|
77 | 77 | "The measurability constraints imposed by the AMSS model are inherited from the restriction that only one-period risk-free bonds\n",
|
78 | 78 | "can be traded.\n",
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79 | 79 | "\n",
|
80 |
| - "Differences between the Ramsey allocations in the two models indicate that at least some of the measurability constraints of the AMSS model of\n", |
| 80 | + "Differences between the Ramsey allocations in the two models indicate that at least some of the **implementability constraints** of the AMSS model of\n", |
81 | 81 | "[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",
|
82 | 82 | "\n",
|
83 |
| - "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", |
| 83 | + "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", |
84 | 84 | "AMSS model are violated at the Ramsey allocation of the Lucas-Stokey model.\n",
|
85 | 85 | "\n",
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86 | 86 | "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",
|
|
90 | 90 | "- The exogenous state variable $ s_t $ is governed by a finite-state Markov chain. \n",
|
91 | 91 | "- With an arbitrary budget-feasible initial level of government debt, the measurability constraints \n",
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92 | 92 | " - bind for many periods, but $ \\ldots $. \n",
|
93 |
| - " - eventually, they stop binding evermore, so $ \\ldots $. \n", |
94 |
| - " - 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", |
| 93 | + " - eventually, they stop binding evermore, so that $ \\ldots $ \n", |
| 94 | + " - 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", |
95 | 95 | "- 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",
|
96 | 96 | " - 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",
|
97 | 97 | " - 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",
|
98 |
| - " - $ \\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", |
| 98 | + " - $ \\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", |
99 | 99 | " - 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",
|
100 | 100 | "- 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",
|
101 | 101 | "- In this special case \n",
|
|
104 | 104 | " - 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",
|
105 | 105 | " - 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",
|
106 | 106 | " - 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",
|
107 |
| - "- 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", |
| 107 | + "- 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", |
108 | 108 | " the same amount of state-contingent debt falls due in all states tomorrow \n",
|
109 |
| - " - 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", |
| 109 | + " - 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", |
110 | 110 | "- This outcome emerges only when we initialize government debt at a particular $ b_0 < 0 $. \n",
|
111 | 111 | "\n",
|
112 | 112 | "\n",
|
|
135 | 135 | "source": [
|
136 | 136 | "## Forces at Work\n",
|
137 | 137 | "\n",
|
138 |
| - "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", |
| 138 | + "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", |
139 | 139 | "\n",
|
140 | 140 | "BEGS provide conditions under which government debt under a Ramsey plan converges to an invariant distribution.\n",
|
141 | 141 | "\n",
|
142 | 142 | "BEGS construct approximations to that asymptotically invariant distribution of government debt under a Ramsey plan.\n",
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143 | 143 | "\n",
|
144 | 144 | "BEGS also compute an approximation to a Ramsey plan’s rate of convergence to that limiting invariant distribution.\n",
|
145 | 145 | "\n",
|
146 |
| - "We shall use the BEGS approximating limiting distribution and the approximating rate of convergence to help interpret outcomes here.\n", |
| 146 | + "We shall use the BEGS approximating limiting distribution and their approximating rate of convergence to help interpret outcomes here.\n", |
147 | 147 | "\n",
|
148 | 148 | "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",
|
149 | 149 | "smooths distortions from the labor tax rate across time and Markov states.\n",
|
150 | 150 | "\n",
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151 |
| - "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", |
152 |
| - "rate rather than fluctuations in par values of debt to insure against shocks to government expenditures.\n", |
| 151 | + "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", |
| 152 | + "rates rather than fluctuations in par values of debt to insure against shocks to government expenditures.\n", |
153 | 153 | "\n",
|
154 |
| - "- 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", |
| 154 | + "- 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", |
155 | 155 | " dynamics that the Ramsey planner uses to facilitate imperfect tax-smoothing across time and states. \n",
|
156 | 156 | "- 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",
|
157 | 157 | " shutting down the stochastic component of debt dynamics. \n",
|
|
186 | 186 | "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",
|
187 | 187 | "an important role in the reverse-engineering calculation to be described below.\n",
|
188 | 188 | "\n",
|
189 |
| - "For that reason, it is helpful to have readily available some key equations underlying a Ramsey plan for the Lucas-Stokey economy.\n", |
| 189 | + "For that reason, it is helpful to have key equations underlying a Ramsey plan for the Lucas-Stokey economy readily available.\n", |
190 | 190 | "\n",
|
191 | 191 | "Recall first-order conditions for a Ramsey allocation for the Lucas-Stokey economy.\n",
|
192 | 192 | "\n",
|
|
206 | 206 | "\n",
|
207 | 207 | "There is one such equation for each value of the Markov state $ s_t $.\n",
|
208 | 208 | "\n",
|
209 |
| - "In addition, given an initial Markov state, the time $ t=0 $ quantities $ c_0 $ and $ b_0 $ satisfy\n", |
| 209 | + "Given an initial Markov state, the time $ t=0 $ quantities $ c_0 $ and $ b_0 $ satisfy\n", |
210 | 210 | "\n",
|
211 | 211 | "\n",
|
212 | 212 | "<a id='equation-ts-barg11b'></a>\n",
|
|
221 | 221 | "$$\n",
|
222 | 222 | "\n",
|
223 | 223 | "In addition, the time $ t=0 $ budget constraint is satisfied at $ c_0 $ and initial government debt\n",
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224 |
| - "$ b_0 $:\n", |
| 224 | + "$ b_0 $\n", |
225 | 225 | "\n",
|
226 | 226 | "\n",
|
227 | 227 | "<a id='equation-eqn-amss2-10'></a>\n",
|
|
1204 | 1204 | "source": [
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1205 | 1205 | "## Long Simulation\n",
|
1206 | 1206 | "\n",
|
1207 |
| - "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", |
| 1207 | + "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", |
1208 | 1208 | "\n",
|
1209 | 1209 | "For the **same** realization of a government expenditure path, the graph reports outcomes for two economies\n",
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1210 | 1210 | "\n",
|
|
1223 | 1223 | "\n",
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1224 | 1224 | "- Notice that this is a martingale-like random process that eventually seems to converge to a constant $ \\bar b \\approx - 1.07 $. \n",
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1225 | 1225 | "- 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",
|
1226 |
| - "- 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", |
| 1226 | + "- 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", |
1227 | 1227 | "\n",
|
1228 | 1228 | "\n",
|
1229 | 1229 | "For the AMSS incomplete markets economy, the marginal tax rate on labor income $ \\tau_t $ converges to a constant\n",
|
|
1285 | 1285 | "source": [
|
1286 | 1286 | "## BEGS Approximations of Limiting Debt and Convergence Rate\n",
|
1287 | 1287 | "\n",
|
1288 |
| - "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", |
| 1288 | + "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", |
1289 | 1289 | "\n",
|
1290 |
| - "[[BEGS17]](https://python-programming.quantecon.org/zreferences.html#begs1) used a slightly different notation to represent a generalization of the AMSS model.\n", |
| 1290 | + "BEGS [[BEGS17]](https://python-programming.quantecon.org/zreferences.html#begs1) used a slightly different notation to represent a generalization of the AMSS model.\n", |
1291 | 1291 | "\n",
|
1292 | 1292 | "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",
|
1293 | 1293 | "that we have used.\n",
|
|
1603 | 1603 | "metadata": {},
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1604 | 1604 | "source": [
|
1605 | 1605 | "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",
|
1606 |
| - "our long simulation above." |
| 1606 | + "our long simulation above.\n", |
| 1607 | + "\n", |
| 1608 | + "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." |
1607 | 1609 | ]
|
1608 | 1610 | }
|
1609 | 1611 | ],
|
1610 | 1612 | "metadata": {
|
1611 |
| - "date": 1619230075.4204001, |
| 1613 | + "date": 1619402673.2683423, |
1612 | 1614 | "filename": "amss2.rst",
|
1613 | 1615 | "kernelspec": {
|
1614 | 1616 | "display_name": "Python",
|
|
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