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Merge pull request #1361 from sidd3888/ChangeCodeLabels
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Replace \texttt{} in notebooks
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alanlujan91 authored Nov 23, 2023
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70 changes: 35 additions & 35 deletions examples/ConsIndShockModel/PerfForesightConsumerType.ipynb
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"source": [
"# Perfect foresight consumption-saving\n",
"\n",
"**The `PerfForesightConsumerType` class**"
"**The** `PerfForesightConsumerType` **class**"
]
},
{
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"\n",
"| Parameter | Description | Code | Example value | Time-varying? |\n",
"| :---: | --- | --- | --- | --- |\n",
"| $\\DiscFac$ |Intertemporal discount factor | $\\texttt{DiscFac}$ | $0.96$ | |\n",
"| $\\CRRA$ |Coefficient of relative risk aversion | $\\texttt{CRRA}$ | $2.0$ | |\n",
"| $\\Rfree$ | Risk free interest factor | $\\texttt{Rfree}$ | $1.03$ | |\n",
"| $1 - \\DiePrb_{t+1}$ |Survival probability | $\\texttt{LivPrb}$ | $[0.98]$ | $\\surd$ |\n",
"|$\\PermGroFac_{t+1}$|Permanent income growth factor|$\\texttt{PermGroFac}$| $[1.01]$ | $\\surd$ |\n",
"|$\\underline{a}$|Artificial borrowing constraint|$\\texttt{BoroCnstArt}$| $None$ | |\n",
"|$(none)$|Maximum number of gridpoints in consumption function |$\\texttt{aXtraCount}$| $200$ | |\n",
"|$T$| Number of periods in this type's \"cycle\" |$\\texttt{T\\_cycle}$| $1$ | |\n",
"|(none)| Number of times the \"cycle\" occurs |$\\texttt{cycles}$| $0$ | |\n",
"| $\\DiscFac$ |Intertemporal discount factor | `DiscFac` | $0.96$ | |\n",
"| $\\CRRA$ |Coefficient of relative risk aversion | `CRRA` | $2.0$ | |\n",
"| $\\Rfree$ | Risk free interest factor | `Rfree` | $1.03$ | |\n",
"| $1 - \\DiePrb_{t+1}$ |Survival probability | `LivPrb` | $[0.98]$ | $\\surd$ |\n",
"|$\\PermGroFac_{t+1}$|Permanent income growth factor|`PermGroFac`| $[1.01]$ | $\\surd$ |\n",
"|$\\underline{a}$|Artificial borrowing constraint|`BoroCnstArt`| $None$ | |\n",
"|$(none)$|Maximum number of gridpoints in consumption function |`aXtraCount`| $200$ | |\n",
"|$T$| Number of periods in this type's \"cycle\" |`T_cycle`| $1$ | |\n",
"|(none)| Number of times the \"cycle\" occurs |`cycles`| $0$ | |\n",
"\n",
"Note that the survival probability and income growth factor have time subscripts; likewise, the example values for these parameters are *lists* rather than simply single floats. This is because those parameters are *time-varying*: their values can depend on which period of the problem the agent is in. All time-varying parameters *must* be specified as lists, even if the same value occurs in each period for this type.\n",
"\n",
"The artificial borrowing constraint can be any non-positive `float`, or it can be `None` to indicate no artificial borrowing constraint. The maximum number of gridpoints in the consumption function is only relevant if the borrowing constraint is not `None`; without an upper bound on the number of gridpoints, kinks in the consumption function will propagate indefinitely in an infinite horizon model if there is a borrowing constraint, eventually resulting in an overflow error. If there is no artificial borrowing constraint, then the number of gridpoints used to represent the consumption function is always exactly two.\n",
"\n",
"The last two parameters in the table specify the \"nature of time\" for this type: the number of (non-terminal) periods in this type's \"cycle\", and the number of times that the \"cycle\" occurs. *Every* subclass of `AgentType` uses these two code parameters to define the nature of time. Here, `T_cycle` has the value $1$, indicating that there is exactly one period in the cycle, while `cycles` is $0$, indicating that the cycle is repeated in *infinite* number of times-- it is an infinite horizon model, with the same \"kind\" of period repeated over and over.\n",
"\n",
"In contrast, we could instead specify a life-cycle model by setting `T_cycle` to $1$, and specifying age-varying sequences of income growth and survival probability. In all cases, the number of elements in each time-varying parameter should exactly equal $\\texttt{T\\_cycle}$.\n",
"In contrast, we could instead specify a life-cycle model by setting `T_cycle` to $1$, and specifying age-varying sequences of income growth and survival probability. In all cases, the number of elements in each time-varying parameter should exactly equal `T_cycle`.\n",
"\n",
"The parameter $\\texttt{AgentCount}$ specifies how many consumers there are of this *type*-- how many individuals have these exact parameter values and are *ex ante* homogeneous. This information is not relevant for solving the model, but is needed in order to simulate a population of agents, introducing *ex post* heterogeneity through idiosyncratic shocks. Of course, simulating a perfect foresight model is quite boring, as there are *no* idiosyncratic shocks other than death!\n",
"The parameter `AgentCount` specifies how many consumers there are of this *type*-- how many individuals have these exact parameter values and are *ex ante* homogeneous. This information is not relevant for solving the model, but is needed in order to simulate a population of agents, introducing *ex post* heterogeneity through idiosyncratic shocks. Of course, simulating a perfect foresight model is quite boring, as there are *no* idiosyncratic shocks other than death!\n",
"\n",
"The cell below defines a dictionary that can be passed to the constructor method for `PerfForesightConsumerType`, with the values from the table here."
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"The $\\texttt{solve}$ method fills in the instance's attribute `solution` as a time-varying list of solutions to each period of the consumer's problem. In this case, `solution` will be a list with exactly one instance of the class `ConsumerSolution`, representing the solution to the infinite horizon model we specified."
"The `solve` method fills in the instance's attribute `solution` as a time-varying list of solutions to each period of the consumer's problem. In this case, `solution` will be a list with exactly one instance of the class `ConsumerSolution`, representing the solution to the infinite horizon model we specified."
]
},
{
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"cell_type": "markdown",
"metadata": {},
"source": [
"Each element of `solution` has a few attributes. To see all of them, we can use the $\\texttt{vars}$ built in function:\n",
"Each element of `solution` has a few attributes. To see all of them, we can use the `vars` built in function:\n",
"\n",
"the consumption functions reside in the attribute $\\texttt{cFunc}$ of each element of `ConsumerType.solution`. This method creates a (time varying) attribute $\\texttt{cFunc}$ that contains a list of consumption functions."
"the consumption functions reside in the attribute `cFunc` of each element of `ConsumerType.solution`. This method creates a (time varying) attribute `cFunc` that contains a list of consumption functions."
]
},
{
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"cell_type": "markdown",
"metadata": {},
"source": [
"The two most important attributes of a single period solution of this model are the (normalized) consumption function $\\texttt{cFunc}$ and the (normalized) value function $\\texttt{vFunc}$. Let's plot those functions near the lower bound of the permissible state space (the attribute $\\texttt{mNrmMin}$ tells us the lower bound of $m_t$ where the consumption function is defined)."
"The two most important attributes of a single period solution of this model are the (normalized) consumption function `cFunc` and the (normalized) value function `vFunc`. Let's plot those functions near the lower bound of the permissible state space (the attribute `mNrmMin` tells us the lower bound of $m_t$ where the consumption function is defined)."
]
},
{
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"cell_type": "markdown",
"metadata": {},
"source": [
"An element of `solution` also includes the (normalized) marginal value function $\\texttt{vPfunc}$, and the lower and upper bounds of the marginal propensity to consume (MPC) $\\texttt{MPCmin}$ and $\\texttt{MPCmax}$. Note that with a linear consumption function, the MPC is constant, so its lower and upper bound are identical.\n",
"An element of `solution` also includes the (normalized) marginal value function `vPfunc`, and the lower and upper bounds of the marginal propensity to consume (MPC) `MPCmin` and `MPCmax`. Note that with a linear consumption function, the MPC is constant, so its lower and upper bound are identical.\n",
"\n",
"### Liquidity constrained perfect foresight example\n",
"\n",
"Without an artificial borrowing constraint, a perfect foresight consumer is free to borrow against the PDV of his entire future stream of labor income-- his \"human wealth\" $\\texttt{hNrm}$-- and he will consume a constant proportion of his total wealth (market resources plus human wealth). If we introduce an artificial borrowing constraint, both of these features vanish. In the cell below, we define a parameter dictionary that prevents the consumer from borrowing *at all*, create and solve a new instance of `PerfForesightConsumerType` with it, and then plot its consumption function."
"Without an artificial borrowing constraint, a perfect foresight consumer is free to borrow against the PDV of his entire future stream of labor income -- his \"human wealth\" `hNrm` -- and he will consume a constant proportion of his total wealth (market resources plus human wealth). If we introduce an artificial borrowing constraint, both of these features vanish. In the cell below, we define a parameter dictionary that prevents the consumer from borrowing *at all*, create and solve a new instance of `PerfForesightConsumerType` with it, and then plot its consumption function."
]
},
{
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"incorrectly_encoded_metadata": "pycharm= [markdown] {\"name\": \"#%% md\\n\"}"
},
"source": [
"At this time, the value function for a perfect foresight consumer with an artificial borrowing constraint is not computed nor included as part of its $\\texttt{solution}$."
"At this time, the value function for a perfect foresight consumer with an artificial borrowing constraint is not computed nor included as part of its `solution`."
]
},
{
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"\n",
"| Description | Code | Example value |\n",
"| :---: | --- | --- |\n",
"| Number of consumers of this type | $\\texttt{AgentCount}$ | $10000$ |\n",
"| Number of periods to simulate | $\\texttt{T\\_sim}$ | $120$ |\n",
"| Mean of initial log (normalized) assets | $\\texttt{aNrmInitMean}$ | $-6.0$ |\n",
"| Stdev of initial log (normalized) assets | $\\texttt{aNrmInitStd}$ | $1.0$ |\n",
"| Mean of initial log permanent income | $\\texttt{pLvlInitMean}$ | $0.0$ |\n",
"| Stdev of initial log permanent income | $\\texttt{pLvlInitStd}$ | $0.0$ |\n",
"| Aggregrate productivity growth factor | $\\texttt{PermGroFacAgg}$ | $1.0$ |\n",
"| Age after which consumers are automatically killed | $\\texttt{T\\_age}$ | $None$ |\n",
"| Number of consumers of this type | `AgentCount` | $10000$ |\n",
"| Number of periods to simulate | `T_sim` | $120$ |\n",
"| Mean of initial log (normalized) assets | `aNrmInitMean` | $-6.0$ |\n",
"| Stdev of initial log (normalized) assets | `aNrmInitStd` | $1.0$ |\n",
"| Mean of initial log permanent income | `pLvlInitMean` | $0.0$ |\n",
"| Stdev of initial log permanent income | `pLvlInitStd` | $0.0$ |\n",
"| Aggregrate productivity growth factor | `PermGroFacAgg` | $1.0$ |\n",
"| Age after which consumers are automatically killed | `T_age` | $None$ |\n",
"\n",
"We have specified the model so that initial assets and permanent income are both distributed lognormally, with mean and standard deviation of the underlying normal distributions provided by the user.\n",
"\n",
"The parameter $\\texttt{PermGroFacAgg}$ exists for compatibility with more advanced models that employ aggregate productivity shocks; it can simply be set to 1.\n",
"The parameter `PermGroFacAgg` exists for compatibility with more advanced models that employ aggregate productivity shocks; it can simply be set to 1.\n",
"\n",
"In infinite horizon models, it might be useful to prevent agents from living extraordinarily long lives through a fortuitous sequence of mortality shocks. We have thus provided the option of setting $\\texttt{T\\_age}$ to specify the maximum number of periods that a consumer can live before they are automatically killed (and replaced with a new consumer with initial state drawn from the specified distributions). This can be turned off by setting it to `None`.\n",
"In infinite horizon models, it might be useful to prevent agents from living extraordinarily long lives through a fortuitous sequence of mortality shocks. We have thus provided the option of setting `T_age` to specify the maximum number of periods that a consumer can live before they are automatically killed (and replaced with a new consumer with initial state drawn from the specified distributions). This can be turned off by setting it to `None`.\n",
"\n",
"The cell below puts these parameters into a dictionary, then gives them to `PFexample`. Note that all of these parameters *could* have been passed as part of the original dictionary; we omitted them above for simplicity."
]
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"source": [
"To generate simulated data, we need to specify which variables we want to track the \"history\" of for this instance. To do so, we set the `track_vars` attribute of our `PerfForesightConsumerType` instance to be a list of strings with the simulation variables we want to track.\n",
"\n",
"In this model, valid arguments to `track_vars` include $\\texttt{mNrm}$, $\\texttt{cNrm}$, $\\texttt{aNrm}$, and $\\texttt{pLvl}$. Because this model has no idiosyncratic shocks, our simulated data will be quite boring.\n",
"In this model, valid arguments to `track_vars` include `mNrm`, `cNrm`, `aNrm`, and `pLvl`. Because this model has no idiosyncratic shocks, our simulated data will be quite boring.\n",
"\n",
"### Generating simulated data\n",
"\n",
"Before simulating, the `initialize_sim` method must be invoked. This resets our instance back to its initial state, drawing a set of initial $\\texttt{aNrm}$ and $\\texttt{pLvl}$ values from the specified distributions and storing them in the attributes $\\texttt{aNrmNow\\_init}$ and $\\texttt{pLvlNow\\_init}$. It also resets this instance's internal random number generator, so that the same initial states will be set every time `initialize_sim` is called. In models with non-trivial shocks, this also ensures that the same sequence of shocks will be generated on every simulation run.\n",
"Before simulating, the `initialize_sim` method must be invoked. This resets our instance back to its initial state, drawing a set of initial `aNrm` and `pLvl` values from the specified distributions and storing them in the attributes `aNrmNow_init` and `pLvlNow_init`. It also resets this instance's internal random number generator, so that the same initial states will be set every time `initialize_sim` is called. In models with non-trivial shocks, this also ensures that the same sequence of shocks will be generated on every simulation run.\n",
"\n",
"Finally, the `simulate` method can be called."
]
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"incorrectly_encoded_metadata": "pycharm= [markdown] {\"name\": \"#%% md\\n\"}"
},
"source": [
"Each simulation variable $\\texttt{X}$ named in $\\texttt{track\\_vars}$ will have the *history* of that variable for each agent stored in the attribute $\\texttt{X\\_hist}$ as an array of shape $(\\texttt{T\\_sim},\\texttt{AgentCount})$. To see that the simulation worked as intended, we can plot the mean of $m_t$ in each simulated period:"
"Each simulation variable `X` named in `track_vars` will have the *history* of that variable for each agent stored in the attribute `X_hist` as an array of shape `(T_sim, AgentCount)`. To see that the simulation worked as intended, we can plot the mean of $m_t$ in each simulated period:"
]
},
{
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"source": [
"A perfect foresight consumer can borrow against the PDV of his future income -- his human wealth -- and thus as time goes on, our simulated agents approach the (very negative) steady state level of $m_t$ while being steadily replaced with consumers with roughly $m_t=1$.\n",
"\n",
"The slight wiggles in the plotted curve are due to consumers randomly dying and being replaced; their replacement will have an initial state drawn from the distributions specified by the user. To see the current distribution of ages, we can look at the attribute $\\texttt{t\\_age}$."
"The slight wiggles in the plotted curve are due to consumers randomly dying and being replaced; their replacement will have an initial state drawn from the distributions specified by the user. To see the current distribution of ages, we can look at the attribute `t_age`."
]
},
{
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"\n",
"One might wonder why HARK requires users to call `initialize_sim` before calling `simulate`: Why doesn't `simulate` just call `initialize_sim` as its first step? We have broken up these two steps so that users can simulate some number of periods, change something in the environment, and then resume the simulation.\n",
"\n",
"When called with no argument, `simulate` will simulate the model for $\\texttt{T\\_sim}$ periods. The user can optionally pass an integer specifying the number of periods to simulate (which should not exceed $\\texttt{T\\_sim}$).\n",
"When called with no argument, `simulate` will simulate the model for `T_sim` periods. The user can optionally pass an integer specifying the number of periods to simulate (which should not exceed `T_sim`).\n",
"\n",
"In the cell below, we simulate our perfect foresight consumers for 80 periods, then seize a bunch of their assets (dragging their wealth even more negative), then simulate for the remaining 40 periods.\n",
"\n",
"The `state_prev` attribute of an AgenType stores the values of the model's state variables in the _previous_ period of the simulation."
"The `state_prev` attribute of an `AgentType` stores the values of the model's state variables in the _previous_ period of the simulation."
]
},
{
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