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Add liquidity constraint to perfect foresight model #35

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Add liquidity constraint to perfect foresight model #35

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255113b
Adjust PF notebook to account for liquidity constraints
mnwhite Jun 2, 2019
444c38d
Slight edits to IndShock and KinkedR dictionaries
mnwhite Jun 2, 2019
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4 changes: 2 additions & 2 deletions notebooks/IndShockConsumerType.ipynb
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Expand Up @@ -141,13 +141,14 @@
},
"outputs": [],
"source": [
"IdiosyncDict={\n",
"IdiosyncDict={ # Click the arrow to expand this parameter dictionary\n",
" # Parameters shared with the perfect foresight model\n",
" \"CRRA\": 2.0, # Coefficient of relative risk aversion\n",
" \"Rfree\": 1.03, # Interest factor on assets\n",
" \"DiscFac\": 0.96, # Intertemporal discount factor\n",
" \"LivPrb\" : [0.98], # Survival probability\n",
" \"PermGroFac\" :[1.01], # Permanent income growth factor\n",
" \"BoroCnstArt\" : 0.0, # Artificial borrowing constraint; imposed minimum level of end-of period assets\n",
" \n",
" # Parameters that specify the income distribution over the lifecycle\n",
" \"PermShkStd\" : [0.1], # Standard deviation of log permanent shocks to income\n",
Expand All @@ -169,7 +170,6 @@
" \"aXtraExtra\" : [None], # Additional values to add to aXtraGrid\n",
" \n",
" # A few other paramaters\n",
" \"BoroCnstArt\" : 0.0, # Artificial borrowing constraint; imposed minimum level of end-of period assets\n",
" \"vFuncBool\" : True, # Whether to calculate the value function during solution \n",
" \"CubicBool\" : False, # Preference shocks currently only compatible with linear cFunc\n",
" \"T_cycle\" : 1, # Number of periods in the cycle for this agent type \n",
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10 changes: 5 additions & 5 deletions notebooks/KinkedRconsumerType.ipynb
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Expand Up @@ -140,12 +140,13 @@
" \"DiscFac\": 0.96, # Intertemporal discount factor\n",
" \"LivPrb\" : [0.98], # Survival probability\n",
" \"PermGroFac\" :[1.01], # Permanent income growth factor\n",
" \"BoroCnstArt\" : None, # Artificial borrowing constraint; imposed minimum level of end-of period assets\n",
" \n",
" # New parameters unique to the \"kinked R\" model\n",
" \"Rboro\" : 1.20, # Interest factor on borrowing (a < 0)\n",
" \"Rsave\" : 1.01, # Interest factor on saving (a > 0)\n",
" \n",
" # Parameters that specify the income distribution over the lifecycle\n",
" # Parameters that specify the income distribution over the lifecycle (shared with IndShockConsumerType)\n",
" \"PermShkStd\" : [0.1], # Standard deviation of log permanent shocks to income\n",
" \"PermShkCount\" : 7, # Number of points in discrete approximation to permanent income shocks\n",
" \"TranShkStd\" : [0.2], # Standard deviation of log transitory shocks to income\n",
Expand All @@ -157,20 +158,19 @@
" \"T_retire\" : 0, # Period of retirement (0 --> no retirement)\n",
" \"tax_rate\" : 0.0, # Flat income tax rate (legacy parameter, will be removed in future)\n",
" \n",
" # Parameters for constructing the \"assets above minimum\" grid\n",
" # Parameters for constructing the \"assets above minimum\" grid (shared with IndShockConsumerType)\n",
" \"aXtraMin\" : 0.001, # Minimum end-of-period \"assets above minimum\" value\n",
" \"aXtraMax\" : 20, # Maximum end-of-period \"assets above minimum\" value\n",
" \"aXtraCount\" : 48, # Number of points in the base grid of \"assets above minimum\"\n",
" \"aXtraNestFac\" : 3, # Exponential nesting factor when constructing \"assets above minimum\" grid\n",
" \"aXtraExtra\" : [None], # Additional values to add to aXtraGrid\n",
" \n",
" # A few other paramaters\n",
" \"BoroCnstArt\" : None, # Artificial borrowing constraint; imposed minimum level of end-of period assets\n",
" # A few other paramaters (shared with IndShockConsumerType)\n",
" \"vFuncBool\" : True, # Whether to calculate the value function during solution \n",
" \"CubicBool\" : False, # Preference shocks currently only compatible with linear cFunc\n",
" \"T_cycle\" : 1, # Number of periods in the cycle for this agent type \n",
" \n",
" # Parameters only used in simulation\n",
" # Parameters only used in simulation (shared with PerfForesightConsumerType)\n",
" \"AgentCount\" : 10000, # Number of agents of this type\n",
" \"T_sim\" : 500, # Number of periods to simulate\n",
" \"aNrmInitMean\" : -6.0, # Mean of log initial assets\n",
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51 changes: 44 additions & 7 deletions notebooks/PerfForesightConsumerType.ipynb
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Expand Up @@ -23,6 +23,7 @@
"from time import clock\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"from copy import copy\n",
"mystr = lambda number : \"{:.4f}\".format(number)"
]
},
Expand Down Expand Up @@ -56,16 +57,19 @@
"\\begin{equation}\n",
"U(C) = \\frac{C^{1-\\CRRA}}{1-\\rho},\n",
"\\end{equation}\n",
"has perfect foresight about everything except whether he will die between the end of period $t$ and the beginning of period $t+1$, which occurs with probability $\\DiePrb_{t+1}$. Permanent labor income $P_t$ grows from period $t$ to period $t+1$ by factor $\\PermGroFac_{t+1}$. The consumer faces no artificial borrowing constraint and is able to borrow against the entire future stream of income he will receive.\n",
"has perfect foresight about everything except whether he will die between the end of period $t$ and the beginning of period $t+1$, which occurs with probability $\\DiePrb_{t+1}$. Permanent labor income $P_t$ grows from period $t$ to period $t+1$ by factor $\\PermGroFac_{t+1}$.\n",
"\n",
"At the beginning of period $t$, the consumer has an amount of market resources $M_t$ (which includes both market wealth and currrent income) and must choose how much of those resources to consume $C_t$ and how much to retain in a riskless asset $A_t$, which will earn return factor $\\Rfree$. The agent's flow of future utility $U(C_{t+n})$ from consumption is geometrically discounted by factor $\\DiscFac$ per period. If the consumer dies, he receives zero utility flow for the rest of time.\n",
"At the beginning of period $t$, the consumer has an amount of market resources $M_t$ (which includes both market wealth and currrent income) and must choose how much of those resources to consume $C_t$ and how much to retain in a riskless asset $A_t$, which will earn return factor $\\Rfree$. The consumer cannot necessarily borrow arbitarily; instead, he might be constrained to have a wealth-to-income ratio at least as great as some \"artificial borrowing constraint\" $\\underline{a} \\leq 0$.\n",
"\n",
"The agent's flow of future utility $U(C_{t+n})$ from consumption is geometrically discounted by factor $\\DiscFac$ per period. If the consumer dies, he receives zero utility flow for the rest of time.\n",
"\n",
"The agent's problem can be written in Bellman form as:\n",
"\n",
"\\begin{eqnarray*}\n",
"V_t(M_t,P_t) &=& \\max_{C_t}~U(C_t) ~+ \\DiscFac (1 - \\DiePrb_{t+1}) V_{t+1}(M_{t+1},P_{t+1}), \\\\\n",
"& s.t. & \\\\\n",
"A_t &=& M_t - C_t, \\\\\n",
"A_t/P_t &\\geq& \\underline{a}, \\\\\n",
"M_{t+1} &=& \\Rfree A_t + Y_{t+1}, \\\\\n",
"Y_{t+1} &=& P_{t+1}, \\\\ \n",
"P_{t+1} &=& \\PermGroFac_{t+1} P_t.\n",
Expand All @@ -79,6 +83,7 @@
"v_t(m_t) &=& \\max_{c_t}~U(c_t) ~+ \\DiscFac (1 - \\DiePrb_{t+1}) \\PermGroFac_{t+1}^{1-\\CRRA} v_{t+1}(m_{t+1}), \\\\\n",
"& s.t. & \\\\\n",
"a_t &=& m_t - c_t, \\\\\n",
"a_t &\\geq& \\underline{a}, \\\\\n",
"m_{t+1} &=& \\Rfree/\\PermGroFac_{t+1} a_t + 1.\n",
"\\end{eqnarray*}"
]
Expand Down Expand Up @@ -107,11 +112,15 @@
"| $\\Rfree$ | 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",
"\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 $\\texttt{float}$, or it can be $\\texttt{None}$ to indicate no artificial borrowing constraint. The maximum number of gridpoints in the consumption function is only relevant if the borrowing constraint is not $\\texttt{None}$; 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 $\\texttt{AgentType}$ uses these two code parameters to define the nature of time. Here, $\\texttt{T_cycle}$ has the value $1$, indicating that there is exactly one period in the cycle, while $\\texttt{cycles}$ 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 $\\texttt{T_cycle}$ 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",
Expand All @@ -125,9 +134,7 @@
"cell_type": "code",
"execution_count": null,
"metadata": {
"code_folding": [
0
]
"code_folding": []
},
"outputs": [],
"source": [
Expand All @@ -137,7 +144,9 @@
" \"Rfree\" : 1.03, # Interest factor on assets\n",
" \"DiscFac\" : 0.96, # Default intertemporal discount factor\n",
" \"LivPrb\" : [0.98], # Survival probability\n",
" \"PermGroFac\" :[1.01], # Permanent income growth factor\n",
" \"PermGroFac\" : [1.01], # Permanent income growth factor\n",
" \"BoroCnstArt\" : None, # Artificial borrowing constraint\n",
" \"aXtraCount\" : 200, # Maximum number of gridpoints in consumption function\n",
" \n",
" # Parameters that characterize the nature of time\n",
" \"T_cycle\" : 1, # Number of periods in the cycle for this agent type\n",
Expand Down Expand Up @@ -231,7 +240,35 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"An element of $\\texttt{solution}$ 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."
"An element of $\\texttt{solution}$ 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",
"\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 $\\texttt{PerfForesightConsumerType}$ with it, and then plot its consumption function."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"LiqConstrDict = copy(PerfForesightDict)\n",
"LiqConstrDict['BoroCnstArt'] = 0.0 # Set the artificial borrowing constraint to zero\n",
"\n",
"LiqConstrExample = PerfForesightConsumerType(**LiqConstrDict)\n",
"LiqConstrExample.cycles = 0 # Make this type be infinite horizon\n",
"LiqConstrExample.solve()\n",
"\n",
"print('Liquidity constrained perfect foresight consumption function:')\n",
"plotFuncs(LiqConstrExample.solution[0].cFunc,0.,10.)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"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 $\\texttt{solution}$."
]
},
{
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