon

BCG_complete_mkts.ipynb

@@ -1406,7 +1406,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444998.0898445,
+  "date": 1618812137.2846944,
   "filename": "BCG_complete_mkts.rst",
   "kernelspec": {
    "display_name": "Python",

BCG_incomplete_mkts.ipynb

@@ -2215,7 +2215,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444998.3490937,
+  "date": 1618812137.5370972,
   "filename": "BCG_incomplete_mkts.rst",
   "kernelspec": {
    "display_name": "Python",

about_lectures.ipynb

@@ -28,7 +28,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444998.4995985,
+  "date": 1618812137.6949542,
   "filename": "about_lectures.rst",
   "kernelspec": {
    "display_name": "Python",

additive_functionals.ipynb

@@ -1498,7 +1498,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444998.5992084,
+  "date": 1618812137.7941382,
   "filename": "additive_functionals.rst",
   "kernelspec": {
    "display_name": "Python",

amss.ipynb

@@ -170,9 +170,9 @@
     "\n",
     "In period $ t $ and history $ s^t $, let\n",
     "\n",
-    "- $ b_{t+1}(s^t) $ be the amount of the time $ t+1 $ consumption good that at time $ t $ the government promised to pay  \n",
+    "- $ b_{t+1}(s^t) $ be the amount of the time $ t+1 $ consumption good that at time $ t $, history $ s^t $ the government promised to pay  \n",
     "- $ R_t(s^t) $ be the gross interest rate on  risk-free one-period debt between periods $ t $ and $ t+1 $  \n",
-    "- $ T_t(s^t) $ be a non-negative lump-sum transfer to the representative household <sup><a href=#fn-a id=fn-a-link>[1]</a></sup>  \n",
+    "- $ T_t(s^t) $ be a non-negative lump-sum *transfer* to the representative household <sup><a href=#fn-a id=fn-a-link>[1]</a></sup>  \n",
     "\n",
     "\n",
     "That $ b_{t+1}(s^t) $ is the same for all realizations of $ s_{t+1} $ captures its *risk-free* character.\n",
@@ -216,7 +216,8 @@
     "                       { u_c(s^{t+1}) \\over u_c(s^{t}) } \\; b_{t+1}(s^t) \\tag{5}\n",
     "$$\n",
     "\n",
-    "Components of $ z(s^t) $ on the right side depend on $ s^t $, but the left side is required to depend on $ s^{t-1} $ only.\n",
+    "Components of $ z(s^t) $ on the right side depend on $ s^t $, but the left side is required to depend only\n",
+    "on $ s^{t-1} $ .\n",
     "\n",
     "**This is what it means for one-period government debt to be risk-free**.\n",
     "\n",
@@ -248,6 +249,9 @@
     "        \\end{aligned} \\tag{6}\n",
     "$$\n",
     "\n",
+    "Notice how the conditioning sets in equation [(6)](#equation-ts-gov-wo3) differ: they are $ s^{t-1} $ on the left side and\n",
+    "$ s^t $ on the right side.\n",
+    "\n",
     "Now let’s\n",
     "\n",
     "- substitute the resource constraint into the net-of-interest government surplus, and  \n",
@@ -264,7 +268,7 @@
     "        -g_t(s_t) - T_t(s^t)\\,. \\tag{7}\n",
     "$$\n",
     "\n",
-    "If we substitute the  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(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",
@@ -297,13 +301,13 @@
    "source": [
     "### Comparison with Lucas-Stokey Economy\n",
     "\n",
-    "The expression on the right side of [(9)](#equation-ts-gov-wo4a) in the Lucas-Stokey (1983) economy would  equal the present value of a continuation stream of government surpluses evaluated at what would be competitive equilibrium Arrow-Debreu prices at date $ t $.\n",
+    "The expression on the right side of [(9)](#equation-ts-gov-wo4a) in the Lucas-Stokey (1983) economy would  equal the present value of a continuation stream of government net-of-interest surpluses evaluated at what would be competitive equilibrium Arrow-Debreu prices at date $ t $.\n",
     "\n",
     "In the Lucas-Stokey economy, that present value is measurable with respect to $ s^t $.\n",
     "\n",
     "In the AMSS economy, the restriction that government debt be risk-free imposes that that same present value must be measurable with respect to $ s^{t-1} $.\n",
     "\n",
-    "In a language used in the literature on incomplete markets models, it can be said that the AMSS model requires that at each $ (t, s^t) $ what would be the present value of continuation government surpluses in the Lucas-Stokey model must belong to  the **marketable subspace** of the AMSS model."
+    "In a language used in the literature on incomplete markets models, it can be said that the AMSS model requires that at each $ (t, s^t) $ what would be the present value of continuation government net-of-interest surpluses in the Lucas-Stokey model must belong to  the **marketable subspace** of the AMSS model."
    ]
   },
   {
@@ -337,7 +341,7 @@
     "\\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",
-    "      \\quad \\forall \\,  s^t \\tag{11}\n",
+    "      \\quad \\forall \\, t,  s^t \\tag{11}\n",
     "$$\n",
     "\n",
     "given $ b_0(s^{-1}) $."
@@ -354,15 +358,15 @@
     "As in the Lucas-Stokey economy, this multiplier is strictly positive when the government must resort to\n",
     "distortionary taxation; otherwise it equals zero.\n",
     "\n",
-    "A consequence of the assumption that there are no markets in state-contingent securities  and that a market exists only in a risk-free security is that we have to attach stochastic processes $ \\{\\gamma_t(s^t)\\}_{t=1}^\\infty $ of\n",
+    "A consequence of the assumption that there are no markets in state-contingent securities  and that a market exists only in a risk-free security is that we have to attach a stochastic process $ \\{\\gamma_t(s^t)\\}_{t=1}^\\infty $ of\n",
     "Lagrange multipliers to the implementability constraints [(11)](#equation-amss-46).\n",
     "\n",
     "Depending on how the constraints  bind, these multipliers can be positive or negative:\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
     "   \\gamma_t(s^t)\n",
-    "   &\\;\\geq\\; (\\leq)\\;\\, 0 \\quad \\text{if the constraint binds in this direction }\n",
+    "   &\\;\\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",
@@ -464,7 +468,7 @@
     "  while the multiplier $ \\Phi $ in the Lucas-Stokey economy is time-invariant.  \n",
     "\n",
     "\n",
-    "We need some code from our [an earlier lecture](https://python-programming.quantecon.org/opt_tax_recur.html)\n",
+    "We need some code from  [an earlier lecture](https://python-programming.quantecon.org/opt_tax_recur.html)\n",
     "on optimal taxation with state-contingent debt  sequential allocation implementation:"
    ]
   },
@@ -895,8 +899,7 @@
     "- 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\n",
-    "  (i.e., $ x_t $ is an exact 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",
@@ -930,7 +933,7 @@
     "expenditure function $ g(s) $ are such that $ g_t $ is perpetually\n",
     "random, $ V_x(x, s) $ almost surely converges to zero.\n",
     "\n",
-    "For quasi-linear preferences, the first-order condition with respect to $ n(s) $ becomes\n",
+    "For quasi-linear preferences, the first-order condition for maximizing [(22)](#equation-eqn-amssapp5) subject to  [(23)](#equation-eqn-amssapp6) with respect to $ n(s) $ becomes\n",
     "\n",
     "$$\n",
     "(1-\\mu(s|s_-) ) (1 - u_l(s)) + \\mu(s|s_-) n(s) u_{ll}(s) =0\n",
@@ -1552,7 +1555,7 @@
     "\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",
+    "- 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",
     "\n",
     "\n",
     "This pattern facilities smoothing tax rates across  states.\n",
@@ -1778,7 +1781,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444998.7623057,
+  "date": 1618874888.5793111,
   "filename": "amss.rst",
   "kernelspec": {
    "display_name": "Python",

amss2.ipynb

@@ -1608,7 +1608,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444998.9261463,
+  "date": 1618812138.1547377,
   "filename": "amss2.rst",
   "kernelspec": {
    "display_name": "Python",

amss3.ipynb

@@ -1647,7 +1647,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444999.130558,
+  "date": 1618812138.3407247,
   "filename": "amss3.rst",
   "kernelspec": {
    "display_name": "Python",

arellano.ipynb

@@ -1037,7 +1037,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444999.4941192,
+  "date": 1618812138.67823,
   "filename": "arellano.rst",
   "kernelspec": {
    "display_name": "Python",

arma.ipynb

@@ -1183,7 +1183,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444999.7153308,
+  "date": 1618812138.9135163,
   "filename": "arma.rst",
   "kernelspec": {
    "display_name": "Python",

black_litterman.ipynb

@@ -1582,7 +1582,7 @@
   }
  ],
  "metadata": {
-  "date": 1618444999.9369135,
+  "date": 1618812139.1406567,
   "filename": "black_litterman.rst",
   "kernelspec": {
    "display_name": "Python",