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QuantEcon committed Apr 26, 2021
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2 changes: 1 addition & 1 deletion BCG_complete_mkts.ipynb
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}
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"date": 1619230074.5343747,
"date": 1619402672.4190016,
"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": 1619230074.8008502,
"date": 1619402672.684707,
"filename": "BCG_incomplete_mkts.rst",
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2 changes: 1 addition & 1 deletion about_lectures.ipynb
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}
],
"metadata": {
"date": 1619230074.956999,
"date": 1619402672.8388484,
"filename": "about_lectures.rst",
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2 changes: 1 addition & 1 deletion additive_functionals.ipynb
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}
],
"metadata": {
"date": 1619230075.0592115,
"date": 1619402672.9367144,
"filename": "additive_functionals.rst",
"kernelspec": {
"display_name": "Python",
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33 changes: 19 additions & 14 deletions amss.ipynb
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" - This term reflects the constraint that\n",
" beginning-of-period government indebtedness must be the same across all\n",
" realizations of next period’s state, a constraint that would not be present if\n",
" government debt could be state contingent. \n",
" government debt could be state-contingent. \n",
" \n",
"1. The Lagrange multiplier $ \\Psi_t(s^t) $ in the first-order condition\n",
" [(14)](#equation-amss-foc-a) may change over time in response to realizations of the state,\n",
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"source": [
"### Anticipated One-Period War\n",
"\n",
"In our lecture on [optimal taxation with state contingent debt](https://python-programming.quantecon.org/opt_tax_recur.html)\n",
"In our lecture on [optimal taxation with state-contingent debt](https://python-programming.quantecon.org/opt_tax_recur.html)\n",
"we studied how the government manages uncertainty in a simple setting.\n",
"\n",
"As in that lecture, we assume the one-period utility function\n",
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"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",
"[optimal taxation with state contingent debt](https://python-programming.quantecon.org/opt_tax_recur.html).\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",
"\n",
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"The following figure plots the Ramsey plan under both 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",
"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",
"\n",
"Paths with circles are histories in which there is peace, while those with\n",
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"\n",
"This pattern facilities smoothing tax rates across states.\n",
"\n",
"The government without state contingent debt cannot do this.\n",
"The government without state-contingent debt cannot do this.\n",
"\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",
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"\n",
"To service the additional debt burden, it raises taxes in all future periods.\n",
"\n",
"The absence of state contingent debt leads to an important difference in the\n",
"The absence of state-contingent debt leads to an important difference in the\n",
"optimal tax policy.\n",
"\n",
"When the Ramsey planner has access to state contingent debt, the optimal tax\n",
"When the Ramsey planner has access to state-contingent debt, the optimal tax\n",
"policy is history independent\n",
"\n",
"- the tax rate is a function of the current level of government spending only,\n",
" given the Lagrange multiplier on the implementability constraint \n",
"\n",
"\n",
"Without state contingent debt, the optimal tax rate is history dependent.\n",
"Without state-contingent debt, the optimal tax rate is history dependent.\n",
"\n",
"- A war at time $ t=3 $ causes a permanent increase in the tax rate. "
"- A war at time $ t=3 $ causes a permanent **increase** in the tax rate. \n",
"- Peace at time $ t=3 $ causes a permanent **reduction** in the tax rate. "
]
},
{
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"model with state-contingent debt.\n",
"\n",
"The figure below plots optimal tax policies for both the economy with\n",
"state contingent debt (circles) and the economy with only a risk-free bond\n",
"state-contingent debt (circles) and the economy with only a risk-free bond\n",
"(triangles)."
]
},
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"metadata": {},
"source": [
"When the government experiences a prolonged period of peace, it is able to reduce\n",
"government debt and set permanently lower tax rates.\n",
"government debt and set persistently lower tax rates.\n",
"\n",
"However, the government finances a long war by borrowing and raising taxes.\n",
"\n",
"This results in a drift away from policies with state contingent debt that\n",
"This results in a drift away from policies with state-contingent debt that\n",
"depends on the history of shocks.\n",
"\n",
"This is even more evident in the following figure that plots the evolution of\n",
"the two policies over 200 periods."
"the two policies over 200 periods.\n",
"\n",
"This outcome reflects the presence of a force for **precautionary saving** that the incomplete markets structure imparts to the Ramsey plan.\n",
"\n",
"In [this subsequent lecture](https://python-programming.quantecon.org/amss2.html) and [this subsequent lecture](https://python-programming.quantecon.org/amss3.html), some ultimate consequences of that force are explored."
]
},
{
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}
],
"metadata": {
"date": 1619230075.2505703,
"date": 1619402673.1077754,
"filename": "amss.rst",
"kernelspec": {
"display_name": "Python",
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44 changes: 23 additions & 21 deletions amss2.ipynb
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"The measurability constraints imposed by the AMSS model are inherited from the restriction that only one-period risk-free bonds\n",
"can be traded.\n",
"\n",
"Differences between the Ramsey allocations in the two models indicate that at least some of the measurability constraints of the AMSS model of\n",
"Differences between the Ramsey allocations in the two models indicate that at least some of the **implementability constraints** of the AMSS model of\n",
"[optimal taxation without state-contingent debt](https://python-programming.quantecon.org/amss.html) are violated at the Ramsey allocation of a corresponding [[LS83]](https://python-programming.quantecon.org/zreferences.html#lucasstokey1983) model with state-contingent debt.\n",
"\n",
"Another way to say this is that differences between the Ramsey allocations of the two models indicate that some of the measurability constraints of the\n",
"Another way to say this is that differences between the Ramsey allocations of the two models indicate that some of the **measurability constraints** imposed by the\n",
"AMSS model are violated at the Ramsey allocation of the Lucas-Stokey model.\n",
"\n",
"Nonzero Lagrange multipliers on those constraints make the Ramsey allocation for the AMSS model differ from the Ramsey allocation for the Lucas-Stokey model.\n",
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"- The exogenous state variable $ s_t $ is governed by a finite-state Markov chain. \n",
"- With an arbitrary budget-feasible initial level of government debt, the measurability constraints \n",
" - bind for many periods, but $ \\ldots $. \n",
" - eventually, they stop binding evermore, so $ \\ldots $. \n",
" - in the tail of the Ramsey plan, the Lagrange multipliers $ \\gamma_t(s^t) $ on the AMSS implementability constraints [(8)](https://python-programming.quantecon.org/amss.html#equation-ts-gov-wo4) converge to zero. \n",
" - eventually, they stop binding evermore, so that $ \\ldots $ \n",
" - in the tail of the Ramsey plan, the Lagrange multipliers $ \\gamma_t(s^t) $ on the AMSS implementability constraints [(8)](https://python-programming.quantecon.org/amss.html#equation-ts-gov-wo4) are zero. \n",
"- After the implementability constraints [(8)](https://python-programming.quantecon.org/amss.html#equation-ts-gov-wo4) no longer bind in the tail of the AMSS Ramsey plan \n",
" - history dependence of the AMSS state variable $ x_t $ vanishes and $ x_t $ becomes a time-invariant function of the Markov state $ s_t $. \n",
" - the par value of government debt becomes **constant over time** so that $ b_{t+1}(s^t) = \\bar b $ for $ t \\geq T $ for a sufficiently large $ T $. \n",
" - $ \\bar b <0 $, so that the tail of the Ramsey plan instructs the government always to make a constant par value of risk-free one-period loans to the private sector. \n",
" - $ \\bar b <0 $, so that the tail of the Ramsey plan instructs the government always to make a constant par value of risk-free one-period loans **to** the private sector. \n",
" - the one-period gross interest rate $ R_t(s^t) $ on risk-free debt converges to a time-invariant function of the Markov state $ s_t $. \n",
"- For a **particular** $ b_0 < 0 $ (i.e., a positive level of initial government **loans** to the private sector), the measurability constraints **never** bind. \n",
"- In this special case \n",
Expand All @@ -104,9 +104,9 @@
" - the **market value** $ \\frac{\\bar b}{R_t(s_t)} $ of government debt at time $ t $ varies as a time-invariant function of the Markov state $ s_t $. \n",
" - fluctuations in the interest rate make gross earnings on government debt $ \\frac{\\bar b}{R_t(s_t)} $ fully insure the gross-of-gross-interest-payments government budget against fluctuations in government expenditures. \n",
" - the state variable $ x $ in a recursive representation of a Ramsey plan is a time-invariant function of the Markov state for $ t \\geq 0 $. \n",
"- In this special case, the Ramsey allocation in the AMSS model agrees with that in a [[LS83]](https://python-programming.quantecon.org/zreferences.html#lucasstokey1983) model in which\n",
"- In this special case, the Ramsey allocation in the AMSS model agrees with that in a Lucas-Stokey [[LS83]](https://python-programming.quantecon.org/zreferences.html#lucasstokey1983) complete markets model in which\n",
" the same amount of state-contingent debt falls due in all states tomorrow \n",
" - it is a situation in which the Ramsey planner loses nothing from not being able to purchase state-contingent debt and being restricted to exchange only risk-free debt debt. \n",
" - it is a situation in which the Ramsey planner loses nothing from not being able to trade state-contingent debt and being restricted to exchange only risk-free debt debt. \n",
"- This outcome emerges only when we initialize government debt at a particular $ b_0 < 0 $. \n",
"\n",
"\n",
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"source": [
"## Forces at Work\n",
"\n",
"The forces driving asymptotic outcomes here are examples of dynamics present in a more general class incomplete markets models analyzed in [[BEGS17]](https://python-programming.quantecon.org/zreferences.html#begs1) (BEGS).\n",
"The forces driving asymptotic outcomes here are examples of dynamics present in a more general class of incomplete markets models analyzed in [[BEGS17]](https://python-programming.quantecon.org/zreferences.html#begs1) (BEGS).\n",
"\n",
"BEGS provide conditions under which government debt under a Ramsey plan converges to an invariant distribution.\n",
"\n",
"BEGS construct approximations to that asymptotically invariant distribution of government debt under a Ramsey plan.\n",
"\n",
"BEGS also compute an approximation to a Ramsey plan’s rate of convergence to that limiting invariant distribution.\n",
"\n",
"We shall use the BEGS approximating limiting distribution and the approximating rate of convergence to help interpret outcomes here.\n",
"We shall use the BEGS approximating limiting distribution and their approximating rate of convergence to help interpret outcomes here.\n",
"\n",
"For a long time, the Ramsey plan puts a nontrivial martingale-like component into the par value of government debt as part of the way that the Ramsey plan imperfectly\n",
"smooths distortions from the labor tax rate across time and Markov states.\n",
"\n",
"But BEGS show that binding implementability constraints slowly push government debt in a direction designed to let the government use fluctuations in equilibrium interest\n",
"rate rather than fluctuations in par values of debt to insure against shocks to government expenditures.\n",
"But BEGS show that binding implementability constraints slowly push government debt in a direction designed to let the government use fluctuations in equilibrium interest\n",
"rates rather than fluctuations in par values of debt to insure against shocks to government expenditures.\n",
"\n",
"- This is a **weak** (but unrelenting) force that, starting from an initial debt level, for a long time is dominated by the stochastic martingale-like component of debt\n",
"- This is a **weak** (but unrelenting) force that, starting from a positive initial debt level, for a long time is dominated by the stochastic martingale-like component of debt\n",
" dynamics that the Ramsey planner uses to facilitate imperfect tax-smoothing across time and states. \n",
"- This weak force slowly drives the par value of government **assets** to a **constant** level at which the government can completely insure against government expenditure shocks while\n",
" shutting down the stochastic component of debt dynamics. \n",
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"Although we are studying an AMSS [[AMSSeppala02]](https://python-programming.quantecon.org/zreferences.html#aiyagari2002optimal) economy, a Lucas-Stokey [[LS83]](https://python-programming.quantecon.org/zreferences.html#lucasstokey1983) economy plays\n",
"an important role in the reverse-engineering calculation to be described below.\n",
"\n",
"For that reason, it is helpful to have readily available some key equations underlying a Ramsey plan for the Lucas-Stokey economy.\n",
"For that reason, it is helpful to have key equations underlying a Ramsey plan for the Lucas-Stokey economy readily available.\n",
"\n",
"Recall first-order conditions for a Ramsey allocation for the Lucas-Stokey economy.\n",
"\n",
Expand All @@ -206,7 +206,7 @@
"\n",
"There is one such equation for each value of the Markov state $ s_t $.\n",
"\n",
"In addition, given an initial Markov state, the time $ t=0 $ quantities $ c_0 $ and $ b_0 $ satisfy\n",
"Given an initial Markov state, the time $ t=0 $ quantities $ c_0 $ and $ b_0 $ satisfy\n",
"\n",
"\n",
"<a id='equation-ts-barg11b'></a>\n",
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"$$\n",
"\n",
"In addition, the time $ t=0 $ budget constraint is satisfied at $ c_0 $ and initial government debt\n",
"$ b_0 $:\n",
"$ b_0 $\n",
"\n",
"\n",
"<a id='equation-eqn-amss2-10'></a>\n",
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"source": [
"## Long Simulation\n",
"\n",
"The following graph shows the par value of government debt and the flat rate tax on labor income for a long simulation for our sample economy.\n",
"The following graph shows the par value of government debt and the flat-rate tax on labor income for a long simulation for our sample economy.\n",
"\n",
"For the **same** realization of a government expenditure path, the graph reports outcomes for two economies\n",
"\n",
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"\n",
"- Notice that this is a martingale-like random process that eventually seems to converge to a constant $ \\bar b \\approx - 1.07 $. \n",
"- Notice that the limiting value $ \\bar b < 0 $ so that asymptotically the government makes a constant level of risk-free loans to the public. \n",
"- In the simulation displayed as well as other simulations we have run, the par value of government debt converges to about $ 1.07 $ afters between 1400 to 2000 periods. \n",
"- In the simulation displayed as well as other simulations we have run, the par value of government debt converges to about $ 1.07 $ after between 1400 to 2000 periods. \n",
"\n",
"\n",
"For the AMSS incomplete markets economy, the marginal tax rate on labor income $ \\tau_t $ converges to a constant\n",
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"source": [
"## BEGS Approximations of Limiting Debt and Convergence Rate\n",
"\n",
"It is useful to link the outcome of our reverse engineering exercise to limiting approximations constructed by [[BEGS17]](https://python-programming.quantecon.org/zreferences.html#begs1).\n",
"It is useful to link the outcome of our reverse engineering exercise to limiting approximations constructed by BEGS [[BEGS17]](https://python-programming.quantecon.org/zreferences.html#begs1).\n",
"\n",
"[[BEGS17]](https://python-programming.quantecon.org/zreferences.html#begs1) used a slightly different notation to represent a generalization of the AMSS model.\n",
"BEGS [[BEGS17]](https://python-programming.quantecon.org/zreferences.html#begs1) used a slightly different notation to represent a generalization of the AMSS model.\n",
"\n",
"We’ll introduce a version of their notation so that readers can quickly relate notation that appears in their key formulas to the notation\n",
"that we have used.\n",
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"metadata": {},
"source": [
"The slow rate of convergence and the implied time of getting within one percent of the limiting value do a good job of approximating\n",
"our long simulation above."
"our long simulation above.\n",
"\n",
"In [a subsequent lecture](https://python-programming.quantecon.org/amss3.html) we shall study an extension of the model in which the force highlighted in this lecture causes government debt to converge to a nontrivial distribution instead of the single debt level discovered here."
]
}
],
"metadata": {
"date": 1619230075.4204001,
"date": 1619402673.2683423,
"filename": "amss2.rst",
"kernelspec": {
"display_name": "Python",
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