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QuantEcon committed Apr 29, 2021
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2 changes: 1 addition & 1 deletion BCG_complete_mkts.ipynb
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}
],
"metadata": {
"date": 1619402672.4190016,
"date": 1619661940.1660957,
"filename": "BCG_complete_mkts.rst",
"kernelspec": {
"display_name": "Python",
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2 changes: 1 addition & 1 deletion BCG_incomplete_mkts.ipynb
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}
],
"metadata": {
"date": 1619402672.684707,
"date": 1619661940.5953467,
"filename": "BCG_incomplete_mkts.rst",
"kernelspec": {
"display_name": "Python",
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2 changes: 1 addition & 1 deletion about_lectures.ipynb
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}
],
"metadata": {
"date": 1619402672.8388484,
"date": 1619661940.744541,
"filename": "about_lectures.rst",
"kernelspec": {
"display_name": "Python",
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2 changes: 1 addition & 1 deletion additive_functionals.ipynb
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}
],
"metadata": {
"date": 1619402672.9367144,
"date": 1619661940.8426106,
"filename": "additive_functionals.rst",
"kernelspec": {
"display_name": "Python",
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87 changes: 44 additions & 43 deletions amss.ipynb
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"\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)."
]
},
{
Expand All @@ -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",
Expand All @@ -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",
Expand All @@ -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",
Expand All @@ -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",
"$$\n",
"\n",
Expand All @@ -258,17 +258,17 @@
"- use the household’s first-order condition $ 1-\\tau^n_t(s^t)= u_{\\ell}(s^t) /u_c(s^t) $ to eliminate the labor tax rate \n",
"\n",
"\n",
"so that we can express the net-of-interest government surplus $ z(s^t) $ as\n",
"so that we can express the net-of-interest government surplus $ z_t(s^t) $ as\n",
"\n",
"\n",
"<a id='equation-amss-44-2'></a>\n",
"$$\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",
"$$\n",
"\n",
"If we substitute appropriate versions of the right side of [(7)](#equation-amss-44-2) for $ z(s^{t+j}) $ into equation [(6)](#equation-ts-gov-wo3),\n",
"If we substitute appropriate versions of the right side of [(7)](#equation-amss-44-2) for $ z_{t+j}(s^{t+j}) $ into equation [(6)](#equation-ts-gov-wo3),\n",
"we obtain a sequence of *implementability constraints* on a Ramsey allocation in an AMSS economy.\n",
"\n",
"Expression [(6)](#equation-ts-gov-wo3) at time $ t=0 $ and initial state $ s^0 $\n",
Expand All @@ -278,7 +278,7 @@
"<a id='equation-ts-gov-wo4'></a>\n",
"$$\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",
"$$\n",
"\n",
"Indeed, it was the *only* implementability constraint there.\n",
Expand All @@ -289,7 +289,7 @@
"<a id='equation-ts-gov-wo4a'></a>\n",
"$$\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",
"$$\n",
"\n",
"Equation [(9)](#equation-ts-gov-wo4a) must hold for each $ s^t $ for each $ t \\geq 1 $."
Expand Down Expand Up @@ -321,7 +321,7 @@
"$$\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",
"$$\n",
"\n",
"where the maximization is subject to\n",
Expand All @@ -330,7 +330,7 @@
"<a id='equation-amss-44'></a>\n",
"$$\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",
"$$\n",
"\n",
"and\n",
Expand All @@ -340,7 +340,7 @@
"$$\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",
"$$\n",
"\n",
Expand Down Expand Up @@ -369,7 +369,7 @@
" &\\;\\geq\\; (\\leq)\\;\\, 0 \\quad \\text{if the constraint binds in the following direction }\n",
" \\\\\n",
" & \\mathbb E_{t} \\sum_{j=0}^\\infty \\beta^j\n",
" { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;z(s^{t+j}) \\;\\geq \\;(\\leq)\\;\\, b_t(s^{t-1})\n",
" { u_c(s^{t+j}) \\over u_c(s^{t}) } \\;z_{t+j}(s^{t+j}) \\;\\geq \\;(\\leq)\\;\\, b_t(s^{t-1})\n",
"\\end{aligned}\n",
"$$\n",
"\n",
Expand Down Expand Up @@ -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",
Expand All @@ -433,7 +433,7 @@
"$$\n",
"\\begin{aligned}\n",
" u_c(s^t)-u_{\\ell}(s^t) &+ \\Psi_t(s^t)\\left\\{ \\left[\n",
" u_{cc}(s^t) - u_{c\\ell}(s^{t})\\right]z(s^{t}) +\n",
" u_{cc}(s^t) - u_{c\\ell}(s^{t})\\right]z_t(s^{t}) +\n",
" u_{c}(s^{t})\\,z_c(s^{t}) \\right\\}\n",
" \\\\\n",
" & \\hspace{35mm} - \\gamma_t(s^t)\\left[\n",
Expand All @@ -449,7 +449,7 @@
"\\mathbb E_{t} \\left[\\gamma_{t+1}(s^{t+1})\\,u_c(s^{t+1})\\right] = 0 \\tag{15}\n",
"$$\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",
Expand Down Expand Up @@ -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",
Expand Down Expand Up @@ -849,7 +848,7 @@
"\\beta V_x(x(s),s) = \\mu(s|s_-) \\tag{26}\n",
"$$\n",
"\n",
"Applying the envelope theorem to Bellman equation [(22)](#equation-eqn-amssapp5) gives\n",
"Applying an envelope theorem to Bellman equation [(22)](#equation-eqn-amssapp5) gives\n",
"\n",
"\n",
"<a id='equation-eqn-amssapp8'></a>\n",
Expand All @@ -864,7 +863,7 @@
"<a id='equation-eqn-amssapp9'></a>\n",
"$$\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",
Expand Down Expand Up @@ -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",
Expand Down Expand Up @@ -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",
Expand Down Expand Up @@ -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."
Expand Down Expand Up @@ -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",
Expand All @@ -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",
Expand Down Expand Up @@ -1786,7 +1787,7 @@
}
],
"metadata": {
"date": 1619402673.1077754,
"date": 1619661941.0074658,
"filename": "amss.rst",
"kernelspec": {
"display_name": "Python",
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}
],
"metadata": {
"date": 1619402673.2683423,
"date": 1619661941.1767492,
"filename": "amss2.rst",
"kernelspec": {
"display_name": "Python",
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}
],
"metadata": {
"date": 1619402673.464876,
"date": 1619661941.3859596,
"filename": "amss3.rst",
"kernelspec": {
"display_name": "Python",
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}
],
"metadata": {
"date": 1619402673.7797248,
"date": 1619661941.6302557,
"filename": "arellano.rst",
"kernelspec": {
"display_name": "Python",
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}
],
"metadata": {
"date": 1619402673.9790573,
"date": 1619661942.0405178,
"filename": "arma.rst",
"kernelspec": {
"display_name": "Python",
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}
],
"metadata": {
"date": 1619402674.204994,
"date": 1619661942.2040126,
"filename": "black_litterman.rst",
"kernelspec": {
"display_name": "Python",
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}
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"metadata": {
"date": 1619402674.4568348,
"date": 1619661942.441115,
"filename": "calvo.rst",
"kernelspec": {
"display_name": "Python",
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}
],
"metadata": {
"date": 1619402674.5947511,
"date": 1619661942.6324031,
"filename": "cattle_cycles.rst",
"kernelspec": {
"display_name": "Python",
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}
],
"metadata": {
"date": 1619402674.8980093,
"date": 1619661942.7952566,
"filename": "chang_credible.rst",
"kernelspec": {
"display_name": "Python",
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